Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform 9783031078224, 9783031078231

Table of contents :
Preface
Book Overview
Book Organization
Acknowledgments
Contents
Part I Foundations of Wireless Sensor Networks
1 General Introduction
1.1 Introduction
1.1.1 Major Tasks
1.1.2 Chapter Organization
1.2 Major Challenges
1.2.1 Limited Resources and Capabilities
1.2.2 Location Management
1.2.3 Sensor Deployment
1.2.4 Time-Varying Network Characteristics
1.2.5 Network Scalability, Heterogeneity, and Mobility
1.2.6 Sensing Application Requirements
1.3 Sample Sensing Applications
1.4 Book Motivations
1.5 Design Requirements
1.6 Book Contributions
1.7 Conclusion
2 Fundamental Concepts, Definitions, and Models
2.1 Introduction
2.1.1 Major Tasks
2.1.2 Chapter Organization
2.2 Terminology
2.3 Deterministic and Stochastic Sensing Models
2.4 Network Connectivity and Fault Tolerance
2.5 Energy Consumption Model
2.6 Percolation Model
2.6.1 Why a Continuum Percolation Model?
2.7 Default Network Model
2.8 Random and Group Mobility Models
2.8.1 Random Waypoint Mobility Model (RWP)
2.8.2 Reference Point Group Mobility Model (RPGM)
2.8.3 Manhattan Mobility Model (MMM)
2.8.4 Why Group and Random Mobility Models?
2.9 Conclusion
Part II Percolation Theory-Based Coverage and Connectivity in Wireless Sensor Networks
3 A Planar Percolation-Theoretic Approach to Coverage and Connectivity
3.1 Introduction
3.1.1 Major Tasks
3.1.2 Chapter Organization
3.2 Phase Transition in Sensing Coverage
3.2.1 Estimation of the Shape of Covered Components
3.2.2 Critical Density of Covered Components
3.2.3 Critical Radius of Covered Components
3.2.4 Characterization of Critical Percolation
3.2.5 Numerical Results
3.3 Phase Transition in Network Connectivity
3.3.1 Integrated Sensing Coverage and Network Connectivity
3.4 Discussion
3.5 Related Work
3.6 Conclusion
4 A Spatial Percolation-Theoretic Approach to Coverage and Connectivity
4.1 Introduction
4.1.1 Major Tasks
4.1.2 Chapter Organization
4.2 Three Percolation Problems
4.2.1 Sensing Coverage Percolation
4.2.2 Network Connectivity Percolation
4.2.3 Coverage and Connectivity Percolation
4.3 Further Discussion
4.3.1 Practicality and Generalizability Issues
4.3.2 Sensor Deployment in Spatial Fields
4.3.3 Relaxations of Assumptions
4.4 Related Work
4.5 Conclusion
Part III Convexity Theory-Based Connected k-Coverage in Wireless Sensor Networks
5 A Planar Convexity Theory-Based Approach for Connected k-Coverage
5.1 Introduction
5.1.1 Major Tasks
5.1.2 Chapter Organization
5.2 Achieving Connected k-Coverage
5.2.1 Connected k-Coverage Problem Modeling
5.2.2 Sufficient Condition to Ensure k-Coverage
5.3 Centralized k-Coverage Protocol
5.3.1 Planar Deployment Field Slicing
5.3.2 Sensor Selection
5.3.3 Slicing Grid Dynamics
5.4 Clustered k-Coverage Protocol
5.4.1 Cluster-Head Selection and Attributed Roles
5.4.2 The T-CRACCk Protocol
5.4.3 The D-CRACCk Protocol
5.5 Triggered-Scheduling Driven k-Coverage
5.5.1 K-Coverage Checking Algorithm and Sensor Selection
5.5.2 State Transition Diagram of Trig-DIRACCk
5.5.3 Ensuring Network Connectivity
5.6 Self-scheduling Based k-Coverage
5.6.1 K-Coverage Candidacy Algorithm
5.6.2 State Transition Diagram of Self-DIRACCk
5.6.3 Tri-DIRACCk Versus Self-DIRACCk
5.7 Relaxation of Assumptions
5.7.1 Relaxing the Unit Disk Model
5.7.2 Relaxing the Sensor Homogeneity Model
5.8 Performance Evaluation
5.8.1 Simulation Settings
5.8.2 Simulation Results
5.8.3 Comparison of Self-DIRACCk with CCP
5.9 Related Work
5.10 Conclusion
6 Planar Convexity Theory-Based Approaches for Heterogeneous, On-Demand, and Stochastic Connected k-Coverage
6.1 Introduction
6.1.1 Major Tasks
6.1.2 Chapter Organization
6.2 Heterogeneous Connected k-Coverage
6.2.1 Random Deployment Approach
6.2.2 Pseudo-random Deployment Approach
6.2.3 Performance Evaluation
6.3 On-Demand Connected k-Coverage
6.3.1 Pseudo-random Sensor Placement
6.3.2 Sensor Mobility for k-Coverage of a Region of Interest
6.3.3 Performance Evaluation
6.4 Stochastic Connected k-Coverage
6.4.1 Stochastic k-Coverage Characterization
6.4.2 Stochastic k-Coverage-Preserving Scheduling
6.4.3 Simulation Results
6.5 Related Work
6.5.1 Sensor Heterogeneity
6.5.2 Sensor Mobility
6.5.3 Probabilistic Sensing Model
6.6 Conclusion
7 Spatial Convexity Theory-Based Approaches for Connected k–Coverage
7.1 Introduction
7.1.1 Major Tasks
7.1.2 Chapter Organization
7.2 Equilateral Spherical Triangle-Based Approach
7.2.1 Problem Analysis: The Curse of Dimensionality
7.2.2 Distributed k-Coverage Protocol
7.2.3 Performance Evaluation
7.3 Reuleaux Tetrahedron-Based Approach
7.3.1 Proposed Solution
7.3.2 Problem Analysis
7.3.3 Optimized Spatial k-Coverage
7.3.4 Using Reuleaux Tetrahedra for Sphere Coverage
7.3.5 Reuleaux Tetrahedron-Based Spatial k-Coverage
7.3.6 Assumption Relaxation
7.3.7 Simulation Results
7.4 Related Work
7.5 Conclusion
Part IV Applied Computational Geometry-Based Connected k-Coverage in Wireless Sensor Networks
8 A Planar Regular Hexagonal Tessellation-Based Approach for Connected k-Coverage
8.1 Introduction
8.1.1 Major Tasks
8.1.2 Chapter Organization
8.2 Study of Planar Pavers
8.2.1 Paving Metric
8.2.2 Planar Regular Convex Paver Quality
8.3 Regular Hexagonal Centroid-Based Connected k-Coverage
8.3.1 Achieving Optimal Coverage
8.3.2 Problems with k-Coverage for k ge2
8.4 Regular Hexagonal Area Stretching-Based Connected k-Coverage
8.4.1 Foundational Study
8.4.2 Random Regular Hexagonal Tessellation
8.4.3 Hexagonal Cone-Based Pseudo-Random k-Coverage
8.4.4 Hexagonal Perimeter-Based Pseudo-Random k-Coverage
8.4.5 Edge Problem
8.4.6 Discussion
8.5 Possible Extensions
8.5.1 Extension 1: Using Non-Deterministic Sensing Model
8.5.2 Extension 2: Heterogenous Sensor Deployment
8.6 Performance Evaluation
8.6.1 Simulation Setup
8.6.2 Simulation Results
8.7 Related Work
8.8 Conclusion
9 A Planar Irregular Hexagonal Tessellation-Based Approach for Connected k-Coverage
9.1 Introduction
9.1.1 Major Tasks
9.1.2 Chapter Organization
9.1.3 Planar Tiling Using Congruent Tiles
9.2 Achieving Planar k-Coverage Using Hexagonal Tiles
9.2.1 Ensuring 1-Coverage
9.2.2 Ensuring k-Coverage
9.3 Achieving Planar k-Coverage Using Irregular Hexagonal Tiles
9.3.1 Irregular Hexagonal Tiling with  IRH( r/2 )
9.3.2 Irregular Hexagonal Tiling with  IRH( r/3 )
9.3.3 Irregular Hexagonal Tiling with  IRH( r/n )
9.3.4 Discussion on Planar Sensor Density
9.4 A k-Coverage Protocol Using Irregular Hexagonal Tiling
9.4.1 Generating Reference Irregular Hexagon and k-Coverage
9.4.2 Expanding Hexagonal Grid and k-Coverage
9.4.3 Example
9.4.4 Problem of Side-Effect
9.5 Performance Evaluation
9.5.1 Simulation Setup
9.5.2 Simulation Results
9.6 Related Work
9.7 Conclusion
10 A Polyhedral Space Filler Tessellation-Based Approach for Connected k-Coverage
10.1 Introduction
10.1.1 Major Tasks
10.1.2 Chapter Organization
10.2 Investigating Polyhedral Space-Fillers
10.2.1 Cubic Space-Filler
10.2.2 Regular Right Hexagonal Prism Space-Filler
10.2.3 Truncated Octahedral Space-Filler
10.2.4 Great Rhombicuboctahedral Space-Filler
10.2.5 Rhombic Dodecahedral Space-Filler
10.2.6 Elongated Dodecahedral Space-Filler
10.2.7 Rhombic Triacontahedral Space-Filler
10.2.8 Sommerville’s Largest Tetrahedral Space-Filler
10.2.9 Baumgartner’s Tetrahedral Space-Filler
10.2.10 Goldberg’s Equilateral Octahedral Space-Filler
10.3 Solving the Connected Coverage Problem
10.3.1 Sensor Selection Algorithm
10.3.2 Performance Evaluation
10.4 Connected k-Coverage Problem
10.4.1 Achieving Spatial k-Coverage
10.4.2 Ensuring Spatial Connected k-Coverage
10.4.3 Discussion
10.4.4 Sensor Selection Protocol
10.5 Performance Evaluation
10.5.1 Simulation Setup
10.5.2 Simulation Results
10.6 Related Work
10.7 Conclusion
Part V Connectivity and Fault-Tolerance Measures of k-Covered Wireless Sensor Networks
11 Planar Unconditional and Conditional Network Connectivity and Fault-Tolerance Measures for k-Covered Wireless Sensor Networks
11.1 Introduction
11.1.1 Major Tasks
11.1.2 Chapter Organization
11.2 Unconditional Fault-Tolerance Measures
11.2.1 Homogeneous Sensors
11.2.2 Heterogeneous Sensors
11.3 Conditional Fault-Tolerance Measures
11.3.1 Homogeneous Sensors
11.3.2 Heterogeneous Sensors
11.4 Related Work
11.5 Conclusion
12 Spatial Unconditional and Conditional Network Connectivity and Fault-Tolerance Measures for k-Covered Wireless Sensor Networks
12.1 Introduction
12.1.1 Major Tasks
12.1.2 Chapter Organization
12.2 Unconditional Connectivity
12.2.1 Homogeneous Sensors
12.2.2 Heterogeneous Sensors
12.2.3 Boundary Effects
12.3 Conditional Connectivity
12.3.1 Homogeneous Sensors
12.3.2 Heterogeneous Sensors
12.4 Discussion
12.4.1 Relaxing the Assumption of k ≥ 4
12.4.2 Sensor Placement Strategy
12.4.3 Sink-Independent Connectivity Measures
12.4.4 Spatial Sensing Applications
12.5 Relaxing the Unit Sphere Model: Convex Model
12.5.1 Homogeneous Sensors
12.5.2 Heterogeneous Sensors
12.6 Underwater Sensor Networks
12.7 Related Work
12.8 Conclusion
Part VI Geographic Data Forwarding, Gathering, and Delivery in Wireless Sensor Networks
13 A Planar Checkpoints-Based Approach for Geographic Forwarding on Always-on Sensors
13.1 Introduction
13.1.1 Major Tasks
13.1.2 Chapter Organization
13.2 The WLDT Protocol
13.2.1 Long-Range Versus Short-Range Forwarding
13.2.2 A Two-Step Data Forwarding Protocol
13.2.3 Illustrative Example
13.3 Analysis of WLDT
13.4 Short-Range Versus Long-Range
13.4.1 Energy Gain
13.4.2 Controlled Short-Range Data Forwarding
13.5 Discussion
13.6 Related Work
13.7 Conclusion
14 A Planar Energy-Delay Trade-off Based Approach for Geographic Forwarding on Always-on Sensors
14.1 Introduction
14.1.1 Major Tasks
14.1.2 Chapter Organization
14.2 A Slicing Approach
14.2.1 Slicing of Communication Range
14.2.2 Selection of Candidate Proxy Forwarders
14.2.3 Uniform Energy Depletion Characterization
14.3 Trading-off Energy with Delay
14.3.1 Simple Analytical Bounds
14.3.2 Multi-objective Optimization Approach
14.3.3 TED Detailed Description
14.4 Relaxation of Several Key Assumptions
14.4.1 Relaxing the Sensor Homogeneity Model
14.4.2 Relaxing the Communication Disk Model
14.4.3 Relaxing the Dense Network Model
14.4.4 Relaxing the Energy Consumption Model
14.4.5 Relaxing the Always-on Sensors Model
14.5 Simulation Results
14.5.1 Simulation Settings
14.5.2 Impact of Selection Space Size
14.5.3 Using the Energy × Delay Metric
14.5.4 Impact of Variability of k
14.5.5 Impact of Sensor Heterogeneity
14.6 Related Work
14.7 Conclusion
15 A Planar Approach for Solving the Energy Sink-Hole Problem with Always-on Sensors
15.1 Introduction
15.1.1 Major Tasks
15.1.2 Chapter Organization
15.2 Energy Sink-Hole Problem Analysis
15.2.1 Base Protocol Average Energy Consumption
15.2.2 Nominal Communication Range–Based Data Forwarding
15.2.3 Adjustable Communication Range-Based Data Forwarding
15.3 Using Heterogeneous Sensors
15.3.1 Multi-tier Architecture
15.3.2 NEAR Performance Evaluation
15.4 Sink Mobility and Energy Aware Voronoi Diagram
15.4.1 Why Energy Aware Voronoi Diagram?
15.4.2 EVEN Detailed Description
15.4.3 EVEN Performance Evaluation
15.5 Related Work
15.5.1 Balancing Energy Consumption
15.5.2 Minimizing Energy Consumption
15.5.3 Mobility-Based Forwarding Protocols
15.6 Conclusion
Part VII Joint k-Coverage and Geographic Data Forwarding and Gathering in Wireless Sensor Networks
16 Planar and Spatial Approaches for Joint k-Coverage and Data Collection Using Homogeneous Duty-Cycled Sensors
16.1 Introduction
16.1.1 Major Tasks
16.1.2 Chapter Organization
16.2 A Planar Approach for Joint k-Coverage and Data Collection
16.2.1 Potential Fields Based Modeling Approach
16.2.2 Data Forwarding Without Aggregation
16.2.3 Data Forwarding with Aggregation
16.2.4 Generalizability of GEFIB
16.2.5 Performance Evaluation
16.3 A Spatial Approach for Joint k-Coverage and Composite Forwarding
16.3.1 First Hybrid Geographic Forwarding
16.3.2 Second Hybrid Geographic Forwarding
16.4 Related Work
16.5 Conclusion
17 A Planar Approach for Joint k-Coverage and Data Collection Using Sparsely Deployed Duty-Cycled Sensors
17.1 Introduction
17.1.1 Major Tasks
17.1.2 Chapter Organization
17.2 Heterogeneous k-Coverage
17.3 Mobile k-Coverage
17.3.1 Four-Tier Sensor Network Architecture
17.3.2 k-Coverage Approach Design Decisions
17.3.3 Achieving Mobile k-Coverage
17.4 Data Gathering Algorithms
17.4.1 Direct Data Gathering
17.4.2 Chain-Based Data Gathering
17.5 Impact of Sensor Heterogeneity
17.6 Performance Evaluation
17.6.1 Simulation Setup
17.6.2 Simulation Results
17.7 Related Work
17.8 Conclusion
18 Planar Approaches for Joint k-Coverage and Data Collection Using Heterogeneous Duty-Cycled Sensors
18.1 Introduction
18.1.1 Major Tasks
18.1.2 Chapter Organization
18.2 Basic Two-Tier Architecture
18.2.1 Impact of the Energy Sink-Hole Problem
18.2.2 Energy Consumption Analysis
18.3 Three-Tier Architecture with Constant Band Width
18.3.1 Proposed Architecture
18.3.2 Joint Mobility and Routing
18.3.3 Architecture 1: 1 Static Sink—1 Mobile Proxy Sink
18.3.4 Architecture 2: 1 Static Sink—N Mobile Proxy Sinks
18.3.5 Architecture 3: N Static Sinks—1 Mobile Proxy Sink
18.3.6 Architecture 4: N Static Sinks – N Mobile Proxy Sinks
18.3.7 Performance Evaluation
18.4 Three-Tier Architecture with Varying Band Widths
18.4.1 Proposed Architecture
18.4.2 Static Data Collection Schemes
18.4.3 Mobile Data Collection
18.4.4 Performance Evaluation
18.5 Conclusion
Part VIII Connected k-Barrier Coverage in Wireless Sensor Networks
19 A Planar Approach for Physical Security Using Connected k-Barrier Coverage
19.1 Introduction
19.1.1 Major Tasks
19.1.2 Chapter Organization
19.2 Tiling-Based k-Barrier Coverage
19.2.1 Intruder’s Abstract Path Counting
19.2.2 Intruder’s Abstract Path Analysis
19.2.3 Square Lattice-Based Sensor Deployment
19.2.4 Hexagonal Lattice-Based Sensor Deployment
19.2.5 Square Lattice Versus Hexagonal Lattice
19.2.6 Discussion
19.3 Generalization
19.4 Source-to-Destination Path Analysis
19.4.1 Square k-barrier Covered Sensor Belt
19.4.2 Rectangular k-barrier Covered Sensor Belt
19.5 Other Possible Generalizations
19.6 Performance Evaluation
19.6.1 Simulation Setup
19.6.2 Simulation Results
19.7 Related Work
19.8 Conclusion
20 A Spatial Approach for Physical Security Through Connected k-Barrier Coverage
20.1 Introduction
20.1.1 Major Tasks
20.1.2 Chapter Organization
20.2 Spatial k-Barrier Coverage Problem Analysis
20.2.1 Simple Cubic Lattice
20.2.2 Body Centered Cubic (BCC) Lattice
20.2.3 Face Centered Cubic (FCC) Lattice
20.2.4 Hexagonal Close-Packed (HCP) Lattice
20.3 Polyhedral Space-Filling Lattice
20.3.1 Intruder’s Path Analysis
20.3.2 Intruder’s Path Representation and Counting
20.4 Performance Evaluation
20.4.1 Simulation Setup
20.4.2 Numerical Versus Simulation Results
20.5 Related Work
20.6 Conclusion
Part IX Applications of Wireless Sensor Networks and Concluding Remarks
21 An Overview of Sensing Hardware, Standards, Operating Systems, Software Development, and Applications and Systems
21.1 Introduction
21.1.1 Major Tasks
21.1.2 Chapter Organization
21.2 Sensing Hardware
21.2.1 Mote Hardware
21.2.2 Sensor Technology
21.2.3 Gateways
21.3 Sensing Software
21.3.1 Industry Standards
21.3.2 Operating Systems
21.4 Sensing Software Development: Challenges and Solutions
21.4.1 Sensing Application Models
21.4.2 Debugging
21.4.3 Memory
21.4.4 Sensing
21.4.5 Protocols and Radio Communication
21.4.6 Security
21.5 Sensing Applications and Systems
21.5.1 Healthcare Industry
21.5.2 Agriculture Industry
21.5.3 Environmental Industry
21.5.4 Industry
21.5.5 Military
21.6 Future Applications and Technologies
21.6.1 Marine Deployments
21.6.2 Smart Homes
21.7 Conclusion
22 Summary and Further Extensions
22.1 Summary of Book Contributions
22.2 Further Extensions
References

Citation preview

Studies in Systems, Decision and Control 214

Habib M. Ammari

Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform

Studies in Systems, Decision and Control Volume 214

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Habib M. Ammari

Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform

Habib M. Ammari Department of Electrical Engineering and Computer Science Texas A&M University-Kingsville Kingsville, TX, USA

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-07822-4 ISBN 978-3-031-07823-1 (eBook) https://doi.org/10.1007/978-3-031-07823-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

The gift of mental power comes from God, Divine Being, and if we concentrate our minds on that truth, we become in tune with this great power. Every living being is an engine geared to the wheelwork of the universe. Though seemingly affected only by its immediate surrounding, the sphere of external influence extends to infinite distance. The desire that guides me in all I do is the desire to harness the forces of nature to the service of mankind. Nikola Tesla (1856–1943)

This book is dedicated To my Creator, Allah—The Most Gracious, the Most Merciful To my Prophet, Mohamed—Peace Be Upon Him To the profound soul of my mother, Mbarka—The reason of my existence “You are gone, but you are always present and engraved in my heart, sentiment, mind, conscience, and memory” To my first teachers: My mother, Mbarka, and my father, Mokhtar To my very best friends: My wife and my children, Leena, Muath, Mohamed-Eyed, Lama, and Maitham To my daughter-in-law, Fatema, and grandson, Malik

To my sisters, Naima, Saloua, Monia, Faouzia, Alia, and Najeh, and my brothers, Mustapha and Lazhar To all my nieces and nephews To the profound souls of my grandparents, Fatma and Abdelkarim, and my uncle, Mahfoudh To my President, Dr. James R. Hallmark, Interim President of Texas A&M University-Kingsville (TAMUK) and Vice Chancellor for Academic Affairs for the Texas A&M University System, for his outstanding support since he was nominated for the above Interim President position starting January 1, 2022. In particular, Dr. Hallmark has been very supportive of both the Honorary Frank H. Dotterweich College of Engineering (CoE) and Electrical Engineering and Computer Science (EECS) Department Distinguished Lecture Series, which I founded in Spring 2022 when its first edition launched; and the Frank H. Dotterweich CoE and EECS Department Distinguished Lecture Series, which I founded in Fall 2019 when its first edition launched. Dr. Hallmark believes that “Guest lectures are an essential component of the university experience.” Although I am aware that he will assume his responsibility as Interim President of TAMUK for a very short period of time (January 1, 2022–August 2022), and I wish he would be our President forever, I am very honored and pleased to dedicate this book to him for both his great appreciation and wonderful support of research.

To my Provost, Dr. Lou Reinisch, Vice President for Academic Affairs, and Professor of Physics, for his outstanding support since he joined TAMUK in August 2020. In particular, Dr. Reinisch has been very supportive of this book project “I am glad to hear that you are working on your sixth book. Keep up your scholarship.” He believes that “It is important that we have faculty who can bring the cutting edge of the subject into the classroom.”, thus, promoting the integration of research and teaching through a research-based model of instruction. To my Associate Vice President for Academic Affairs, Dr. Jaya S. Goswami, Professor of Bilingual Education in Department of Teacher and Bilingual Education at TAMUK, for her outstanding support since I joined TAMUK in August 2019. To my Interim Dean, Dr. Robert Diersing, Professor Emeritus of Computer Science, and HPCC System Administrator, Frank H. Dotterweich College of Engineering at TAMUK, for his outstanding support since I first met with him in Corpus Christi on Thursday, March 20, 2014, during my visit for my first interview at TAMUK for the position of Full Professor of Computer Science. Now, he is Interim Vice-President for Research and Graduate Studies. To my Dean, Dr. Heidi A. Taboada, Professor of Mechanical and Industrial Engineering, Frank H. Dotterweich College of Engineering at TAMUK, for her outstanding support since she joined TAMUK in January 2022. In

particular, Dr. Taboada has been very supportive of my teaching activities “It truly shows how committed you are to student learning and student success.” She is the first female dean in the over 85-year history of the university’s College of Engineering. To my Chair, Dr. Scott C. Smith, Professor of Electrical Engineering in the Department of Electrical Engineering and Computer Science in the Frank H. Dotterweich College of Engineering at TAMUK, for his outstanding support since we joined TAMUK in August 2019. To my friend and colleague, Dr. Rajab Challoo, P.E. and Professor of Electrical Engineering in the Department of Electrical Engineering and Computer Science in the Frank H. Dotterweich College of Engineering at TAMUK, for his wonderful friendship and outstanding support since I first met with him at TAMUK on Friday, March 21, 2014, during my visit for my first interview at TAMUK for the position of Full Professor of Computer Science. To all my friends and colleagues at TAMUK, and, particularly, those in the Frank H. Dotterweich College of Engineering, and, especially, those in the Department of Electrical Engineering and Computer Science, and, exceptionally, Drs. Reza Nekovei and Amit Verma, for their wonderful friendship and outstanding support since I joined TAMUK in August 2019.

To all my undergraduate and graduate students, and particularly, WiSeMAN-IoT-ACE Undergraduate and Graduate Research Assistants. Last but not least, to the profound soul of the Serbian-American genius inventor, electrical engineer, mechanical engineer, futurist, and father of electricity, Nikola Tesla.

Preface

I am credited with being one of the hardest workers and perhaps I am, if thought is the equivalent of labour, for I have devoted to it almost all of my waking hours. But if work is interpreted to be a definite performance in a specified time according to a rigid rule, then I may be the worst of idlers. Nikola Tesla (1856–1943)

This Preface gives an overview of this book. Then, it outlines the book’s structure consisting of nine parts: (1) Foundations of Wireless Sensor Networks, (2) Percolation Theory-Based Coverage and Connectivity in Wireless Sensor Networks, (3) Convexity Theory-Based Connected k-Coverage in Wireless Sensor Networks, (4) Applied Computational Geometry-Based Connected k-Coverage in Wireless Sensor Networks, (5) Connectivity and Fault-Tolerance Measures of k-Covered Wireless Sensor Networks, (6) Data Forwarding and Gathering in Wireless Sensor Networks, (7) Joint k-Coverage and Data Forwarding and Gathering in Wireless Sensor Networks, (8) Connected k-Barrier Coverage in Wireless Sensor Networks, and (9) Applications of Wireless Sensor Networks and Concluding Remarks. Also, it provides a brief summary of each of the chapters in every part. In addition, it acknowledges all the author’s friends and colleagues who have contributed their support directly or indirectly to this work.

Book Overview Wireless sensor networks have received significant attention because of their important role and many conveniences in our lives. Indeed, the recent and fast advances in inexpensive sensor technology and wireless communications have made the design and development of large-scale wireless sensor networks cost effective and appealing to a wide range of mission-critical situations, including civilian, natural, industrial,

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Preface

and military applications, such as health and environmental monitoring, seism monitoring, industrial process automation, and battlefields surveillance, respectively. A wireless sensor network consists of a large number of tiny, low-powered devices, called sensors, which are randomly or deterministically deployed in a field of interest while collaborating and coordinating for the successful accomplishment of their mission. These sensors suffer from very scarce resources and capabilities, such as bandwidth, storage, CPU, battery power (or energy), sensing, and communication, to name a few, with energy being the most critical one. The major challenge in the design process of this type of network is mainly due to the limited capabilities of the sensors, and particularly, their energy, which makes them unreliable. This book aims to develop a reader’s thorough understanding of the opportunities and challenges of two types of networks: The first category is called k-covered wireless sensor networks, where every point in a deployment field is covered (or sensed) by at least k sensors. The second category is called k-barrier covered wireless sensor networks, where every path in a deployment field intersects with the sensing range (disk or sphere) of at least k sensors. Following René Descartes’ most elegant methodology of dividing each difficulty into as many parts as might be possible and necessary to best solve it (“Discourse on Method,” which is a translation of “Discours de la Méthode,” René Descartes, 1637), this book presents a variety of theoretical studies based on percolation theory, convexity theory, and applied computational geometry, as well as the algorithms and protocols that lead to the design and development of k-covered and k-barrier covered wireless sensor networks. In particular, as it can be implied from its title, this book focuses on the “Cover/Sense/Inform” paradigm, which helps build a unified framework, where connected k-coverage (or connected k-barrier coverage), sensor scheduling, and geographic data forwarding, gathering, and delivery are jointly considered. I have written this book given the tremendous interest of numerous researchers in k-covered and k-barrier covered wireless sensor networks, which has been expressed by their very active and productive research for the last 23 years and until now. Indeed, several protocols have been proposed to solve problems related to the design and implementation of energy-efficient k-covered and k-barrier covered wireless sensor networks that span a variety of topics, such as sensor deployment, network connectivity, sensing coverage, sensor scheduling (or duty cycling), and geographic data forwarding, collection, and delivery. This book is mainly based on my research work that has been focused so far on the study of k-covered and k-barrier covered wireless sensor networks. These are two of the major topics covered in both the introductory and advanced courses on wireless sensor networks that I have taught at Hofstra University and the University of Michigan-Dearborn. This book is useful to senior undergraduate and graduate students in computer science, computer engineering, electrical engineering, information science, information technology, mathematics, and any related discipline. It is also of interest to computer scientists, researchers, and practitioners in both academia and industry with interest in k-covered and k-barrier covered wireless sensor networks from their deployment until data gathering and delivery.

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Book Organization This book consists of nine parts, each of which has either two to three chapters.1 Next, we present the structure (or organization) of this book in terms of its nine parts along with a short description of each part through a brief summary of the contents of each of the chapters forming it. Part I, titled Foundations of Wireless Sensor Networks, consists of two chapters. Chapter 1 introduces wireless sensor networks and states several challenges related to their design and development. Also, it briefly presents their applications and the design requirements of the algorithms and protocols discussed in this book. Moreover, it lists the eleven research problems investigated in this book and gives a brief summary of their solutions. Chapter 2 introduces all the key terms and presents several models, namely percolation model, energy consumption model, default network model, deterministic and stochastic sensing models, and mobility models, which are used throughout this book. Moreover, it discusses some graph models, such as Voronoi diagram and Delaunay triangulation. Furthermore, it describes a new concept of network connectivity, called conditional connectivity, which seems to be appropriate for wireless sensor networks. Part II, titled Percolation Theory-Based Coverage and Connectivity in Wireless Sensor Networks, is composed of two chapters. Chapter 3 focuses on the problems of almost sure integrated coverage and connectivity in planar wireless sensor networks using percolation theory. The goal is to compute the critical sensor density above which the network is almost surely connected and the deployment field is almost surely covered. It also studies the integrated problem of connectivity and coverage based on a suitable integration model. Chapter 4 investigates the above problems in spatial wireless sensor networks using an approach different from its counterpart used in planar wireless sensor networks. Part III, titled Convexity Theory-Based Connected k-Coverage in Wireless Sensor Networks, has three chapters. Chapter 5 discusses the problem of connected kcoverage in densely deployed planar wireless sensor networks. It considers static, homogeneous, and duty-cycled sensors with a goal to achieve k-coverage of a planar field, while ensuring connectivity among all these sensors. The work presented in Chap. 6 is a generalization of the one presented in Chap. 5. Precisely, Chap. 6 focuses on the same problem in heterogeneous planar wireless sensor networks, where the sensors may not possess the same capabilities. In addition, it discusses the connected k-coverage problem in mission-oriented mobile planar wireless sensor networks, which may have more than one monitoring task. Both deterministic and stochastic sensing models are considered. Chapter 7 studies the problem of connected k-coverage in duty-cycled and homogeneous spatial wireless sensor networks, while considering densely deployed static sensors along with a deterministic sensing model.

1

If you only knew the magnificence of the 3, 6, and 9, then you would have the key to the universe. Nikola Tesla (1856–1943)

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Part IV, titled Applied Computational Geometry-Based Connected k-Coverage in Wireless Sensor Networks, contains three chapters. Chapter 8 investigates the connected k-coverage problem in planar wireless sensor networks using a hexagonal tiling-based approach. The latter is based on a special category of irregular hexagons, which are capable of tiling the Euclidean plane. Chapter 9 presents an in-depth study of the problem of connected k-coverage in planar wireless sensor networks using an approach based on slicing a planar field into convex regular hexagons. To this end, it formulates a multi-objective optimization problem, which computes an optimum solution to the planar k-coverage problem. Chapter 10 studies the problem of connected k-coverage in spatial wireless sensor networks using a tiling-based approach. To this end, it suggests a polyhedral framework using a variety of convex polyhedral space-fillers. The goal is to find the largest enclosed convex polyhedron space-filler in the sensing sphere of the sensors, with a goal to maximize their utilized sensing volume. Part V, titled Connectivity and Fault-Tolerance Measures of k-Covered Wireless Sensor Networks, has two chapters. Chapter 11 presents measures of unconditional (or traditional) and conditional connectivity and fault tolerance of planar k-covered wireless sensor networks. The latter measures are based on the concept of forbidden faulty set and seem to be more realistic. Chapter 12 computes the connectivity and fault tolerance of spatial homogeneous and heterogeneous k-covered wireless sensor networks. It also calculates conditional network connectivity and fault-tolerance measures for these planar k-covered wireless sensor networks. Part VI, titled Geographic Data Forwarding, Gathering, and Delivery in Wireless Sensor Networks, includes three chapters. Chapter 13 investigates the problem of energy-efficient data forwarding in always-on planar wireless sensor networks. It investigates short-range and long-range data forwarding schemes on always-on sensors. The proposed solution is based on Delaunay triangulation. Chapter 14 focuses on the trade-offs between energy and delay in data forwarding in planar wireless sensor networks. It formulates this trade-off as a multi-objective optimization problem whose solution is an input to the proposed data forwarding scheme. Chapter 15 discusses the energy sink-hole problem in static and always-on planar wireless sensor networks. It presents a heterogeneous sensor deployment strategy, where the sensors do not possess the same energy reserve. The goal is to let all the sensors deplete their energy at the same time. The proposed solution exploits sensor mobility and a new concept of Voronoi diagram, called energy-aware Voronoi diagram. Part VII, titled Joint k-Coverage and Geographic Data Forwarding and Gathering in Wireless Sensor Networks, consists of three chapters. Chapter 16 analyzes the problem of joint k-coverage, sensor scheduling, and geographic forwarding in planar and spatial wireless sensor networks. It studies geographic forwarding in dutycycled, k-covered wireless sensor networks and describes protocols for geographic forwarding on duty-cycled sensors with and without data aggregation. These protocols are based on the concept of virtual potential fields. Chapter 17 investigates the problem of joint k-coverage and geographic forwarding in duty-cycled and sparsely deployed planar wireless sensor networks. Chapter 18 focuses on the problem of joint

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k-coverage and data collection in heterogeneous planar wireless sensor networks. It investigates the best mobility strategy of mobile proxy sinks to minimize the total energy consumption for data collection. Part VIII, titled Connected k-Barrier Coverage in Wireless Sensor Networks, is composed of two chapters. Chapter 19 analyzes the k-barrier coverage problem in planar wireless sensor networks from a tiling perspective using two planar deterministic sensor deployment strategies, namely square lattice and hexagonal lattice. Chapter 20 investigates the problem of spatial k-barrier coverage from a tiling viewpoint through the analysis of various spatial deterministic sensor deployment methods, namely simple cubic, body-centered cubic, face-centered cubic, and hexagonal close-packed lattices. Also, it considers polyhedral space-fillers to solve the k-barrier coverage problem in spatial wireless sensor networks. Part IX, titled Applications of Wireless Sensor Networks and Concluding Remarks, contains two chapters. Chapter 21 proposes a classification of applications of wireless sensor networks based on their application domains, namely health care, agriculture, environmental, industry, and military. It provides an overview of a variety of applications of wireless sensor networks for each application domain. Chapter 22 summarizes our contributions made in various areas in planar and spatial wireless sensor networks, such as connected k-coverage, connectivity, connected k-barrier coverage, sensor scheduling, geographic forwarding, and data collection. Also, it highlights potential future research work to solve some of the (open) research problems in a more effective and efficient way. April 2022

Habib M. Ammari, Ph.D. (CSE), Ph.D. (CS) Founding Director Wireless Sensor and Mobile Ad-hoc Networks Internet of Things and Applied Cryptography Engineering (WiSeMAN-IoT-ACE) Research Lab Department of Electrical Engineering and Computer Science Frank H. Dotterweich College of Engineering Texas A&M University-Kingsville Kingsville, TX, USA

Acknowledgments

This book, titled Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, provides a detailed description of all the contributions I made in the areas of coverage, connectivity, connected k-coverage, connected k-barrier coverage, sensor duty cycling (or sensor scheduling), geographic forwarding, and data collection and delivery in both planar and spatial wireless sensor networks. And, it is really a great pleasure and an honor for me to work in the above areas for the past 17 years (precisely, since January 2005 until now). Now, it is the time to acknowledge all the people who are directly or indirectly involved in this book project. Indeed, throughout this book writing project, I have been generously and tremendously supported by several outstanding people whom I would like to cordially recognize. They have been a continuous source of inspiration for me and have been helping me finish this book and make it a reality. I am very grateful and privileged to be surrounded by these people without whom it would not be possible to complete this book and make it available to all the researchers and practitioners who are interested in the foundations of wireless sensor networks and particularly k-covered and k-barrier covered wireless sensor networks. First and foremost, I am sincerely and permanently grateful to Allah—the Most Gracious, the Most Merciful—for everything He has been providing me with. In particular, I would very much love to thank Him for giving me the golden opportunity to put together this book and for helping me publish it within over one year. I am extremely happy and so excited to dedicate this modest book to Him and very much hope that He would kindly accept it and put His blessings in it. His saying “And of knowledge, you (mankind) have been given only a little” has an endless, pleasant echo in my heart and always reminds me that our knowledge is much less than a drop in the ocean. I would like to express my sincere gratitude to Dr. Thomas F. La Porta, IEEE Fellow, Evan Pugh Professor and William E. Leonhard Professor in the Department of Computer Science and Engineering, and Director of the School of Electrical Engineering and Computer Science at Penn State University, for his great foreword, kindness, and endless support to me in several ways.

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I started this project on Tuesday, October 13, 2020, at 10:04 PM when I contacted Publishing Editor, Dr. Thomas Ditzinger, who approved my proposal for this book. I made all chapters for this book as well as the foreword was uploaded on the website of Springer and made accessible to Project Coordinator, Mr. Prasanna Kumar Narayanasamy, on March 21, 2022. Hence, this project lasted about a year and a half. I hope that the readers will appreciate all my modest efforts and love all the materials in this book. I have devoted a considerable amount of time to finish this book, and I hope that all of these efforts will be paid off in the future. I would like to acknowledge all of my family members who have provided me with an excellent source of support and constant encouragement over the course of this project. First of all, I am extremely grateful to both of my first teachers, my mother, Mbarka, who passed away on January 31, 2021, because of COVID-19, and my father, Mokhtar, for their sincere prayers, love, support, and encouragement, and for always teaching me and reminding me of the value of knowledge and the importance of family. I owe them a lot and cannot find my words to thank them enough for everything they have done to make me who I am now. In particular, I am very grateful to my mother for all she taught me since I became a small part of her pure soul and blessed body, until the last moment before she left this universe. In addition, I would like to express my hearty gratitude to my lovely and beautiful wife and children, Leena, Muath, Mohamed-Eyed, Lama, and Maitham, for their endless love, support, and encouragement. They have been one of my greatest joys, very patient, and understanding. Also, I am most grateful to my very best friend, companion, and wife for her genuine friendship and for being extremely supportive and unboundedly patient while I was working on this book. They all have been a wonderful inspiration to me and very patient throughout the life of this project. Without their warm love and care, this project would never even have been started. Furthermore, my special thanks and gratitude go to all of my sisters, brother, nieces, and nephews for their love, thoughtful prayers, concern, and valuable support all the time. This project could not have been completed without the great support of the people around me who made this experience successful and more than enjoyable. I would like to thank all my friends and colleagues at Texas A&M University-Kingsville (TAMUK), and, particularly, my fellows in the Frank H. Dotterweich College of Engineering, and especially in the Department of Electrical Engineering and Computer Science (EECS), for the collegial and very friendly atmosphere they provided me with to finish this book. In particular, I am very grateful to my Interim Dean, Dr. Robert Diersing, Professor and Executive Director, High-Performance Computing Center, for his kindness, continuous encouragement, and outstanding support to WiSeMANIoT-ACE Research Laboratory since I joined the EECS Department at TAMUK in August 2019. Also, I am very thankful to my Dean, Dr. Heidi A. Taboada, Professor of Mechanical and Industrial Engineering, for her wonderful support since she joined the Frank H. Dotterweich College of Engineering at TAMUK in January 2022. In addition, I am very thankful to Dr. Mahesh Hosur, Professor and Associate Dean for Graduate Affairs and Research, and Dr. Afzel Noore, Professor and Associate Dean for Undergraduate Affairs in the Frank H. Dotterweich College of Engineering

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at TAMUK, for their humbleness and outstanding support to me in several ways. Furthermore, I would like to express my profound gratitude to all my colleagues in the EECS Department at TAMUK, especially Drs. Rajab Challoo, Reza Nekovei, Sung-won Park, Scott C. Smith (EECS Department Chair), and Amit Verma; and Mr. G.R. Benavides, Laboratory Coordinator; for their friendship, kindness, extended support, and encouragement. In addition, I am very thankful to my colleagues in the Frank H. Dotterweich College of Engineering, Dr. Breanna M. Bailey, Professor and Interim Chair of the Department of Civil and Architectural Engineering, and Dr. Selahattin Ozcelik, Professor and Mechanical Engineering Graduate Coordinator in the Department of Mechanical and Industrial Engineering, for their friendship and great support. Moreover, I would like to convey my warm thanks and profound appreciation to all my colleagues, the staff members in the Frank H. Dotterweich College of Engineering Dean’s Office who have been very helpful and supportive, namely Dr. Cynthia Alvarado-Stinson, Outreach Coordinator; Ms. Tamara Denise Guillen, Executive Assistant II to the Dean; Ms. Annie Hinojosa, Business Administrator; Ms. Laura F. Salinas, Administrative Associate IV; Mr. Austin McCoy, Director of Javelina Engineering Student Success Center; Mr. Eric S. Hollingshead, End User Support Specialist IV; and Ms. Emily Morin, Administrative Coordinator II. Also, I am very thankful to my colleagues, Dr. Michelle R. Johnson-Vela, Professor of Spanish in the Department of Language and Literature in the College of Arts and Sciences, and previously Interim Director of the Center for Teaching Effectiveness; Dr. Jingbo Louise Liu, Professor of Chemistry in the Department of Chemistry in the College of Arts & Sciences, and Director of the Center for Teaching Effectiveness; Ms. Rose Anna Gomez in the Accommodations Counselor Disability Resource Center, Student Health & Wellness Center, and previously Administrative Associate IV in the Frank H. Dotterweich College of Engineering Dean’s Office; and Ms. Abigail De La Mora, Administrative Coordinator II in the Office of Academic Affairs, for their wonderful support and encouragement. In addition, I am very thankful and grateful to all my colleagues, the staff members in the Office of Research & Graduate Studies, namely Dr. Robert Diersing, Interim Vice President for Research and Graduate Studies; Ms. Diana Polendo Luna, Director, Contracts & Grants/Authorized Organizational Representative; Mr. Jeffrey Garza, Proposal Administrator I; Ms. Lizette Gonzales, Proposal Administrator I; and Ms. Martha Alegria, Graduate Coordinator; and in the Office of Admission, namely Dr. Cheri D. Shipman, Director of Strategic Enrollment Management, for their great support. This work is partially supported by the National Science Foundation (NSF) grants 0917089 and 1054935. Last but not least, I would like to convey my special thanks to all the people at Springer who have been involved in this book project. In particular, I would like to express my deep appreciation and gratitude to Dr. Thomas Ditzinger, Editorial Director, Interdisciplinary Applied Sciences, and Publishing Editor; Dr. Janusz Kacprzyk, Series Editor; Mr. Holger Schaepe, Senior Editorial Assistant; Ms. Sabine Gutfleisch, Editorial Assistant; Mr. Prasanna Kumar Narayanasamy and Ms. Sylvia Schneider, Project Coordinators for Books Production; Ms. Daniela Brandt, Project Coordinator for Publishing; Mr. Sooryadeepth Jayakrishnan, Project Manager for Book Production. It was a great pleasure to work with all of them. I would like to

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acknowledge the publisher, Springer, for the professionalism, patience, and the high quality of their typesetting team as well as their timely publication of this book. April 2022

Habib M. Ammari, Ph.D. (CSE), Ph.D. (CS) Founding Director, Wireless Sensor and Mobile Ad-hoc Networks, Internet of Things, and Applied Cryptography Engineering (WiSeMAN-IoT-ACE) Research Lab, Department of Electrical Engineering and Computer Science, Frank H. Dotterweich College of Engineering, Texas A&M University-Kingsville, Kingsville, TX, USA

Contents

Part I

Foundations of Wireless Sensor Networks

1

General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Major Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Limited Resources and Capabilities . . . . . . . . . . . . . . . . 1.2.2 Location Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Sensor Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Time-Varying Network Characteristics . . . . . . . . . . . . . . 1.2.5 Network Scalability, Heterogeneity, and Mobility . . . . 1.2.6 Sensing Application Requirements . . . . . . . . . . . . . . . . . 1.3 Sample Sensing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Book Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Book Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 5 5 6 6 7 7 8 8 9 10 12 14 20

2

Fundamental Concepts, Definitions, and Models . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Deterministic and Stochastic Sensing Models . . . . . . . . . . . . . . . . 2.4 Network Connectivity and Fault Tolerance . . . . . . . . . . . . . . . . . . . 2.5 Energy Consumption Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Percolation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Why a Continuum Percolation Model? . . . . . . . . . . . . . .

21 21 21 22 22 36 38 39 40 41

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2.7 2.8

2.9 Part II 3

4

Default Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random and Group Mobility Models . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Random Waypoint Mobility Model (RWP) . . . . . . . . . . 2.8.2 Reference Point Group Mobility Model (RPGM) . . . . . 2.8.3 Manhattan Mobility Model (MMM) . . . . . . . . . . . . . . . . 2.8.4 Why Group and Random Mobility Models? . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 43 43 44 44 45 46

Percolation Theory-Based Coverage and Connectivity in Wireless Sensor Networks

A Planar Percolation-Theoretic Approach to Coverage and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Phase Transition in Sensing Coverage . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Estimation of the Shape of Covered Components . . . . . 3.2.2 Critical Density of Covered Components . . . . . . . . . . . . 3.2.3 Critical Radius of Covered Components . . . . . . . . . . . . . 3.2.4 Characterization of Critical Percolation . . . . . . . . . . . . . 3.2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Phase Transition in Network Connectivity . . . . . . . . . . . . . . . . . . . 3.3.1 Integrated Sensing Coverage and Network Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Spatial Percolation-Theoretic Approach to Coverage and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Three Percolation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Sensing Coverage Percolation . . . . . . . . . . . . . . . . . . . . . 4.2.2 Network Connectivity Percolation . . . . . . . . . . . . . . . . . . 4.2.3 Coverage and Connectivity Percolation . . . . . . . . . . . . . 4.3 Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Practicality and Generalizability Issues . . . . . . . . . . . . . 4.3.2 Sensor Deployment in Spatial Fields . . . . . . . . . . . . . . . 4.3.3 Relaxations of Assumptions . . . . . . . . . . . . . . . . . . . . . . . 4.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 51 51 52 54 58 59 60 62 63 67 67 69 71 71 72 73 73 73 79 83 87 87 88 88 91 92

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Part III Convexity Theory-Based Connected k-Coverage in Wireless Sensor Networks 5

6

A Planar Convexity Theory-Based Approach for Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Achieving Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Connected k-Coverage Problem Modeling . . . . . . . . . . . 5.2.2 Sufficient Condition to Ensure k-Coverage . . . . . . . . . . 5.3 Centralized k-Coverage Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Planar Deployment Field Slicing . . . . . . . . . . . . . . . . . . . 5.3.2 Sensor Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Slicing Grid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Clustered k-Coverage Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Cluster-Head Selection and Attributed Roles . . . . . . . . . 5.4.2 The T-CRACCk Protocol . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 The D-CRACCk Protocol . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Triggered-Scheduling Driven k-Coverage . . . . . . . . . . . . . . . . . . . . 5.5.1 K-Coverage Checking Algorithm and Sensor Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 State Transition Diagram of Trig-DIRACCk . . . . . . . . . 5.5.3 Ensuring Network Connectivity . . . . . . . . . . . . . . . . . . . . 5.6 Self-scheduling Based k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 K-Coverage Candidacy Algorithm . . . . . . . . . . . . . . . . . 5.6.2 State Transition Diagram of Self-DIRACCk . . . . . . . . . 5.6.3 Tri-DIRACCk Versus Self-DIRACCk . . . . . . . . . . . . . . . 5.7 Relaxation of Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Relaxing the Unit Disk Model . . . . . . . . . . . . . . . . . . . . . 5.7.2 Relaxing the Sensor Homogeneity Model . . . . . . . . . . . 5.8 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Comparison of Self-DIRACCk with CCP . . . . . . . . . . . . 5.9 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planar Convexity Theory-Based Approaches for Heterogeneous, On-Demand, and Stochastic Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Heterogeneous Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Random Deployment Approach . . . . . . . . . . . . . . . . . . . .

95 95 96 98 98 99 99 103 103 104 105 108 109 109 111 113 114 116 117 119 120 120 121 122 122 123 124 124 124 128 132 135

137 138 138 140 140 141

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6.3

6.4

6.5

6.6 7

6.2.2 Pseudo-random Deployment Approach . . . . . . . . . . . . . 6.2.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . On-Demand Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Pseudo-random Sensor Placement . . . . . . . . . . . . . . . . . . 6.3.2 Sensor Mobility for k-Coverage of a Region of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Stochastic k-Coverage Characterization . . . . . . . . . . . . . 6.4.2 Stochastic k-Coverage-Preserving Scheduling . . . . . . . . 6.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Sensor Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Sensor Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Probabilistic Sensing Model . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Spatial Convexity Theory-Based Approaches for Connected k–Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Equilateral Spherical Triangle-Based Approach . . . . . . . . . . . . . . . 7.2.1 Problem Analysis: The Curse of Dimensionality . . . . . 7.2.2 Distributed k-Coverage Protocol . . . . . . . . . . . . . . . . . . . 7.2.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Reuleaux Tetrahedron-Based Approach . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Problem Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Optimized Spatial k-Coverage . . . . . . . . . . . . . . . . . . . . . 7.3.4 Using Reuleaux Tetrahedra for Sphere Coverage . . . . . 7.3.5 Reuleaux Tetrahedron-Based Spatial k-Coverage . . . . . 7.3.6 Assumption Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142 145 147 148 148 152 156 156 160 162 167 169 170 172 174 177 177 180 181 181 182 186 187 189 189 190 192 196 198 203 206 211 212

Part IV Applied Computational Geometry-Based Connected k-Coverage in Wireless Sensor Networks 8

A Planar Regular Hexagonal Tessellation-Based Approach for Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Study of Planar Pavers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 215 216 217 218

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8.3

8.4

8.5

8.6

8.7 8.8 9

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8.2.1 Paving Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Planar Regular Convex Paver Quality . . . . . . . . . . . . . . . Regular Hexagonal Centroid-Based Connected k-Coverage . . . . . 8.3.1 Achieving Optimal Coverage . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Problems with k-Coverage for k ≥ 2 . . . . . . . . . . . . . . . . Regular Hexagonal Area Stretching-Based Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Foundational Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Random Regular Hexagonal Tessellation . . . . . . . . . . . . 8.4.3 Hexagonal Cone-Based Pseudo-Random k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Hexagonal Perimeter-Based Pseudo-Random k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Edge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Extension 1: Using Non-Deterministic Sensing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Extension 2: Heterogenous Sensor Deployment . . . . . . Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A Planar Irregular Hexagonal Tessellation-Based Approach for Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Planar Tiling Using Congruent Tiles . . . . . . . . . . . . . . . . 9.2 Achieving Planar k-Coverage Using Hexagonal Tiles . . . . . . . . . . 9.2.1 Ensuring 1-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Ensuring k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Achieving Planar k-Coverage Using Irregular Hexagonal Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Irregular Hexagonal Tiling with I R H (r/2) . . . . . . . . . . 9.3.2 Irregular Hexagonal Tiling with I R H (r/3) . . . . . . . . . . 9.3.3 Irregular Hexagonal Tiling with I R H (r/n) . . . . . . . . . . 9.3.4 Discussion on Planar Sensor Density . . . . . . . . . . . . . . . 9.4 A k-Coverage Protocol Using Irregular Hexagonal Tiling . . . . . . 9.4.1 Generating Reference Irregular Hexagon and k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Expanding Hexagonal Grid and k-Coverage . . . . . . . . . 9.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 221 224 226 226 226 227 236 237 237 238 238 239 239 243 248 248 248 261 265 267 267 270 271 272 273 274 275 277 277 280 285 293 295 296 297 298

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9.4.4 Problem of Side-Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 299 300 300 306 308

10 A Polyhedral Space Filler Tessellation-Based Approach for Connected k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Investigating Polyhedral Space-Fillers . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Cubic Space-Filler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Regular Right Hexagonal Prism Space-Filler . . . . . . . . 10.2.3 Truncated Octahedral Space-Filler . . . . . . . . . . . . . . . . . 10.2.4 Great Rhombicuboctahedral Space-Filler . . . . . . . . . . . . 10.2.5 Rhombic Dodecahedral Space-Filler . . . . . . . . . . . . . . . . 10.2.6 Elongated Dodecahedral Space-Filler . . . . . . . . . . . . . . . 10.2.7 Rhombic Triacontahedral Space-Filler . . . . . . . . . . . . . . 10.2.8 Sommerville’s Largest Tetrahedral Space-Filler . . . . . . 10.2.9 Baumgartner’s Tetrahedral Space-Filler . . . . . . . . . . . . . 10.2.10 Goldberg’s Equilateral Octahedral Space-Filler . . . . . . 10.3 Solving the Connected Coverage Problem . . . . . . . . . . . . . . . . . . . 10.3.1 Sensor Selection Algorithm . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Connected k-Coverage Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Achieving Spatial k-Coverage . . . . . . . . . . . . . . . . . . . . . 10.4.2 Ensuring Spatial Connected k-Coverage . . . . . . . . . . . . 10.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Sensor Selection Protocol . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309 309 312 314 314 316 316 317 318 319 320 321 322 323 324 325 328 331 333 334 343 346 347 347 347 348 349 351

9.5

9.6 9.7

Part V

Connectivity and Fault-Tolerance Measures of k-Covered Wireless Sensor Networks

11 Planar Unconditional and Conditional Network Connectivity and Fault-Tolerance Measures for k-Covered Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

355 355 357 358

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11.2 Unconditional Fault-Tolerance Measures . . . . . . . . . . . . . . . . . . . . 11.2.1 Homogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Heterogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Conditional Fault-Tolerance Measures . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Homogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Heterogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

358 358 362 367 368 370 373 373

12 Spatial Unconditional and Conditional Network Connectivity and Fault-Tolerance Measures for k-Covered Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Unconditional Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Homogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Heterogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Boundary Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Conditional Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Homogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Heterogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Relaxing the Assumption of k ≥ 4 . . . . . . . . . . . . . . . . . 12.4.2 Sensor Placement Strategy . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Sink-Independent Connectivity Measures . . . . . . . . . . . 12.4.4 Spatial Sensing Applications . . . . . . . . . . . . . . . . . . . . . . 12.5 Relaxing the Unit Sphere Model: Convex Model . . . . . . . . . . . . . . 12.5.1 Homogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Heterogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Underwater Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

375 375 377 378 378 379 383 384 385 385 387 390 391 391 391 392 393 393 394 394 395 396

Part VI

Geographic Data Forwarding, Gathering, and Delivery in Wireless Sensor Networks

13 A Planar Checkpoints-Based Approach for Geographic Forwarding on Always-on Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The WLDT Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Long-Range Versus Short-Range Forwarding . . . . . . . . 13.2.2 A Two-Step Data Forwarding Protocol . . . . . . . . . . . . . . 13.2.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 399 400 401 401 401 404 408

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13.3 Analysis of WLDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Short-Range Versus Long-Range . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Energy Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Controlled Short-Range Data Forwarding . . . . . . . . . . . 13.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

408 416 416 419 422 423 423

14 A Planar Energy-Delay Trade-off Based Approach for Geographic Forwarding on Always-on Sensors . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 A Slicing Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Slicing of Communication Range . . . . . . . . . . . . . . . . . . 14.2.2 Selection of Candidate Proxy Forwarders . . . . . . . . . . . 14.2.3 Uniform Energy Depletion Characterization . . . . . . . . . 14.3 Trading-off Energy with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Simple Analytical Bounds . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Multi-objective Optimization Approach . . . . . . . . . . . . . 14.3.3 TED Detailed Description . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Relaxation of Several Key Assumptions . . . . . . . . . . . . . . . . . . . . . 14.4.1 Relaxing the Sensor Homogeneity Model . . . . . . . . . . . 14.4.2 Relaxing the Communication Disk Model . . . . . . . . . . . 14.4.3 Relaxing the Dense Network Model . . . . . . . . . . . . . . . . 14.4.4 Relaxing the Energy Consumption Model . . . . . . . . . . . 14.4.5 Relaxing the Always-on Sensors Model . . . . . . . . . . . . . 14.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Impact of Selection Space Size . . . . . . . . . . . . . . . . . . . . 14.5.3 Using the Energy × Delay Metric . . . . . . . . . . . . . . . . . . 14.5.4 Impact of Variability of k . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.5 Impact of Sensor Heterogeneity . . . . . . . . . . . . . . . . . . . . 14.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425 425 426 427 427 428 429 430 431 431 437 442 452 452 453 453 453 454 454 455 455 456 460 461 462 463

15 A Planar Approach for Solving the Energy Sink-Hole Problem with Always-on Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Energy Sink-Hole Problem Analysis . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Base Protocol Average Energy Consumption . . . . . . . . 15.2.2 Nominal Communication Range–Based Data Forwarding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15.2.3 15.3

15.4

15.5

15.6 Part VII

Adjustable Communication Range-Based Data Forwarding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using Heterogeneous Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Multi-tier Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 NEAR Performance Evaluation . . . . . . . . . . . . . . . . . . . . Sink Mobility and Energy Aware Voronoi Diagram . . . . . . . . . . . . 15.4.1 Why Energy Aware Voronoi Diagram? . . . . . . . . . . . . . . 15.4.2 EVEN Detailed Description . . . . . . . . . . . . . . . . . . . . . . . 15.4.3 EVEN Performance Evaluation . . . . . . . . . . . . . . . . . . . . Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Balancing Energy Consumption . . . . . . . . . . . . . . . . . . . 15.5.2 Minimizing Energy Consumption . . . . . . . . . . . . . . . . . . 15.5.3 Mobility-Based Forwarding Protocols . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

472 478 478 481 483 484 484 487 490 491 492 492 493

Joint k-Coverage and Geographic Data Forwarding and Gathering in Wireless Sensor Networks

16 Planar and Spatial Approaches for Joint k-Coverage and Data Collection Using Homogeneous Duty-Cycled Sensors . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 A Planar Approach for Joint k-Coverage and Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Potential Fields Based Modeling Approach . . . . . . . . . . 16.2.2 Data Forwarding Without Aggregation . . . . . . . . . . . . . . 16.2.3 Data Forwarding with Aggregation . . . . . . . . . . . . . . . . . 16.2.4 Generalizability of GEFIB . . . . . . . . . . . . . . . . . . . . . . . . 16.2.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 A Spatial Approach for Joint k-Coverage and Composite Forwarding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 First Hybrid Geographic Forwarding . . . . . . . . . . . . . . . 16.3.2 Second Hybrid Geographic Forwarding . . . . . . . . . . . . . 16.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A Planar Approach for Joint k-Coverage and Data Collection Using Sparsely Deployed Duty-Cycled Sensors . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Heterogeneous k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Mobile k-Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Four-Tier Sensor Network Architecture . . . . . . . . . . . . . 17.3.2 k-Coverage Approach Design Decisions . . . . . . . . . . . .

497 498 499 500 500 501 502 503 508 510 512 513 518 522 524 525 526 528 530 530 532 532 533

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17.3.3 Achieving Mobile k-Coverage . . . . . . . . . . . . . . . . . . . . . 17.4 Data Gathering Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Direct Data Gathering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Chain-Based Data Gathering . . . . . . . . . . . . . . . . . . . . . . 17.5 Impact of Sensor Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Planar Approaches for Joint k-Coverage and Data Collection Using Heterogeneous Duty-Cycled Sensors . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Basic Two-Tier Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Impact of the Energy Sink-Hole Problem . . . . . . . . . . . . 18.2.2 Energy Consumption Analysis . . . . . . . . . . . . . . . . . . . . . 18.3 Three-Tier Architecture with Constant Band Width . . . . . . . . . . . . 18.3.1 Proposed Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Joint Mobility and Routing . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Architecture 1: 1 Static Sink—1 Mobile Proxy Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.4 Architecture 2: 1 Static Sink—N Mobile Proxy Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.5 Architecture 3: N Static Sinks—1 Mobile Proxy Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.6 Architecture 4: N Static Sinks – N Mobile Proxy Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.7 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Three-Tier Architecture with Varying Band Widths . . . . . . . . . . . . 18.4.1 Proposed Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Static Data Collection Schemes . . . . . . . . . . . . . . . . . . . . 18.4.3 Mobile Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

534 542 542 543 548 550 550 552 557 559 561 562 563 565 565 565 566 568 568 569 571 578 582 584 585 589 590 590 595 599 602

Part VIII Connected k-Barrier Coverage in Wireless Sensor Networks 19 A Planar Approach for Physical Security Using Connected k-Barrier Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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19.2 Tiling-Based k-Barrier Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Intruder’s Abstract Path Counting . . . . . . . . . . . . . . . . . . 19.2.2 Intruder’s Abstract Path Analysis . . . . . . . . . . . . . . . . . . 19.2.3 Square Lattice-Based Sensor Deployment . . . . . . . . . . . 19.2.4 Hexagonal Lattice-Based Sensor Deployment . . . . . . . . 19.2.5 Square Lattice Versus Hexagonal Lattice . . . . . . . . . . . . 19.2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Source-to-Destination Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . 19.4.1 Square k-barrier Covered Sensor Belt . . . . . . . . . . . . . . . 19.4.2 Rectangular k-barrier Covered Sensor Belt . . . . . . . . . . 19.5 Other Possible Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

610 610 615 618 619 622 626 626 630 630 641 646 648 648 648 649 652

20 A Spatial Approach for Physical Security Through Connected k-Barrier Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Spatial k-Barrier Coverage Problem Analysis . . . . . . . . . . . . . . . . . 20.2.1 Simple Cubic Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2 Body Centered Cubic (BCC) Lattice . . . . . . . . . . . . . . . . 20.2.3 Face Centered Cubic (FCC) Lattice . . . . . . . . . . . . . . . . 20.2.4 Hexagonal Close-Packed (HCP) Lattice . . . . . . . . . . . . . 20.3 Polyhedral Space-Filling Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Intruder’s Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2 Intruder’s Path Representation and Counting . . . . . . . . . 20.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.2 Numerical Versus Simulation Results . . . . . . . . . . . . . . . 20.5 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

655 656 656 657 658 658 660 664 667 669 672 675 679 680 680 681 682

Part IX Applications of Wireless Sensor Networks and Concluding Remarks 21 An Overview of Sensing Hardware, Standards, Operating Systems, Software Development, and Applications and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Major Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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21.2 Sensing Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Mote Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Sensor Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Gateways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Sensing Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Industry Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Operating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Sensing Software Development: Challenges and Solutions . . . . . 21.4.1 Sensing Application Models . . . . . . . . . . . . . . . . . . . . . . . 21.4.2 Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.3 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.4 Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.5 Protocols and Radio Communication . . . . . . . . . . . . . . . 21.4.6 Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Sensing Applications and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5.1 Healthcare Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5.2 Agriculture Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5.3 Environmental Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5.4 Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5.5 Military . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Future Applications and Technologies . . . . . . . . . . . . . . . . . . . . . . . 21.6.1 Marine Deployments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.2 Smart Homes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

687 687 688 693 698 698 699 702 702 702 704 705 705 706 707 707 712 713 716 720 724 724 725 726

22 Summary and Further Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 22.1 Summary of Book Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 22.2 Further Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

Part I

Foundations of Wireless Sensor Networks

Chapter 1

General Introduction

If you want to find the secrets of the universe, think in terms of energy, frequency and vibration. Nikola Tesla (1856–1943)

Overview This chapter gives a brief introduction to wireless sensor networks and presents the major challenging problems in their analysis, design, and development. Moreover, it describes a sample of their potential applications as well as a key set of design requirements of the algorithms and protocols proposed in this book. In addition, it states the eleven research problems being investigated in this book along with a brief description of their possible solutions.

1.1 Introduction Recent advances in miniaturization, low-cost and low-power circuit design, and wireless communications have led to the development of low-cost, low-power, and physically small communication devices, called sensors. Like nodes (or computers, laptops, etc.) in traditional wireless networks, such as mobile ad hoc networks, the sensors have data storage, processing, and communication capabilities. Unlike those nodes, the sensors have an extra functionality related to their sensing capability. These sensors can be engaged in a variety of sensing tasks, such as temperature, sound, vibration, light, humidity, to name a few. This sensor technology made the design and development of large-scale wireless sensor networks possible for a variety of real-world applications dealing with monitoring (e.g., health and environmental monitoring, seism monitoring), control (object detection and tracking, industrial process automation), and surveillance (battlefields surveillance). The popularity of wireless sensor networks is due to the fact that they are cost-effective and appealing to a wide range of mission-critical situations. A wireless sensor network consists of low-powered sensors that have the capability of sensing the physical environment, collecting and processing data, and communicating with each other to accomplish certain tasks. Also, wireless sensor networks are commonly characterized by the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_1

3

4

1 General Introduction

presence of a central gathering point, called the sink, where all the data collected by the sensors reside. These sensors sense specific environment phenomenon and may also perform in-network processing on the sensed data before sending their results to the sink for further processing and analysis. Indeed, all the decision-making process is run by the sink based on the collected data. In this type of network, sensors communicate with each other (possibly) through multi-hop, wireless communication links and forward sensed data on behalf of others so the sink can receive them on-time. The major challenge in the design of wireless sensor networks is mainly due to the inherent constraints imposed on the storage, processing, sensing, and communication capabilities of the sensors. In particular, compared to personal computers, for instance, the sensors have severely constrained power supplies, making them unreliable. In fact, they become faulty due to improper hardware functioning as well as low battery power (or energy). Under the assumption that sensors’ faults are caused only by insufficient energy reserves, the latter is the unique crucial resource that should be considered in the design and implementation of this type of network for their correct operation and longevity. For excellent surveys on sensor networks, the interested reader is referred to [10, 11, 379]. Next, we briefly describe the major tasks we want to accomplish in this chapter. Moreover, we briefly state how to achieve each one of them.

1.1.1 Major Tasks Compared to traditional wireless networks, such as mobile ad hoc wireless networks, wireless sensor networks have several inherent characteristics. First, given that sensors’ energy is the most crucial resource, the sensors are usually densely deployed in a field so as to extend the network lifetime. Indeed, using a large number of sensors facilitates multi-hop communication between them, and hence the sensors can save their energy by transmitting or forwarding their sensed data through short distances. Liu et al. [273] proposed a distributed information dissemination and retrieval system for wireless sensor networks, where each sensor is autonomous and can be viewed as a data source as well as a data sink. They suggested two protocols that allow the information at a data source to reach any interested sensor in this information in an energy-efficient manner. While the first protocol is based on the quorum scheme, the second one uses the home agent scheme. Second, the network topology may change very frequently as sensors join and/or leave the network. Thus, protocols designed for wireless sensor networks should account for all these features, which are inherent to these types of networks so they remain operational as longer as possible. For localized algorithms for coordination among the sensors, and directed diffusion, the interested reader is referred to the work by Estrin et al. [153]. We should mention that the design of this type of network requires taking into consideration several parameters, which may depend on the type of wireless sensing application or system to be developed. Therefore, the network designer should consider some key tasks for a successful design, analysis, and implementation of their target wireless sensing

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5

applications or systems. Next, we specify these tasks and give our corresponding plan of actions. First, we want to know all the challenges that face the design and development of wireless sensor networks. We consider the inherent characteristics of this type of network. Then, we discuss other issues related to sensor location management, sensor deployment, time-varying network features, network scalability, and sensor heterogeneity and mobility. Second, we are interested in all applications of wireless sensor networks. We focus on various areas, such as health, home, environmental, and military. We briefly list potential applications of wireless sensor networks spanning these areas. Third, we want to investigate the major design requirements for wireless sensor networks. We briefly discuss various key requirements, such as energy awareness, on-demand connected k-coverage, autonomous and purposeful mobility, fault tolerance, heterogeneity, and dimensionality, to name a few, for the design, analysis, and development of algorithms and protocols for a special class of wireless sensor networks, called k-covered wireless sensor networks.

1.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 1.2 gives the challenges that face the design of wireless sensor networks while Sect. 1.3 describes a sample of their potential applications. Section 1.4 presents the motivations of the work in this book. Section 1.5 describes the major requirements driving the design of the algorithms and protocols that we propose in this book. Section 1.6 discusses the major contributions of this book by stating the eleven research problems that are addressed in this book along with a brief overview of the proposed solutions. Section 1.7 concludes the chapter.

1.2 Major Challenges The design of network protocols for wireless sensor networks, including those for coverage configuration and data dissemination, is a challenging problem due to several constraints. Next, we describe these constraints that are imposed not only by the characteristics of the individual sensors, the behavior of the network, and the nature of physical environments (or deployment fields), but also by the requirements of the sensing applications in terms of some desirable metrics.

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1.2.1 Limited Resources and Capabilities Because of their inherent characteristics, the design of wireless sensor networks for different applications running in different deployment fields is facing several challenges. First of all, energy efficiency is the primary concern in the design of wireless sensor networks [148]. Indeed, the sensors forming a network suffer the limitations of several resources, such as storage, CPU, bandwidth, communication, sensing, and battery power (or energy). In fact, Woo and Culler [413] proposed an energyefficient media access control (MAC) and adaptive transmission rate control scheme for sensor networks. This MAC protocol is not only energy efficient, but also allows fair bandwidth allocation to the infrastructure for all nodes in the network. Also, Rodoplu and Meng [339] proposed an optimized distributed network protocol whose goal was to achieve the minimum energy for randomly deployed ad hoc networks. This is a position-based algorithm that helps set up and maintain a minimum energy network between users (or nodes), which are randomly deployed over an area and are allowed to move with random velocities. In addition, Li and Halpern [258] proposed a minimum-energy communication network protocol, which computes a subnetwork G’ from a given network G such that, for every pair of nodes (u, v), which are connected in the original network G, there is a minimum-energy path between u and v in the subnetwork G’. That is, this path enables message transmission between u and v with a minimum use of energy. As it can be seen, energy is the most crucial resource for mobile ad hoc networks, and, particularly, wireless sensor networks. In fact, energy determines the lifetime of the sensors and hence the lifetime of the entire network. Energy poses a serious problem for network designers especially in hostile environments, such as battlefields, where it is difficult or even impossible to access the sensors and recharge or renew their batteries. Furthermore, when the energy of the sensors reaches a certain threshold, they become unreliable (or faulty) and would not be able to function properly. As a consequence, the behavior of those faulty sensors will have a major impact on the network performance. Thus, network protocols and algorithms designed to be run by the sensors should be as energyefficient as possible to extend their lifetime and hence prolong the network lifetime while guaranteeing good performance overall. We should mention that it is important to quantify the remaining energy distribution within a sensor network. Indeed, Zhao et al. [452] proposed sensor network scans, which help inform on the remaining energy distribution within a sensor network. These residual energy scans can help notify users of resource depletion or abnormal activities.

1.2.2 Location Management Sensor location management is another major challenge in the design of deployment strategies to achieve a certain degree of coverage. In most of the protocols designed for wireless sensor networks, the sensors are aware of their locations through either

1.2 Major Challenges

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the use of global positioning system (GPS) receivers or some localization technique, such as the ones proposed by Bulusu et al. [97] and Ji and Zha [216]. On the one hand, a GPS receiver-based solution provides the sensors with highly accurate locations but is not cost-effective for densely deployed sensors given that each sensor should be equipped with a GPS receiver. On the other hand, the use of a localization technique does not require any additional cost but may not guarantee high sensor location accuracy.

1.2.3 Sensor Deployment As mentioned earlier, a deployment field may cause a problem not only to access the sensors for replacing and/or recharging their batteries but also for their deployment. Thus, a deterministic sensor deployment strategy is not always possible. Such a strategy would help cover the field appropriately and minimize the total number of sensors required to achieve the specific needs of sensing applications in terms of their expected type of coverage. Indeed, an application may demand partial coverage where only a certain percentage of the field is covered; full coverage, where the entire field is covered; or redundant coverage, where every location in the field is covered by multiple sensors simultaneously. In the case where the sensors cannot be deployed deterministically because of the field nature, random deployment is the only remaining strategy. However, there is no guarantee that the coverage required by the application would be satisfied. There might be some areas that are not covered well or even not covered at all and this would lead to a problem, known as coverage hole. Moreover, all the deployed sensors are not guaranteed to be connected to each other or to the sink. This would lead to another problem, known as connectivity hole. These are two of the reasons why most of time wireless sensor networks are designed with densely deployed sensors. Thus, the nature of the field has an influence on the network and this is a challenge for the designer and the investing party at least costwise. As discussed in the next section, one of the most widely used assumption in the design of routing and data dissemination protocol is highly sensor density. Although highly dense deployed wireless sensor networks involve more than necessary sensors, they help guarantee network connectivity and achieve the coverage demanded by the application.

1.2.4 Time-Varying Network Characteristics The topology of a wireless sensor network, which is defined by the sensors and communication links between them, changes frequently due to sensor addition and deletion. When new sensors decide to join the network, the neighbor sets of some sensors have to be updated. Indeed, it may seem necessary to add more sensors to

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maintain certain properties of coverage of the deployment field and network connectivity. Similarly, when the sensors deplete all their energy, they are considered faulty and no longer belong to the network. Thus, the neighbor sets of the fault sensors’ neighbors should be updated. Also, in mobile wireless sensor networks, the network topology gets updated as the sensors move in the deployment field. Consequently, any topology change in the network will have an impact on the communication paths (or routes) between the sensors in the network. Therefore, routing and data dissemination paths should consider network topology dynamics due to limited energy and mobility of the sensors as well as increasing the size of the network to maintain specific application needs in terms of coverage and connectivity. It is worth noting that connectivity to the sink is very important. In fact, coverage would be meaningless if the sensed data cannot reach the sink, i.e., there is no communication paths between the source sensors (or data generators) and the sink. Thus, connectivity between all source sensors and the sink either directly or indirectly should be guaranteed for the correct operation of the network.

1.2.5 Network Scalability, Heterogeneity, and Mobility The design of protocols for connected k-covered wireless sensor networks should consider network scalability. In other words, these protocols should scale with the number of sensors in the network, sensor mobility, and the size of the deployment field. For large-size networks, the number of sensors could be on the order of thousands. Also, the sensors may not necessarily be static given that sensor mobility helps achieve better quality of coverage [138, 270]. Furthermore, the sensors could be heterogeneous with regard to their storage, processing, communication, sensing, and energy capabilities and resources. In real-world applications, wireless sensor networks are composed of heterogeneous sensors that have a potential to increase the network lifetime and reliability without causing significant increase in its cost [433]. Indeed, deploying heterogeneous sensors helps reduce the probability of simultaneous failure of the entire neighbor set of a sensor [46].

1.2.6 Sensing Application Requirements In most sensing applications, the sensed data should be as accurate as possible to assure better decision making by the sink. Moreover, the sensed data needs to reach the sink in a timely manner. Thus, the delay metric should also be considered in the design process of wireless sensor networks; otherwise, the underlying network may not be useful. Also, for several sensing applications, data redundancy is desirable in that it increases data accuracy. For instance, in an intruder detection and tracking application, multiple sensors should be active at the same time to gather enough information about the intruder and track its motion accurately. Therefore, the design

1.3 Sample Sensing Applications

9

of coverage configuration and routing and data dissemination protocols should guarantee data delivery and accuracy so the sink can gather the required knowledge about the physical phenomenon on time. Furthermore, the sensors may deplete their energy before expected and become faulty. As discussed earlier, a deployment field may not be accessible and thus replacing those faulty sensors would be impossible. Hence, a wireless sensor network should tolerate the presence of faulty sensors and remain functional in spite of those failures. The degree of fault tolerance of a network depends on the underlying sensing application. Thus, coverage configuration and routing and data dissemination protocols for wireless sensor networks should be fault-tolerant for this type of sensor failure. It is worth noting that the link and sensing unit failures may also occur during the operation of a wireless sensor network. While sensing unit failure are due to imperfections in manufacturing or aging, link failures are caused by sensor failures and sensor mobility. In this book, however, we only consider sensor failures due to low battery.

1.3 Sample Sensing Applications The design of wireless sensor networks should also be guided by the very specific requirements of the target applications. The knowledge gathered about the underlying application would help a network designer deploy more appropriate types of sensors and develop algorithms and protocols that meet the needs of the application. In this section, we describe some potential applications of wireless sensor networks spanning health, home, environmental, and military areas [10, 11, 61]. • Tracking and monitoring a hospital: Sensors may be attached to patients and doctors. For a patient, specific sensors are used to perform a particular task. For instance, to detect the heart rate, a special sensor needs to be used. Also, to detect the blood pressure, another specific type of sensor has to be used. For a doctor, sensors may be used to track their locations in the hospital to facilitate their mission. • Smart environment: One of the home applications is the design and development of a smart home (or environment), where a wireless sensor network can be deployed to satisfy the specific needs of habitants. The sensors could be embedded anywhere in a room (or apartment) and communicate with each other to offer services desired by habitants. For instance, for saving energy, the light and temperature in a room could be controlled by the sensors. In this case, the light is on only when the habitants are in the room and the temperature should be set to appropriate value depending on the time and season, for instance. The goal of this type of network is to provide habitants with the level of comfort they wish to have without any human intervention. • Forest fire detection: The sensors could be randomly and densely deployed in a forest to detect the origin of a fire and report this information in a timely fashion

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to the end users to act accordingly before the fire spreads. This helps avoid catastrophic situations that may result. In this type of application, the sensors may be used for a long period of time, and hence have to be equipped with continuous source of energy, such as solar cells. Furthermore, the sensors need to collaborate with each other in their sensing activity to overcome several problems, such as obstacles. Also, the sensors should be densely deployed for a quick and accurate detection. • Intruder detection and tracking: Business stores, for instance, could be covered with special sensors to detect and track the motion of intruders. To achieve high accuracy of detection and tracking of an intruder, sensor redundancy is desirable and hence a dense network should be deployed. When an intruder is detected by some sensors, several other sensors become awake to cover the trajectories of the intruder. The collected information about an intruder is reported to end users for analysis and processing. • Battlefield surveillance: A wireless sensor network can be deployed in a battlefield for performing detection and tracking of target objects, such as tanks and vehicles, and sending real-time information about the enemy mobility to a central control unit. Precisely, a network should be able to detect and classify multiple targets, such as vehicles and troop movements, using sensors that are capable of sensing acoustic and magnetic signals generated by different target objects.

1.4 Book Motivations There are several critical applications, such as intruder detection and tracking, where wireless sensor networks need to be deployed in a planar or spatial field in such a way that every point is sensed (or covered) by at least one sensor. In particular, it is sometimes desirable to deploy a dense wireless sensor network to achieve redundant coverage of a deployment field, where every point in the field is guaranteed to be covered by at least k sensors simultaneously and we say that the network is configured to provide k-coverage. Our main interest in writing this book on k-covered wireless sensor networks is motivated by at least the following three applications, which require a degree of coverage that is at least equal to three, i.e., k ≥ 3. First, in order to cope with the problem of sensor failures due their fragility, the design of sensor networks for planet exploration [370] should be as reliable as possible since failed sensors in space cannot be easily diagnosed and replaced. Sun et al. [370] simulated a Confidence Weighted Voting technique on top of a k-cover deployment strategy. They showed that k-cover deployment with k ≥ 3 is necessary to guarantees data redundancy, which improves data reliability and fault tolerance of sensing applications. Indeed, high coverage degree helps achieve higher sensing accuracy and stronger robustness against sensor failures. Second, multiple-sensor data fusion [234] was found to be useful for at least a three-sensor system, i.e., system whose degree of coverage is k ≥ 3. Klein [234] discussed how this type of system helps detect, classify, and track the target objects.

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Since at least all the three sensors participate in the decision, it is unlikely that a false target would be detected as a true target. Third, the design of triangulation-based positioning systems [312] requires that each point in a target field be covered by at least three sensors. Nicules and Nath [312] showed that this type of positioning system helps increase the accuracy of the positions of the sensors. The design of network configuration protocols for wireless sensor networks faces a challenging problem, namely energy conservation, due to the constrained battery power of the sensors. Several energy conservation protocols for wireless sensor networks have been proposed at the MAC and network layers, such as the ones suggested by Biswas and Morris [83, 84], Casari et al. [104], and Zorzi and Rao [458, 459]. Moreover, a variety of energy-efficient coverage configuration schemes have been suggested, such as the ones proposed by Wang et al. [401], Xing et al. [425], and Zhang and Hou [447]. In general, the sensors are deployed with high density, and hence the design of network configuration protocols should benefit from this fact to provide k-coverage. It is well known that the best approach to save the energy of the sensors is duty-cycling so the sensors remain operational for as long as possible. Using a duty-cycling approach, the sensors can be turned on (i.e., active) or off (i.e., inactive) according to some sleep-wakeup scheduling protocol while guaranteeing k-coverage all the time. Achieving k-coverage becomes difficult especially in hostile environments, such as battlefields, where access to the sensors is not feasible or even impossible. This implies that the sensors cannot be always on but rather duty-cycled. Otherwise, they will deplete their energy and die quickly. Also, k-coverage of a field should use as minimum number of active sensors as possible to extend the network lifetime. Before all, the main goal of sensor deployment is to monitor a field and report data to the sink for further analysis and processing. Hence, the sensors should also be able to forward data on behalf of each other. More importantly, the load of data forwarding should be evenly distributed among all the active sensors, which currently k-cover the field, so all the sensors have the same chance to relay data for others. This implies that the network of active sensors should be connected. Otherwise, the sensed data will not reach the sink. Indeed, network connectivity is required for data routing and information dissemination. Therefore, it is important that the network provide k-coverage while maintaining connectivity between all active sensors. In addition, it is well known that geographic forwarding is an energy-efficient and practical scheme for wireless sensor networks. Indeed, the sensors are not required to maintain global and detailed information on the topology of the entire network. The sensors need only maintain local knowledge on their one-hop neighbors. Thus, for more effective sensor deployment, the load of k-coverage and data forwarding should be evenly distributed among all the sensors to maximize the network lifetime.

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1.5 Design Requirements In this section, we summarize the major requirements driving the design, analysis, and development of protocols for k-covered wireless sensor networks. Particularly, we identify those requirements that are necessary for building our unified framework, where coverage, duty-cycling, and geographic forwarding are jointly considered. These requirements are summarized as follows: • Energy Awareness: The sensors have severe limitations in terms of storage, computational, communication, and sensing capabilities. In particular, the sensors have scarce battery power (or energy). Thus, prolonging the network lifetime is the major challenge in the design and development of wireless sensor networks. With this in mind, it is essential that we design energy-aware protocols in all facets of the network operations, including sensor deployment, coverage and connectivity, scheduling, and data routing and dissemination. • On-Demand Connected k-Coverage: Several sensing applications prefer collecting redundant data to guarantee the most accurate decision-making process. Indeed, intruder detection and tracking applications may require more than one sensor to be active to collect information about the intruder (or malicious node) and track its motion accurately. Also, the sensors may die quickly because of their low energy, and thus, the network should be fault-tolerant. Unlike most related work that focused on connected k-coverage with static sensors and fixed degree k of coverage, where each point in a field is covered (or sensed) by at least k sensors, our proposed framework supports mobile connected k-coverage, where a region of interest is k-covered using mobile sensors and k may change over time. Also, the degree k of coverage of a region of interest may not be the same for another region in the field. We call this on-demand connected k-coverage. Our framework adopts centralized and distributed strategies to ensure mobile connected k-coverage of a region of interest in a deployment field while maintaining network connectivity. • Autonomous and Purposeful Mobility: In addition to their mobility, the sensors are autonomous, and thus are able to make their own decision based on the information they receive from the rest of the network. Particularly, these sensors should be able to move to designated locations in a region of interest whenever necessary to ensure its k-coverage and accomplish the target mission that has been determined by the sink. In order to account for mission-oriented wireless sensor networks, where the mobility of the sensors is controlled by the underlying mission, our framework enables purposeful mobility of the sensors and this mobility must be traded-off against the goals of the missions. • Situation Awareness and Intelligent Collaboration: Situation awareness is an essential and critical foundation for successfully accomplishing all missions, and this requires intelligent collaboration between the sensors. Thus, all the sensors should be aware of any mission that has to be accomplished. This situationawareness assumes that there is a central entity, such as the sink, which decides

1.5 Design Requirements

•

•

•

•

•

13

the type of mission that has to be accomplished and in which region in the deployment field. This information should be propagated within the network while minimizing the total energy consumption needed to advertise this information about the mission. Self-Organization: The sensor mobility and scheduling may affect the topology of the network, which could possibly result in a connectivity-hole problem and/or a coverage-hole problem. One of the goals of our proposed framework is to make the sensors self-organizing and adaptive so they guarantee network connectivity, which is required for communication between the sensors, data forwarding from the sensors to the sink, and data dissemination from the sink to the sensors. Also, it enables the sensors to achieve any degree of coverage needed by an application for a mission. Fault Tolerance: In most sensing applications, the sensed data should be accurate to ensure better decision making by the sink. Also, data redundancy is desirable in that it increases data accuracy. For instance, in the intruder detection and tracking application, multiple sensors should be active to gather enough information about the intruder (or malicious node) and track its motion accurately. Redundant coverage (or k-coverage) ensures higher data redundancy and accuracy. Indeed, sensor nodes may deplete their energy and die. Hence, a network should tolerate the presence of faulty sensors and remains functional. Thus, kcoverage provides a high degree of fault tolerance of the network, where the value of k depends on the requirements of the application in terms of coverage. Heterogeneity: Unlike most related work that considered k-coverage with homogeneous sensors, our framework focuses on heterogeneous sensors, which do not necessarily have the same capabilities in terms of computation, storage, sensing and communication ranges, and initial energy. In real-world applications, wireless sensor networks are composed of heterogeneous sensors that have a potential to increase the network lifetime and reliability without causing significant increase in its cost. Indeed, as mentioned earlier, the use of heterogeneous sensors helps reduce the probability of simultaneous failure of all the neighbors of a sensor [55]. Dimensionality: In the literature, most of the works on wireless sensor networks dealt with planar (or two-dimensional Euclidean) settings, where sensors are deployed in a planar field. However, there exist several applications that cannot be effectively modeled in the Euclidean plane. For instance, sensors deployed on the trees of different heights in a forest, or in a building with multiple floors, or underwater applications [8, 9] require the design in the three-dimensional Euclidean space (or simply space). Moreover, oceanographic data collection, pollution monitoring, offshore exploration, disaster prevention, and assisted navigation are typical applications of underwater sensor networks [9], which have to be designed using spatial (or three-dimensional) settings, which represent more accurately the network design for real-world applications. Stochastic Features: Although the majority of studies on wireless sensor networks considered the disk sensing model, where all sensor readings are assumed to be precise and have no uncertainty, the signal attenuation and the presence of noise associated with sensor readings require the use of a more realistic sensing model

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that reflects the real properties of the sensors. Precisely, the sensing capability of a sensor must be modeled as the probability of successful detection of an event, and hence should depend on the distance between it and the event as well as the type of propagation model being used (free-space vs. multi-path). Indeed, it has been showed that the probability that an event in a distributed detection application can be detected by an acoustic sensor depends on the distance between the event and the sensor [139]. Thus, the protocols designed for wireless sensor networks should account for the probabilistic nature of the sensor capabilities, and particularly, their sensing and communication ranges. • Data Delivery: The main goal of the design of wireless sensor networks is to monitor a phenomenon in a field of interest and collect data related to that specific phenomenon. It is very important that the sensed data reach the sink for further processing. The accuracy of the decisions and actions taken by the sink depend on the availability of the data. Thus, the sensors should be able to use robust protocols that will enable them to deliver the data they have collected to the sink with high data delivery ratio. This issue depends on the data forwarding and dissemination capabilities of the sensors, which are in turn strongly dependent on the efficiency of the design of the corresponding routing protocols. • Delay: For some time-critical sensing applications, such as forest fire detection and tracking, the sink should receive the sensed data collected by the sensors in a timely manner to avoid any undesirable consequence. It is necessary that the design of protocols for these types of sensing applications be conducted under the delay constraint so that the sensed data reach the sink within a certain time bound. In case the sensors are always on (or active), the data forwarding and dissemination protocol are responsible for meeting this delay constraint. However, when the sensors are duty-cycled, the duty-cycling protocols are also responsible to meet this delay bound.

1.6 Book Contributions This book aims at investigating the following eleven research problems, which are not totally disjoint. Each problem is introduced by a statement, which is accompanied by a brief solution statement. • Research Problem Statement 1: Almost Sure Connected Coverage What is the critical sensor density above which a deployment field (respectively, network) is almost surely covered (respectively, connected)? Solution Statement: We propose continuum percolation-based approaches to study phase transitions in coverage and connectivity in static wireless sensor networks in an integrated fashion. Precisely, we propose probabilistic approaches to compute the critical sensor density above which a field is almost surely covered and the network is almost surely connected. Our proposed solutions consider both planar and spatial deployment of the sensors. These solutions help the network designers

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achieve full coverage of a field with a minimum number of connected, active sensors, thus maximizing the network lifetime. • Research Problem Statement 2: Connected k-Coverage What is a sufficient condition of the sensor density for full k-coverage of a planar (respectively, spatial) deployment field, where each point in the field is guaranteed to be covered by at least k sensors while maintaining network connectivity using static homogeneous sensors only under the assumption of a deterministic sensing model? Solution Statement: In order to solve this problem, thus supporting different applications and environments with diverse requirements in terms of coverage and connectivity using static sensors only, we extend our above analysis to kcoverage using a deterministic approach so the network self-configures to meet these requirements while considering a deterministic sensing model. To this end, we compute the active sensor density that is necessary to achieve full k-coverage of a field while guaranteeing connectivity between all active sensors. Our first analysis is based on Helly’s Theorem [85] and the geometric properties of the Reuleaux triangle (respectively, Reuleaux tetrahedron) in planar (respectively, spatial) deployment fields. Using this analysis, we design randomized centralized protocols, pseudo-distributed protocols, as well as fully-distributed protocols for connected k-coverage configurations in wireless sensor networks. Our second analysis uses computational geometry-based techniques. Indeed, we find that the problem of coverage of a planar (respectively, spatial) field of interest has some similarity with the tiling problem in the Euclidean plane (respectively, space), which is a fundamental problem in a much more concrete branch of mathematics, called elementary geometry. Precisely, we provide an approximation of the sensing range of the sensors based on the analysis of several planar convex tiles (respectively, space filling polyhedra) in planar (respectively, spatial) wireless sensor networks. We introduce appropriate metrics to assess the quality of coverage of planar (respectively, spatial) deployment fields caused by the underlying planar convex tiles (respectively, space filling polyhedra) in planar (respectively, spatial) wireless sensor networks. Both studies help reduce significantly the planar (respectively, spatial) sensor density to achieve connected k-coverage in planar (respectively, spatial) deployment fields. • Research Problem Statement 3: Heterogeneous Connected k-Coverage Given a field to be monitored, a positive integer k ≥ 3, and a set S of heterogeneous sensors, select a minimum subset of sensors S’ ⊆ S to stay on (or active) such that each point in the field is k-covered while the network induced by all the sensors in S’ is guaranteed to be connected. Solution Statement: We exploit the results obtained with the homogeneous model to solve the connected k-coverage problem for heterogeneous wireless sensor networks, where the sensors do not necessarily have the same sensing range, communication range, and initial energy. We show that while it is possible to design distributed protocols to guarantee connected k-coverage of a field using

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heterogeneous sensors while achieving good performance overall, it is impossible that a centralized protocol could be designed efficiently due to sensor deployment randomness and sensor heterogeneity. Thus, we propose a pseudo-random deployment approach, where the sensors are deployed in different layers in a circular deployment field with respect to the sink according to the strengths of their sensing and communication ranges as well as their initial energy. Based on this deployment strategy, we propose centralized and distributed protocols for generating energy-efficient connected k-coverage configurations using heterogeneous sensors. • Research Problem Statement 4: Mobile Connected k-Coverage How to guarantee connected k-coverage in mission-oriented mobile wireless sensor networks under the following requirements? (i) On-demand k-coverage: A region of interest in a field should be k-covered whenever needed, where k ≥ 3. Consequently, a region of interest to be k-covered does not have to be the same all the time and hence may change. (ii) Network connectivity: The sensors should be maintained connected for the correct network operation. (iii) Sensor mobility: The sensors should be able to move to designated locations in a region of interest to ensure its k-coverage whenever necessary. Solution Statement: We divide the problem of k-coverage in mission-oriented mobile wireless sensor networks into two sub-problems, namely sensor placement and sensor selection. The sensor placement problem is to compute the minimum number of sensors and their locations in a region of interest so that this region is k-covered. The sensor selection problem is to determine which sensors should move to the above-computed locations in the region while minimizing the total energy consumption due to the mobility of the sensors and their communication. Specifically, we propose centralized and distributed approaches to solve the kcoverage problem in mission-oriented mobile wireless sensor networks. In the centralized approach, the sink designates a set of sensors to move toward specific locations in the region of interest to be k-covered. These locations are computed by the sink. In the distributed approach, the sensors compute the target locations, where the sensors should move to, and coordinate between themselves to k-cover a region of interest with as small number of sensors as possible (or simply small number of sensors). Our approach enables the sensors to move toward a region of interest and k-cover it while minimizing their mobility energy based on their closeness to the target locations in the region and the availability of other sensors. • Research Problem Statement 5: Stochastic Connected k-Coverage Find a tight sufficient condition so that every point in a field is probabilistically covered by at least k sensors with a probability no less than pth , called threshold probability, under a stochastic sensing model and compute the required number of sensors. Then, select and schedule the sensors while providing stochastic kcoverage of a planar deployment field as well as connectivity between all the selected sensors.

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Solution Statement: We adapt the results of the sensor scheduling problem for k-coverage of a planar deployment field under the deterministic sensing model, which is discussed earlier, to solve the sensor scheduling problem for stochastic kcoverage under probabilistic (or stochastic) sensing and communication models. It has been found that a stochastic sensing mode is more realistic than a deterministic (or binary) sensing model. Indeed, the former takes into account not only the distance between the sensors and the target locations but also the type of propagation model being used, i.e. free-space model or multi-path model. Under a stochastic sensing model, a point p in a field is said to be probabilistically kcovered if the detection probability of an event occurring at p by at least k sensors is at least equal to some threshold probability, 0 < pth < 1. We propose a distributed stochastic k-coverage protocol, where each sensor runs a k-coverage candidacy algorithm to check whether it is eligible to turn itself on (or active). This helps us design a global framework for k-coverage in wireless sensor networks that considers both of the deterministic and stochastic sensing models. • Research Problem Statement 6: Geographic Forwarding on Always-on Sensors How can data be forwarded in always-on wireless sensor networks while minimizing the total energy consumption of the sensors, thus maximizing the network lifetime? Solution Statement: There is an ongoing debate on short-range versus long-range data forwarding in multi-hop wireless networks. This book supports the shortrange data forwarding strategy for wireless sensor networks, where energy should be given the highest priority. More precisely, we propose an energy-efficient data forwarding protocol for wireless sensor networks so they remain operational as long as possible. Our protocol, called Weighted Localized Delaunay Triangulation-based data forwarding (WLDT), uses 1-lookahead scheme to guarantee data delivery to the sink. WLDT aims to minimize the average energy consumption of the sensors during data forwarding towards the sink. It exploits the geometric properties of the Delaunay triangulation [67] to build an energyefficient path between a source and the sink as a sequence of sub-paths whose endpoints are called checkpoints. These checkpoints are selected based on their locations in the field and their remaining energy. A sub-path between a pair of checkpoints consists of a series of forwarders, which are the endpoints of short Delaunay edges and are selected based on their locations and remaining energy to forward the data between their corresponding checkpoints. • Research Problem Statement 7: Energy-Delay Trade-off in Geographic Forwarding How to achieve minimum energy consumption of the sensors while ensuring uniform battery power (or energy) depletion of the sensors and meeting the required delay constraints in geographic forwarding in static wireless sensor networks? Solution Statement: We propose a communication range slicing-based approach to trade-off between conflicting objectives of sensing applications, namely

18

1 General Introduction

minimum energy consumption, minimum delay, and uniform energy depletion. Our approach aims to slice the communication range of the sensors into concentric circular bands and classify them with a goal to satisfy specific requirements of sensing applications in terms of energy consumption, delay, and energy depletion. First, we formulate the trade-off between these three conflicting goals as a multi-objective optimization problem which is solved using a weighted scaleuniform-unit sum approach. Then, we propose a data forwarding protocol for wireless sensor networks, which exploits our solution to the multi-objective optimization problem to find an optimum trade-off between three conflicting goals. For tractability, we consider a unit-disk communication model, where the communication ranges of the sensors are supposed to be circular and have the same radius. Then, we will discuss ways of relaxing these assumptions. • Research Problem Statement 8: Energy Sink-Hole How and to what extent can a uniform energy depletion of all the sensors be guaranteed so as to avoid the energy sink-hole problem in always-on, static wireless sensor networks, where the sensors nearer the sink are heavily used in forwarding data to the sink on behalf of all other sensors, thus depleting their energy very quickly compared to all other sensors in the network? How can this problem be addressed in homogeneous, always-on wireless sensor networks? Solution Statement: We show that static wireless sensor networks suffer from the energy sink-hole problem regardless of how efficient a geographic forwarding protocol is. We propose a solution to this problem by enabling sensors to adjust their transmission range when sending/forwarding sensed data to the sink. However, we prove that this solution imposes a severe restriction on the size of the deployment field. To alleviate this shortcoming, we propose another solution that suggests a sensor deployment strategy exploiting energy heterogeneity with a goal that the sensors in the network deplete their energy uniformly. When all the sensors have the same initial energy, we propose greedy, localized protocol, called energy aware Voronoi diagram-based data forwarding (EVEN), which exploits sink mobility and uses our proposed new concept of Voronoi diagram, called energy aware Voronoi diagram, where the locations of the sensors are time-varying and are locally and virtually computed based on their remaining energy. Furthermore, we propose a third solution based on the use of mobile proxy sinks in order to change the neighbors of a sink over time. These mobile proxy sinks collect data from source sensors and drop them off at an immobile sink. Precisely, we propose a three-tier architecture that has immobile source sensors, immobile sinks, and mobile proxy sinks. We investigate the best mobility strategy of mobile proxy sinks to minimize the total energy consumption for data collection. Then, we propose joint mobility and routing schemes in this type of network based on the numbers of immobile sinks and mobile proxy sinks, and analyze their performance. • Research Problem Statement 9: Geographic Forwarding on Duty-Cycled Sensors

1.6 Book Contributions

19

How to design energy-efficient geographic forwarding protocols with and without data aggregation in duty-cycled, k-covered wireless sensor networks in planar and spatial deployment fields? Solution Statement: We design a unified framework for geographic forwarding on duty-cycled sensors. More specifically, we propose different approaches for both cases of planar and spatial sensor deployment. In the former case, using a potential fields-based approach, we propose energy-efficient clustering-based geographic forwarding protocols for duty-cycled, k-covered wireless sensor networks with different levels of data aggregation. In the latter case, we focus on finding a tradeoff between uncertainty due to duty-cycling with deterministic forwarding, and contention due to opportunistic forwarding. Then, we propose a hybrid forwarding approach based on this trade-off. Indeed, in deterministic forwarding, a next best forwarder is determined a priori. Hence, duty-cycling introduces uncertainty at the sender side which is not totally certain that its selected next best forwarder would remain awake after data is being forwarded. In opportunistic forwarding, however, a next best forwarder is decided on-the-fly and after the data is transmitted. Thus, several active sensors may hear the transmitted data, thus creating high contention at the receiver side to select a next best forwarder. • Research Problem Statement 10: Network Connectivity and Fault-Tolerance What are the unconditional and conditional network connectivity and faulttolerance of k-covered wireless sensor networks that are deployed in planar and spatial fields? Solution Statement: We benefit from our characterization of k-coverage in wireless sensor networks stated earlier and compute the traditional (or unconditional) connectivity of k-covered wireless sensor networks. Furthermore, we compute the conditional connectivity of k-covered wireless sensor networks based on the concept of forbidden faulty set. The latter shows that the classical connectivity used to capture network fault tolerance underestimates the resilience of large-scale networks, such as k-covered wireless sensor networks. Our measures consider both planar and spatial deployment fields. We show that our measures of connectivity for the latter case are not a straightforward generalization of those for the former case. • Research Problem Statement 11: Connected k-Barrier Coverage How can the sensors be deterministically placed in a planar (respectively, spatial) belt so that every crossing path of this belt is connected k-barrier covered, i.e., every path intersects with at least k connected sensors, and what is the minimum number of sensors to achieve connected k-barrier coverage of this sensor belt? What is the total number of all possible (or abstract) paths of an intruder crossing a connected k-barrier covered sensor belt, and how can these abstract paths be represented? Solution Statement: We analyze the connected k-barrier coverage problem in planar wireless sensor networks from a tiling perspective, where the sensors’

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1 General Introduction

sensing disks are tangential to each other. We study two deterministic sensor deployment strategies, which yield square lattice and hexagonal lattice wireless sensor networks, respectively. We compute the number of sensors deployed over a k-barrier covered sensor belt region for both lattices. Also, we determine the ratio of the communication range to the sensing range of the sensors to ensure connectivity of the resulting k-barrier covered network configurations. Then, we introduce the concept of intruder’s abstract paths along a k-barrier covered sensor belt region, and compute their number. In addition, we propose a polynomial representation of all abstract paths. We extend this analysis to spatial wireless sensor networks, where we consider various sensor deployment methods based on four well-known spatial lattices, namely simple cubic, body centered cubic, face centered cubic, and hexagonal close-packed lattices. We prove that none of these lattices guarantee k-barrier coverage. In order to remedy this problem, we consider various space-filling polyhedra, which satisfy the k-barrier coverage property.

1.7 Conclusion In this chapter, we review the main challenges in the analysis, design, and implementation of wireless sensor networks as well as their potential applications. Then, we describe the design metrics that drive the design process of these types of networks. Moreover, we present the motivations behind writing this book. Furthermore, we give an overview of the main contributions of this book by stating the eleven research problems being investigated along with a brief description of our proposed solutions.

Chapter 2

Fundamental Concepts, Definitions, and Models

The scientists from Franklin to Morse were clear thinkers and did not produce erroneous theories. The scientists of today think deeply instead of clearly. One must be sane to think clearly, but one can think deeply and be quite insane. Nikola Tesla (1856–1943)

Overview This chapter introduces the terminology and background that are necessary for the description of all the algorithms and protocols discussed in this book. Precisely, it gives some key definitions and describes a percolation model to study coverage and connectivity in planar and spatial wireless sensor networks. In addition, it presents an energy consumption model and defines the default network model as well as deterministic and stochastic sensing models. Moreover, it discusses a few mobility models, such as random waypoint mobility model, reference point group mobility model, and Manhattan mobility model.

2.1 Introduction 2.1.1 Major Tasks In this book, we use some terminology and different models, such as Voronoi diagram model, energy model, sensing model, and continuum percolation model, to describe our proposed approaches and protocols for connected k-coverage, duty-cycling, and geographic forwarding in wireless sensor networks. Furthermore, our work is based on a specific network model. The goal of this chapter is to present the different terms and models we use in this book.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_2

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2 Fundamental Concepts, Definitions, and Models

2.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 2.2 presents the key definitions and fundamental concepts that are used in this book. Section 2.3 presents deterministic and stochastic sensing models. Section 2.4 discusses different types of network connectivity and fault tolerance. Section 2.5 describes the energy model while Sect. 2.6 presents the percolation model. Section 2.7 presents the default network model that we use in the design of our energy-efficient framework for joint k-coverage, duty-cycling, and geographic forwarding in wireless sensor networks. Section 2.8 describes two classes of mobility models: Random mobility and group mobility models. Section 2.9 concludes the chapter.

2.2 Terminology In this section, we give the key definitions and describe some fundamental concepts used throughout this book. Definition 2.1 (Sensing Range): The sensing range (or detection range) of a sensor si , denoted by SR(si ), is a region where every event that takes place in this region can be detected by si . The sensing neighbor set of a sensor si , denoted by SN(si ), is the ∎ set of all the sensors that are located in its sensing range SR(si ). Definition 2.2 (Communication Range): The communication range of a sensor si is a region such that si can communicate with any sensor located in this region. The communication neighbor set of a sensor si , denoted by CN(si ), is a set of all the sensors that are located in its communication range. ∎ As it can be seen, the sensing range is related to the sensing capability of a sensor to sense (or detect) events. However, the communication range concerns the communication capability of the sensor to communicate (or interact) with other sensors. Both of these spaces are totally orthogonal as they associated with two different concepts, namely sensing and communication capabilities of the sensors. Throughout this book, we use communication range and transmission range interchangeably. To illustrate the following definitions, we assume that the sensing ranges and communication ranges of the sensors in a planar (respectively, spatial) wireless sensor network are represented by disks (respectively, spheres). Furthermore, the sensors are assumed to have the same sensing range and the same communication range. Figure 2.1 shows the sensing and communication ranges of the sensors of radii r and R, respectively. Definition 2.3 (Collaborating Sensors): Two sensors si and sj in a planar (respectively, spatial) wireless sensor network are said to be collaborating if the Euclidean distance between the centers of their sensing disks (respectively, spheres) satisfies |ξi − ξj | ≤ 2r, where r is the radius of the sensing disks (respectively, spheres) of

2.2 Terminology

23

Fig. 2.1 Schematic of overlapping concentric disks (respectively, spheres)

the sensors. Intuitively, the two sensing disks (respectively, spheres) centered at ξi and ξj are either tangential or overlapping (see Fig 2.2a). The collaborating set of a sensor si , denoted by Col (si ), includes all the sensors it can collaborate with, i.e., Col(si ) = { sj : |ξi − ξj | ≤ 2r}. ∎ Definition 2.4 (Communicating Sensors): Two sensors si and sj in a planar (respectively, spatial) wireless sensor network are said to be communicating if the Euclidean distance between the centers of their communication disks (respectively, spheres) satisfies |ξi − ξj | ≤ R, where R is the radius of the communication disks (respectively, spheres) of the sensors (see Fig 2.2b). The communicating set of a sensor si is the set of sensors it can communicate with, i.e., Com(si ) = { sj : |ξi − ξj | ≤ R}. ∎ Definition 2.5 (Coordinating Sensors): Two sensors si and sj in a planar (respectively, spatial) wireless sensor network are said to be coordinating if and only if they

Fig. 2.2 a Collaborating, b communicating, and c coordinating sensors

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2 Fundamental Concepts, Definitions, and Models

both collaborate and communicate (Fig 2.2c). The coordinating set of a sensor si is the set of sensors it can collaborate and communicate with at the same time. ∎ Definition 2.6 (Covered k-Component): A covered component (or covered region) in a planar (respectively, spatial) wireless sensor network is a maximal set of sensing disks (respectively, spheres), i.e., not included in any other subset except when it is equal to the original entire set of sensing disks (respectively, spheres) whose corresponding sensors are collaborating directly or indirectly (Fig 2.3a shows three covered components). A covered k-component, denoted by CCk , is a covered component having k sensing disks (respectively, spheres). ∎ Definition 2.7 (Connected k-Component): A connected component in a planar (respectively, spatial) wireless sensor network is a maximal set of communication disks (respectively, spheres) whose corresponding sensors are communicating

Fig. 2.3 a Covered, b connected, and c coordinated components

2.2 Terminology

25

directly or indirectly (Fig 2.3b shows three connected components). A connected k-component, denoted by CCk . is a connected component with k communication disks (respectively, spheres). ∎ Definition 2.8 (Coordinated Component): A coordinated component in a planar (respectively, spatial) wireless sensor network is a maximal set of concentric sensing and communication disks (respectively, spheres) whose corresponding sensors are coordinating directly or indirectly (Fig 2.3c shows three coordinated components). ∎ Definition 2.9 (Connectivity): A wireless sensor network is said to be connected if there is a communication path between any pair of sensors. In other words, any pair of sensor can communicate with each other either directly or indirectly. ∎ Definition 2.10 (Largest Enclosed Disk/Sphere): The largest enclosed disk (respectively, sphere) of closed convex area (respectively, volume) A is a disk (respectively, sphere) that lies inside A and whose diameter is equal to the minimum distance between any pair of points on the boundary of A. ∎ Definition 2.11 (Homogeneity vs. Heterogeneity): A planar (respectively, spatial) wireless sensor network is said to be homogeneous if all of it sensors have the same storage, processing, battery power, sensing, and communication capabilities. In particular, all deployed sensors have the same radius r of their sensing disks (respectively, spheres) and the same radius R of their communication disks (respectively, spheres). Otherwise, the network is said to be heterogeneous. ∎ Definition 2.12 (Sparseness): A wireless sensor network is said to be sparse if it is impossible to k-cover any region of interest using static sensors only, and k-cover the whole planar (respectively, spatial) deployment field using static and mobile sensors. ∎ Definition 2.13 (Width): The width of a closed convex planar shape (or area) A is the maximum distance between any pair of parallel lines that bound the area A. ∎ Definition 2.14 (Breadth): The breadth of closed convex spatial shape (or volume) A is the maximum distance between tangential planes on opposing faces or edges of the volume A. ∎ The Voronoi diagram [67], also known as Dirichlet tessellation, represents one of the most fundamental data structures in computational geometry. It has interesting mathematical and algorithmic properties and potential applications. Definition 2.15 (Voronoi Diagram): Let S = {s1 , … , sn } be a finite set of n sites (or points) in the plane. The Voronoi diagram of S, denoted by Vor(S), is a subdivision of the plane containing S into n cells VC(si ), 1 ≤ i ≤ n, such that each cell VC(si ) includes only one site si with the property that any point p located in VC(si ) is closer to si than any other site in S. The cell VC(si ) corresponding to site si is called the

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Voronoi cell of si , which is a (possibly unbounded) open convex polygonal region. The edges of a Voronoi cell are called Voronoi edges and its endpoints are called Voronoi vertices. The Voronoi diagram of S is the union of the Voronoi cells of all sites in S. The Delaunay triangulation, denoted by DT (S), is the dual of the Voronoi diagram Vor(S) [67]. A DT (S) graph has an edge between two sites si and sj if and only if their Voronoi cells VC(si ) and VC(sj ), respectively, share a common edge. ∎ Notice that DT (S) is a planar graph whose Delaunay edges are orthogonal to their corresponding Voronoi edges. Figure 2.4 shows a Voronoi diagram while Fig 2.5 shows a Voronoi diagram and its dual, i.e., Delaunay triangulation. Let CN (si ) be the communication neighbor set of a sensor si , which are located in its communication disk whose radius is equal to Ri . From CN (si ), the sensor si considers only a subset of sensors, denoted by SCN (si , sm ), located between si and the sink sm to act as data forwarders to the sink.

Fig. 2.4 Voronoi diagram

Fig. 2.5 The Delaunay triangulation (bold lines) on top of the Voronoi diagram (dotted lines) of a wireless sensor network

2.2 Terminology

27

Definition 2.16 (Localized Voronoi Diagram): Let sj ∈ SCN (si , sm ). The Voronoi diagram computed by si , denoted by Vor({si , sm } ∪ SCN (si , sm )), is said to be localized. A Voronoi cell VC(sm ) of the sink sm is said to be adjacent to the sensor sj if VC(sm ) and VC(sj ) of sm and sj , respectively, have at least one common Voronoi edge. ∎ Definition 2.17 (Localized Delaunay Triangulation): A localized Delaunay triangulation of a sensor si , denoted by LDT (si ), is the Delaunay triangulation computed ∎ by si with respect to SCN(si ) ∪ {si , sm }. Assume that a source s0 wishes to disseminate its data to the sink sm . Figure 2.6 shows the localized Voronoi diagram of s0 , where SCN (s0 , sm ) = {si |1 ≤ i ≤ 18}. Notice that CF(s0 , sm ) = {s9 , s10 , s15 , s17 , s18 }, where the Voronoi cell of each of those sensors shares one Voronoi edge with that of sm . These Voronoi cells are shaded in green. Definition 2.18 (Reference Sensor): Let sref ∈ SCN (si , sm ). A sensor sref is said to be a reference sensor of si if sref has the highest remaining energy among all sensors ∎ in SCN (si , sm ). Definition 2.19 (Candidate Checkpoints): The candidate checkpoints of a sensor si , denoted by CCP(si , sm ), are the sensors that are adjacent to the sink sm in the ∎ localized Delaunay triangulation of si , LDT (si ).

Fig. 2.6 Localized Voronoi diagram

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2 Fundamental Concepts, Definitions, and Models

Definition 2.20 (Candidate Forwarder): Let sj ∈ SCN (si , sm ). A sensor sj is said to be a candidate forwarder of si if VC(sm ) is adjacent to sj , where VC(sm ) ∈ Vor({si , sm } ∪ SCN (si , sm )). The set of candidate forwarders of si is denoted by CF(si , sm ). ∎ Definition 2.21 (Mobile Proxy Sink): A mobile proxy sink is a mobile sensor that only collects data from source sensors and delivers it to an immobile sink when it comes close by. ∎ Definition 2.22 (Data Forwarding Path): A data forwarding path is a path that is traversed by a data packet originated from a source sensor (or simply source) and destined to the sink. It includes all the sensors (including the source) that forwarded the data packet to the sink on behalf of the source. ∎ Definition 2.23 (Long-Range and Short-Range Forwarding): A forwarding scheme is said to be long-range if each sensor in any data forwarding path can use at most one of its one-hop neighbors to forward a data packet toward its ultimate destination. A forwarding scheme is said to be short-range if each sensor in any data forwarding path can use multiple one-hop neighbors to forward a data toward its destination. ∎ First, we introduce the terminology that is essential to our study of the problem of connected coverage in planar (respectively, spatial) wireless sensor networks. Particularly, we define the concept of quality of coverage that distinguishes the various planar (respectively, spatial) convex polygons (respectively, polyhedral space-fillers) from one another, which we study in this chapter to cover a planar (respectively, spatial) FoI. From now on, “space” refers to “Euclidean space of three dimensions”. Definition 2.24 (Congruent Regular Polygons): A polygon is a planar geometric shape that is bounded by a closed chain of straight line segments, called edges. The polygon’s vertices are the points at which the polygon’s edges meet. A polygon is said to be regular if all of its angles are equal and all of its edges are equal. A set P of regular polygons are said to be congruent if they have the same number of sides, all their corresponding sides are the same length, and all their corresponding interior angles are the same measure. ∎ Definition 2.25 (Planar and Spatial Paver/Tile): A planar (respectively, spatial) convex polygon (respectively, polyhedron) is said to be a planar (respectively, spatial) paver (or tile) if it can pave a planar (respectively, spatial) field through replication of its congruent copies without any overlaps or gaps between any pair of congruent copies. ∎ In this book, we use the words paver and tile interchangeably. Definition 2.26 (Regular Polygonal Tiling): A polygonal tiling consists of covering the Euclidean plane with polygons without creating any gaps or overlaps. A regular polygonal tiling is a polygonal tiling that uses congruent regular polygons. ∎ Definition 2.27 (Polygonal k-Coverage—k-Cover Set): A polygonal coverage consists of covering the Euclidean plane with polygons without creating any gaps,

2.2 Terminology

29

while allowing (but not necessarily requiring) overlaps between adjacent polygons. In this case, every point in the Euclidean plane is covered by at least one polygon. A cover set is a set of convex polygons to achieve polygonal coverage. A polygonal k-coverage is a polygonal coverage, where every point in the Euclidean plane is covered by at least k polygons. A k-cover set is a set of convex polygons to ensure polygonal k-coverage. ∎ Notice that polygonal tiling is a special case of polygonal coverage. Indeed, the latter is more general than the former as it may or may not have overlap between adjacent polygons. In this book, we focus on polygonal k-coverage using polygonal tiling. Next, we introduce the concept of coverage quality of a planar (respectively, spatial) convex tile. First, we define the rate of planar (respectively, spatial) overlap of a sensor’s sensing disk (respectively, sphere) with those of its neighbors. Definition 2.28 (Rate of Planar Overlap): Let T be a planar convex tile whose area is denoted by A(T ), and SDr the sensors’ sensing disk whose radius is equal to r and area is A(SDr ) = π r 2 . The rate of planar overlap of the sensing disk of sensor si with those of its sensing neighbor set is denoted by RO(si ) and given by: RO(si ) =

A(SDr ) − A(T ) A(SDr )

∎

As it can be seen, theoretically, RO(si ) reaches its minimum, 0, when V (SSr ) = V (P), and its maximum, 1, when V (P) = 0. In other words, we have: 0 ≤ RO(si ) ≤ 1 Definition 2.29 (Polyhedron—Convex Polyhedral Space-Filler): A polyhedron is a spatial solid whose faces join at their edges. A convex polyhedron is said to be a convex polyhedral space-filler if it can fill (or cover) the space through replication of its congruent copies, while not allowing any overlap of the interiors of any pair of adjacent copies or leaving any gap between any pair of adjacent copies. ∎ Definition 2.30 (Rate of Spatial Overlap): Let P be a polyhedral space-filler whose volume is denoted by V (P), and SS r be the sensing sphere of the sensors whose radius is equal to r and whose volume is V (SS r ). The rate of spatial overlap of the sensing sphere of sensor si with those of its sensing neighbor set is denoted by RO(si ) and given by: RO(si ) = where V (SSr ) = 4π r 3 /3.

V (SSr ) − V (P) V (SSr ) ∎

Definition 2.31 (Sensor Quality of Coverage—k-Cover Set Quality of Coverage): The quality of coverage of sensor si , denoted by QoC(si ), is linearly proportional to its rate of overlap, and is computed as follows:

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QoC(si ) = 1 − RO(si ) Let S be a k-cover set and si a sensor of this k-cover set (i.e., si ∈ S). The quality of coverage of S, denoted by QoCavg (S), is the average quality of coverage over all the sensors in the k-cover set S. That is QoCavg (S), is given by: QoCavg (S) =

1 Σ QoC(si ) |S| si ∈S

where |S| stands for the cardinality (or size) of the set S.

∎

Given the above lower and upper bounds on the rate of overlap RO(si ), we obtain the following bounds on the quality of coverage for si' s and the cover set S: 0 ≤ QoC(si ) ≤ 1 0 ≤ QoC(S) ≤ 1 Given that the sensors’ sensing range is a disk (respectively, sphere), it is impossible to cover a planar (respectively, spatial) field with a set of non-overlapping sensing disks (respectively, spheres). Ideally, we want to have QoC(si ) = 1 and, consequently, QoC(S) = 1. However, neither QoC(si ) nor QoC(S) can be equal to 1. Thus, we allow the overlap among the sensing disks (respectively, spheres) of a cover setS. Hence, our goal is to maximize the quality of coverage of each individual sensor, QoC(si ), as much as possible, and, thus, maximizing the quality of coverage of the cover setS, i.e., QoC(S). To this end, we propose to restrict the sensing disk (respectively, sphere) of the sensors to a planar tile (respectively, convex polyhedral space-filler). Our goal is to identify a planar tile (respectively, convex polyhedral space filler) that possesses the maximum quality of coverage QoC(si ), which helps maximize the quality of coverage QoC(S) of the k-cover setS. In order to find the best planar tile (respectively, convex polyhedral space-filler), our approach is as follows: For each candidate planar tile (respectively, convex polyhedral space-filler), we compute the maximum area (respectively, volume) that can be enclosed in the sensors’ sensing disk (respectively, sphere) whose radius is r. To this end, we determine the corresponding relationship that exists between r and the length of the edge of each of these candidate planar tiles (respectively, convex polyhedral space-fillers). Then, we compute the corresponding rate of the overlap RO(si ) and derive the associated quality of coverage QoC(si ) and QoC(S). Recall the relationship among these three attributes: QoC(S) = QoC(si ) = 1 − RO(si ). We investigate various planar tiles (respectively, convex polyhedral space-fillers) in order to identify the best one with respect to the above-mentioned quality of coverage metric. The literature has three regular polygons, which have been recognized as planar tiles: Equilateral triangle, square, and equilateral hexagon, which are given in Fig 2.7. Also, it has several convex polyhedral that have been known as

2.2 Terminology

31

Fig. 2.7 Equilateral triangle, square, and equilateral hexagon

space-fillers, namely the cube [125], regular right hexagonal prism [125], truncated octahedron [292], great rhombicuboctahedron [292], rhombic dodecahedron [292], elongated dodecahedron [292], rhombic Triacontahedron [292], Sommerville’s tetrahedra [345], and Goldberg’s equilateral octahedron [173], which are shown in Fig 2.8.

Cube

Regular right hexagonal prism

Truncated octahedron

Great rhombicuboctahedron

Rhombic dodecahedron

Elongated dodecahedron

Rhombic Triacontahedron

Sommerville’s largest tetrahedron

Goldberg’s equilateral octahedron

Fig. 2.8 Convex polyhedral space-fillers

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2 Fundamental Concepts, Definitions, and Models

Fig. 2.9 a Square lattice and b hexagonal lattice

In our study of the planar physical security problem based on planar k-barrier coverage, we study two deterministic sensor deployment strategies, which yield square lattice and hexagonal lattice wireless sensor networks (see Definition 2.32 below), which are shown in Fig 2.9. Likewise, in the case of spatial physical security problem, we consider spatial k-barrier coverage, which accounts for the abovementioned convex polyhedral space-fillers (see Fig 2.8) as well as various deterministic spatial sensor deployment strategies leading to the following spatial lattice wireless sensor networks (see Fig 2.10 and Definition 2.33 below): • • • • •

Simple cubic lattice Body centered cubic lattice Face centered cubic lattice Hexagonal close-packed lattice Space filler lattice.

Definition 2.32 (Square and Hexagonal Lattice Wireless Sensor Networks): A square lattice wireless sensor network is a network whose sensors are deployed according to a square lattice. A hexagonal lattice wireless sensor network is a network ∎ whose sensors are deployed using a hexagonal lattice (see Fig 2.9). Definition 2.33 (Spatial Lattice Wireless Sensor Networks and Polyhedral SpaceFiller Lattice): A simple cubic lattice wireless sensor network is a wireless sensor

Simple cubic lattice

Body centered cubic lattice

Fig. 2.10 Spatial lattice wireless sensor networks

Face centered cubic lattice

Hexagonal closepacked lattice

2.2 Terminology

33

network whose sensors are deployed according to a simple cubic lattice. A body centered cubic lattice wireless sensor network is a wireless sensor networks whose sensors are deployed using a body centered cubic lattice. A face centered cubic lattice wireless sensor network is a wireless sensor network whose sensors are deployed according to a face centered cubic lattice. A hexagonal close-packed lattice wireless sensor network is a wireless sensor network whose sensors are deployed using a hexagonal close-packed lattice (see Fig 2.10). A polyhedral space-filler lattice wireless sensor network is a wireless sensor network whose sensors are deployed using polyhedral space-fillers. In addition, we define the following terms, which are used in our study of both the planar and spatial k-barrier coverage problems in planar and spatial wireless sensor networks, respectively. Definition 2.34 (Stealthy Sensor): A sensor is said to be stealthy if no intruder is aware of its geographic location. ∎ Definition 2.35 (Planar and Spatial Sensor Belt Region): A planar (respectively, spatial) sensor belt region is a planar (respectively, spatial) belt region that has a set of sensors deployed in it. ∎ Definition 2.36 (Barrier): A barrier is an obstacle (or fence) that prevents any movement from an accessible region to an inaccessible (critical or protected) region, such as an international border. ∎ From Definitions 2.35 and 2.36, it is clear that a planar (respectively, spatial) sensor belt region defines a barrier. The latter will prevent any intruder’s attempt to cross it and access a critical region. Definition 2.37 (k-Barrier Covered Path): A planar (respectively, spatial) path is said to be k-barrier covered if it intersects with the sensing disks (respectively, spheres) ∎ of at least k sensors, which are deployed in a barrier, where k ≥ 1. Definition 2.38 (k-Barrier Covered Planar/Spatial Sensor Belt Region): A planar (respectively, spatial) sensor belt region is k-barrier covered if any path crossing it k-barrier covered. ∎ It is worth noting that the concept of k-barrier coverage is different from that of k-coverage [53]. A path is k-covered if every point along this path is covered by at least k sensors. However, From Definition 2.37a path is k-barrier covered if some or all of its points are covered by at least k sensors deployed in a barrier. Definition 2.39 (Planar and Spatial abstract paths): An intruder’s planar (respectively, spatial) path can be represented by a planar (respectively, spatial) abstract path, denoted by IAP = (N , E), where the node set N represents the set of sensing disks (respectively, spheres), and the edge set E stands for transitions between the sensors’ sensing disks (respectively, spheres). ∎ Definition 2.40 (Structural k-Node Line): A structural k-node line, denoted by lk , with k ≥ 1, is a line that has k nodes such that no two of them are located at the same level, i.e., each level contains only one node. ∎

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2 Fundamental Concepts, Definitions, and Models

Fig. 2.11 Sample of structural 3-node, 4-node, 7-node lines

Figure 2.11 shows a sample of structural k-node lines. Notice that for a square lattice, each node, except the leaf node, has one left node, right node, or vertical node. Definition 2.41 (Weakly, Strongly, and Mildly Planar/Spatial k-Barrier Covered Paths): Let k ≥ 1 be a natural number. A weakly planar (respectively, spatial) kbarrier covered path is a planar (respectively, spatial) k-barrier covered path whose O(k) of its points intersect with the sensing disks (respectively, spheres) of at least k sensors deployed in a barrier. A strongly planar (respectively, spatial) k-barrier covered path is a planar (respectively, spatial) k-barrier covered path such that each of its points intersects with the sensing disks (respectively, spheres) of at least one sensor among k sensors deployed in a barrier. A mildly planar (respectively, spatial) k-barrier covered path is a planar (respectively, spatial) k-barrier covered path that is not weakly planar (respectively, spatial) k-barrier covered or strongly ∎ planar (respectively, spatial) k-barrier covered (see Fig 2.12). Next, we define the concept of intruder’s abstract path observability. Planar Move): An intruder’s Definition 2.42 (Progressive ( ) ) planar move from location ( (xi , yi ) to location xj , yj , denoted by (xi , yi ) ↓ xj , yj , is said to be progressive if yj < yi . ∎ Definition 2.43 (Progressive ( Spatial ) Move): An intruder’s spatial ( move) from location (xi , yi , zi ) to location xj , yj , zj , denoted by (xi , yi , zi ) ↓ xj , yj , zj , is said to be progressive if one of the two following conditions is true: (C 1 ) yj < yi , (C 2 ) zj < zi . ∎ Definition 2.44 (Intruder’s Abstract Path Observability): The observability of an intruder’s abstract path IAP along a sensor belt region SBRw,l , denoted by OIAP , is defined as the minimum cardinality of the intersection set among all possible intersection sets between an intruder’s abstract path IAPi and SBRw,l . Formally, OIAP is computed as follows: |{ }| OIAP = min| IAPi ∩ SBRw,l |IAPi ∈ IAP |

2.2 Terminology

35

Fig. 2.12 Weakly, mildly, and strongly k-barrier covered paths

where IAP is the set of all intruder’s abstract paths.

∎

Intuitively, the observability of an intruder’s abstract path IAP measures the percentage of IAP being observed (or sensed) by those sensors that are able to detect the intruder. Notice that OIAP reaches its maximum value 1 for a planar (respectively, spatial) strongly k-barrier covered path whose all of its points intersect with the sensing disks (respectively, spheres) of the sensors, and its minimum value εk for a weakly planar (respectively, spatial) k-barrier covered path, which intersect with exactly k points of the sensing disks (respectively, spheres) of the sensors. That is, we have the following equality: εk ≤ OIAP ≤ 1. Definition 2.45 (Sensing Area Usage Rate): Let RCP be a regular convex paver whose area is denoted by A(RCP). The sensing area usage rate of the sensors’ sensing disk using RCP is denoted by SAUR(RCP) and is given by: SAUR(RCP) =

A(RCP) A(SDr )

where SDr stands for the sensors’ sensing disk whose radius is equal to r and area is ∎ given by A(SDr ) = π r 2 . Definition 2.46 (Unit Cell Covered Volume Ratio): Let UC be a unit cell whose volume is Vuc . The unit cell covered volume ratio, denoted by ϑ, is computed as

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follows: ϑ=

Vcov (O) Vuc

where Vcov (O) is the total volume of UC covered (or occupied) by object O.

∎

Definition 2.47 (Forbidden Neighboring Sensor Set): A forbidden neighboring sensor set of a sensor si , denoted by FNSS(si ), consists of all the sensors sj that are sensing neighbors tosi , have low battery power (i.e., their residual energy is below a certain threshold, denoted by ethreshold ), and/or have been so far selected for ∎ a certain number ncr of consecutive rounds.

2.3 Deterministic and Stochastic Sensing Models In the deterministic sensing model (also known as binary), a point (or event) p in a planar (respectively, spatial) field is sensed/covered (or detected) by a sensor si based on the Euclidean distance δ(p, si ) between p and si . Throughout this chapter, we use “coverage of a point” and “detection of an event” interchangeably. Formally, the coverage Cov(p, si ) of a point p by a sensor si under the deterministic sensing model is defined as follows: ⎧ 1 if δ(p, si ) ≤ r Cov (p, si ) = (2.3.1) 0 otherwise As it can be seen, the deterministic sensing model for planar (respectively, spatial) wireless sensor networks considers the sensing range of a sensor as a disk (respectively, sphere), and hence all sensor readings are precise and have no uncertainty. However, it has been found that the communication range of the radios is highly probabilistic and irregular [456]. Thus, the deterministic sensing model does not reflect the real behavior of the sensing units of the sensors, which are irregular in nature. Hence, given the signal attenuation and the presence of noise associated with sensor readings, it is necessary to consider a more realistic sensing model by defining the coverage Cov(p, si ) using some probability function. In other words, the sensing capability of a sensor needs to be modeled as the probability of successful detection of an event. Specifically, the sensor’s sensing capability should depend on the distance between it and the event as well as the type of propagation model being used (free-space vs. multi-path). Indeed, Duarte and Hu [139] demonstrated that the probability that an event in a distributed detection application can be detected by an acoustic sensor depends on the distance between the event and the sensor. Cao et al. [101] presented a realistic sensing model for passive infrared (PIR) sensors that reflects their non-isotropic range. Indeed, they verified by simulations the sensing

2.3 Deterministic and Stochastic Sensing Models

37

irregularity of PIR sensors, thus, reflecting the non-isotropic nature of PIR sensors’ sensing range. Thus, in a stochastic sensing model, the coverage Cov(p, si ) is defined as the probability of detection P(p, si ) of an event occurring at point p by sensor si as follows: ⎧ −βδ(ξ,s )α i if δ(p, si ) ≤ r e P (p, si ) = (2.3.2) 0 otherwise where β represents the physical characteristic of the sensing units of the sensors and 2 ≤ α ≤ 4 is the path-loss exponent. Precisely, β measures the uncertainty introduced by the sensing unit of the sensors. Also, for the free-space model, we have α = 2 and for the multi-path model, 2 < α ≤ 4. Our stochastic sensing model is motivated by the one introduced by Elfes [146], where the sensing capability of a sonar sensor is modeled by a Gaussian probability density function. Moreover, a probabilistic sensing model for coverage and target localization in wireless sensor networks is proposed in [461]. This sensing model is similar to ours except that it considers δ(p,si ) − (r − r e ) instead of δ(p,si ), where r is the detection range of the sensors and r e < r is a measure of uncertainty in the sensor detection capability. Our stochastic sensing model is also similar to the one in [460], except that ours uses α, and reduces to the deterministic sensing model if we set β = 0. Definition 2.48 (Deterministic k-Coverage): Under the deterministic sensing model, a point p in a planar (respectively, spatial) field is said to be k-covered if it belongs to the intersection of the sensing ranges of at least k sensors. ∎ As stated earlier, using a deterministic sensing model, a point q in a FoI is guaranteed to be k-covered if it is within the intersection of at least k sensors’ sensing disks. In addition, an area A is said to be k-covered if every point in A is k-covered. Definition 2.49 (Probabilistic k-Coverage): Under the stochastic sensing model, a point p in a planar (respectively, spatial) field is said to be probabilistically kcovered if the detection probability of an event occurring at p by at least k sensors is at least equal to some threshold probability pth , where 0 < pth < 1. An area A is probabilistically k-covered if each point in A is probabilistically k-covered. ∎ Definition 2.50 (k-Coverage): For both deterministic and probabilistic sensing models, a region A is said to be k-covered if every point p ∈ A is k-covered. A k-covered wireless sensor network is a network that fully k-covers a field. ∎ Definition 2.51 (Degree of Coverage): We call degree of coverage provided by a wireless sensor network the maximum value of k such that a planar (respectively, spatial) field is fully k-covered. ∎ Definition 2.52 (Planar and Spatial Sensor Density): The planar (respectively, spatial) sensor density is the number of sensors per unit area (respectively, volume). ∎

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Definition 2.53 (Connected k-Coverage): A given k-coverage is said to be connected if the network induced by all the sensors that are selected to achieve k-coverage is connected. ∎ Definition 2.54 (Energy-Efficient Connected k-Coverage): A region of interest A is said to be energy-efficiently k-covered if a small number of sensors is selected to k-cover A. ∎

2.4 Network Connectivity and Fault Tolerance Definition 2.55 (Communication Graph, Connectivity, and Fault Tolerance): A communication graph of a homogeneous (respectively, heterogeneous) wireless sensor network is an undirected (respectively, directed) graph, G = (S, E), where S is a set of sensors and E is a set of edges (respectively, arcs) between them such that for all si , sj ∈ S, (si , sj ) ∈ E if δ(pi , pj ) ≤ Ri , where pi and Ri stand for the location and radius of the communication disk of sensor si , respectively. The vertex-connectivity (or simply, connectivity) of G is equal to K if G can be disconnected by the removal (or failure) of at least K nodes. The fault tolerance of the underlying network is equal to K − 1. ∎ Definition 2.56 (Forbidden Faulty Set): A forbidden faulty set of a graph G = (S, E) of a wireless sensor network, is a set of sensors F ⊂ S that cannot fail at the same time. ∎ Let P be “The faulty set cannot include the neighbor set of any sensor”, F P ⊂ S a faulty set satisfying property P, and G = (S,E) a communication graph representing a wireless sensor network. According to our conditional fault-tolerance model, a faulty sensor set F P is given by F P = {U ⊂ S | ∀ si ∈ S: CN(si ) /⊂ U}, where CN(si ) is the communication neighbor set of sensor si . Thus, the communication neighbor set of a sensor cannot fail simultaneously, and hence it is a forbidden faulty set. More precisely, any faulty sensor set that includes the communication neighbor set of at least one sensor si is considered as a forbidden faulty set. Definition 2.57 (Conditional Connectivity): The conditional connectivity of G = (S, E) with respect to the property P, denoted by κ(G : P), is the minimum size of F P such that the resulting graph Gdis = (S − F P , E dis ) is disconnected into components each satisfying P. ∎ Another generalization of connectivity, called restricted connectivity, was proposed by Esfahanian [150] in which the restriction is on the faulty set (i.e., set of nodes that can fail). Restricted connectivity uses the concept of forbidden faulty set in which the entire neighbor set of a node cannot be faulty at the same time. Definition 2.58 (Conditional Fault-Tolerance): The conditional fault-tolerance of G = (S, E) with respect to the property P is given by η(G : P) = κ(G : P) − 1. ∎

2.5 Energy Consumption Model

39

2.5 Energy Consumption Model The energy consumption of the sensors is dominated by data transmission and reception. Let si and sj be two neighboring sensors. According to the energy consumption model specified by Heinzelman et al. [195], when data is sent from a transmitter to a receiver, there is energy consumption incurred at both ends (i.e., transmitter and receiver). While at the receiver end, the energy consumption is due to only one component, called the transceiver, the energy consumption at the transmitter end depends on two components, namely the transceiver and the transmitter amplifier. The energy consumed by the latter component depends on the size of the data packet that has been sent, the distance between the transmitter and the receiver, and another constant, called transmitter amplifier and denoted by ε. The value of this constant depends on whether the free space or the multi-path model is being considered. Formally, according to [195], the energy spent in data transmission and reception are given by Etx (si , sj ) = a (Eelec + ε δ α (si , sj )) and Erx (si ) = a Eelec , respectively, where δ is the Euclidean distance function, a is data size in bits, Eelec is the electronics energy and is equal to 50 nJ/bit, ε ∈ {εfs , εmp } is the transmitter amplifier in the freespace (εfs = 10 pJ/bit/m2 ) or the multi-path (εmp = 0.013 pJ/bit/m2 ) model, and 2 ≤ α ≤ 4 is the path-loss exponent. When the identities of sender and receiver are not important, we simply write Etx = a (Eelec + ε d α ) and Erx = a Eelec , where d stands for the transmission distance used by a sender. We consider static wireless sensor networks with constant data reporting to the sink, where all the source sensors and sink are static. We assume that a constant rate of energy drain will incur during sensor mobility [398]. Let emove be the energy cost for a sensor to move one-unit distance, and l the distance traveled by a sensor. The energy spent by a sensor due to its mobility, denoted by Emob (dmob ), is computed as Emob (dmob ) = emove dmob where emove is the energy cost for a mobile sensor to move one-unit distance, and dmob is the total distance traveled by the mobile sensor. We assume κ = 256. We assume that a constant rate of energy drain will incur during sensor mobility [396]. More precisely, the energy spent by a sensor during its mobility is computed as Emv (l) = emove × l where emove is the energy cost for a sensor to move one-unit distance, l is the distance traveled by a sensor, and emove is randomly selected between 0.008 J and 0.012 J [396]. Definition 2.59 (Delay): The delay is defined as the time elapsed between the departure of the data from a source s0 and its arrival to the sink sm . This delay is given

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2 Fundamental Concepts, Definitions, and Models

by D(s0 , sm ) = (qd + td + pd )Nf (s0 , sm ) where qd is the average queuing delay per intermediate forwarder, td is the average transmission delay, pd is the average propagation delay, and Nf (s0 , sm ) is the number ∎ of intermediate forwarders between s0 and sm . Given that the size of the planar (respectively, spatial) field is in the order of a few miles, the average propagation delay is negligible. Thus, the delay D(s0 , sm ) is proportional to N f (s0 , sm ), i.e., D(s0 , sm ) = c Nf (s0 , sm ) ∝ Nf (s0 , sm ) where c = qd + td . A similar result can be found in [253].

2.6 Percolation Model Assume that the sensing and communication ranges of the sensors in a planar (respectively, spatial) wireless sensor network are modeled by disks (respectively, spheres). Let Xλ = { ξi : i ≥ 1 } be a planar (respectively, spatial) homogeneous Poisson point process of density λ, where ξi represents the location of a sensor si . Let Xλ (A) be a random variable representing the number of points in an area (respectively, volume) A. The probability that there are k points inside A is computed as P (Xλ (A) = k) =

λk |A|k −λ|A| e k!

(2.3.3)

for all k ≥ 0, where |A| is the size of the area (respectively, volume) of A. Definition 2.60 (Covered Area/Volume Fraction): In a planar (respectively, spatial) wireless sensor network, the covered area (respectively, volume) fraction of a Poisson Boolean model (Xλ , {Bi (r) : i ≥ 1}) given by A(r) = 1 − e−bλ [189] is the mean fraction of area (respectively, volume) covered by the sensing disks (respectively, spheres) Bi (r), for i ≥ 1, in a region of unit area (respectively, volume), where b = π r 2 (respectively, b = 4π r 3 /3) is the area (respectively, volume) of the sensing disks (respectively, spheres) of the sensors and λ is the density of the Poisson point ∎ process Xλ . Assume that λ is not a constant as the sensors could appear and disappear independently of one another. We want to compute the density λc , called critical percolation density (or critical density) such that there exists an infinite covered component when

2.6 Percolation Model

41

λ > λc , and hence the Boolean model (Xλ , {Di (r) : i ≥ 1}) is said to be percolating. Otherwise, there is no infinite covered component and hence the Boolean model (Xλ , {Di (r) : i ≥ 1}) does not percolate. Definition 2.61 (Critical Covered Area/Volume Fraction): The critical covered area (respectively, volume) fraction of (Xλ , {Di (r) : i ≥ 1 }), computed as Ac (r) = 1−e−a λc , is the fraction of area (respectively, volume) covered at critical percolation, where λc is the associated density of Xλ . ∎ Percolation processes were introduced by Broadbent and Hammersley [96] to model the random flow of a fluid through a medium. Because of their simplicity of description and display of critical behavior, where the behavior of a model changes abruptly (phenomenon known as phase transition) as the value of a parameter crosses a threshold, percolation models are attractive in several areas of mathematics, physical science, and engineering. A percolation model can be viewed as an ensemble of points distributed in space, where some pairs are adjacent (or connected) [152]. We consider a Boolean model [293] which is defined below. Definition 2.62 (Boolean Model): A Boolean model consists of two components, namely point process Xλ and connection function h. The set Xλ = {ξi : i ≥ 1} is a homogeneous Poisson point process of density λ in the Euclidean plane IR2 (respectively, Euclidean space IR3 ), where the elements of Xλ are the locations of the sensors used to cover a planar (respectively, spatial) field. The connection function, h, is defined such that two points ξi and ξj are adjacent with probability h(|ξi − ξj |) = 1 independently of all other points if |ξi − ξj | ≤ d and h(|ξi − ξj |) = 0 if |ξi − ξj | > d , where d ≥ 0 and |ξi − ξj | is the Euclidean distance between ξi and ξj . In other words, h ( |ξi − ξj | ) given by ⎧ h(|ξi − ξj |) =

1 if|ξi − ξj | ≤ d 0 otherwise

∎

2.6.1 Why a Continuum Percolation Model? We consider a continuum percolation model rather than a discrete percolation model for the following reason. In discrete percolation [179], also known as lattice model, the sites, which are randomly occupied in a discrete lattice, may have different configurations, namely square, triangle, honeycomb, etc. In continuum percolation [293], the positions of the sites are randomly distributed and thus there is no need to have different analysis for each of these regular lattices. Precisely, we consider a continuum percolation model, which consists of homogeneous disks (respectively, spheres) whose centers representing the locations of the sensors are randomly distributed in planar (respectively, spatial) field, according to a spatial Poisson point process of density λ. In percolation theory, we are interested in the critical density λc above

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2 Fundamental Concepts, Definitions, and Models

which an infinite cluster of overlapping disks (respectively, spheres) first appears. The density λc is the critical value for the density λ such that there exists no infinite cluster of overlapping disks (respectively, spheres) almost surely when λ < λc , but there is an infinite cluster of overlapping disks (respectively, spheres) almost surely when λ > λc and we say that percolation occurs.

2.7 Default Network Model In this section, we specify the default network model used in this book unless stated otherwise. Assumption 2.1 (Planar and Spatial Static, and Isotropic Networks): All the sensors are deployed in a planar (respectively, spatial) wireless sensor network, and are static and isotropic. In other words, all the sensors do not move and have the same (or homogeneous) sensing and communication ranges. ∎ Assumption 2.2 (Unit Disk/Sphere Model): All the sensors follow the unit disk model (respectively, sphere model). That is, the sensing range of a sensor si is modeled by a disk (respectively, sphere) whose radius is r, and its communication range is represented by a disk (respectively, sphere) with radius equal to R. Moreover, both of the sensing and communication ranges of the sensor si are centered at pi , i.e., the location of the sensor si . Therefore, each sensor si is characterized by two concentric disks (respectively, spheres) associated with its sensing and communication ranges, ∎ respectively, as shown in Fig 2.1. Assumption 2.3 (Always-On Sensors): All the sensors are always-on, meaning that they constantly report their sensed data to a single static sink. Hence, the sensors cannot be turned off while monitoring a physical phenomenon. ∎ Assumption 2.4 (Sensor ID’s and Location and energy Awareness): Each sensor has a unique id (an integer, for instance) and is aware of its own location information through GPS (Global Positioning System) or some localization technique [216]. The sensors advertise their location information only once when they start their sensing task. In addition, each sensor advertises its remaining energy by piggybacking it on the data sent to the sink. ∎ Assumption 2.5 (Random, Uniform, and Dense Sensor Deployment): The sensors are randomly, uniformly deployed in a planar (respectively, spatial) field whose size is much larger than that of the sensing and communication ranges of the sensors. Moreover, the sensors are supposed to be densely deployed. As indicated in Chap. 1, the limited battery power of the sensors and the difficulty of replacing and/or recharging batteries on the sensors in hostile environments require that the sensors be deployed with high density in order to extend the network lifetime.

2.8 Random and Group Mobility Models

43

Assumption 2.6 (Transmit-Control Power and Communication Link Reliability): Each sensor has transmit-power control and hence can adjust its transmission distance so it can transmit its data over a distance that is less than or equal to the radius of its communication range. Also, the communication links between the sensors are perfectly reliable while the sensors can fail or die independently due to low battery power. ∎ Assumption 2.7 (Absence of Obstacles): Any region of interest in the planar (respectively, spatial) deployment field does not contain any obstacles. ∎ Observation 2.1 (Assumption Relaxation): We should mention that some of these assumptions will be further relaxed so as to promote the use of our proposed protocols in real-world sensing applications. These relaxations will be discussed later.

2.8 Random and Group Mobility Models In this section, we consider two classes of mobility models, namely random mobility and group mobility models. More specifically, we account for a random mobility model, called random waypoint mobility model (RWP) [218]. In the literature, there are two well-known group mobility models, namely reference point group mobility model (RPGM) [168] and Manhattan mobility model (MMM) [297]. While MMM has a more restricted mobility pattern and requires a certain deployment field, RPGM is more general and can be applied to various sensor deployment scenarios. Therefore, we focus on RPGM in order to study its impact on the k-coverage problem in heterogeneous and sparsely deployed wireless sensor networks using mobile sensors. It is worth noting that the reference point group mobility model has several similarities with the community-based mobility model [307] in the way groups and communities are formed. All of these mobility models were originally proposed for mobile ad-hoc networks [310]. In this chapter, we use these models to support sensor mobility in order to ensure k-coverage of a region of interest in a deployment field. The goal of our study is to show the appropriateness and effectiveness of group mobility models compared to random mobility models. Next, we briefly describe these random and group mobility models.

2.8.1 Random Waypoint Mobility Model (RWP) The random waypoint mobility model [218] is the most widely used mobility model in the simulation of protocols for mobile wireless network systems. As its name suggests, this mobility model allows a mobile node to randomly select a destination, known as waypoint, in a specific simulation area and move towards it using a straight path at a constant speed from [0, vmax ], where vmax is the maximum speed. Once

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it has reached the target waypoint, the mobile node remains stationary for a time interval, known as pause time, and randomly selects new waypoint and speed to continue its movement. This process iterates until the end of the simulation. Both of the maximum speed and pause time are parameters of this mobility model. As it can be seen, this mobility model exhibits no correlation between the speed and direction of any pair of motions, i.e., it does not capture any spatial or temporal dependency between mobile node motions. This high degree of randomness makes this mobility model unrealistic and inappropriate for environments that require certain degree of collaboration between mobile nodes when accomplishing a common target task. In order to accommodate this random waypoint mobility model to our study, both of the target waypoint and pause time parameters must be chosen appropriately.

2.8.2 Reference Point Group Mobility Model (RPGM) The reference point group mobility model (RPGM) [168] defines the mobility behavior for a group of nodes with respect to a special node, called reference point (or leader). The speed and motion direction of mobile nodes are derived from those of their reference point. Thus, the mobile nodes of a group follow the leader’s movements although they have their own individual random motion behavior. In this mobility model, a group leader’s movement is defined by a motion vector that specifies its speed and its motion direction. The members of a group define their own general motion through a deviation from the group leader’s general motion vector as follows: vnode (t) = vcenter (t) + random() × SDR × vmax qnode (t) = qcenter (t) + random() × ADR × qmax where vnode (t) is the speed of a mobile node, vcenter (t) stands for the speed of its center (or reference point), SDR denotes the speed deviation ratio, ADR is the angle deviation ratio, vmax represents the maximum speed of the center, θ max equals 180°, and 0 ≤ SDR, ADR ≤ 1, where SDR and ADR are set to 0.1. It is worth noting that those group leaders are selected randomly. Also, each mobile node moves at a constant speed between 0 and vmax until it reaches its destination in the region of interest in the deployment field.

2.8.3 Manhattan Mobility Model (MMM) The Manhattan mobility model [297] is based on the notion of street maps, which are used for generating node mobility. As shown in Fig 2.13, a street map includes horizontal lanes, vertical lanes, and their intersections. Initially, all the mobile nodes

2.8 Random and Group Mobility Models

45

Fig. 2.13 Manhattan map

are placed randomly in those horizontal and vertical lanes. Based on the nature of the lane, a mobile node would move either vertically or horizontally. In particular, when a mobile node reaches an intersection, a decision would be made probabilistically on whether to keep moving straight, turn left, or turn right. The speed of mobile nodes depends on the direction of their previous movements. In order to use this mobility model, these horizontal and vertical lanes must be defined appropriately.

2.8.4 Why Group and Random Mobility Models? The reason we selected these two classes of mobility models rather than others is that they are orthogonal regarding their inherent properties. While RWP is based on randomness and non-collaboration between nodes’ motion, RPGM has autocorrelation properties between successive values of speed and direction. Also, mobile node movements are not totally independent. Rather, they are dependent on their group leaders’ motion. These properties are desirable in collaborative environments, such as mobile deployed wireless sensor networks. In fact, in mission-critical applications, such as rescue fields, sensor nodes gather to accomplish common tasks, and thus coordination between them is unavoidable. Thus, there must be spatial and temporal dependency in their motion. Moreover, a deployed wireless sensor network can be split into multiple groups, each of which can be led by a leader to coordinate the movements of its mobile sensors. Although randomness is still present in RPGM, the derivation of speed and direction from previous ones is highly dependent on the history of mobile sensors’ motions. All these features make RPGM more realistic than RWP and useful in collaborative situations, where the notion of teams of mobile sensors is an appealing solution to accomplish a collaborative task.

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2.9 Conclusion In this chapter, we define useful terms that are used throughout this book. Also, we discuss the notion of unconditional (or traditional) connectivity as well as conditional connectivity, and derived the network connectivity and fault-tolerance. Moreover, we describe the energy consumption and percolation models. In addition, we specify our default network model, which is used in the design and description of all the protocols proposed in this book. Furthermore, we discuss a subset of well-known mobility models.

Part II

Percolation Theory-Based Coverage and Connectivity in Wireless Sensor Networks

Chapter 3

A Planar Percolation-Theoretic Approach to Coverage and Connectivity

My brain is only a receiver, in the Universe there is a core from which we obtain knowledge, strength and inspiration. I have not penetrated into the secrets of this core, but I know that it exists. Nikola Tesla (1856–1943)

Overview This chapter addresses the problems of almost sure integrated coverage and connectivity in planar wireless sensor networks from the perspective of percolation theory. Specifically, it focuses on finding the critical sensor density above which the network is almost surely connected and the planar deployment field is almost surely covered. It proposes our solution to this problem using a probabilistic approach. Precisely, each of the above problems is discussed separately. Then, both are investigated in an integrated manner using a suitable integration model.

3.1 Introduction In wireless sensor networks, sensing coverage reflects the surveillance quality provided by active sensors in a field, while network connectivity enables active sensors to communicate with each other in data forwarding to a central gathering node, called the sink. For the correct operation of the network, it is necessary that both sensing coverage and network connectivity be maintained. Assuming perfectly reliable wireless links, both sensing coverage and network connectivity are affected by the planar sensor density. In this chapter, we compute the value of this sensor density to provide sensing coverage and network connectivity. Next, we briefly describe the major tasks we want to accomplish in this chapter. Moreover, we briefly state how to achieve each one of them.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_3

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3.1.1 Major Tasks This chapter focuses on the problems of phase transition in sensing coverage and network connectivity in planar wireless sensor networks. It is well known that the nature of phase transition is a central topic in percolation theory of Boolean models. The process of the ground getting wet during a period of rain [293] helps us understand the concept of phase transition. A circular wet patch forms whenever a point of the ground is hit by a raindrop. At the start of the rain, one can see a small wet area within a large dry area. After some time and as many raindrops continue to hit the ground, the situation suddenly changes and one can see a small dry area within a large wet area. This phase transition phenomenon occurs at a given density of the raindrops. This example gives us a better analogy with the phase transition problems that occur with respect to sensing coverage and network connectivity, respectively, which are discussed below. Moreover, it helps us approach these two problems of phase transition in sensing coverage and network connectivity from a perspective of continuum percolation. Next, we specify these tasks and give our corresponding plan of actions. First, given a planar field that is initially uncovered, as more and more sensors are continuously added to the network, the size of the partial covered areas increases. At some point, the situation abruptly changes from small fragmented covered areas to a single large covered area in the field. We call this abrupt change as the sensingcoverage phase transition (SCPT) [48]. The SCPT problem can be stated as follows: Given a planar field that is initially uncovered, we want to compute the sensor density corresponding to the first appearance of a single large covered component that spans the entire network. Second, likewise, given a network that is originally disconnected, the number of connected components changes with the addition of sensors such that the network suddenly becomes connected at some point. We call this sudden change in the network topology as the network-connectivity phase transition (NCPT) [48]. The NCPT problem can be expressed as follows: Given a network that is initially disconnected, we want to compute the sensor density corresponding to the first appearance of a single large connected component that spans the entire network. In order to accomplish the above two tasks and compute the corresponding sensor density for SCPT and NCPT, respectively, we proceed as follows: We propose a probabilistic approach to compute the covered area fraction at critical percolation for both of the SCPT and NCPT problems. Then, we derive the corresponding critical sensor density for each of these problems taken separately. In addition, we compute the critical sensor density when both of these problems are considered together in an integrated manner. In [44], we propose a different percolation theory-based approach to study coverage and connectivity for spatial wireless sensor networks. This study is discussed in Chap. 4. As discussed in the next section, the specific connection function used in the NCPT problem has not been studied before and hence no bound on the critical covered area

3.2 Phase Transition in Sensing Coverage

51

fraction is known. Furthermore, given that sensing coverage and network connectivity are not totally orthogonal [55, 425], we propose a new model for percolation in wireless sensor networks, called correlated disk model, which allows network connectivity and sensing coverage to be studied together in an integrated fashion. We show that the SCPT and NCPT problems have the same solution (i.e., same critical covered area fraction). Precisely, we solve the SCPT and NCPT problems together, where the radii of the sensing disks (r) of the sensors and the radii of their transmission disks (R) are related by R = αr with α ≥ 1. We show that networkconnectivity phase transition occurs provided that sensing-coverage phase transition arises and the ratio R/r has certain value. This study is of practical use for wireless sensor network designers to build up more reliable sensing applications in terms of their sensing coverage and network connectivity. For several real-word sensing applications, and in particular, intruder detection and tracking, it is required that each location in a target planar field be covered (or sensed) by at least one sensor. This would definitely imply better information gathering about the intruder, thus leading to accurate analysis and processing of the situation. On the one hand, sensing coverage is associated with all locations in a target planar field, and hence would guarantee that any event about the intruder is sensed by sensors. On the other hand, network connectivity would enable gathered data about an intruder to reach a central control unit for further analysis and processing. Thus, both sensing coverage and network connectivity should be maintained for high intruder detection and tracking accuracy. Network connectivity, however, depends on sensing coverage. Thus, it is necessary to compute the critical planar sensor density above which the target planar field is almost surely guaranteed to be covered and the network is almost surely guaranteed to be connected. It is worth noting that the exact value of the critical density at which an infinite (or single large) cluster of overlapping disks first appears is still an open problem, and its approximation is either predicted by simulations [322, 338, 386] or computed analytically [162]. From now on, “infinite” means “single large”.

3.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 3.2 solves the SCPT problem. Section 3.3 solves the NCPT problem. Section 3.4 discusses our results. Section 3.5 reviews related work. Section 3.6 concludes the chapter.

3.2 Phase Transition in Sensing Coverage This section discusses the sensing-coverage phase transition (SCPT) problem and solves it using a percolation-theoretic approach.

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Let X λ = { ξi : i ≥ 1 } be a planar homogeneous Poisson point process of density λ, where ξi represents the location of a sensor si . Given an initially uncovered planar field, the SCPT problem is to compute the probability of the first appearance of an infinite (or single large) covered component that spans the entire network. In particular, we are interested in the limiting case of an infinite planar field, where there exists no single large covered component for sufficiently small density λ and it suddenly appears at a critical percolation density λc .

3.2.1 Estimation of the Shape of Covered Components Each covered k-component CCk as defined earlier in Chap. 2 (Sect. 2.2, Definition 2.6) is characterized by a reference point, called center and denoted by ξ(k). Figure 3.1 shows various covered components of different sizes. Using the Poissonness argument stated in [189] (pp. 200–202), as the centers {ξi : i ≥ 1} form a Poisson process with density λ, the centers of all covered k-components also form a Poisson process with density λ(k). In other words, the covered components are randomly and independently distributed according to a Poisson process with a density of λ(k) centers per unit area.

Fig. 3.1 Schematic of overlapping disks (three covered components of size 1, two of size 2, one of size 3, and one of size 4)

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53

Fig. 3.2 Shape of a covered component

We want to determine the smallest shape enclosing a covered k-component. In fact, the shape of the covered components varies depending on the number of its overlapping sensing disks. For tractability of the problem, we assume that the geometric form that encloses a covered k-component is a circle (Fig. 3.2), which tends to minimize the area of uncovered region around the covered component. Indeed, the circle is the most compressed shape. Let Rk be the radius of a circle, denoted by C(Rk , k), which encloses a covered k-component. Thus, there is no other sensing disk that could overlap with the boundary of the circle. In other words, the concentric circular band of width r, denoted by CC B(r ) and which surrounds the circle, should not include any other sensing disk. Hence, the annulus between radii Rk and Rk + r around the center ξ(k) must be empty. Let P(k) be the conditional probability that the circle encloses only one covered k-component. This probability is given by P(k) = Prob[C(Rk , k)|CC B(r )is empty] By definition, this conditional probability is computed as P(k) =

Prob [C(Rk , k) ∧ CC B(r )empty] Prob[CC B(r )empty]

(3.1)

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3 A Planar Percolation-Theoretic Approach …

where Prob[C(Rk , k) ∧ CC B(r )empty] can be interpreted as the probability that the circle of radius Rk + r encloses only one covered k-component. Thus, Prob[C(Rk , k) ∧ CC B(r )empty] = Prob[C(Rk + r, k)] Using Eq. (2.3) shown earlier in Chap. 2, we obtain the following results: Prob [ C (Rk + r, k) ] =

(λ π (Rk + r )2 )k −λ π (Rk +r )2 e k!

Prob[CC B(r )empty] = e−λπ((Rk +r )

2

−r 2 )

Therefore, Eq. (3.1) becomes P(k) =

(λπ(Rk + r )2 )k −λπ Rk2 e k!

(3.2)

It is worth mentioning that the analysis of SCPT and NCPT problems is based on the form of conditional probability given in Eq. (3.2).

3.2.2 Critical Density of Covered Components Although there exist a few definitions of the average distance between clusters (i.e., covered components), one of them is more appropriate. It is defined as the average of the minimum distance between all pairs of sensing disks, each from one covered component. Indeed, two covered components could be merged together into a single one if there is at least a pair of sensing disks, one from each covered component, such that the distance between their centers is at most equal to 2r. Lemma 3.1 computes the mean distance between neighboring covered k-components at critical percolation. Lemma 3.1 (Mean Distance Between Neighboring Covered k-Components): Let {CCk } be a set of covered k-components with density λ(k) and Y a random vari1 between two able representing distances between them. The mean distance davg neighboring covered components at critical percolation is computed as follows:

1 1 davg = √ 2 λc (k)

(3.3)

where λc (k) is the density of {CCk } at critical percolation. Proof Let ωk be the mean number of covered k-components in a planar circular field of radius R. Denote by p(σ ) the probability that there is a covered component

3.2 Phase Transition in Sensing Coverage

55

whose center is located at a distance upper bounded by σ from the center, say ξ(k), of a given covered component. We denote by P(σ )dσ the probability that a nearest center of a covered component to a given center ξ(k) is located at a distance between σ and σ + dσ . Hence, P(σ )dσ can be viewed as the probability that there exists one of the ωk − 1 covered components at a distance between σ and σ + dσ from the center ξ(k) and the other ωk − 2 covered components are at a distance larger than σ from ξ(k). Thus, (

) ωk − 1 ∂ p(σ ) dσ (1 − p(σ ))(ωk −2) P(σ )dσ = 1 ∂σ ∂ p(σ ) = (ωk − 1) dσ (1 − p(σ )) (ωk −2) ∂σ

(3.4)

) dσ stands for the probability that there is a covered component whose where ∂ ∂p(σ σ center lies within a circular band located at a distance σ from the center ξ(k) and whose width is dσ. Notice that p(σ ) can be computed as the ratio of the number of covered components within the circle of radius σ to the total number of covered components within the field. Thus, we obtain:

p(σ ) =

λk π σ 2 σ2 = 2 2 λk π R R

and 2σ ∂ p(σ ) = 2 ∂σ R

(3.5)

Substituting Eq. (3.3) in Eq. (3.4) gives )(ωk −2) ( 2σ σ2 P(σ )dσ = (ωk − 1) 2 dσ 1 − 2 R R where ωk = λk π R2 . We assume that the planar circular field contains all covered components. Now, the mean distance between two covered k-components can be computed as ∫R E[Y ] =

σ P(σ )dσ 0

2 (ωk − 1) = R2

∫R 0

( )(ωk −2) σ2 σ2 1 − 2 dσ R

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Using the variable change T =

σ2 , R2

we obtain

E[Y ] = (ωk − 1)R

∫1 √

T (1 − T )(ωk −2) dT

0

Recall that the beta function [476] is defined by ∫1 B(m, n) =

u m−1 (1 − u)n−1 du 0

Thus, E[Y ] = (ωk − 1)RB(3/2,ωk − 1)

(3.6)

where B(m, n) =

Γ(m)Γ(n) Γ(m + n)

Γ(m) = (m − 1)Γ(m − 1) = (m − 1)! Hence, Eq. (3.6) becomes E[Y ] = (ωk − 1)R

Γ(3/2) Γ(ωk ) Γ(3/2)Γ(ωk − 1) =R Γ(ωk + 1/2) Γ(ωk + 1/2)

However, Graham et al. [178] proved that ( ) √ 1 Γ(x + 1/2) 1 5 21 = x 1− + + − + ··· Γ(x) 8x 128x 2 1024x 3 32768x 4 Thus, √ Γ(x + 1/2) = x x→∞ Γ(x) lim

Notice that at critical percolation, the value of ωk should be large enough (ωk → ∞)√so an infinite covered component spanning the network could form. Since 1 Γ(3/2) = 2π and ωk = λk π R2 , the mean distance davg between two neighboring covered k-components at critical percolation is given by 1 1 1 davg = lim E[Y ] = R Γ(3/2) √ = √ ωk →∞ ωk 2 λc (k)

3.2 Phase Transition in Sensing Coverage

57

where λc (k) is the critical density of covered k-components.

∎

Lemma 3.2 computes the average distance between neighboring covered kcomponents at critical percolation using another approach. As it can be seen later, Lemma 3.2 helps us compute the density of covered k-components at critical percolation. Lemma 3.2 (Mean Distance Between Neighboring Covered k-Components): Let {CCk } be a set of covered k-components with density λ(k), and Y a random variable 2 between two associated with the distances between them. The mean distance davg neighboring covered components at critical percolation is computed as 2 davg =

√ √ 2 er f (2 λc πr ) − 4 λc r e−4λc πr √ 2 λc

(3.7)

where λc is the density of a set of sensing disks {Di (r ) : i ≥ 1} at critical percolation. Proof For a homogeneous Poisson point process, the probability that there is no 2 neighbor within distance σ of an arbitrary point is given by e−λ π σ [126]. Therefore, the probability that the distance between a point and its neighbor is less than or equal to σ is given by P[Y ≤ σ ] = 1 − e−λπ σ

2

Hence, the corresponding probability density function is given by f (Y |Y ≤ σ ) = 2λπ σ e−λπ σ

2

2 The mean distance davg between two neighboring covered k-components of {CCk } at critical percolation is obtained when the distance σ between two sensing disks, say Di (r ) and D j (r ), each from one covered component, belongs to the interval [0, 2r ]. Therefore,

∫2r 2 davg

= E[Y |Y ≤ 2r ] =

σ × f (Y |Y ≤ σ )dσ 0

√ √ 2 er f (2 λc πr ) − 4 λc r e−4λc πr = √ 2 λc where er f (x) is the error function [478].

∎

Lemmas 3.3, which follows from Lemmas 3.1, Lemmas 3.2, computes the density of covered k-components at critical percolation.

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Lemma 3.3 (Density of Covered k-Components): The critical density of a set of covered k-components {CCk } is computed as follows: λc λc (k) = ( )2 √ √ er f (2 λc πr ) − 4 λc r e−4λc πr 2

(3.8)

where λc is the density of sensing disks at critical percolation and er f (x) is the error function [478]. Proof From Lemma 3.1 (Eq. 3.3) and Lemma 3.2 (Eq. 3.7), the mean distance between two covered k-components at critical percolation should verify the following 1 2 = davg , which implies that the density of covered k-components at equality davg critical percolation λc (k) is given by λc λc (k) = ( )2 √ √ er f (2 λc πr ) − 4 λc r e−4λc πr 2

∎

3.2.3 Critical Radius of Covered Components There is a particular value of the radius Rk of the circular shape enclosing a covered component that almost surely guarantees the formation of special class of covered k-components, called critical covered k-components. Any non-empty circle of radius 2r should enclose a covered k-component. In other words, regardless of the number of sensing disks of radius r located in a circle of radius 2r, these sensing disks should definitely form a covered k-component. Moreover, this covered k-component is a complete graph in that each pair of sensors, say si and sj whose sensing disks are included in this circle of radius 2r are collaborating given that |ξi − ξ j |max ≤ 2r . Lemma 3.4 computes the density of critical covered k-components at critical percolation. Lemma 3.4 (Density of Critical Covered k-Components): At critical percolation, the density of covered k-components, which are enclosed in circles whose radii is equal to 2r, is given by λc (k) = λc

(9λc π r 2 )k −4λc π r 2 e k!

(3.9)

where λc and λc (k) are the densities of sensing disks and covered k-components at critical percolation, respectively. Proof Let N be the total number of sensing disks that are randomly deployed in a planar circular field of radius R according to a spatial Poisson process with density equal to

3.2 Phase Transition in Sensing Coverage

59

λ=

N π R2

(3.10)

Using ωk = λ(k) π R2 , which represents the mean number of covered kcomponents in the planar circular field, and Eq. (3.10), we obtain λ(k) = λ

ωk N

(3.11)

We can approximate ωNk by the probability P[rad(CCk ) = 2r ] of finding a covered k-component whose radius is equal to 2r. Hence, we have P[rad(CCk ) = 2r ] =

ωk N

(3.12)

Substituting Eq. (3.12) in Eq. (3.11) gives λ(k) = λ P[rad(CCk ) = 2r ]

(3.13)

Following the same reasoning as in Sect. 3.2.1, P[rad(CCk ) = 2r ] is the conditional probability of finding k sensing disks enclosed in a circle with radius 2r and centered at ξ(k) such that the annulus between circles of radii 2r and 2r + r around the center ξ(k) is empty. Substituting Rk = 2r into Eq. (3.2) gives P(k) =

(9λ π r 2 )k −4λ π r2 e k!

and hence Eq. (3.13) becomes λc (k) = λc

(9λc π r 2 )k −4λc π r 2 e k!

where λc and λc (k) are the critical densities of sensing disks and covered k-components, respectively. ∎

3.2.4 Characterization of Critical Percolation Now, we generate an equation that characterizes a set of covered k-components at critical percolation. By equating Eqs. (3.8) and (3.9), we obtain a new equation g1 (λc , r, k) = 0, where

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√ √ 2 g1 (λc , r, k) = (er f (2 λc π r ) − 4 λc r e−4λc π r )2 ×

(9λc π r 2 )k −4λc π r 2 e −1 k!

(3.14)

Instead of focusing on finding the critical value of the density λc of sensing disks at which an infinite covered component first appears, we consider a dimensionless metric, i.e., the covered area fraction at critical percolation given by A c (r ) = 1 − e−λc π r

2

The benefits of using A c (r ) instead of λc are two-fold: first the number of unknown parameters is reduced to two, namely A c (r ) and k, thus removing any direct dependency of g1 (λc , r, k) on r. Hence, the parameter r does not have any direct impact on the critical percolation density. Second, we know the exact domain of A c (r ) is [0, 1], which helps us study exactly the entire behavior of the function g1 (λc , r, k) = 0 for all values of A c (r ). Substituting A c (r ) into Eq. (3.14) and let μ = − log(1 − Ac (r )) gives a new function g1 (Ac (r ), k) given by ( )2 √ 4 μe−4μ √ g1 ( Ac (r ), k) = er f (2 μ) − √ π (9μ)k −4μ e × −1 k!

(3.15)

3.2.5 Numerical Results Figures 3.3, 3.4 and 3.5 plot the function g1 ( Ac (r ), k) given in Eq. (3.15) with respect to different values of k and Ac (r ). Notice that for k < 4, the function g1 (Ac (r ), k) cannot be equal to zero (Figs. 3.3 and 3.4). Thus, percolation first occurs at k = 4 and Ac (r ) = 0.575 (Fig. 3.5), which is a bit smaller than the values 0.688 of Vicsek and Kertesz [386], 0.68 of Pike and Seager [322], and 0.62 of Roberts [338] (all predicted by Monte Carlo experiments), and the value 0.67 as calculated by Fremlin [162] for studying the percolation of overlapping homogeneous disks. Thus, when the number of collaborating sensors of a sensor is larger than four (k ≥ 5), it is almost surely that an infinite covered component that spans the entire network will appear for the first time.

3.2 Phase Transition in Sensing Coverage

Fig. 3.3 No critical percolation at k = 2

Fig. 3.4 No critical percolation at k = 3

61

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3 A Planar Percolation-Theoretic Approach …

Fig. 3.5 Critical percolation at k = 4 and Ac (r) = 0.575

3.3 Phase Transition in Network Connectivity Let X λ = {ξi : i ≥ 1} be a planar homogeneous Poisson point process of density λ, where ξi represents the location of sensor si . Given a network that is originally disconnected, the network-connectivity phase transition (NCPT) problem is to compute the planar sensor density corresponding to the first appearance of an infinite (or single large) connected component that spans the network. Notice that both of the SCPT and NCPT problems have similar structure although the difference of the concepts of collaboration (SCPT) and communication (NCPT) between the sensors in the SCPT and SCPT problems, respectively, as stated earlier in Chap. 2 (Sect. 2.2). In the SCPT problem, two sensing disks belong to the same covered component if the distance between them is at most equal to one diameter (2r). However, the NCPT problem requires that two communication disks be at a distance of at most half their diameter (i.e., R) from each other so they belong to the same connected component, where r and R stand for the radii of the sensing and communication disks of the sensors, respectively. To our knowledge, the connection function of the NCPT problem has not been studied previously in the literature. Some sensing applications require that every location in the planar field be covered by at least one sensor and that the active sensors be also able to communicate with each other so the sensed data could reach the sink. Indeed, sensed data would be meaningless if connectivity between the sensors is not maintained. Thus, we are mainly

3.3 Phase Transition in Network Connectivity

63

interested in the formation of an infinite (or single large) connected covered component that spans the entire network. Next, we study the SCPT and NCPT problems together using percolation theory.

3.3.1 Integrated Sensing Coverage and Network Connectivity We propose a new model for percolation in wireless sensor networks, called correlated disk model. Each sensor is associated with two concentric disks of radii r and R representing the radii of its sensing and communication disks, respectively. This kind of structure reveals a double behavior of the sensors that can be described by their collaboration and communication. The collaboration between sensors depends on the relationship between the radii of their sensing disks, whereas communication is related to the relationship between the radii of their communication disks. Previous studies by Wang et al. [401] and Ammari and Das [55] showed the existence of certain dependency between the concepts of sensing coverage and network connectivity. Our proposed correlated disk model allows us to study these two concepts together from a percolation-theoretic viewpoint to account for their correlation. This problem can be viewed as a correlated continuum percolation problem. Next, we study the simultaneous percolation of the sensing and communication disks of the sensors based on the ratio R/r .

3.3.1.1

Simultaneous Phase Transitions When R ≥ 2r

As discussed in the next section, Wang et al. [401] proved that if a wireless sensor network is configured to be covered and the radius R of the communication disk of the sensors is at least double the radius r of their sensing disk, then the network is guaranteed to be connected. Ammari and Das [55] provided a tighter relationship between R and r, while achieving network connectivity provided that sensing coverage is guaranteed. In fact, the “worst-case” behavior is when the sensing disks of the sensors are tangential, i.e., the distance between their corresponding centers is equal to 2r . Hence, when R ≥ 2 r, there is a dependency between sensing coverage and network connectivity in that the former implies the latter. In other words, collaboration between the sensors leads to their communication. In this case, the SCPT and NCPT problems are equivalent, and thus have the same critical covered area fraction. Thus, a set of communication disks percolates at k = 4 with a covered area fraction Ac (R) = 0.575 at critical percolation. Therefore, when the number of communicating sensors of a given sensor is larger than four (k = 4), an infinite connected component spanning the network will almost surely form.

64

3.3.1.2

3 A Planar Percolation-Theoretic Approach …

Simultaneous Phase Transitions When r ≤ R < 2r

The interesting case is when the radii of the sensing and communication disks of the sensors are related by R = α r, where 1 ≤ α < 2. Precisely, we focus on the study of the percolation of the sensing disks of the sensors, where two sensors collaborate if and only if the distance between the centers of their sensing disks is equal to α r, where 1 ≤ α < 2. The communication disks of the sensors also percolate given that R = αr . Thus, our goal is to compute the critical covered area fraction above which both the sensing and communication disks of the sensors percolate when r ≤ R < 2r . It is a valid assumption that the radius of the communication disks of the sensors cannot be less than the radius of their sensing disks as shown in Tables I and II [446] for a wide spectrum of sensor devices. We consider the previous analysis in Sect. 3.2, where we replace 2 r by α r, with 1 ≤ α < 2. Without repeating those details, we obtain a new equation that characterizes a set of covered k-components at critical percolation, which is given by g2 (λc , r, α, k) = 0, where √ √ 2 2 g2 (λc , r, α, k) = (er f ( λc π αr ) − λc α 2r e−λc πα r )2 ×

(9λc π α 2r 2 /4)k −λc πα2 r 2 e −1 k!

Let μ = − log(1 − Ac (r )). We substitute A c (r ) in g2 (λc , r, α, k) to obtain a new function g2 ( Ac (r ), α, k) given by (

)2 √ 2 α 2 μe−α μ g2 ( Ac (r ), α, k) = er f (α μ) − √ π ×

√

(9α 2 μ/4)k −α2 μ −1 e k!

(3.16)

Figures 3.6, 3.7, 3.8 and 3.9 show the plots of the function g2 (A c (r ), α, k) given in Eq. (3.16) for different values of k and α, where 2 ≤ k ≤ 5 and 1 ≤ α < 2. As it can be seen from Figs. 3.6 and 3.7, the function g2 ( A c (r ), α, k) cannot be equal to zero for k < 4, regardless of the value of α. Furthermore, a set of sensing disks percolates (which occurs when g2 ( A c (r ), α, k) = 0) faster for large values of α. For instance, when α = 1 (which corresponds to R = r ), critical percolation occurs at k = 5 and A c (r ) = 0.925 (Fig. 3.9). Thus, when α = 1, it is almost surely that an infinite covered component spanning the entire network will appear when the number of collaborating sensors of a sensor is larger than five (k ≥ 6). However, when α = 1.5 (i.e., R = 1.5 r ), critical percolation occurs at k = 4 and A c (r ) = 0.580 (Fig. 3.8). Finally, for α = 1.25 (i.e., R = 1.25 r), critical percolation occurs at k = 4 and A c (r ) = 0.760 (Fig. 3.8). For the last two cases (α = 1.25 and α = 1.5), it is almost surely that an infinite covered component that spans the entire network will appear when the number of collaborating sensors of a sensor is larger than four

3.3 Phase Transition in Network Connectivity

65

Fig. 3.6 Plot of the function g2 (A c (r ), α, k) for different values of α (1 ≤ α < 2). No critical percolation occurs at k = 2

Fig. 3.7 Plot of the function g2 (A c (r ), α, k) for different values of α (1 ≤ α < 2). No critical percolation occurs at k = 3

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3 A Planar Percolation-Theoretic Approach …

Fig. 3.8 Plot of the function g2 (A c (r ), α, k) for different values of α (1 ≤ α < 2). For k = 4, critical percolation depends on the value of α

Fig. 3.9 Plot of the function g2 (A c (r ), α, k) for different values of α (1 ≤ α < 2). For k = 5, critical percolation depends on the value of α

3.5 Related Work

67

(k ≥ 5). Given the connection function defined for the collaboration between the sensing disks, percolation should be quicker for large disks than for smaller ones. In all cases, the value of the corresponding critical covered area fraction will almost surely guarantee the appearance of an infinite connected component that spans the underlying network provided that an infinite covered component arises and spans the entire network. Moreover, the value of critical covered area fraction depends on the ratio R/r .

3.4 Discussion It is worth noting that both values of critical covered area fractions for sensing coverage and network connectivity represent only lower bounds. In other words, if the actual covered area fraction is higher than Ac (r ), it is almost surely that there exists an infinite (or single large) covered component that spans the entire network. That is, a large portion of the planar deployment field is guaranteed to be covered. Otherwise, there are only a few small fragmented regions of the planar field that are covered. However, there is no guarantee on the size of the region of the planar field being covered. Similarly, if the actual covered area fraction is higher than Ac (R), it is almost surely that there exists an infinite connected component that spans the entire network. Otherwise, it is almost surely that the network is disconnected. However, there is no guarantee neither on the number of nodes being connected in this infinite component nor whether the sink belongs to the infinite connected component. There appears to be little disagreement between our theoretical calculation of the critical covered area fraction (Ac (r ) = 0.575) compared to the values previously obtained by approximate calculation and Monte Carlo simulation (between 0.62 and 0.688). Our analysis of phase transitions in both sensing coverage (respectively network connectivity) is mainly based on an estimation of the smallest shape enclosing a covered (respectively connected) k-component. We assume that this shape is a circle. Although it may not be always true that a circle is the smallest shape enclosing covered (and connected) k-component, we use it to simplify the analysis enough and make it mathematically tractable. Also, we consider this shape as an ellipse with minor axis ak and major axis bk . Maximizing k P(k) = (λ π (ak +rk!) (bk +r )) e−λ π ak bk , the probability that an ellipse encloses a covered k-component, leads however to a unique solution ak = bk representing a degenerate ellipse or circle.

3.5 Related Work Adlakha and Srivastava [5] showed that the number of sensors required to cover an area of size A is in the order of O (A/ˆr 22 ), where rˆ 2 is a good estimate of the radius

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3 A Planar Percolation-Theoretic Approach …

r of the sensing disk of the sensors. Specifically, r lies between rˆ 1 and rˆ 2 , where rˆ 1 overestimates the total number of sensor required to cover an area of size A, while rˆ 2 underestimates it. Franceschetti et al. [161] investigated the number of disks of given radius r, centered at the vertices of an infinite square grid, which are required to entirely cover an arbitrary disk of radius r placed on the plane. Their result depends on the ratio of r to the grid spacing. Kumar et al. [243] proved that for random deployment with uniform distribution, if there exists a slowly growing function φ(np) such that npπr 2 ≥ log(np) + k log log(np) + φ(np), then a square unit area is k-covered with high probability when n sensors are deployed in it, where p is the probability that a sensor is active. It is worth noting that n also represents the planar sensor density given that the area of the square region is equal to 1. Hence, the above inequality can be written as , which means that the minimum sensor density required np ≥ log(np)+k logπrlog(np)+φ(np) 2 for k-coverage of a unit square region is equal to log(np)+k logπrlog(np)+φ(np) . If we set 2 . Recently, Balister p = 1 (i.e., every sensor is active), we obtain log(n)+k logπrlog(n)+φ(n) 2 et al. [74] computed the sensor density necessary to achieve both sensing coverage and network connectivity in finite region, such as thin strips (or annuli) whose lengths are finite. Balister et al. [74] applied this result to achieve barrier coverage [242] and connectivity in thin strips. In this type of deployment, the sensors act as a barrier to ensure that any moving object or phenomenon that crosses the barrier of sensors will be detected. Zhang and Hou [446, 448] proved that the required density for k-coverage of a planar square field, where sensors are distributed according to a Poisson point process and always active, depends on both the side length of the field and k. Precisely, Zhang and Hou [446, 448] found that a necessary and sufficient condition of complete kcoverage of a planar square field with side length l is that the sensor density is equal to λ = log l 2 + (k + 1) log log l 2 + c(l), where c(l) → +∞ as l → ∞. Also, given a wireless sensor network deployed as a Poisson point process with density λ and every sensor is active, Zhang and Hou [445] provided a sufficient condition for k-coverage of a square region with area A. Precisely, they proved that assuming λ = log A + 2 k log log A + c(A), if c(A) → ∞ as A → ∞, then the probability of k-coverage of the square region approaches 1. Furthermore, Zhang and Hou [445] provided the same result in the case where sensors are deployed according to a uniformly random distribution. Both results are based on the following statement: √ the square region is divided into square grids with side length s = log2rA , where r stands for the radius of the sensing range of the sensors. For a grid i to be completely k-covered, it is sufficient that there are at least k sensors within a disk centered at the center of the grid and with radius (1 − u) r, denoted by Bi ((1 − u)r ), where u = 1/ log A. Wan and Yi [389] showed that with boundary effect, the asymptotic (k + 1)coverage of a square with area s by Poisson point process with unit-area coverage range requires that the sensor density be equal to log s + 2(k + 1) log log s + ξ(s) with lims→∞ ξ(s) = ∞. Without the boundary effect, however, the asymptotic (k +

3.6 Conclusion

69

1)-coverage requires that the sensor density be computed as log s+(k+2) log log s+ ξ(s) with lims→∞ ξ(s) = ∞. The concept of continuum percolation originally due to Gilbert [171], is to find the critical density of a Poisson point process at which an unbounded connected component almost surely appears so the network can provide long-distance multihop communication. For random plane networks, Gilbert claimed that the filling factor, representing the critical value of the expected number of points in a circle of radius R should be around 3.2. Since then, Gilbert’s model has become the basis for studying continuum percolation in wireless networks. Booth et al. [86] discussed different classes of covering algorithms and determined the critical density of a Poisson point process (centers of spheres of radius r) above which an unbounded connected component arises. They also discussed the almost sure existence of an unbounded connected component based on the ratio of the connectivity range of the base stations to the connectivity range of the clients. Bertin et al. [81] proved the existence of site percolation and bond percolation in the Gabriel graph [163] for both Poisson and hard-core stationary point processes. The critical bounds corresponding to the existence of a path of opens sites and a path of open bonds were found by simulation. Glauche et al. [172] proposed a distributed protocol, which guarantees strong connectivity almost surely of ad hoc nodes. They translated the problem of finding the critical communication range of mobile devices to that of determining the critical node neighborhood degree above which an ad hoc network graph is almost surely connected. To achieve a little above this degree, each node needs to adjust its transmission power locally. Jiang and Bruck [217] proposed the concept of monotone percolation based on the local adjustment of the communication radii of the nodes for efficient topology control of the network. Their proposed algorithms guarantee the existence of relatively short paths between any pair of source and destination nodes, which makes monotonic progress. Liu and Towsley [276] considered both Boolean and general sensing models, each with a variety of network scenarios, to characterize fundamental coverage properties of large-scale sensor networks, namely area coverage, node coverage, and detectability. According to their simulation setting, the critical density is about 3.53 × 10−3 . Khanjary et al. [230] extended the work presented in this chapter to account for fixed-orientation directional sensor networks.

3.6 Conclusion In this chapter, we investigate two phase-transition problems for sensing-coverage and network-connectivity in wireless sensor networks using a probabilistic approach [48]. Our goal is to determine when an infinite covered component (SCPT problem) and an infinite connected component (NCPT problem) could form for the first time. To achieve this objective, we compute the covered area fraction for SCPT and NCPT problems at critical percolation. The problem of overlapping disks has been studied extensively in percolation theory and is similar to the SCPT problem. We find that the value of the covered area fraction is close to the one found by Monte Carlo

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3 A Planar Percolation-Theoretic Approach …

simulations. The specific connection function of the Boolean model associated with the SCPT problem, however, has not been studied before and hence no bound exists in the literature. We propose a correlated disk model in order to study SCPT and NCPT problems in an integrated way from a continuum percolation perspective. Precisely, we consider the physical correlation between them, which is based on the ratio of the radius of the communication disks of the sensors to the radius of their sensing disks. Thus, when an infinite covered component arises for the first time, an infinite connected component will almost surely appear based on the ratio α = R/r .

Chapter 4

A Spatial Percolation-Theoretic Approach to Coverage and Connectivity

The day science begins to study non-physical phenomena, it will make more progress in one decade than in all the previous centuries of its existence. Nikola Tesla (1856–1943)

Overview This chapter focuses on the problem of almost sure integrated coverage and connectivity in spatial wireless sensor networks. Precisely, it investigates the critical spatial sensor density above which coverage percolation and connectivity percolation in spatial deployment fields will almost surely occur. It discusses our solution to this problem using an approach that is totally different from the one proposed for planar deployment fields. Also, it addresses the problem of integrated coverage and connectivity in a spatial field.

4.1 Introduction Although most of the studies on coverage and connectivity in wireless sensor networks considered planar settings, such networks can in reality be accurately modeled in the space. While extensive work has been done on coverage and connectivity in planar wireless sensor networks, the analysis for spatial wireless sensor networks has hardly received any sustained attention. In particular, only a few references on spatial wireless sensor networks exist in the literature [12, 324, 326, 327, 328, 336] although there are several scenarios of real-world set up where the sensors are deployed in the space rather than in a planar field. For instance, sensor networks deployed on the trees of different heights in a forest, in a building with multiple floors, and underwater, require a design in the space rather than in the plane. While coverage creates collaboration between the sensors in covering a target spatial field for monitoring specific phenomena, connectivity enables communication between them. The concepts of continuum percolation theory best fit the problem of connectivity in wireless sensor networks that aims to find out whether the network can provide long-distance multi-hop communication [293]. We say the network exhibits © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_4

71

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4 A Spatial Percolation-Theoretic Approach …

coverage percolation when a giant covered region almost surely spans the entire spatial deployment field for the first time. Also, we say the network presents connectivity percolation when a giant connected component almost surely spans the entire network for the first time. While coverage depends on the sensing capability of the sensors, connectivity depends on their communication capability. However, these two concepts are not totally independent of one another and thus can be studied in combination from a perspective of percolation theory. Indeed, it has been proved that when a wireless sensor network is configured to provide coverage and the radius of the communication ranges of the sensors is at least double the radius of their sensing ranges, the network is guaranteed to be connected [425]. Next, we present the major tasks we want to accomplish in this chapter. In addition, we briefly discuss how to accomplish each one of them.

4.1.1 Major Tasks As described earlier in Chap. 3, a few approaches for studying coverage and connectivity in wireless networks, including planar wireless sensor networks, and using percolation theory have been proposed [81, 86, 171, 172, 217, 276]. This chapter addresses the problems of coverage percolation, connectivity percolation, and integrated coverage and connectivity percolation in spatial wireless sensor networks using a continuum percolation theory-based approach [44]. Precisely, it investigates the critical spatial density as well as the corresponding critical network degree for percolation in coverage and connectivity in spatial wireless sensor networks. Next, we specify these tasks and give our corresponding plan of actions. First, we want to compute the critical spatial density above which a giant covered region of a spatial field will almost surely appear for the first time. Also, we want to determine the corresponding critical network degree. Second, we want to calculate the critical spatial density above which a giant connected component will almost surely appear for the first time. In addition, we want to compute the corresponding critical network degree. Third, we want to compute the critical spatial density above which a giant covered region of a spatial field and a giant connected component will almost surely appear simultaneously for the first time. Moreover, we determine the corresponding critical network degree. In order to answer these questions, our proposed approach is based on Baxter’s factorization [79] of Ornstein-Zernike equation [317] and the pair-connectedness theory [122]. It is worth mentioning that both methods were initially proposed to understand the structure and dynamics of simple liquids.

4.2 Three Percolation Problems

73

4.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 4.2 introduces our integrated-concentric-sphere model, discusses coverage percolation, connectivity percolation, and integrated coverage and connectivity percolation problems in spatial wireless sensor networks, and computes their corresponding critical spatial density. It also computes their corresponding network degree. Section 4.3 discusses the results in Sect. 4.2 and generalizes them by relaxing some widely used assumptions in coverage and connectivity in wireless sensor networks. Section 4.4 reviews related work. Finally, Sect. 4.5 concludes the chapter.

4.2 Three Percolation Problems In this section, we discuss coverage percolation, connectivity percolation, and integrated coverage and connectivity percolation problems in spatial wireless sensor networks. Precisely, we compute their corresponding critical densities and network degrees. Let X λ = {ξi : i ≥ 1} be a homogeneous Poisson point process of density λ in IR3 , where ξi represents the center of two concentric spheres corresponding to the sensing and communication spheres of a sensor si . Assume that λ is not a constant as the sensors could appear and disappear independently of one another.

4.2.1 Sensing Coverage Percolation Problem formulation: The coverage percolation (COVP) problem can be stated as follows: Given a spatial field that is initially uncovered (or consists of small fragmented covered areas), compute the spatial density λcov c , called critical percolation spatial density (or critical spatial density), such that there surely exists a giant covered region that spans the entire spatial deployment field when λ > λcov c , and hence the Boolean model (X λ , {Bi (r ) : i ≥ 1}) percolates. Otherwise, there is no giant covered region and hence (X λ , {Bi (r ) : i ≥ 1}) does not percolate. Precisely, we are interested in the limiting case of an infinite (or very large) spatial field, where at a spatial density lower than some density, called critical percolation spatial density λc , there are several small covered regions in a spatial deployment field and that at a spatial density higher than λcov c , there appears a giant covered region. In this problem, we only consider the sensing spheres of the sensors of radius r. The concept of physical clustering of particles was first investigated by Hill [201]. In this section, the sensors will be identified with particles. According to Hill [201], the probability of connectedness and disconnectedness depends on the Boltzmann

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4 A Spatial Percolation-Theoretic Approach …

factor e(σ ) that is separated into connected and disconnected parts e+ (σ ) and e∗ (σ ) respectively, such that e(σ ) = e+ (σ ) + e∗ (σ ) = e−β u(σ )

(4.1)

where ⎧

e+ (σ ) = e−β u e∗ (σ ) = e−β u

+

∗

(σ )

(σ )

⎧ + u (σ ) : interaction potential for connected pair of ⎪ ⎪ ⎨ particles separated by a distance, σ, apart ∗ ⎪ (σ ) : interaction potential for disconnected pair of u ⎪ ⎩ particles separated by a distance, σ, apart

(4.2)

(4.3)

and β = 1/(k B T ), where T is the temperature and k B the Boltzmann constant. Following Hill’s definition of physical clusters based on inter-particle distance [201], we set up the interaction potentials u+ (σ ) and u∗ (σ ) as follows: ⎧

u + (σ ) = u(σ ) and u ∗ (σ ) = +∞ for 0 < σ < 2r u + (σ ) = +∞ and u ∗ (σ ) = u(σ ) for σ > 2 r

(4.4)

Hill’s work was the basis for other theories, such as theory of percolation in liquids [134, 135] and theory of pair connectedness [122]. Coniglio et al. [121] extended Hill’s work with a general theory of the equilibrium distribution of physical clusters, which was then extended with a theory of the pair connectedness. The pair-connectedness function [122] H (σ ) is defined so that λ2 H (σ )d 2 σ is the probability of finding a pair of particles (or sensors) in the same cluster (or covered component) and separated by a distance σ apart, where λ is the particle density. Our correlated continuum percolation-based approach is driven by the concept of physical clustering [201] and can be formulated in terms of an integral equation of OrnsteinZernike [317]. Hence, we need to compute the pair-connectedness function H (σ ). It is worth mentioning that the Ornstein-Zernike relation was originally proposed for a homogeneous fluid to deal with inter-particle correlation in order to find out how the position of a particle affects that of its neighbors by virtue of inter-particle interaction [200, 201]. Coniglio et al. [122] considered another metric, namely connectedness, and proposed an Ornstein-Zernike equation for the pair-connectedness function in analogy with that for the pair-correlation function [317]. According to Coniglio et al. [122], the Ornstein-Zernike relation can be established from the pair-connectedness function H (σ ) and the direct pair-connectedness function C + (σ ) as follows: +

H (σ ) = C (σ ) + λ

∫

| | C + (σ ' )H (| σ − σ ' | )dσ '

(4.5)

4.2 Three Percolation Problems

75

Following Baxter’s transformation [79], Eq. (4.5) can be formulated as ~+ (k) H ~(k) ~(k) = C ~+ (k) + λ C H

(4.6)

where ~ H(k) = 4π k−1

∫∞ dσ sin(kσ )σ H(σ ) 0

and + ~ C (k) = 4π k−1

∫∞

dσ sin(kσ )σ C + (σ )

0

are the Fourier transforms of H(σ ) and C + (σ ), respectively. Equation (4.6) implies that + + ~ C (k) H(k) (1 − λ ~ C (k)) = ~

and + ~ ~ ~ C (k) (1 + λ H(k)) = H(k)

Thus, it follows that ~ 1 + λ H(k) =

1 + 1−λ~ C (k)

(4.7)

Using Baxter’s factorization [79], we introduce a new function Q(σ ) such that + ~ Q(k) ~ Q(−k) = 1 − λ ~ C (k)

(4.8)

where ~ Q(k) = 1 − 2π λ

∫2r dσ e i kσ Q(σ ) 0

and Q(σ ) = 0 for σ > 2r

(4.9)

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4 A Spatial Percolation-Theoretic Approach …

Hence, the Ornstein-Zernike relation for the pair-connectedness function (Eq. 4.5) can be written as follows [79]: ⎧ ⎨ ⎩ ⎧ ⎨ ⎩

) + 2π λ σ C + (σ ) = − ∂ Q(σ ∂σ

for 0 < σ < 2r

) + 2π λ σ H (σ ) = − ∂ Q(σ ∂σ

for σ > 0

∫2r

∫2r σ

dt ∂ Q(t) Q(t − σ ) ∂t

dt(σ − t)H (|σ − t|)Q(t)

0

(4.10)

(4.11)

where Q(σ ) = 0 for σ < 0 and σ > 2r Notice that the direct pair-connectedness function C + (σ ) vanishes outside the associated range parameter (C + (σ ) = 0) for σ > 2 r (2r is the diameter of sensing spheres). Furthermore, Coniglio et al. [122] showed that H(σ ) = e−β u

+

(σ )

g(σ )eβ u(d) + e−β u

∗

(σ )

(H(σ ) − C + (σ ))

(4.12)

Substituting Eq. (4.4) for σ ≤ 2r in Eq. (4.12) implies H (σ ) = g(σ )

(4.13)

where g(σ ) = gs (σ ) = 1−ωs [122] stands for the Percus-Yevick approximation [80] of the pair-connectedness function, and ωs is the minimum overlap volume fraction between two sensing spheres whose range is given by 0 < ωs < 1. Indeed, two collaborating sensing spheres need only intersect. Thus, H (σ ) = 1−ωs . Substituting the value of H (σ ) in Eq. (4.7) for 0 < σ < 2 r, it follows that ∂ Q(σ ) + 2π λ σ (1 − ωs ) = − ∂σ

∫2r d t(σ − t)(1 − ωs ) Q(t)

(4.14)

σ

for 0 < σ < 2r That is, ⎛ ⎞ ∫2r ∂ Q(σ ) ⎝ = (ωs − 1) + 2π λ(1 − ωs ) Q(t)dt]⎠σ ∂σ 0

∫2r + 2π λ(ωs − 1)

t Q(t)dt = aσ + b 0

(4.15)

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77

where a and b are constant determined by Q(σ ) and given by ⎧ ∫2r ⎪ ⎪ ⎨ a = (ωs − 1) + 2π λ(1 − ωs ) Q(t)dt ∫2r ⎪ ⎪ ⎩ b = 2π λ(ωs − 1) t Q(t)dt

0

(4.16)

0

The integration of Eq. (4.15) under the assumption that Q(σ ) = 0 for σ < 0 and σ > 2 r gives Q(σ ) =

a 2 (σ − 4r 2 ) + b(σ − 2r) 2

(4.17)

Substituting Eq. (4.17) in Eq. (4.16) for a and b, and solving them, produces a unique solution given by ⎧

a= b=

−12(1−ωs )(3−λπ(1−ωs )(2r )3 ) λ2 π 2 (1−ωs )2 (2r )6 +12λπ(1−ωs )(2r )3 +36 −9λπ (1−ωs )2 (2r )4 λ2 π 2 (1−ωs )2 (2r )6 +12λπ(1−ωs )(2r )3 +36

(4.18)

According to Coniglio et al. [122], the mean cluster (or covered component) size is computed as ∫ MC S = 1 + λ

H (σ )dσ

(4.19)

~(k) of the pair connectedness, we obtain Using the Fourier transform H ~(0) MC S = 1 + λ H

(4.20)

According to Eq. (4.7), MC S =

1 ~+ (0) 1 − λC

(4.21)

At critical percolation, the cluster mean size diverges (i.e., MC S → ∞) and this occurs only when λc c˜+ (0) = 1, where λc is the percolation critical spatial density. Using Eq. (4.8), we find that MC S =

1 ~ Q (0) 2

(4.22)

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4 A Spatial Percolation-Theoretic Approach …

Substituting Eq. (4.9) in Eq. (4.22) and using Eqs. (4.17) and (4.18), we find that ⎛ MC S = ⎝1 − 2π λ ( =

∫2r

⎞−2 Q(σ )dσ ⎠

0

λ2 π 2 (1 − ωs )2 (2r )6 + 12λπ(1 − ωs )(2r )3 + 36 −12λπ(1 − ωs )(2r )3 + 36

)2 (4.23)

Hence, the critical percolation spatial density λcov c (r, ωs ) of the sensing spheres of radius r that corresponds to the divergence of MC S (i.e., MC S → ∞ or −12 λ π (1− ωs ) (2 r )3 + 36 = 0) is given by λcov c (r, ωs ) =

0.119 (1 − ωs ) r 3

(4.24)

where 0 < ωs < 1 is the minimum overlap volume fraction between sensing spheres. Notice that λc is a spatial variable. Theorem 4.1 (Critical Spatial Sensor Density for Giant Covered Component): The critical spatial sensor density λcov c (r, ωs ) above which a giant covered region that span a spatial deployment field will almost surely form is given by λcov c (r, ωs ) =

0.119 (1 − ωs ) r 3

where 0 < ωs < 1 is the minimum overlap volume fraction between the sensing spheres of the sensors of radius r. ∎ To remove the dependability on r, we consider the function μc (ωs ) = 0.119 3 = 1−ω . Figure 4.1 shows the plot of the function, μc (ωs ) for λcov c (r, ωs ) r s 0 < ωs < 1. As it can be seen, the percolation spatial density depends monotonically on the minimum overlap volume fraction ωs between the sensing spheres of the sensors. Higher overlap volume fraction requires high percolation spatial density and vice versa. Notice that the percolation spatial density on Fig. 4.1 corresponds to the minimum spatial density above which a giant covered component almost surely appears for the first time. However, Kong and Yeh [235] found that for spatial Poisson random geometric graphs (RGGs), the analytical lower bound for the critical spatial density is 0.4494. Although their analysis is elegant, we believe that the critical spatial density should depend on the coverage capability of the sensors (i.e., sensing range). Let us now compute the critical network degree ηc , which in this case coincides with the average size of the collaborating sets of the sensors (Chap. 2, Sect. 2.2, Definition 2.3) at the percolation spatial density. According to Hill [200], the average network degree at percolation (or critical network degree) is given by

4.2 Three Percolation Problems

79

Fig. 4.1 Plot of the function μc (ωs ) for 0 < ωs < 1

∫2r ηc = avg{|C ol(s i )|c : i ≥ 1} = 4 π λc

σ 2 g(σ )dσ

(4.25)

0

where g(σ ) = gs (σ ) = 1 − ωs . Using Eqs. (4.24) and (4.25), we find that ηc = 4. Thus, when the sensing spheres of the sensors percolate, every sensor’s sensing sphere overlaps with exactly other four sensing spheres. However, Dall and Christensen [127] found that in random geometric graphs (RGGs), the lowest connectivity at which the fraction of vertices in the largest cluster is larger than 0 in the macroscopic limit is equal to 2.74, which is lower than ours.

4.2.2 Network Connectivity Percolation Problem formulation: The connectivity percolation (CONP) problem can be stated as follows: Given a network that is originally disconnected (or consists of multiple connected components), compute the critical percolation spatial density λcon c , such that there exists a giant connected component that spans the entire network when λ > λcon c , and hence the Boolean model (X λ , {Bi (R) : i ≥ 1}) percolates. Otherwise, there is no giant connected component and hence (X λ , {Bi (R) : i ≥ 1}) does not percolate.

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4 A Spatial Percolation-Theoretic Approach …

Specifically, we are interested in the limiting case of an infinite spatial field, where at a spatial density lower than λcon c , there are several small connected components and that at a spatial density higher than λcon c , a giant connected component appears. Now, we consider the communication spheres of radius R. The main difference between the COVP and CONP problems is the connection function (Definitions 2.3 and 2.4, Sect. 2.2, Chap. 2), which has an impact on the overlap volume fraction between the sensing spheres (ωs ) and between the communication spheres (ωt ) of the sensors. For the COVP problem, it is sufficient that a pair of sensing spheres intersect for collaboration to occur. Connectivity between a pair of sensors, however, requires that at least the halves of their communication spheres overlap. Let ωt be the minimum overlap volume fraction between two communication spheres so their corresponding sensors communicate with each other. Lemma 4.1 computes this minimum overlap volume fraction. Lemma 4.1 (Minimum Overlap Volume Fraction): Two sensors communicate with each other if the minimum overlap volume fraction of their communication spheres is equal to Vmin = (5/12) × π × R 3 where R is the radius of their communication spheres. Proof Let us consider the diagram in Fig. 4.2, which corresponds to the minimum overlap volume fraction of the communication spheres of two communicating sensors si and s j when they are located at a distance R from each other. Without loss of generality, assume that one of the two spheres is located at the center (0, 0, 0) of the Euclidean space IR3 , while the other one is located at (R, 0, 0). Recall that the

Fig. 4.2 Minimum overlap volume for communication

4.2 Three Percolation Problems

81

equations of these two spheres are given by x 2 +y 2 +z 2 = R 2 and (x − R)2 +y 2 +z 2 = R 2 , respectively. Substituting the result of the first equation into the second one gives (x − R)2 + x 2 = 0 whose unique solution is given by x0 = R/2. It means that the intersection of the two spheres is a curve, which lies in a plane parallel to the yz plane at a single x coordinate. Substituting the value of x0 into the first equation of the sphere gives y 2 + z 2 = R 2 − R 2 /4 = 3R 2 /4 √ which corresponds to a circle whose radius is R0 = 3R/2. The intersection volume of the two spheres is the sum of the two spherical caps, where the distance between the center of the first sphere and the center of the intersection volume (or base of the left spherical cap) is x0 and the distance between the center of the second sphere and base of the right spherical cap is R − x0 = x0 too. Hence, the height of these spherical caps is equal to R − x0 = x0 and thus the volume of these two spherical caps is given by V = 2 × (1/3) × π × (R/2) 2 × (3R − R/2) = (5/12) × π × R 3 Hence, the minimum intersection volume of the communication spheres of two ∎ communicating sensors is equal to Vmin = (5/12) × π × R 3 . From Lemma 4.1, it follows that the minimum overlap volume fraction of the communication spheres of two communicating sensors is equal to ωt = Vmin /V0 = (5/12) × (3/4) = 0.3125 where V 0 = (4/3) × π × R3 is the volume of a communication sphere of radius R. Hence, the range of ωt is given by 0.3125 ≤ ωt < 1. Following the same analysis as in the COVP problem, which is discussed in Sect. 4.2.1, we find that the critical spatial density λcon c (R,ωt ) for the CONP problem is given by λcon c (R,ωt ) =

0.955 (1 − ωt ) R 3

(4.26)

where 0.3125 ≤ ωt < 1. Theorem 4.2 (Critical Spatial Sensor Density for Giant Connected Component): The critical spatial sensor density λcon c (R, ωt ) above which a giant connected component that span a spatial wireless sensor network will almost surely form is computed as λcon c (R, ωt ) =

0.955 (1 − ωt ) R 3

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4 A Spatial Percolation-Theoretic Approach …

where 0.3125 ≤ ωt < 1 is the overlap volume fraction between the communication spheres of the sensors of radius R. ∎ 0.955 3 Figure 4.3 shows the plot of the function μc (ωt ) = λcon c (R, ωt )R = 1−ωt , for 0.3125 ≤ ωt < 1. Notice that the percolation spatial density in the case of the CONP problem is higher than its counterpart for the COVP problem. This is mainly due to the difference in the concepts of collaboration and communication between the sensors. Recall that two collaborating sensors require that the distance between their sensing spheres be at most equal to 2r, whereas the distance between the communication spheres of two communicating sensors should be at most equal to R. Thus, it is expected that the network that results from connectivity percolation be denser than the network which results from coverage percolation. Also, the critical network degree, which in this case coincides with the average size of the communicating sets of the sensors (Definition 2.4, Sect. 2.2, Chap. 2) at the percolation spatial density, which is given by

∫R ηc = avg{|C om(s i )|c : i ≥ 1 } = 4π λc

σ 2 g(σ )dσ 0

is equal to 4 where g(σ ) = gt (σ ) = 1 − ωt . In other words, the communication sphere of a sensor overlaps with four other communication spheres at the percolation spatial density so that communications between it and the corresponding four sensors is guaranteed.

Fig. 4.3 Plot of the function μc (ωt ) for 0.3125 ≤ ωt < 1

4.2 Three Percolation Problems

83

4.2.3 Coverage and Connectivity Percolation Problem formulation: The integrated coverage and connectivity percolation (ICCP) problem can be expressed as follows: Given a spatial field that is initially uncovered and a network that is originally disconnected, compute the critical percolation spatial such that there almost surely exists a giant coordinated component density λcov−con c that span the entire spatial deployment field and network when λ > λcov−con , and c hence the Boolean model (X λ , {Bi (r ) : i ≥ 1} ∪ {Bi (R) : i ≥ 1}) percolates. Otherwise, there is no giant coordinated component and hence (X λ , {Bi (r ) : i ≥ 1} ∪ {Bi (R) : i ≥ 1}) does not percolate. In this problem, we are interested in the limiting case of an infinite spatial field, , there are several small coordinated where at a spatial density lower than λcov−con c , there appears a giant components and that at a spatial density higher than λcov−con c coordinated component.

4.2.3.1

Two-Concentric-Sphere Model

The model that we propose to address coverage and connectivity in spatial wireless sensor networks in an integrated fashion is called two-concentric-sphere (TCS). Recall that each sensor is represented by two concentric spheres of radii r and R, respectively, namely sensing sphere and communication sphere. In this section, we study a two-phase transition problem, where both coverage percolation and connectivity percolation are considered together. A dependency between both problems stems from the fact that coverage percolation is capable of leading to connectivity percolation when specific conditions, which will be discussed later, hold. Our proposed TCS model accounts for this inherent relationship. Precisely, we investigate integrated coverage and connectivity percolation of the sensing and communication spheres of the sensors based on their ratio R/r. Our TCS model is different from existing models for studying two-phase systems, such as the permeable sphere model [79], the adhesive sphere model [80], and the penetrable-concentric-shell (PCS) model [382]. DeSimone et al. [134, 135] proposed a model of extended spheres, which is very similar to the penetrable-concentric-shell model. The TCS and PCS models have some similarity except that the notion of hard-core, which exists in the PCS model, does not exist in the TCS model. Indeed, according to our TCS model, any pair of sensing spheres of the sensors can be penetrable, i.e., the volume of their overlap region can be greater than zero.

4.2.3.2

Integrated Continuum Percolation

Our TCS model exhibits in general two types of interactions: one between the sensing spheres of the sensors and another one between their communication spheres. Also, these interactions depend on the distance between the centers of the concentric

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spheres of the sensors. Lemma 4.2 states the condition under which a pair of sensor coordinates. As discussed in the next section, this lemma will be the basis of our study of percolation of integrated sensing coverage and network connectivity in spatial wireless sensor networks. Lemma 4.2 (Condition for Sensor Coordination): A pair of sensors, say si and s j , coordinates if |ξi − ξ j | ≤ τ r, where τ = min{2, R/r }. Furthermore, the coordination process depends on the ratio R/r of the radius of communication spheres R to the radius of sensing spheres r of the sensors. Proof By definition, two sensors si and s j collaborate if |ξi − ξ j | ≤ 2r and they communicate if |ξi − ξ j | ≤ R. Thus, si and s j coordinate if |ξi − ξ j | ≤ min{2r, R}, i.e., |ξi − ξ j | ≤ min{2, R/r } × r, where the values of R correspond to varying degrees of sensing sphere overlap. For instance, if R = 2r, then the coordination between two sensors requires that the latter be at most 2 r apart from each other. Hence, it would be sufficient if the sensing spheres of the sensors are tangential. Now, if R = r, the overlap area between the sensors increases since the distance between them decrease as the coordination requires |ξi − ξ j | ≤ r . As it can be seen, the overlap volume fraction between the sensing spheres of the sensors depends on the ratio R/r. When this ratio is high, the size of overlap volume of the sensing spheres is small and increases when this ratio decreases. ∎ By Lemma 4.2, a solution to the problem of percolation of integrated coverage and connectivity in spatial wireless sensor networks would depend on the ratio R/r of the radius of communication spheres to the radius of sensing spheres of the sensors. Without loss of generality and because of the physical properties of the sensors, it is reasonable to assume that the radius of the communication spheres of the sensors is at least equal to the radius of their sensing spheres, i.e., R ≥ r. As a matter of fact, it is always the case that the radius of the communication range of the sensors is higher than that of their sensing range, according to the data sheet regarding the specification of the architectural features of the sensors, such as those manufactured by CrossBow [467]. By Lemma 4.2, the integrated coverage and connectivity percolation of the network will depend on the distance between the centers of the concentric spheres as well as the ratio R/r. For the general case, we can set up R = α r, where α ≥ 1. Xing et al. [425] proved that when the network is configured to provide sensing coverage and R ≥ 2 r, the network is guaranteed to be connected. Since we are interested in the co-appearance of a giant covered component and a giant connected component, we consider two ranges of values of α, namely 1 ≤ α < 2 and α ≥ 2. Notice that the relationship R = 2 r corresponds to the worst-case behavior, where the sensing spheres of a pair of sensors are tangential. As studied earlier, the distance between a pair of overlapping sensing spheres cannot exceed 2 r. Hence, if we apply the same reasoning to the case where α ≥ 2, we obtain a critical percolation spatial density λcov−con (r, ωs ) equal to c λcov−con (r, ωs ) = c

0.119 (1 − ωs ) r 3

4.2 Three Percolation Problems

85

where 0 < ωs < 1. In the sequel, we only focus on the case R = α r, where 1 ≤ α < 2. Thus, the maximum distance between the centers of overlapping sensing spheres is equal to α r with 1 ≤ α < 2. The following analysis will consider only the sensing spheres to compute the percolation spatial density. With this in mind, the Ornstein-Zernike relation for the pair-connectedness function can be expressed as follows: ⎧ αr ⎨ σ C + (σ ) = − ∂ Q(σ ) + 2π λ ∫ dt ∂ Q(t) Q(t − σ ) ∂σ ∂t σ ⎩ for 0 < σ < αr

⎧ αr ⎨ σ H (σ ) = − ∂ Q(σ ) + 2π λ ∫ dt(σ − t)H (|σ − t|) Q(t) ∂σ 0 ⎩ for σ > 0

(4.27)

(4.28)

where Q(σ ) = 0 for σ < 0 and σ > α r. We also have H (σ ) = g(σ ) with g(σ ) = g s (σ ) = 1 − ωs for σ < α r. Following the same analysis as in Sect. 4.2.1, we find that the minimum volume fraction of the intersection of two sensing spheres whose centers are located at a distance α r from each other is equal to Vmin = (1/12) π (4 + α) (2 − α) r 3 . Therefore, the minimum overlap volume fraction of the sensing spheres of the sensors is equal to ϑ(α) = Vmin /V0 = (4 + α ) (2 − α)/16, where V0 = (4/3) π r 3 is the volume of a sensing sphere of radius equal to r, and hence, ϑ(α) ≤ ωs < 1. Substituting the value of H (σ ) in Eq. (4.24) for 0 < σ < α r, it follows that ∂ Q(σ ) + 2π λ σ (1 − ωs ) = − ∂σ

∫α r d t(σ − t)(1 − ωs ) Q(t)

(4.29)

0

for 0 < σ < αr Similarly, we find

∂ Q(σ ) ∂σ

= a σ + b whose integration gives

Q(σ ) =

a 2 (σ − (α r)2 ) + b(σ − α r) 2

(4.30)

with Q(σ ) = 0 for σ < 0 and σ > α r, where ⎧ α ∫r ⎪ ⎪ ⎨ a = (ωs − 1) + 2π λ (1 − ωs ) Q(t)d t 0

α ∫r ⎪ ⎪ ⎩ b = 2π λ (ωs − 1)t Q(t)d t 0

(4.31)

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Fig. 4.4 Plot of the function μc (ωs , α) for ϑ(α) ≤ ωs < 1 and 1 ≤ α < 2

Using the same procedure as earlier, solving the two integral Eq. (4.31) with two unknowns, namely a and b, defines the function Q(σ ) given in Eq. (4.30). Using Eqs. (4.9), (4.22), (4.30) and (4.31), we find that the critical percolation spatial density λcov−con (r, ωs , α) for integrated coverage and connectivity is given by c λcov−con (r, ωs , α) = c

0.955 (1 − ωs ) α 3 r 3

(4.32)

where ϑ(α) ≤ ωs < 1, with ϑ(α) = (4 + α ) (2 − α)/16 being the minimum overlap volume fraction between sensing spheres, and 1 ≤ α < 2. Theorem 4.3 (Critical Spatial Sensor Density for Giant Coordinated Component): Under the assumption that R = α r , a giant coordinated component spans the entire spatial wireless sensor network at a critical percolation spatial sensor density λcov−con equal to c λcov−con (r, ωs , α) = c

0.955 (1 − ωs ) α 3 r 3

where ωs is the overlap volume fraction between sensing spheres of the sensors with ϑ(α) ≤ ωs < 1 and ϑ(α) = (4 + α ) (2 − α)/16, 1 ≤ α < 2, and r is the radius of the sensing spheres of the sensors. ∎

4.3 Further Discussion

87

0.955 Figure 4.4 plots μc (ωs , α) = λcov−con (r ,ωs , α) r 3 = (1−ω 3 depending on the c s) α value of α (1 ≤ α < 2), which represents the ratio of the radius R of the communication spheres to the radius r of the sensing spheres of the sensors, and the value of the overlap volume fraction between the sensing spheres ωs , where ϑ(α) ≤ ωs < 1. We find that the percolation spatial density increases with the overlap volume fraction between the sensing spheres. More importantly, we notice that the percolation spatial density decreases when the ratio α = R/r increases. Indeed, when α increases, the radius of the communication spheres increases, and hence the percolation of the communication spheres of the sensors will almost surely occur for lower percolation spatial density. Furthermore, we find that the critical network degree, which is in this case corresponds to the average size of the collaborating sets of the sensors at percolation and given by

∫α r ηc = avg{|C ol(s i )|c : i ≥ 1} = 4 π λc

σ 2 g(σ )dσ 0

where g(σ ) = 1 − ωs , is equal to 4.

4.3 Further Discussion In this chapter, we consider the problem of wireless sensor networks that need to provide sensing coverage and network connectivity in a spatial field of interest. Further, rather than the placement (or deployment) problem of finding constraints on where sensors could be placed to provide the necessary coverage and connectivity, we address a more theoretical question: Given that the sensors are randomly distributed in a spatial field, what is the necessary minimum spatial sensor density that almost surely guarantees that the sensors will provide the desired coverage properties?

4.3.1 Practicality and Generalizability Issues Concerning the practicality and generalizability of our results, we make the following points. First, as mentioned earlier, there are some situations where the planar deployment assumption is not valid anymore. The design of wireless sensor networks in three dimensions is indeed more realistic than that in two dimensions. For instance, wireless sensor networks deployed underwater, require design in the space. Oceanographic data collection, disaster prevention, assisted navigation, offshore exploration, and pollution monitoring are typical applications of underwater wireless sensor networks. The success of these applications depends on how well the spatial volume under consideration is covered. In particular, every point in the spatial volume should

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be covered. Hence, it is important to investigate the situation when full coverage of a spatial volume would be achieved. The critical spatial density that we compute corresponds to the density above which coverage percolation in spatial wireless sensor networks will almost surely occur. In this case, the underlying spatial wireless sensor network would be able to detect any object that would penetrate the spatial volume. Second, the measures of critical percolation spatial density that we provide in this chapter for sensing coverage and network connectivity are more general than the ones corresponding to the definitions of collaboration and communication between the sensors. Indeed, for the sensing coverage, it is required that sensors collaborate. By Definition 2.3 given earlier in Chap. 2 (Sect. 2.2), this implies that the sensing spheres of these two sensors must be at least tangential. So, the degree of overlap between the sensing spheres tends towards 0 given that these spheres must be at least tangential. Therefore, if we set ws = 0, we obtain a critical spatial density for sensing coverage that depends only on the radius of the sensing spheres of the sensors. Also, by Definition 2.4 presented earlier in Chap. 2 (Sect. 2.2), if we set wt = 0.3125, we also obtain a critical spatial density for network connectivity that depends only on the radius of the communication spheres of the sensors.

4.3.2 Sensor Deployment in Spatial Fields In the case of wireless sensor networks deployed on the trees of different heights in a forest, sensors could be seen almost everywhere in the space. Also, Pompili et al. [328] proposed different deployment strategies for planar and spatial communication architectures for underwater acoustic sensor networks, where the sensors are anchored at the bottom of the ocean for the planar design and float at different depths of the ocean to cover the entire spatial region. Indeed, oceanographic data collection, pollution monitoring, offshore exploration, disaster prevention, and assisted navigation are all typical applications of underwater sensor networks [8, 9]. For wireless sensor networks deployed in buildings with multiple floors, sensors are placed on the ground and/or the wall, but the networks seldom contain sensors floating in the middle of the room. The first examples show that our proposed spatial model is valid and can be applied to choose the spatial sensor density in practical problems. The last example, however, shows the limited validity of our model due to the restriction imposed on the placement of sensors inside buildings or rooms.

4.3.3 Relaxations of Assumptions For practical applicability of our results, we provide the following relaxations of the assumptions that we make, namely the unit sphere model for sensing and communication ranges of the sensors (Assumption 2.2, Sect. 2.7, Chap. 2), and homogeneous sensor model (Assumption 2.7, Sect. 2.2, Chap. 2). Although these two assumptions

4.3 Further Discussion

89

are the basis for most of coverage and connectivity protocols for wireless sensor networks, they may not be valid in practice and hence need to be relaxed.

4.3.3.1

Relaxing the Unit Sphere Model

Zhao and Govindan [451] found that the communication range of MICA motes is asymmetric and environment-dependent. Also, Zhou et al. [456] found that the communication range of radios is highly probabilistic and irregular. In this section, we consider convex sensing and communication models, where the sensors have the same sensing and communication ranges. Corollaries 4.1, 4.2, and 4.3 correspond to Theorems 4.1, 4.2, and 4.3, respectively. Their proof is the same as that in Sects. 4.2.1, 4.2.2, and 4.2.3, respectively, using the notion of largest enclosed sphere. Let Rled and rled be the radii of the largest enclosed spheres of the sensing and communication ranges of the sensors, respectively. Corollary 4.1 (Critical Spatial Sensor Density for Giant Covered Component): The critical spatial sensor density λcov c (rled ,ωs ) above which a giant covered component that span a spatial deployment field will almost surely form is computed as λcov c (rled ,ωs ) =

0.119 3 (1 − ωs ) rled

where 0 < ωs < 1 is the overlap volume fraction of the largest enclosed spheres of the sensors’ convex sensing ranges. ∎ Corollary 4.2 (Critical Spatial Sensor Density for Giant Connected Component): The critical spatial sensor density λcon c (Rled ,ωt ) above which a giant connected component that span a spatial wireless sensor network will almost surely form is given by λcon c (Rled ,ωt ) =

0.955 3 (1 − ωt ) Rled

where 0.3125 ≤ ωt < 1 is the overlap volume fraction of the largest enclosed spheres of the convex communication ranges of the sensors. ∎ Corollary 4.3 (Critical Spatial Sensor Density for Giant Coordinated Component): Under the assumption that Rled = α rled , a giant coordinated component spans the entire spatial wireless sensor network at a critical percolation spatial sensor density equal to λcov−con (rled ,ωs , α) = c

0.955 3 (1 − ωs ) α 3 rled

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4 A Spatial Percolation-Theoretic Approach …

where ωs is the overlap volume fraction of the largest enclosed spheres of the homogeneous sensors’ convex sensing ranges of with ϑ(α) ≤ ωs < 1, ϑ(α) = ∎ (4 + α ) (2 − α)/16, and 1 ≤ α < 2.

4.3.3.2

Relaxing the Homogeneous Sensor Model

Real-world applications may require heterogeneous sensors in terms of their sensing and communication capabilities in order to enhance the reliability of the network and extend its lifetime [433]. Even sensors equipped with identical hardware may not always have the same sensing model. Now, we relax this assumption by considering heterogeneous sensors with different yet convex sensing and communication ranges. Corollaries 4.4, 4.5, and 4.6 correspond to Theorems 4.1, 4.2, and 4.3, respectively. Their proof is literally the same as that in Sects. 4.2.1, 4.2.2, and 4.2.3 using the notion of largest enclosed sphere. Our reasoning is based on the least powerful sensors in min min and rled be the terms of their sensing and communication capabilities. Let Rled minimum radii of the largest enclosed spheres of the sensing and communication ranges of the sensors, respectively. Corollary 4.4 (Critical Spatial Sensor Density for Giant Covered Component): The min critical spatial sensor density λcov c (rled ,ωs ) above which a giant covered component that span a spatial deployment field will almost surely form is computed as min λcov c (rled ,ωs ) =

0.119 min 3 (1 − ωs ) rled

where 0 < ωs < 1 is the overlap volume fraction between the smallest largest enclosed spheres of the convex sensing ranges of the sensors. ∎ Corollary 4.5 (Critical Spatial Sensor Density for Giant Connected Component): min The critical spatial sensor density λcon c (Rled ,ωt ) above which a giant connected component that span a spatial wireless sensor network will almost surely form is given by min λcon c (Rled ,ωt ) =

0.955 min 3 (1 − ωt ) Rled

where 0.3125 ≤ ωt < 1 is the overlap volume fraction between the smallest largest enclosed spheres of the convex communication ranges of the sensors. ∎ Corollary 4.6 (Critical Spatial Sensor Density for Giant Coordinated Component): min min = α rled , a giant coordinated component spans the Under the assumption that Rled entire spatial wireless sensor network at a critical percolation spatial density equal to

4.4 Related Work

91 min λcov−con (rled , ωs , α) = c

0.955 min 3 (1 − ωs ) α 3 rled

where ωs is the overlap volume fraction of the smallest largest enclosed spheres of the heterogeneous sensors’ convex sensing ranges with ϑ(α) ≤ ωs < 1, ϑ(α) = (4 + α ) (2 − α)/16, and 1 ≤ α < 2. ∎ Thus, the assumptions of the sphere model for the sensing and communication ranges of the sensors and the homogeneous sensor model can be relaxed with slight updates to the results in Theorems 4.1, 4.2, and 4.3 using the notion of largest enclosed sphere. We should mention that because of these relaxations, the critical spatial density may be a bit higher than necessary in some regions of a spatial deployment field.

4.4 Related Work In this section, we review related work on coverage and connectivity in spatial wireless sensor networks. We also summarize existing works that are based on percolation theory. The study of coverage and connectivity in spatial wireless sensor networks, such as underwater sensor networks [8], has gained relatively less attention in the literature compared to that of planar wireless sensor networks. Alam and Haas [12] investigated coverage and connectivity issues in spatial wireless networks. They proposed a placement strategy based on Voronoi tessellation of the space, which creates truncated octahedral cells. This strategy uses a minimum number of nodes to guarantee 100% coverage of the space and the minimum ratio of the communication range to the sensing range of such a placement strategy. Huang et al. [208] addressed the coverage problem in spatial wireless sensor networks by reducing the geometric problem from the space to the plane, and proposed a polynomial-time algorithm to solve the α-Ball-Coverage (α-BC) problem whose goal is to check α-coverage of a spatial region, i.e., all locations in the region are α-covered or not. Pompili et al. [328] proposed a deployment strategy for spatial communication architecture for underwater acoustic sensor networks, where sensors float at different depths of the ocean to cover the entire spatial region. Poduri et al. [324] discussed some difficulties encountered in the design of spatial wireless sensor networks, such as ensuring network connectivity in the case of uniform random deployment and restrictions imposed by the environment structure on sensor deployment. In particular, they discussed a few issues concerning the geometry of spatial wireless sensor networks and possible extensions of existing planar designs for deployment and configuration to spatial design. In [336], Ravelomanana investigated fundamental properties of randomly deployed spatial wireless sensor networks for connectivity and coverage, such as the required sensing range to guarantee certain degree of coverage of a region,

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the minimum and maximum network degrees for a given communication range, and the network hop-diameter.

4.5 Conclusion In this chapter, we consider spatial wireless sensor networks and investigate the critical percolation densities above which a giant covered component almost surely spans a spatial deployment field (coverage percolation problem) and a giant connected component almost surely span the entire network (connectivity percolation problem) [44]. In order to deal with the problem of integrated coverage and connectivity percolation, i.e., coverage percolation problem (COVP) and connectivity percolation problems (CONP) at the same time, we propose a two-concentric-sphere (TCS) model, where every sensor is modeled by two concentric spheres, namely sensing sphere and communication sphere. For each of these problems, we compute the corresponding critical spatial sensor density as well as the network degree. We find that the critical spatial density corresponding to the connectivity percolation is higher than its counterpart for coverage percolation. This difference is due to the inherent properties of collaboration and communication between the sensors. Indeed, collaboration between the sensors requires that their sensing spheres be at a distance at most equal to 2 r from each other, while communication between the sensors necessitates that their communication spheres be at a distance at most equal to R. For the integrated coverage and connectivity percolation (ICCP) problem, we consider a more general analysis, where R = α r with 1 ≤ α < 2 and α ≥ 2. We find that the percolation spatial density depends monotonically on α, which represents the ratio R/r and depends on the overlap volume fraction of the sensing spheres. We believe that these findings can be useful for the design of energy-efficient topology control protocols for spatial wireless sensor networks in terms of coverage and connectivity.

Part III

Convexity Theory-Based Connected k-Coverage in Wireless Sensor Networks

Chapter 5

A Planar Convexity Theory-Based Approach for Connected k-Coverage

The scientific man does not aim at an immediate result. He does not expect that his advanced ideas will be readily taken up. His work is like that of the planter – for the future. His duty is to lay the foundation for those who are to come, and point the way. Nikola Tesla (1856–1943)

Overview This chapter focuses on the problem of connected k-coverage in randomly and densely deployed planar wireless sensor networks. Several applications require k-coverage, where each point in a planar field is covered by at least k sensors simultaneously. This concept of k-coverage helps increase data availability to ensure better data reliability. However, it is also well-known that coverage alone in wireless sensor networks is not sufficient because data originated from source sensors are not guaranteed to reach a central gathering node (i.e., sink) for further analysis and processing. While coverage is a metric that measures the quality of surveillance provided by a network, connectivity provides a means for the source sensors to report their sensed data to the sink. This chapter investigates how to achieve k-coverage of a planar field with a minimum number of sensors while maintaining network connectivity. It considers static and homogeneous sensors, while adopting a deterministic sensing model. Because wireless sensor networks suffer scarce energy resources, where the sensors have limited battery power, a more practical deployment strategy requires that all the sensors be duty-cycled to save energy. With duty-cycling, sensors can be turned on or off according to some scheduling protocol, thus reducing the number of active sensors required for k-coverage and helping all the sensors deplete their energy as slowly and uniformly as possible. This chapter proposes centralized, pseudodistributed, and fully distributed sensor scheduling protocols, where the sensors are duty-cycled, while guaranteeing connected k-coverage configurations all the time.

5.1 Introduction Sensing coverage (or simply coverage) is an essential functionality of wireless sensor networks. It is one of the important research problems in the area of wireless sensor © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_5

95

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5 A Planar Convexity Theory-Based Approach for Connected k-Coverage

networking. Indeed, the effectiveness of any protocol designed to be used by the sensors is measured by its potential impact on increasing the network survivability. The latter depends on the quality of coverage provided by the network. It is one of the fundamental concepts in the design of wireless sensor networks in the sense that the monitoring quality of a phenomenon depends on the quality of service provided by the sensors in terms of how well a planar field of interest is covered. It enables the sensors to detect any event that may occur in the field, thus, meeting the applicationspecific requirements. Some real-world applications, such as intrusion detection, may require a high degree of coverage, and hence a large number of deployed sensors in order to enable accurate tracking of intruders. Indeed, the limited battery power of the sensors and the difficulty of replacing and/or recharging batteries on the sensors in hostile environments require that the sensors be deployed with high density in order to extend the network lifetime. To enhance network reliability, k-coverage (or redundant coverage) is an appealing solution, where each point in a planar deployment field is covered by at least k sensors at the same time, where k ≥ 1 is a natural number. As it can be seen, k-coverage helps improve sensed data availability, which would enable the sink to make the best decision using data redundancy provided that the sensors are connected to each other and to the sink. Hence, network connectivity should be guaranteed so the source sensors can be connected to the sink, thus allowing the underlying network to function properly. Thus, it is necessary that k-coverage and connectivity be studied together in a unified framework. This fundamental problem in wireless sensor networks is referred to as connected k-coverage with k ≥ 1. In wireless sensor networks, coverage and connectivity have been jointly addressed in an integrated framework. In addition, for such highly dense deployed and energyconstrained sensors, it is necessary to duty-cycle them to save energy. Thus, the design of k-coverage protocols for wireless sensor networks should minimize the number of active sensors to ensure the degree of coverage of a planar field, which is required by the sensing application, while maintaining connectivity between all active sensors. Hence, the first challenge is the determination of the number of sensors required to remain active to k-cover a planar field of interest. The second challenge is the design of an efficient scheduling protocol that decides which sensors to turn on (active) or off (inactive) for k-coverage of a planar field. Next, we briefly describe the major tasks we want to accomplish in this chapter. Moreover, we state how to achieve each one of them.

5.1.1 Major Tasks This chapter focuses on the study of the connected k-coverage problem because of the existence of several applications and systems, such as multiple-sensor data fusion [234], triangulation-based positioning systems [312], and Space and planet exploration [370], to name a few. Undoubtedly, there are several other applications, such as intruder detection and tracking as stated earlier, which require that each location in a planar field be sensed by at least one sensor. In these types of applications,

5.1 Introduction

97

the decision making process’ accuracy depends on the availability of the data gathered during monitoring. With connected k-coverage, the sink would be able to gather enough data for a successful decision regarding any event that would occur in the monitored planar field. As stated by Xing et al. in their seminal work [425], a k-covered wireless sensor network is guaranteed to be connected if R ≥ 2r, where r and R stand for the radii of the sensing and communication ranges of the sensors, respectively. In this chapter, we assume that the above relationship is satisfied although we show that we can obtain a less rigid one, where R < 2r. Hence, we focus more on the k-coverage problem. Also, we study duty-cycling to achieve both k-coverage and connectivity in highly dense deployed wireless sensor networks [39, 52], where each location in a planar deployment field is covered by at least k active sensors while maintaining connectivity between all active sensors, with k ≥ 3. Next, we specify these tasks and give our corresponding plan of actions. First, we want to compute the minimum planar sensor density to achieve kcoverage of a planar field. To this end, we conduct an in-depth analysis of the kcoverage problem in planar wireless sensor networks. Given the shape of the sensing range of the sensors, which is assumed to be a disk, we aim at computing a sufficient condition of the planar sensor density for complete k-coverage of a planar field. Indeed, an important problem in the design of such network configurations is computing the minimum active planar sensor density required to guarantee kcoverage of a planar field. For tractability of the problem, we analyze the intersection of k sensing disks so we can characterize k-coverage provided by a planar wireless sensor network Based on this analysis, we derive a tight sufficient condition of the planar density of active sensors to achieve complete k-coverage of a planar field. Second, we aim to find the minimum ratio of the communication range to the sensing range of the sensors to guarantee connectivity between all sensors while maintaining k-coverage of a planar field. We need to find the relationship between the sensing and communication ranges of the sensors based on the above characterization of k-coverage. Third, we want to design duty-cycling protocols for highly dense deployed planar wireless sensor networks in order to guarantee k-coverage of a planar field with a minimum number of active sensors that are constantly connected to each other in every round. For this purpose, we propose five connected k-coverage protocols for planar wireless sensor networks. The first protocol, called centralized randomized connected k-coverage (CERACCk ), is executed under the control of the sink, which is responsible for selecting the required number of sensors to guarantee k-coverage of a planar field while maintaining connectivity between all active selected sensors in every round. Both of the second and third protocols, called T-clustered randomized connected k-coverage (T-CRACCk ) and D-clustered randomized connected kcoverage (D-CRACCk ), respectively, are performed under the control of the sink and a subset of sensors, called cluster-heads, in every round. These cluster-heads are selected by the sink and are responsible for selecting a subset of neighboring sensors to k-cover their underlying cluster while remaining connected to each other. Each of the two protocols, T-CRACCk and D-CRACCk considers a different degree

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5 A Planar Convexity Theory-Based Approach for Connected k-Coverage

of network clustering. The fourth protocol, called distributed randomized connected k-coverage (DIRACCk ), is executed by all the sensors, which are required to coordinate among themselves to k-cover a planar field while being mutually connected in every round. Precisely, DIRACCk has two variants, namely Trig-DIRACCk , and SelfDIRACCk , While the former allows sensors to trigger others to become active, the latter enables sensors to become active based on the local information they possess about their sensing neighbors so as to k-cover their sensing range. Fourth, we want to assess the performance of all the five proposed connected kcoverage protocols in planar wireless sensor networks. We provide extensive simulations of the above-mentioned protocols, CERACCk , T-CRACCk , D-CRACCk , TrigDIRACCk , and Self-DIRACCk . We use several performance metrics to study their efficiency.

5.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 5.2 discusses the connected k-coverage problem in wireless sensor networks and shows how to solve it. Sections 5.3, 5.4, 5.5, and 5.6 present our connected k-coverage configuration protocols. Section 5.7 enhances the applicability of these protocols by relaxing some widely used assumptions in coverage protocols. Section 5.8 presents simulation results of our protocols and compares them with another existing one. Section 5.9 reviews related work. Section 5.10 concludes the chapter.

5.2 Achieving Connected k-Coverage Although the problem of planar k-coverage has been well-studied in the literature, only a few elegant approaches characterized k-coverage of a planar field [425, 447]. However, none of them guarantees the deployment of a minimum number of sensors to achieve k-coverage of a planar field and hence they would result in shorter operational network lifetime. Previous work [425, 447] only characterized k-coverage. According to [425], a planar field is k-covered if all intersection points between the boundaries of sensing ranges of the sensors and all those between the boundaries of sensing ranges of the sensors and the boundary of a planar field are k-covered. This is a generalization of the result for 1-coverage [189]. Thus, if two sensing ranges intersect, one more is needed to cover their intersection point. A point in a planar field that coincides with an intersection point would be 3-covered instead of 1-covered. Hence, more than enough sensors are required to k-cover a field. However, our approach for characterizing k-coverage of a planar field is different from the ones proposed in [425, 447]. Precisely, our approach is able to quantify the minimum planar density of active sensors to fully k-cover a planar field, thus computing the corresponding minimum number of sensors.

5.2 Achieving Connected k-Coverage

99

In this section, we propose our approach for obtaining connected k-coverage configurations in planar wireless sensor networks. First, we model the connected k-coverage problem in planar wireless sensor networks. Then, we derive a tight sufficient condition of the active planar sensor density such that a planar field is fully k-covered during the network lifetime while all active sensors are being connected to each other. In Sects. 5.3, 5.4, 5.5, and 5.6, we propose five protocols to solve the problem of connected k-coverage in planar wireless sensor networks based on the results of this section.

5.2.1 Connected k-Coverage Problem Modeling Solving the connected k-coverage problem in wireless sensor networks requires finding a sensor deployment strategy such that each location in a planar field is covered by at least k active sensors while all active sensors are connected. Our approach solution to the connected k-coverage problem in wireless sensor networks consists of decomposing it into two sub-problems, namely planar deployment field slicing and sensor selection, and solving them. The planar deployment field slicing problem is to slice a planar field into small regions of particular shape (to be defined later), each of which is guaranteed to be k-covered provided that at least k sensors are randomly deployed in it. The sensor selection problem is to select a minimum subset of sensors to remain active and connected such that each location in a planar field is guaranteed to be k-covered. Besides selecting a minimum number of active sensors, all selected sensors should have the maximum remaining energy. Because the problem of selecting a minimum subset of sensors to k-cover a planar field is NPhard [455], we propose efficient centralized, clustered, and distributed approximation algorithms to solve the connected k-coverage problem.

5.2.2 Sufficient Condition to Ensure k-Coverage We want to compute the minimum active planar sensor density required to k-cover a planar field. To do so, we should compute the maximum size of a convex area A of a planar field so that A is k-covered with exactly k sensors. Intuitively, the distance between any point in A and each of the k sensors should be at most equal to the radius of their sensing disks. Lemma 5.1 gives an upper bound on the width of such a convex area. We omit its proof since it is trivial. Lemma 5.1 (k-Covered Convex Area Width) Let r be the radius of the sensing disks of the sensors and k ≥ 3. A convex area A is guaranteed to be k-covered when exactly k homogeneous sensors are deployed in it, if the width of A does not exceed r. ∎ Now, we present Helly’s Theorem [85, p. 90], a fundamental result of convexity theory, which characterizes the intersection of convex sets.

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Helly’s Theorem [85] (Convexity Theory Helly’s Theorem): Let E be a family of convex sets in Rn such that for m ≥ n + 1 any m members of E have a non-empty intersection. Then, the intersection of all members of E is non-empty. ∎ Theorem 5.1, which is an instance of Helly’s Theorem [85], will help us compute the minimum planar sensor density required to k-cover a planar field. More specifically, Helly’s Theorem [85] together with a geometric structure, called Reuleaux triangle [481], will be used to characterize k-covered wireless sensor networks. Theorem 5.1 (Planar Instance of Helly’s Theorem) Let k ≥ 3. The intersection of k sensing disks is not empty if and only if the intersection of any three of those k sensing disks is not empty. ∎ Following Theorem 5.1, Lemma 5.2 states a sufficient condition for complete k-coverage of a planar field. Lemma 5.2 (Guaranteed Planar k-Coverage) Let r be the radius of the sensing disks of the sensors and k ≥ 3. A planar field is k-covered if any Reuleaux triangle region of width r in the field contains at least k active sensors. Proof First, we compute the maximum area that is k-covered with exactly k sensors. Let A be the intersection area of the sensing disks of k sensors. From Lemma 5.1, the width of A should be upper-bounded by r so that any point in A is k-covered by these k sensors. Let us first consider the case of three sensors. Using the Venn diagram given in Fig. 5.1, the maximum size of the intersection area of the sensing disks of the sensors s1 , s2 , and s3 , so that the distance between any pair of sensors is at most equal to r, is obtained when s1 , s2 , and s3 are symmetrically located from each other. This area, called Reuleaux triangle [481], is denoted by RT (r) and has a constant width equal to r as shown in Fig. 5.2. Given that the intersection area of k sensing disks is at most equal to that of three sensing disks such that the maximum distance between any pair of sensors is at most equal to r, the maximum size of A is equal to the area of RT (r), which we call slice (see Fig. 5.3). Thus, any point in A is k-covered with exactly k active sensors deployed in A. Since this applies to any RT (r) region (or slice) in a planar field, it is guaranteed that the field is k-covered. ∎ As discussed in the next sections, our sensor selection scheme exploits the overlap between adjacent slices in a planar field to select a minimum number of active sensors for full k-coverage of the field. Precisely, two adjacent slices intersect in a region shaped as a lens (also known as the fish bladder) so that the sides of their associated regular triangles fully coincide as shown in Fig. 5.4. Note that k sensors located in the lens of two adjacent slices, say C1 and C2 , k-cover the area associated with their union. Indeed, the distance between any of these k sensors located in the lens and any point in the area of the union of both C1 and C2 is at most equal to r. Lemma 5.3 states this result. Lemma 5.3 (Adjacent Slices k-Coverage) k active sensors located in the lens of two adjacent slices surely k-cover both slices. ∎

5.2 Achieving Connected k-Coverage

Fig. 5.1 Symmetric intersection of three sensing disks Fig. 5.2 Reuleaux triangle

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Fig. 5.3 Three lenses of a slice

Fig. 5.4 Adjacent slices

Theorem 5.2, which exploits the results of Lemma 5.3, refines the result of Lemma 5.2 by stating a tighter sufficient condition for complete k-coverage of a planar field. Theorem 5.2 (Guaranteed Planar k-Coverage—Tight Condition) Let k ≥ 3. A planar field is guaranteed to be k-covered if for any slice in the field, there is at least one adjacent slice such that their lens contains at least k active sensors. ∎ Theorem 5.3, which exploits the result of Theorem 5.2, computes the minimum planar sensor density required for complete k-coverage of a planar field. Theorem 5.3 (Active Planar Sensor Density) Let k ≥ 3. The active planar sensor density required to guarantee k-coverage of a planar field is given by λ(r, k) =

6k √ (4π − 3 3)r 2

where r is the radius of the sensing disks of the sensors.

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Proof It is easy to check that the area ||Area(r )|| of the union of two adjacent slices is computed as ||Area(r )|| = 2 A1 + 4 A2 = (4 π − 3

√

3) r 2 /6

√ where A1 = √3 r 2 /4 is the area of the central equilateral triangle of side r and A2 = (π/6 − 3/4) r 2 is the area of each of the three curved regions α (Fig. 5.1). By Theorem 5.2, k sensors should be deployed in the lens of two adjacent slices to k-cover both of them. Thus, the minimum planar sensor density that guarantees k-coverage of a planar field is equal to √ λ(r, k) = k/||Ar ea(r )|| = 6k/(4π − 3 3) r 2 It is worth noting that Adlakha and Srivastava [5] showed that the number of sensors required to cover an area of size A is in the order of O(A/ˆr 22 ), where rˆ 2 is a good estimate of the radius r of the sensing disks of the sensors. Precisely, r lies between rˆ 1 and rˆ 2 ; rˆ 1 overestimates the number of sensors required to cover A, while rˆ 2 underestimates it. One may suggest that the maximum area that is guaranteed to be k-covered with exactly k sensors is a circle of radius r/2. Fortunately, it is easy to check that our density λ(r, k) is smaller than the one corresponding to the configuration where k active sensors √ are deployed in a circle of radius r/2. In other words, λ(r, k) = 6 k/(4 π − 3 3) r 2 < 4 k/π r 2 .

5.3 Centralized k-Coverage Protocol In this section, we present our centralized randomized connected k-coverage (CERACCk ) protocol to fully k-cover a planar field while maintaining connectivity between active sensors. According to the connected k-coverage model mentioned in Sect. 5.2.1, the protocol has two main steps. First, we slice a planar field into regions whose shape helps characterize the k-coverage property of the field, and thus leads to compute the corresponding minimum number of active sensors. Then, we select an appropriate subset of sensors to guarantee k-coverage of each slice, and hence k-cover the entire planar field based on the geometric characteristics of those regions.

5.3.1 Planar Deployment Field Slicing This section provides a solution to the planar deployment field slicing problem, where all sensors have the same sensing and communication disks whose radii are r and

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Fig. 5.5 Slicing grid of a planar square field

R, respectively. The goal of this slicing phase is to decompose the sensing range of a sensor into smaller, congruent regions such that each of them is guaranteed to be k-covered when exactly k sensors are deployed in it. Let us consider a planar square field and k ≥ 3. Based on the result of Theorem 5.2, it is easy to check whether a given network can k-cover the field. For this purpose, we propose a slicing scheme of a planar field by dividing it into overlapping Reuleaux triangles of width r, called slices as shown in Fig. 5.3, such that two adjacent slices intersect in a region shaped as a lens (also known as the fish bladder) as shown in Fig. 5.4. This implies that the field is mainly sliced into regular triangles of side r. The result of this slicing operation is called slicing grid. Figure 5.5 shows a slicing grid of a planar field.

5.3.2 Sensor Selection In this section, we propose a centralized algorithm to select a minimum subset of active sensors that k-covers a planar field. The purpose of the selection phase is to specify which sensors turn on, how, and when. We assume that the sink is responsible for this selection process. Using Lemma 5.3, we start first by selecting sensors located in the three lenses of a given slice as shown in Fig. 5.3 (a slice overlaps with at most three other slices). At every selection, we check whether we already selected k sensors to k-cover the underlying slice. At the same time, we update the degree of coverage of the other adjacent slices. We repeat this process until we visit all slices in a planar field. We assume that each slice has a unique id, such as an integer.

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Theorem 5.4 is a consequence of Theorem 5.2. Theorem 5.4 (Minimum Planar k-Coverage) Let k ≥ 3. A planar field is guaranteed to be k-covered with a minimum number of sensors if all the sensors are selected from lenses of adjacent slices. ∎ The order in which the slices are treated is critical. It can be easily shown at the end of the sensor selection phase that if the slices are processed randomly, there is no guarantee that each slice is k-covered with a minimum number of sensors. Thus, the entire planar field is not guaranteed to be k-covered using a minimum total number of sensors. In order to avoid this problem, it is imperative that slices of a slicing grid be processed in a particular order. Assume that we initially picked slice Ci for processing and let Ci1 , Ci2 , and Ci3 be its adjacent slices. We use a FIFO (First-InFirst-Out) data structure, called NYP (Not Yet Processed), to keep track of the slices whose degrees of coverage have been updated but not yet processed. Hence, when we process slice Ci , we store the id’s i 1 , i 2 , and i 3 in NYP. When slice Ci has been processed, we consider slice Ci1 as the next one to be processed. After that, we pick Ci2 followed by Ci3 followed by the slices adjacent to Ci1 and so on. The sensor selection algorithm for k-coverage of a planar field is given in Fig. 5.6. The sensor selection algorithm described earlier generates only one subset of active sensors to k-cover a planar field. If this algorithm is executed in each round for the same slicing pattern of the field, sensors located in the lenses will be suffering from a severe battery power depletion problem. Hence, those sensors will die very quickly and possibly disconnect the network. Recall that the sensors have several limited resources with energy being the most critical one. Thus, it would be more efficient if in each round a different subset of sensors is selected for k-coverage of the planar field so all the sensors are given the same chance to be active. Our objective is to balance the load of k-coverage on all sensors so they deplete their energy uniformly. Next, we describe an approach to achieve this goal.

5.3.3 Slicing Grid Dynamics Our goal is to select different subsets of sensors Si , i ≥ 1 such that each subset Si is selected to remain active in the i th round to k-cover a planar field. Notice that to achieve a better load balancing among sensors, we could add a restriction so that selected subsets of sensors are mutually disjoint, i.e., Si ∩ S j = ∅, ∀ i /= j. However, the disjointness constraint yields a small number of mutually disjoint minimum subsets of sensors. Thus, we only require partially disjoint subsets of sensors. Given that our selection criterion is based on the remaining energy of sensors, it is guaranteed that sensors with low energy level would be avoided. Furthermore, it is rarely that the same sensors participate in several successive rounds to k-cover a planar field. The question that we want to address now is: How would minimum subsets of sensors be selected, each of which k-covers a planar field?

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Fig. 5.6 Sensor selection for k-coverage of a planar field

To address this question, we consider the dynamics of slicing grid from one round to another. Recall that the result of slicing a planar field into slices is called slicing grid. The selection of different minimum subsets of active sensors will be determined based on the obtained slicing grids. Since our scheme for selecting active sensors highly prioritizes the ones located in lenses of all slices, it is important that those lenses be able to scan the entire planar field, and hence include distinct subsets of sensors in different rounds. Thus, it is necessary that the sink be able to generate a slicing grid randomly at each round. Our objective is to obtain as (partially) disjoint minimum subsets of selected sensors as possible. For each obtained slicing grid,

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the sink applies the selection algorithm in Fig. 5.5. Thus, the slicing grid undergoes some dynamics to achieve balanced load of k-coverage among all sensors during the operation of CERACCk . The question that we want to address now is: How would a slicing grid of a planar field be randomly generated in each round? First, the sink randomly generates a point p1 in a planar field as shown in Fig. 5.7. Point p1 is temporarily considered as the center of the Euclidean plane. To randomly determine a second point p2 , the sink generates a random angle 0 ≤ θ ≤ 2π so that the segment [p1 , p2 ] forms an angle θ with the x-axis centered at p1 and the length of [p1 , p2 ] is r. Then, the sink deterministically finds a third point p3 to form the first regular triangle (p1 , p2 , p3 ), called reference triangle, as shown in Fig. 5.7. As its name indicates, all other regular triangles of the slicing grid are computed based on the reference triangle. Figure 5.7 shows a randomly generated slicing grid of a planar square field. Using the Reuleaux Triangle model described earlier and based on the structure of the clusters of slices, Theorem 5.5 states a condition under which k-coverage implies connectivity. Theorem 5.5 (Network Connectivity Condition) Let k ≥ 3. Under the assumption of a centralized or clustered k-coverage protocol, a k-covered wireless sensor network is guaranteed to be connected if the radius R of the communication disks of the sensors is at least equal to the radius r of their sensing disks, i.e., R ≥ r. Proof First, each cluster is connected if R ≥ r. Also, cluster-heads are connected to each other via active sensors. Thus, A k-covered wireless sensor network is guaranteed to be connected if R ≥ r. ∎

Fig. 5.7 Random slicing grid

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Theorem 5.6 states that CERACCk is a minimum-energy connected k-coverage configuration protocol that guarantees maximum network lifetime. Theorem 5.6 (CERACCk k-Coverage Protocol Performance) CERACCk fully kcovers a planar field with a minimum number of sensors in each round and maintains connectivity between them. Hence, it consumes minimum energy. Proof The sink guarantees that each slice of a planar field is covered by exactly k sensors. Therefore, by Theorem 5.2, each slice of the field is k-covered, and hence the entire field is fully covered. Moreover, sensors are selected from lenses so all adjacent slices are k-covered with a minimum number of active sensors. Thus, by Theorem 5.4, CERACCk guarantees that a planar field is k-covered using a minimum total number of active sensors in each round, and hence, it consumes a minimum amount of energy in each round. By Theorem 5.5, a k-covered wireless sensor network is connected, assuming that R ≥ r. Given that CERACCk prioritizes sensors with the highest remaining energy to remain active in each round, and given that it is based on the dynamics of slicing grid, necessarily all sensors deplete their energy slowly and uniformly, thus leading to a maximum network lifetime. ∎ In general, the sink is connected to an infinite source of energy, such as a wall outlet, and thus can be viewed as a line-powered node [433] that has no energy constraint. Hence, if node failure is due only to low battery power, the problem of single-point failure does not arise at all in this type of centralized wireless sensor network architecture. Also, under the centralized control of the sink, no coordination between sensors is required to select a minimum number of active sensors to k-cover a planar field. Given that sensors have limited energy, this approach would save them energy. Indeed, a schedule that determines which sensors should remain active in each round to k-cover a planar field is computed by the sink and forwarded to the selected ones. Thus, such a centralized approach is intended to gain insight into a lower bound on the number of sensors required for complete k-coverage of a planar field, and hence an upper bound on the network lifetime. In Sect. 5.7, we show how to relax the centralized approach to implement CERACCk in a fully distributed manner.

5.4 Clustered k-Coverage Protocol In this section, we propose a family of clustered k-coverage protocols, called clustered randomized connected k-coverage (CRACCk ), in which part of the duties of the sink in the centralized protocol CERACCk is delegated to a subset of sensors, called cluster-heads. These protocols differ by their degree of granularity of network clustering, and hence produce clusters of different shapes and sizes.

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5.4.1 Cluster-Head Selection and Attributed Roles As in the centralized protocol CERACCk , the family of protocols CRACCk allows the sink to randomly generate in each round a slicing grid, which consists of adjacent overlapping Reuleaux triangles. In contrast with CERACCk , the sink randomly designates for each cluster a particular sensor, called cluster-head, which is responsible for k-coverage of its assigned cluster during a given round. Precisely, each cluster-head is located within the cluster it is in charge of selecting some of its sensing neighbors to k-cover it. The sink advertises a packet, called ClusterHeadList, including all sensors’ ids that have been selected as cluster-heads. When a sensor receives ClusterHeadList, it checks whether its id is included. If so, it removes its id from the list and forwards the updated ClusterHeadList packet. Otherwise, it just forwards the original packet it has received. The CRACCk family of protocols requires that each cluster-head coordinates its activity with its adjacent cluster-heads to k-cover its cluster, and hence select a total minimum number of sensors to k-cover a planar field. To achieve this goal, sensors that would remain active in each round to k-cover a cluster should be selected from lenses (i.e., intersection areas of adjacent Reuleaux triangles). For a better balance, a cluster-head attempts to select k/3 sensors from each lens so any Reuleaux triangle in a cluster contains exactly k sensors as per Theorem 5.2. The size of a cluster and the number of its adjacent ones depend on the type of clustering being used. As it can be seen, the protocols of the CRACCk family are pseudo-distributed in the sense that the selection of active sensors for complete k-coverage of a planar field is not under the control of the sink. Next, we describe two protocols of the CRACCk family, namely T-CRACCk and D-CRACCk , for connected k-coverage in wireless sensor networks based on their degree of clustering.

5.4.2 The T-CRACCk Protocol In T-CRACCk (“T” for Reuleaux triangle), a cluster is a slice in the obtained slicing grid and a cluster-head is called slice-head. In each round, the sink is responsible for randomly generating a slicing grid of a planar field. Given that each slice has at most three adjacent slices (Fig. 5.3), the T-CRACCk protocol requires that each slicehead coordinates its activity with its adjacent slice-heads to select a minimum total number of sensors to k-cover a planar field. Figure 5.8 shows slice-head sh 0 sharing three lenses with slice-heads sh 1 , sh 2 , and sh 3 . For instance, sh 0 could k-cover its slice by selecting sensors located in its three lenses. Then, it communicates the numbers n 1 , n 2 , and n 3 of sensors selected from lenses Lens 1, Lens 2, and Lens 3, respectively, to its adjacent slice-heads sh 1 , sh 2 , and sh 3 , respectively. Slice-head sh 1 would need to select k − n 1 more sensors from its lenses to k-cover its slice. It would definitely coordinate with its adjacent slice-heads to k-cover its slice and so does each slice-head.

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Fig. 5.8 Adjacent cluster-heads

Theorem 5.7 states that T-CRACCk yields minimum-energy connected kcoverage. Theorem 5.7 (T-CRACCk k-Coverage Protocol Performance) The T-CRACCk protocol fully k-covers a planar field. It also is a minimum-energy connected k-coverage protocol. Proof Each slice-head ensures that each slice of a planar field is k-covered by exactly k sensors by coordinating with each of its three adjacent slice-heads. Also, the sink assigns a slice-head to each slice in a planar field. Thus, by Theorem 5.2, the entire field is fully k-covered. Moreover, active sensors are selected only from lenses. Therefore, by Theorem 5.4, T-CRACCk guarantees that a planar field is k-covered with a minimum number of active sensors, and hence consumes a minimum amount of energy in each round. By Theorem 5.5, a k-covered wireless sensor network is connected, assuming that R ≥ r. Given that T-CRACCk favors sensors with highest remaining energy in each round and benefits from slicing grid dynamics, all sensors are equally likely to be selected for k-coverage of a planar field in each round. Thus, all sensors deplete their energy slowly and uniformly, thus leading to a maximum network lifetime. ∎

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5.4.3 The D-CRACCk Protocol The D-CRACCk (“D” for Disk) protocol has a higher network clustering granularity compared to T-CRACCk . Precisely, each cluster consists of six adjacent slices forming a disk (Fig. 5.9). Notice that for ease of representation, Fig. 5.9 represents each cluster by six regular triangles instead of six Reuleaux triangles. Next, we describe all the phases of the D-CRACCk protocol in detail.

5.4.3.1

Planar Deployment Field Clustering

In addition to slicing a planar field, we assume that in each round, the sink is also responsible for forming clusters of slices from the randomly obtained slicing grid. Precisely, each cluster consists of at most six adjacent slices forming a disk. Because of the random generation of slicing grids and the geometry of the planar field, some clusters consist of an entire disk, and hence called interior clusters, while others are formed by a portion of a disk, and hence called boundary clusters. Figure 5.9 shows a clustered planar deployment field. Moreover, for each cluster, the sink selects a sensor, called cluster-head, which is located as near as possible to the center of its cluster. The random generation of slicing grid ensures that all sensors are equally likely to act as cluster-heads in each round. Each cluster is defined by one point, i.e., (x, y) coordinates, representing its center and at most six other points defining its slices (or slice portions for a non-complete cluster). These seven points define the slicing information of a cluster, which the sink would broadcast to its corresponding cluster-head. Next, we define interior and boundary lenses.

Fig. 5.9 Clustering for D-CRACCk

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Definition 5.1 (Interior Lens and Boundary Lens) An interior lens of a disk is a lens that is shared with no other adjacent disk and a boundary lens is shared by two adjacent disks. ∎ Notice that each cluster overlaps with at most six others as shown in Fig. 5.9. By Lemma 5.3 and Theorem 5.4, sensors located in the boundary lenses of a given cluster should be selected first in order to minimize the necessary total number of active sensors to achieve full k-coverage of a planar field. However, this requires certain coordination between cluster-heads.

5.4.3.2

Cluster-Heads Coordination and Sensor Selection

Each cluster-head is in charge of selecting some of its sensing neighbors to k-cover its cluster based on its slicing information. Precisely, each cluster-head exploits the overlap between the slices of its cluster as well as the overlap between its slices and those of its adjacent cluster-heads to select a minimum number of its sensing neighbors to k-cover its cluster. We assume that each sensor advertises its remaining energy to its sensing neighbors at the start of a round when it turns itself on. Each cluster-head sch i maintains a list E rem _List(sch i ) = {E rem (j):sj ∈ SN(sch i )} of remaining energy of its sensing neighbors, where E rem (j) is the remaining energy of sj . It uses this list to select the ones with high remaining energy to stay active by sending a SELECT message including the cluster-head’s id as well as the id’s of all selected sensors. This would avoid those ones with low remaining energy and help the sensors deplete their energy as slowly and uniformly as possible. We assume that at the beginning of each round, all the sensors are active. Those ones which are selected by their corresponding cluster-heads would remain active during the underlying round, while the others turn themselves off (or go to sleep). For the sensor selection, each cluster-head assigns priorities to sensors located in boundary lenses, interior lenses, and middle of slices in descending order. That is, sensors located in boundary lenses have high priority to be selected based on Theorems 5.2 and 5.4. Given that each cluster has at most six slices, each cluster-head manages at most six interior lenses and at most six boundary lenses. On the one hand, each cluster-head is responsible for selecting sensors from its interior lenses without any coordination with its adjacent clusterheads. On the other hand, each cluster-head coordinates with at most six adjacent cluster-heads to select sensors from its boundary lenses in order to k-cover its cluster with a minimum number of sensors. For instance, in the case of a disk, its clusterhead, say sch0 (Fig. 5.10), would advertize the subsets S 1 , S 2 , S 3 , S 4 , S 5 , S 6 of sensors selected from its six boundary lenses to its adjacent cluster-heads sch1 , sch2 , sch3 , sch4 , sch5 , sch6 , respectively. Theorem 5.8 states that D-CRACCk yields minimum-energy connected kcoverage. Theorem 5.8 (D-CRACCk k-Coverage Protocol Performance) The D-CRACCk protocol fully k-covers a planar field. It also is a minimum-energy connected k-coverage protocol.

5.5 Triggered-Scheduling Driven k-Coverage

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Fig. 5.10 Clustering for D-CRACCk

Proof In each round, each disk-head uses exactly k sensors to k-cover each of its six slices by coordinating with its six adjacent disk-heads. Also, the sink guarantees that each disk is under the responsibility of a disk-head. Thus, by Theorem 5.2, the whole planar field is k-covered. Moreover, each disk-head k-covers its disk with sensors selected only from interior and boundary lenses. Therefore, by Theorem 5.4, D-CRACCk uses a minimum total number of active sensors in each round such that a planar field is guaranteed to be k-covered. Thus, D-CRACCk consumes a minimum amount of energy in each round. By Theorem 5.5, a k-covered wireless sensor network is connected, assuming that R ≥ r. Moreover, D-CRACCk selects sensors with maximum remaining energy, thus helping all sensors deplete their energy slowly and ∎ uniformly. Hence, D-CRACCk guarantees maximum network lifetime. Notice that T-CRACCk requires more coordination between cluster-heads than D-CRACCk and hence has more overhead to k-cover a planar field. This is due to the difference of their cluster sizes. Thus, D-CRACCk is more energy-efficient than T-CRACCk .

5.5 Triggered-Scheduling Driven k-Coverage In this section, we propose a fully distributed k-coverage protocol, called distributed randomized connected k-coverage (Trig-DIRACCk ). The centralized protocol CERACCk presented in Sect. 5.3 does not rely heavily on global information. Thus, it can be redesigned in a fully distributed fashion based on the local information sensors have about their one-hop neighbors with regard to their physical locations and remaining energy. Also, Trig-DIRACCk design requires coordination among sensors

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to achieve k-coverage of a planar field. Indeed, Trig-DIRACCk is a scheduling scheme in which the sensors can be triggered by other sensors to become active. Clearly, this scheduling scheme assumes a high cooperation between sensors in the sense that sensors satisfy the requests of their sensing neighbors when they are solicited to turn on. Next, we describe Trig-DIRACCk . First, we describe our algorithm that enables a sensor to check whether it should turn active and/or select some of its sensing neighbors to turn active in order to k-cover its sensing range. Then, we present the state diagram of Trig-DIRACCk .

5.5.1 K-Coverage Checking Algorithm and Sensor Selection A sensor runs a k-coverage checking algorithm, which is given in Fig. 5.12, in order to find out whether its sensing disk is k-covered. To do so, each sensor slices it sensing disk into six overlapping slices as shown in Fig. 5.11 such that two adjacent slices intersect in a lens. Thus, the slicing grid in Trig-DIRACCk consists of exactly six complete slices. Similarly, the k-coverage checking algorithm exploits the overlap between adjacent slices as in the case of centralized protocol CERACCk . Each sensor guarantees that each of the six Reuleaux triangles forming its sensing range (Fig. 5.11) is k-covered. To do so, it randomly decomposes its sensing disk into six Reuleaux triangles and checks whether each one of them is k-covered based on Theorem 5.4. A sensor starts first by selecting the sensors located in the three lenses (Fig. 5.3) to k-cover a given Reuleaux triangle. The selection algorithm exploits the overlap between Reuleaux triangles to select a minimum number of sensors to remain active. It selects its sensing neighbors to become active based on their remaining energy and locations in its sensing disk. Then, it activates them by sending AWAKE messages. When a sensor receives an AWAKE message, it becomes active

Fig. 5.11 Slicing grids of the sensing disk of a sensor

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ALGORITHM 2: k-COVERAGE-SELECTION(r, k) (* This code is run by each sensor *) Begin /* Sensing disk slicing */ 1. Randomly decompose a sensing disk into six overlapping Reuleaux triangles RTi(r), i=1..6 /* Sensors selection */ 2. n_k_covered_RT = 0 3. For each Reuleaux triangle RTi(r) Do 4. If RTi(r) contain k active sensors Then 5. n_k_covered_RT = n_k_covered_RT + 1 6. Else 7. Select sensors with high remaining energy and in the lenses such that RTi(r) is k-covered 8. Activate the selected sensors. 9. End 10. End 11. If n_k_covered_RT = 6 Then /* k-covered disk */ Return(“non-candidate”) 12. End 13. Return(“candidate”) End Fig. 5.12 k-Coverage checking algorithm of Trig-DIRACCk

and broadcasts a NOTIFICATION message to inform all its neighbors that it has become active. Every sensor keeps track of its active neighbors. Moreover, a sensor will turn itself active if one of the Reuleaux triangles is not k-covered. Using Lemma 5.3, each sensor checks whether each of the six slices forming its sensing disk is k-covered. For each slice, a sensor checks whether the number of active sensors in the three lenses (Fig. 5.3) including itself is equal to k. Otherwise, it checks whether the number of active sensors located the entire slice, i.e., three lenses and middle of the slice (Fig. 5.3), is equal to k. For each slice, a sensor computes its degree of coverage and save the result in an array variable Cov DegSlices. Based on the content of Cov DegSlices, a sensor activates a necessary number of its sensing neighbors to k-cover its sensing disk. If a sensor is unable to k-cover its sensing area when it runs the k-coverage checking algorithm, this means that the minimum planar sensor density required for k-coverage is not satisfied, and hence the planar field cannot be fully k-covered.

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5.5.2 State Transition Diagram of Trig-DIRACCk At any time, a sensor can be in one of the three states, namely Ready, Wwaiting, or Running. A state transition diagram associated with Trig-DIRACCk and indicating the three possible states of a sensor and transitions between them is shown in Fig. 5.13. • Ready: A sensor is listening to AWAKE messages and thus is ready to switch to the Running state. • Waiting: A sensor is neither communicating with other sensors nor sensing a planar field, and thus its radio is turned off. However, after some fixed time interval, it switches to the Ready state to receive AWAKE messages if its neighbors decide to do so for achieving k-coverage. • Running: A sensor can communicate with other sensors and sense the environment. • At the start of the monitoring task, all sensors are in the Ready state except one that is in the Running state. The single sensor in the Running state is one of the communication neighbors of the sink that is chosen randomly to activate some of its sensing neighbors to achieve k-coverage of its sensing disk. Those selected sensors will in turn run their selection algorithm to k-cover their sensing disks. This chain of sensor activations continues until the entire planar field is k-covered. As mentioned earlier, when a given sensor is selected by any other sensor, it will send out a NOTIFICATION message to inform all its neighbors. • While in the Ready state, a sensor keeps track of its sensing neighbors that are in the Running state. If it finds out that its sensing area is k-covered, it will switch to the Waiting state. It is not cost-effective to guarantee that a sensor

Fig. 5.13 State diagram of Trig-DIRACCk

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is not selected more than once during one round. Indeed, guaranteeing disjoint subsets of selected sensors requires much coordination between the sensors, thus introducing unnecessary overhead. • For energy savings, a sensor may wish to switch from the Running state to the Waiting state. For this purpose, a sensor broadcasts a RUNNING-to-WAITING message and waits for some transit time t transit . If t transit expires and it has not received any Running-to-Waiting message, it switches to the Waiting state and sends a NOTIFICATION-WAITING message, where it can stay there for t wait time. When t wait expires, it switches to the Ready state, where it can stay in this state for t ready time. • If a sensor finds out that its sensing area is not k-covered, it will broadcast a READY-to-RUNNING message and wait for some transit time t transit before switching to the Running state. If after t transit it has not received any other READY-to-RUNNING message, it will switch to the Running state and sends a NOTIFICATION-RUNNING message; otherwise, it will switch to the Waiting state.

5.5.3 Ensuring Network Connectivity Theorem 5.9 states the necessary relationship between the sensing and communication ranges of the sensors so that k-coverage implies network connectivity. Theorem 5.9 (Network Connectivity Condition for Trig-DIRACCk ): Let k ≥ 3. Under the assumption of Trig-DIRACC k protocol, a k-covered wireless sensor √ network is connected if R ≥ 3 r, where r and R are the radii of the sensing and communication ranges of the sensors, respectively. Proof Consider either configuration in Fig. 5.14. To compute the minimum communication range for the network to be connected, assume all sensors are located at one of the extreme points of a lens. With a little algebra, it is easy to check that the distance between two extreme points, say p and q, of two adjacent slices is equal to √ δ ( p, q) = 3 r. Thus, connectivity of k-covered wireless sensor networks requires that R ≥ r. ∎ Our result in Theorem 5.9 improves on the one given in [425], which requires R ≥ 2r for connectivity in k-covered wireless sensor networks. Theorem 5.10 states that Trig-DIRACCk is a minimum-energy distributed connected k-coverage protocol. Theorem 5.10 (Trig-DIRACCk k-Coverage Protocol Performance) The TrigDIRACCk protocol fully k-covers a planar field. It also is as minimum-energy protocol as D-CRACCk . Proof We proceed by contradiction. Assume that the total area A of a planar field is not fully k-covered by active sensors. Hence, A can be decomposed into a k-covered

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Fig. 5.14 Maximum distance between sensors in two lenses

area Ac and a non-k-covered area Anc , i.e., A = Ac ∪ Anc . Thus, there is at least one sensor si whose sensing disk intersects the area Anc , i.e., S D(ξi , ri ) ∩ Anc /= ∅. In particular, the sensing disk of si is not fully k-covered. Hence, sensor si is not active. Precisely, si is in the Waiting state. According to Trig-DIRACCk , however, sensor si must be in the Running state (i.e., active) given that its sensing disk is not fully k-covered. This contradicts our assumption. Thus, the total area A of a planar field is fully k-covered by active sensors. Let us now show that Trig-DIRACCk uses as minimum number of active sensors as D-CRACCk to k-cover a planar field. Using Trig-DIRACCk , each sensor checks whether its sensing area is k-covered. In order to k-cover each of its six slices, each sensor makes sure that there are exactly k active within its sensing disk. Each sensor favors active sensors that are located in the interior and boundary lenses of its sensing disk. Given that there is no preslicing of the entire planar field into adjacent slices as is the case with the three other protocols, there is no guarantee that all active sensors belong to the lenses of each sensor. Therefore, it may happen that active sensors located in some lenses of the sensing disk of a sensor are located in the lenses and/or the middle of slices of other sensors. Definitely, these sensors will consider those active sensors located in the middle of their slices so they k-cover their sensing disks with as minimum number of active sensors as possible. By Theorems 5.2 and 5.4, it follows that TrigDIRACCk could use a few more sensors than D-CRACCk for full k-coverage of a planar field. Hence, Thus, Trig-DIRACCk consumes as minimum amount of energy as D-CRACCk during the operational network lifetime. Similarly, Trig-DIRACCk

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selects sensors with highest remaining energy to remain active, thus helping all sensors deplete their energy as slowly and uniformly as possible, thus extending the network√lifetime. By Theorem 5.9, a k-covered wireless sensor network is connected ∎ if R ≥ 3 r .

5.6 Self-scheduling Based k-Coverage In Sect. 5.5, we propose a sensor scheduling protocol, called Trig-DIRACCk , which allows a sensor to trigger a necessary number of its sensing neighbors to become active in order to achieve k-coverage of its sensing range. This protocol can be classified as a triggered-scheduling driven k-coverage. In this section, we suggest another connected k-coverage protocol using a different scheduling approach. In this protocol, each sensor turns itself on based on the local information it has about its sensing neighbors in order to k-cover its sensing range. This protocol can be classified as a self-scheduling driven k-coverage and is denoted by Self-DIRACCk , In Sect. 5.8, we evaluate the performance of Trig-DIRACCk and Self-DIRACCk and compare between them. First, we present our k-coverage candidacy algorithm, which is shown in Fig. 5.15 and enables a sensor to check its candidacy to become active. Then, we present a state transition diagram describing the possible states a sensor can be in and transitions between those states. Fig. 5.15 k-Coverage candidacy algorithm of self-DIRACCk

ALGORITHM 1: k-COVERAGE-CANDIDACY(r, k) (* This code is run by each sensor *) Begin /* Sensing disk decomposition */ 1. Randomly slice a sensing disk into six Reuleaux triangles, RTi(r) 1≤ i ≤ 6 /* Localized k-coverage candidacy checking */ 2. For each Reuleaux triangle RTi(r) Do 3. If RTi(r) contains k active sensors Then 4. Skip /* i.e., do nothing */ 5. Else 6. Return (“candidate”) 7. End 8. End 9. Return (“non-candidate”) End

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5.6.1 K-Coverage Candidacy Algorithm Using a self-scheduling scheme, each sensor decides by itself whether to turn on or off . This decision is based solely on the local information it has about the status (on vs. off ) of its sensing neighbors, i.e., sensors located in its sensing range. A sensor turns active if its sensing disk is not k-covered. Based on Theorem 5.2, a sensor randomly decomposes its sensing disk into six overlapping Reuleaux triangles of width r (or slices) and checks whether each one of them contains at least k sensors. Each sensor should know the status of its sensing neighbors only to decide whether it is candidate to turn active. If any of the six slices does not have k active sensors, a sensor is a candidate to become active. Otherwise, it is not.

5.6.2 State Transition Diagram of Self-DIRACCk Figure 5.16 shows a state transition diagram for Self-DIRACCk . At any time, a sensor can be in one of Ready, Waiting, and Running states: • Ready: A sensor listens to messages and checks its k-coverage candidacy to switch to the Running state. • Running: A sensor is active and can communicate with other sensors and sense a planar field. • Waiting: A sensor is neither communicating with other sensors nor sensing a planar field, and thus its radio is turned off . However, after some fixed time

Fig. 5.16 State transition diagram of self-DIRACCk

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•

•

•

•

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interval, it switches to the Ready state to check its candidacy for k-coverage (Fig. 5.15) and receive messages. At the beginning of the monitoring task, all sensors are in the Running state. Moreover, each sensor chooses randomly and independently of all other sensors a value t check between 0 and t check-max after which it runs the k-coverage candidacy algorithm (Fig. 5.15) to check whether it stays active or switches to the Waiting state. Our intuition behind this random selection of t check is to avoid higher or lower coverage of any region in a planar field. When a sensor runs the k-coverage candidacy algorithm and finds out that it is a candidate, it sends out a NOTIFICATION-RUNNING message to inform all its neighbors. While in the Ready state, a sensor keeps track of its sensing neighbors that are in the Running state. If it finds out that its sensing area is k-covered, it will switch to the Waiting state. To save energy, a sensor may wish to switch from Running state to Waiting state. It broadcasts a RUNNING-to-WAITING message and waits for t transit transit time. If t transit expires and it has not received any RUNNING-to-WAITING message, it switches to the Waiting state to stay t wait time, and sends out a NOTIFICATIONWAITING message. When t wait expires, it switches to the Ready state, where it can stay t ready time. When a sensor in the Ready state receives a RUNNING-toWAITING message from its sensing neighbor, it runs the k-coverage candidacy algorithm to check whether it should be active. If a sensor in the Ready state finds out that it has not been active for some t inactive time, it will broadcast a READY-to-RUNNING message and wait for some t transit time. If t transit expires without receiving any READY-to-RUNNING message, it will send out a NOTIFICATION-RUNNING message and switch to the Running state. Else, it stays in the Ready state. A sensor in the Ready state would also apply the same process if it finds out that it has not heard from one of its sensing neighbors within some t active time, i.e., this neighbor depleted its entire energy. Thus, each sensor in the Running state should broadcast an ALIVE message after each t active time. We assume that each sensor remains active for at least t active time.

5.6.3 Tri-DIRACCk Versus Self-DIRACCk As it can be seen, Tri-DIRACCk benefits well from the result of Theorem 5.4 in the sensor selection process. Indeed, each sensor prioritizes sensors located in the lenses of its sensing disk, and hence the latter is guaranteed to be k-covered with a minimum number of sensors. However, Self-DIRACCk does not exploit Theorem 5.4 since each sensor only checks whether each of the six Reuleaux triangles forming its sensing disk is k-covered. Thus, we can conclude that Self-DIRACCk would require more active sensors than Trig-DIRACCk to fully k-cover a planar field. On the other hand, Tri-DIRACCk requires total coordination between sensors in the sense that when a sensor receives a message from another sensor to become active, it should

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satisfy its request and turn itself on. Therefore, Trig-DIRACCk introduces more control overhead than Self-DIRACCk . We anticipate that in general Trig-DIRACCk outperforms Self-DIRACCk with respect to the total energy consumption to achieve complete k-coverage of a planar field, and hence yields higher network lifetime.

5.7 Relaxation of Assumptions The design of our connected k-coverage protocols are based on the unit disk model and homogeneous sensor model. Although these assumptions are the basis for most of coverage and connectivity protocols in wireless sensor networks, they may not be valid in real-world wireless sensor network platforms. In this section, we relax these assumptions to promote the use of our protocols in real-world applications.

5.7.1 Relaxing the Unit Disk Model Cao et al. [101] found that the sensing range of the sensors is non-isotropic. Also, it was found that the communication range of MICA motes is asymmetric and depends on the environments [451] and that the communication range of radios is highly probabilistic and irregular [456]. For problem tractability, we consider a convex model, where the communication and sensing ranges of sensors are homogeneous and convex but not necessarily circular. Lemmas 5.4, 5.5 correspond to Theorems 5.2, 5.3, respectively. Their proof is literally the same as that in Sect. 5.2.2 by exploiting the notion of the largest enclosed disk. Let rled be the radius of the largest enclosed disk of the sensing range of the sensors. Lemma 5.4 (Guaranteed k-Coverage Using Convex Sensing Model) Let k ≥ 3. A planar field is guaranteed to be k-covered if for any slice of width rled in the field, there is at least one adjacent slice of width rled such that their lens contains at least k active sensors. ∎ Lemma 5.5 (Minimum k-Coverage Using Convex Sensing Model) Let k ≥ 3. The minimum planar sensor density required to fully k-cover a planar field by homogeneous sensors with convex sensing ranges but not necessarily circular is equal to λ ( rled , k) =

6k √ 2 (4 π − 3 3) rled

∎

Now, we discuss how CERACCk , T-CRACCk , D-CRACCk , Trig-DIRACCk , and Self-DIRACCk can be implemented using the convex sensing model. The unit of

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slicing, i.e., Reuleaux triangle, has a width equal to rled . Any other processing remains the same for each of those five protocols. Hence, the assumption of the unit disk model can be easily relaxed with the help of the notion of the largest enclosed disk of the sensing range of the sensors.

5.7.2 Relaxing the Sensor Homogeneity Model Real-world sensing applications [204] may require heterogeneous sensors in terms of their sensing and communication capabilities to enhance the reliability of the network and extend its lifetime [433]. Even sensors equipped with identical hardware may not always have the same sensing model. In this section, we consider heterogeneous sensors with different yet convex sensing and communication ranges. Lemmas 5.6 and 5.7 correspond to Theorems 5.2 and 5.3, respectively. Lemma 5.6 (Guaranteed k-Coverage Using Sensor Heterogeneity) Let k ≥ 3. A min in the field, planar field is guaranteed to be k-covered if for any slice of width rled min there is at least one adjacent slice of width rled such that their lens contains at least k active sensors. ∎ Lemma 5.7 (Planar Sensor Density for k-Coverage Using Sensor Heterogeneity) Let k ≥ 3. The planar sensor density that is necessary for k-coverage of a planar field by heterogeneous sensors with convex sensing ranges is given by min λ (rled , k) =

6k √ min 2 (4 π − 3 3) rled

min where rled is the minimum radius of the largest enclosed disks of the sensing ranges of the sensors. ∎

In the case of CERACCk , T-CRACCk , and D-CRACCk , the sink has to slice a min and apply the same processing as in Sect. 5.2.2. planar field into slices of width rled For Trig-DIRACCk and Self-DIRACCk , each sensor needs to consider its largest enclosed disk and run the same steps as in Sect. 5.2.2. Thus, the assumption of homogeneous sensors can also be relaxed with slight updates to our protocols. min Both protocols should consider rled instead of r when the sensors decompose their sensing ranges to k-cover them. Notice that if a single sensor with a very small sensing range is deployed, the network would have a large number of active sensors. Thus, it is necessary that Trig-DIRACCk and Self-DIRACCk adapt the planar sensor density to the sensing ranges of sensors in the area. This is a challenging problem that should be addressed adequately in order to better exploit sensor heterogeneity, where these connected k-coverage protocols need to be more adaptive.

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5.8 Performance Evaluation In this section, we present the simulation results of our protocols using a high-level simulator written in the C programming language. First, we specify the simulation environment as well as the energy consumption model that we use. Then, we present the simulation results with respect to several parameters.

5.8.1 Simulation Settings We consider a planar square field of side length 1000 m. We use the energy model given in [435], where the sensor energy consumption in transmission, reception, idle, and sleep modes are 60 mW, 12 mW, 12 mW, and 0.03 mW, respectively. Following [447], the energy required for a sensor to stay idle for 1 s is equivalent to one unit of energy. We assume that the initial energy of each sensor is 60 J enabling a sensor to operate about 5000 s in reception/idle modes [435]. All simulations are repeated 200 times and the results are averaged.

5.8.2 Simulation Results In this section, we present the simulation results of our protocols. Figure 5.17 shows the planar sensor density versus the coverage degree k, where the radius r of the sensing range of sensors is fixed to r = 25 m. The planar sensor density increases with k for a fixed r, as expected. As it can be seen, the five protocols yield a planar sensor density closer to the one given in Theorem 5.3. Also, CERACCk outperforms all other protocols while Trig-DIRACCk uses more sensors than these protocols due to its distributed nature. Figure 5.18 plots the planar sensor density versus r with k = 3. We observe that the planar sensor density decreases with r for a fixed k, Likewise, the five protocols require a planar sensor density closer to the one computed in Theorem 5.3. Figures 5.19 and 5.20 show the number of active sensors versus the total number of deployed sensors for Trig-DIRACCk . In Fig. 5.19, we consider different values of k, while in Fig. 5.20, we consider different values of r. For higher values of k, more sensors need to be active to achieve the required coverage. However, for higher values of r, a smaller number of sensors is needed to k-cover a planar field. In both experiments, the number of active sensors does not depend on the number of deployed sensors but only on k and r. Figure 5.21 shows the degree k of coverage versus the total number na of active sensors for Trig-DIRACCk . Notice that k increases with na . Also, for the same na , k increases quickly as r increases as a larger region of the planar field would be covered.

5.8 Performance Evaluation

Fig. 5.17 λ(r, k) versus k

Fig. 5.18 λ(r, k) versus r

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Fig. 5.19 Number of active sensors versus number of deployed sensors (k variable)

Fig. 5.20 Number of active sensors versus number of deployed sensors (r variable)

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Fig. 5.21 k versus number na of active sensors

Figure 5.22 shows that the total remaining energy of the sensors in the four protocols decreases smoothly (in this experiment, the number of deployed sensors is 16,000). Notice that the centralized protocol CERACCk consumes less energy

Fig. 5.22 Remaining energy versus time

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Fig. 5.23 Trig-DIRACCk versus self-DIRACCk (planar sensor density vs. coverage degree)

than all other protocols while the distributed protocol for Trig-DIRACCk consumes the highest amount of energy. Thus, CERACCk yields longer network lifetime than Trig-DIRACCk . This shows the advantage of our centralized protocol (CERACCk ) over the distributed one (Trig-DIRACCk ). Indeed, the number of messages needed by CERACCk to distribute the optimal schedule to the selected sensors may be less than that required by Trig-DIRACCk due to the periodic messages exchanged by sensors to coordinate their mission for k-coverage of a planar field. As discussed earlier in Sect. 5.6.3, all the plots in Figs. 5.23, 5.24, and 5.25 match our expectation. They show that Trig-DIRACCk is slightly more energy-efficient than Self-DIRACCk in terms of the necessary total number of active sensors to fully k-cover a planar field (Figs. 5.23 and 5.24) as well as the network lifetime (Fig. 5.25).

5.8.3 Comparison of Self-DIRACCk with CCP In this section, for fair comparison with the CCP protocol [425], we compare SelfDIRACCk . Indeed, both of CCP and Self-DIRACCk share the same idea in the sense that the sensors decide whether they turn active based on the result of their coverage eligibility algorithm (for CCP) and k-coverage candidacy algorithm (for Self-DIRACCk ). The CCP protocol provides different degrees of full coverage of a convex region. CCP was the first protocol that discussed k-coverage and connectivity within a unified framework. It was proved that coverage implies connectivity when R ≥ 2r [425]. Hence, no other mechanism would be necessary to guarantee connectivity. However, CCP was integrated with a topology maintenance protocol, called

5.8 Performance Evaluation

Fig. 5.24 Trig-DIRACCk versus self-DIRACCk (planar sensor density vs. sensing range)

Fig. 5.25 Trig-DIRACCk versus self-DIRACCk (total remaining energy vs. time)

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SPAN [112, 113], to provide both coverage and connectivity guarantees when R < 2r. Recall that a convex region A is k-covered in CCP if all intersection points between sensing disks of sensors and between sensing disks of sensors and A’s boundary are at least k-covered. Figure 5.26 plots the degree k of coverage versus the number n a of active sensors for Self-DIRACCk as compared to CCP. It shows that Self-DIRACCk requires less active sensors than CCP to achieve the same degree of coverage, thus yielding significant energy savings. This is due not only to a higher number of active sensors required by CCP, and hence an additional energy spent in sensing, but also to the communication overhead caused by the exchange of messages between active sensors running CCP to coordinate among themselves and provide the requested k-coverage service. Thus, CCP consumes more energy than Self-DIRACCk as shown in Fig. 5.27. Note that while CCP requires SPAN to provide connectivity between active sensors when R < 2 r, Self-DIRACCk does not need such a topology maintenance protocol as all it requires is R ≥ r, thus providing connectivity when k-coverage is guaranteed. Indeed, Self-DIRACCk is based on the analysis of sensors’ sensing range to provide k-coverage. Figure 5.28 plots n a versus R with r = 30 m, while Fig. 5.29 plots n a versus r for different ratios α = R/r, where α ≥ 1. In both cases, we fix k = 3. Given that α ≥ 1, any increase in the communication range of sensors would not have any impact on the performance of Self-DIRACCk . It would, however, affect the performance of CCP. As it can be observed, n a decreases as R increases. Indeed, SPAN would require a smaller number of sensors to maintain connectivity between active sensors as R increases. However, at some point, (surprisingly enough, this point corresponds to

Fig. 5.26 Self-DIRACCk compared to CCP (k vs. na )

5.8 Performance Evaluation

Fig. 5.27 Self-DIRACCk compared to CCP (remaining energy vs. time)

Fig. 5.28 Self-DIRACCk compared to CCP (n a vs. R)

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Fig. 5.29 Self-DIRACCk compared to CCP (n a vs. r)

R ≥ 2 r ), the number n a of active sensors required for k-coverage does not decrease any further. Indeed, when R ≥ 2 r, SPAN is not needed at all as both k-coverage and R ≥ 2 r would guarantee connectivity. Similarly, the performance of CCP improves as the ratio α increases (Fig. 5.29), i.e., R increases. That is, a smaller number of active sensors is needed to provide k-coverage and connectivity.

5.9 Related Work A variety of configuration protocols for coverage and connectivity in wireless sensor networks have been proposed in the literature with a goal to extend the network lifetime. In this section, we review a sample of these configuration protocols. Adlakha and Srivastava [5] addressed the issue of determining the required number of sensors to achieve full coverage of a desired region. Precisely, they proposed an exposure-based model to find the planar sensor density based on the physical characteristics of the sensors and the properties of the target. Kumar et al. [243] showed that the minimum number of sensors needed to achieve k-coverage with high probability is approximately the same regardless of whether the sensors are deployed deterministically or randomly, if the sensors fail or sleep independently with equal probability. Shakkottai et al. [348, 349] gave necessary and sufficient conditions for 1-covered, 1-connected wireless sensor grid network. Also, they proposed a variety of algorithms to maintain connectivity and coverage in large wireless sensor networks. Sohrabi et al. [357, 358] proposed two self-assembly mechanisms for wireless sensor

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networks based on clustering, dual network clustering and Rendezvous clustering algorithm, at the link layer for efficient interconnection of clusters and, hence, end-toend connectivity. Koushanfar et al. [236] proposed an adaptive coordination scheme of the sensors’ sleep schedules. This scheme requires the development of models to predict the measurement of one sensor based on the data from other sensors. These prediction models are used to solve the sleeping coordination problem with an integer linear programming solver. Also, it considers the creation of a maximal number of subgroups of disjoint nodes, each of whose data is sufficient to recover the measurements of the entire sensor network. Ai and Abouzeid [6] proposed a directional sensors-based approach for network coverage, where the coverage region of a sensor depends on its location and orientation. Li et al. [263] proposed efficient distributed algorithms to optimally solve the best-coverage problem with the least energy consumption. Megerian et al. [294, 296] proposed optimal polynomial time worst and average case algorithms for coverage calculation based on the Voronoi diagram and graph search algorithms. Also, Vieira et al. [387] exploited the concept of Voronoi diagram to solve the coverage problem in wireless sensor networks with high density. They proposed a mechanism that helps control the density of densely deployed wireless sensor networks. Their mechanism is based on a criterion, which decide the nodes to be turned on. In addition, they presented a management function, which helps take the sensor nodes out of service (or off ) temporally. Their proposed solution is based on the concept of Voronoi Diagram, which decomposes the space into regions around each node. Then they utilized the area of the Voronoi cell to determine which nodes to stay awake. Huang and Tseng [206, 207] presented polynomial-time algorithms, in terms of the number of sensors, for the coverage problem formulated as a decision problem. Abrams et al. [4] proposed a distributed algorithm to partition a wireless sensor network into k covers, each of which contains a subset of sensors that is activated in a round-robin fashion such that as many areas are monitored as frequently as possible. Comprehensive surveys of a variety of approaches on energy-efficient coverage as well as connectivity can be found in [103, 169, 306, 384]. Xing et al. [425] proposed the first combined study on k-coverage and connectivity and proved that if the radius of the communication ranges of sensors is double the radius of their sensing ranges, the network is connected provided that sensing coverage is guaranteed. Also, they computed the network connectivity based on whether the disconnected node is boundary or interior, and proposed a network configuration protocol based on the degree of coverage of the sensing application. Yang et al. [429] formalized the k-coverage set and the k-connected coverage set problems in terms of linear programming and proposed two non-global solutions for them. Also, Nakamura et al. [308] provided a linear programming formulation of the coverage problem whose objective function is to minimize the network energy consumption and whose constraints assure the quality of service requirements, such as coverage and connectivity, with respect to nodes energy restrictions. Specifically, they presented a dynamic mixed integer linear programming model in order to solve the coverage and connectivity dynamic problems in flat wireless sensor networks. The model solution determines an optimal node scheduling scheme, which indicates

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the network topology in pre-defined time periods so as to assure the area coverage and network connectivity at each period while minimizing the energy consumption. Bai et al. [69] proposed an optimal deployment strategy to achieve both full coverage and 2-connectivity regardless of the relationship between communication and sensing radii of the sensors. Huang et al. [210] studied the relationship between coverage and connectivity of wireless sensor networks and proposed distributed protocols to guarantee both their coverage and connectivity. Zhang and Hou [447] proposed a distributed algorithm to keep a small number of active sensors in a network regardless of the relationship between sensing and communication ranges. Tian and Georganas [378] proved that if the original network is connected and the identified active nodes can cover the same region as all the original nodes, then the network formed by the active nodes is connected when the communication range is at least twice the sensing range. Yener et al. [436] proposed a probabilistic Markov model to solve the problem of minimizing power consumption in each sensor while ensuring coverage and connectivity. Lee et al. [254] proposed a new coverage measure of sensor networks, which considers arbitrary source–destination pairs and extends the concept of the best and worst-case path-based coverage. This measure helps evaluate the coverage of a network, while accounting for arbitrary paths. They considered two problems with respect to the support and the breach of the sensor network. While the evaluation problem is to compute the value of the support/breach of a given sensor network, the deployment problem is to find optimal locations for k additional sensors to best improve the support/breach for a given integer k. Also, they presented several centralized and localized algorithms to solve these two problems. Gupta et al. [182, 186] proposed centralized and distributed algorithms for connected sensor cover so the network can self-organize its topology in response to a query and activate the necessary sensors to process the query. Datta, et al. [129] proposed two self-stabilizing algorithms to the problem of minimal connected sensor cover. Zhou et al. [454] proposed a distributed and localized algorithm using the concept of the k th -order Voronoi diagram to provide fault tolerance and extend the network lifetime, while maintaining a required degree of coverage. Cortes et al. [124] designed control and coordination algorithms for a multi-vehicle network with limited sensing and communication capabilities. Also, Liu et al. [270] proposed adaptive, distributed, and asynchronous coverage algorithms for mobile sensing networks. Moreover, they proved that mobility can be used to improve coverage in wireless sensor networks. Bin Malek et al. [288] formulated the k-coverage problem, which determines the orientation of minimal directional sensors so that each target is covered at least k times, as an integer linear programming problem. They discussed the incapability of integer linear programming in coverage balancing. They provided an integer quadratic programming and integer non-linear programming formulation of the above k-coverage problem, which accounts for coverage balancing. They proposed a centralized greedy k-coverage algorithm in order to approximate the formulations. Gupta et al. [183] focused on coverage and connectivity, which are essential for data forwarding from each target to a remote base station in targetbased wireless sensor networks. To satisfy both coverage and connectivity, which is an NP-complete problem, they attempted to find a minimum number of potential

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positions to place sensor nodes when given a set of target points. They proposed a genetic algorithm-based scheme to solve this problem, which provides k-coverage to all targets and m-connectivity to each sensor node. Luo et al. [280] investigated the maximum coverage sets scheduling problem, where each coverage set of a coverage set collection covers all targets or the whole detection area. The goal is to find a feasible scheduling for the coverage set collection that maximizes the network lifetime. They formulated this coverage problem as an integer linear programming problem, and proposed both a greedy algorithm and an approximation algorithm for solving it.

5.10 Conclusion In this chapter, we study the problem of connected k-coverage in planar wireless sensor networks, where each location in a planar field is covered by at least k active sensors while all active sensors are being connected [39, 52]. First, we characterize k-coverage of a planar field based on a geometric analysis of the intersection of sensing disks of k sensors. We prove that k-coverage implies connectivity between active sensors when the communication range of the sensors is at least equal to their sensing range. By looking at real sensor node platforms, it is always the case that the communication range of the sensors is higher than their sensing range, and hence our argument is always valid. Indeed, Tables I and II given in [447] show the communication range of Berkeley motes is much higher than the sensing range of several typical sensors, that and hence support our argument. We also prove that a sufficient condition of k-coverage of a planar field is that the minimum planar sensor density depends only on k and the sensing range of the sensors. Moreover, we proposed centralized (CERACCk ), pseudo-distributed (T-CRACCk and D-CRACCk ), and fully distributed (Trig-DIRACCk ) protocols to solve the connected k-coverage problem in wireless sensor networks. Also, we propose a fully distributed, self-scheduling protocol, called Self-DIRACCk , where each sensor turns itself on if its sensing range is not k-covered. This approach is different from that of Trig-DIRACCk , where each sensor selects a necessary number of sensors to become active in order to k-cover its sensing range. Despite the fact that Trig-DIRACCk requires coordination between the sensors to k-cover their sensing ranges, thus introducing additional control overhead, we find through simulations that Trig-DIRACCk outperforms Self-DIRACCk with regard to the necessary number of active sensors for full k-coverage of a planar field and network lifetime. We also extend our analysis by relaxing several widely used assumptions in k-coverage configurations in planar wireless sensor networks. Precisely, we relax the assumptions of the unit sensing disk model and homogeneous sensor. These relaxations have helped us handle the convex sensing model and heterogeneous wireless sensor networks, and hence promote the use of our protocols in real-world applications. Our simulation results show that Self-DIRACCk is more energy-efficient than CCP [425], with respect to the number of active sensors required for k-coverage and network lifetime.

Chapter 6

Planar Convexity Theory-Based Approaches for Heterogeneous, On-Demand, and Stochastic Connected k-Coverage My method is different. I do not rush into actual work. When I get an idea, I start at once building it up in my imagination. I change the construction, make improvements and operate the device in my mind. It is absolutely immaterial to me whether I run my turbine in thought or test it in my shop. I even note if it is out of balance. There is no difference whatever – the results are the same. In this way, I am able to rapidly develop and perfect a conception without touching anything. When I have gone so far as to embody in the invention every possible improvement I can think of and see no fault anywhere, I put into concrete form this final product of my brain . Nikola Tesla (1856–1943)

Overview This chapter introduces our solution to the problem of connected kcoverage in heterogeneous planar wireless sensor networks, where the sensors do not necessarily have the same capabilities. Also, it discusses our solution to the same problem in mission-oriented mobile wireless sensor networks, which are deployed in planar fields that could have more than one monitoring task, i.e. mission, to be accomplished. Specifically, it focuses on how to fully k-cover a region of interest in a planar field using a minimum number of mobile sensors while minimizing the total energy consumption, which is due to the mobility of the sensors as well as their communication in order to successfully accomplish their specific mission. The above studies consider a deterministic sensing model. Furthermore, this chapter addresses the problem of stochastic connected k-coverage in planar wireless sensor networks using a more realistic, probabilistic sensing model instead of a deterministic sensing model. These studies are generalizations of the work described earlier in Chap. 5. Our goal is to enhance the practicality of our work presented in the previous chapter and extend its scope of applicability.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_6

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6.1 Introduction In real-world applications, wireless sensor networks are composed of heterogeneous sensors which do not necessarily have the same capabilities in terms of their sensing range, communication range, initial energy, computation, storage, etc. These networks have a potential to increase the network lifetime and reliability without causing significant increase in their cost [433]. For instance, Intel deployed two types of sensors, namely line-powered and battery-powered sensors, in the design of a pilot application of sensor nets in order to monitor the health of mechanical equipment in its fabrication plants [240]. While line-powered sensors can be attached to pumps and motors in the fabrication plant, battery-power sensors can be used to reduce installation cost and complexity. Indeed, Yarvis et al. [433] presented several analytical, simulation, and real testbed results showing the potential benefit and impact of energy and link heterogeneity on sensor nets, where all the sensors report their sensed data to a single sink. In addition to sensor heterogeneity, sensor mobility is another interesting feature that enables the design of mission-oriented mobile wireless sensor networks. These types of networks can be viewed as distributed and dynamic systems, where the sensors are mobile, autonomous, and interacting with each other to accomplish a specific mission in a region of interest in a planar field. In this type of network, these distributed sensors should be continuously self-organizing, especially for connected coverage of a region of interest and sensed data gathering, so they can achieve the goals of the mission in a dynamic and collaborative manner while minimizing their energy consumption. Also, sensor mobility should be purposeful and traded off against the goals of the mission. In particular, this mobility should be energy-aware given the scarce energy resources of the individual sensors. Next, we present the major tasks we want to accomplish in this chapter. In addition, we briefly describe how to achieve each one of them.

6.1.1 Major Tasks Although the problem of coverage, and particularly, connected k-coverage, in wireless sensor networks has been studied well and extensively in the literature [39, 69, 136, 182, 186, 210, 263, 370], most existing studies focused on homogeneous wireless sensor networks, where all the sensors have the same capabilities with regards to their storage, computational power, sensing range, communication range, initial energy, etc. Similarly, the problem of k-coverage-preserving scheduling (or sensor duty-cycling) in homogeneous wireless sensor networks has gained considerable attention [52, 182, 186, 210, 398, 425, 429, 447, 460]. However, this type of sensors poses a severe restriction on the design of real-life sensing applications as all the sensors are required to have the same power with respect to all of their features, including sensing, communication, and energy. In addition, several existing works

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on k-coverage in wireless sensor networks assumed a perfect sensing model (also known as deterministic sensing model), where a point in a planar field is guaranteed to be covered by a sensor provided that this point is within the sensor’s sensing range [447]. While some approaches focused on coverage only [5], others considered both coverage and connectivity in an integrated framework to ensure the correct operation of the network [425, 447]. Indeed, coverage deals with all locations in a planar field, and hence informs how well a phenomenon in the field is monitored, whereas connectivity is related to the locations of the sensors, and hence quantifies how well the active sensors communicate with each other and forward sensed data on behalf of each other to the sink. We addressed this problem in Chap. 5, where centralized, pseudo-distributed, and distributed protocols are proposed. A few approaches, however, considered a more realistic sensing model (also known as stochastic sensing model) in the design of sensor scheduling protocols while preserving either full coverage [276, 295, 296, 447] or k-coverage of a planar field [398, 425, 447], where a point is covered by a sensor with some probability. In this chapter, we consider three fundamental problems in planar wireless sensor networks, namely heterogeneous connected k-coverage, on-demand connected kcoverage, and stochastic connected k-coverage. Next, we specify these tasks and give our corresponding plan of actions. First, we want to k-cover a planar field in the presence of heterogeneous sensors, while keeping all the sensors connected. To this end, we investigate the problem of connected k-coverage in heterogeneous wireless sensor networks [60], where the sensors that mainly differ by their sensing and communication ranges as well as their initial energy. We propose to divide a planar circular field into concentric circular bands of strictly decreasing widths starting from the inmost band. Our goal is to achieve k-coverage of any region of interest while alleviating the energy sink-hole problem, which is inherent in static wireless sensor networks, and extending the network operational lifetime. Second, we aim at k-covering any region of interest in a planar field. We plan to address the problem of on-demand connected k-coverage in mission-oriented wireless sensor networks under the following three requirements [30, 43]: A region of interest in a planar field should be k-covered, where k ≥ 3, whenever needed. Consequently, a region of interest to be k-covered does not have to be the same all the time, and hence may change. Also, all the involved sensors in the k-coverage process should be maintained connected for the correct network operation. In addition, the sensors should be able to move to designated locations in a region of interest to ensure its k-coverage whenever necessary. Third, we want to account for a non-deterministic sensing model, where the sensor’s sensing capability is probabilistic and does not follow the disk model. We focus on the design of stochastic connected k-coverage configuration protocols for planar wireless sensor networks [34]. More specifically, we decompose this problem into two sub-problems: Stochastic k-coverage characterization problem and stochastic k-coverage-preserving scheduling problem. Specifically, the first problem is to find a sufficient condition so that every point in a planar field is covered by at least k sensors with a probability no less than pth , called threshold probability, under

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our stochastic sensing model, which is introduced earlier in Chap. 2 (Sect. 2.3), and compute the corresponding minimum number of sensors. Fourth, we want to assess the performance of the proposed connected k-coverage protocols for the three classes of wireless sensor networks, namely heterogeneous, mission-oriented, and stochastic wireless sensor networks. We provide several simulation results of the corresponding connected k-coverage protocols based on several performance metrics.

6.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 6.2 solves the problem of connected k-coverage in heterogeneous planar wireless sensor networks. Section 6.3 investigates the problem of mobile connected k-coverage in mission oriented wireless sensor networks. Section 6.4 focuses on the problem of stochastic connected kcoverage in wireless sensor networks. Section 6.5 reviews related work. Section 6.6 concludes the chapter.

6.2 Heterogeneous Connected k-Coverage In real-world applications, sensor nets may have sensors with different capabilities, thus increasing the network reliability and lifetime [433]. In this section, we focus on heterogeneous k-covered sensor nets. More precisely, we exploit the results of Chap. 5 for generating energy-efficient connected k-coverage configurations using heterogeneous sensors. Using a geometric approach, we characterize k-coverage of a planar field with homogeneous sensor nets and derive a tight condition to ensure k-coverage. Then, we use this characterization for heterogeneous sensor nets. Our sensor net could have more than two types of sensors, which could be either randomly or pseudo-randomly deployed in a planar circular field. Also, we propose centralized (whenever possible) and distributed connected k-coverage protocols for heterogeneous sensor nets. To the best of our knowledge, this chapter provides the first analysis and design of centralized and distributed protocols for connected k-coverage in heterogeneous sensor nets. More importantly, our proposed framework is more general in the sense that it can be applied to both homogeneous and heterogeneous sensor nets.

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6.2.1 Random Deployment Approach In this section, we assume that all the sensors are randomly deployed in a planar circular field. Next, we discuss the feasibility of centralized and distributed connected k-coverage protocols.

6.2.1.1

Centralized Connected k-Coverage Protocol

The design of centralized protocols for k-coverage in heterogeneous sensor nets, where the sensors are randomly deployed, poses a major challenge. Indeed, a central gathering node, such as the sink, could be responsible for designating a subset of sensors to remain active in a given scheduling round (or simply round) to achieve kcoverage of a planar field. However, it is almost impossible to select a small number of sensors to k-cover a planar field. In fact, there is no unique solution to obtain a slicing grid of a planar field (Chap. 5, Fig. 5.5), i.e., slice a planar field into overlapping Reuleaux triangles, given that the sensors are heterogeneous. In particular, there are two extreme schemes for slicing a planar field: Scheme 1: It accounts for the least powerful sensors in terms of their sensing range. Thus, a planar field would be sliced into overlapping Reuleaux triangles of width r min , the smallest radius of the sensing ranges of the sensors. This solution, however, would lead to over k-coverage situation, where some regions in the planar field are denser with sensors than others, and hence more than k-covered, i.e., using more than enough sensors. This is highly likely to occur when most of the selected sensors have sensing ranges whose radii are greater than r min . Scheme 2: It accounts for the most powerful sensors, thus slicing a planar field into overlapping Reuleaux triangles of width r max , the largest radius of the sensing ranges of the sensors. Unfortunately, this scheme would lead to under k-coverage situation, where some regions in the planar field are less than k-covered. Likewise, this is highly likely to arise when most of the selected sensors have sensing ranges whose radii are less than r max . Both problems in the design of centralized connected k-coverage protocols are due to pure deployment randomness and sensor heterogeneity. Thus, we focus on the design of a distributed protocol for heterogeneous sensors (R-Het-DCCk ).

6.2.1.2

Distributed Connected k-Coverage Protocol (R-Het-DCCk )

Overview: Each sensor si randomly slices its sensing range into six overlapping Reuleaux triangles of width equal to r i , the radius of si ’s sensing range. Based on the result of Theorem 5.2 in Chap. 5 (Sect. 5.2.2), a sensor si randomly picks one of the three-lens flowers (as shown in Fig. 6.1) and checks whether its sensing range is k-covered. A sensor starts first by choosing the sensors located in the three lenses of

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Fig. 6.1 Three-lens flowers of si

the selected three-lens flower to remain active and k-cover its sensing range based on their remaining energy and the radii of their sensing ranges. Specifically, a sensor si selects k sensors from each of the lenses of the three-lens flower whose radii of their sensing ranges is at least equal to r i , the radius of the sensing range of si . Then, it activates them by sending AWAKE messages. When a sensor receives an AWAKE message, it becomes active and broadcasts a NOTIFICATION message to inform all its neighbors. Every sensor keeps track of its active neighbors. Also, a sensor will turn itself active if its sensing range is not k-covered. Next, we propose an efficient deployment strategy, where the sensors are pseudorandomly deployed in a planar field, in order to exploit the benefits of heterogeneity.

6.2.2 Pseudo-random Deployment Approach In [60], we investigated the energy sink-hole problem in static sensor nets, where sensors located nearer a static sink are heavily used in forwarding data to it, thus suffering from severe energy depletion. To alleviate this problem, we considered a planar circular field and proposed a deployment approach using heterogeneous sensors with regard to their initial energy only. These sensors are assigned to the bands of the planar deployment field in such a way that they all deplete their initial energy at the same time. Each of these bands has a width equal to R, the radius of the communication range of the sensors, since we focused on data forwarding. Also, each band has homogeneous sensors. But, any two bands in the planar field have heterogeneous sensors. Moreover, the difference between the amounts of energy in two adjacent bands should verify certain ratio so all the sensors in the network deplete their initial energy at the same time. A fine description of our multi-tier architecture is in [60]. In this section, we focus on the problem of connected k-coverage, and we further assume that the sensors have different radii of sensing and communication ranges.

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As demonstrated in [60], the most powerful sensors should be located nearer the sink. Thus, we slice a planar circular field (C D ) into n bands of strictly decreasing widths starting from the inmost band. Assume that the minimum and maximum radii of the sensing range of the sensors are r min and r max , respectively, and that the radius of the planar field is D. We also assume that the difference in width between any two adjacent bands is d w . More specifically, the inmost band, denoted by b1 , has a width equal to r max while the outmost band, denoted by bn , has a width equal to r min as shown earlier in Fig. 6.2. It is important to design energy-efficient connected k-coverage protocols that help extend the network lifetime when sensed data routing is considered. That is, these protocols should be designed to be used later on when a network designer wishes to build a uniform framework that jointly considers sensor duty-cycling for connected k-coverage, and data forwarding on duty-cycled sensors. In fact, the ultimate goal of the design of sensor nets is to monitor an environment and generate sensed data to be forwarded to the sink for further analysis and processing. As stated in the beginning of this section, to alleviate the problem of the energy sink-hole problem in static sensor nets, it is necessary that the most powerful sensors should be deployed nearer the sink as they are heavily used in data forwarding to the sink. The motivation behind decomposing a planar field into bands with such widths is the fundamental result stated by Xing et al. [425]. Precisely, they proved that a sensor Fig. 6.2 A planar field decomposed into circular bands

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net is surely connected when the radius of the communication range of the sensors is at least equal to double the radius of their sensing range provided that coverage is guaranteed. In our heterogeneous sensor deployment strategy, the width wj of a given band bj is computed such that the radius Rj of the communication range of any sensor sj placed in bj should be at least equal to the sum of the width of bj and its adjacent predecessor band bj-1 . That is, Rj ≥ wj + wj−1 , where j ≥ 2, wj = r j and wj−1 = r j−1 , = r j + d w , with r j−1 being the radius of the sensing range of any sensor sj−1 located in band bj−1 and d w is the difference in width between bands bj−1 and bj as discussed earlier. We should recall that the width of the inmost band b1 is r max and that of the outmost band bn is r min . Figure 6.2 illustrates this discussion. This design ensures network connectivity to enable communication between the sensors and facilitate forwarding of the sensed data generated by the sensors toward the sink. This issue becomes clearer when a geographic forwarding protocol is built on top of our connected k-coverage protocols. Our sensor deployment approach is hierarchical due to the presence of multiple bands (or layers) forming the planar deployment field. Moreover, it is pseudo-random in the sense that the sensors are supposed to be deployed densely and randomly within each band while ensuring that the sensors in any pair of bands in the planar circular field are heterogeneous. Our sensor deployment approach is designed in a way that all the sensors placed in the band bj are homogeneous and hence have the same sensing range whose radius is equal to r max – (j – 1) × d w , where d w > 0 and r max > r min , such that the following system of equations is satisfied, where n designates the number of bands forming the planar circular field, and hence the number of types of sensors in the heterogeneous sensor net: rmin + (n − 1) × dw = rmax

(6.1)

n × rmin + n × (n − 1) × dw /2 = D

(6.2)

with n and d w being the only unknowns in the above equations. This is a valid assumption since we know only the capabilities of the sensors in terms of their sensing range, namely r max and r min , and the radius D of the planar circular field. Therefore, knowing r max , r min , and D, it is easy to check that there is a unique solution (n, d w ) to the above equations. For instance, assume that we have r min = 1, r max = 3, and D = 16, we get n = 8 and d w = 2/7. In the following sections, we describe in detail both of our centralized and distributed connected k-coverage configuration protocols which exploit the characteristics of the sensors in their respective bands.

6.2.2.1

Centralized Connected k-Coverage Protocol (PR-Het-CCCk )

Generally, the sink is attached to an infinite source of energy, such as a wall outlet, and thus has no energy limitation. Thus, the single-point failure problem does not exist when the sink plays any particular role as the battery depletion problem for the

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sink cannot arise. In our centralized protocol, we assume that the sink is responsible for randomly decomposing each of these bands into overlapping Reuleaux triangles. This implies that the sink is aware of the locations of all the sensors in the network. Moreover, we assume that every sensor knows the id of the band it belongs to. Then, the sink applies the result of Theorem 5.2 in Chap. 5 to select a subset of sensors to k-cover each of these bands with a small number of sensors. The sensor will have to send a request to each band, where it specifies a band id and the ids of a subset of sensors located in that band so they remain active to k-cover the underlying band. We assume that each sensor is uniquely identified by an id, i.e., an integer. The sink performs these actions at the beginning of each round under the assumption that all the sensors are awake at the beginning of a round and during some time interval t awake . This time t awake should be large enough so that any sensor in the outmost band would be able to receive any message (or request) sent by the sink and destined to this band. This would help each of the sensors in the outmost band to check whether it has been designated by the sink to participate in the k-coverage of its band.

6.2.2.2

Distributed Connected k-Coverage Protocol (PR-Het-DCCk )

We reuse the same distributed connected k-coverage protocol described earlier in Chap. 5 in Sect. 5.5. Recall that all the sensors located in the same band have the same capabilities, including their sensing range. Therefore, a sensor sj located in a band bj randomly decomposes its sensing range into six overlapping Reuleaux triangle of width r j = r max – (j – 1) × d w , the radius of sj ’s sensing range (i.e., the width of the band bj ). The only difference with the protocol proposed in Sect. 5.5 in Chap. 5 is that a sensor sj would select from its three-lens flower only the sensors that are located either in its band bj or in its adjacent band bj−1 to k-cover its sensing range. Indeed, the sensors located in bj−1 have higher sensing range than sj and thus will be able to participate in the k-coverage of sj ’s sensing range when they are selected. This means that a sensor sj would select sensors from its band bj which have the same power as sj (especially, their sensing range) or more powerful than sj (i.e., from the band bj−1 ) to ensure k-coverage of its sensing range.

6.2.3 Performance Evaluation In this section, we present the simulation results of our protocols using a high-level simulator written in C. We consider a planar circular field of radius D = 1000 m. We use the energy model in [435], where the sensor energy consumption in transmission, reception, idle, and sleep modes are 60 mW, 12 mW, 12 mW, and 0.03 mW, respectively. Following [447], the energy required for a sensor to stay idle for 1 s is equivalent to one unit of energy. We assume that the initial energy of each sensor is

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Fig. 6.3 R-Het-DCCk versus R-Hom-DCCk

60 J enabling it to operate about 5000 s in reception/idle modes [435]. All simulations are repeated 100 times and the results are averaged. First, we compare both of our distributed connected k-coverage protocols for homogeneous sensor nets, denoted by R-Hom-DCCk , where the sensors are homogeneous and randomly deployed, and R-Het-DCCk (where the sensors are heterogeneous and randomly deployed). Indeed, R-Hom-DCCk is similar to R-Het-DCCk except that the sensors use the same sensing range r. For R-Hom-DCCk , we assume that the sensing range of the sensors is r = 25 m. On the other hand, for R-HetDCCk , the sensing range of the sensors is between r min = 25 m and r max = 50 m. As it can be seen from Fig. 6.3, R-Het-DCCk outperforms R-Hom-DCCk as it requires a smaller number of sensors for any coverage degree k. Indeed, the presence of more powerful sensors helps ensure k-coverage of a planar field with a smaller number of active sensors. We obtain the same result with regard to energy expenditure, i.e., R-Het-DCCk consumes less energy than R-Hom-DCCk . Due to space limitation, we have not presented it here. Figure 6.4 shows that our centralized protocol for pseudo-random deployment of heterogeneous sensors (PR-Het-CCCk ) outperforms our distributed protocol for pseudo-random deployment of heterogeneous sensors (PR-Het-DCCk ). Also, PRHet-DCCk presents better results than our distributed protocol for random deployment of heterogeneous sensors (R-Het-DCCk ). This shows the impact of slicing the planar field into concentric circular bands and deploying sensors in those bands based on their capabilities. Indeed, it is expected that our centralized protocol PR-Het-CCCk has the best performance compared to all other protocols, namely PR-Het-DCCk and

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Fig. 6.4 Comparing PR-Het-CCCk , PR-Het-DCCk , and R-Het-DCCk

R-Het-DCCk . This is mainly due to the absence of coordination between adjacent sensors to achieve k-coverage of their sensing ranges. The sink only needs to keep track of the locations of all the sensors as well as their remaining energy. Although there is some energy consumed to forward all requests of the sink to their destination sensors, it is less than the energy needed for the coordination between the sensors to select a small number of sensors to k-cover the planar field. Next, we focus on the problem of on-demand connected k-coverage in homogeneous planar wireless sensor networks.

6.3 On-Demand Connected k-Coverage In this section, we focus on connected k-coverage in mission-oriented mobile wireless sensor networks, where a region of interest needs to be fully k-covered while maintaining network connectivity with the goal to minimize the total number of sensors necessary to achieve k-coverage and minimize the total energy consumption due to sensor mobility and communication between all the sensors in the network. Indeed, the closest work to ours is the one proposed by Wang and Tseng [398], and hence we compare our protocols to theirs. First, we present our pseudo-random placement strategy of the mobile sensors to k-cover a region of interest.

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Fig. 6.5 Decomposition of a square region of interest into Reuleaux triangles, where mobile sensors should be located in lenses to k-cover a region of interest

6.3.1 Pseudo-random Sensor Placement In this type of wireless sensor network, there must be a node that is aware of the mission objectives of the network in the monitored planar field. We assume that the sink is aware of any region of interest in the planar field that needs to be k-covered and hence is responsible for computing the locations that should be occupied by the sensors in order to k-cover a region of interest. Moreover, it has been proved that the optimum location of the sink in terms of energy-efficient data gathering is the center of the planar field [281]. Based on Theorem 5.2 in Chap. 5, the sink randomly decomposes a region of interest into overlapping Reuleaux triangles of width r (or slices) such that two adjacent slices intersect in a region shaped as a lens as shown in Fig. 5.3 in Chap. 5. Thus, each region of interest to be k-covered is sliced into regular triangles of side r as shown in Fig. 6.5. The sink exploits the result of Theorem 5.2 in Chap. 5 so that a region of interest is k-covered by a small number of active sensors. Hence, it identifies the necessary lenses where the active sensors should be located and broadcast this information into the network. Precisely, each lens is uniquely identified by a pair of its extreme points. Figure 6.5a, b shows the lenses that should be occupied by mobile sensors to achieve energy-efficient k-coverage of the region of interest.

6.3.2 Sensor Mobility for k-Coverage of a Region of Interest In the previous section, we show how the sink computes the areas (or lenses) of a region of interest that should be occupied by mobile sensors to minimally k-cover the region. In this section, we describe two centralized and distributed protocols that

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decide which sensors move to these selected areas (or lenses) while minimizing the total energy consumption due to sensors movement. Next, we discuss both protocols in details.

6.3.2.1

Centralized Approach for Mobile Sensor Selection (CAMSEL)

In addition to slicing the region of interest and determining its lenses where sensors should be placed, the sink selects the sensors that should move to occupy these lenses. To do so, the sink needs to be aware of the current locations of all the sensors. Also, the sink has to keep track of the remaining energy of each sensor in the network. Thus, the sink is required to maintain a database for all the sensors where each entry contains a sensor’s id, its current location, and remaining energy. This information can be gathered by the sink under the assumption that the sensors move only when they are requested by the sink. Indeed, knowing the current positions of the sensors and their closest target locations in the lenses, the sink can compute the energy consumption required to reach these target locations and update their remaining energy accordingly based on the energy model [398, 435] that is stated below in Sect. 6.3.3. Also, the sink can estimate the energy consumed by each sensor while monitoring a region of interest based on the same energy model [398, 435]. Using this approach, the sink would be able to select a small number of sensors to fully k-cover a region of interest. Moreover, the sink would be able to minimize the total energy consumption of the sensors introduced by their mobility by choosing the sensors closer to the target region of interest to k-cover it. However, the network should guarantee that all the sensors selected by the sink to move to their target lenses receive the queries originated from the sink. For energy savings purposes, all queries are sent individually to all selected sensors. The sink will broadcast as many queries as the number of sensors necessary to k-cover a region of interest. Each query includes an id of a selected sensor and its target location (x, y)target in one of the lenses of a region of interest. Precisely, a query has the following structure: query = ⟨ id,(x, y)target ⟩. When a sensor receives a query, it will decide whether to forward the query toward the selected sensor based on the location of the sink and the target location (x, y)target in the query. Our goal is to reduce the amount of unnecessary data transmission that would flood the network to save the energy of individual sensors, thus extending the network lifetime.

6.3.2.2

Distributed Approach for Mobile Sensor Selection (DAMSEL)

In the centralized approach, the sink is supposed to be aware of the status of all the sensors in the network with regard to their location and energy consumption. Although our centralized approach helps obtain the best schedule in terms of energyefficient k-coverage and minimum energy consumption due to sensor movements, it would incur delay in the sensor selection phase especially for a large network that require a huge database for maintaining the current status of all the sensors in

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the network. In this section, we propose a distributed approach to remedy to this shortcoming. In this approach, the sensors will cooperate with each other to move to the region of interest to k-cover it with a small number of sensors while minimizing the energy consumption introduced by their mobility. The sink will only specify the region of interest to be k-covered, which supposed to be a square that is characterized by its center (x 0 ,y0 ) and side length a. Thus, the sink will broadcast a unique query that has the following structure: query = ⟨ (x 0 , y0 ),a⟩. Moreover, we assume that all the sensors generate the same slicing grid of the region of interest. This means that all the sensors deterministically generate the same reference triangle whose center coincides with that of the region of interest. This would enable all the sensors to have the same set of lenses, each of which should be occupied with at least k sensors. Also, each sensor is supposed to be moving at a constant speed until it reaches its destination in the lens it selected in the region. State Transition Diagram of DAMSEL. At any time, a sensor can be in one of the three states: WAITING, WILLING, MOVING, SENSING/COMMUNICATING, or SLEEPING. A state transition diagram associated with DAMSEL and indicating the five possible states of a sensor and transitions between them is shown in Fig. 6.6 above. Next, we describe each of these five states of a sensor using the protocol DAMSEL:

Fig. 6.6 State diagram of DAMSEL

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151

• Waiting: At the beginning of each round for mobile sensor selection, all the sensors are waiting for a query for k-coverage of a region of interest. Upon receipt of a k-coverage query from the sink, a sensor switches to the Willing state provided that its remaining energy would allow it to move. If a sensor does not receive any query within time t waiting , it will switch to the Sleeping state. • Willing: When a sensor receives a query, it computes the slicing grid for the region of interest and chooses the closest lens to which it intends to move. Then, it broadcasts a data packet, called WILLING_Msg, including its id, current location (x, y)current , and target location (x, y)target in the selected lens in the region of interest, thus expressing its willingness to move to this lens. Precisely, this message has the following structure: WILLING_Msg = ⟨ id,(x, y)current ,(x, y)target ⟩. While in the Willing state, if a sensor si receives at least k WILLING _Msg originated from k distinct sensors that are closer to the target lens than si , the sensor si will simply discard its request and selects another target lens. If after some time t willing a sensor finds itself unable to move to any of the first three closest lenses, it will simply give up and automatically switch to the Sleeping state. • Moving: If after some waiting time t wait the sensor that broadcasted a WILLING_Msg does not receive at least k MOVING_Msg originated from k distinct sensors that would surely move to the underlying lens, it will decide to move to its selected lens and broadcast a MOVING_Msg, thus expressing its final decision to move to this lens. While moving, if a sensor si receives a WILLING_Msg from another sensor sj that is closer to the target lens than si , the sensor si will simply give up and let sj move to this target lens. Then, the sensor si switches to the Sleeping state. This would guarantee that each lens is k-covered by the closest k sensors, thus avoiding wasting more energy due to sensor mobility. • Sensing/Communicating: When a mobile sensor arrives at its target location in the selected lens, it will switch to the Sensing/Communicating state, where it would start sensing and communicating its sensed data to the sink. At the end of its sensing and communication activity in a region of interest, a sensor will move to the Available state for a new mobile sensor selection round. • Sleeping: In this state, a sensor is neither communicating with any other sensor nor sensing a planar field, and thus its radio is turned off. However, after some time t sleeping , it switches to the Waiting state in order to receive query messages. 6.3.2.3

How to Ensure Network Connectivity?

In this type of mission-oriented wireless sensor network, sensor mobility is necessary so any region of interest could be k-covered with a small number of sensors, where the degree of coverage k depends on the mission to be accomplished. However, this mobility may not guarantee that the network remains connected all the times. Although the sensors coordinate between themselves to k-cover a region of interest, there is no coordination between them with regard to their locations so they remain connected during the network operation. To alleviate this problem, the sensors select some of them, called dMULEs, based on their remaining energy to act as data MULEs

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[347] to keep the network connected. The dMULEs will transport messages on behalf of others and disseminate them to the other sensors in the network, thus enabling continuous communications between all the sensors including the sink. These messages could be either originated from the sink as queries to k-cover a region of interest or initiated from the sensors to coordinate their activity to achieve k-coverage of a region of interest. These dMULEs will be used efficiently in the centralized approach for mobile sensor selection to gather information about the current locations of the sensors and inform the sink accordingly so it selects the closest ones to the region of interest to be k-covered. This helps minimize the necessary energy consumption for k-coverage due to sensor mobility.

6.3.3 Performance Evaluation In this section, we present the simulation results of CAMSEL and DAMSEL using a high-level simulator written in the C language. We use the same energy model given in Sect. 6.2.3. For fair comparison with Wang and Tseng’s approach [398], we consider a planar square field of side length 600 m and a square region of interest of side length 300 m. Moreover, the energy spent in moving is given by E mv (d) = emove × d, where emove is the energy cost for a sensor to move one unit distance, and d is the total distance traveled by a sensor [398]. As stated in [398], emove is randomly selected in [0.8 J, 1.2 J] and the moving speed of each sensor is 1 m/s. Also, the sensing range of the sensors is equal to 20 m. All the simulations are repeated 100 times and the results are averaged. Figures 6.7 and 6.8 plot the number of sensors needed to k-cover a region of interest, where r = 20 m, for both CAMSEL and DAMSEL protocols. As it can be seen from Figs. 6.7 and 6.8, CAMSEL and DAMSEL require a number of sensors for k-coverage that is close to its corresponding theoretical value computed in Theorem 5.3 in Chap. 5. Thus, both protocols perform well compared to the theoretical result given in Theorem 5.3 (Chap. 5). However, CAMSEL slightly outperforms DAMSEL as shown in Fig. 6.9. Indeed, with CAMSEL, the sink has a global view of the network and hence computes a small number of sensors to k-cover a region of interest, while DAMSEL helps the sensors make their decision to participate in k-coverage based on their local knowledge of their neighbors, which varies due to sensor mobility. Also, DAMSEL requires higher energy consumption than CAMSEL (Fig. 6.10). With DAMSEL, more sensors are willing to move to ensure k-coverage of a region of interest. Due to the distributed nature of DAMSEL, a moving sensor will not always be able to know that other sensors, which are not currently located in its neighborhood, decided to participate in k-coverage of the underlying region of interest. Figures 6.11 and 6.12 show that our protocol DAMSEL outperforms the protocol Competition [398] with respect to the number of sensors required for k-coverage as well as the corresponding total moving energy. Indeed, our approach provides a fine analysis for ensuring k-coverage of a region of interest with a small number of

6.3 On-Demand Connected k-Coverage

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Fig. 6.7 CAMSEL compared to the result of Theorem 5.3 in Chap. 5

Fig. 6.8 DAMSEL compared to the result of Theorem 5.3 in Chap. 5

sensors based on Helly’s Theorem [85] and the geometric properties of the Reuleaux triangle [481]. In addition, the continuous collaboration between the sensors enables only a necessary number of sensors to move toward a region of interest to k-cover it.

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Fig. 6.9 CAMSEL compared to DAMSEL (number of sensors)

Fig. 6.10 CAMSEL compared to DAMSEL (total remaining energy)

6.3 On-Demand Connected k-Coverage

Fig. 6.11 DAMSEL compared to Competition (Number of sensors)

Fig. 6.12 DAMSEL compared to Competition (Total remaining energy)

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Next, we investigate the problem of stochastic connected k-coverage in homogeneous planar wireless sensor networks.

6.4 Stochastic Connected k-Coverage Although the approach on k-coverage in wireless sensor networks proposed in [425] is elegant and considers both deterministic and probabilistic sensing models, it does not provide any proof on whether its k-coverage eligibility algorithm would yield a minimum number of selected sensors to k-cover a planar field. In this section, we propose a solution to the stochastic connected k-coverage problem stated earlier using our stochastic sensing model, which reflects the real behavior of the sensing units of the sensors that are irregular in nature.

6.4.1 Stochastic k-Coverage Characterization In this section, we exploit the results found earlier in Sect. 5.2 of Chap. 5 in order to characterize k-coverage in wireless sensor networks based on the stochastic sensing model described earlier. It is also similar to the stochastic sensing model in [460], except that ours accounts for the type of propagation model, i.e., 2 < α ≤ 4. Precisely, we use the Reuleaux triangle model to compute the minimum k-coverage probability, denoted by pk,min , such that every point in a planar field is k-covered. Theorem 6.1 computes pk,min . Theorem 6.1 (Minimum k-Coverage Probability) Let r be the radius of the nominal sensing range of the sensors and k ≥ 3. The minimum k-coverage probability pk,min so that each point in a planar field is probabilistically k-covered by at least k sensors under the stochastic sensing model defined in Eq. (2.2) in Chap. 2 is approximately computed as ( pk,min ≈ 1 − 1 − e

−β

)

√r 3

) α )k

(6.3)

Proof First, we identify the least k-covered point in a planar field so we can compute pk,min . By looking at the Reuleaux triangle corresponding to three sensors and given in Fig. 6.13, it is easy to check that the center ξ of the Reuleaux triangle is the least 3-covered point as shown in Fig. 6.14. Indeed, ξ is located close to the boundaries of the sensing ranges of the sensors si , s j , and sl . By Lemma 5.2 stated earlier in Chap. 5, k sensors should be deployed in each Reuleaux triangle regions of a planar field to achieve k-coverage with a minimum number of sensors. Thus, on the average, we can claim that the center ξ is also the least k-covered point in a planar field. Note that ξ is equidistant from the sensors si , s j , and sl . Using the configuration in Fig. 6.14,

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157

a little algebra√shows that the distance between ξ and each of these three sensors is equal to r/ 3. Let Sk = {s1 , . . . , sk } ⊆ S be a set of sensors that k-cover ξ . As per the above observation, we can approximate the distance between any sensor in √ √ Sk and ξ by r/ 3, i.e., δ(si , ξ ) ≈ r/ 3, for each sensor si ∈ Sk . Therefore, the minimum k-coverage probability (or detection probability) for the least k-covered point ξ by exactly k sensors (s1 , .., sk ) under our stochastic sensing model is given by pk,min = 1 −

k Π

( ) ) α )k −β √r3 (1 − p(ξ, si )) ≈ 1 − 1 − e

i=1

∎

The stochastic k-coverage problem is to select a minimum subset Smin ⊆ S of sensors such that each point in a planar field is k-covered by at least k sensors and that the minimum k-coverage probability of each point is at least equal to some given threshold probability pth , where 0 < pth < 1 This helps us compute the sensing range r s , which provide full k-coverage of a planar field with a probability no less than pth : pk,min ≥ pth ⇒ rs ≤

( )1/α √ 1 3 − log (1 − (1 − pth )1/k ) β

Lemma 6.1 computes the value of the stochastic sensing range of the sensors.

Fig. 6.13 Reuleaux triangle

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Fig. 6.14 Location of a least k-covered point

Lemma 6.1 (Stochastic Sensing Range) Let k ≥ 3. The stochastic sensing range rs of the sensors that is necessary to fully k-cover a planar field with a minimum number of sensors and with a probability no lower than 0 < pth < 1 is given by rs =

( )1/α √ 1 3 − log (1 − (1 − pth )1/k ) β

(6.4)

where β is the physical characteristic of the sensors’ sensing units and 2 ≤ α ≤ 4 is the path-loss exponent, which depends on the propagation model (free space model versus multi-path model). ∎ The upper bound on the stochastic sensing range rs of the sensors computed in Eq. (6.4) is used as one of the input parameters to the k-coverage candidacy algorithm, which is presented in Sect. 6.4.2.1. Figures 6.15, 6.16, and 6.17 show rs for different values of pth and k while considering the free-space model (α = 2) (Fig. 6.15) and the multi-path model (α = 3, 4) (Figs. 6.16 and 6.17). As it can be seen, rs decreases as pth and α increase. However, it increases with k. Indeed, to achieve higher degree of coverage, the stochastic sensing range of the sensors should increase. Lemma 6.2 states a sufficient condition for full k-coverage of a planar field while Lemma 6.3 states a sufficient condition to guarantee connectivity between sensors under our stochastic sensing model when both pth and k are known.

6.4 Stochastic Connected k-Coverage

Fig. 6.15 Upper bound of rs versus k for α = 2

Fig. 6.16 Upper bound of rs versus k for α = 3

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Fig. 6.17 Upper bound of rs versus k for α = 4

Lemma 6.2 (Sufficient Condition for Probabilistic Full Planar k-Coverage) Let k ≥ 3. A planar field is probabilistically fully k-covered with a probability no lower than 0 < pth < 1 if any Reuleaux triangle region of width rs in the planar field contains at least k sensors. ∎ Lemma 6.3 (Condition for Probabilistic Network Connectivity) Let k ≥ 3. The sensors that are selected to k-cover a planar field with a probability no less than 0 < pth < 1 under our stochastic sensing model defined earlier in Chap. 2 in Sect. 2.3 are guaranteed to be connected if the radius of their communication range ∎ is at least equal to rs , the stochastic sensing range of the sensors.

6.4.2 Stochastic k-Coverage-Preserving Scheduling In this section, we focus on the design of a distributed sleep-wakeup scheduling protocol for stochastic k-coverage (SCPk ) of a planar field. The same approach could be applied for k-coverage under the deterministic sensing model by replacing rs by r. We exploit the results of Chap. 5 to present a distributed approach for the selection of a minimum number of active sensors to k-cover a planar field under the stochastic sensing model defined in Eq. (2.2) in Chap. 2 in Sect. 2.3. Recall the two approaches for sensor scheduling presented in Chap. 5. We use the second approach which is different from the first one in the sense that each sensor decides whether it is eligible to turn itself active. This decision is based on the degree of coverage

6.4 Stochastic Connected k-Coverage Fig. 6.18 k-Coverage candidacy algorithm

161

ALGORITHM 1: k-COVERAGE-CANDIDACY(rs,k) (* This code is run by each sensor *) Begin /* Sensing range slicing */ 1. Randomly decompose sensing range into six overlapping Reuleaux triangles RT(rs)i, 1≤ i ≤ 6 /* Localized k-coverage candidacy checking */ 2. For each Reuleaux triangle RT(rs)i Do 3. If RT(rs)i contains k active sensors Then 4. Skip /* i.e., do nothing */ 5. Else 6. Return (“candidate”) 7. End 8. End 9. Return (“non-candidate”) End

of its sensing range. More specifically, each sensor runs our k-coverage candidacy algorithm, which is given in Fig. 6.18, before it takes this decision.

6.4.2.1

k-Coverage Candidacy Algorithm

A sensor turns active if its sensing disk is not k-covered. Precisely, a sensor randomly slices its sensing range into six overlapping Reuleaux triangle of width rs and checks whether each one of them contains at least k sensors. Each sensor should know the status of its sensing neighbors only to decide whether it is candidate to turn active or not. If any of the six overlapping Reuleaux triangles of width rs of the sensing range of a sensor si does not have k active sensors, the sensor si is a candidate to become active. Figure 6.18 shows the pseudo-code of our k-coverage candidacy algorithm.

6.4.2.2

State Transition of SCPk

The state transition diagram associated with our stochastic k-coverage protocol (SCPk ) is similar to the one given in Fig. 5.16 in Chap. 5 by replacing r with rs . In this case, however, each sensor decides whether to turn itself on by running the k-coverage candidacy algorithm given in Fig. 6.18.

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6.4.3 Simulation Results In this section, we present the simulation results of SCPk using a high-level simulator written in the C programming language. We consider a planar square field of side length 1000 m. We use the energy model given in [435], where the sensor energy consumption in transmission, reception, idle, and sleep modes are 60 mW, 12 mW, 12 mW, and 0.03 mW, respectively. Following [447], the energy required for a sensor to stay idle for 1 s is equivalent to one unit of energy. We assume that the initial energy of each sensor is 60 J enabling a sensor to operate about 5000 s in reception/idle modes [435]. All simulations are repeated 100 times and the results are averaged. In Figs. 6.19, 6.20, and 6.21, we plot the planar sensor density as a function of the degree of coverage k for different values of the threshold probability pth and path-loss exponent α. As expected, the planar sensor density increases with pth and α. Indeed, as we increase pth , more sensors would be needed to achieve the same degree of coverage k. Recall that the width of the Reuleaux triangle that is guaranteed to be covered with exactly k sensors is equal to the stochastic sensing range of the sensors given earlier in Eq. (6.4) and hence decreases as pth and α increase. On the other hand, the planar sensor density required for full k-coverage of a planar field is inversely proportional to the area of this Reuleaux triangle as stated implicitly in Lemma 6.2. Figures 6.22, 6.23, and 6.24 plot the achieved degree of coverage k versus the total number of deployed sensors. Moreover, we vary both pth and α. Definitely, higher number of deployed sensors would yield higher coverage degree. Here also, any increase in pth and α would require a larger number of deployed sensors to

Fig. 6.19 Planar sensor density versus degree of coverage k for α = 2

6.4 Stochastic Connected k-Coverage

163

Fig. 6.20 Planar sensor density versus degree of coverage k for α = 3

Fig. 6.21 Planar sensor density versus degree of coverage k for α = 4

provide the same degree of coverage. Both experiments show a good match between simulation and analytical results. Figures 6.25, 6.26, and 6.27 show that the number of active sensors required to provide 3- coverage increases with the characteristic of the sensors β used in the definition of our stochastic sensing model presented earlier in Chap. 2 in Sect. 2.3.

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Fig. 6.22 Degree of coverage k versus number of deployed sensors for α = 2

Fig. 6.23 Degree of coverage k versus number of deployed sensors for α = 3

6.4 Stochastic Connected k-Coverage

Fig. 6.24 Degree of coverage k versus number of deployed sensors for α = 4

Fig. 6.25 Number n a of active sensors versus. β for k = 3 and α = 2

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Fig. 6.26 Number n a of active sensors versus β for k = 3 and α = 3

Fig. 6.27 Number n a of active sensors versus β for k = 3 and α = 4

6.5 Related Work

167

Fig. 6.28 Total remaining energy versus time for k = 3, α = 2, and pth = 0.7

Recall that β measures the uncertainty of the sensing units of the sensors. This result is expected given the definition of the stochastic sensing range computed in Eq. (6.4). Figures 6.28, 6.29, and 6.30 show the impact of pth on the operational lifetime of the network to provide 3-coverage. As mentioned earlier, higher values of pth require larger numbers of active sensors, and hence more energy consumption. To the best of our knowledge, the work in [425] is the only one on probabilistic k-coverage. However, the probabilistic sensing model used in [425] is totally different from ours. While our stochastic sensing model quantifies the detection probability of a sensor by an exponential function, the one in [425] only assigns it a constant value. Therefore, it is impossible to provide a fair quantitative comparison between our SCPk protocol and the one in [425].

6.5 Related Work In this section, we describe approaches and protocols that have been proposed for planar heterogeneous as well as mobile wireless sensor networks. In addition, we present sample approaches on stochastic coverage in planar wireless sensor networks. A special focus is on those on coverage and connectivity.

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Fig. 6.29 Total remaining energy versus time for k = 3, α = 2, and pth = 0.8

Fig. 6.30 Total remaining energy versus. time for k = 3, α = 2, and pth = 0.9

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169

6.5.1 Sensor Heterogeneity While coverage and connectivity in homogeneous wireless sensor networks have been studied well [46, 425], heterogeneous wireless sensor networks have received little attention [31]. Wang et al. [399] proposed a fine analysis of coverage using two types of sensors with different capabilities and discussed the impact of heterogeneous sensing and communication ranges of the sensors on coverage and broadcast reachability. Duarte-Melo and Liu [140] focused on heterogeneous sensors equipped with different battery power in their analysis of a clustering approach. Precisely, they considered two sensor deployment strategies, where the first one includes type-I sensors, while the second one includes an overlay of type-II sensors, which are more powerful but fewer in number. Then, they estimated the average network lifetime and quantified the optimal number of these type-II sensors, which act as cluster-heads. Moreover, they showed how to allocate the energy between the overlay sensors and normal sensors. Lazos and Poovendran [251, 252] discussed the coverage problem in heterogeneous sensor networks. They formulated the coverage problem as a set intersection problem and derived analytical expressions, which quantify the coverage achieved by stochastic coverage. Moreover, Lazos et al. [253] addressed the problem of detecting mobile targets using sensors that have heterogeneous sensing areas of arbitrary shape with a goal to increase the robustness of target detection. Specifically, they considered a deterministic sensor deployment strategy to maximize the probability of target detection. Ma et al. [284] proposed a resource-oriented protocol that implements a network topology according to the number of heterogeneous sensors as well as their specific resources and characteristics, such as radio coverage, power capacity, processing capabilities, and mobility attributes. Du and Lin [136] proposed a differentiated coverage algorithm for heterogeneous sensor nets. The motivation behind providing different degrees of sensing coverage for different regions is that different network areas do not have the same importance and hence some areas require higher coverage degree than others. Yuce et al. [441] proposed a heterogeneous sensor network system for monitoring physiological parameters from multiple patient bodies. Their system provides patients with mobility, and medical staff with flexibility in obtaining physiological data on the patients on-demand through the Internet. Mhatre et al. [298] considered a heterogeneous sensor net with two types of nodes which differ by their intensity (or average number per unit area) (λ0 , λ1 ) and their battery energy (E 0 , E 1 ). While type 1 nodes perform sensing, type 2 nodes perform sensing and act as cluster heads. Their study consists of computing the optimum node intensities (λ0 , λ1 ) and node energies (E 0 , E 1 ) that ensure connected coverage of the surveillance area with a high probability while guaranteeing a network lifetime of at least T units and minimizing the overall cost of the network. Hanh et al. [191] investigated the problem of coverage maximization with sensors with heterogeneous sensing ranges, which is known to be NP-hard. They proposed a metaheuristic in the form of a genetic algorithm, which enables the deployment of heterogeneous sensors so as to maximize their area coverage in wireless sensor networks that collect and transfer environmental data from a predefined region to a base station. Their

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proposed genetic algorithm includes a heuristic population initialization procedure, the proposed exact integral area calculation for the fitness function, and a combination of Laplace Crossover and Arithmetic Crossover Method operators. For a comprehensive summarization and classification on the data fusion-based coverage optimization problem and techniques, the interested reader is referred to the work by Deng et al. [131].

6.5.2 Sensor Mobility Recently, significant efforts have been focused on studying coverage using mobile wireless sensor networks [30]. Indeed, sensor mobility has been recognized as an efficient way to guarantee better sensor deployment. In this regard, Liu et al. [270] proved that mobility of the sensors can be used to improve coverage. Their study showed that dynamic aspects of the coverage using mobile sensors depends on the process of sensor movement and that new stationary configurations that help improve coverage can be obtained once the sensors move to their desired locations. They characterized area coverage as a function of time and the detection time of a randomly located target. Wang et al. [394] proposed solutions to two related deployment problems in wireless sensor networks, namely sensor placement and sensor dispatch, in the presence of obstacles. For the first problem, they proposed a solution that considers arbitrary-shaped obstacles as well as arbitrary relationship between the communication and sensing ranges of the sensors. For the second problem, they proposed centralized and distributed approaches. While the centralized one exploits the results for sensor placement and converts the dispatch problem to the maximumweight maximum-matching problem whose objective function is minimizing the total energy consumption due to mobility or maximizing the average remaining energy of the sensors after movement, the distributed solution enables the sensors to compute their moving directions autonomously. Furthermore, Wang et al. [398] generalized their solutions to the sensor selection and dispatch problems [394] by considering multi-level coverage without obstacles. Yang and Cardei [428] dealt with the Movement-assisted Sensor Positioning (MSP) problem with a goal to increase the network lifetime. First, they proposed a solution to address the energy-hole problem caused around the sink by computing the desired non-uniform sensor density in the monitored area. Then, they proposed a centralized algorithm to relocate mobile sensors while satisfying the sensor density requirement with minimum cost. Wu and Yang [420] proposed a method, called Scan-based Movement-Assisted sensoR deploymenT (SMART), to achieve a balance state by balancing the workload of the sensors while avoiding the communication-hole problem in wireless sensor networks. SMART uses clustering, where a planar rectangular deployment field is partitioned into 2D mesh, which also partitioned into 1D arrays by rows and columns, and each square area is assigned a cluster-head. These rows and columns are scanned to determine the overload and underload in clusters so the load is shifted from overloaded clusters to underloaded clusters in order to achieve a balance state. Wu and Yang

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171

[419] took a step further and proposed an optimal but centralized approach based on the Hungarian method to minimize the total moving distance. They also proposed non-optimal distributed solutions for the same purpose based on the scan-based approach in [420]. Rao and Kesidis [334] investigated mobility in mission-oriented wireless sensor networks, where a sensor moves to a location so it can perform any one or all the tasks better, and hence the notion of purposeful mobility. With this type of controlled mobility, Cao et al. [99] proposed techniques for mobility assisted sensing and routing while considering the computation complexity, network connectivity, the energy consumption due to both mobility and communication, and the network lifetime. Wang et al. [393] addressed the problem of how to meet sensing coverage requirements using mobile sensors. They proposed a Grid-Quorum solution that locate the closest redundant sensors with low message complexity and relocate them in a timely, efficient and balanced way using cascaded movement. Also, they proposed a Voronoi diagram-based approach to detect coverage holes and three approaches, namely VEC (VECtor-based), VOR (VORnoi-based) and Minmax, each of which enables the sensors to move from densely deployed areas to sparsely deployed areas using less time, movement distance, and message complexity. Du and Lin [138] proposed an approach to improve the performance of wireless sensor networks in terms of coverage, connectivity, and routing by introducing a few mobile sensors in addition to the static ones, which constitute the majority of the sensors in the network. Moreover, they proposed several schemes for effective sensor mobility that helps achieve the above-mentioned goals. Wang et al. [391] proposed a proxybased approach that allows the sensors to move directly to their target locations and not a zig-zag way with a goal to provide satisfactory coverage. In harsh environments, such as battlefields, sensor mobility helps ensure the required coverage, where mobile sensors can reach areas that cannot be reached by static sensors. Heo and Varshney [197] proposed distributed, energy-efficient deployment algorithms that employ a synergetic combination of cluster-structuring and a peer-to-peer deployment scheme. Also, they proposed an energy-efficient Voronoi diagram-based deployment algorithm. Elhoseny et al. [147] proposed a model using a genetic algorithm for optimizing the coverage requirements in wireless sensor networks. The goal is to provide continuous monitoring of specified targets for longest possible time with limited energy resources, where the mobile sensors use a continuous and variable speed movement in order to keep all the targets covered all times. Liu et al. [278] proposed a virtual molecular force algorithm for node deployment and mobile coverage using mobile sensor networks. This algorithm assumes the existence of interactions among the sensors, which yield forces that drive the sensors to move to the corresponding locations to repair the monitoring of the blind area, so as to maximize the coverage of the network. Li and Liu [261] proposed a monitoring area coverage optimization algorithm based on perceived probability around nodes in order to solve the problem of poor coverage effect of existing algorithms. Also, they designed a reasonable scheme of sensor mobility, which helps gradually disperse the sensors and improve the coverage effect in the monitored area. Li et al. [262] focused on the problem of coverage maximizing mobile sensor deployment with directional and arbitrarily oriented sensors, and proposed two algorithms: Concurrent rotation

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and motion control (first algorithm) and staged rotation and motion control (second algorithm). While the first algorithm aims at reaching local maximum, the second one decouples rotation and motion controls in order to reduce the computation complexity with slight sacrifice in optimality. Kuo et al. [247] proposed a mechanism for adaptive trap coverage in area coverage, which relies on the cooperation of the mobile sensors to shrink or expend the trap. This adaptive trap coverage can be explored in an efficient way by the energy consumption of sensor mobility and the number of data communications among sensors. Etancelin et al. [154] presented a decentralized approach to build a connected dominating set coupled with attractive and repulsive forces for sensor mobility with a goal to achieve coverage and maximize its lifetime, while maintaining the network connectivity. In addition, they introduced a metric, called speed of coverage, which helps evaluate the balance between coverage and lifetime. Fang et al. [155] proposed two sensor deployment schemes in mobile wireless sensor networks in order to find the target locations for sensors to heal the coverage holes efficiently. While the first one is blind-zone centroid-based scheme, the second one is disturbed centroid-based scheme. In the first scheme, the Voronoi blind-zone polygon of each sensor is computed based on the positions of its neighbors, and its centroid is the target location of each sensor if the coverage can increase. According to the second scheme, the sensors find coverage holes according to the first scheme in each round. Then, they move to the target locations under the local perturbation and local reconstruction operators. Gorain and Mandal [176] investigated the sweep coverage problem, which aims to minimize the number of sensors required in order to guarantee sweep coverage for a given set of points of interest on a plane using both static and mobile sensors, while minimizing the energy consumption of the sensors. They proposed an 8-approximation algorithm to solve this NP-hard problem, and a 2-approximation algorithm to deal with a special case. Moreover, they focused on an energy restricted sweep coverage problem whose objective is to find the minimum number of mobile sensors to guarantee sweep coverage, while the energy consumption of a mobile sensor in a given time period is bounded. They proposed a (5 + 2/α)-approximation algorithm to solve this NP-hard problem, where α is a bounded integer. Younis and Akkaya [437] provided a comprehensive survey on node placement in wireless sensor networks. Also, for a comprehensive survey on coverage in mobile wireless sensor networks, the interested reader is referred to the work by Mohamed et al. [301].

6.5.3 Probabilistic Sensing Model Lazos and Poovendran [252] formulated the coverage problem in heterogeneous wireless sensor networks as a set intersection problem and derived analytical expressions, which quantify the coverage achieved by stochastic coverage. Megerian et al. [295] studied the exposure in wireless sensor networks, which is related to the quality of coverage provided by these networks, based on a general sensing model, where the

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sensing signal of a sensor at an arbitrary point by a function that is inversely proportional to the distance between the sensor and point. Liu and Towsley [276] studied three coverage measures, namely area coverage, node coverage, and detectability, using the general sensing model defined in [295]. Liu et al. [277] presented a joint scheduling scheme based on a randomized algorithm for providing statistical sensing coverage and guaranteed network connectivity. This scheme does not make any assumption on the relationship between sensing and transmission ranges, and works without the availability of per-node location information. Zou and Chakrabarty [460] proposed a distributed approach for the selection of active sensors to fully cover a planar field based on the concept of connected dominating set. This approach is based on a probabilistic sensing model, where the probability of the existence of a target is defined by an exponential function that represents the confidence level the received sensing signal. Wang and Tseng [398] proposed solutions to the k-coverage sensor deployment problem using both deterministic and probabilistic sensing models. These solutions compute the minimum number of sensors required to k-cover a planar field as well as their locations, and schedule the sensors to move to these locations. In the first solution, the sink computes those locations and the sensors bid for their closest locations. The second solution enables the sensors to derive the target locations by themselves. Xing et al. [425] extended their CCP protocol to provide probabilistic coverage guarantee based on a probabilistic coverage model, where the sensors may have non-uniform and irregular communication and sensing regions. According to this model, a point in a convex coverage area is guaranteed to be k-covered with a probability no lower than β. CCP provides probabilistic coverage via a mapping of the (k, β)-coverage requirement to a pseudo coverage degree k  , which is computed analytically. Megerian et al. [294, 296] combined computational geometry (Voronoi diagram) and graph theoretic techniques (graph search algorithms), to solve the best and worst coverage problems using optimal polynomial-time algorithms. Huang and Tseng [207] formulated the coverage problem as a decision problem, and solved it using polynomial-time algorithms in terms of the number of sensors. Kumar et al. [243] proposed a probabilistic approach to compute the minimum number of sensors to achieve k-coverage with high probability. They showed that this number is approximately the same for both deterministic and random sensor deployment, if the sensors fail or sleep independently with equal probability. Li et al. [263] proposed distributed algorithms to optimally solve the best-coverage problem with the least energy consumption. They considered a general sensing model, where the sensing ability of the sensors diminishes as the distance increases. Shakkottai et al. [348, 349] provided necessary and sufficient conditions for 1covered, 1-connected wireless sensor grid network. Also, they suggested various algorithms for connectivity and coverage in large wireless sensor networks. Liu et al. [277] proposed a scheduling scheme to provide statistical sensing coverage and network connectivity using a randomized algorithm. Yener et al. [436] proposed a probabilistic Markov model to ensure connectivity and coverage, while minimizing the sensors’ power consumption. Wang et al. [400] gave an overview of

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non-deterministic coverage problems (or coverage problems with uncertain properties), and provided a thorough summary of relevant models, such as detection models, network models, and deployment models. They analyzed the characteristics of partial or probabilistic coverage problems, and compared them with full coverage problems.

6.6 Conclusion In this chapter, first, we study the problem of energy-efficient connected k-coverage in mission-oriented mobile wireless sensor networks [30]. First, we exploit the characterization of k-coverage of a region of interest, which is introduced earlier in Chap. 5 and is based on the Helly’s Theorem [85] and the geometric properties of the Reuleaux triangle [481]. Then, we propose centralized and distributed approaches for achieving k-coverage of a region of interest using mobile sensors. In the centralized approach, the sink is responsible for slicing a region of interest into slices and designating a set of sensors to be moving to the selected lenses of that region in order to k-cover it. In the distributed approach, the sink sends only a query including the coordinates of a region of interest to be k-covered and its degree of coverage k while the sensors coordinate between themselves to ensure energy-efficient k-coverage of the region. However, to maintain network connectivity, we use a few sensors as data MULEs. With regard to our proposed k-coverage protocols, namely CAMSEL and DAMSEL, simulation results show that CAMSEL outperforms DAMSEL. Also, we find that DAMSEL in turn outperforms the k-coverage protocol, called Competition [398], in terms of the number of sensors and the total moving energy required to k-cover a region of interest. Second, we exploit the results of the homogeneous model, which are discussed earlier in Chap. 5, to solve the connected k-coverage problem for heterogeneous sensor nets, where the sensors do not have the same sensing range, communication range, and initial energy [60]. Precisely, we propose centralized and distributed protocols for generating energy-efficient connected k-coverage configurations in heterogeneous sensor nets. First, we consider a random deployment approach, where the sensors are distributed in a random fashion in the planar field. We show that while it is possible to design distributed protocols to guarantee connected k-coverage of a planar field using heterogeneous sensors, it is impossible that a centralized protocol could be designed efficiently due to deployment randomness and sensor heterogeneity. Thus, we suggest a pseudo-random deployment approach, where the sensors are deployed in different layers in a planar circular deployment field with respect to the sink according to their strengths with regard to sensing range, communication range, and initial energy, and propose energy-efficient centralized and distributed connected k-coverage protocols. Through simulations, we show that our proposed approach for connected k-coverage in homogeneous sensor nets outperforms an existing one in terms of the number of sensors necessary to fully k-cover a planar field, and network lifetime. Moreover, we find that our pseudo-random sensor deployment approach outperforms our random deployment approach with respect to the above-mentioned

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metrics. Indeed, our multi-tier sensor deployment architecture is energy efficient and can be exploited as a solid basis for the design of a geographic forwarding protocol to solve the energy sink-hole problem in static sensor nets. Third, we propose a distributed approach to solve the scheduling problem in stochastic k-covered wireless sensor networks [34], where the sensing ability of the sensors is represented by a probability function. Indeed, stochastic sensing models are more realistic than the deterministic sensing model, which does not capture the probabilistic nature of the sensors’ characteristics. Our methodology is based on a geometric analysis using the Reuleaux triangle model. For problem tractability, we consider a deterministic sensing model and then extend the analysis to a stochastic sensing model. First, we characterize k-coverage in wireless sensor networks and provide a necessary and sufficient condition to achieve k-coverage with a minimum number of sensors. Then, we present our k-coverage-preserving scheduling protocol (SCPk ) based on this characterization. Precisely, sensors activate themselves by running a k-coverage candidacy algorithm to ensure that their sensing ranges are k-covered. We find a good match between simulation and analytical results.

Chapter 7

Spatial Convexity Theory-Based Approaches for Connected k–Coverage

Throughout space there is energy. Is this energy static or kinetic? If static our hopes are in vain; if kinetic – and this we know it is, for certain – then it is a mere question of time when men will succeed in attaching their machinery to the very wheelwork of nature. Nikola Tesla (1856–1943)

Overview This chapter investigates the problem of connected k-coverage in homogeneous spatial wireless sensor networks while considering densely deployed static sensors and a deterministic sensing model. Precisely, it presents two localized (i.e., based on local information of one-hop neighbors), pseudo-distributed (i.e., not fully dis-tributed) approaches to achieve k-coverage of a spatial field with a reduced number of active sensors, while ensuring connectivity between them. All the sensors are scheduled (or duty-cycled) to save energy, thus, extending the network life-time, while satisfying connected k-coverage of a spatial field. Both approaches consider static and homogeneous sensors, while using a deterministic sensing model. While the first one is based on the concept of the equilateral spherical tri-angle, the second approach uses the Reuleaux tetrahedron. We find that the sec-ond approach outperforms the first one as it requires a smaller number of sensors to k-cover a spatial field, while ensuring network connectivity.

7.1 Introduction Until now, we believe that the theory of coverage in the Euclidean space is not sufficiently developed. In particular, the problem of k-coverage in spatial wireless sensor networks has not been adequately investigated. In fact, most existing work on coverage and connectivity considers the Euclidean plane, where the sensors are deployed in a planar field. However, there are several cases, where the two dimensions assumption is not valid for the design of those types of wireless sensor networks, such as underwater sensor deployment and also when the sensors are deployed on the trees of different heights in a forest. In this chapter, we investigate the problem © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_7

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of k-coverage in spatial wireless sensor networks, where each point in a spatial field is covered by at least k sensors simultaneously. The concept of dimensionality is of significant importance in the study of wellknown and interesting geometric research problems, such as k-coverage. Considering a real Euclidean space with two or three dimensions, it has been found that those problems are more tractable in the Euclidean plane, whereas their counterparts in the Euclidean space are challenging yet more practical. There are several challenging issues in the design of spatial wireless sensor networks, such as coverage and connectivity, to name a few, which received little attention compared to their counterparts in planar wireless sensor networks. In particular, the problem of sensor duty-cycling (or scheduling) in spatial wireless sensor networks, where the sensors are capable of switching on and off on an as-needed basis, has not been decently addressed in the literature. As discussed in Chap. 4, spatial settings reflect more accurately network design for real-world applications than their more traditional, planar counterparts. The assumption of two dimensions is well accepted by the sensor network community while that of three dimensions has been receiving little attention due to the challenges imposed by the design of spatial wireless sensor networks [9]. Indeed, most (if not all) of the work on the design of protocols for wireless sensor networks, and particularly, those on deployment, have focused on the Euclidean plane, where the sensors are deployed in a planar field. As mentioned by Poduri et al. [324], there is a tendency of ignoring the extension of protocols initially designed for planar to spatial wireless sensor networks either because it is simple or straightforward. Furthermore, Poduri et al. [324] showed that there are a few properties in planar wireless sensor networks that cannot generalize at all to spatial wireless sensor networks. In general, the design of connected k-coverage configuration protocols for wireless sensor networks in the Euclidean space is more challenging than their counterparts in the Euclidean plane. As indicated earlier in Chap. 2, we use the terms “plane” (respectively, “space”) to refer to the “Euclidean plane” (respectively, “Euclidean space”) in the discussion below. Our focus on the above problems is motivated by the following four main observations in the design, analysis, and development of spatial wireless sensor networks. These observations deal with the use of a planar model rather than a more general one, i.e., spatial model; redundant coverage; the deployment of always-on sensors instead of duty-cycled ones; and the extensibility of the analysis of problems related to the design of wireless sensor networks, such as k-coverage, from the Euclidean plane to the Euclidean space. Next, we discuss each of these issues in details. • Observation 7.1 (Planar Sensor Deployment): Most existing work on coverage and connectivity considers the Euclidean plane [39, 74, 210, 243, 389], where the sensors are deployed in a planar field. However, there are several cases where the two dimensions assumption is not valid for the design of wireless sensor networks. For instance, wireless sensor networks deployed on the trees of different heights in a forest, and in a building with multiple floors, require design in the space rather than in the plane. Also, it is well known that underwater and underground sensor

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networks [8] are spatial in nature and, thus, the planar model cannot be applied to the design of those two types of sensor networks. With regard to their geometric properties, structure-wise, the sensors are spatial objects, i.e., they are spatial and not planar objects. Now, concerning their physical properties, the sensing and communication ranges of the sensors have a spatial rather than a planar shape. Therefore, in general, those geometric and physical properties of the sensors make it more realistic to deploy them in the space. • Observation 7.2 (Redundant Coverage): For their efficacy, the monitoring systems for environments, such as underwater and forests, require high fault-tolerance given that the sensors may be physically damaged and/or lose their battery power. It is clear that k-coverage would be an appealing solution to provide the required level of fault tolerance for these types of environment monitoring systems. • Observation 7.3 (Always-On Sensor Deployment): It is commonly assumed in most of the work on the problem of geographic forwarding in wireless sensor networks that all the sensors are always active during the network operational lifetime, and, particularly, during data forwarding. However, this type of design is neither practical nor efficient for the sensors whose energy is crucial and limited. Thus, this assumption is not realistic for most sensing applications, where the sensors must be duty-cycled to save energy. Notice that the network size is application-dependent. Indeed, there exists a wide spectrum of spatial sensing applications that can be implemented only when the sensors are densely deployed in a spatial field of interest. For example, intruder detection and tracking is one of those sensing applications that require significant levels of data accuracy, and, thus, higher degree of coverage. Also, the sensors are generally randomly deployed because of the nature of the monitored environment. For instance, in harsh environments, such as battlefields, it is not always easy or even impossible to access the field in order to place the sensors at specific locations. However, randomness may create one of the two problems: Coverage holes, where some regions in the field are not covered, and connectivity holes, where at least one set of sensors is disconnected from the rest of the network. To cope with these two problems, the sensors should be densely deployed. Thus, because of the type of sensing application and/or the nature of the deployment field, a large number of sensors should be deployed. However, if all the sensors are continuously maintained active, they will definitely run into a battery power depletion problem and die quickly. To remedy to this problem, all the sensors must be duty-cycled to save their energy so they can remain operational for as long as possible, while satisfying the main sensing application’s requirement in terms of its degree of coverage. This sensor duty-cycling process will help reduce the total number of active sensors and, thus, prolong the network lifetime. Therefore, it is essential that sensor duty-cycling be considered in geographic forwarding in order to account for real-world scenarios and wireless sensor network platforms. • Observation 7.4 (Planar-to-Spatial Analysis Extensibility): It is always believed that the study and analysis of any research problem begin with special results and lead gradually up to generalities. However, as discussed in Sect. 7.3 below,

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the process of computing the sensor density to ensure k-coverage in spatial wireless sensor networks using the Reuleaux tetrahedron model is not a straightforward generalization of the process that is used earlier in Chap. 5 to compute the sensor density for planar k-covered wireless sensor networks [39]. Not surprisingly, in our study of the k-coverage problem in the space, we find that some interesting geometric properties that hold for the plane cannot generalize to the space. This makes the extension of the analysis of k-coverage from planar wireless sensor networks to spatial wireless sensor networks not straightforward. Particularly, we find that the Reuleaux tetrahedron model [480]—the basis of our kcoverage theory in the space—does not have the same features as the Reuleaux triangle model [481], which we have exploited to achieve k-coverage in the plane [39]. Indeed, it has been found that the Reuleaux triangle has a constant width [481], whereas its spatial counterpart, Reuleaux tetrahedron (or spherical tetrahedron), does not have a constant breadth [480]. Notice that the width (respectively, breadth) of a planar (respectively, spatial) closed convex region C is the maximum distance between parallel tangential lines (resp., tangential planes on opposing faces or edges) that bound C. Next, we present the major tasks we want to accomplish in this chapter. In addition, we briefly describe how to achieve each one of them.

7.1.1 Major Tasks In this chapter, we address the problem of connected k-coverage in spatial wireless sensor networks using a deterministic sensing model of the sensors, and focus on the problem of sensor scheduling for spatial k-coverage [22–24, 32, 53]. Not surprisingly, we show that the extension of our analysis of planar wireless sensor networks to spatial wireless sensor networks is really not straightforward due to the non-preserving nature of some of the properties for the Euclidean plane when we consider the Euclidean space. Specifically, we investigate the problem of k-coverage in duty-cycled spatial wireless sensor networks. Precisely, we consider a spatial deployment field, where each point is covered by at least k active sensors simultaneously while all the active sensors are connected to each other. Next, we describe these tasks and briefly present our corresponding plan of actions. First, we want to compute the spatial sensor density that is needed to k-cover a spatial field using densely deployed wireless sensor networks, where k ≥ 4 is a natural number that stands for the degree of coverage requested by a sensing application. We use the equilateral spherical triangle as a model to k-cover the sensing sphere of a sensor. Then, we compute the corresponding spatial sensor density to achieve k-coverage of a spatial field. Second, we want to determine the shape of the best tetrahedron variant and the number of overlapping tetrahedron variant of this shape that are needed to fill the sensing sphere of a sensor, where each tetrahedron variant has one vertex at the

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sphere’s center. We analyze the k-coverage problem and find out that the Reuleaux tetrahedron is the best variant that can be used to k-cover the sensors’ sensing sphere. Third, we aim to select sensors to remain active and k-cover a spatial field so their number does not exceed the one corresponding to the spatial density of active sensors, which is computed in the first (based on the equilateral spherical triangle model) and second (using the Reuleaux tetrahedron model) tasks. We propose two connected k-coverage approaches for spatial wireless sensor networks: The first approach is called distributed randomized connected k-coverage protocol (3DIRACCk ), while the second one is called, localized pseudo-distributed k-coverage protocol (LDCk ). Fourth, we want to assess the performance of the proposed 3DIRACCk and LDCk connected k-coverage protocols for spatial wireless sensor networks. We provide extensive simulation results of both protocols based on various performance metrics.

7.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 7.2 discusses the first approach to solve the problem of connected k-coverage in spatial wireless sensor networks. It also presents simulation results of our proposed protocol. Section 7.3 presents a second solution to the same problem and gives simulation results of our proposed protocol. Section 7.4 reviews related work. Section 7.5 concludes the chapter.

7.2 Equilateral Spherical Triangle-Based Approach While coverage in planar wireless sensor networks have been well studied, spatial wireless sensor networks have gained relatively less attention in the literature. In this section, analyze the k-coverage problem in spatial wireless sensor networks and propose a distributed k-coverage protocol. Then, we relax some widely used assumptions to promote the use of our k-coverage protocol in real-world scenarios. Finally, we evaluate the performance of our protocol. Our work is complementary to existing ones, especially those few works which dealt with spatial wireless sensor networks. First, we propose an energy-efficient k-coverage protocol for spatial wireless sensor networks. Indeed, Alam and Haas [12] considered only 1-coverage and proposed deterministic sensor placement strategies to achieve full coverage of the space. However, 1-coverage is not always enough given that sensors are not highly reliable and some applications, such intruder detection and tracking, require high coverage of a target spatial field. First, we show the problem that we encounter when we attempt to extend our analysis of connected k-coverage in planar to spatial wireless sensor networks. We refer to this problem as the curse of dimensionality, which is due to the fact that some

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properties that are valid for the plane cannot hold for the space. Then, we propose an energy-efficient connected k-coverage protocol for spatial wireless sensor networks.

7.2.1 Problem Analysis: The Curse of Dimensionality First, we state an optimized version of the space-coverage problem we want to solve. Then, we show the various issues that arise when analyzing this problem. Problem (Minimum spatial connected k-coverage): Let S be a set of sensors and k ≥ 4. We want to select a minimum subset S’ ⊆ S of sensors such that each point in a spatial field is k-covered and all selected sensors in S’ are connected. Lemma 7.1 states a sufficient condition for k-coverage of a spatial convex region. Lemma 7.1 (Breadth-Based Sufficient Condition for Spatial k-coverage): A spatial convex region C k is k-covered with exactly k sensors if its breadth does not exceed r, where r is the radius of the sensing spheres of the sensors. Proof We proceed using a proof by contradiction. Assume that the breadth of a spatial convex region C k is less than or equal to r and C k is not k-covered when exactly k sensors are deployed in it. Notice that each of these k sensors is located inside or on the boundary of C k . Thus, there must be at least one location l ∈ C k that is not k-covered. In other words, there is at least one sensor si located at ξ ’such that the Euclidean distance δ(ξ, l) > r, which contradicts our hypothesis that the breadth ∎ of C k is less than or equal to r. From Lemma 7.1, we can deduct that the deployment of k sensors in a spatial convex region C k cannot ensure that C k is k-covered if the breadth of C k exceeds r, where r is the radius of the sensing spheres of the sensors. To illustrate this claim, consider two points pi and pj in C k such that one sensor si is located at pi and δ (pi , pj ) = b > r, where b is the breadth of C k and δ(pi , pj ) is the Euclidean distance between pi and pj . Given that we have b > r, it is impossible for si to sense any event that occurs at pj . Thus, there is at least one sensor (i.e., si ) among those k sensors, which cannot cover pj . Thus, C k cannot be k-covered since its breadth b exceeds r (i.e., b > r). Next, we investigate the shape of a spatial convex region C k that is k-covered with exactly k sensors. To this end, as in the case of the Euclidean plane discussed earlier in Chap. 5, our approach is based on Helly’s Theorem [85], which is a fundamental result in the convexity theory. It characterizes the intersection of m convex sets in an n-dimensional space, where m ≥ n + 1. Helly’s Theorem (Intersection of Convex Sets) [85]: Let C be a family of convex sets in Rn such that for m ≥ n + 1 any m members of C have a non-empty intersection. Then the intersection of all members of C is not empty. ∎ Next, we give an instance of Helly’s Theorem in the space (i.e., n = 3).

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Instance of Helly’s Theorem (Necessary and Sufficient Condition for Spatial kCoverage): Let k ≥ 4. A spatial convex region C k is k-covered by k sensors if ∎ and only if C k is 4-covered (i.e., m = 4) by any four of those k sensors. Now, we analyze the above problem from the perspective of the shape of a spatial region C k in a spatial field corresponding to minimum k-coverage. In other words, we want to determine the shape of C k so that it is guaranteed to be k-covered when exactly k sensors are deployed in it. Clearly, the breadth of C k should be less than or equal to the radius r of the sensing spheres of sensors so that each location in C k is within the sensing spheres of these k sensors. Since our goal is to achieve k-coverage of a spatial field with a minimum number of sensors, the volume of C k should be maximum, and hence the breadth of C k must be equal to r. Therefore, our problem reduces to find the shape of this spatial region C k that has a constant breadth equal to r. Let C k be the intersection of k sensing spheres. Using Helly’s Theorem [85], the maximum volume of the intersection of these k sensing spheres is equal to that of four spheres since k ≥ 4. However, the maximum intersection (or overlap) volume of four sensing spheres such that the maximum distance between any pair of sensors is equal to r, corresponds to the configuration where the center of each sensing sphere is at distance r from the centers of all other three ones. In this configuration, the edges between the centers of these four spheres form a regular tetrahedron and the shape of their intersection volume is known as the Reuleaux tetrahedron [480] (Fig. 7.1). In [40, 47, 52], we use Helly’s Theorem [85] in our analysis of the k-coverage problem and exploited the geometric properties of the Reuleaux triangle to derive a sufficient condition to fully k-cover a planar field. Note that the Reuleaux triangle of side r, which represents the intersection of three symmetric, congruent disks of radius r, consists of a central regular triangle of side r and three curved regions. More importantly, it has a constant width equal to r [481]. We find that a Reuleaux triangle region of width r of a planar field is guaranteed to be k-covered with exactly k sensors, where r is the radius of the sensing range of the sensors [52]. Also, the regular triangle allows a perfect tiling of the plane. Based on this characterization,

Fig. 7.1 a Intersection of four spheres, b their Reuleaux tetrahedron (or c “Inflated” tetrahedron with 4 curved edges) [480]

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we design an energy-efficient k-coverage configuration protocol for planar wireless sensor networks [40, 47, 52]. Now, we provide some facts why the Reuleaux tetrahedron is not an appropriate solution to our minimum connected k-coverage problem. First of all, the Reuleaux tetrahedron does not have a constant breadth whose value is slightly larger than the radius r of the corresponding spheres [480]. In contrast to the regular triangle, the regular tetrahedron does not allow a perfect filling of the space (i.e., covering the space without gap or overlap). Indeed, Conway and Torquato showed that the dihedral angle of a regular tetrahedron is equal to 70.53°, which is not sub-multiple of 360° [123]. They also gave two arrangements of regular tetrahedra such that five regular tetrahedra packed around a common edge would result in a small gap of 7.36° as shown in Fig. 7.2 (left diagram), and that twenty regular tetrahedra packed around a common vertex yield gaps that amount to a solid angle of 1.54 steradians as shown in Fig. 7.2 (right diagram) [123]. This shows that some properties that hold for the plane are not valid for the space. Thus, the extension of the analysis of k-coverage in the plane [52] to the space is not straightforward, and hence another approach should be used. More precisely, we want to address the following question: What is the “closest shape” to the Reuleaux tetrahedron that will guarantee energy-efficient k-coverage of the space?

To address this question efficiently, we consider two halves of the sensing sphere of the sensors, i.e., the top half and bottom half. Note that in planar wireless sensor networks, we divide the sensing disk of the sensors into six overlapping Reuleaux triangles [52]. By analogy with the planar analysis given in [52], we divide each of the halves of a sensing sphere into six congruent spatial regions, called slices, each of which has three flat faces and one curved face representing an equilateral spherical triangle. Unfortunately, the distance between the point B at the top of a slice and all the points E on the edge of any spherical triangle is larger than r. Thus, a sensor located at B cannot cover any point E and a sensor located at any point E cannot

Fig. 7.2 Five regular tetrahedra about a common edge and twenty regular tetrahedra about a shared vertex [123]

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Fig. 7.3 Planar projection of a slice

cover B. Any sensor located in the region < A, C, D > is able to cover the whole slice as shown in Fig. 7.3. However, sensors located in the regions < A, B, C > and < A, D, E > cannot cover the entire slice. Thus, if k1 = k, then k2 = k3 = 0; otherwise, k2 = k3 = k − k1 . In other words, to guarantee k-coverage of a slice and hence a sensing sphere with a minimum number of sensors, it is necessary that active sensors should be located in the region < A, C, D >, thus efficiently solving the minimum connected k-coverage problem. Theorem 7.1, which follows from the above analysis, states a tight sufficient condition for k-coverage of a spatial field. Theorem 7.1 (Tight Sufficient Condition for Spatial k-Coverage): Let k > 1. A spatial field is k-covered if any slice of the field contains at least k active sensors. ∎ Theorem 7.2, which follows from Theorem 7.1, computes the spatial sensor density necessary to fully k-cover a spatial field. Theorem 7.2 (Spatial Sensor Density for Spatial k-Coverage): Let r be the radius of the sensing spheres of sensors and k > 1. The minimum spatial sensor density required to guarantee k-coverage of a spatial field is computed as λ(r, k) =

9k π r3

Proof The volume of a slice is vol(slice) = π r 3 /9. By Theorem 7.1, each slice should contain at least k sensors. Thus, k-covering a spatial field with a minimum number of sensors requires that every slice in the field contain exactly k sensors. Thus, the minimum spatial sensor density to k-cover a spatial field is equal to k/vol(slice). ∎

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Using Theorem 7.1, Theorem 7.3 states a sufficient condition to maintain connectivity in spatial k-covered wireless sensor networks. Theorem 7.3 (Sufficient Condition for Spatial Connected k-Coverage): Let k > 1. A spatial k-covered wireless sensor network is connected if R ≥ r , where r and R stand for the radii of the sensing and communication spheres of sensors, respectively. ∎

7.2.2 Distributed k-Coverage Protocol In this section, we describe our distributed randomized connected k-coverage protocol (3DIRACCk ) for spatial wireless sensor networks. First, we present our algorithm that enables a sensor to check its candidacy to turn active. k-Coverage-candidacy algorithm: A sensor turns active if its sensing sphere is not k-covered. Based on Lemma 5.2 in Chap. 5, a sensor randomly decomposes its sensing sphere into twelve slices of side r and checks whether each one of them contains at least k sensors. Each sensor should know the status of its sensing neighbors only to decide whether it is candidate to turn active or not. If any of the twelve slices does not have k active sensors, a sensor is a candidate to become active. Else, it is not. Figure 7.4 shows the pseudo-code of our k-Coverage-Candidacy algorithm. State Transition Diagram of 3DIRACC k : We use the same state transition diagram described earlier in Chap. 5 (Sect. 5.6.2, Fig. 5.16) for the Self-DIRACCk protocol. Fig. 7.4 k-Coverage-Candidacy algorithm

Algorithm 1: k-Coverage-Candidacy (* This code is run by each sensor *) Begin /* Sensing sphere slicing */ 1. Randomly decompose a sensing sphere into twelve slices /* Localized k-coverage candidacy checking */ 2. For each slice Do 3. If it contains k awake sensors Then 4. Skip /* i.e., do nothing */ 5. Else 6. Return (“candidate”) 7. End 8. End 9. Return (“non-candidate”) End

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7.2.3 Performance Evaluation We consider a cubic deployment field of side length 1000 m where all sensors are randomly and uniformly deployed. We use the same energy model used earlier in Chap. 5 (Sect. 5.8.1). All simulations are repeated 100 times and the results are averaged. Figure 7.5 plots the spatial sensor density λ(r, k) versus the coverage degree k, where the radius r of the sensing range of sensors is equal to 30 m. We observe a close to perfect match between our simulation and analytical results. Notice that λ(r, k) increases with k for a fixed r. Indeed, higher coverage degree of a spatial spatial field would require more active sensors. Figure 7.6 plots λ(r, k) versus the radius r of the sensing range of sensors, where the degree of coverage k is equal to 3. We observe that λ(r, k) decreases with r for a fixed k. In fact, sensors with larger sensing range would cover more areas and hence less number of active sensors is required to achieve a certain coverage degree k of a spatial field. Figure 7.7 plots λ(r, k) versus the radius R of the communication range of sensors for different values of the radius r of their sensing range, where the degree k of coverage is equal to 3. Notice that λ(r, k) does not increase with R. Indeed, as we find in Theorem 7.3, the spatial sensor density λ(r, k) of active sensors to k-cover a spatial field depends only on the radius r of the sensing range of sensors. Also, our k-coverage protocol 3DIRACCk is based on the sensing range of sensors in the sense that each sensor guarantees that its sensing range only is k-covered.

Fig. 7.5 λ(r, k) versus k

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Fig. 7.6 λ(r, k) versus r

Fig. 7.7 λ(r, k) versus R

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7.3 Reuleaux Tetrahedron-Based Approach Now, we investigate the use of the Reuleaux tetrahedron model to cover the space, while enabling overlap between adjacent Reuleaux tetrahedra. As a major curse of dimensionality, Conway and Torquato [123] found out that the regular tetrahedron cannot perfectly fill the space although its planar counterpart, the regular triangle, can be used to ensure a perfect tiling of the plane. We compute the number of Reuleaux tetrahedra of side length r 0 , denoted by τRT (r 0 ), to cover a sphere of radius r 0 while allowing overlap between them. Also, based on the above results, we present a localized, pseudo-distributed (i.e., not fully distributed) k-coverage protocol for spatial wireless sensor networks, where each point in a spatial field is k-covered, where k ≥ 4. Precisely, each sensor ensures that its sensing range is k-covered with a necessary number of active sensors. In addition, we compute the minimum ratio of the communication range R to the sensing range r of the sensors so that spatial k-coverage guarantees spatial network connectivity. Furthermore, we discuss how to generalize this approach by relaxing some of those widely used assumptions that we adopt in our study, such as sensor homogeneity model and sensing and communication sphere models. This helps promote the applicability of our proposed protocol to various sensor plate-forms and real-world sensing applications. We show that these relaxations are reasonable and yield good overall performance of our joint protocol for duty-cycled spatial wireless sensor networks. Finally, we provide a sample of simulation results of our k-coverage protocol in duty-cycled spatial wireless sensor networks in order to corroborate our theoretical analysis. We show a close to perfect match between our theoretical and simulation results.

7.3.1 Proposed Solution This work, which is an extension of our previous work in [32], takes into account kcoverage and sensor duty-cycling (or scheduling) in spatial wireless sensor networks. We provide an in-depth analysis of the use of the Reuleaux tetrahedron to solve the k-coverage problem in spatial wireless sensor networks. Precisely, we compute lower and upper bounds on the number of Reuleaux tetrahedra to k-cover a sensing sphere. Moreover, we discuss some relaxations of some widely used assumptions in the study of wireless sensor networks, with a goal to make the proposed study more general and realistic. Also, the work discussed in this chapter complements some existing approaches [12, 53]. Alam and Haas [12], which focused on the problem of 1-coverage and proposed deterministic sensor placement strategies to cover a spatial field. Ammari and Das [53] addressed the problem of k-coverage in spatial wireless sensor networks, where k > 1. However, this work targets wireless sensor networks that require denser coverage, i.e., k ≥ 4. Moreover, this work provides a fine analysis of the k-coverage problem with k ≥ 4, while using a smaller number of sensors compared to the result found in [53]. Liu and Ma [274] computed the spatial sensor

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density to cover a mountainous region using stochastic sensor deployment, where sensors are deployed on spatial rolling surfaces. As stated earlier, Zhang et al. [443] considered only 1-coverage, which is not always enough as sensors are not highly reliable. Moreover, there are various sensing applications, such intruder detection and tracking, which require high coverage of a target spatial field. We should emphasize that the nature of the problem of k-coverage in the space requires a dense sensor deployment, given that k ≥ 4. Otherwise, there may not be even enough number of sensors to fully k-cover a spatial field. In this case, we would not be able to meet the k-coverage requirement of the underlying sensing application. Therefore, it is important to satisfy the requirement that the sensors should be densely deployed in order to solve the k-coverage problem in spatial wireless sensor networks.

7.3.2 Problem Analysis In this section, we give an overview of the fundamental problem of covering the space with polyhedra. Then, we define and analyze an optimized variant of the spacecoverage problem, called k-coverage in spatial wireless sensor networks. Also, we present our pseudo-distributed spatial k-coverage protocol. There are several polyhedron types that vary by their geometry and mathematical properties. Platonic are most widely used convex polyhedral frameworks in spherical subdivision. Their edges are line segments of equal length between two neighboring vertices. The edges of their spherical counterparts are geodesic arcs of equal length. The platonic solids are regular polyhedra whose faces are of the same type, namely equilateral triangles, squares, or pentagons. The platonic solids are of particular interest to our study, and include the tetrahedron, octahedron, hexahedron, icosahedrons, and dodecahedron. The space-filling problem is to find out which solid can be used to fill the space without gap or overlap. Because of its geometric structure, it is trivial that the cube is a space-filler. Moreover, two or more different types of polyhedra can be used together to fill the space. Indeed, it was proved that a combination of regular octahedra and tetrahedra fill the space [407]. Precisely, six octahedra and eight tetrahedra can fill the space about a point that in a way can be extended indefinitely [407]. In addition, the truncated octahedron, the tetrahedron with beveled edges, and a combination of tetrahedra and truncated tetrahedra are the other space-fillers [175]. Also, the truncated cuboctahedra, truncated octahedra, and cubes in the ratio 1:1:3, can fill the space. In our study of the problem of k-coverage in spatial wireless sensor networks, we focus on a variant of the regular tetrahedron, called Reuleaux tetrahedron [480]. Our decision is based on the following three arguments: • In this chapter, we consider an optimized version of the space-coverage problem, which is described earlier. Specifically, we investigate the problem of covering the space using a minimum number of solids while allowing overlap between them.

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• The regular tetrahedron is the basic spatial convex solid as it has the minimum number of faces, i.e., only four faces. This makes the above-mentioned problem more tractable and less computationally intensive compared to other Platonic solids with more faces. Consequently, the regular tetrahedron yields more energy savings for the sensors whose battery power (or energy) is crucial and limited. Indeed, because of its four faces, the shape of the regular tetrahedron lends itself well to Helly’s Theorem [85]. This helps us quantify the spatial sensor density that is needed to k-cover the space. This will lead to minimize the energy consumption of the sensors, thus, extending their lifetime and that of the whole network. • Given that the regular tetrahedron cannot fill the space [123], we find that the Reuleaux tetrahedron has a nice geometric structure that can be exploited to solve the above problem efficiently. Indeed, the curved faces of adjacent Reuleaux tetrahedra can overlap, thus, coping with the shortcomings of the regular tetrahedron [123]. This problem is discussed in details in Sect. 7.3. Now, let us address the first query: What is the shape of a spatial convex area that is k-covered with exactly k sensors? Theorem 7.4 provides an answer to this query. Theorem 7.4 (Tight Condition for k-Coverage of Reuleaux tetrahedra): Let k ≥ 4. Any Reuleaux tetrahedron with a maximum breadth equal to r 0 = r/ 1.066 is guaranteed to be k-covered with exactly k sensors, where r stands for the radius of the sensing spheres of the sensors. Proof Let C k be the intersection of k sensing spheres and assume that their centers do not coincide in a spatial field. From Lemma 7.1, it follows that C k is guaranteed to be k-covered by exactly k sensors if its breadth does not exceed the radius r of the sensing spheres of sensors. Thus, the maximum volume of C k is obtained when its breadth is equal to r. From Helly’s Theorem [85], it follows that the intersection of k sensing spheres is not empty if the intersection of any four of these k spheres is not empty. On the other hand, the maximum intersection volume of these k sensing spheres is equal to that of four spheres provided that the maximum distance between any pair of these k sensors does not exceed r. Let us focus on the analysis of four sensing spheres. The maximum overlap volume of four sensing spheres such that every point in this overlap volume is 4-covered, corresponds to the following configuration: The center of each sensing sphere is at distance r from the centers of all other three sensing spheres. Precisely, the sensing sphere of each of the four sensors passes through the centers of the other three sensing spheres as shown in Fig. 7.1a [480]. The edges between the centers of these four spheres form a regular tetrahedron and the shape of the intersection volume of these four spheres is known as the Reuleaux tetrahedron (Fig. 7.1b [480]) and denoted by RelT (r). The latter is an “inflated” tetrahedron with four curved edges (Fig. 7.1c [480]). Notice that the configuration provided in this proof indicates the maximum intersection volume of the four sensing spheres, under the assumption that each of the four sensors is capable of sensing any event that may occur at any of the locations of the three other sensors (i.e., centers of their sensing spheres). In other words, we intend to show the “worst” case, where the four

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sensing spheres are separated from each other as much as possible (i.e., the distance between any pair of two spheres’ centers is equal to r, where r stands for the sensing range of the sensors), while any point in their maximum intersection would be fourcovered. Unfortunately, it was proved that the Reuleaux tetrahedron does not have a constant breadth [480]. Indeed, while the distance between some pairs of points on the boundary of the Reuleaux tetrahedron RelT (r) is equal to r, the maximum distance between other pairs of points on the boundary of RelT (r) is equal to 1.066 r [480]. This implies that the Reuleaux tetrahedron RelT (r) cannot be k-covered with k sensors as the distance between some pairs of points on the boundary of RelT (r) is larger than r. Thus, the Reuleaux tetrahedron that can be k-covered with exactly k ∎ sensors should have a side length equal to r 0 = r/1.066. It is worth mentioning that the word “maximum” refers to the space that could be sensed by the four sensors at the same time, while each sensor can still sense the locations of the other three sensors. Our goal is to maximize (not minimize) the space that could be k-covered simultaneously by the sensors. This would help us minimize the total number of sensors to k-cover a spatial region of interest (or volume). Given that the energy consumption depends on the number of deployed sensors, minimizing their number would minimize the energy consumption of the sensors. Consequently, this would help extend the operational network lifetime. Here, we briefly discuss the results found by Sommerville [361], Goldberg [175], and Conway and Torquato [123] with respect to filling the space (or covering the space without gap or overlap). It was believed for several centuries that the regular tetrahedron is one of the space-fillers among other regular solids, such as the cube. However, this was proved to be wrong [175, 361]. Sommerville [361] suggested four different tetrahedra that fill the space, while Goldberg [175] provided a survey of all known space-filling tetrahedra. In contrast to the regular triangle, the regular tetrahedron does not allow a perfect filling of the space as it is discussed above based on the findings of Conway and Torquato [123]. In the sequel, we want to find out how we can benefit from Theorem 7.4 and exploit the structure of the Reuleaux tetrahedron to k-cover a spatial field using a necessary number of sensors. We should mention that a fundamental question in sensor network coverage is the minimum number of required sensors to achieve a certain level of coverage. In this chapter, the analysis is conducted based on the Reuleaux tetrahedron model. However, under which spatial solid model we can obtain the optimal analytical results is still an open problem that requires more investigation and efforts from the research community.

7.3.3 Optimized Spatial k-Coverage Based on the results of Lemma 7.1 stated in Sect. 7.2.1 and Theorem 7.4 given in Sect. 7.3.2, the Reuleaux tetrahedron that is guaranteed to be k-covered with exactly k sensors should have a side length equal to r 0 = r/1.066. The volume of the Reuleaux

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tetrahedron RT (r 0 ) is given by [480] vol(r0 ) = (3

√

√ 2 − 49 π + 162 tan−1 ( 2)r03 /12 ≈ 0.422 r03

Thus, RT (r 0 ) is the maximum volume that can be k-covered by exactly k sensors, where k ≥ 4. We conclude that the maximum volume of C k , denoted by volmax (C k ), is equal to volmax (C k ) = 0.422r 0 3 . Given that volmax (C k ) has to contain k sensors to k-cover C k , we conclude that the spatial sensor density per unit volume, denoted by λ(r, k), which is required to fully k-cover a spatial field is computed as follows: λ(r, k) = k/volmax (Ck ) k λ(r, k) = 0.422 r03 where r 0 = r/1.066. It is worth emphasizing that the value of λ(r, k) is tight in the sense that it is minimum given that k sensors should be located within volmax (C k ), such that C k is guaranteed to be k-covered with exactly these k sensors. Also, notice that λ(r, k) depends only on the coverage degree k dictated by a sensing application and the radius r of the sensing range of sensors. Moreover, λ(r, k) increases with k. Indeed, high coverage degree k requires more sensors to be deployed, and hence a densely deployed wireless sensor network is necessary. Also, λ(r, k) decreases as r increases. When the sensing range gets larger, a fewer number of sensors is needed to fully kcover a spatial field. Both of these observations reflect the real behavior of the sensors. Thus, λ(r, k) does not depend on the size of the field as opposed to the result of the stochastic approach in [446]. As it can be seen, the above result considers an isolated spatial convex region. However, a spatial deployment field is, in general, larger than a Reuleaux tetrahedron RelT (r 0 ) with side length r 0 and whose volume is given by [480]: vol(RelT (r0 )) ≈ 0.422 r03 It is worth mentioning that it is impossible to tile a spatial deployment field with non-overlapping Reuleaux tetrahedra. In other words, because of their geometric structure, those Reuleaux tetrahedra must overlap in order to avoid any gap, thus, covering the entire spatial field. Theorem 7.5 investigates the maximum volume that can be k-covered with exactly k sensors. Theorem 7.5 (Maximum k-Covered Volume): The maximum volume that can be k-covered with exactly k sensors is equal to the volume of two adjacent Reuleaux tetrahedra of side r 0 , where r 0 = r/ 1.066 and k ≥ 4, with r being the radius of the sensing spheres of the sensors.

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Fig. 7.8 Two adjacent Reuleaux tetrahedra and their lens [480]

Proof Let r 0 = r/1.066 and k ≥ 4, where r is the radius of the sensing spheres of the sensors. Consider two Reuleaux tetrahedra of side r 0 that are adjacent. That is, their regular tetrahedra have one coinciding triangular face, as shown in Fig. 7.8. Thus, the overlap volume of two adjacent Reuleaux tetrahedra of side r 0 , called volumetric lens, is twice the volume of a curved region. By Theorem 7.4, a Reuleaux tetrahedron with maximum breadth r 0 = r/1.066 is guaranteed to be k-covered with k sensors. In particular, deploying k sensors in the volumetric lens of two adjacent Reuleaux tetrahedra can k-cover both of them. Thus, the maximum volume that can be k-covered with exactly k sensors is equal to the volume of two adjacent Reuleaux tetrahedra. ∎ We should emphasize that our result stated in Theorem 7.5 is based on the one given in Theorem 7.4. Moreover, our conclusion stated in Theorem 7.5 can be reached only when k sensors are deployed in and selected from the volumetric lens of two adjacent Reuleaux tetrahedra of side r 0 , where r 0 = r/1.066, r denotes the radius of the sensing spheres of the sensors, and k ≥ 4. That is, only in that case that those k sensors can k-cover both of those two adjacent Reuleaux tetrahedra. Thus, the maximum volume that can be k-covered by at least k sensors correspond to the volume of two adjacent Reuleaux tetrahedral of side r 0 = r/1.066. An optimal k-covering consists to use a minimum number of congruent Reuleaux tetrahedra by minimizing the overlap volume between them. Theorem 7.6 states a tight sufficient condition for k-coverage of a spatial field using the overlap between Reuleaux tetrahedra. Theorem 7.6 (Tight Sufficient Condition for Spatial k-Coverage): Let k ≥ 4 and r 0 = r/ 1.066, where r is the radius of the sensing spheres of the sensors. A spatial field is guaranteed to be k-covered if any volumetric lens of two adjacent Reuleaux tetrahedra of side r 0 in the field contains at least k sensors. Proof Let k ≥ 4 and r 0 = r/1.066, where r is the radius of the sensing spheres of the sensors. Consider two Reuleaux tetrahedra of side r 0 that are adjacent. By Theorem 7.5, deploying k sensors in the volumetric lens of two adjacent Reuleaux tetrahedra can k-cover both of them. If we apply this result to all volumetric lenses between all adjacent Reuleaux tetrahedra in the spatial deployment field, the entire field is k-covered. ∎

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Based on Theorems 7.5 and 7.6, Theorem 7.7 computes the spatial sensor density that is needed to k-cover a spatial field. Theorem 7.7 (Spatial Sensor Density for Spatial k-Coverage): Let k ≥ 4 and r 0 = r/ 1.066, with r being the radius of the sensing spheres of the sensors. The spatial sensor density that is required to k-cover a spatial field is given by ) ( λ(k, r0 ) = k/ 0.692 r03 Proof Let k ≥ 4 and r 0 = r/1.066, where r is the radius of the sensing spheres of the sensors. As stated above, the volume vol(RelT (r 0 )) of a Reuleaux tetrahedron RelT (r 0 ) of side length r 0 is equal to vol(RelT (r0 )) ≈ 0.422 r03 Also, the volume vol(RegT (r 0 )) of the regular tetrahedron RegT (r 0 ) associated with its Reuleaux tetrahedron RelT (r 0 ) is given by [367] √ vol(RegT (r0 )) = ( 2/12)r03 ≈ 0.117 r03 As shown in Fig. 7.1b, a Reuleaux tetrahedron RelT (r 0 ) consists of a regular tetrahedron RegT (r 0 ) and four curved regions, each of which denoted by CurReg(r 0 ). Thus, the volume vol(CurReg(r 0 )) of a curved region CurReg(r 0 ) is given by vol(Cur Reg(r0 )) = (vol(RelT (r0 )) − vol(RegT (r0 )))/4 Thus, we have: vol(Cur Reg(r0 )) ≈ 0.076 r03 As described above (Proof of Theorem 7.5), the volumetric lens of two adjacent Reuleaux tetrahedra of side r 0 as shown in Fig. 7.8, has twice the volume of a curved region. Thus, the volume vol(AdjRelT (r 0 )) of two adjacent Reuleaux tetrahedra of side r 0 is equal to vol( Ad j RelT (r0 )) = 2 vol(RelT (r0 )) − 2 vol(Cur Reg(r0 )) Thus, we obtain: vol( Ad j RelT (r0 )) ≈ 0.692 r03 Using Theorem 7.5, k sensors should be deployed in the volumetric lens of two adjacent Reuleaux tetrahedra to k-cover their above volume vol(AdjRelT (r 0 )). Using Theorem 7.6, the spatial sensor density that is necessary to k-cover a spatial field is equal to

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) ( λ(k, r0 ) = k/ 0.692 r03

∎

As it can be seen, based on the Reuleaux tetrahedron model, the spatial sensor density, λ(k, r 0 ), that is needed to k-cover the space is proportional to the coverage degree k and inversely proportional to the sensing range of the sensors r given that r 0 = r/1.066. It is clear that λ(k, r 0 ) improves on the spatial sensor density λ(k, r) = k / (0.422 r 3 ), which is computed in [41]. Indeed, we have λ(k, r 0 ) < λ(k, r). Next, Corollary 7.1 states a necessary condition to ensure connected k-coverage configurations in spatial wireless sensor networks. Corollary 7.1 (Necessary Condition for Spatial Connected k-Coverage): Let k ≥ 4 be the coverage degree. A spatial k-covered wireless sensor network is connected if the radii of the sensing and communication ranges of the sensors satisfy the relationship: R ≥ 1.066 r0 Proof As stated in the proof of Theorem 7.4, the maximum distance between pairs of points in a Reuleaux tetrahedron of side length r 0 corresponds to that between a pair of points on the boundary of the Reuleaux tetrahedron. Moreover, that distance ∎ [480] is found to be equal to 1.066 r 0 .

7.3.4 Using Reuleaux Tetrahedra for Sphere Coverage The major problem that should be addressed now is to find out the minimum number of Reuleaux tetrahedra that is necessary to cover a sphere whose radius is equal to r 0 = r/1.066. Our objective is to ensure k-coverage of this sphere while allowing overlap between adjacent Reuleaux tetrahedra. Recall that as stated by Conway and Torquato [123], it is impossible to pack a sphere (and the space, in general) with non-overlapping regular tetrahedra. However, in our study, we allow overlap between Reuleaux tetrahedra. This is a valid design decision for two reasons. First, the sensing sphere of a sensor is not solid. Second, our focus is on covering and not packing a sphere. To solve this sphere covering problem, we benefit from the four curved faces of the Reuleaux tetrahedron, which lie on the four planar faces of its corresponding regular tetrahedron. Our ultimate goal is to minimize the number of Reuleaux tetrahedra to cover a sphere of radius r 0 = r/1.066. It is worth noting that the problem of packing a sphere with regular tetrahedra is still an open problem that is being addressed in the literature. It was proved that the number of regular tetrahedra of side length 1 that may be packed into a sphere of radius 1 without overlap, except on the sides, is between 20 and 22, where each tetrahedron has one vertex at the center of the sphere [408]. Next, we focus on the problem of covering the space with Reuleaux tetrahedra and compute the lower and upper bounds on the number of Reuleaux tetrahedra necessary to cover a sphere. Given that

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the volume of the Reuleaux tetrahedron is larger than that of its corresponding regular tetrahedron, we expect both of these bounds to be smaller than their counterparts for the regular tetrahedron, i.e., 20 and 22. Theorem 7.8 (Number of Reuleaux Tetrahedra for Sphere Coverage): Let τ RT (r 0 ) be the number of Reuleaux tetrahedra of side length r 0 to cover a sphere of radius r 0 while allowing overlap between adjacent Reuleaux tetrahedra. We have the following lower and upper bounds for τ RT (r 0 ): 10 ≤ τ RT (r0 ) ≤ 18 Proof A lower bound for τRT (r 0 ), denoted by τ–RT (r 0 ), can be obtained by dividing the volume of a sphere of radius r 0 , i.e., (4/3) π r 0 3 , by the volume of a Reuleaux tetrahedra of side length r 0 , which is given by 0.422 r 0 3 . Thus, we get τ−

RT (r 0 )

= (4/3)π r03 /0.422 r03 = 9.926 ≈ 10

Now, to compute an upper bound for τRT (r 0 ), denoted by τ + RT (r 0 ), we consider a Reuleaux tetrahedra deprived of its three curved faces, called augmented regular tetrahedron (i.e., the corresponding regular tetrahedron augmented with only one curved face that lies on one of its four triangular faces). We want to compute the number of augmented regular tetrahedra to fill the sphere, such that each of those augmented regular tetrahedra has one vertex at the sphere’s center. To this end, we propose to compute the number of times the area of this unique curved face of an augmented regular tetrahedron, denoted by ηRT (r 0 ), which can be tiled on the surface of the sphere of radius r 0 . The value of τ + RT (r 0 ) should be equal to this number. Thus, the upper bound τ + RT (r 0 ) is defined as the ratio of the area of a sphere of radius r 0 , which is equal to 4 π r 0 2 , to the area of the curved face ηRT (r 0 ) of an augmented regular tetrahedron of side length r 0 as shown in Fig. 7.9. It is easy to check that this area ηRT (r 0 ) is computed as follows [480]: ) ( η RT (r0 ) = 2π − (9/2) cos−1 (1/3) r02 First, we compute the number of Reuleaux tetrahedra of side r 0 , denoted by nRT (r 0 ), which is necessary to cover half a sphere (or hemisphere). We find: n RT (r0 ) = 2 π r02 /η RT (r0 ) = 8.447 ≈ 9 Thus, to cover a sphere (two hemispheres), we obtain: τ+

RT (r 0 )

= 2 × n RT (r0 ) = 2 × 9 = 18

∎

As it is expected, both of these lower and upper bounds (10 and 18, respectively) for sphere-coverage with Reuleaux tetrahedra are smaller than their counterparts (20 and 22, respectively) for sphere-coverage with regular tetrahedra. Our computation

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Fig. 7.9 Area of a curved face of a Reuleaux tetrahedron [480]

process of the lower and upper bounds, τ–RT (r 0 ) and τ + RT (r 0 ), respectively, of τRT (r 0 ) considers two extremes cases. Indeed, the value of τ–RT (r 0 ) takes into account the entire volume of the Reuleaux tetrahedron of side length r 0 , while that of τ + RT (r 0 ) accounts for the volume of the augmented regular tetrahedron of side length r 0 . We conclude that there are at least 10 Reuleaux tetrahedra that pack a sphere without overlap. However, there are at most 18 Reuleaux tetrahedra that can cover a sphere with overlap. In the latter case, we implicitly consider certain overlap between adjacent Reuleaux tetrahedra with respect to the three curved faces out of their four curved faces. This is in line with our optimized k-coverage approach of a spatial field, which is discussed in Sect. 7.3 and stated in Theorem 7.6.

7.3.5 Reuleaux Tetrahedron-Based Spatial k-Coverage In this section, we discuss our localized, pseudo-distributed k-coverage (LDCk ) protocol for spatial wireless sensor networks using the results of Theorems 7.6 and 7.8. First, we present our algorithm that enables a sensor to check its candidacy to remain or become on (or active).

7.3 Reuleaux Tetrahedron-Based Approach Fig. 7.10 k-Coverage-Candidacy algorithm

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Algorithm: k-Coverage-Candidacy /* This code is run by each sensor si in every round */ Begin /* Sensing sphere decomposition */ 1. By Theorem 7.8, si randomly decomposes its sensing sphere of radius r0 into eighteen slices /* Localized k-coverage candidacy checking */ 2. For each pair of adjacent slices Do 3. 4. 5. 6.

If their volumetric lens has k active sensors Then By Theorem 7.6, Skip /* i.e., do nothing */ Else Return (“Candidate”)

7. Return (“Non-candidate”) End

7.3.5.1

k-Coverage Candidacy Algorithm

We assume that the network operational lifetime is divided into rounds of equal time interval. Let us analyze the behavior of a sensor in a given round. The ideal case is to have disjoint subsets of active sensors for all rounds. However, this is not always possible due to the sensor random deployment, and also the randomness in the selection of the sensors to be active. As it can be seen in the pseudo-code of our k-Coverage Candidacy algorithm, which is given in Fig. 7.10, a sensor si decides to remain or become active if its sensing sphere is not yet k-covered. By Theorem 7.8, si randomly divides its sensing sphere into eighteen Reuleaux tetrahedra of side r 0 = r/1.066, where r is the radius of the sensing range of the sensors. In other words, based on our discussion in the proof of Theorem 7.4, we restrict the radius of the sensing range of the sensors to r 0 = r/1.066, instead of r. From now on, we assume that r 0 is the actual radius of the sensing range of the sensors. Then, it checks whether each of the volumetric lenses has at least k sensors. If so, si is not a candidate to remain or turn itself active. Otherwise, si is a candidate to remain or become active. From now on, a Reuleaux tetrahedra of side r 0 = r/1.066 is referred to as slice.

7.3.5.2

Spatial Localized k-Coverage Protocol (LDCk )

Now, we discuss our proposed k-coverage protocol for spatial wireless sensor networks in a duty-cycled model. In our spatial localized, pseudo-distributed kcoverage (LDCk ) protocol, we assume that all the sensors are active at the start of each round so they can receive messages from the sink as well as other sensors. In particular, we assume that each sensor (including the sink) is aware of the locations of the sensors in its neighborhood. This can be achieved through an exchange of messages

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among the sensors at the beginning of the sensor deployment. A typical message would include a sensor’s id along with its physical location. The sensors should be able to obtain their physical locations through a GPS or some localization technique [216]. This location awareness is essential for the working of our proposed 3D-kCovComFor framework. Next, we describe the state transition diagram associated with our LDCk protocol. As stated earlier, LDCk is a pseudo-distributed protocol. Indeed, the sink will have to intervene only once at the start of the sensor deployment to trigger the k-coverage process. This is the reason why our spatial k-coverage protocol is not fully distributed. We assume that at the beginning of sensor deployment to k-cover a spatial field (i.e., first round only), the sink randomly designates q sensors, known as INITIATORS, to initiate the k-coverage process as discussed above. To this end, the sink broadcasts a TRIGGER packet that includes the id’s of these designated sensors. That is, TRIGGER = , where trig indicates the type of message (i.e., TRIGGER), id 0 is the sink’s id, and […, id u , id v , id w , …] is a list of id’s of the sensors selected as INITIATORS. Upon receiving a TRIGGER packet, a sensor si checks whether it is selected by the sink as an INITIATOR. If not, si simply forwards the TRIGGER packet. Otherwise, si runs the following three-step procedure. First, si removes its id from the TRIGGER packet and forwards it only if it has more id’s. Second, based on Theorem 7.8, si decomposes its sensing sphere of radius r 0 , denoted by SS(r 0 ), into eighteen slices. Third, based on Theorem 7.6 stated above, si activates the necessary number of sensors from the volumetric lenses to ensure k-coverage of SS(r 0 ). To this end, si builds an ACTIVE packet that includes a list of id’s of the sensors to become active. That is, ACTIVE = , where act represents the type of message (i.e., ACTIVE), id i is the id of sensor si , and […, id r , id s , id t , …] is a list of id’s of the sensors selected to be active. Then, it forwards it to its one-hop sensing neighbors, using time-to-live set to 1. When a sensing neighbor sj of si receives an ACTIVE packet, it checks whether it has been selected by si to become active. If so, sj repeats the same procedure, which is run by si earlier, to k-cover its sensing range of radius r 0 as given in Theorem 7.4. Otherwise, sj just drops the ACTIVE packet. This process continues until the whole spatial field is k-covered. We distinguish three states of a sensor, namely Com-Sens, No-Com-Sens, and Listen, which we describe next. • Com-Sens state: In this state, a sensor is capable of communicating with other sensors and sensing a spatial field. A sensor si remains in this state for time t 2 that is at least t round and at most 2 × t round without interruption, where t round is the time of one round of k-coverage. • No-Com-Sens state: Any sensor that is in this state should be idle. That is, it cannot communicate with any other sensor in the network nor sense a spatial field. Therefore, its radio is off. In order to avoid a forever idle state, a sensor is allowed to switch to the Listen state, which is described below, after a certain

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time interval to check whether it is a candidate for k-coverage by running the kCoverage-Candidacy algorithm described earlier. Notice that a sensor si remains in this state for some time t 3 that is at most equal to t round without interruption. • Listen state: This state enables a sensor to listen to messages from other sensors. Also, a sensor checks its k-coverage candidacy whether it needs to switch to the Com-Sens state. We assume that a sensor remains in the Listen state for a listening time period t 1 = t listen . We assume that all the sensors are in the Listen state at the beginning of their deployment in a spatial field. While in this state, a sensor decides randomly and independently of all other sensors, when to run the k-Coverage-Candidacy algorithm. This randomness helps avoid that all the sensors check their candidacy at the same time, which may yield an over k-coverage problem, where some regions in the field are more than k-covered. Figure 7.11 shows the corresponding state transition diagram of a sensor si . Next, we describe all those state transitions. • Listen state to Com-Sens state transition: A sensor si undergoes this transition if one of the three following situations arises. First, when the sink designates si as one of the INITIATORS to initiate the k-coverage process. Second, if si is selected by an already active sensor sj to k-cover sj ’s sensing range. Third, when sj runs the k-Coverage Candidacy algorithm, and finds out that it should remain active. In the first two cases, si must receive a TRIGGER packet (during the first round) or an ACTIVE packet (during the second round and subsequent ones) before the expiration of t listen . If this transition occurs, si broadcasts an I-AM-ACTIVE packet, which includes its id, to its sensing neighbors and moves to the Com-Sens state. That is, I-AM-ACTIVE = , where iam-act stands for the type of message (i.e., I-AM-ACTIVE) and id i is sensor si ’id. Fig. 7.11 State transition diagram of LDCk protocol

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• Listen state to No-Com-Sens state transition: This transition occurs during the first round when si has not been designated by the sink in its TRIGGER packet to start the k-coverage process of the spatial deployment field. Also, this transition happens if si is not selected by any one of its sensing neighbors to k-cover their sensing ranges, or when it finds out that it is not a candidate for k-coverage after running the k-Coverage Candidacy algorithm. Thus, si waits for some t listen time. If si does not receive any TRIGGER packet or any ACTIVE packet, it moves from the Listen state to the No-Com-Sens state. • No-Com-Sens state to Listen state transition: While it is inactive, si switches to the Listen state and waits for t listen to receive ACTIVE packets from its sensing neighbors. If so, it sends out an I-AM-ACTIVE packet including its id to its sensing neighbors, and moves to the Com-Sens state, i.e., I-AM-ACTIVE = , where id i is the id of sensor si . Otherwise, it goes back to its No-Com-Sens state. • Com-Sens state to No-Com-Sens state transition: For fairness among all the sensors, each sensor cannot be in the Com-Sens state for more than two consecutive rounds. In case si has been active for two rounds successively, at the end of second round, it broadcasts an I-AM-INACTIVE packet that includes its id to its sensing neighbors, and moves to the No-Com-Sens state. That is, I-AMNACTIVE = , where iam-inact stands for the type of message (i.e., I-AM-INACTIVE) and id i is sensor si ’id. Next, we provide a proof that each point in a spatial field is at least k-covered after our LDCk protocol is executed. Recall that at the start of each round, we assume that all the sensors are active so they are able to receive messages from the sink and their neighboring sensors. Our proof is based on the state transitions a sensor may undergo in its function during its lifetime and the network operation. Theorem 7.9 (LDC k k-Coverage Protocol): Our spatial localized, pseudodistributed k-coverage (LDC k ) protocol ensures that each point in a spatial field is at least k-covered. Proof We proceed using a proof by cases based on the above-mentioned state transitions. • Case 1: State transition from Listen to Com-Sens First, at the beginning, q sensors are selected and designated by the sink as INITIATORS to initiate the k-coverage process. Each of these sensors would select some of their neighbors to remain active and participate in k-covering their sensing range whose actual radius is r 0 . Consequently, each of these INITIATORS would ensure that any point in its sensing range is k-covered. Second, when a sensor is selected either by an INITIATOR or one of its neighboring sensors to k-cover its sensing range, it would also check whether its sensing range is k-covered. If not, it would select some of its neighbors to k-cover it. Third, when a sensor runs the k-Coverage Candidacy algorithm, it would ensure that its sensing range is k-covered by activating the required number of sensors. As it can be seen from

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these three scenarios, every point in the sensing range of each of these sensors is k-covered. Thus, each point in the spatial field is at least k-covered. • Case 2: State transition from Listen to No-Com-Sens A sensor would undergo this type of state transition only when it is not needed to k-cover its sensing range or the sensing range of any of its neighbors. In other words, all the sensors in the network in such a situation would take the same state transition when the sensing ranges of their neighbors are k-covered. This implies that each point in the spatial field is guaranteed to be at least k-covered. Moreover, this ensures that k-coverage of the whole field is achieved using the necessary number of active sensors, which helps extend the network lifetime. • Case 3: State transition from No-Com-Sens to Listen First, whenever a sensor moves to the Com-Sens state, it has been selected to k-cover the sensing range of one of its neighbors. Under the assumption that all the sensors are willing to cooperate with the rest of the sensor in the network, this would ensure that each sensor’s neighbor’s sensing range is k-covered. Second, if a sensor moves to the No-Com-Sens state, it implies that none of the sensing ranges of its neighbors is less than k-covered. In either case, one can conclude that each point in the spatial field is at least k-covered. • Case 4: State transition from Com-Sens to No-Com-Sens Owing to these I-AM-INACTIVE messages, every sensor becomes aware of its neighbors that would be willing to be active in any round. This would allow sensors to avoid sensors that have been active for two consecutive rounds and select other ones to k-cover their sensing ranges. In any case, given that the sensors are densely deployed and that each sensor randomly slices its sensing range of radius r 0 into 18 slices (i.e., Reuleaux tetrahedra), each sensor would be able to select the appropriate sensors to remain active in any given round and k-cover its sensing range. Thus, each point in the spatial field is at least k-covered. Regardless of the state transition any sensor may undergo during its operation, each point in a spatial field is guaranteed to be at least k-covered using our LDCk protocol. ∎

7.3.6 Assumption Relaxation Here, we relax some widely used assumptions, which are stated earlier in our Default Network Model in Chap. 2 (Sect. 2.7) and discussed below. These relaxations help generalize our LDCk protocol and promote its use in real-world wireless sensor network plate-forms. Also, we discuss the impact of relaxing the other assumptions, namely time synchronization among the sensors, sensors’ location information, and communication reliability among the sensors.

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Sensing Orthogonality

In this section, we consider non-homogeneous sensors with different spherical sensing and spherical communication ranges. In this case, each sensor accounts for a sensing neighbor with the smallest sensing sphere. In other words, each sensor reduces its sensing sphere to that of its sensing neighbor that has the smallest radius, say rmin . Then, it decomposes its sensing sphere into eighteen Reuleaux tetrahedra of side rmin to check its candidacy to turn active. All other processing remains unchanged. It was proved that heterogeneity helps improve the wireless sensor networks performance. In Sect. 7.3.7, we study the impact of using heterogeneous sensors on our LDCk protocol.

7.3.6.2

Sensing Asymmetry and Convexity

In this section, we consider a convex model, where the sensing and communication ranges of the sensors are convex, but not necessarily spherical. Indeed, previous work found that the communication range of MICA motes is asymmetric and environmentdependent [451] while others found that the communication range is probabilistic and irregular [456]. Each of those non-spherical sensing ranges of the sensors for the experiments given in Sect. 7.3.7 is generated by two random numbers associated with its minimum and maximum breadth, respectively. Here, we assume a convex and heterogeneous model, where the sensing range of each sensor is a closed convex volume and those sensing ranges are not necessarily identical. In order to apply our Reuleaux tetrahedron-based k-coverage approach, we propose that each sensor reduce its sensing range to the largest enclosed sphere. The latter is a sphere that lays inside the sensor’s convex sensing volume and whose diameter is equal to the minimum distance between any pair of points on the boundary of this volume. Thus, each sensor uses its largest enclosed sphere as its actual sensing range when checking its candidacy to turn itself active. The rest of the steps of our LDCk protocol remain the same. In Sect. 7.3.7, we show some simulation results about this relaxation. It is worth noting that several sensors may not have a convex sensing range. This is due to a few factors, such as antenna direction, obstacles, and interference. Here, also, the above relaxation approach could be still applied to extract the largest enclosed sphere in this non-convex (i.e., irregular) sensing range. However, given the irregularity of the sensors’ sensing range, we may end up using more than enough number of sensors to achieve k-coverage of a spatial deployment field. While the relaxation approach described earlier is still applicable, it may not be energy efficient when the sensing range of the sensors is randomly irregular.

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Perfect Time Synchronization

In our study, we assume a perfect time synchronization among the sensors given that each sensor should know the duty cycles of others. However, it is well known that perfect time synchronization is unrealistic. If we relax this assumption, a sensor may not be able to know the duty cycles of its neighbors. In order to have such knowledge about their neighbors, the sensors have to broadcast their schedules either at the beginning of their deployment, in case they have a fixed schedule, or whenever they have a change in their duty cycle schedule, if it is variable. As it can be seen, either scheme would introduce a considerable amount of overhead due to schedule communication among the sensors. Moreover, this process would cause delay, which may have significant impact on the k-coverage process, especially, for those sensing applications that have hard deadlines, which should be met.

7.3.6.4

Location Information Accuracy

Our proposed approach assumes an accurate location information of the sensors. However, even with a GPS, there is no perfect location information. Consequently, inaccurate locations would affect the working of all proposed protocols. However, the inaccuracy introduced by a GPS is very small and almost negligible [216], and, thus, can be tolerated by our proposed protocols. Also, given that the sensors would be chosen from volumetric lenses of adjacent Reuleaux tetrahedra, the latter are large enough as the sensing range of the sensors is a few tens of meters. Therefore, even though some volumetric lenses may be deviated from their accurate locations, all the sensors selected from those lenses would be able to help k-cover the whole spatial deployment field. In some cases, certain sensors may not located in the sensing range of a given sensor si due to this problem of location accuracy, but may be selected by si . Similarly, other sensors may belong to the sensing range of si , but may not be selected by si . This is true for those sensors whose locations are closer to the boundary of the sensing range of si . As it can be seen, this problem of location accuracy may have impact on the efficiency of k-coverage. In order to ensure kcoverage of its sensing range, it is essential that si wait for a reply from the sensors it selected to make sure that they would participate to the k-coverage process. This type of “hand shaking” activity would introduce some delay, which may have effect on those sensing applications with hard deadlines. Furthermore, positional errors may have an impact on the routing of data and control packets. However, given that our 3D-kCov-ComFor framework uses a hybrid method based on both deterministic and opportunistic forwarding schemes, those packets will always be received by some sensors. This will help ensure delivery of those packets to their final destinations.

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Communication Reliability

In this work, we suppose that all the types of communications, including the transmission and reception of sensed data packets, control packets, and acknowledgments, are reliable. However, in real-case scenarios, some of those communications may fail. In order to recover those data and control packets lost, it is necessary to use a “hand shaking” activity, where sensors acknowledge data and control packet transmissions. This may result in additional transmissions to ensure that all of the data and control packets have been received successfully. These data and control retransmissions would cause more overhead, which require more energy consumption, and incur additional delay. Clearly, all these have an impact on the whole k-coverage process. In Sect. 7.3.7, we provide simulation results with respect to the first two relaxations. We plan to address all the issues with the other relaxations, which are discussed in Sects. 7.3.6.3, 7.3.6.4, and 7.3.6.5, in our future work.

7.3.7 Simulation Results In this work, we provide a rigorous analysis of the problem of joint k-coverage and geographic forwarding in duty-cycled spatial wireless sensor networks. With respect to geographic forwarding, we find that the closest work to ours is related to ExOR [84] and GeRaF [458, 459] although they were mainly proposed for planar wireless sensor networks. Given that ExOR is also an opportunistic routing protocol, we provide a performance comparison of our unified framework against our proposed k-coverage protocol combined with ExOR routing protocol, which is denoted by CovExORk . However, with respect to coverage, we cannot provide a fair comparison between our approach and the ones in [12, 274], which considered 1-coverage only, while our proposed algorithms deal with a degree of coverage that is at least equal to 4 (i.e., k ≥ 4). In this section, we evaluate the performance of our framework with a high-level simulator written in the C language.

7.3.7.1

Simulation Setup

We consider a cubic field of side length 300 m and 10,000 deployed sensors, where all the sensors are placed using a random deployment. Moreover, we assume that the sensors are densely deployed in this cubic field. Therefore, it is always possible to find a k-cover for every deployment. We leave the case of sparse sensor deployment as one of our future work. In such a deployment scheme, the probability of not finding a k-cover for some sensor deployments will be a non-zero value. We will investigate this problem in depth in the future. We use the energy model used in [435], where the energy consumption in transmission, reception, idle, and sleep modes are 60 mW, 12 mW, 12 mW, and 0.03 mW, respectively. Following [435], one unit of energy

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is defined as the energy required for a sensor to stay idle for 1 s. We assume that the initial energy of each sensor is 60 J enabling a sensor to operate about 5000 s in reception/idle modes. All simulations are repeated 100 times and the results are averaged.

7.3.7.2

Simulation Results

In Fig. 7.12, we plot the spatial sensor density λ(r, k) as a function of the degree of coverage k while considering a radius the sensing range of sensors r = 30 m. As it can be seen, there is a close to perfect match between the results of simulations and those analytical ones. Not surprisingly, the density λ(r, k) increases linearly with k given that r is fixed. This is in line with the result of Theorem 7.7 stated in Sect. 7.3.3. In fact, more sensors should be deployed to achieve a higher coverage degree k of the cubic field. As it can be observed from Fig. 7.12, the spatial sensor deployment density for k = 4 and r = 30 is equal to 2.5 × 10–4 , and the corresponding number of active sensors is 6750 in a cubic volume of 27 × 106 m3 . Recall that the side length of the cubic field is 300 m. The plot in Fig. 7.13 shows the impact of the radius r of the sensors’ sensing range on the spatial sensor density λ(r, k), where the coverage degree is k = 4. As it is expected, for a fixed value of k, we find that λ(r, k) decreases as r increases. Similarly, there is an expected behavior given that λ(r, k) is inversely proportional to r as stated in Theorem 7.7. In fact, as the sensors’ sensing ranges increase, the

Fig. 7.12 λ(r,k) versus k

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Fig. 7.13 λ(r,k) versus r

sensors are able to cover larger areas in the monitored field. Thus, a lesser number of sensors is needed to ensure k-coverage of the whole cubic deployment field. In Fig. 7.14, we plot the spatial sensor density λ(r, k) versus the radius R of the communication range of sensors. Indeed, this figure summarizes the results of several experiments, each of which considered a given value of the communication range of the sensors R and three values of their sensing ranges r. That is, in these experiments, the degree of coverage k is fixed to 4 whereas the radius of the sensing range r of the sensors varies from 20 to 30 m, while their communication range is between 30 and 210 m. As shown in Theorem 7.7, the spatial sensor density λ(r, k) is independent of R. It depends only on the radius r of the sensing range of sensors and the degree of coverage k. Through this experiment, we want to show that the resulting spatial density of active sensors is not affected in any way by their communication range R. Furthermore, our k-coverage protocol accounts for the sensing range of the sensors to meet any sensing application coverage requirement. We observe conformity between the results of the plot in Fig. 7.14 and our theoretical results. The plots in Figs. 7.15 and 7.16 show the impact of sensor heterogeneity and sensing range irregularity on our unified framework for joint k-coverage and composite forwarding in duty-cycled spatial wireless sensor networks. The plot in Fig. 7.15 shows the impact of deploying sensors, which are not necessarily homogeneous. Here, we assume that the sensing range of the sensors is between r min = 20 m and r max = 40 m with mean value equal to 30 m, and the initial energy of each sensor is between 40 and 80 J with mean value is equal to 60 J. Notice that our k-coverage protocol using heterogeneous sensors, denoted by Het-LDCk , requires less number of sensors than the one using homogeneous sensors, denoted by Hom-LDCk . Indeed,

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Fig. 7.14 λ(r, k) versus R

Fig. 7.15 Impact of sensor heterogeneity

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Fig. 7.16 Impact of sensing range irregularity

the presence of more powerful sensors helps achieve k-coverage with less number of active sensors. In Fig. 7.16, we study the impact of the irregularity of the sensing range of the sensors whose minimum breadth is 20 m and maximum breadth does not exceed 40 m. We find that our k-coverage protocol using irregular sensing ranges, denoted by Irg-LDCk , needs higher number of sensors than Hom-LDCk , which uses identical, spherical sensing range with radius is r = 30 m. As it is stated in Sect. 7.3.6.2, each sensor reduces its sensing range to the largest enclosed sphere. Thus, each sensor does not exploit all of its sensing range. Hence, it is expected that more sensors are needed to k-cover a spatial field. It is clear that our proposed conservative strategy for an irregular radio/sensing model may result in serious waste of resources. Indeed, the energy consumption depends on the total number of deployed sensors to fully k-cover a spatial field of interest. However, we found that this strategy requires more sensors to solve the problem of k-coverage in spatial wireless sensor networks. Thus, it would surely have an impact on the network operational lifetime. That is, this conservative strategy may result in serious waste of resources. More specifically, this waste may be significant for higher degree of coverage k of a large spatial field.

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7.4 Related Work In this section, we present sample approaches on coverage and connectivity in spatial wireless sensor networks. Only a few approaches have dealt with the coverage problem in spatial wireless sensor networks. Alam et al. [12] used the Voronoi Tessellation of the space to create truncated octahedra cells to ensure optimal coverage and connectivity. Their sensor placement approach is based on Kelvin Conjecture, where sensors are placed in the middle of truncated octahedra. They proved that octahedron-based placement approach is valid if the ratio of the transmission range to the sensing range of the sensors is at least 1.7889. On the other hand, they showed that sensor placement based on hexagonal prism or rhombic dodecahedron can be adopted when the above ratio is 1.4142–1.7889. In order to achieve full coverage of a spatial field, each Voronoi cell must have the maximal volume for a given radius of sensing range. This can be obtained when the radius of circumsphere is equal to the radius of the sensing range of the sensors. Ammari and Das [41] provided several measures of connectivity and fault-tolerance in spatial k-covered wireless sensor networks. In particular, they computed the spatial sensor density, denoted by λ(k, r), to k-cover a spatial field, and found that it is equal to λ(k, r) = k / (0.422 r 3 ), where r stands for the radius of the sensing range of the sensors. Also, Ammari and Das [44] computed the critical spatial sensor density above which a spatial field is almost surely covered. While the work in [32] and [41] addressed the problem of k-coverage (with k ≥ 4) in spatial wireless sensor networks using a deterministic approach, the study provided in [44] focused on the problem of 1-coverage (i.e., k = 1) in spatial wireless sensor networks using a probabilistic approach. Ravelomanana [336] investigated fundamental properties of randomly deployed spatial wireless sensor networks for connectivity and coverage, such as the required sensing range to ensure a coverage degree of a region, the minimum and maximum network degrees for a given communication range. Pompili et al. [326] proposed a deployment strategy for spatial communication architecture for underwater acoustic sensor networks, where sensors float at different depths of the ocean to cover the entire spatial region. Poduri et al. [324] discussed some difficulties encountered in the design of spatial wireless sensor networks, such as ensuring network connectivity in the case of uniform random deployment and restrictions imposed by the environment structure on sensor deployment. Zhang et al. [443] presented a set of patterns to cover a spatial field while guaranteeing k-connectivity, where there are at least k disjoint paths between any pair of sensors in the wireless sensor network with k ≤ 4. Yu et al. [440] gave a survey of spatial ocean sensor networks with a focus on several issues, such as deployment, localization, topology design, and position-based routing.

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7.5 Conclusion In this chapter, we investigate the problem of connected k-coverage in duty-cycled spatial wireless sensor networks. We find that our model for k-coverage in planar wireless sensor networks does not generalize to their spatial counterpart. This is due to the inherent characteristics of the Reuleaux tetrahedron that totally differ from those of its planar Reuleaux triangle counterpart. We find that it is necessary to compute the spatial sensor density to k-cover a spatial field. First, we consider a new model, namely the equilateral spherical triangle, and derive the spatial sensor density to guarantee full k-coverage of a spatial field. Also, we propose a distributed randomized connected k-coverage protocol (3DIRACCk ) for a spatial field and evaluate its performance. Second, our theory shows that the Reuleaux tetrahedron model can be used to characterize k-coverage of the space. Moreover, we determine the number of Reuleaux tetrahedra that can fill (or cover) a sphere with overlap such that each Reuleaux tetrahedron has one vertex at the center of the sphere. Precisely, we derive lower and upper bounds of this number. Based on these sphere-coverage results, we propose an energy-efficient k-coverage protocol, where each point in a spatial field is covered by at least k sensors. Also, we relax some of the widely used assumptions in the design of wireless sensor networks, such as spherical sensing model and sensor homogeneity, to make our spatial k-coverage protocol more general. We present several simulation results to validate our proposed protocol. We believe that our results can be used in the development of spatial wireless sensor networks for applications with various coverage degrees and different sensing and communications ranges. We find that our simulation results match their counterpart theoretical ones.

Part IV

Applied Computational Geometry-Based Connected k-Coverage in Wireless Sensor Networks

Chapter 8

A Planar Regular Hexagonal Tessellation-Based Approach for Connected k-Coverage

The idea of atomic energy is illusionary but it has taken so powerful a hold on the minds, that although I have preached against it for twenty-five years, there are still some who believe it to be realizable. Nikola Tesla (1856–1943)

Overview This chapter investigates the problem of connected k-coverage in planar wireless sensor networks using a regular hexagonal tessellation-based approach. The latter assimilates the sensors’ sensing disk to the regular convex hexagon with respect to the proposed metric, sensing range usage rate. It proposes an energy-efficient connected k-coverage protocol based on hexagonal slicing to k-cover a planar field using a minimum number of sensors so as to maximize the network lifetime. Thus, it formulates a multi-objective optimization problem, which computes an optimum solution to the planar k-coverage problem that meets two requirements: First, it maximizes the size of the k-covered area, Ck , in order to minimize the planar sensor density to k-cover a planar field. Second, it maximizes the area of the sensor locality, L k , where at least k sensors are located to k-cover Ck , with a goal to minimize the interference between the sensors. Also, it presents various simulation results to substantiate the theoretical analysis.

8.1 Introduction The planar k-coverage problem in wireless sensor networks is still open. Precisely, determining the minimum planar sensor density to k-cover a planar field, where every point in the field is covered by at least k sensor simultaneously, has been investigated by several researchers. It is worth mentioning that the k-coverage operation is done appropriately if the collected data by the sensors can reach the sink for further analysis and processing. Therefore, in order to accomplish the k-coverage task successfully, it is essential that network connectivity during the entire network operation, be maintained among all of those sensors that are selected to k-cover a planar field. Next, we © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_8

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describe the major tasks we want to accomplish in this chapter and how to achieve each one of them.

8.1.1 Major Tasks In this chapter, we attempt to address the problem of planar connected k-coverage, where a planar field is k-covered, while the wireless sensor network being formed is connected. However, there are two major challenges to solve this problem. The first one is due the scarce energy resources of the sensors. Hence, it is essential to minimize the total energy consumption of the sensors, which depends on the number of sensors. Thus, it is important to solve the planar connected k-coverage problem using a minimum number of sensors in order to maximize the lifetime of the individual sensors. The second challenge is due to the geometric characteristics of the sensing range of the sensors, which is generally supposed to be a disk in planar wireless sensor networks. It is important that the area of the sensing range actually utilized be maximized in order to minimize the total number of active sensors to k-cover a planar field. The main goal is to extend the lifetime of the whole network, while enabling the underlying sensing application to accomplish its target mission successfully. We believe that there is a tight relationship between the concepts of coverage and paving. While the former generally yield gaps given the geometric shape of the sensors’ sensing range (i.e., disk), and, thus, allows overlap to remove any gaps, the latter does not yield any overlaps or gaps. In order to solve the connected koverage problem in planar wireless sensor networks, we propose a regular hexagonal tessellation-based approach. This allows a planar field to be sliced into regular convex hexagons, which have an interesting geometric property for paving a planar field. In this chapter, we attempt to accomplish the following tasks in order to solve the connected k-coverage problem in planar wireless sensor network. Next, we list these tasks and specify our corresponding plan of actions. First, we want to determine the convex polygonal shape that can best assimilate the sensors’ sensing range so as to maximize its actual area being utilized for paving a planar field. To this end, we investigate three well-known regular planar pavers: Equilateral triangle, square, and convex regular hexagon. This analysis is based on a metric, called sensing area usage rate, which is introduced earlier in Chap. 2 (Definition 2.45, Sect. 2.2) and helps identify the best convex polygon among the above three ones: Convex regular hexagon. Thus, we propose to model the sensors’ sensing range using the convex regular hexagon. Second, we want to k-cover a planar field efficiently by using a minimum number of sensors. To achieve this goal, we investigate the relationship that may exist between the area of the region to be k-covered by at least k sensors, which is denoted by C k , and the area of the region where the set of at least k sensors are located to k-cover C k . This area is called sensor locality and denoted by L k .

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217

Third, we want to compute the minimum planar sensor density that is necessary to ensure k-coverage of a planar field, where k ≥ 1. In order to quantify this planar sensor density, we formulate a multi-objective optimization problem and solve it with a goal to maximize both areas of C k and L k . Fourth, we want to solve the problem of connected k-coverage of a planar field, where all k-coverage configurations are guaranteed to be connected. To do so, we need to find the relationship that should exist between the communication range and the sensing range of the sensors. Based on the solution to the above-mentioned multi-objective optimization problem, we determine the ratio of the communication range to the sensing range of the sensors to guarantee connected k-coverage of a planar field, where k ≥ 1. Fifth, we want to identify the most energy-efficient sensor selection approach that enables connected k-coverage of a planar field using a minimum number of sensors, where k ≥ 1. To account for this, we exploit the solution to this multi-objective optimization problem to design an energy-efficient k-coverage protocol based on regular hexagonal tessellation of a planar field. Sixth, we aim to generalize the above analysis of the connected k-coverage problem in planar wireless sensor networks and the corresponding protocols. Consequently, we account for a probabilistic sensing model, which does not assume that the sensing range of the sensors is not necessarily represented by a disk. Moreover, this extended study considers sensor heterogeneity, which does not require the sensors to possess the same characteristics in terms of their sensing range, communications range, and initial energy reserve.

8.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 8.2 studies regular convex pavers and determine the best regular convex polygonal shape that paves a planar field based on the sensing area usage rate metric. Section 8.3 discusses the problem of connected coverage (i.e., k = 1), whereas Sect. 8.4 investigates the connected k-coverage problem in planar wireless sensor networks. Each of Sects. 8.3 and 8.4 computes the planar sensor density that is required to k-cover a planar field. Then, it derives the relationship that should exist between the communication range and the sensing range of the sensors in order to maintain network connectivity in planar k-covered wireless sensor networks. Section 8.5 provides a generalization of the proposed solution to the connected k-coverage in planar wireless sensor networks using a probabilistic sensing model and sensor heterogeneity. Section 8.6 shows some simulation results. Section 8.7 reviews existing approaches for coverage in planar wireless sensor networks. Section 8.8 concludes the chapter.

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8.2 Study of Planar Pavers According to Kershner [227], a regular n-gon can pave the plane only if n = 3, 4, or 6. Indeed, the vertex angle of a regular n-gon is ((n − 2)/n) × π . Thus, an n-gon can pave the plane only if m of these vertices can meet at a point to fill 2π . That is, we should have m × ((n − 2)/n) × π = 2π , or mn = 2(n + m). Therefore, the only planar regular convex shapes that are capable of pavinf the Euclidean plane without overlaps or gaps, are the equilateral triangle, square, and regular hexagon. Indeed, he only positive integer (n, m) solutions of this equation are (n = 3, m = 6) for the equilateral triangle (it has 3 vertices, n = 3; 6 vertices meet at a vertex to fill 2π, m = 6), (n = 4, m = 4) for the square (it has 4 vertices, n = 4; 4 vertices meet at a vertex to fill 2π, m = 4), and (n = 6, m = 3) for the regular hexagon (it has 6 vertices, n = 6; 3 vertices meet at a vertex to fill 2π, m = 3). Next, we compute the quality of coverage (or paving) provided by each of the above-mentioned planar regular convex pavers: Equilateral triangle, square, and regular convex hexagon. These are the only shapes that are capable of tiling the Euclidean plane, as shown in Fig. 8.1 above. Indeed, for each of these three shapes, the sum of the interior angles of their corresponding polygons at the common points at which they meet is 360°. This property is not true neither for the pentagon whose interior angles are 108°, nor for the heptagon whose interior angle is 128.6°. Likewise, the interior angle of the octagon is 135°. This is the reason why the latter three shapes cannot tile the Euclidean plane. Table 8.1 given above summarizes these results. We will exploit these results to investigate the problem of connected k-coverage in planar wireless sensor networks. To this end, we leverage a paving metric, called sensing area usage rate (see Chap. 2, Definition 2.45), to identify the best planar regular convex paver.

(a) Planar Tiling with squares

(b) Planar Tiling with equilateral triangles

Fig. 8.1 Tiling with two-dimensional regular convex polygons

(c) Planar Tiling with regular hexagons

Interior angle

60°

90°

120°

108°

Regular polygon

Equilateral triangle

Square

Regular hexagon

Pentagon

Table 8.1 Characteristics of two-dimensional regular convex polygons

108° × 3 /= 360°

120° × 3 = 360°

90° × 4 = 360°

60° × 6 = 360°

Sum of interior angles

Gap

Yes

No

No

No

(continued)

8.2 Study of Planar Pavers 219

Interior angle

128.6°

135°

Regular polygon

Heptagon

Octagon

Table 8.1 (continued)

135° × 3 /= 360°

128.6° × 3 /= 360°

Sum of interior angles

Yes

Yes

Gap

220 8 A Planar Regular Hexagonal Tessellation-Based Approach …

8.2 Study of Planar Pavers

221

8.2.1 Paving Metric According to our default network model defined earlier in Chap. 2 in Sect. 2.7, the sensing range of the sensors is represented by a disk, which cannot pave the Euclidean plane without gaps or overlaps. In order to cope with this problem, we determine the “best” shape that assimilates the disk so we can pave the plane using the possible smallest number of sensing ranges of the sensors. In other words, we want to find the regular convex paver that maximizes the area of the sensing disk being utilized in paving the plane. To this end, we use the concept of sensing area usage rate, which is defined earlier in Chap. 2 (see Definition 2.45, Sect. 2.2).

8.2.2 Planar Regular Convex Paver Quality We want to measure the quality of paving of each of the three planar regular convex pavers: Equilateral triangle, square, and regular convex hexagon. Theorem 8.1 below specifies the “best” regular convex paver in terms of the sensing area usage rate paving metric. Theorem 8.1 (Best Planar Regular Convex Paver): The regular convex hexagon is the best planar regular convex paver. Proof We prove that the regular convex hexagon provides the maximum sensing area usage rate compared to the equilateral triangle and square. First, we need to compute the greatest regular convex paver, which can be included in the sensing disk of the sensors. In this case, all the vertices of the underlying planar regular convex paver must touch the perimeter of the sensing disk SDr of the sensors. Next, we compute the maximum area of each of these three planar regular convex pavers that can be contained in SDr . • Equilateral triangle: Let us consider Fig. 8.2a. First, we compute the edge length a of the greatest equilateral triangle EQT contained in SDr . According to Fig. 8.2a, we have:

Fig. 8.2 Greatest regular convex pavers contained in a disk

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8 A Planar Regular Hexagonal Tessellation-Based Approach …

√ 3 2(b + r ) (b + r ) sin α = sin 60 = = ⇒a= √ 2 a 3 Also, we have: a 1 b ⇒b= √ tan β = tan 30 = √ = a/2 3 2 3 Thus, the edge length a of EQT is computed as: a=

√

3r

and its corresponding maximum area A(EQT ) and sensing area usage rate SAUR(EQT ) are, respectively, given by: √ 3 2 3 3 2 a = r A(E QT ) = 4 4 √

S AU R(E QT ) =

A(E QT ) = 0.41 A(S Dr )

This means that less than half the sensing area (or disk) of the sensors is used, if the sensors’ sensing disk is reduced to an equilateral triangle. • Square: We consider Fig. 8.2b. We should compute the edge length a of the greatest square SQR that can be contained in SDr . According to Fig. 8.2b, we have:

a 2 + a 2 = (2r )2 ⇒ a =

√

2r

and its corresponding maximum area A(SQR) and sensing area usage rate SAUR(SQR) are, respectively, given by: A(S Q R) = a 2 = 2r 2 S AU R(S Q R) =

A(S Q R) = 0.64 A(S Dr )

In this case, a little more than half the sensing disk of the sensors is exploited. Clearly, the square provides a slightly better performance than the equilateral triangle. But, the corresponding paving quality is still low.

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223

• Regular Convex Hexagon: Let us consider Fig. 8.2c. We want to compute the edge length a of the greatest regular convex hexagon RCH included in SDr . According to Fig. 8.2c, we have: a = r . and its corresponding maximum area A(RCH) and sensing area usage rate SAUR(RCH) are, respectively, given by: √ √ 3 3 2 3 3 2 a = r A(RC H ) = 2 2 S AU R(RC H ) =

A(RC H ) = 0.83 A(S Dr )

Based on the above results, the regular convex hexagon has the highest sensing area usage rate compared to the equilateral triangle and the square. Thus, we have: S AU R(RC H ) = max(S AU R(RC P) : RC P ∈ {E QT , S Q R, RC H }) In other words, the regular hexagon is the best regular convex paver as it outperforms both of the equilateral triangle and square. ∎ Based on Theorem 8.1, we conclude that the regular convex hexagon yields the best quality of coverage (or paving). Therefore, the sensing disk of the sensors can be assimilated to a regular convex hexagon. That is, the sensing area of the sensors has to be reduced to a regular convex hexagon of edge length r, where r stands for the radius of the sensing disk of the sensors. We compute the planar sensor density that is required to k-cover a planar field by reducing the sensors’ sensing disk to a regular convex hexagon of edge length r. Next, we focus on the problem of connected k-coverage of a planar field. More precisely, we investigate two sensor placement and selection approaches to k-cover a planar field. First, we slice a planar field randomly into tangential congruent regular convex hexagons to produce a planar regular hexagonal tessellation (see Fig. 8.3). For each k-coverage approach, we perform the following two steps: • We compute the planar sensor density to achieve k-coverage of a planar field. Fig. 8.3 Planar regular hexagonal grid

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• We determine the corresponding ratio of the communication range R to the sensing range r of the sensors in order to guarantee network connectivity for all planar k-coverage configurations. In the sequel, we present a thorough analysis of the connected k-coverage problem in planar wireless sensor networks. This analysis leads to solve a multi-objective optimization problem. Then, we discuss three approaches to address the problem of planar connected k-coverage, where k ≥ 1. The first approach, which is presented in Sect. 8.3, is optimal for a degree of coverage k = 1. However, it is not practical and not even feasible for k ≥ 2. The last two approaches investigate the connected k-coverage problem, where k ≥ 2. Both approaches leverage the solution of this multi-objective optimization problem. We describe the proposed two approaches in detail in Sect. 8.4.

8.3 Regular Hexagonal Centroid-Based Connected k-Coverage We consider one particular sensor relocation approach. First, the edge length of each regular convex hexagon forming the regular hexagonal grid is equal to r , where r is the radius of the sensing disk of the sensors. In this approach, k sensors have to be located at the same center O of each regular convex hexagon of edge length r , denoted by RC H (O, r ). It is obvious that every RC H (O, r ) is k-covered. Indeed, the maximum distance between the center Oi and any point located in RC H (O, r ) is r . That is, every point in RC H (O, r ) is within the sensing range of k sensors. Thus, the whole planar field is surely k-covered. In order to k-cover a planar field in every round during the operational network lifetime, the centroid of every regular convex hexagon of edge length r (i.e., RC H (O, r )) of the hexagonal grid should have exactly k sensors. It is essential that different sets of sensors be selected in different rounds. This would help distribute the k-coverage load fairly among all the sensors deployed in a planar field. These selected sets for different rounds may not be totally disjoint. For sensor relocation to take place, the sensors should have a mobility support so they can move from their current locations to the centroids of all of those regular convex hexagons that constitute the hexagonal grid of a planar field. Lemma 8.1 computes the planar sensor density corresponding to this sensor placement approach. Lemma 8.1 (Planar Sensor Density): The planar sensor density, denoted by ρ1 (k, r ), which is necessary to k-cover a planar field according to the hexagonal single-point sensor placement approach describe above, can be computed as follows: ρ1 (k, r ) =

0.38k r2

8.3 Regular Hexagonal Centroid-Based Connected k-Coverage

225

where r is the radius of the sensing disks of the sensors. Proof According to Definition 2.52 (see Chap. 2, Sect. 2.2), the planar sensor density ρ1 (k, r ), which corresponds to hexagonal single-point sensor placement approach, is given by: ρ1 (k, r ) =

k A(RC H (O, r ))

where A(RC H (O, r )) stands for the area of the regular convex hexagon of edge length r, which is denoted by RCH(O, r), and is computed as: √ 3 3 2 r A(RC H (O, r )) = 2 This implies: ρ1 (k, r ) =

0.38k r2

∎

Theoretically speaking, A(RC H (O, r )) refers to the maximum k-covered area by at least k sensors as stated Lemma 8.1. We will use this maximum area in Sect. 8.3.2 in formulating our multi-objective optimization problem to maximize two utility functions, which is specified in the next sections. Then, we solve it to compute the optimum solution for the k-coverage problem. Next, we establish the relationship that should exist between the radii of sensor’s communication range and sensing range r and R, respectively, to ensure network connectivity. Corollary 8.1 shown below states the ratio of the communication range to the sensing range of the sensors for network connectivity. Corollary 8.1 (Network Connectivity): Given k-coverage, the underlying network is connected only if we have the following inequality: R ≥ 2r Proof The farthest two sensors from each other need to be connected in order to guarantee network connectivity. In other words, the sensors’ communication range should be at least equal to the diameter of RC H (O, r ), which is equal to 2r , i.e., R ≥ 2r . ∎

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8.3.1 Achieving Optimal Coverage According to Lemma 8.1, the planar sensor density, which is necessary to cover a planar field, reaches its optimum value when every sensor is placed at the centroid of a different regular convex hexagon of radius r , i.e., RC H (O, r ), in the hexagonal grid of a planar field. This type of sensor relocation approach to achieve 1-coverage of a planar field yields the minimum number of sensors to cover the entire planar field, which helps extend the lifetime of the individual sensors. This will in turn help maximize the operational network lifetime.

8.3.2 Problems with k-Coverage for k ≥ 2 First, this hexagonal centroid sensor relocation approach is neither realistic nor practical when the degree of coverage is k ≥ 2. In fact, given the physical properties of the sensors, it is impossible to place all k sensors at the same location. In addition, this type of sensor placement approach may cause other major problems, such as interference, which may arise due to sensor closeness to each other. Furthermore, the sensors deployed according to this approach do not possess any mobility support. It is essential to account for a feasible and a more practical approach for sensor selection to ensure k-coverage of a planar field. To this end, we need to stretch the hexagonal centroid to a larger area, which is not confined to a single point. This stretching approach allows the k sensors to be deployed anywhere in this area and also to be mobile so they can be located at any point in it. Next, we discuss our approach for stretching a hexagonal centroid to a planar area. Specifically, we compute the minimum planar sensor density that is required to k-cover a planar field of interest. Moreover, we compute the relationship between the radius of the communication range and the radius of the sensing range of the sensors so all the sensors participating in the k-coverage process are mutually connected. In addition, based on the above analysis, we propose two optimized, energy-efficient random sensor selection approaches to ensure connected k-coverage of a planar field using a small number of deployed sensors.

8.4 Regular Hexagonal Area Stretching-Based Connected k-Coverage In this section, we present our proposed hexagonal area stretching approach to achieve k-coverage of a planar field.

8.4 Regular Hexagonal Area Stretching-Based Connected k-Coverage

227

8.4.1 Foundational Study First, we define the following notation. Let Ck denote the largest region, which can be k-covered by at least k sensors, and |Ck | be the size of its area. Also, let L k stand for the sensor locality, which is associated with Ck , and |L k | be the size of its area. Precisely, those k sensors k-covering the region Ck should be located in L k . In other words, our hexagonal area stretching approach stretches a hexagonal centroid (i.e., point) to L k (i.e., area). Lemma 8.2 below shows the relationship that exists between Ck and L k , and their respective areas, |Ck | and |L k |. Lemma 8.2 (Areas of Sensor Locality and k-Covered Region): The size of the area of the sensor locality L k is upper bounded by that of the k-covered region Ck , i.e., |L k | ≤ |Ck |. Moreover, the area L k is subset of the region Ck , i.e., L k ⊆ Ck . Proof Assume that O and σ are the centroid and diameter of the k-covered region Ck , respectively. Also, suppose that there are k sensors placed in the area L k . However, Ck is the largest k-covered region with k sensors. Then, the diameter σ should be at least equal to r, with r being the radius of the sensors’ sensing disk, i.e., σ ≥ r . Indeed, because Ck is k-covered, any point in Ck should be within the sensing range of each of those k sensors. Now, since Ck is the largest k-covered region, the maximum distance between any of those k sensors and any location in Ck cannot exceed r. The minimum and maximum diameters of Ck are σmin = r and σmax = 2r , respectively. Let us proceed using a proof by contradiction, while considering the following three cases. Case 1: σ = σmin = r Let si be the sensor that is located outside Ck and lies on the line segment that coincides with the diameter uv of Ck . Then, the distance between si and the farthest point u exceeds r. It implies that u is not covered by si , and, consequently, the area Ck is not k-covered. This is contradictory to our assumption that C is k-covered. Case 2: σmin = r < σ < σmax = 2r Suppose that sensor si is located either at the perimeter of Ck or outside Ck and lies on the line segment that coincides with the diameter uv of Ck . In either case, the distance between si and u is greater than r, and, thus, u is not covered by si . Consequently, Ck is not k-covered. This violates our assumption that Ck is k-covered. Case 3: σ = σmax = 2r This case is trivial. Assume that there is a sensor si whose location is not the centroid O of Ck and is located on the line segment that coincides with the diameter uv of Ck . Clearly, at least the centroid O cannot be covered. Thus, Ck is not k-covered. This contradicts our assumption that Ck is k-covered. Based on these three cases, it is clear that there is no such a sensor si that is located ∎ outside of Ck . Hence, we have L k ⊆ Ck , and, therefore, |L k | ≤ |Ck |. According to Lemma 8.2, we can conclude that the maximum area |L k |max of the sensor locality L k should be equal to the area of Ck , i.e., |L k |max = |Ck |. In this case, a

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8 A Planar Regular Hexagonal Tessellation-Based Approach …

planar field is sliced to form a regular hexagonal grid, which is formed of congruent regular convex hexagon RC H (O, r/2) whose edge length e2 is equal to r/2, i.e., e2 = r/2. In other words, the k-covered region Ck corresponds to the regular convex hexagon RC H (O, r/2), i.e., Ck = RC H (O, r/2). Therefore, the maximum area |L k |max of the sensor locality L k , which coincides with the area of RC H (O, r/2), denoted by A(RC H (O, r/2)), is computed as follows: |L k |

max

√ √ 3 3 2 3 3 ( r )2 r = = A(RC H (O, r/2)) = 2 2 8

We should mention that the exact shape of L k should be a disk of radius is r/2 and centered at O. With this type of shape of the sensor locality, the region Ck = RC H (O, r/2) is still surely k-covered. However, because the unit of slicing a planar field of interest to form a grid is a convex regular hexagon, we decide that the shape of the sensor locality L k be a convex regular hexagon instead of a circle. In order to achieve k-coverage of a planar field, k sensors should be selected from each RC H (O, r/2). Given that the diameter of RC H (O, r/2) is 2 × (r/2) = r , every point inside RC H (O, r/2) is within the sensing range of those k sensors. Thus, each regular convex hexagon RC H (O, r/2) is k-covered, hence, k-covering an entire planar field. Lemma 8.3 below calculates the corresponding planar sensor density ρ2 (k, r ). Lemma 8.3 (Planar Sensor Density): Let r be the radius of the sensing range of the sensors, and k be a natural number, with k ≥ 2. Using a random hexagonal-area sensor selection approach, the planar sensor density ρ2 (k, r ), which is necessary to guarantee k-coverage of a planar field, is calculated as follows: ρ2 (k, r ) =

1.54k r2

Proof Let Ck = RC H (O, r/2) be a regular convex hexagon of edge length r/2. Its area |Ck | = |RC H (O, r/2)| is given by: |Ck | = |RC H (O, r/2)| =

√ 3 3 2 r 8

Consequently, the planar sensor density ρ2 (k, r ) is computed as follows: ρ2 (k, r ) =

1.54k k = |Ck | r2

∎

Our goal is to minimize the total number of sensors to k-cover a planar field of interest. To this end, we need to maximize the area of the region Ck to be k-covered by at least k sensors. Next, we compute the planar sensor density corresponding to the area of the k-covered region Ck , which we increase gradually. Clearly, we should decrease the area of the sensor locality L k . Let us discuss the following sample

8.4 Regular Hexagonal Area Stretching-Based Connected k-Coverage

229

Fig. 8.4 C k and L k for e(L k ) = r/3 and e(C k )= 2r/3

configurations in order to better understand the relationship between the areas Ck and L k . • Configuration 1: L k =RC H(O, r/3). Given the hexagonal shape of L k whose edge length is r/3, it is easy to determine the exact hexagonal shape of Ck , which should be a regular convex hexagon of edge length e3 = 2r/3. In other words, Ck = RC H (O, 2r/3). Figure 8.4 shows both areas of Ck and L k . As it can be seen, one can easily verify that Ck is guaranteed to be k-covered using k sensors placed in the sensor locality L k . Given the size |Ck | of the area Ck , which contains k sensors, the underlying planar sensor density is given by: ρ3 (k, r ) =

0.87k k = |Ck | r2

• Configuration 2: L k =RC H(O, r/4) We decrease the size of the area of sensor locality L k . Now, given that L k = RC H (O, r/4), we can easily determine that the edge length of Ck should be e4 = 3r/4. That is, Ck has to be a regular convex hexagon of edge length 3r/4, i.e., Ck = RC H (O, 3r/4). Notice that the maximum distance between any sensor in L k and any point in Ck is still equal to r. Therefore, the area Ck is ensured to be k-covered with k sensors located in their sensor locality L k . Thus, the planar sensor density, which corresponds to the above configuration, can be computed as follows: ρ4 (k, r ) =

0.68k k = |Ck | r2

• Configuration 3: L k =RC H(O, r/5) We decrease more the size of the area of sensor locality L k whose edge length is r/5. Similarly, one can check that Ck is guaranteed to be k-covered using k sensors

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8 A Planar Regular Hexagonal Tessellation-Based Approach …

located in their sensor locality L k provided that the edge length of Ck is equal to e5 = 4r/5. In other words, Ck = RC H (O, 4r/5). Given this configuration, the corresponding planar sensor density is given by: ρ5 (k, r ) =

0.60k k = |Ck | r2

Notice that the planar sensor density decreases as the size of the area of sensor locality L k gets smaller and, consequently, as the area of the k-covered region Ck gets bigger. That is, ρ5 (k, r ) < ρ4 (k, r ) < ρ3 (k, r ). Generalization: L k = RC H( O, r/i ), where i ≥ 2 Now, we generalize the above analysis and focus a more general case of the shapes of both areas L k and Ck , where the edge length of the sensor locality L k is r/i. The natural number i ≥ 2 is called the stretching factor. Lemma 8.4 below computes the edge length of the regular hexagon representing the k-covered region Ck . Lemma 8.4 (Edge length of the hexagonal k-covered region Ck ): Let e(L k ) be the edge length of the sensor locality L k = RC H (O, r/i ), with i ≥ 2. Then, the edge length e(Ck ) of the hexagonal k-covered region Ck is computed as follows: e(Ck ) = ei =

(i − 1)r i

Proof Let r be the sensors’ sensing disk radius, and i ≥ 2 be the stretching factor. Figure 8.5 shows the sensor locality L k = RC H (O, r/i ) and the k-covered region Ck = RC H (O, (i − 1)r/i ). The area Ck should be k-covered by at least k sensors. Thus, the maximum distance between any sensor in L k and any point in Ck should not exceed the radius r of the sensing disk of the sensors. That is, the Euclidean distance between the two farthest points U in L k and V in Ck should satisfy the following equation: δ(U, V ) = r

Fig. 8.5 C k = RC H( O, (i − 1)r/i ) and L k = RC H( O, r/i )

8.4 Regular Hexagonal Area Stretching-Based Connected k-Coverage

231

Also, we have: δ(U, V ) = δ(U, O) + δ(O, V ) where δ(U, O) = e(L k ) =

r i

δ(O, V ) = e(Ck ) Finally, we obtain: e(Ck ) = δ(O, V ) = δ(U, V ) − δ(U, O) = r −

r (i − 1)r = i i

∎

One can clearly observe that the edge of the hexagonal k-covered region Ck is upper-bounded by the radius of the sensing range of the sensors, thus, satisfying the following inequality: e(Ck ) = ei =

(i − 1)r 0, w2 > 0, and w1 + w2 = 1. Proof Let Ak (i ) = (|Ck |, |L k |)T be our multi-objective function to be maximized. We apply the weighted scale-uniform-unit sum approach to investigate our k-coverage problem. Recall that the values of the weighting coefficients w1 and w2 , where 0 < w1 , w2 < 1 with w1 + w2 = 1, reflect the relative importance of each of the above-mentioned two objective functions. Now, we formulate our unconstrained multi-objective optimization problem to trade-off the two objective functions described earlier, namely maximizing the area of the k-covered region, denoted by Ck = RC H (O, (i − 1)r/i ) and maximizing the area of the sensor locality, denoted by L k = RC H (O, r/i ). Maximi ze Ak (i ) = w1 c1 |Ck | + w2 c2 |L k | Subject to i ≥ 2 where μ |Ck |max μ c2 = |L k |max c1 =

where |Ck |max and |L k |max stand for the maximum k-covered region, and maximum sensor locality, respectively. Using Lemma 8.1 and Lemma 8.2, and as stated earlier, we have: √ 3 3 2 max |Ck | = A(RC H (O, r )) = r 2√ 3 3 2 |L k |max = A(RC H (O, r/2)) = r 8 This helps compute the values of μ, c1 , and c2 as follows: √ { } 3 3 2 max max max r μ = max |Ck | , |L k | = = |Ck | 2 c1 = 1 and c2 = 4

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8 A Planar Regular Hexagonal Tessellation-Based Approach …

Therefore, the function Ak can be defined as follows: √ ( ) 4 2 3 3 (i − 1)2 w1 Ak (i ) = + w2 2 r 2 i2 i with w1 ≥ 0 and w2 ≥ 0, where w1 + w2 = 1 Assume that we set the values of w1 and w2 as follows: w1 = w2 = 0.5. In this case, maximizing both of the size |Ck | of the area of the k-covered region Ck , and the size |L k | of the area of the sensor locality L k have the same importance. If w1 > w2 , the network designer prioritizes the minimization of the total energy consumption through maximizing |Ck | over the maximization of the space for sensor mobility via maximizing |L k |. If w2 > w1 , the network designer is more interested in the maximization of |L k | than the maximization of |Ck |. Now, we focus on the study of our multi-objective function Ak , which is defined by Ak (i ) as shown above. Let i ∗ denote a solution to our unconstrained multi-objective optimization problem stated earlier. We compute the derivative of Ak (i ) to obtain the following: ∂ Ak (i ) w2 = 0 ⇒ i∗ = 1 + 4 ∂i w1 Clearly, we should have w1 /= 0 as there is always a need to maximize the size |Ck | of the area of Ck = RC H (O, (i − 1)r/i ). Now, let us show that i ∗ corresponds to the maximum value of Ak (i ). To this end, we compute the second derivative of Ak (i ). It is easy to check that the following inequality ∂ 2 Ak (i ) >0 ∂ 2i holds whenever the following inequality is satisfied: i < iˆ =

w2 6w1 + 24w2 = 1.5 + 6 4w1 w1

ˆ Thus, the value of i ∗ shown above corresponds As it can be seen, we have i ∗ < i. to the maximum of Ak (i ∗ ). In addition, if we vary the weights w1 and w2 within their interval of definition [0, 1] such that w1 + w2 = 1, we obtain various solution values ∎ of the maximum of Ak (i ∗ ). Based on the above analysis and the optimum value i ∗ of the stretching factor i, Theorem 8.3 computes the minimum planar sensor density, which is necessary to achieve k-coverage of a planar field. Theorem 8.3 (Minimum Planar Sensor Density): The minimum planar sensor density, denoted by ρi ∗ (k, r ), which is required to k-cover a planar field, can be computed as follows:

8.4 Regular Hexagonal Area Stretching-Based Connected k-Coverage

235

2i ∗ k ρi ∗ (k, r ) = √ 3 3(i ∗ − 1)2 r 2 2

Proof Let r be the radius of the sensing disk of the sensors and k ≥ 1. Using Definition 2.52 (Chap. 2, Sect. 2.2) and the optimum value of |Ck |, denoted by |Ck |* , which is computed based on the optimum value i ∗ of the stretching factor i, the planar sensor density ρi ∗ (k, r ) that is required to ensure k-coverage of a planar field is given by: 2i ∗ k k ρi ∗ (k, r ) = ∗ = √ |Ck | 3 3(i ∗ − 1)2 r 2 2

∎

We should mention that the planar sensor density for k-coverage of a planar field, which we find earlier in Chap. 5, was computed as follows [26]: λ(k, r ) = 0.81k/r 2 Without loss of generality, if both objective functions are assigned an equal weight, i.e., w1 = w2 , we obtain the following: ρi ∗ (k, r ) = 0.6k/r 2 < λ(k, r ) yielding a decent improvement of 1 − 0.6/0.81 = 26%. Next, we investigate the ratio of the communication range R to the sensing range r of the sensors so as to guarantee connectivity among all the sensors participating in the process of k-covering a planar field. Lemma 8.5 below computes this ratio. Lemma 8.5 (Communication range and Sensing Range Relationship): Let r and R be the radiu of the sensing disk and communication disk of the sensors, respectively. Under the assumption that k-coverage of a planar field is achieved, network connectivity is guaranteed if R and r are related by the following inequality: R≥

2 r i∗

Proof As discussed earlier, all the sensors that are selected to k-cover the area Ck are located inside the area L k . Therefore, the pair of sensors that are placed as far as possible from each other are located on the line segment that coincides with the diameter of L k . In other words, the maximum distance between any pair of sensors that are selected to k-cover Ck is equal to the diameter 2e∗ (L k ) of the sensor locality L k . Thus, in order to maintain network connectivity, it is necessary that the communication range of the sensors be at least equal to 2e∗ (L k ), where e* (L k ) = r/i* . Hence, we have the following inequality: R ≥ 2e∗ (L k ) =

2 r i∗

∎

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8 A Planar Regular Hexagonal Tessellation-Based Approach …

Next, we discuss two sensor selection approaches for k-coverage of a planar field [18]. In both approaches, the sensors are selected pseudo-randomly to k-cover a planar field. First, we describe the fundamental operation (i.e., preprocessing) that should take place at the beginning of every k-coverage round. Then, the sensors would be selected according to one of the two approaches to k-cover a planar field.

8.4.2 Random Regular Hexagonal Tessellation First, we assume that at the beginning of each k-coverage round, all the sensors are in listening mode (or awake). Also, we suppose that the sink has an infinite source of energy as it can be connected to a power outlet. Furthermore, we assume that the sink generates a random regular hexagonal tessellation at the beginning of each k-coverage round. That is, the sink randomly slices a planar field into regular convex hexagons of edge length e∗ (Ck ), which is given by: e∗ (Ck ) = ei ∗ =

(i ∗ − 1)r i∗

This random regular hexagonal tessellation adds dynamics to the proposed kcoverage protocols, which are discussed below. This type of dynamics gives an opportunity to all the sensors to be equally selected to participate in k-covering a planar field. It is achieved by randomly generating the first regular convex hexagon, RC H (O, (i ∗ − 1)r/i ∗ ), which is called reference hexagon. The entire random regular hexagonal slicing grid will be produced based on this reference hexagon. Second, recall that the k-covered region Ck is represented by a regular convex hexagon, denoted by RC H (O, (i ∗ − 1)r/i ∗ ), whereas the sensor locality L k is modeled by a regular convex hexagon, denoted by RC H (O, r/i ∗ ). We require that both of the regular convex hexagons RC H (O, (i ∗ − 1)r/i ∗ ) and RC H (O, r/i ∗ ) be centered at the same point O in order to ensure that every point in RC H (O, (i ∗ − 1)r/i ∗ ) is k-covered by those k sensors located inside RC H (O, r/i ∗ ). First, the sink randomly selects a set of k sensors from each sensor locality RC H (O, r/i ∗ ), which is randomly generated at the center O of its corresponding RC H (O, (i ∗ − 1)r/i ∗ ). This randomness aims at ensuring that all the deployed sensors participate in the k-coverage process of a planar field. Once the sensor selection process terminates, the sink broadcasts the schedule for the current round, which includes the ID’s of the sensors that will participate in k-covering a planar field in this round. When it receives the schedule, a sensor checks the presence of its ID to decide whether its status should be active (on state) or inactive (off state). Then, it forwards this schedule (without its ID, if the received schedule contains the sensor’s ID) to all its one-hop neighbors. As it can be seen, the two k-coverage approaches of a planar field consider a degree of coverage is k ≥ 2. Moreover, they are based on the concepts of hexagonal sensing range, k-covered region Ck , and sensor locality L k .

8.4 Regular Hexagonal Area Stretching-Based Connected k-Coverage

237

Fig. 8.6 Slicing L k into k congruent cones centered at O

8.4.3 Hexagonal Cone-Based Pseudo-Random k-Coverage Compared to the k-coverage approach, which is described in Sect. 8.3, the one proposed in this section is energy efficient, yet practical for k ≥ 2. This approach exploits the geometric properties of the regular convex hexagon as well as the solution to the multi-objective optimization problem discussed earlier. First, it divides the regular convex hexagon into several regions of equal area. Precisely, the sink starts by randomly dividing the sensor locality L ∗k = RC H (O, r/i ∗ ) into k congruent cones, each of which is centered at O, has an angle equal to 2π/k, and is denoted by ∇i (O, 2π/k), where 1 ≤ i ≤ k. This cone-based k-coverage approach aims at providing all the sensors inside L ∗k = RC H (O, r/i ∗ ) with an equal opportunity to be selected to k-cover the region Ck∗ = RC H (O, (i ∗ − 1)r/i ∗ ). It is important that all sets of sensors, which are identified in all the k-coverage rounds to guarantee kcoverage of a planar field, be as disjoint as possible. Furthermore, the sink randomly selects one sensor from every cone. It may happen that a given cone ∇ j (O, 2π/k) does not have any sensors located in it. In this case, the sink should designate a sensor from the closest cone to ∇ j (O, 2π/k) in order to ensure that every region Ck∗ = RC H (O, (i ∗ − 1)r/i ∗ ) is k-covered by exactly k sensors. Figure 8.6 shows all the cones ∇ j (O, 2π/k), where 1 ≤ j ≤ k, which form a sensor locality L ∗k for various values of k, where k = 2, 3, 6.

8.4.4 Hexagonal Perimeter-Based Pseudo-Random k-Coverage According to this second k-coverage approach, the sink selects k sensors located on the perimeter of every regular convex hexagon associated with the sensor locality L ∗k = RC H (O, r/i ∗ ). The ideal case would be to select k/6 sensors from each of the six edges of L ∗k = RC H (O, r/i ∗ ). This helps achieve a better load balancing among all the sensors in all the regions of a planar field. Also, this type of scenario allows the selected sensors to be distant from each other, which in turn helps overcome some potential interference problems. However, this may not always be achievable and some areas crossed by some or all of the six edges of the sensor locality

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L ∗k = RC H (O, r/i ∗ ) may have less than k/6 sensors or may even not have sensors at all. In this type of situation, the remaining sensors should be selected from the closest areas to those edges. Consequently, it is essential that those remaining sensors move toward those edges in order to k-cover their corresponding hexagons Ck∗ = RC H (O, (i ∗ − 1)r/i ∗ ), thus, achieving k-coverage of the entire planar field. Thus, the sensors should have a mobility support, which would cost an additional energy consumption that would have an effect on the operational network lifetime. The simulation results, which are presented in Sect. 8.6.2, show the significant difference in energy consumption incurred by the two k-coverage approaches: Cone-based k-coverage and perimeter-based k-coverage. Specifically, the latter approach costs more energy consumption than the former one and this is due to sensor mobility to be located on the perimeter of the sensor locality L ∗k = RC H (O, r/i ∗ ).

8.4.5 Edge Problem As it is expected, the number of active sensors that is needed to k-cover a planar is higher than the one, which is computed based on the theoretical planar sensor density given earlier in Sect. 8.4.1. This additional number of active sensors is due to the slicing process of a planar field into regular convex hexagons. Indeed, as shown in Fig. 8.2, it will always yield several incomplete (or truncated) regular convex hexagons around the edges of the field. In other words, there is a partial existence of several regular convex hexagons Ck∗ = RC H (O, (i ∗ − 1)r/i ∗ ) closer to the edges of a planar field. That is, only portions of regular convex hexagons appear closer to the edges of the planar field. However, these “truncated” regular convex hexagons should be k-covered regardless of their respective sizes. According to the above-mentioned analysis k sensors must be located in every regular convex hexagon regardless of whether it is complete or truncated. This situation is unavoidable due to the geometric shapes of the planar field and the regular convex hexagons. Although our goal is to reduce the number of selected sensors to k-cover a planar field so as to extend the lifetime of the individual sensors and that of the entire network, this extra number of sensors is necessary for full k-coverage of this planar field. Otherwise, the planar field would be partially k-covered, leaving some regions less than k-covered or even not covered at all. This situation, which is known as the edge problem, will always have an impact on the network span.

8.4.6 Discussion In Chap. 5, we investigate the connected k-coverage problem in planar wireless sensor networks using a well-known result of convexity theory: Helly’s theorem [85]. In addition, we consider the geometric properties of the Reuleaux triangle [481] in order to compute the planar sensor density, which is necessary to achieve k-coverage of

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a planar field. In this chapter, we leverage the tiling property of the regular convex hexagon, and introduce two different concepts: k-Covered region, denoted by Ck , which can be k-covered by k sensors, and sensor locality, denoted by L k , which stands for mobility space of the sensors to ensure k-coverage of Ck . Our theoretical analysis shows that the proposed regular hexagonal tessellationbased k-coverage approach in this chapter is more practical and even more energyefficient than the convexity theory-based k-coverage method presented earlier in Chap. 5. Also, Sect. 8.6 presents various simulation results, including the comparison between the above k-coverage approaches discussed in this chapter and in Chap. 5, respectively. These results are in line with the theoretical results. Thus, our regular hexagon-based k-coverage approaches in this chapter, outperform our k-coverage method previously described in Chap. 5. Our comparison study is based on two dimensions: The size of the k-covered area (dimension 1) and the size of the mobility space (dimension 2) assigned to the sensors. In fact, the size of the k-covered region Ck is larger than the size of the two adjacent Reuleaux triangle, which form the k-covered area in Chap. 5, and, consequently, a smaller planar sensor density to k-cover a planar field. Moreover, the mobility space L k given to the deployed sensors to k-cover Ck according to the k-coverage approach in this chapter is bigger than the one defined in the Reuleaux triangle-based k-coverage approach, which is discussed in our previous work in Chap. 5. Indeed, the size of the area of the sensor locality L k is larger than the size of the lens, i.e., the intersection (or overlap area) of two adjacent Reuleaux triangles, which define the space mobility of the k-coverage approach introduced in Chap. 5. As discussed earlier, sensor mobility is necessary for the perimeter-based k-coverage approach. This mobility is much easier in the sensor locality L k than in the lens. Also, sensor mobility helps overcome the interference problem among the sensors.

8.5 Possible Extensions In this section, we propose to generalize our study of the connected k-coverage problem in planar wireless sensor networks. To this end, we relax a few widely used assumptions. First, we use a stochastic sensing model instead of a deterministic sensing model. Second, we account for the deployment of heterogeneous sensors instead of homogeneous sensors. Next, we discuss these two relaxations in order to make our proposed approaches more general.

8.5.1 Extension 1: Using Non-Deterministic Sensing Model We consider a deterministic sensing model in the above investigation of the kcoverage problem, where the coverage capability of a sensor depends on the Euclidean distance between the location of the sensor and the point in a planar

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field. According to this deterministic sensing model, the sensing range of the sensors is represented by a disk, where the sensors’ readings are computed with certainty. However, the signal may be attenuated and the sensors’ readings are noisy. To account for all these issues, we need to consider a non-deterministic (or probabilistic) sensing model, which is a more realistic than its deterministic counterpart, where the coverage of a point q by a sensor si , denoted by Cov(q, si ), depends on a certain probability function. In other words, the coverage Cov(q, si ) can be viewed as the detection probability of an event at location q by sensor si , denoted by Prob(q, si ), which is computed as indicated in Sect. 2.3 (see Chap. 2). Now, we exploit the results established in Sect. 8.4 to investigate the probabilistic k-coverage problem. Theorem 8.4 below computes the minimum k-coverage probability Probk, min based on the adopted probabilistic sensing model. Theorem 8.4 (Minimum k-Coverage Probability): Let k ≥ 1 and r be the radius of the nominal sensing range of the sensors, and consider the probabilistic sensing model, which is discussed in Chap. 2 in Sect. 2.3. Also, let Probmin, k be the minimum kcoverage probability, where every point in a planar field is probabilistically k-covered by at least k sensors. Probkmin is computed as follows: ( α )k Pr obmin,k = 1 − 1 − e−βr Proof As stated earlier in Lemma 8.4, the area Ck (or RC H (O, (i − 1)r/i )) is kcovered if k sensors are located in the sensor locality L k (or RC H (O, r/i )). In order to compute the minimum k-coverage probability, denoted by Probk, min , we should identify the least k-covered point in a planar field. Based on the above k-coverage approach discussed earlier, only the sensors located in L k can be selected to k-cover Ck . The least k-covered point in a planar field corresponds to the farthest point in this field from all k sensors. These k sensors could be positioned on the perimeter of L k . It is clear that each of the points located on the perimeter of Ck is the least k-covered point. For instance, as shown on Fig. 8.5, the point V located on Ck is the farthest one from the point U on L k . Thus, the point V is the least k-covered point provided that all the selected k sensors (s1 , .., sk ) are located around the point U. Thus, the distance dist(V , (s1 , s2 , . . . , sk )) between V and each of these k sensors, is equal to the radius of Ck plus the radius of L k . That is, dist(V , (s1 , s2 , . . . , sk )) = (i − 1)r/i +r/i = r . The minimum k-coverage probability, Probmin, k , corresponds to the k-coverage of the least k-covered point V by k sensors using the underlying probabilistic sensing model. We conclude that Probmin, k can be calculated using the following expression: Pr obmin,k = 1 −

k Π

(1 − Pr ob(V , si ))

i=1

( α )k = 1 − (1 − Pr ob(V , si ))k = 1 − 1 − e−βr

∎

Next, we proceed to solve the probabilistic k-coverage problem. Let S be the set of all deployed sensors. We want to select a minimum subset of sensors Smin ⊆ S such

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241

that every point in a planar field is k-covered by at least k sensors, and the minimum k-coverage probability of each point is at least equal to a certain threshold probability Pth , where 0 < Pth < 1. This will help us compute the probabilistic sensing range, r s , which provides probabilistic k-coverage of a planar field with a probability no less than Pth . Lemma 8.6 below computes this probabilistic sensing range r s . Lemma 8.6 (Probabilistic Sensing Range): Let k be the degree of coverage, where k ≥ 1, the physical characteristic of the sensing units of the sensors be denoted by β, and 2 ≤ α ≤ 4. The probabilistic sensors’ sensing range r s , which is required to achieve probabilistic k-coverage of a planar field with a minimum number of sensors and with a probability at least equal to Pth with 0 < Pth < 1, verifies the following inequality: ) ( ( ) 1/α 1 rs ≥ − log 1 − (1 − Pth )1/k β Proof From the above statement, we give the following inequality: Pr obmin,k ≥ Pth In other words, we have: ( α )k 1 − 1 − e−βr ≥ Pth ( α )k ⇒ 1 − e−βr ≤ 1 − Pth α

⇒ 1 − e−βr ≤ (1 − Pth )1/k Therefore, the probabilistic sensing range r s of the sensors is defined by the following inequality: ) ( ( ) 1/α 1 1/k rs ≥ − log 1 − (1 − Pth ) β

∎

From Lemma 8.6, we conclude that the minimum probabilistic sensing range, denoted by rsmin , is computed as follows: rsmin

) ( ( ) 1/α 1 1/k = − log 1 − (1 − Pth ) β

From now on, we refer to this lower bound on probabilistic sensing range r s of the sensors when we attempt to solve the probabilistic connected k-coverage problem based on the probabilistic sensing model discussed above. That is, rsmin will serve as one of the input parameters to this problem.

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The respective counterpart results to Lemma 8.4, Theorem 8.3, and Lemma 8.5 are Lemma 8.7, Theorem 8.5, and Lemma 8.8, which account for the probabilistic nature of the sensing model under consideration. We state these results without their proofs as they are very similar to the ones associated with the results Lemma 8.4, Theorem 8.3, and Lemma 8.5. Lemma 8.7 (Sensor Locality Edge Length): Let r s be the probabilistic sensing range of the sensors, and e(L k ) be the edge length of the sensor locality L k = RC H (O, r/i ), where i ≥ 2. The edge length e(Ck ) of the hexagonal k-covered region Ck = RC H (O, (i − 1)r/i ) is given by e(Ck ) = ei =

(i − 1)rs i

Theorem 8.5 (Minimum Planar Sensor Density): Let k be the degree of coverage, where k ≥ 1, and r s be the probabilistic sensing range of the sensors. The minimum planar sensor density ρi ∗ (k, rs ) for k-coverage of a planar field is computed as follows: 2i ∗ k ρi ∗ (k, rs ) = √ 3 3(i ∗ − 1)2 rs2 2

where i ∗ is given by: i∗ = 1 + 4

w2 w1

with w1 > 0, w2 > 0, and w1 + w2 = 1.

∎

Lemma 8.8 (Relationship between Communication and Sensing Ranges): Let Rs be the probabilistic radius of the communication range of the sensors and rs the probabilistic radius of their sensing range. Under the assumption that probabilistic k-coverage is achieved, the network connectivity is guaranteed if Rs and rs are related by the following inequality: Rs ≥

2 rs i∗

where i ∗ is given by: i∗ = 1 + 4 with w1 > 0, w2 > 0, and w1 + w2 = 1.

w2 w1 ∎

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243

8.5.2 Extension 2: Heterogenous Sensor Deployment Heterogeneous sensors may not have the same features, including their sensing range, communication range, and energy reserves. It is clear that the deployment of heterogeneous sensors is a more realistic scenario. Indeed, in general, real-world networked sensing systems are intrinsically heterogeneous. While sensor heterogeneity may have a positive effect on the global network performance compared to the case of homogeneous sensor deployment, it may also cause some severe problems. Next, we discuss some of the potential challenges, which may arise from heterogeneous sensor deployment, along with the resulting opportunities.

8.5.2.1

Challenges with k-Coverage

It is important to consider joint network connectivity and sensing k-coverage (or simply k-coverage) to solve rigorously and more efficiently the problem of connected k-coverage in planar wireless sensor networks. While network connectivity depends on the radius of the communication range of the sensors, k-coverage depends on the radius of their sensing range. However, joint network connectivity and k-coverage is closely related to both of the radii of the sensing and communication ranges of the sensors. In order to solve the problem of k-coverage of a planar field using a minimum number of sensors, we need to focus on their sensing range. However, the sensors may not necessarily have the same communication range, sensing range, and initial energy supply. In particular, because the sensors may not have identical sensing ranges, we should focus on two extreme cases. In the first case, we consider the smallest radius of the sensing range of the sensors, whereas in the second case, we account for their largest sensing range. Next, we discuss the challenging k-coverage problems that may be caused by these two cases. Case 1: Smallest Sensing Range • Challenging Problem 1: More than k-Coverage The sink slices a planar field into adjacent regular hexagons whose edge length is r min , where r min stands for the smallest radius of the sensors’ sensing ranges. That is, the sink accounts for the least powerful sensors in the deployment field of interest. Consequently, the sensors with a radius of sensing range that is higher than r min would be able to cover other regions in a planar field that are already k-covered by other sensors. Indeed, the presence of more powerful sensors with larger sensing range leads to covering some areas of a planar field, which are already k-covered by other sensors. That is, there areas are not only k-covered by those k selected sensors by the sink, but also by these more powerful sensors, which are not part of this specific set of k selected sensors. In other words, these areas are more than k-covered, thus, yielding a phenomenon of over k-coverage. While this type of scenario provides a guarantee of achieving k-coverage of a planar field, this may have an effect on the performance of the entire network. Also, this effect

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can be aggravated for large planar fields. Indeed, the preprocessing becomes a time-consuming process due to the slicing process of a planar field into a large number of small regular hexagons of edge length r min . Given that each of these regular hexagons should have k selected sensors located in it, the total number of sensors that is necessary to k-cover a planar field is large. However, all k-coverage rounds have always the same number of sensors to ensure k-coverage of a planar field. As it can be seen, heterogeneity becomes a major problem for the whole network performance due to the presence of these least powerful sensors, i.e., sensors whose sensing range has radius equal to r min . Precisely, the operational network lifetime may be shortened due to this increased number of deployed sensors to achieve k-coverage. Case 2: Largest Sensing Range • Challenging Problem 2: Less than k-Coverage The sink slices a planar field into adjacent regular hexagons of edge length r max , where r max is the largest radius of the sensing ranges of the sensors. In other words, the sink takes into consideration the most powerful sensors in the deployment field of interest. In order to achieve k-coverage of a planar field, it is essential that each regular hexagon contains k active sensors. However, given that a set of k sensors may include some least power sensors and/or sensors whose radius of their sensing range is less than r max , some areas in a planar field may not be k-covered. That is, the planar field may have several areas that are less than k-covered as the regular hexagons associated with them cannot be reached by all the sensors located in these regular hexagons, thus, leading to a phenomenon of under k-coverage. This is due to the presence of these sensors, which do not belong to the category of most powerful sensors and, therefore, are unable to cover the whole regular hexagons in which they are located. The only way to k-cover these under k-covered areas is to add more sensors to the underlying regular hexagons. Furthermore, all rounds may not necessarily need the same total number of deployed of sensors to ensure k-coverage of a planar field during the operational network lifetime. Comparison: Over k-Coverage versus Under k-Coverage As it can be observed, the first problem (i.e., over k-coverage) may be more desirable than the second one (i.e., under k-coverage). Although some regions in a planar field are over k-covered, there is a guarantee that the entire field is k-covered using the same number of active sensors in every regular hexagon of edge length r min . Also, a large number of sensors may be needed to guarantee k-coverage of a planar field. However, with the under k-coverage problem, a planar field is not guaranteed to be k-covered by selecting k sensors in each regular hexagon of edge length r max . In fact, there is a need to select more sensors in some of those regular hexagons that contain at least one sensor whose radius of sensing range is less than r max . Consequently, a larger number of sensors is indispensable and all k-coverage rounds may not necessarily require the same number of selected sensors. This additional number of sensors in each round can shorten the lifetime of the sensors as they may be involved more often to cope with the presence of those under k-covered regions

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in a planar field. This will in turn shorten the lifetime of the whole network. The major issue with the under k-coverage problem is that the sink should always check whether every regular hexagon is k-covered, which may degrade the overall network performance.

8.5.2.2

Challenges with Network Connectivity

Challenging Problem 3: One-Way Communications Because of the connected k-coverage problem being addressed in this chapter, both coverage and network connectivity must be jointly considered. Providing k-coverage is not useful if network connectivity is not guaranteed. The latter is needed as it allows the sensors to interact with each other for the proper operation of the whole network. Specifically, the network is deployed to accomplish a certain mission, which can be achieved through the sensed data collection. Given that the sensors are heterogeneous, the radii of their communication ranges may not be the same. These differences may not enable the sensors to communicate with each other. In other words, some sensors cannot establish paths for mutual communications. We should mention that the radii of the sensors’ sensing and communication ranges are proportional to each other. That is, smaller radii of the sensing ranges of the sensors implies smaller radii of their communication ranges. We distinguish the following two cases based on the slicing unit of a planar field into regular hexagons, and study whether the sensors can communicate properly. • Case 1: Slicing Unit = Regular Hexagons of Edge Length rmin When the sink slices a planar field into adjacent regular hexagons of edge length r min , it is guaranteed that all the selected sensors are able to communicate with each other. In other words, they are symmetrically connected to each other so that any pair of selected sensors in any regular hexagon of edge length r min are mutually connected to each other regardless of the radii of their sensing ranges. Any pair of selected sensors in every regular hexagon are able to communicate directly with each other. That is, the communication links between any two sensors, which are selected from every regular hexagon to k-cover a planar field, are symmetric. In particular, the sensors whose sensing range’s radius is r min are capable of communicating directly with all of the sensors in their respective regular hexagon of edge length r min . In summary, the set of sensors, which are selected from all regular hexagons in order to achieve k-coverage of a planar field, form a connected network, where the communication paths between any pair of them are bidirectional. • Case 2: Slicing Unit = Regular Hexagons of Edge Length rmin < r ≤ r max Now, when the sink decides to slice a planar field into a set of regular hexagons of edge length r, where r min < r ≤ r max , there is a major problem. Indeed, there is no guarantee that all the sensors, which are selected to k-cover a planar field, are mutually connected to each other. Precisely, in some regular hexagons, some of the selected sensors whose sensing range’s radius is smaller than r max may

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8 A Planar Regular Hexagonal Tessellation-Based Approach …

not be able to connect directly with other sensors with higher sensing range’s radius. Assume there are two sensors si and sj whose radii of communication range are Ri and Rj , respectively, such that Ri > Rj . Also, assume that sj is within the communication range of si , whereas si is outside the communication range of sj . Thus, while sensor si can send directly messages to sensor sj , sensor sj cannot transmit directly messages to si . To sum up, the communication links between some pairs of selected sensors can be asymmetric. 8.5.2.3

Opportunities

Although heterogeneity may cause a few problems as discussed above, it has several benefits, which help boost the overall network performance. Next, we present the various opportunities, such as extending the network lifetime, avoiding coverage and connectivity holes, and leveraging clustering, which can be offered through the deployment of heterogeneous sensors in order to ensure k-coverage of a planar field. Opportunity 1: Extending the Network Lifetime We can show that the deployment of heterogeneous sensors with respect to their energy reserves helps extend the operational lifetime of the network. It is well known that a low battery power of the sensors may cause their failure. Indeed, the sensors may fail to work properly due to their low battery power. The sensors with high initial energy supplies (or powerful sensors) would be able to stay operational for a longer period of time compared to the other sensors. This would help the network stay functional and extend its lifetime. Precisely, this would delay the possibility of occurrence of sensor failures because of their insufficient energy supplies. Furthermore, there is a high probability that these powerful sensors would be selected more often than the other sensors whose energy reserves are low so they participate in k-covering a planar field. This would help accomplish the mission of the sensing application in a timely manner without any further delay. In other words, avoiding those sensors with low battery power during the selection process of the sensors to k-cover a planar field can reduce the chances of failures of the sensors due to their low energy reserves. Opportunity 2: Coverage/Connectivity-Hole Avoidance It is clear that over k-coverage can overcome the problem of having some areas in a planar field hat are not k-covered. This is due to the fact that the deployment of these powerful sensors helps reach farther areas in a planar field and cover them thanks to their large sensing range. As stated earlier, those selected sensors with low battery power to k-cover a planar field could fail, thus, leaving some regular hexagons in the field less than k-covered. However, the presence of those powerful sensors to participate in the k-coverage process of certain regular hexagons could still help k-cover those less than k-covered regular hexagons. That is, these powerful sensors with respect to their sensing range can help cope with the coverage-hole problem, which in turn may yield a connectivity-hole problem. Now, regarding their

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communication range, these powerful sensors can also shorten the length of the communication paths among other sensors participating in the k-coverage process of a planar field. In other words, a small number of sensors would be included in those communication paths in order to establish connectivity among the sensors kcovering a planar field. Consequently, less energy consumption would be incurred, thus, prolonging the network lifetime. Opportunity 3: Clustering-Based k-Coverage As discussed above, the deployment of heterogeneous sensors has several advantages compared to the use of homogeneous ones. In particular, the presence of powerful sensors provides more opportunities to the whole network. In fact, thanks to their sensing capability, these powerful sensors can help enhance coverage efficiency. For instance, a small number of these powerful sensors are capable of ensuring kcoverage of a planar field compared to the deployment of a large number of less powerful sensors. Achieving k-coverage using fewer sensors yields a significant amount of energy savings, which in turn helps extend the network span. Also, due to their differences in their sensing range capabilities, it may not be always possible to design a centralized k-coverage protocol, which could be run by a central powerful node, such as the sink. Indeed, we show the two problems, i.e., under k-coverage and over k-coverage, which may arise due to the presence of heterogeneous sensors in the same planar field. We believe that this type of heterogeneous network architecture can be better exploited by the design of a pseudo-distributed k-coverage protocol. Specifically, it is more practical to design a clustering-based k-coverage protocol, where the whole planar field is guaranteed to be fully k-covered. In this type of protocol, those powerful sensors could be selected to act as leaders (or cluster heads), where each cluster head would be responsible for k-covering a certain area of a planar field. Clearly, with respect to the region it is responsible for k-covering it, a cluster head can play the role of the sink in a centralized k-coverage protocol. However, all these cluster heads need cooperate and coordinate with each other in order to accomplish their target mission (i.e., guarantee k-coverage of a planar field) successfully. Also, because of their higher communication ranges, these cluster heads should be able to communicate directly with each other in order to ensure full k-coverage of the entire planar field. Thus, this type of clustering-based k-coverage protocol introduces less amount of overhead compared to the one caused by a centralized k-coverage protocol, which would be run by the sink for all of the deployed sensors in a planar field. This helps minimize further the energy consumption of the sensors involved in the kcoverage process. In fact, using the concept of clustering, each cluster head would be in charge of only a small subset of sensors. Our theoretical study above shows that sensor heterogeneity can be presented in three different forms: Differences in their sensing ranges, communication ranges, and energy supplies. Each of these three forms helps reduce the energy consumption of the individual sensors, which helps elongate their lifetime as well as the operational network lifetime. Next, in our simulation study, we account for these various forms of sensor heterogeneity to investigate their effect on the performance of the abovementioned connected k-coverage protocols for planar wireless sensor networks.

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8.6 Performance Evaluation In this section, we specify the simulation setup. Then, we discuss the simulation results of our proposed approaches to the problem of connected k-coverage in wireless sensor networks, i.e., cone-based and perimeter-based approaches.

8.6.1 Simulation Setup Without loss of generality, our planar field is a square of edge length 300 m. We assume that there are 1000 sensors randomly deployed in this field. Also, we assume that the radii of the communication range and sensing range of the sensors are 50 m and 25 m, respectively. Moreover, we use the energy model developed by Ye et al. [435], where the energy consumption by a sensor due to data transmission, data reception, idle mode, and sleep mode are estimated to be equal to 60 mW, 12 mW, 12 mW, and 0.03 mW, respectively. Furthermore, all simulations are repeated 100 times and the obtained results are averaged, using a high-level simulator written in C. We suppose that the initial energy of each sensor is 70 J. We use the IEEE 802.11 distributed coordinated function with CSMA/CA as the underlying MAC protocol. Furthermore, we consider a radio interference model given the pervasiveness of other 2.4 GHz radio sources. As stated in the energy model, which is discussed in Chap. 2 (Sect. 2.5), we take into consideration the four forms of energy dissipation, namely data transmission, data reception, data sensing, sensor mobility, active state (a sensor is in on state, but, not it is not involved in any of the above four activities), and control messages. The latter are indispensable for the correct operation of both of our proposed k-coverage protocols, namely cone-based and perimeter-based sensor selection approaches. We provide simulation results to compare these two approaches with respect to some evaluation metrics, such as the total number of sensors required for k-coverage and network lifetime.

8.6.2 Simulation Results 8.6.2.1

Cone-Based versus Perimeter-Based k-Coverage

As it can be seen in Fig. 8.7, both of our cone-based and perimeter-based k-coverage approaches require the same number of sensors to k-cover a planar field. In fact, both curves coincide. However, Fig. 8.8 show that our cone-based k-coverage approach outperforms our perimeter-based sensor selection approach. Indeed, the perimeterbased method yields higher energy consumption. This extra energy consumed is due to sensor mobility so sensors are located on the edges of the sensor locality L k . In Fig. 8.9, we set the coverage degree k = 3, the radius of the sensors’ sensing

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Fig. 8.7 ρ i ∗ (k, r) versus r

Fig. 8.8 Remaining energy

disk r = 25, and vary the stretching factor i as a function of w1 and w2 , where w1 + w2 = 1. It shows that the planar sensor density ρi ∗ (k, r ) = k/|Ck |∗ decreases for larger values of i. Indeed, increasing i leads to increasing the area of the k-covered region Ck , thus, decreasing ρi ∗ (k, r ). Fig. 8.10 shows that ρi ∗ (k, r ) increases linearly as k increases. Indeed, more sensors are would be needed for higher values of k. However, Fig. 8.11 shows that ρi ∗ (k, r ) decreases as the radius r of the sensing disk increases. In fact, a larger area can be k-covered. Thus, a smaller number of sensors would be needed. These figures show a close-to-perfect match between our simulation and theoretical results. The slight differences are due to the above edge problem.

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Fig. 8.9 ρ i ∗ (k, r) versus i

Fig. 8.10 ρ i ∗ (k, r) versus k

8.6.2.2

Comparison with Existing Approaches

As discussed above, our cone-based k-coverage approach outperforms the perimeterbased k-coverage approach (see Fig. 8.8). In the sequel, we compare the least performing one of the above k-coverage approaches, namely perimeter-based kcoverage approach, against the coverage configuration protocol (CCP) proposed by Xing et al. [425], which is the k-coverage closest protocol to ours. Figs. 8.12, 8.13, and 8.14 show that our perimeter-based k-coverage approach outperforms the k-coverage protocol in [26]. Indeed, Figs. 8.12 and 8.13 show that the total number of sensors needed to k-cover a planar field using our perimeter -based approach is smaller than the one needed by our Reuleaux triangle-based approach

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Fig. 8.11 ρ i ∗ (k, r) versus r

Fig. 8.12 Comparing planar sensor density using r (against Helly-based k-coverage protocol)

[26]. Thus, based on Fig. 8.8 as discussed earlier, this implies that our cone-based kcoverage approach has better performance than the one discussed in Chap5. Also, as shown in Fig. 8.14, given that the total energy consumption depends on the number of sensors that participate to k-cover a planar field, our perimeter-based approach helps extend more the network lifetime compared to its counterpart, which is proposed in Chap. 5. Notice that as our previous approach discussed earlier in Chap. 5 requires a degree of coverage k that is at least 3, i.e., k ≥ 3, we set k = 3 in our comparison study. First, we should mention that our k-coverage protocol presented in Chap. 5, which is based on Helly’s theorem [85] and the geometric properties of the Reuleaux triangle [481], outperforms CCP [425]. Here also, we find that our perimeter-based k-coverage approach outperforms the CCP k-coverage protocol [425] as it can be seen

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Fig. 8.13 Comparing planar sensor density using k (against Helly-based k-coverage protocol)

Fig. 8.14 Comparing remaining energy (against Helly-based k-coverage protocol)

in Figs. 8.15 and 8.16. In fact, the plot in Fig. 8.15 shows that our k-coverage protocol, which is described in Chap. 5, yields smaller planar sensor density compared to the CCP protocol [425]. Also, Fig. 8.16 shows that it consumes less energy compared to the CCP protocol [425], thus leading to more extended operational network span.

8.6.2.3

Using Probabilistic Sensing Model

Figs. 8.17, 8.18, and 8.19 show the impact of the degree of coverage k on the minimum probabilistic sensing range rsmin for different values of the path loss exponent (α = 2, 3, 4), and the detection probability (pth = 0.7, 0.8, 0.9). As it can be seen, rsmin increases

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Fig. 8.15 Comparing planar sensor density using k (against configuration coverage protocol “CCP”)

Fig. 8.16 Comparing remaining energy (against configuration coverage protocol “CCP”)

with k. Indeed, the sensors need to be more powerful in terms of their sensing range to k-cover a planar field. Moreover, rsmin decreases as the detection probability increases. In fact, to achieve the same degree of coverage k, higher detection probability requires less powerful sensors, and, thus, smaller radius of their sensing range. Furthermore, rsmin tends to decrease as α increases. In other words, the free space model (i.e., α = 2), which is characterized by the absence of any obstacles, offers the highest sensing power of the sensors. Next, we provide more results based on the minimum probabilistic sensing range rsmin , the path loss exponent α, the sensor characteristic β, and the detection probability pth , whenever possible. Figs. 8.20, 8.21, and 8.22 show the changing behavior of the probabilistic planar sensor density ρi ∗ (k, rs ) based on the degree of coverage k, threshold probability, pth ,

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Fig. 8.17 Probabilistic sensing range rs versus k (α = 2)

Fig. 8.18 Probabilistic sensing range rs versus k (α = 3)

and path-loss exponent, α. The plot shows that the planar sensor density increases with pth and α. In fact, any increase in pth requires more active sensors in order to provide the same degree of coverage k. We should mention that the proposed probabilistic k-coverage does not seem to be adequate for a degree of coverage k = 1 for a path loss exponent α = 2 regardless of the value of the detection probability, and also for α = 3 and pth = 0.9. As it can be seen from Fig. 8.20, the planar sensor density for α = 2 and k = 1 is the highest one compared to the other cases, where the degree of coverage k = 2, … , 7. Also, Fig. 8.21 shows that the planar sensor density for α = 3 and pth = 0.9 is higher than that for k = 2, 3. However, for α = 4, the trend of the plot in Fig. 8.22 is normal and meets our expectations. In fact, as

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Fig. 8.19 Probabilistic sensing range rs versus k (α = 4)

Fig. 8.20 Probabilistic planar sensor density ρi ∗ (k, rs ) versus k (α = 2)

the degree of coverage k increases, the probabilistic planar sensor density increases proportionally to k. The plots in Figs. 8.23, 8.24, and 8.25 show the impact of the probabilistic sensing range rs on the corresponding planar sensor density ρi ∗ (k, rs ), where the degree of coverage k takes on the values 1, 2, 3, … , 7. Recall that rs depends on k. As stated in Theorem 8.5 above, the probabilistic planar sensor density ρi ∗ (k, rs ) that is necessary for full k-coverage of a planar field is inversely proportional to the probabilistic sensing range rs of the sensors. However, rs is higher for smaller detection probability. Therefore, it is expected that ρi ∗ (k, rs ) is smaller for smaller detection probability regardless of the value of the path loss exponent α. When α = 2, the planar sensor density seems to be higher for k = 1 compared to α = 2, … , 7. This shows that our

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Fig. 8.21 Probabilistic planar sensor density ρi ∗ (k, rs ) versus k (α = 3)

Fig. 8.22 Probabilistic planar sensor density ρi ∗ (k, rs ) versus k (α = 4)

proposed probabilistic connected k-coverage approach does not handle 1-coverage appropriately (i.e., k = 1). Also, for each of the values of the detection probability for α = 2, ρi ∗ (k, rs ) has a minimum value for a specific value of its probabilistic sensing range. For instance, when pth = 0.9, ρi ∗ (k, rs ) reaches its minimum for r s = 8 m. When α = 3, the probabilistic planar sensor density ρi ∗ (k, rs ) increases with the probabilistic sensing range rs , except for pth = 0.9, where a coverage degree k = 1 requires more sensors compared to k = 2, 3. For α = 4, ρi ∗ (k, rs ) increases proportionally to rs , and reaches its minimum for k = 1. The plots in Figs. 8.26, 8.27, and 8.28 below show the achieved coverage degree k versus the total number of deployed sensors, while we vary both of the threshold probability pth , and path-loss exponent α. As it is expected, a larger number of

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Fig. 8.23 Probabilistic planar sensor density ρi ∗ (k, rs ) versus rs (α = 2)

Fig. 8.24 Probabilistic planar sensor density ρi ∗ (k, rs ) versus rs (α = 3)

deployed sensors yields a higher degree of coverage k. Likewise, any increase in pth and α needs a higher number of sensors to achieve the same degree of coverage k. Both experiments show a good match between simulation and analytical results. The plots in Figs. 8.29, 8.30, and 8.31 consider various values of the detection probability pth , and path loss exponent α. They indicate that the number of active sensors that is necessary to ensure 3-coverage increases with the physical characteristic of the sensors’ sensing units β, which is used in the definition of our probabilistic sensing model in Chap. 2 in Sect. 2.3. The plots in Figs. 8.32, 8.33, and 8.34 shown below consider the same degree of coverage k = 3, and path-loss exponent α = 2, while it varies the threshold probability pth , i.e., pth = 0.7, 0.8, and 09. It shows the impact of pth on the network lifetime to provide 3-coverage of a planar field. As it is discussed earlier, larger values of

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Fig. 8.25 Probabilistic planar sensor density ρi ∗ (k, rs ) versus rs (α = 4)

Fig. 8.26 Achieved degree of coverage versus number of deployed sensors (α = 2)

pth necessitate higher numbers of active sensors to k-cover a planar field, which in turn yields higher energy consumption for the correct network operation regarding satisfying the k-coverage requirement.

8.6.2.4

Deploying Heterogeneous Sensors

Here, we study the impact of heterogeneity on our solution to the connected kcoverage problem. The plots in Figs. 8.35, 8.36, and 8.37 provide a comparison between two versions of each of our proposed k-coverage protocols, namely conebased k-coverage approach and perimeter-based k-coverage approach, where all the

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Fig. 8.27 Achieved degree of coverage versus number of deployed sensors (α = 3)

Fig. 8.28 Achieved degree of coverage versus number of deployed sensors (α = 4)

sensors are randomly deployed. While the first version uses homogeneous sensors, the second one deploys heterogeneous sensors. For the homogeneous version, we assume that the sensing range of all the sensors is r = 25 m and that the initial energy of each sensor is 70 J. However, for the heterogeneous version, we assume that the sensing range of the sensors is between r min = 20 m and r max = 30 m with mean value equal to 25 m, and the initial energy of each sensor is between 60 and 80 J with mean value equal to 70 J. As it can be seen from Fig. 8.35, the heterogeneous version of the cone-based k-coverage approach outperforms its homogeneous counterpart as it requires a smaller number of sensors for any coverage degree k. Obviously, the presence of more powerful sensors helps ensure k-coverage of a planar field with

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Fig. 8.29 Number of active sensors versus physical characteristic of sensors’ sensing units β (k = 3, α = 2)

Fig. 8.30 Number of active sensors versus physical characteristic of sensors’ sensing units β (k = 3, α = 3)

a smaller number of active sensors. Also, Fig. 8.36 shows that the heterogeneous version of the perimeter-based k-coverage approach outperforms its homogeneous counterpart. In fact, the former needs a smaller planar sensor density compared to the latter to k-cover a planar field. Moreover, as it is the case for the homogeneous case, Fig. 8.37 shows that the heterogeneous version of the cone-based k-coverage approach outperforms its counterpart of the perimeter-based k-coverage approach. This show that sensor heterogeneity-based deployment has better results than sensor homogeneity-based deployment.

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Fig. 8.31 Number of active sensors versus physical characteristic of sensors’ sensing units β (k = 3, α = 4)

Fig. 8.32 Total remaining energy versus time (pth = 0.7, α = 2)

8.7 Related Work In this section, we review existing approaches for coverage and k-coverage in planar wireless sensor networks. In addition, we present the unique features of our proposed study in this chapter, compared to our work described in Chap. 5. Xing et al. [425] addressed the k-coverage problem in wireless sensor networks, and proposed a protocol, named connected coverage protocol. Huang et al. [210]

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Fig. 8.33 Total remaining energy versus time (pth = 0.8, α = 2)

Fig. 8.34 Total remaining energy versus time (pth = 0.9, α = 2)

proposed distributed protocols for coverage and connectivity. Bai et al. [69] developed an optimal deployment strategy to ensure coverage and 2-connectivity independently of the relationship between the radii of the sensors’ sensing and communication ranges. Gupta et al. [182] provided centralized and distributed algorithms for connected sensor cover, where the network can self-organize its topology when a query is issued, and activate the sensors that are necessary to process the query. Zhang and Hou [447] suggested an optimal geographical density control protocol to keep a small number of sensors active to cover a planar field regardless of the ratio of the sensors’ communication range to their sensing range. Zhou et al. [454] proposed a distributed algorithm using kth-order Voronoi diagram to provide fault

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Fig. 8.35 Cone-based k-coverage approach

Fig. 8.36 Perimeter-based k-coverage approach

tolerance, while ensuring coverage. Abrams et al. [4] proposed a distributed algorithm to partition a wireless sensor network into k covers (i.e., sets of sensors), each of which covers an area. Zou and Chakrabarty [460] investigated the problem of sensor selection for sensing and connectivity, using a connected dominating set-based approach. Huang et al. [206, 207] and Tezcan et al. [377] provided the conditions under which k-coverage implies network connectivity. Abbasi et al. [1] proposed a coverage control method for continuous and potentially long regions and passages. A group of autonomous mobile sensors move within boundaries to ensure optimal coverage. Deng et al. [132] addressed the problem of coverage hole in hybrid industrial wireless sensor networks, and proposed a healing approach using static and

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Fig. 8.37 Cone-based k-coverage approach versus perimeter-based k-coverage approach

mobile sensors. The static sensors detect the coverage holes, whereas the randomly scattered mobile sensors are selected and dispatched to repair them. Gupta et al. [185] investigate the problem of coverage in spatial mobile sensor networks for bounded fields of interest, which yield a border effect problem. They discuss the impact of border effects on the number of sensors for the desired coverage under several mobility models, such as random walk, waypoint mobility models. Hoyingcharoen and Teerapabkajorndet [205] computed the expected detection probability at any arbitrary point, and the expected degree of sink connectivity for any sensor node that cannot directly transmit to the sink. They show that their proposed study can be used to predict the levels of coverage and connectivity. Qin and Chen [331] proposed an area coverage algorithm based on differential evolution with a goal to obtain a given coverage ratio ε. The proposed algorithm exploits binary differential evolution to search for an improved node subset in order to meet the coverage demand. Sun et al. [371] suggested a k-degree coverage algorithm based on optimization nodes deployment. Wei et al. [406] proposed an energy balance and coverage control algorithm to address the problem of network congestion. The latter is due to the generation of redundant data when a target is covered by k sensors. Yu et al. [439] exploited the Reuleaux triangle-based k-coverage approach proposed by Ammari [26] and introduced the concept of contribution coverage area to k-cover a planar field of interest using a small number of sensors. For optimal planar node deployment scenarios for coverage and connectivity in wireless sensor networks, the interested reader is referred to the work done by Wang et al. [403], Gupta et al. [184], Yun et al. [442], and Bai et al. [70]. Tripathi et al. [363] provided a detailed survey regarding the issues of coverage and connectivity in wireless sensor networks. Kumar et al. [243] proposed a probabilistic approach to compute the minimum number of sensors to achieve k-coverage with high probability. They showed that this number is approximately the same for both deterministic and random sensor deployment, if the sensors fail or sleep independently with equal probability. Shakkottai

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et al. [348, 349] provided necessary and sufficient conditions for 1-covered, 1connected wireless sensor grid network. Also, they suggested various algorithms for connectivity and coverage in large wireless sensor networks. Moreover, Xue et al. [427] and Liu [269] investigated the problem of finding the critical node density for preserving connectivity. Li et al. [263] proposed distributed algorithms to optimally solve the best-coverage problem with the least energy consumption. Megerian et al. [294] combined Voronoi diagram and graph search algorithms to solve the best and worst coverage problems using optimal polynomial-time algorithms. Liu et al. [277] proposed a scheduling scheme to provide statistical sensing coverage and network connectivity using a randomized algorithm. Yener et al. [436] proposed a probabilistic Markov model to ensure connectivity and coverage, while minimizing the sensors’ power consumption. While Kumar et al. [243] computed the minimum number of sensors to achieve k-coverage with high probability, we should mention that none of the existing approaches were able to quantify the exact sensor density that is required to k-cover a planar field in a deterministic way (i.e., not a probabilistic way). To the best of our knowledge, only both of our current work in this chapter, and the one in Chap. 5 attempt to compute the sensor density deterministically for planar k-coverage [18, 26]. Furthermore, while the work done by Liu [269] and Xue et al. [427] is to find the critical node density for preserving connectivity, our work investigates the relationship that should exist between the sensing and communication ranges of the sensors for maintaining network connectivity. Our proposed approach in this chapter is different from our previous one presented in Chap. 5. Specifically, this work takes into account the size of the region to be kcovered, denoted by C k , as well as the region where the sensors should be located, denoted by L k , to k-cover C k . We show that there is a correlation between C k and L k . However, in Chap. 5, our focus was on C k only.

8.8 Conclusion In this chapter, we focus on the connected k-coverage problem in planar wireless sensor networks based on the use of regular convex hexagons. To this end, we investigate all planar regular convex pavers in order to identify the best paver, which maximizes the area of the sensing range of the sensors being utilized. Our study is based on our proposed sensing range usage rate metric. Among all he studied planar pavers, we find that the regular convex hexagon is the “best” one as it possesses the maximum sensing range usage rate. Then, we investigate the problem of how to achieve full k-coverage of a planar field and compute the corresponding planar sensor density based on the “best” paver, i.e., regular convex hexagon. Our study considers two parameters, namely the area of the k-covered region, denoted by Ck , and the area of the sensor locality, denoted by L k . We prove that these two parameters are dependent on each other. Furthermore, these two parameters are conflicting. Thus, we find that it is necessary to solve a multi-objective optimization problem,

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which attempts to maximize the areas of Ck and L k , in order to compute the optimum planar sensor density to k-cover a planar field. Precisely, we use a weighted scaleuniform-unit sum approach to compute the optimum values of the edge lengths of Ck and L k , which represent the best trade-off between maximizing Ck and L k at the same time. These two parameters, Ck and L k , are respectively associated with the weights w1 and w2 , which define the levels of interest in maximizing Ck and L k , respectively. Moreover, we determine the corresponding ratio of the communication range R to the sensing range r of the sensors to guarantee network connectivity among all the sensors, which are selected to k-cover a planar field. In addition, we propose an energy-efficient sensor selection approach, which forms the basis of a connected k-coverage protocol for a planar field. Achieving k-coverage using planar wireless sensor networks is based on two steps: First, we slice the planar field into congruent regular convex hexagons Ck . Second, we slice the sensor locality L k into k congruent cones. Furthermore, using a few relaxations of some widely used assumptions, we extend our proposed study to solve the connected k-coverage problem in planar wireless sensor networks. First, we consider a non-deterministic (or probabilistic) sensing model, where the sensors’ sensing capability does not the binary sensing model, where the sensing range of the sensors is represented by a disk. We calculate the minimum k-coverage probability Probkmin , where every point in a planar field is probabilistically k-covered by at least k sensors. Then, we compute the corresponding lower bound on the probabilistic sensing range. Second, we consider sensor heterogeneity, where the sensors do not necessarily have the same features with respect to their sensing ranges, communication ranges, and battery power. We discuss the various challenging problems, which are caused by the deployment of heterogeneous sensors to achieve k-coverage of a planar field. Also, we present the opportunities offered, such as energy savings and clustering, which help extend the network lifetime through deploying heterogeneous sensors for the same goal.

Chapter 9

A Planar Irregular Hexagonal Tessellation-Based Approach for Connected k-Coverage

The mind is sharper and keener in seclusion and uninterrupted solitude. No big laboratory is needed in which to think. Originality thrives in seclusion free of outside influences beating upon us to cripple the creative mind. Be alone, that is the secret of invention; be alone, that is when ideas are born. Nikola Tesla (1856–1943)

Overview This chapter focuses on the connected k-coverage problem in planar (or two-dimensional) wireless sensor networks using a hexagonal tiling-based approach. First, it investigates a planar convex tile that best approximates the sensing range of the sensors, while maximizing the percentage being used of this sensing range. Then, it presents several sensor placement strategies, based on the degree of coverage k, where k is a non-zero natural number. Also, it computes the planar sensor density for each of these sensor placement strategies. In addition, it discusses a more general sensor placement strategy based on our introduced family of irregular hexagons, which are capable of tiling the Euclidean plane. Moreover, it derives the corresponding planar sensor density. In addition, it computes the relationship that should exist between the sensing range r and communication range R of the sensors for each of those sensor placement strategies. Furthermore, it presents various simulation results to corroborate our analysis.

9.1 Introduction Regardless of the target sensing applications, coverage remains one of the crucial tasks in the design, analysis, and development of wireless sensor networks. There are various classes of coverage problems, each of which has its own algorithms and protocols. This classification of coverage problems along with their required algorithms and protocols is based on the combination of four attributes, namely extent (partial vs. full), multiplicity (single vs. redundant), type (deterministic vs. probabilistic), and dimensionality (planar vs. spatial). The appropriate coverage is guided by the nature of the underlying sensing applications, which may have different © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_9

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coverage requirements, and may need combinations of these attributes. In addition, on-demand coverage may require the sensors to have some mobility support so they can move to specific regions of interest in a planar field and cover them. In general, sensor mobility is required for sparsely deployed wireless sensor networks, where the number of deployed sensors is not enough to cover the entire field. However, regardless of the class of the coverage problem being addressed and the particular requirements of the underlying sensing application, it is imperative to minimize the total number of sensors to successfully achieve the requested coverage service. As discussed earlier in Chap. 1, it is well known that the resources of the sensors are crucial, and, especially, their sensing range, communication range, battery power (or energy), processing power, and storage space. In particular, the battery power of the sensors is the most crucial resource as it is a determinant factor of their individual lifetime. The latter decisively affects the operational lifetime of the whole network. The problem of coverage of a planar field is an interesting one, and has similarity with the tiling problem in the Euclidean two-dimensional space [167, 180]. The latter is a fundamental yet challenging problem in a much more concrete branch of mathematics, called elementary geometry, in Euclidean space of two dimensions. The main question related to tiling in a Euclidean plane can be stated as follows: How can a Euclidean plane be tiled by replicas of a set (or tiles)?

It is worth mentioning that this is an instance of the second part of Hilbert’s eighteenth problem [300], which he formulated as follows: Is there a polyhedron that admits only an anisohedral tiling in three dimensions, i.e., a polyhedron that tiles a three-dimensional Euclidean space, but does not admit an isohedral (tile-transitive) tiling?

In this chapter, we restrict our attention to planar convex tiles, and consider covering a planar field such that the interiors of the tiles are pairwise disjoint. In addition, these planar convex tiles have the property of guaranteeing an edge-toedge tiling of a planar field. In other words, the intersection of any pair of planar tiles is either empty (for non-adjacent tiles) or an edge of each (for adjacent tiles). Also, we focus on planar monohedral tiling. That is, we assume that the planar convex tiles are the same in terms of their shape as well as the size of their area. Furthermore, a planar field refers to a finite two-dimensional area that has two finite dimensions, namely length and width. As mentioned above, these low-power devices (or sensors) have limited amount of energy (or battery power), which is critical to their lifetime. Consequently, it is essential to investigate the planar convex tile that best approximates the sensors’ sensing range. This will help us solve the problem of coverage in planar wireless sensor networks the most efficient way, therefore, maximizing the operational lifetime of the whole network. In particular, hexagonal tiling (or hexagonal tessellation) provides a tiling of the Euclidean plane, where three adjacent hexagons meet at each vertex. This property holds regardless of whether the underlying hexagonal tile is either regular or irregular. It is worth mentioning that hexagonal tiling is the densest way to pack circles on the Euclidean plane, where each circle is tangent to six other

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ones, thus, minimizing the total gap area of the plane. We should mention that the problem of covering a set of points with circles was initially introduced by Kershner [228]. The interested reader is referred to the book of Pach and Sharir [319] (Chap. 8) and the work of Toth [381] on k-fold coverings for a rigorous analysis. Fortunately, we find that the above property is reversible for regular hexagonal tiles. In fact, our analysis identifies the regular hexagon as the geometric shape that can be inscribed in a circle while occupying the maximum size of its area, compared to the other well-known regular convex planar tiles, i.e., square and equilateral triangle. We give a detailed discussion of this analysis in one of the next sections. In this chapter, we exploit the concept of hexagonal tiling in order to solve the problem of coverage in the Euclidean plane [16, 19]. In Chap. 5, we provide a thorough study of the connected k-coverage problem and compute a bound on the planar sensor density (i.e., number of sensors per unit area) that is required to k-cover a planar field. More precisely, we benefit from Helly’s Theorem [85], which is a powerful theorem of convexity theory, and exploit the geometric properties of a structure, called Reuleaux triangle [481], in order to quantify the planar sensor density that ensures k-coverage of a planar field. However, we should exploit the optimality guaranteed by hexagonal tiling of the Euclidean plane, which is not considered in our study presented earlier in Chap. 5. We believe that it is important that the k-coverage problem in planar wireless sensor networks be studied and analyzed using a hexagonal tiling-based approach in order to find an efficient yet practical solution to this problem. In this chapter, we study and analyze the problem of connected k-coverage in planar wireless sensor networks using a hexagonal tiling-based approach of the Euclidean plane. This work is motivated by the lack of such study and analysis in the literature as well as in the author’s previous work to k-cover a planar field. We should mention that there are several lines of research, including the art gallery problem and the watchman route problem, which are also within the category of deterministic approaches of sensor deployment for coverage. Next, we provide an overview of each of these two problems and compare them with the k-coverage problem addressed in this chapter. The Art Gallery Problem: It aims at finding the smallest number of guards necessary to cover an art gallery, which has the shape of a polygon with n vertices. In other words, the art gallery problem can be formulated in geometry as computing the minimum number of guards required to be placed in an n-vertex polygon so that every interior point is visible. That is, the line segment joining a guard and an interior point lies inside the polygon (i.e., both of the guard and interior point are mutually visible). It is worth mentioning that [n/3] guards are occasionally necessary and always sufficient to cover a polygon with n vertices. It is worth mentioning that art gallery problem was one of the earliest and most influential problems in sensor placement. The Watchman Route Problem: It consists of finding a route within a polygon so that every interior point in the polygon is visible from at least one point along the

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route. This problem can also be defined as designing a route that a watchman follows to ensure that all is in place. The goal is to minimize the route length. As it can be seen, there is a similarity between the art gallery problem and the k-coverage problem. However, the latter is more general than the former in the sense that every point in a planar field, regardless of its shape, can be “visible” or sensed by at least k sensors simultaneously. Indeed, a planar field can be circular. Here, the sensor’s visibility is defined in terms of the distance between a sensor and a point in the planar field. This distance cannot exceed the radius of the sensor’s sensing range to guarantee that a point can be sensed by this sensor. If our focus should be on the sensed data collection, we need to have special sensors, called mobile proxy sinks, which will be responsible for gathering the sensed data from the source sensors and their delivery to the static sink. In this case, a mobile proxy sink should find the shortest route to collect the sensed data from the source sensors in order to minimize its energy consumption due to its mobility. This shows the commonality between the watchman route problem and the k-coverage problem. Next, we provide a short description of the major tasks we want to accomplish in this chapter. In addition, we briefly state how to achieve each one of them.

9.1.1 Major Tasks As discussed earlier, we identify several instances of the coverage problem in wireless sensor networks based on the combination of the above-mentioned attributes: Extent, multiplicity, type, and dimensionality. In this chapter, we focus on full, redundant, deterministic, and planar coverage in densely deployed wireless sensor networks. More specifically, we want to investigate the k-coverage problem in planar wireless sensor networks, where each point in a planar field is k-covered (i.e., covered by at least k sensors simultaneously), where k ≥ 1 stands for the degree of coverage. In addition, we assume that the deployed sensors are homogeneous, i.e., having the same characteristics in terms of their sensing range, communication range, and initial amount of energy. Although we have a homogeneous sensor model, this problem is still challenging because of the geometric properties of the sensors’ sensing range, which is represented by a disk. This is due to the fact that it is impossible to tile a planar field with disks placed tangentially to each other. This would lead to some gap areas between adjacent disks. Indeed, the regular triangle, square, and regular hexagon are the only regular polygons that can tile a two-dimensional Euclidean plane [102]. In this chapter, we aim at efficiently solving the connected k-coverage problem in planar wireless sensor networks by accomplishing certain key tasks. Next, we state these tasks and present our corresponding plan of actions. First, we want to identify the largest shape that can be contained in the sensing range of the sensors and that is capable of tiling a planar field. To achieve this goal, we benefit from the process of tiling the Euclidean plane to solve the k-coverage problem in planar wireless sensor networks. Our study shows that hexagonal tiling

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271

is an essential component to find an efficient solution to the k-coverage problem. Recall that in Chap. 8, we consider regular hexagon-based approaches to achieve kcoverage of a planar field. Contrary to the study presented in Chap. 8, in this chapter, we focus on the concept of irregular hexagon so as to k-cover a planar field with a small number of sensors. Second, our goal is to determine the “best” placement method of the sensors in a planar field so that every point in the field is k-covered using a small number of sensors, where k ≥ 1. To this end, we suggest a few sensor placement strategies based on the degree of coverage k that is required by sensing applications. Then, we propose a more general one using a specific shape of irregular hexagon, which we denote by I R H (r/n), where r stands for the radius of the sensing disk of the sensors, and n ≥ 2 is a natural number, which will be specified later on. We prove by mathematical induction that I R H (r/n) is capable of tiling the Euclidean plane. Third, we should compute the planar sensor density (i.e., number of sensors per unit area) that is required to ensure k-coverage of a planar field. We consider each of the proposed sensor placement strategies and compute the corresponding planar sensor density. Also, we derive the planar sensor density of the above-mentioned more general sensor placement strategy that is based on the concept of irregular hexagon, denoted by R H (r/n), where n ≥ 2. We want to determine the relationship that must hold between the sensing range and the communication range of the sensors so that k-coverage implies network connectivity. We compute this relationship for each of the above sensor placement strategies. Fourth, we want to assess the performance of the theoretical study of the connected k-coverage in planar wireless sensor networks. Thus, we provide various simulation results of the proposed connected k-coverage protocol to corroborate our theoretical analysis.

9.1.2 Chapter Organization The remainder of this chapter is structured as follows: Sect. 9.2 studies and analyzes various planar geometric convex polygons that are capable of tiling the Euclidean plane. Section 9.3 analyzes and attempts to solve the k-coverage problem using hexagonal tiling, and shows the limitations of the proposed solution. Based on this analysis, Sect. 9.4 proposes to use a more efficient and practical solution to the k-coverage problem based on irregular hexagonal tiling of a planar field. Precisely, it suggests and investigates a few sensor placement strategies to solve the k-coverage problem in planar wireless sensor networks, where k ≥ 1, and computes the corresponding planar sensor density. Also, it determines the relationship that must hold between the sensors’ sensing and communication ranges. Section 9.5 discusses the corresponding k-coverage protocol, called I R H Ck . Section 9.6 presents various simulation results to assess the performance of I R H Ck to k-cover a planar field. Section 9.7 provides

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an overview of the literature that solves the problem of k-coverage of a planar field. Finally, Sect. 9.8 concludes the chapter.

9.1.3 Planar Tiling Using Congruent Tiles We want to identify, among well-known shapes, the convex polygonal tile that covers the maximum space of the sensors’ sensing range, i.e., disk of radius r . This convex polygonal tile is used in our analysis of the k-coverage problem in planar wireless sensor networks. To this end, it is essential to compute the maximum area of a convex polygon that could be enclosed in the sensors’ sensing disk. Theorem 9.1 given below states the necessary condition to obtain this maximum area. Theorem 9.1 (Enclosed Convex Polygon’s Maximum Area): The width of the largest enclosed convex polygon, denoted by w(L EC P), in the sensing disk of the sensors must be equal to the diameter of this disk, i.e., w(L EC P) = 2r . Proof We can proceed by contradiction. Let T be a convex polygonal tile to be enclosed in the sensing disk S Dr of the sensors. Assume that the width of T is less than 2r , i.e., w(L EC P) < 2r . This implies that the distance between the farthest points (or vertices) on T is less than 2r . That is, some of the vertices of T do not touch the boundary of the sensing disk S Dr . Thus, it is possible to stretch T so it occupies more area in this disk. This contradicts our assumption made above. Hence, ∎ we should have w(L EC P) = 2r . Lemma 9.1 computes the number of sensors required to cover a planar field with congruent regular polygons, which can tile the Euclidean plane. Lemma 9.1 (Required Number of Sensors): The total number of sensors that are needed to cover a planar field, where the sensing range of the sensors is restricted to congruent regular polygonal tiles (CRPT), is denoted by n tot (C R P T ) and computed as: n tot (C R P T ) =

A(FoI ) A(C R P T )

where A(FoI ) is for the area of a planar field, and A(C R P T ) is the area of CRPT. Proof Based on our assumption, we restrict the sensors’ sensing range to convex polygonal tiles that have the same shape and size. Given that these convex polygons do not create any overlap or any gaps, the total number of sensors, n tot (C R P T ), which is necessary to tile a planar field should be equal to the area of this field divided by the area of a convex polygonal tile. That is, we have: n tot (C R P T ) =

A(FoI ) A(C R P T )

∎

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In Chap. 8, we study and analyzed a set of planar convex, geometric shapes (or convex polygons) that are capable of tiling (or covering without overlap or gap) the Euclidean plane. The study and analysis of these planar convex tiles are essential for the study of the k-coverage problem in planar wireless sensor networks. Precisely, we focus on the three planar regular convex polygonal tiles, namely equilateral triangle, square, and regular hexagon. For each one of them, we derive the associated quality of coverage, which is measured in terms of the metric, called sensing area usage rate. Also, to investigate these three planar convex tiles, we introduce a similar metric, called quality of coverage (Definition 2.31, Sect. 2.2, Chap. 2), which is linearly proportional to the overlap rate of a sensor’s sensing disk with those of its neighbors [19]. Both studies aim at finding the convex polygonal tile among the above-mentioned three planar convex polygons that has the maximum quality of coverage. In Chap. 8, we prove that the regular hexagon is the best tile (or paver) compared to the triangle and the square. In the sequel, we consider both of regular and irregular hexagons to study the k-coverage problem in planar homogeneous wireless sensor networks. In Sect. 9.4, we show that there are irregular hexagons with specific shapes that are also capable of tiling the Euclidean plane.

9.2 Achieving Planar k-Coverage Using Hexagonal Tiles In the context of this chapter, the sensing range of the sensors in the Euclidean plane is modeled by a convex polygon. The latter is discussed below in detail. In this section, we restrict the sensing range of the sensors to the hexagonal shape because it possesses the maximum coverage quality as stated above. Our goal is to investigate the most efficient sensor placement strategy to achieve k-coverage of a planar field, and derive the corresponding planar sensor density. Our analysis is mainly based on the concept of hexagonal tiling. Precisely, we consider two different methods for tiling a planar field. In the first method, we use regular hexagons, whereas in the second one, we use irregular hexagons. Consequently, an instance of our k-coverage problem in planar wireless sensor networks can be formulated as follows: k-Coverage Problem Instance: Given a grid of regular or irregular hexagons representing a planar field, and a set of sensors, how should those sensors be placed on this grid so that every hexagon is k-covered with at least k sensors?

We should mention that those specific hexagons are built in such a way that they are, indeed, k-covered by the sensors arranged in a specific way. Both of the shapes of regular and irregular hexagons as well as the sensor placement strategy will be discussed in detail in this section (i.e., Sect. 9.3) and Sect. 9.4. Next, we attempt to solve this problem using each of the above-mentioned hexagonal tiling methods. We want to minimize the total number of sensors to k-cover a planar field, while maximizing the network operational lifetime.

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Fig. 9.1 Tiling with congruent regular hexagons

We consider two cases based on the value of k: k = 1 and k ≥ 2. For each case, we specify the tiling approach of a planar field along with the sensor placement strategy that is suitable to k-cover it, while using as small number of sensors as possible. Figure 9.1 shows a tiling grid using regular hexagons of edge length r .

9.2.1 Ensuring 1-Coverage We slice a planar field into regular hexagons (rh) whose side length is r and area is √ Ar h (r ) = 3 2 3 r 2 , where r is the radius of the sensors’ sensing range. Each hexagon contains exactly one sensor located at its center. Figure 9.2 shows one sensor positioned at point A, which is the center of the corresponding hexagon (colored in blue). This type of sensor placement ensures coverage of each hexagon, thus, covering the dep whole field. The corresponding deployed planar sensor density, denoted by τr h (r ), is given by: dep

τr h (r ) =

2 0.38 1 = √ = 2 2 Ar h (r ) r 3 3r

The six yellow curved areas shown in Fig. 9.2 represent the portion of the sensors’ sensing range that is wasted. As discussed earlier, this is the overlap area of a sensor’s sensing range with those of its six adjacent ones. In reality, a sensor should be able to cover the entire circle whose area is πr 2 . Thus, theoretically, the actual planar

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Fig. 9.2 Area 1-covered—regular hexagon of edge length r

sensor density, denoted by τract h (r ), should be: τract h (r ) =

0.32 1 = 2 2 πr r

It is clear that the reduction of the sensors’ sensing range from a disk of radius r to a regular hexagon of edge length r incurred an extra planar sensor density, denoted by Δ r h (r ), which is given by: dep

Δ r h (r ) = τr h (r ) − τract h (r ) =

0.07 r2

As it can be seen in Fig. 9.2, the distance between any pair of neighboring sensors is exactly 2r . Thus, the network is guaranteed to be connected if the radius R of the communication range of the sensors satisfies the following inequality: R ≥ 2r

9.2.2 Ensuring k-Coverage Now, we consider a degree of coverage k ≥ 2. We divide a planar field into regular hexagons whose side length is r/2, where r is the radius of the sensing range of the sensors. Thus, the width of each of these hexagons is equal to r . As shown in Fig. 9.3, each regular hexagon consists of six (6) equilateral√triangles whose edge length is ( )2 equal to r/2, and has an area equal to Ar h (r ) = 3 2 3 r2 . Given that the width of a hexagon is r , we can place k sensors anywhere inside a hexagon to k-cover it, thus, covering the whole planar field. This type of sensor placement gives a deployed dep planar sensor density, denoted by σr h (r, k), which is equal to:

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Fig. 9.3 Area k-covered—regular hexagon of edge length r/2

dep

σr h (r, k) =

8k 1.54 × k k = √ = Ar h (r ) r2 3 3r 2

Here also, there is a portion of the sensors’ sensing range not being exploited. More precisely, those k sensors can also k-cover the six (6) yellow curved regions of equal size, each of which is associated with an equilateral triangle, as shown in Fig. 9.3. In other words, those k sensors can actually k-cover a circular area of radius r/2. Thus, theoretically, the actual planar sensor density, denoted by σract h (r, k), should be: σract h (r, k) =

4k 1.27 × k k = ( r )2 = 2 πr r2 π 2

The corresponding extra planar sensor density, Δ r h (r, k), can be calculated as follows: dep

Δ r h (r, k) = σr h (r, k) − σract h (r, k) =

0.27 r2

In order to maintain network connectivity, any pair of sensors in two adjacent hexagons should be able to communicate with each other. As shown in Fig. 9.3, in the worst case, the two farthest sensors from each other in two adjacent hexagons should be 4 × (r/2) = 2r distant from each other. Consequently, this type of k-coverage configurations ensures network connectivity if the relationship below between r and R holds: R ≥ 2r Next, we study the problem of k-coverage using planar tiling with irregular hexagons. We show that the use of irregular hexagons is more efficient than that of

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regular hexagons. More specifically, we prove that achieving k-coverage using irregular hexagons helps reduce the planar sensor density by more than half compared to regular hexagons.

9.3 Achieving Planar k-Coverage Using Irregular Hexagonal Tiles The tiling problem becomes more challenging if a hexagon does not have to be regular. Let us investigate the case of tiling the plane with irregular hexagons (irh). First, we present our approach for sensor placement. Sensor placement approach: Our proposed solution is to place the sensors around the centers of the irregular hexagons of the tiling grid so they cover more space. To this end, the sensors should be placed on the edges of the equilateral triangles forming the irregular hexagons and located closer to their centers.

Next, we investigate some specific sensor placement strategies to k-cover a planar field, and compute the corresponding planar sensor density. Then, we study a more general sensor placement strategy, and derive the associated planar sensor density.

9.3.1 Irregular Hexagonal Tiling with I R H(r/2) Our goal is to determine the shape of the irregular hexagon that is k-covered with k sensors. To this end, we consider the following analysis. Here, we divide a planar field into regular hexagons whose side length is r/2, where r is the radius of the sensors’ sensing range. Each of those regular hexagons is dissected into six equilateral triangles of edge length r/2, as shown in Fig. 9.4. We can achieve a better bound on the planar sensor density to ensure k-coverage of a planar field if we place k sensors on the common edge of two adjacent regular hexagons of side length r/2. This Fig. 9.4 Area k-covered—I R H (r/2)

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procedure is conform to our sensor placement approach presented earlier. Clearly, those k sensors are able to k-cover an area that consists of those two adjacent regular hexagons (i.e., colored in green and orange) and the four equilateral triangles of edge length r/2, which are located above and below them (i.e., colored in blue). This entire k-covered area (i.e., two adjacent regular hexagons and four equilateral triangles) with k sensors forms an irregular hexagon, denoted by I R H (r/2). Each of those two regular hexagons consists of six equilateral triangles of edge length r/2. Thus, the region that is k-covered with exactly k sensors, i.e., I R H (r/2), consists of 16 equilateral triangles of edge length r/2. Figure 9.5 shows the shape of the irregular hexagon, I R H (r/2). After identifying the shape of the irregular hexagon I R H (r/2) that is k-covered with k sensors, we slice a planar field into I R H (r/2). Figure 9.6 shows a tiling grid using I R H (r/2). As stated above, this irregular hexagon has 16 equilateral triangles Fig. 9.5 Our irregular hexagon I R H (r/2)

Fig. 9.6 Tiling grid using I R H (r/2)

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√ ( ) 2 of edge length r/2, each of which has an area equal to Aet (r ) = 43 r2 . Thus, dep the corresponding deployed planar sensor density, denoted by ψir h (r, k), which is associated with this sensor placement strategy, is given by: dep

ψir h (r, k) =

k 0.58 × k k =√ = 2 16 × Aet (r ) r2 3r

To be precise, those k sensors k-cover also those six yellow curved regions as shown in Fig. 9.4. Four of these curved regions have the same shape and area, while the other two ones are smaller and have the same shape and size. In other words, theoretically, the k-covered area is the intersection area of those two circles CG and CH centered at G and H , respectively (i.e., the sensing disks of the sensors si and s j located at G and H , respectively). The area of this k-covered area, denoted by Ak−cov (r, k, δ(G, H )), is computed as follows [410]: Ak−cov (r, δ(G, H )) = Ak−cov (r ) ( ) / δ(G, H ) 2 −1 δ(G, H ) − = 2r cos 4r 2 − δ(G, H )2 2r 2 where cos−1 is the inverse cosine function, and δ(G, H ) stands for the Euclidean distance between points G and H . Given that δ(G, H ) = r/2, we obtain: Ak−cov (r ) = 2.15 × r 2 Thus, the actual planar sensor density, denoted by ψiracth (r, k) and corresponding to this sensor placement strategy, is given by: ψiracth (r, k) =

k Ak−cov (r )

=

0.46 × k r2

The corresponding extra planar sensor density, Δ i r h (r, k), is given by: dep

Δ i r h (r, k) = ψir h (r, k) − ψiracth (r, k) =

0.12 × k r2

Consider the irregular hexagon shown in Fig. 9.7. According to Kershner [227], such an irregular hexagon can tile the plane if and only if one of the three following conditions holds, as stated in his theorem given below: Kershner’s Theorem (Planar Tiling with Irregular Hexagon) [227]: An irregular hexagon can pave the plane if and only if it is one of the three following types: • Hexagon of Type 1: ∠A + ∠B + ∠C = 2π, a = d. • Hexagon of Type 2: ∠A + ∠B + ∠D = 2π, a = d, c = e. • Hexagon of Type 3: ∠A = ∠C = ∠E = 2π/3, a = b, c = d, e = f .

∎

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Fig. 9.7 Irregular hexagon

It is worth mentioning that a substantial and complete work on hexagons was done by Reinhardt in his doctoral thesis [337]. He found that there were exactly three types of irregular hexagons that tile the plane, which are described above in Kershner’s Theorem. Lemmas 9.2 and 9.3 show that our irregular hexagon I R H (r/2) can tile the Euclidean plane. Lemma 9.2 (Planar Tiling with I R H (r/2)): Our irregular hexagon I R H (r/2) is a planar tile. Proof Consider Fig. 9.8 showing our irregular hexagon I R H (r/2). The latter is a hexagon of type 1 according to Kershner’s Theorem. First, all the 16 triangles forming our irregular hexagon I R H (r/2) are equilateral. Therefore, we have: ∠A = ∠B = ∠C = 120◦ , i.e., ∠A + ∠B + ∠C = 2π . Second, the length of each edge of this equilateral triangle is r/2. Thus, we have a = d = r . Consequently, our irregular ∎ hexagon I R H (r/2) tiles the plane. Lemma 9.3 (Irregular Hexagon I R H (r/2)): Our irregular hexagon I R H (r/2) is also a hexagon of type 2 as defined by Kershner’s Theorem above. Proof Let us consider our irregular hexagon I R H (r/2) shown in Fig. 9.8. If it undergoes a counter-clockwise rotation of an angle of π/3, we obtain a hexagon of type 2 as shown in Fig. 9.9. In fact, we have ∠A + ∠B + ∠C = 2π , a = d = 2r/3, ∎ and c = e = r . Thus, I R H (r/2) is a type 2 hexagon.

9.3.2 Irregular Hexagonal Tiling with I R H(r/3) Now, we divide a planar field into regular hexagons of side length is r/3. In this case, we propose to place k sensors on the edge of an equilateral triangle whose side length is r/3 as shown in Fig. 9.10. Following the same analysis given above, the total area that is being k-covered with those k sensors is an irregular hexagon,

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Fig. 9.8 Our irregular hexagon I R H (r/2) (type 1)

Fig. 9.9 Our irregular hexagon I R H (r/2) after a counter-clockwise rotation of π/3 (Type 2)

denoted by I R H (r/3), which consists of all those 42 equilateral triangles of edge √ ( ) 2 length r/3, each of which has an area equal to Aet (r ) = 43 r3 . Figures 9.11 and 9.12 show the structure of I R H (r/3) and a tiling grid using I R H (r/3), respectively. dep The deployed planar sensor density, denoted by ψir h (r, k), which corresponds to this sensor placement strategy is given by: dep ψir h (r, k)

√ 2 3k 0.49 × k k = = = 2 42 × Aet (r ) 7r r2

As discussed earlier, theoretically, the actual total k-covered area is the intersection area of the sensing disks CG and CG of sensors si and s j located at the centers G and H , respectively, as shown in Fig. 9.10. This k-covered area has a size, denoted by

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Fig. 9.10 Area k-covered—I R H (r/3) Fig. 9.11 I R H (r/3)

Fig. 9.12 Tiling grid using I R H (r/3)

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283

Ak−cov (r ), which is computed as follows [410]: Ak−cov (r ) = 2r cos 2

−1

(

δ(G, H ) 2r

)

/ δ(G, H ) − 4r 2 − δ(G, H )2 2

Given that δ(G, H ) = r/3, we obtain: Ak−cov (r ) = 2.48 × r 2 Thus, the actual planar sensor density, denoted by ψiracth (r, k), which corresponds to this sensor placement approach is calculated as: ψiracth (r, k) =

k Ak−cov (r )

=

0.40 × k r2

The corresponding excess of planar sensor density, Δ i r h (r, k), is given by: dep

Δ i r h (r, k) = ψir h (r, k) − ψiracth (r, k) =

0.09 × k r2

Lemma 9.4 computes the relationship between the radii of the sensing and communications ranges of the sensors to maintain network connectivity. Lemma 9.4 (Connected Hexagonal k-Covered Wireless Sensor Networks Using I R H (r/3)): A hexagonal k-covered wireless sensor network using I R H (r/3) is connected if the following relationship between the radii R and r holds: √ 2 7 r R≥ 3 Proof Let us consider the configuration given in Fig. 9.13. To compute the radius of the sensors’ communication range that is required to maintain connectivity among all the sensors in the network, it is necessary to compute the farthest distance between any pair of sensors in two neighboring irregular hexagons. Let G 1 and H1 be the locations of sensors si and s j in the irregular hexagon I R H1 . The farthest sensor sl in I R H3 from si must be located at position H3 . Using Pythagore’s Theorem, the Euclidean distance between G 1 and H3 , denoted by δ(G 1 , H3 ), can be computed as follows: / δ(G 1 , H3 ) = δ(G 1 , I1 )2 + δ(I1 , H3 )2 √ √ Given that δ(G 1 , I1 ) = 3h = 3r/2 and δ(I1 , H3 ) = 3r/2, where h = r/2 3 is the height of an equilateral triangle whose side length is r/3, we obtain: δ(G 1 , H3 ) =

√

3r = 1.732r

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Fig. 9.13 Distance for connectivity of a hexagonal k-covered wireless sensor network using I R H (r/3)

Also, the farthest sensor sm in I R H4 from si must be located at position H4 . Thus, the Euclidean distance between G 1 and H4 , δ(G 1 , H4 ), is given by: δ(G 1 , H4 ) = 6h =

√

3r = 1.732r

Now, the farthest sensor s p in I R H2 from s j should be located at position G 2 . Likewise, using Pythagore’s Theorem, the Euclidean distance between H1 and G 2 , δ(H1 , G 2 ), is computed as follows: / δ(H1 , G 2 ) =

δ(H1 , I2 )2 + δ(I2 , G 2 )2

However, δ(H1 , I2 ) = 43 r and δ(I2 , G 2 ) = 4h =

√2 r , 3

we get:

√ 2 7 r = 1.764r δ(H1 , G 2 ) = 3 In order to maintain network connectivity, the radius R of the communication range of the sensors should satisfy: √ 2 7 r R ≥ max{δ(G 1 , H3 ), δ(G 1 , H4 ), δ(H1 , G 2 )} = δ(H1 , G 2 ) = 3 dep

∎

Notice that the deployed planar sensor density ψir h (r, k) associated with our irregular hexagon I R H (r/3) is smaller than its counterpart associated with I R H (r/2). The same observation holds for both of the actual planar sensor density ψiracth (r, k), and the excess in planar sensor density Δ i r h (r, k). Likewise, we find that the results obtained with I R H (r/4) are better than those produced using I R H (r/3), and the

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285

results associated with I R H (r/5) outperform those associated with I R H (r/4). Consequently, it is essential to study the more general case of our irregular hexagon, denoted by I R H (r/n), to solve more efficiently the k-coverage problem in planar wireless sensor networks. Thus, our goal is to investigate ) the optimum value of n, ( denoted by n opt , which is associated with I R H r/n opt that yields the deployed planar sensor density. This is discussed in detail in Sect. 9.4.3.

9.3.3 Irregular Hexagonal Tiling with I R H(r/n) dep

It is worth noting that the planar sensor density ψir h (r, k) depends on the Euclidean distance δ(G, H ) between points G and H . As mentioned above, those k sensors are to be placed on the line segment connecting G and H , and whose length is δ(G, H ). Therefore, δ(G, H ) is a key parameter of the k-coverage problem. Let δ(G, H ) = r/n. As noticed from the two previous cases (i.e., n = 2 and n = 3), our irregular hexagon I R H (r/n) has a specific structure in terms of the length of each of its six edges, i.e., (E, F), (F, A), (A, B), (B, C), (C, D), and (D, E), the number of its rows, and the number of equilateral triangles per row. Table 9.1a shows the structure of our irregular hexagon I R H (r/n), and Table 9.1a, b gives the number of equilateral triangles per each of its rows, both for n = 2, 3, 4, 5. Based on the results listed in Table 9.1a, b given, we can extend them to the more general case of our irregular hexagon I R H (r/n), where the edge length of the equilateral triangle is r/n. Table 9.2a shows those general results for I R H (r/n). Notice that Table 9.2b is symmetric with respect to the n th row of I R H (r/n). First, let us prove that I R H (r/n) can tile the Euclidean plane. Theorem 9.2 states and proves this result.

Table 9.1 Structure of our irregular hexagon I R H (r/n) for some values of n (a) n

(E, F) (F, A) ( A, B) (B, C) (C, D) (D, E) # Rows

#EquilateralT riangles

2

r/2

r

r

r/2

r

r

3

16

3

2r/3

r

r

2r/3

r

r

7

42

4

3r/4

r

r

3r/4

r

r

9

80

5

4r/5

r

r

4r/5

r

r

11

130

Row6

Row7

(b) n

Row1

Row2

Row3

2

5

6

5

3

7

9

4

9

11

5

11

13

Row4

Row5

10

9

7

13

14

13

11

9

15

17

18

17

15

Row8

Row9

13

11

286

9 A Planar Irregular Hexagonal Tessellation-Based Approach …

Table 9.2 General structure of our irregular hexagon I R H (r/n) (a) (E, F)

(F, A)

( A, B)

(B, C)

(C, D)

(D, E)

# Rows

(n − 1)r/n

r

r

(n − 1)r/n

r

r

2n − 1

(b) Row1

Row2

2n + 1 2n + 3

. . . Row n−1

Row n

Row n + 1

. . . Row 2n − 2 Row 2n − 1

2n+ 2n+ 2n+ 2(n − 1) − 1 2(n − 1) 2(n − 1) − 1

2n + 3

2n + 1

Theorem 9.2 (Generalization: Irregular Hexagon I R H (r/n)): Our irregular hexagon I R H (r/n) is a planar tile, where n ≥ 2. Proof To prove this result, let P(n) be the following statement: P(n) : “I R H (r/n) can tile the Euclidean plane, where n ≥ 2”

We can prove that P(n) is true using a proof by mathematical induction on n. Basis step: Let us prove that P(n) is true for n = 2, i.e., P(2) is true. According to Lemma 9.2, I R H (r/2) is a planar tile, thus, proving P(2) is true. Inductive step: We assume that P(m) is true, i.e., our irregular hexagon I R H (r/m) is a planar tile, for m ≥ 2. First, the edge length of each equilateral triangle is r/m. That is, we have: δ I R H (r/m) (F', A') = δ I R H (r/m) (C', D') = r . We want to prove that P(m + 1) is true. First, let us show how to construct I R H (r/m + 1) starting from I R H (r/m). This construction process, which is shown in Fig. 9.14, has two steps: • Step 1: Change the edge length of each equilateral triangle in I R H (r/m) from r/m to r/m +1. We denote the resulting irregular hexagon by I R H (r/m → r/m + 1). Fig. 9.14 Construction of I R H (r/m + 1) from I R H (r/m)

9.3 Achieving Planar k-Coverage Using Irregular Hexagonal Tiles

287

• Step 2: Add a closed belt of equilateral triangles of edge length r/m + 1 around I R H (r/m → r/m + 1) so as to form our irregular hexagon I R H (r/m + 1). From Step 1, one can infer that I R H (r/m → r/m + 1) has the same structure as that of I R H (r/m). In particular, I R H (r/m → r/m + 1) has the same interior angles as I R H (r/m), which is a planar tile, by the inductive hypothesis. Therefore, the anterior angles of I R H (r/m → r/m + 1) satisfy the first condition given in Kerhsner’s Theorem given above. That is, ∠A' + ∠B ' + ∠C ' = 2π . Moreover, given that the closed belt added around I R H (r/m → r/m + 1) includes congruent, equilateral triangles of edge length r/m + 1, the anterior angles of R H (r/m + 1) are the same as those in I R H (r/m → r/m + 1). That is, the equality ∠A + ∠B + ∠C = 2π remains valid. In addition, the length of each of those six edges of I R H (r/m → r/m + 1) increases by r/m + 1. Thus, we have: ( ) δ I R H (r/m+1) (F, A) = δ I R H (r/m→r/m+1) F ' , A' + r/m + 1 ( ) δ I R H (r/m+1) (C, D) = δ I R H (r/m→r/m+1) C ' , D ' + r/m + 1 ( ) ( ) Given that δ I R H (r/m→r/m+1) F ' , A' = δ I R H (r/m→r/m+1) C ' , D ' , we conclude that δ I R H (r/m+1) (F, A) = δ I R H (r/m+1) (C, D) ( ) Indeed, for I R H (r/m), we have: δ I R H (r/m) F ' , A' = m × mr = r . Thus, based on the construction process of I R H (r/m + 1) from I R H (r/m) given above, we have: r r + =r m+1 m+1 r r + =r δ I R H (r/m+1) (C, D) = m × m+1 m+1

δ I R H (r/m+1) (F, A) = m ×

Consequently, R H (r/m + 1) satisfies both conditions of “Hexagon of Type 1” given in Kershner’s Theorem [227]. Thus, I R H (r/m + 1) can tile the Euclidean plane, meaning that the statement P(m + 1) is true. Thus, the statement P(n) is true, for all n ≥ 2. Indeed, we have the following inference rule: (P(2) ∧ (∀m ≥ 2, P(m) → P(m + 1))) → P(n), ∀n ≥ 2

∎

Theorem 9.3 computes the planar sensor density to k-cover a planar field using our irregular hexagon I R H (r/n). Theorem 9.3 (Deployed Planar Sensor Density for k-Coverage): Let r be the sensing range of the sensors, and k ≥ 1 and n ≥ 2 be two natural numbers. The deployed dep planar sensor density, denoted by ψir h (r, k, n), which is required to achieve kcoverage of a planar field using I R H (r/n), is computed as follows:

288

9 A Planar Irregular Hexagonal Tessellation-Based Approach …

4nk dep ψir h (r, k, n) = √ 3(6n − 4)r 2 Proof First, let us compute the size of the k-covered area, i.e., the total number N (n) of equilateral triangles of edge length r/n and whose area is Aet (r, n) = √ tot,et ( ) 3 r 2 , which form our irregular hexagon I R H (r/n). Based on Table 9.2b, we 4 n have: Ntot,et (n) = (2n + 1) + (2n + 3) + · · · + (2n + 2(n − 1) − 1) + (2n + 2(n − 1)) + (2n + 2(n − 1) − 1) + · · · + (2n + 3) + (2n + 1) = 2((2n + 1) + (2n + 3) + · · · + (2n + 2(n − 1) − 1)) + (2n + 2(n − 1)) = 2(2n + 2n + · · · + 2n + (1 + 3 + · · · + (2(n − 1) − 1))) + (2n + 2(n − 1))

However, the term '' 2n '' repeats (n − 1) times, and the sum of the first (n − 1) odd numbers is given by: 1 + 3 + . . . + (2(n − 1) − 1) = (n − 1)2 It is easy to prove the above result about the sum of the first positive odd numbers using a proof by mathematical induction on n. Therefore, we get: ) ( Ntot,et (n) = 2 2n × (n − 1) + (n − 1)2 + (2n + 2(n − 1)) = 6n 2 − 4n We can check that Ntot,et (n) = 6n 2 −4n is correct and produces the same numbers of equilateral triangles in our irregular hexagon I R H (r/n) as shown in Table 9.1a. To prove this result, let Q(n) be the following statement: Q(n) : “The number of equilateral triangles of edge length r/n in I R H (r/n) is 6n 2 − 4n, for n ≥ 2”

We can proceed using a proof by mathematical induction on n to prove that the statement Q(n) is true. Basis step: Let us prove that Q(n) is true for n = 2. Indeed, as shown in Figs. 9.4 and 9.5, our irregular hexagon I R H (2) has 16 equilateral triangles of side length r/2. If we substitute n = 2 into 6n 2 − 4n, we get 6 × 22 − 4 × 2 = 16. Thus, Q(2) is true. Inductive step: We assume that Q(m) is true, and prove that Q(m + 1) is also true. The statement Q(m) is true means that our irregular hexagon I R H (r/m) has 6m 2 −4m equilateral triangles of edge length r/m, for m ≥ 2. We construct our irregular hexagon I R H (r/m + 1) from I R H (r/m) as discussed earlier in the proof of Theorem 9.2 (Fig. 9.14). Recall that as we move from I R H (r/m) to I R H (r/m + 1), the length of each of those six edges of I R H (r/m → r/m + 1) increases by r/m +1 to form the new irregular hexagon, I R H (r/m + 1). Also, there is a belt of equilateral triangles of edge length r/m + 1 that is added around I R H (r/m → r/m + 1), thus,

9.3 Achieving Planar k-Coverage Using Irregular Hexagonal Tiles Table 9.3 (a) I R H (r/m) and I R H (r/m → r/m + 1) ( ' ') ( ' ') ( ' ') E ,F F ,A A,B

(

B', C '

289

)

( ' ') C ,D

(

D' , E '

I R H (r/m)

2m − 3

2m − 1

2m − 1

2m − 3

2m − 1

2m − 1

I R H (r/m → r/m + 1)

2m − 3

2m − 1

2m − 1

2m − 3

2m − 1

2m − 1

)

(b) I R H (r/m + 1) I R H (r/m + 1)

(E, F)

(F, A)

( A, B)

(B, C)

(C, D)

(D, E)

2m − 1

2m + 1

2m + 1

2m − 1

2m + 1

2m + 1

yielding I R H (r/m + 1). Specifically, this belt contains those equilateral triangles located on the new six edges (E, F), (F, A), (A, B), (B, C), (C, D), and (D, E) of I R H (r/m + 1). In particular, the number of equilateral triangles of edge length r/m + 1 along each of those six edges of I R H (r/m → r/m + 1) increases by 2 to obtain I R H (r/m + 1). Table 9.3a summarizes the numbers of equilateral triangles of edge length r/m along the six edges of I R H (r/m), whereas Table 9.3b specifies those of equilateral triangles of edge length r/m + 1 along the six edges of I R H (r/m → r/m + 1) and I R H (r/m + 1). Notice that those numbers are equal for I R H (r/m) and I R H (r/m → r/m + 1) given that the latter is a simple transformation of the former as discussed in Step 1 of the construction process above. Both of Tables 9.3a and Tables 9.3b are shown above. Let Ntot,et (m) and Ntot,et (m + 1) be the total numbers of equilateral triangles of edge length r/m and r/m +1 in I R H (r/m) and I R H (r/m + 1), respectively. Also, −−−→ we denote by Ntot,et (m + 1) and Nbelt (m + 1) the numbers of equilateral triangles of edge length r/m + 1 in I R H (r/m → r/m + 1) and the new belt, respectively, with: −−−→ Ntot,et (m + 1) = Ntot,et (m) Nbelt (m + 1) = (2m − 1) + (2m + 1) + (2m + 1) + (2m − 1) + (2m + 1) + (2m + 1) = 12m + 2 Therefore, we have: −−−−→ Ntot,et (m + 1) = Ntot,et (m + 1) + Nbelt (m + 1) ( ) = Ntot,et (m) + 12m + 2 = 6m 2 − 4m + 12m + 2 ( ) = 6m 2 − 4(m + 1) + 4 + 12m + 2 = 6m 2 + 12m + 6 ( ) − 4(m + 1) = 6 m 2 + 2m + 1 − 4(m + 1) = 6(m + 1)2 − 4(m + 1)

That is, Ntot,et (m + 1) = 6(m + 1)2 − 4(m + 1). It implies that the statement Q(m + 1) is true. Thus, the statement Q(n) is true, for all n ≥ 2. Indeed, we have the inference rule:

290

9 A Planar Irregular Hexagonal Tessellation-Based Approach …

(Q(2) ∧ (∀m ≥ 2, Q(m) → Q(m + 1))) → Q(n), ∀n ≥ 2 Now, given that there are Ntot,et (n) equilateral triangles of edge length r/n forming the actual k-covered area Aact,k−cov (r, k, n), the latter is given by: Aact,k−cov (r, k, n) = Ntot,et (n)Aet (r, n)

√ ( 2 ) ) 3 ( r )2 ( 2 = 6n − 4n Aet (r, n) = 6n − 4n 4 n dep

The corresponding deployed planar sensor density, ψir h (r, k, n), is given by: dep

ψir h (r, k, n) =

k 4nk =( ) √3 ( r )2 = √ Aact,k−cov (r, k, n) 3(6n − 4)r 2 6n 2 − 4n 4 n k

∎

Theoretically, the k-covered area Ak−cov (r, k, n) is given by: ( ) ) ( 1 1 √ − 2 4n 2 − 1 r 2 Ak−cov (r, k, n) = 2cos −1 2n 2n Thus, the actual planar sensor density, denoted by ψiracth (r, k, n), is computed as: ψiracth (r, k, n) =

k k ) =( √ ( ) 1 Ak−cov (r, k, n) 2cos −1 2n − 2n1 2 4n 2 − 1 r 2

Table 9.4 summarizes the value of ϕiracth (r, k, n) as a function of n. Question: How Large is n (i.e., what is the optimum value of n)? As shown in Table 9.4, when n increases, the length of the line segment [A, B] on which the k sensors are located, decreases. In particular, as n tends toward infinity (i.e., n → ∞), the two points A and B coincide (i.e., r/n → 0), and the k sensors appear to be placed at the dep

Table 9.4 ψir h (r, k, n) and

n

ψiracth (r, k, n)

ψir h (r, k, n)

ψiracth (r, k, n)

2

0.58×k r2 0.49×k r2 0.46×k r2 0.44×k r2 0.43×k r2 0.42×k r2 0.38×k r2 0.38×k r2

0.46×k r2 0.40×k r2 0.38×k r2 0.36×k r2 0.36×k r2 0.35×k r2 0.32×k r2 0.32×k r2

n

as a function of

3 4 5 6 7 100 ∞

dep

9.3 Achieving Planar k-Coverage Using Irregular Hexagonal Tiles

291

same point. The latter coincides with the center of the sensing range of the sensors, and, thus, those k sensors k-cover a circle of radius r . This scenario yields a planar sensor density that is equal to k/πr 2 = 0.31830988618 × k/r 2 , which is conform to the last entry in Table 9.4. The k-covered area Ak−cov (r, n) has two parameters, namely r and n, where the radius of the sensors’ sensing range r is supposed to be known a priori before any sensor deployment. Therefore, the function Ak−cov (r, n) depends only on n, and reaches its maximum when n → ∞. However, the latter corresponds to an unrealistic sensor deployment, where all k sensors are placed at just one same location. A reasonable value for n should enable communication among those k sensors as well as with other sensors in the network without much interference. Given that those k sensors are located on a line segment [G, H ], it is essential to find the minimum distance dmin between any pair of consecutive sensors on [G, H ] so it can accommodate the deployment of k sensors. Theorem 9.4 computes the planar sensor density to k-cover a planar field. Theorem 9.4 (Deployed Planar Sensor Density): Let r be the sensing range of the sensors, k ≥ 1 a natural number, and dmin the minimum distance between any pair of neighboring sensors. The deployed planar sensor density, denoted by dep ψir h (r, k, dmin ), to achieve k-coverage of a planar field, is computed as follows: 4k

dep

ψir h (r, k, dmin ) = √

3(6r − 4(k − 1)dmin )r

Proof Let d be the distance between any pair of consecutive sensors out of those k sensors, which are located on the segment [G, H ] of length r/n. Based on the above discussion, the value of n should meet the equality below: (k − 1)d =

r n

In fact, there are (k − 1) pairs of consecutive sensors on [G, H ], each of which is separated by a distance d. As stated earlier, we should maximize the value of n dep in order to maximize the value of the k-covered area Ak−cov (r, n), thus, yielding the ( ) min,dep deployed planar sensor density, ψir h r, k, n opt . Based on Theorem 9.3, If we dep substitute n = n opt into ψir h (r, k, n), we get: ) 4n opt k dep ( ψir h r, k, n opt = √ ( ) 3 6n opt − 4 r 2 where n opt is the optimum value of n. Using the above equality, and let dmin be the minimum distance between two sensors, we can derive n opt as follows: n opt =

r (k − 1)dmin

292

9 A Planar Irregular Hexagonal Tessellation-Based Approach … min,dep (

) r, k, n opt given above, we obtain:

If we substitute n = n opt into ψir h

4k

dep

ψir h (r, k, dmin ) = √

3(6r − 4(k − 1)dmin )r

∎

Likewise, the maximum value of Aact k−cov (r, k, n), denoted by ( ) theoretically, max,act n = A Amax,act is given by: k, d (r, ), opt min k−cov k−cov ( Amax,act k−cov (r, k, dmin )

= 2cos

−1

(

1 2n opt

)

) 1 / 2 4n opt − 1 r 2 − 2 2n opt

Hence, the actual planar sensor density, ψiracth (r, k, dmin ), is: k Amax,act k−cov (r, k, dmin ) k / =( ( ) ) 2cos −1 2n1opt − 2n12 4n 2opt − 1 r 2

ψiracth (r, k, dmin ) =

opt

Lemma 9.5 generalizes the result given in Lemma 9.4. Lemma 9.5 (Generalization: Connected Hexagonal k-Covered Wireless Sensor Networks Using I R H (r/n)). A hexagonal k-covered wireless sensor network using I R H (r/n), where n ≥ 2, is connected if the radii R and r meet the following inequality: R ≥ max

⎧ √

⎞ √ 7(n + 1)r 3r, 2n

Proof Let us consider the configuration shown in Fig. 9.13, and assume that each of those four irregular hexagon is I R H (r/n), with n ≥ 2. On the one hand, √ given that we have: δ(H1 , I2 ) =√ (n + 1)r/n and δ(I2 , G 2 ) = (n + 1)h, with h = 3r/2n, we obtain: √ we have δ(G 1 , I1 ) = √ δ(H1 , G 2 ) = 7(n + 1)r/2n. On the other hand, nh = √3r/2 and δ(I1 , H3 ) = 3r/2, we get: δ(G 1 , H3 ) = 3r . Also, δ(G 1 , H4 ) = 2nh = 3r . Thus, we have the following inequality: ⎧

√ 3r, R ≥ max

⎞ √ 7(n + 1)r 2n

∎

In their work on the connected k-coverage problem in planar wireless sensor networks, Xing et al. [425] require R ≥ 2r to ensure }connectivity. However, {√ network √ 7(n+1)r < 2r . Thus, our work our proposed approach necessitates R ≥ max 3r, 2n requires less powerful sensors than those in [425]. Table 9.5 shows the relationship between R and r for various I R H (r/n), with n ≥ 2.

9.3 Achieving Planar k-Coverage Using Irregular Hexagonal Tiles

293

Table 9.5 Relationship between sensing range r and communication range R {√ √ ( {√ √ ( √ ( ) } ) } ) 7 n+1 R ≥ max 3r, 27 n+1 max 3r, 27 n+1 n n r n r 2 n r 2

1.98 × r

1.98 × r

R ≥ 1.98 × r

3

1.76 × r

4

1.65 × r

5

1.59 × r

7

1.51 × r

1.76 × r √ 3r √ 3r √ 3r

R ≥ 1.76 × r √ R ≥ 3r √ R ≥ 3r √ R ≥ 3r

9.3.4 Discussion on Planar Sensor Density dep

As noticed from Table 9.4 given, the deployed planar sensor density ψir h (r, k, n), which is associated with our irregular hexagon I R H (r/n), decreases as n increases. Here, for a fair comparison with existing work, and practical sensor deployment, we consider the one that is associated with our irregular hexagon I R H (r/3), to ensure k-coverage of a planar field. That is, dep

ψir h (r, k) =

0.49 × k r2

We compare it with four similar results using different sensor placement strategies. • First result: One trivial solution to the problem of k-coverage consists of deploying k sensors anywhere within a circle of diameter r. Notice that this circle has a constant width equal to r. This type of sensor placement yields a planar sensor density σ (r, k) that is equal to: σ (r, k) =

1.27 × k k 4k = ( r )2 = 2 πr r2 π 2

dep

It is clear that ψir h (r, k) < σ (r, k). Thus, our sensor placement and planar sensor density are more appropriate. • Second result: In Chap. 5, we propose a bound on the planar sensor density λ(r, k) to k-cover a FoI, where k ≥ 3. Precisely, we exploit an elegant, fundamental result of convexity theory, also known as Helly’s Theorem [85], and the properties of a geometric structure, called Reuleaux triangle, in order to compute λ(r, k). We find the following result: 6k 0.81 × k λ(r, k) = ( √ ) = r2 2 4π − 3 3 r

294

9 A Planar Irregular Hexagonal Tessellation-Based Approach … dep

It is easy to check that λ(r, k) = 1.64 × ψir h (r, k). On the one hand, compared to the results presented earlier in Chap. 5, we reduce the planar sensor density by more dep than half (or ψir h (r, k) = 0.61 × λ(r, k)). Clearly, this is a significant improvement on our previous result. On the other hand, our current study is more general than the previous one. In fact, in this work, we consider any degree of coverage, i.e., k ≥ 1, while our previous one presented earlier in Chap. 5 accounts for k ≥ 3. • Third result: In [228], it was suggested that the best approach to cover an area with circles with radius r is to place the centers of the circles on the vertices of a lattice of equilateral triangles. The minimum number of circles covering a set is given by Kershner’s Theorem [228] as follows: Kershner’s Theorem (Set Covering) [228]: Let M denote a bounded plane point set and N(r) N(r)N(r)N(r) be the minimum number of circles of radius r which can cover M. Then, we have: √ 2π 3 ' M lim πr N (r ) = r →0 9 2

where M ' denotes the closure of M. This corresponds to a planar sensor density of √ 2 3 0.38 ρ(r ) = = 2 2 9r r Now, we can extend the above result in order to k-cover an area. That is, we can place k sensors on each vertex of this lattice of equilateral triangles. Consequently, the planar sensor density corresponding to this sensor placement to achieve k-coverage is given by: ρ(r, k) = dep

0.38 × k r2

As it can be seen, ψir h (r, k) and ρ(r, k). are almost equal. However, the sensor placement associated with ρ(r, k) does not seem to be realistic in any real-world sensor deployment scenarios. Indeed, it is not possible to place k sensors at the same point of a triangular lattice, where k > 1. This is due to at least the following three reasons. First, the sensors are physical devices, and, thus, have a three-dimensional structure, which is described with three parameters, namely length, width, and height. Therefore, these sensors occupy a certain space in a field, and it is not practical to reduce this physical structure to just one point in a two-dimensional space. Second, these sensors have typical technology characteristics, such as sensing and communication capabilities. Thus, placing k sensors at each triangular lattice point is unfeasible given the non-tolerable amount of interference this sensor placement would introduce. This would make it almost impossible for the sensors to communicate with one another, given the type of their communication medium. Third, these sensors are

9.4 A k-Coverage Protocol Using Irregular Hexagonal Tiling

295

fragile and unreliable. In fact, they may not be able to work properly due to a software failure, hardware failure, and/or low residual energy level. When this happens, the sensing application’s requirement in terms of coverage may not be satisfied as some areas may be left less than k-covered (i.e., under-coverage problem). It is well known that sensor mobility can be exploited to cope with this problem. Unfortunately, this type of placement does not support sensor mobility from their associated positions on the triangular lattice. However, our proposed sensor placement strategy is practical dep and feasible, and so is our new bound on the planar sensor density ψir h (r, k) for k-coverage of a planar field. According to our k-coverage approach, k sensors are placed on an edge of a triangle of side length r/3, which belongs to a non-regular hexagon as shown in Fig. 9.10 shown above. That is, these sensors could be located anywhere on this edge. Also, in case of an under-coverage problem, the sensors have the freedom to move to an edge within a certain non-regular hexagon and k-cover it. Thus, our sensor placement approach and planar sensor density are appealing solutions for the k-coverage problem in planar wireless sensor networks. • Fourth result: Another solution to the problem of k-coverage is to place k sensors at the center of a circle of radius r . This leads to a planar sensor density μ(r, k) that is equal to: μ(r, k) =

0.32 × k k = πr 2 r2

dep

It is true that μ(r, k) < ψir h (r, k). However, similarly to the third result discussed above, this solution does not seem to be realistic. Indeed, putting k sensors at the same location is not possible at all for the same reasons mentioned above. While both dep of the two planar sensor density measures μ(r, k) and ψir h (r, k) are comparable, our measure is realistic and more practical.

9.4 A k-Coverage Protocol Using Irregular Hexagonal Tiling The irregular hexagon-based k-coverage protocol, denoted by I R H Ck , which we propose is almost fully distributed (or pseudo-distributed). Indeed, the sink gets involved only at the beginning of each round to compute the first irregular hexagon of the hexagonal grid, called reference irregular hexagon and denoted by I R Hr e f (r/n), and ensure its k-coverage. More precisely, our irregular hexagon-based k-coverage protocol, I R H Ck , has two major steps, namely reference irregular hexagon generation and k-coverage (step 1), and hexagonal grid expansion and k-coverage (step 2). Our I R H Ck protocol applies in every round of k-coverage of a planar field, with k ≥ 1. Next, we describe the above two steps of our I R H Ck protocol in detail. Then, we present an illustrative example of this hexagonal grid expansion process. We find

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that the latter causes an avoidable side-effect problem, which is due to the hexagonal tiling at the edges of the sensor field. We discuss this problem and show its impact on the planar sensor density.

9.4.1 Generating Reference Irregular Hexagon and k-Coverage As its first task, the sink randomly selects the first point E in a circle of radius r and centered at the center of the FoI. The reason we make this design decision is that we want the reference irregular hexagon I R Hr e f (r/n) to be located closer to the center of the FoI. In fact, the construction of the whole hexagonal grid will expand from I R Hr e f (r/n) and it would be more efficient if the sensors that are involved in this expansion process do the same amount of work. This issue will become clearer when we discuss the generation of the rest of the hexagonal grid starting from I R Hr e f (r/n). Once point C has been chosen, the sink sequentially computes the remaining five points, namely A, B, C, D, and F, of I R Hr e f (r/n) as follows: • It picks randomly point F on the perimeter of half a circle centered at E and whose radius is (n − 1)r/n. • It picks point A deterministically such that δ(F, A) = r and ∠F = 2π/3. • It picks point B deterministically such that δ(A, B) = r and ∠A = 2π/3. • It picks point C deterministically such that δ(B, C) = (n − 1)r/n and ∠B = 2π/3. • It picks point D deterministically such that δ(C, D) = r and ∠C = 2π/3. Next, as its second (and last) task, the sink designates one sensor located in the reference irregular hexagon I R Hr e f (r/n), and denoted by schl , as a cluster-head leader, and provides it with the x and y coordinates of the six points A, B, C, D, E, and F defining the reference irregular hexagon I R Hr e f (r/n). This cluster-head leader schl will be responsible for k-covering I R Hr e f (r/n). Specifically, based on these x and y coordinates, schl selects k sensors inside I R Hr e f (r/n) to be positioned along the line segment [G, H ] whose length is r/n. These k selected sensors should not be part of the forbidden neighboring sensor set, which is defined earlier in Chap. 2 (Definition 2.47, Sect. 2.2). It is worth mentioning that the energy threshold ethr eshold has to be recomputed for each round. It should be a strictly decreasing function from one round to another. Moreover, the value of n cr should be computed for each round, with n cr ≥ 2. However, toward the end of sensor deployment, we should have n cr = 1. This means that no sensor would be allowed to participate in two consecutive rounds of k-coverage. In fact, as time progresses, the energy level of a sensor decreases. Consequently, a sensor cannot participate in several consecutive rounds. Thus, n cr could be initialized to a certain value α. Then, this value can be decremented from one round to another, i.e., n cr = α → n cr = α − 1 → n cr = α − 2 → · · · → n cr = 1.

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That is, a sensor is allowed to be selected in at most α − 1 consecutive rounds. In addition, those k selected sensors should have the highest residual energy levels among all the sensors located in I R Hr e f (r/n), where n ≥ 2. Indeed, it is more beneficial to avoid those sensors with low energy levels. The cluster-head leader schl sends out an invitation packet, denoted by I nv P, which includes the id’s of those k selected sensors (i.e.,id 1 , id2 , …, idk ) along with the x and y coordinates of points G and H , denoted by (x G , yG ) and (x H , y H ), respectively. In other words, we have I nv P = id1 , id2 , . . . , idk |(x G , yG )|(x H , y H ), where idi is the id of the selected sensor si , with 1 ≤ i ≤ k. Upon receiving an invitation packet, I nv P, each sensor checks whether it is concerned with it. If not, it discards it. Otherwise, it moves to the line segment [G, H ] and gets located there with the k − 1 other sensors.

9.4.2 Expanding Hexagonal Grid and k-Coverage The expansion of the hexagonal grid and its k-coverage take place as follows. Initially, the hexagonal grid consists of only the reference irregular hexagon I R Hr e f (r/n). This expansion process has the following steps, which are run sequentially: • Step 2.0: First, the cluster-head leader schl computes the x and y coordinates of the six points A, B, C, D, E, and F for each of its six neighboring irregular hexagons (or clusters). Then, it selects a cluster-head for each one of them, and provides each one of them with the x and y coordinates corresponding to its cluster. The selection process of sensors to act as cluster-heads is based on their residual energy levels. In other words, preference is given to those sensors with high battery power. Each selected cluster-head is responsible for k-coverage of its corresponding irregular hexagon. It acts exactly the same way as schl in the selection of the sensors, which are located in its irregular hexagon, so they can move to the line segment [G, H ] and k-cover it. Part of Fig. 9.15 shows this configuration, while considering a tiling grid using our irregular hexagon I R H (r/2). Those six selected cluster-heads are numbered 2, 3, 4, 5, 6, and 7, as shown in Fig. 9.15. • Step 2.1: Now, we have six clusters with their cluster-heads. These clusters are surrounded by a belt that consists of 12 clusters, as shown in Fig. 9.15. Thus, each of these six cluster-heads is responsible for two neighboring clusters. First, each cluster-head computes the x and y coordinates of those six points defining each of the two neighboring clusters. Second, it selects a cluster-head for each one of them. As described above, each of those 12 selected cluster-heads is responsible for k-covering its corresponding cluster. • Step 2.2: The outer belt of the hexagonal grid has 12 clusters, each of which with its own cluster-head. Also, these clusters are surrounded by 18 clusters (Fig. 9.15). In this case, six of these 12 cluster-heads are responsible for selecting one clusterhead for a given cluster, while the six other cluster-heads select two cluster-heads for two given clusters. For fairness and energy efficiency, the second group of six

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Fig. 9.15 Hexagonal grid expansion based on tiling using I R H (r/2)

cluster-heads should have higher residual energy levels than the first group of the six other cluster-heads. • Step 2.i: Here, we consider a more general configuration based on the results from the previous steps, 1, 2, 3, . . . , i −1. Now, we have 6×i cluster-heads surrounded by 6× (i + 1) clusters. Similarly, there are six of these 6×i cluster-heads, each of which is responsible for two neighboring clusters, while each other cluster-head is responsible for only one cluster. Likewise, the first group of six cluster-heads is more powerful than the other remaining 6 × (i − 1) cluster-heads in terms of their residual energy levels.

9.4.3 Example Let us consider the configuration shown in Fig. 9.15. It displays a hexagonal tiling of a planar field using our irregular hexagon I R H (r/2). The cluster-head leader schl has id = 1 and selects six cluster-heads whose id’s are 2, 3, 4, 5, 6, and 7, respectively. These cluster-heads are responsible for k-covering their corresponding irregular hexagons. Also, each of them designates two cluster-heads for two neighboring irregular hexagons. For instance, cluster-head with id = 2 selects cluster-heads with id = 8 and id = 9 for two of its adjacent irregular hexagons, which have not been assigned yet any cluster-head. Likewise, cluster-head with id = 3 selects clusterheads with id = 10 and id = 11 for two of its neighboring irregular hexagons.

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However, cluster-head with id = 8 selects cluster-heads with id = 20 and id = 21, while cluster-head with id = 9 selects only one cluster-head with id = 22. Recall that except for the six cluster-heads of the first belt of irregular hexagons around the cluster-head leader, any other cluster-head can select one or two cluster-heads for one or two of its adjacent irregular hexagons, respectively.

9.4.4 Problem of Side-Effect We should mention that this expansion process may introduce a side-effect problem. Indeed, the latter may arise with those irregular hexagons that are located at the edges of a planar field as they are not complete. Figure 9.15 shows a hexagonal tiling grid using our irregular hexagon I R H (r/2), where the ones numbered 20, 35, 36, and 37 are not fully positioned inside a planar field, and, thus, are incomplete. Unfortunately, this scenario is unavoidable given the shape of our irregular hexagon I R H (r/n) as well as the shape of a planar field. In fact, it may not be possible to find the line segment [G, H ] inside the planar field and along which those k sensors should be positioned. In order to cope with this problem, those k sensors have to be placed as close as possible to the center of each incomplete irregular hexagon. Although there are only small portions of those irregular hexagons at the edges of a planar field, it is still necessary to place k sensors inside each one of them so they are surely k-covered. We expect this side-effect problem to activate more sensors, and its impact to be dependent on the shape of the sensor field. Another way to solve this side-effect problem is to adapt the edges (or perimeter) of the planar field so that it is shaped according to our proposed irregular hexagon, I R H (r/n). This solution is practical and feasible by extending the area of the planar field so it includes all the missing parts of those irregular hexagons located at the perimeter of the field.

9.5 Performance Evaluation In this section, we present the results of the performance evaluation of our proposed irregular hexagon-based k-coverage protocol, I R H Ck , as compared to those sensor placement strategies to achieve k-coverage in planar wireless sensor networks based on some metrics. First, we describe the simulation environment. Then, using a highlevel simulator written in C, we discuss a few simulation results of our hexagonal tiling-based approaches to compute the planar sensor density and place the sensors to k-cover a planar field.

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9.5.1 Simulation Setup Our irregular hexagon-based k-coverage protocol, I R H Ck , can be applied to any planar field regardless of its geometric shape whether it is a circle, square, rectangle, or any polygon. However, the impact of the side-effect problem discussed earlier may be of different magnitude depending on the shape of the sensor deployment field. In this chapter, we consider two shapes of a planar field, namely square and circle. In the first set of experiments, we use a square field whose side length is 300 m. However, in the second set of experiments, we use a circular field whose radius is 169.25 m, thus, having almost a similar size as the square field above. Indeed, for a fair comparison between the two experiments in order to investigate the impact of the shape of the sensor field on the side-effect problem described above, both of the square and circular fields of interest should have the same area size. We assume that the total number of sensors is 1000, and that those sensors are randomly and uniformly deployed in the underlying deployment field. We suppose that the initial energy of each sensor is 70 J. We use the IEEE 802.11 distributed coordinated function with CSMA/CA as the underlying MAC protocol. Also, we consider a radio interference model given the pervasiveness of other 2.4 GHz radio sources. We account for several forms of energy consumption, namely data sensing, data transmission, data reception, sensor mobility, and control messages, which are required for the correct operation of our irregular hexagon-based k-coverage protocol, I R H Ck . To this end, we refer to the energy consumption model presented earlier in Chap. 2 in Sect. 2.5. For the energy spent by the sensors due to a phenomenon sensing, we refer to the energy model suggested by Ye et al. [435]. A sensor consumes 0.012 J when it is in the idle mode, and 0.0003 J when it is in the sleep mode. Also, the energy cost for a sensor to move one-unit distance, emove , is randomly selected between 0.008 J/m and 0.012 J/m [398]. Also, we assume that the radii of the sensing and communication ranges of all the sensors are 30 m and 50 m, respectively. We use the IEEE 802.11 distributed coordinated function with CSMA/CA as the underlying MAC protocol. Also, we consider a radio interference model given the pervasiveness of other 2.4 GHz radio sources. For each of the two experiments, all simulations are repeated 100 times and the results are averaged.

9.5.2 Simulation Results As stated in Sect. 9.6.1, there are two sets of experiments. The results of the first set of experiments are shown in Figs. 9.16, 9.17, 9.18, 9.19, 9.20, and 9.21 (theoretical vs. simulation results), whereas those related to the second experiment set are presented in Figs. 9.22, 9.23, and 9.24 (comparison with existing work). In our first set of experiments, a comparison of all the planar sensor density bounds discussed in Sect. 9.4.4 is given in Fig. 9.16, where k = 3 and n = 3, while the radius r of the sensors’ sensing range takes on various values, i.e., r =

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Fig. 9.16 ψir h (r, k, n) for k = 3 and n = 3

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Fig. 9.17 ψir h (r, k, n) vs. ψiracth (r, k, n)

20 m, 25 m, 30 m, 35 m, 40 m, 45 m, 50 m. In Chap. 5, we investigate the problem of k-coverage in planar wireless sensor networks and consider a degree of coverage k ≥ 3. Thus, we set k = 3 for a comparison between our currently proposed planar sensor density and our previously suggested one in Chap. 5. The difference between both bounds is visible, thus, showing the remarkable improvement made by our current work. As stated earlier, there is no significant difference between dep ψiracth (r, k) and ρ(r, k). We notice that the value of ψir h (r, k, n) decreases inversely proportionally to r . Figure 9.17 shows the difference between the deployed planar dep sensor density, ψir h (r, k, n), and the actual planar sensor density, ψiracth (r, k, n), for k = 3 and r = 25 m, and various values of n, i.e., n = 2, 3, 4, 5, 6, 7, 8. The dep value of ψir h (r, k, n) decreases as n increases. Indeed, the area of the k-covered

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Fig. 9.18 ψir h (r, k, n) vs. ψirsim h (r, k, n) for k = 3 and r = 25m

dep

Fig. 9.19 ψir h (r, k, n) vs. ψirsim h (r, k, n) for n = 3 and r = 25m

region increases proportionally to n, thus, yielding a smaller number of sensors to k-cover it. Thus, is the irregular hexagon I R H (r/n) that has a higher value of n leads to a smaller planar sensor density to achieve k-coverage of a planar field of dep interest. Figure 9.18 shows the plots of ψir h (r, k, n) and ψirsim h (r, k, n) for k = 3 and r = 25 m, and various values of n, i.e., n = 2, 3, 4, 5, 6, 7, 8. Figure 9.19 shows dep that ψir h (r, k, n) increases proportionally to k, where n = 3 and r = 25m. This is expected as higher level of degree of coverage requires more sensors. From now on, ψirsim h (r, k, n) denotes the planar sensor density obtained by simulation. The plots in Figs. 9.18, 9.19 and 9.20 compare between the theoretical and simulation results regarding the planar sensor density, while varying those three key

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Fig. 9.20 ψir h (r, k, n) vs. ψirsim h (r, k, n) for k = 3 and n=3

Fig. 9.21 ψirsim h (r, k, n) for square/circle for k = 3 and r = 25 m

parameters, namely n, k, and r . As it can be seen from those figures, there is a close dep match between the values of ψir h (r, k, n) and ψirsim h (r, k, n). The difference that exists between them is due to the side-effect problem, which is discussed in Sect. 9.5.4. The purpose of Figs. 9.21, 9.22, and 9.23 is to study the impact of the unavoidable side-effect problem on the process of k-coverage. To this end, we consider two shapes of the sensor deployment field, i.e., square and circle. We find that the results obtained for a square field outperform those associated with a circular field. These outcomes, which are presented on Figs. 9.21, 9.22, and 9.23, seem to be realistic. In fact, with a circle, there are more incomplete irregular hexagons on its edge (or boundary) compared to a square. Each of those incomplete irregular hexagons needs to be k-covered regardless of its area. That is, k sensors should be placed in any of

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Fig. 9.22 ψirsim h (r, k, n) for square/circle for n = 3 and r = 25 m

Fig. 9.23 ψirsim h (r, k, n) for square/circle for k = 3 and n=3

those portions of irregular hexagons as close as possible to their centers. Thus, the process of k-covering a square deployment field requires a smaller number of sensors than that of k-covering a circular deployment field. In our second set of experiments, our simulation results show that our irregular hexagon-based k-coverage protocol, I R H Ck , outperforms the CCP k-coverage protocol [425]. Figure 9.24 plots the remaining energy versus the time for I R H Ck as compared to our distributed k-coverage protocol, denoted by DIRACCk , which is discussed in detail in Chap. 5. We should mention that in Chap. 5, we already show that DIRACCk outperforms the CCP k-coverage protocol [425]. Figure 9.24 shows that our current k-coverage protocol, I R H Ck , incurs less energy consumption than DIRACCk . This is mainly due to the significant improvement in terms of the number

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Fig. 9.24 Remaining energy vs. time (compared to our previous work [39])

of active sensors that are needed to k-cover a planar field as indicated in Fig. 9.16. dep Recall that we have ψir h (r, k) = 0.61 × λ(r, k), as discussed in Sect. 9.4.4. Given that the total energy consumption depends on the number of active sensors, I R H Ck yields significant energy savings compared to DIRACCk . In fact, a higher number of sensors would cause more energy consumption due to sensing and the communication overhead that is introduced by the control messages exchanged among the active sensors to ensure the coverage degree, k, of the underlying sensing application. Compared to Chap. 5, the work discussed in this chapter has much better performance. Indeed, our proposed k-coverage protocol, I R H Ck , requires a smaller number of sensors to achieve k-coverage of a planar field, and, thus, less energy consumption than our previous k-coverage protocol, DIRACCk in Chap. 5. The latter uses the Reuleaux triangle RT r as a basis to k-cover a planar field. However, the Reuleaux triangle is not a planar tile, whereas the irregular hexagon I R H (r/n) used by I R H Ck can tile the Euclidean plane. Our suggested irregular hexagon I R H (r/n) maximizes the total area being used out of the sensing disk of the sensors. As a consequence, the planar sensor density (i.e., number of sensors deployed per unit area) to ensure k-coverage of a planar field using I R H (r/n) is smaller than its counterpart using RT r . Thus, the total energy consumed by DIRACCk is higher than the one consumed by I R H Ck . This difference in energy consumption is due to the sensing task of the sensors as well as the communication that is needed among the selected sensors so that they can successfully accomplish their mutual sensing task. As expected, the network lifetime associated with our proposed k-coverage protocol I R H Ck is larger than the one corresponding to previous k-coverage protocol DIRACCk , as shown in Fig. 9.24.

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9.6 Related Work In this section, we provide an overview of sample deterministic approaches to investigate the coverage as well as the k-coverage problems in planar wireless sensor networks. Abrams et al. [4] proposed a distributed algorithm to partition a wireless sensor network into k covers, where each cover is a set of sensors to cover an area. These covers are activated in a round-robin fashion such that as many areas are monitored as frequently as possible. Zhou et al. [455] proposed a greedy algorithm for k-coverage with a minimum set of connected sensors. Zhang and Hou [447] suggested an optimal geographical density control protocol to keep a small number of sensors active to cover a sensor filed regardless of the ratio of the sensors’ communication range to their sensing range. Zou and Chakrabarty [460] investigated the problem of sensor selection for sensing and connectivity, using a connected dominating setbased approach. Zhou et al. [457] proposed a distributed algorithm using kth-order Voronoi diagram to provide fault tolerance, while ensuring coverage. Xing et al. [425] addressed the k-coverage problem in planar wireless sensor networks, and proposed a protocol, named connected coverage protocol (CCP). In addition, they provided a relationship R ≥ 2r between the communication range, R, and sensing range, r , of the sensors to maintain network connectivity. Yang et al. [429] suggested a non-global solution to the k-connected coverage set problem that they formulated using a linear programming-based approach. Bai et al. [69] developed an optimal deployment strategy to ensure coverage and 2-connectivity independently of the relationship between the radii of the sensors’ sensing and communication ranges. Gupta et al. [186] provided centralized and distributed algorithms for connected sensor cover, where the network can self-organize its topology when a query is issued, and activate the sensors that are necessary to process the query. Huang et al. [210] proposed distributed protocols to guarantee coverage and connectivity of wireless sensor networks, while enabling an arbitrary relationship between the sensors’ communication and sensing ranges. Yu et al. [438] proposed a connected k-coverage working sets construction algorithm based on Euclidean distance to k-cover the sensing region while minimizing the number of working sensors. Abo-Zahhad et al. [3] proposed a centralized deployment algorithm based on mixing the multi-objective immune algorithm and Voronoi diagram with a goal to maximize coverage and lifetime of wireless sensor networks using binary and probabilistic models. Chenait et al. [118] proposed a stable and predictive energy-aware coverage scheduling protocol in order to reduce the scheduling energy waste by removing useless transitions from the scheduling strategy and preventing the run of unnecessary eligibility executions, as well as extend the coverage lifetime. More and Raisinghani [303] proposed an optimized discharge-curve-based coverage protocol to deal with the situation, where multiple active nodes monitor the same area, which would result in energy wastage, as well as sudden failure of nodes, which may result in coverage gaps within the sensing area. Movassagh and Aghdasi [305] proposed a distributed game theory-based node scheduling for coverage control in order to select a minimum number of nodes as

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active while keeping others in sleep mode to preserve energy and extend network lifetime. Roselin et al. [340] proposed an energy efficient connected coverage scheduling algorithm to maximize the network lifetime through forming non-disjoint cover sets using remaining energy, coverage and connectivity of every sensor. Sharmin et al. [351] formulated the problem of maximizing area coverage with minimum number of active nodes as a mixed-integer linear programming optimization problem for clustered directional sensor networks. They suggested a distributed, greedy alternate solution, namely α-overlapping area coverage, enabling cluster heads to determine the active member nodes and their sensing directions, where sensing node have at most α% coverage overlapping with their neighbors. Song and Fan [362] addressed the problem of coverage control for mobile sensor networks with limited communication ranges. Their goal was to minimize a coverage cost function that indicates the largest arrival time from the mobile sensor network to the points on a circle. They developed a distributed coverage control law using low gain feedback. Sun et al. [372] provided a shortest path connectivity and coverage algorithm for wireless sensor networks to reduce the energy consumption and extend the network lifetime. This algorithm allows redundant nodes to enter into the ready-to-sleep mode, and sleepless nodes to enter into the active mode. Also, redundant nodes on the shortest path become active to ensure network connectivity. Vatankhah and Babaie [385] proposed a coverage enhancing algorithm based on Delaunay triangulation with adjustable rages with a purpose to reduce the mobility dependency of the network in order to improve the coverage of hybrid wireless sensor networks. The latter includes both stationary and mobile sensor nodes. Xu et al. [426] considered the coverage optimization problem in wireless sensor networks with three objectives: Minimizing the energy consumption, maximizing the coverage rate, and maximizing the equilibrium of energy consumption. They proposed two improved hybrid multi-objective evolutionary algorithms, namely Hybrid-MOEA/D-I and Hybrid-MOEA/D-II, in order to effectively optimize sub-problems of the multi-objective optimization problem in wireless sensor networks. Boukerche and Sun [92] provided a survey and classification of the state-of- the-art algorithms and techniques that address the connectivitycoverage issues in the wireless sensor networks. Nguyen et al. [311] addressed the problem of placement of sensor nodes and relay nodes to guarantee target coverage and connectivity. They suggested an integer linear programming model to solve the target coverage problem, and proposed two approximation algorithms to tackle the network connectivity problem using the minimum group Steiner tree to minimize the placed relay nodes, and clustering and spanning tree approaches. Chen et al. [111] proposed a coverage- and energy-aware protocol with intra- and intercluster methods, while considering the sensor node density and coverage overlapping. Binh et al. [82] proposed two metaheuristics, namely Genetic Algorithm and Particle Swarm Optimization, to maximize area coverage of a given network in the presence of obstacles with the connectivity constraints to guarantee a feasible solution. Chakraborty et al. [106] proposed a Monte-Carlo Markov chain simulation approach to evaluate the coverage-reliability index, which was introduced to satisfy the application-specific coverage area requirement with reliable data delivery to the mobile sink.

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9.7 Conclusion In this chapter, we investigate the k-coverage problem in planar wireless sensor networks using a hexagonal tiling-based approach. More specifically, we address the sensor placement problem and the corresponding planar sensor density to ensure k-coverage of a planar field. Indeed, the network lifetime depends on the total energy consumption of the sensors, which in turn depends on the number of active sensors that are necessary to k-cover a planar field. Therefore, it is important to compute the planar sensor density and specify how those selected sensors should be placed in a planar field to k-cover it. We find that there is a correlation between the kcoverage and tiling problems in planar wireless sensor networks. To this end, we study planar geometric shapes, namely regular and irregular hexagons, to solve the k-coverage problem in planar wireless sensor networks. We determine the exact shape of this irregular hexagon, denoted by I R H (r/n), and prove that it is capable of covering the Euclidean plane without any overlap or gaps between any pair of them. We propose a few sensor placement strategies based on the degree of coverage k required by the underlying sensing application. For each of these sensor placement approaches, we compute the corresponding planar sensor density. Also, we calculate the relationship that should exist between the sensing and communication ranges of the sensors to guarantee connectivity among all the sensors participating in the k-coverage process. Moreover, we specify the planar sensor density associated with ) ( I R H r/n opt . However, the value of n opt has not been yet computed and needs to be further investigated. We focus on determining the value ofn opt , which is necessary to compute the exact value of the planar sensor density for k-coverage of a planar field. To this end, we should compute the minimum distance dmin between any pair of adjacent sensors. This may be achieved through intensive simulation/experiments or theoretical analysis. It is worth mentioning that none of the above approaches, including the work by Yu et al. [438], quantified the planar sensor density to achieve k-coverage of a planar field. The work in this chapter remedies to this lack and computes a bound on this planar sensor density for ensuring k-coverage in planar wireless sensor networks using as small number of sensors as possible. To the best of our knowledge, it is only our work (i.e., the previous one and this one) that computed the planar sensor density to k-cover a planar field. As discussed earlier in Sect. 9.4, our current bound on this planar sensor density in this chapter outperforms our previous one, and shows a significant improvement. Also, the approach proposed in this chapter to address the k-coverage problem in planar wireless sensor networks is totally different from the one we suggest earlier in Chap. 5. While the latter is based on a fundamental result of convexity theory, known as Helly’s Theorem [85], the former exploits the concept of hexagonal tiling with congruent irregular hexagons, which is proposed in this chapter. We discuss irregular hexagons, which have a specific shape, and prove that they are capable of tiling the Euclidean plane.

Chapter 10

A Polyhedral Space Filler Tessellation-Based Approach for Connected k-Coverage

I am trying to awake the energy contained in the air. These are the main sources of energy. What is considered as empty space is just a manifestation of matter that is not awakened. Nikola Tesla (1856–1943)

Overview This chapter addresses the problem of connected k-coverage in spatial wireless sensor networks using a tiling-based approach. Indeed, the coverage problem of a spatial field has similarity with the tiling problem in the same space, which can be formulated as follows: How can a three-dimensional space be tiled by replicas of tiles? This is an instance of the second part of Hilbert’s eighteenth problem [300], which is stated as follows: “What convex polyhedra exist for which a complete filling of all space is possible by juxtaposition of congruent copies?” In this chapter, we propose a polyhedral framework to investigate the connected k-coverage problem in spatial wireless sensor networks. First, we restrict the sensors’ sensing sphere to a variety of convex polyhedral space-fillers. Our study aims at finding the largest enclosed convex polyhedral space-filler in the sensing sphere of the sensors, with a goal to maximize their utilized sensing volume. Second, based on this analysis, we select a minimum number of sensors to k-cover a spatial field using deterministic and random sensor deployment strategies. Third, we compute the ratio of the communication range to the sensing range of the sensors to ensure network connectivity. We start by solving the connected coverage in spatial wireless sensor networks (i.e., k = 1). Then, we generalize our study to address the connected k-coverage in the same space, where k ≥ 1, while considering several sensor placement strategies based on the best convex polyhedral space-filler and the required degree of coverage k. Fourth, we corroborate our analysis with various simulation results.

10.1 Introduction The design and development of wireless sensor networks face challenges due to several key attributes, including deployment field accessibility, such as battlefields; © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_10

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limited sensor capabilities, such as battery power; and dimensionality. Given all these limitations, it is essential that all the protocols designed for the sensors be localized and energy-efficient. In particular, dimensionality has been a major issue in the design, analysis, and implementation of wireless sensor networks. The problem of coverage in spatial wireless sensor networks is challenging and hard. This is due to the geometric shape of the sensing range of the sensors, which is supposed to be a sphere. The spatial connected k-coverage problem, where every point in a spatial field is covered by at least k sensors and all the active sensors are connected, is even more challenging and harder. Indeed, in their work on the problem of deployment and configuration of sensor networks and how to construct efficient network topologies in the space, Poduri et al. [324] stated several problems, which are due to dimensionality, in the design of protocols for wireless sensor networks. Particularly, the problem of coverage in spatial wireless sensor networks is hard and challenging. It is even more challenging than its counterpart in planar wireless sensor networks due to these two reasons. First, there is only one regular polyhedron, i.e., cube, which tiles the three-dimensional Euclidean space [125], whereas there are several regular polygons, such as regular triangle, square, and hexagon, which tile the two-dimensional Euclidean plane. Second, some solutions in the two-dimensional Euclidean plane (or simply, the plane) may not be applicable to the three-dimensional Euclidean space (or simply, the space). In particular, it is not always possible to extend a solution to the problem of coverage in the plane to its counterpart in the space. For instance, the equilateral triangle can be used to cover a planar field without any overlap or any gap, whereas its counterpart in the three-dimensional Euclidean space, i.e., the regular tetrahedron, cannot cover a spatial field. The k-coverage problem is even harder and more challenging. In this chapter, we focus on the problem of connected k-coverage of a spatial field, where every point is covered by at least k sensors, while the network formed by those sensors is connected. In the discussion of his list of twenty-three unsolved problems in mathematics, Hilbert [300] raised the question “What convex polyhedra exist for which a complete filling of all space is possible by juxtaposition of congruent copies?” that is known as Hilbert’s eighteenth problem. This is one of the oldest and most challenging geometric problems in the history of mathematics. A solution to this problem is not trivial although it has been approached extensively by several well-known researchers. We should notice that the same above question of which tetrahedra fill a three-dimensional space was inadvertently raised by Aristotle. Indeed, this problem was originally investigated by Aristotle [64] several centuries ago. He mistakenly suggested a solution to it and thought it was correct. Next, we state Aristotle’s erroneous assertion regarding covering the space with regular tetrahedra [248], which was not supported by any evidence. Aristotle’s Assertion (Space tessellation with regular tetrahedra): Regular tetra∎ hedra tessellate the space. Aristotle’s conjecture means that it is possible to perfectly fill the Euclidean space with congruent regular tetrahedra when they are arranged face-to-face. Because of the high level of deference owed to Aristotle’s assertion, it was believed for several centuries that the regular tetrahedron is one of the space-fillers among other regular

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Fig. 10.1 a Five regular tetrahedra and b twenty regular tetrahedra

solids, such as the cube. However, this was proved to be wrong [345, 383]. In fact, the dihedral angle of a regular tetrahedron is equal to 70.53º, which is not sub-multiple of 360º [107]. Particularly, Conway and Torquato [123] provided two configurations showing the gap that would result from the use of the regular tetrahedron. In the first configuration given in Fig. 10.1a, five regular tetrahedra form a gap of 7.36° when they are packed around a common edge. This gap cannot be filled by a regular tetrahedron. In the second configuration, twenty regular tetrahedra yield an uncovered solid angle of 1.54 steradians when they are packed around a common vertex, as shown in Fig. 10.1b. Similarly, this gap cannot be filled by a regular tetrahedron. To solve Hilbert’s eighteenth problem, Sommerville [361] suggested four different tetrahedra that fill the space, three of which were independently rediscovered by Davies [130]. Moreover, Baumgartner [78] found four polyhedral space-fillers, one of which is different from the ones discovered by Sommerville [360]. Goldberg [174] provided a survey of all known tetrahedral space-fillers. Interestingly, the problem of coverage in spatial wireless sensor networks resembles to the eighteenth Hilbert’s problem [198] stated above. However, the tile that we consider in this study is not a polyhedron. But, it is a sphere that represents the sensors’ sensing range, as per our network model presented earlier in Chap. 2 in Sect. 2.7. This is the reason why we decide to investigate the spatial coverage problem using a polyhedral framework-based approach, where we examine various spatial convex polyhedral space-fillers. Precisely, we restrict the sensing sphere of the sensors to a spatial convex polyhedral space-filler, with a goal to find the one that enables filling a spatial field using as small number of congruent copies of this polyhedron space-filler as possible. That is, we want to minimize the total number of sensors to cover a spatial field, while maximizing the operational network lifetime. Notice that filling the space with a convex polyhedron space-filler means covering the space without any gap or overlap of any adjacent pair of this convex polyhedral space-filler. The literature has several convex polyhedra, which have been recognized as space-fillers. These nine space-fillers are the cube [125], regular right hexagonal prism [125], truncated octahedron [292], great rhombicuboctahedron [292], rhombic dodecahedron [292], elongated dodecahedron [292], rhombic Triacontahedron [292], Sommerville’s tetrahedra [345], and Goldberg’s equilateral octahedron [173], which are shown in Fig. 10.2 above and are discussed below.

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Cube

Regular right hexagonal prism

Truncated octahedron

Rhombic dodecahedron

Elongated dodecahedron

Great rhombicuboctahedron

Rhombic Triacontahedron

Sommerville’s largest tetrahedron

Goldberg’s equilateral octahedron

Fig. 10.2 Convex polyhedral space-fillers

Next, we present the major tasks we want to accomplish in this chapter. Also, we briefly describe how to achieve each one of them.

10.1.1 Major Tasks In this chapter, we investigate the problem of connected k-coverage in spatial wireless sensor networks from a computational geometry perspective. Our study is based on polyhedral space-fillers, such as Platonic and Archimedean solids. It is worth mentioning that the connected k-coverage problem in spatial wireless sensor networks is more challenging than its counterpart in planar wireless sensor networks, which was discussed earlier in Chaps. 8 and 9. Indeed, the literature has a few studies that dealt with the problem of placement of the sensors in a spatial field. The closest work to the one discussed in this chapter is the work done by Alam and Haas [12], which investigated this problem based on the volumetric quotient of a few polyhedra. Our work investigates the quality of coverage, which is defined earlier in Chap. 2 (Definition 2.31, Sect. 2.2), for a variety of convex polyhedral space-fillers. Also, there is no work that quantifies the spatial sensor density to cover a spatial field. The work presented in this chapter is motivated by the lack of such a study

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313

in the literature. In this chapter, we want to realize the following tasks, which help solve the connected k-coverage problem in spatial wireless sensor network using a computational geometry-based approach. Next, we specify these tasks and state our corresponding plan of actions. First, we want to determine the best convex polyhedral space-filler, which can be used to assimilate the sensing range of the sensors in order to maximize their volume being utilized to cover a spatial field. In other words, we should investigate the largest enclosed convex polyhedral space-filler in the sensing sphere of the sensors so as to minimize the total number of deployed sensors to cover a spatial field. To this end, we propose an approach that is based on the use of polyhedral convex space-fillers. Precisely, we represent the sensors’ sensing range by various convex polyhedra, which are known as space-fillers. Our goal is to study various convex polyhedral space-fillers and identify the one that maximizes the volume of the sensing range of the sensors that is utilized to k-cover a spatial field. More specifically, our study considers and analyzes the coverage capability of a spatial field of each of the abovementioned nine convex polyhedral space fillers. This analysis is based on our metric, called quality of coverage, which is introduced earlier in Chap. 2 (Definition 2.31, Sect. 2.2) to find the best convex polyhedral space-filler to cover a spatial field. Second, we need to investigate all possible sensor placement strategies to achieve k-coverage of a spatial field, where k ≥ 1. In fact, we propose a few sensor placement strategies that ensure k-coverage of a spatial field based on the outcome of the first task, which attempts to identify the best convex polyhedral space-filler to k-cover a spatial field. Third, we want to compute the spatial sensor density that corresponds to each of these sensor placement strategies, which is required to k-cover a spatial field, where k ≥ 1. Based on the above analysis, we calculate the minimum spatial sensor density that is necessary to k-cover a spatial field for each of these sensor placement strategies. Then, we derive the corresponding minimum number of sensors that are needed to k-cover a spatial field. Fourth, we should precise how the sensors are selected to k-cover a spatial field. For this purpose, we propose an energy-efficient algorithm that aims to select a minimum subset of sensors to cover a spatial field, where the sensors are randomly deployed. This selection algorithm for spatial k-coverage is based on the best convex polyhedral space-filler. Fifth, we should determine the relationship that should exist between the communication range and the sensing range of the sensors to ensure connected k-coverage of a spatial field. To this end, based on the best convex polyhedral space-filler, for each sensor placement strategy, we compute the ratio of the communication range to the sensing range of the sensors, which is needed to guarantee connected k-coverage of a spatial field. Sixth, we support our theory to solve the connected k-coverage problem in spatial wireless sensor networks. That is, we corroborate our analysis of these sensor placement methods and their associated spatial sensor density with various simulation results.

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10.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 10.2 investigates various convex polyhedral space-fillers, and identifies the best one in terms of quality of coverage. Section 10.3 analyzes the connected coverage problem in spatial homogeneous wireless sensor networks using a polyhedral framework. It computes the minimum spatial sensor density for each sensor placement strategy to cover a spatial field, and the corresponding number of sensors to cover a spatial field. Also, it derives the corresponding ratio of the communication range to the sensing range of the sensors for connected coverage of a spatial field. Section 10.4 extends the above analysis to the connected k-coverage problem in spatial homogeneous wireless sensor networks using the same polyhedral framework. It considers several sensor placement strategies to ensure k-coverage configurations and to guarantee connectivity among all the sensors in each of these k-coverage configurations. For each of these sensor placement methods, it computes the required spatial sensor density as well as the ratio of the communication range to the sensing range of the sensors. Also, it describes our sensor selection algorithm for coverage of a spatial field. Section 10.5 shows various simulation results. Section 10.6 reviews existing work on the spatial coverage problem. Section 10.7 concludes the chapter.

10.2 Investigating Polyhedral Space-Fillers In this chapter, we use a polyhedral framework in order to analyze the connected k-coverage problem in spatial homogeneous wireless sensor networks, where all the sensors have the same sensing range, communication range, and battery power. This analysis is based on various spatial convex polyhedral space-fillers. Next, we will compute the value of the quality of coverage of sensor si , denoted by QoC(si ), for each of the convex polyhedral space-fillers we investigate in this study. To this end, we propose to cover a spatial field with a set S of congruent, tangential (or kissing), but non-overlapping polyhedra. Precisely, we restrict the sensors’ sensing range to each of the nine convex polyhedral space-fillers, which are shown on Fig. 10.2, and derive the corresponding quality of coverage, QoC(S). In the sequel, we investigate each of these nine space-fillers in order to identify the best one with respect to the quality of coverage metric. Our study aims to find, among all the convex polyhedral space-fillers, the best polyhedron that provides the maximum quality of coverage, QoC(S), which is defined in Chap. 2 (Definition 2.31, Sect. 2.2). Thus, it is necessary to compute the maximum volume of a polyhedral space-filler that could be enclosed in the sensing sphere of the sensors whose radius is r. Theorem 10.1 given below states the required condition to achieve this maximum volume.

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315

Theorem 10.1 (Enclosed Polyhedron’s Maximum Volume) The maximum volume of any enclosed polyhedron P in the sensing sphere SS r of the sensors can be obtained when the breadth of P is equal to the diameter of this sphere, i.e., 2r. Proof We can proceed by contradiction. Let P be a polyhedral space-filler to be enclosed in the sensing sphere SSr of the sensors whose radius is r. Assume that the maximum distance between tangential planes on opposing faces or edges that bound P is less than 2r. This implies that the distance between the farthest points (or precisely, vertices) on P is less than 2r. That is, some of the vertices of P do not touch the boundary of the sensing sphere of the sensors. Thus, it is possible to stretch P so it occupies more space in this sphere. This contradicts our assumption above. Thus, the above-mentioned maximum distance should be equal to 2r. ∎ Based on Theorem 10.1, we require the distance between the farthest pair of vertices on this polyhedral space-filler P to be equal to 2r . Hence, we use the result of Theorem 10.1 stated above, and adopt it as our strategy in computing the quality of coverage of the corresponding convex polyhedral space-filler. Our results in the following sections are based on Lemma 10.1 stated below. Lemma 10.1 (Quality of Coverage) Let S be a cover set, and si ∈ S. The quality of coverage of S for covering a spatial field using congruent polyhedral space-fillers is computed as follows: QoC(S) = QoC(si ) = 1 − R O(si ) Proof Given that all the sensors are homogeneous, they all play the same role, and there is no sensor that is more privileged (or more critical) than others. That is, all the polyhedral space-fillers are congruent. Thus, the rate of overlap and quality of coverage of a sensor si are the same for all the sensors in the cover set S to cover a spatial field. That is, we have: ( ) QoC(si ) = QoC s j |si ∈ S, s j ∈ S, and i /= j Thus, we obtain: QoC(S) =

|S| × QoC(si ) 1 Σ QoC(si ) = = QoC(si ) |S| s ∈S |S| i

∎

Next, we use both Definitions 2.30 (Rate of Spatial Overlap, Chap. 2, Sect. 2.2) and 2.31 (Sensor Quality of Coverage, Chap. 2, Sect. 2.2), and exploit the result in Lemma 10.1 to study the quality of coverage QoC(S) using each of the above congruent, nonoverlapping polyhedral space-fillers (Fig. 10.2). This metric (i.e., quality of coverage) will help us compare these convex polyhedral space-fillers and identify the most efficient (or optimal) one to cover a spatial field.

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10.2.1 Cubic Space-Filler The sensors’ sensing range is restricted to a cube. Here, the farthest pair of vertices on a cube (c) concerns those opposite vertices A and B that do not share a common face. Using Theorem 10.1, let the length of the line segment that connects A and B be equal to 2r as shown in Fig. 10.2. Also, we denote by ac the length of a side of the cube, which is the unknown parameter to compute. Using Pythagore’s Theorem, we have: δ(A, E) = δ(E, D) = δ(B, D) = ac δ 2 (E, B) = δ 2 (E, D) + δ 2 D δ 2 (A, B) = δ 2 ( A, E) + δ 2 (B, E) (2r )2 = ac2 +

(√ )2 2ac

With a little algebra, the length ac of a side of the cube is: 2 ac = √ r 3 The volume of the cube is given by: 8 V (Cube) = ac3 = √ r 3 3 3 The rate of overlap R Oc (si ) and the quality of coverage QoCc (S) are computed as follows: R Oc (si ) =

2 V (SSr ) − V (Cube) =1− √ = 0.63244740305 V (SSr ) 3π

QoCc (S) = QoCc (si ) = 1 − R Oc (si ) = 0.36755259695

10.2.2 Regular Right Hexagonal Prism Space-Filler A hexagonal prism is a polyhedron that has 8 faces, 18 edges, and 12 vertices as shown in Fig. 10.2. While two of its faces are hexagons, the remaining faces are rectangles. The two hexagonal faces are on the top and bottom. When all the edges

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317

have the same length, this polyhedron is called regular right hexagonal prism (hp). The latter is a polyhedral space-filler whose six rectangular faces are squares. We need to compute the largest regular right hexagonal prism that could be enclosed in a sphere of radius r (i.e., sensors’ sensing sphere). Thus, the Euclidean distance between the farthest two vertices on this hexagonal prism should be equal to the diameter of this sphere. Let A and B be these two vertices, δ(A, B) the Euclidean distance between them, and ahp the edge length of this prism. As it can be seen from Fig. 10.2, ∠BG D = 60◦ . Also, we have: δ(A, E) = δ(E, D) = δ(B, G) = ahp δ 2 (E, B) = δ 2 (E, D) + δ 2 (B, D) δ(B, D) = 2δ(B, G) sin ∠BG D =

√ 3ahp

) ( 2 2 2 2 = 5ahp δ 2 (A, B) = (2r )2 = δ 2 (A, E) + δ 2 (E, B) = ahp + ahp + 3ahp 2 ⇒ ahp = √ r 5 The volume of this regular right hexagonal prism [125] is given by: √ 12 3 3 3√ 3 3ahp = √ r V (Regular right hexagonal prism) = 2 5 5 The rate of overlap R Ohp (si ) and the quality of coverage QoC hp (S) are computed as follows: V (SSr ) − V (Regular right hexagonal prism) V (SSr ) √ 9 3 =1− √ = 0.55618880027 5 5π

R Ohp (si ) =

QoC hp (S) = QoC hp (si ) = 1 − R Ohp (si ) = 0.44381119973

10.2.3 Truncated Octahedral Space-Filler A regular octahedron has 8 faces and 6 vertices. A truncated octahedron (also, called cubo-octahedron or orthic tetrakai-decahedron) can be formed from a regular octahedron by truncating all the vertices of a regular octahedron, which leaves its original

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faces as hexagons. Thus, the number of faces of a truncated octahedron (to) is equal to that of a regular octahedron augmented by 6 square faces that resulted from the deletion of those right square pyramids of a regular octahedron. The edges of this fourteen-sided polyhedron have the same length. Also, this truncation replaces each of the 6 vertices of a regular octahedron by 4 new vertices. This fourteen-sided polyhedron has 24 vertices as shown in Fig. 10.2. The largest truncated octahedron that could be enclosed in a sphere of radius r is defined such that each of its vertices touches the surface of√this sphere. The length of the edge of this truncated octahedron is equal to ato = 10r/5 [477]. Also, its volume [477] is given by: √ √ 3 =8 2 V (T r uncated Octahedr on) = 8 2ato

(√

10r 5

)3

32 = √ r3 5 5

The rate of overlap R Oto (si ) and the quality of coverage QoCto (S) are computed as follows: 48 V (SS R ) − V (T r uncated Octahedr on) =1− √ V (SS R ) 10 5π = 0.31670795832

R Oto (si ) =

QoCto (S) = QoCto (si ) = 1 − R Oto (si ) = 0.68329204168

10.2.4 Great Rhombicuboctahedral Space-Filler This space-filler has other alternate names, including rhombitruncated cuboctahedron, truncated cuboctahedron, omnitruncated cube (or cantitruncated cube), or omnitruncated cuboctahedron. This is a cubic Archimedean polyhedron. It has 48 vertices, 72 edges, and 26 faces, namely 12 squares, 8 hexagons, and 6 octagons). As it can be seen from Fig. 10.2, all the vertices of the great rhombicuboctahedron (gr) are equivalent as each one of them is incident to the same three types of faces (i.e., square, hexagon, and octagon). The length agr of the edge of the great rhombicuboctahedron [479] is given by: agr = √

2

√ r 13 + 6 2

Also, its volume [479] is given by: ( √ ) 3 V (Gr eat Rhombicuboctahedr on) = 22 + 14 2 agr

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319

)3 ( √ ) 2 = 22 + 14 2 √ r3 √ 13 + 6 2 (

The rate of overlap R Ogr (si ) and the quality of coverage QoC gr (S) are computed as follows: V (SSr ) − V (Gr eat Rhombicuboctahedr on) V (SSr ) )3 ( ( √ ) 2 √ 3 22 + 14 2 √ 13+6 2 =1− 4π = 0.19840389947

R Ogr (si ) =

QoC gr (S) = QoC gr (si ) = 1 − R Ogr (si ) = 0.80159610053

10.2.5 Rhombic Dodecahedral Space-Filler As shown in Fig. 10.2, the rhombic dodecahedron has 12 rhombic faces, 24 edges, and 14 vertices. Maclean [285] showed that the radius r of the circumsphere of the rhombic dodecahedron (rd) is related to the length of its edge, say ar d , by the following formula: 2 r = √ ar d 3 This implies that the length of the edge of the rhombic dodecahedron is computed as: √ ar d =

3 r 2

The volume of this rhombic dodecahedron [285] is given by: 16 √ 3 3ar d 9 ( √ )3 3 16 √ = r3 3 2 9

V (Rhombic Dodecahedr on) =

The rate of overlap R Or d (si ) and the quality of coverage QoCr d (S) is computed as follows:

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V (SSr ) − V (Rhombic Dodecahedr on) V (SSr ) 3 = 0.52253517072 =1− 2π

R Or d (si ) =

QoCr d (S) = QoCr d (si ) = 1 − R Or d (si ) = 0.47746482928

10.2.6 Elongated Dodecahedral Space-Filler This polyhedron is also known as the extended rhombic dodecahedron and rhombohexagonal dodecahedron. It is obtained from the rhombic dodecahedron by elongating it along one of its four-fold axes. This elongated dodecahedral (ed) space-filler has 18 vertices, 28 edges, and eight rhombic and four equilateral hexagonal faces, as shown in Fig. 10.3. First, let us compute the length aed of the edge of the elongated dodecahedron. Based on Fig. 10.3 [125], the farthest pair of vertices of the elongated dodecahedron are V8 (on the right of Fig. 10.3) and V8 (on the left of Fig. 10.3). The distance between them should correspond to the diameter of the circumsphere (or sensing sphere), 2r , of the elongated dodecahedron, i.e., δ(V8 , V8 ) = 2r . It is easy to check that this distance is equal to the distance between vertices V7 and V9 , plus the distance between V8 (on the right of Fig. 10.3) and the center of the segment connecting V2 and V5 (denoted by V2 V5 ), plus the same distance between V8 (on the left of Fig. 10.3) and the center of the segment connecting V2 and V5 , where δ(V7 , V9 ) = 2aed and δ(V8 , center (V2 V5 )) = aed cos 60 = aed /2. Thus, we have: δ(V8 , V8 ) = 2r = 2aed + 2 × aed /2 = 3aed We find that the length aed of the edge of the elongated dodecahedron is given by: Fig. 10.3 Elongated dodecahedron [125]

10.2 Investigating Polyhedral Space-Fillers

aed =

321

2 r 3

The volume of this elongated dodecahedron [477] is given by: 3 V (Elongated Dodecahedr on) = 6aed =

16 3 r 9

The rate of overlap R Oed (si ) and the quality of coverage QoCed (S) is computed as follows: 4 V (SSr ) − V (Elongated Dodecahedr on) =1− V (SSr ) 3π = 0.57558681842

R Oed (si ) =

QoCed (S) = QoCed (si ) = 1 − R Oed (si ) = 0.42441318157

10.2.7 Rhombic Triacontahedral Space-Filler Maclean [285] provided a fine analysis of the rhombic triacontahedron (rt). In particular, he proved that the length of the segment, O ' V of the rhombic triacontahedron, as shown in Fig. 10.4, and the length of its edge, say ar t , are related as follows: O ' V = ϕ × ar t

Fig. 10.4 Rhombic triacontahedron [285]

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where ϕ = 1.61802994469. However, O ' V corresponds to the radius of the circumsphere, r , of the rhombic triacontahedron, i.e., O ' V = r . This implies that the length of the edge of the rhombic triacontahedron is computed as: ar t =

r 1.61802994469

The volume of this rhombic triacontahedron [285] is given by: / √ V (Rhombic T riacontahedr on) = 4 5 + 2 5ar3t / )3 ( √ 1 r3 =4 5+2 5 1.61802994469 The rate of overlap R Or t (si ) and the quality of coverage QoCr t (S) is computed as follows: V (SSr ) − V (Rhombic T riacontahedr on) V (SSr ) √ √ ( )3 1 3 5 + 2 5 1.61802994469 =1− π = 0.30619778962

R Or t (si ) =

QoCr t (S) = QoCr t (si ) = 1 − R Or t (si ) = 0.69380221038

10.2.8 Sommerville’s Largest Tetrahedral Space-Filler As stated earlier, Sommerville [361] suggested four tetrahedra to cover a threedimensional space. These four tetrahedral space-fillers are shown in Fig. 10.5. In our study, we focus on the largest tetrahedra that would be enclosed in the sensing sphere of a sensor whose radius is equal to R. As it can be seen in Fig. 10.5, tetrahedra of types (b), which is also shown in Fig. 10.2, and (c) are the largest ones and have the same volume. In fact, it was stated by Senechal [345] that each of those tetrahedra can be obtained by joining two tetrahedra of type (a). More precisely, two tetrahedra of type (a) can be obtained by bisecting tetrahedron of type (b) along the cube face. Also, joining two tetrahedra of type (a) along a common face through a vertex of the cube, center of a face, and center of a cube would result in tetrahedron of type (c). The volume of Sommerville’s largest tetrahedral (slt) space-filler is computed as follows. Let us use tetrahedron of type (ii). The distance between the farthest pair of vertices is equal to 2r . This implies that the side length of the cube should be 2r . However, according to Senechal [345], the cube can be dissected into 24 congruent

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323

Fig. 10.5 Sommerville tetrahedra [149]

tetrahedra of type (a). Hence, the volume of tetrahedron of type (a) is given by: V (T etrahedr on o f T ype(a)) =

r3 (2r )3 = 24 3

Given that tetrahedron of type (b) is obtained from two tetrahedra of type (a), we have: V (T etrahedr on o f T ype(b)) =

2r 3 3

The rate of overlap R Oslt (si ) and the quality of coverage QoCslt (S) is computed as follows: 1 V (SSr ) − V (T etrahedr on o f T ype(b)) =1− V (SSr ) 2π = 0.8408450569

R Oslt (si ) =

QoCslt (S) = QoCslt (si ) = 1 − R Oslt (si ) = 0.1591549431

10.2.9 Baumgartner’s Tetrahedral Space-Filler Baumgartner [77] proposed an elementary, asymmetric tetrahedral space-filler, which cannot be subdivided into other congruent tetrahedra. Baumgartner’s tetrahedron (bt) can fill space by replication. It is half of Sommerville’s tetrahedron of type (a) shown

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in Fig. 10.5. Indeed, the latter can be subdivided into two Baumgartner’s congruent tetrahedra. The rate of overlap R Obt (si ) and the quality of coverage QoCbt (S) is computed as follows: ) ( V (SSr ) − V Baumgar tner ' s T etrahedr on R Obt (si ) = V (SSr ) V ( Sommer ville' s T etrahedr on o f T ype(a)) V (SSr ) − 2 = V (SSr ) ) ( 2 × V (SSr ) − V Sommer ville' s T etrahedr on o f T ype(a) = 2 × V (SSr ) 1 =1− = 0.92042252845 4π QoCbt (S) = QoCbt (si ) = 1 − R Obt (si ) = 0.07957747155

10.2.10 Goldberg’s Equilateral Octahedral Space-Filler As stated earlier, it is well known that the only Platonic polyhedral space-filler is the cube [125]. Particularly, the regular octahedron is not a polyhedral space-filler. Indeed, Minkowski [190] proved that octahedra can be packed together to cover only 18/19 of unbounded three-dimensional space. However, Goldberg [173] discovered an octahedron-based polyhedral space-filler, called Goldberg’s equilateral octahedron (geo), as shown in Fig. 10.6. The latter can be formed by joining two square Fig. 10.6 Equilateral octahedron enclosed in a sphere

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325

pyramids at the common face. Notice that six pairs of square pyramids can be obtained from a cube by cutting planes from its center to its edges. Each joined pair of those square pyramids at their square faces give rise to an equilateral octahedral space-filler. Let us compute the largest equilateral octahedron that could be enclosed in the sensing sphere of the sensors whose radius is r . As shown in Fig. 10.6, the six vertices of the equilateral octahedron touch the surface of the sphere. Thus, the height, say h, of a square pyramid is equal to r . Also, the base edge, say ageo , of the square face is equal to r . Hence, the volume of one square pyramid of height r and base edge r , and, consequently, the volume of the corresponding equilateral octahedron are, respectively, given by: 2 V (Squar e P yramid) = ageo

r3 h = 3 3

V (Equilateral Octahedr on) =

2r 3 3

The rate of overlap R Ogeo (si ) and the quality of coverage QoC geo (S) is computed as follows: V (SSr ) − V (Equilateral Octahedr on) V (SSr ) 1 =1− = 0.8408450569 2π

R Ogeo (si ) =

QoC geo (S) = QoC geo (si ) = 1 − R Ogeo (si ) = 0.1591549431

10.3 Solving the Connected Coverage Problem In this section, we analyze the connected coverage problem in spatial homogeneous wireless sensor networks using a polyhedral framework, where all the sensors have the same sensing range, communication range, and battery power. This analysis is based on the optimal spatial convex polyhedral space-filler. O’Keeffe and Hyde [315] found the great rhombicuboctahedron as the largest polyhedron with equivalent vertices and cubic symmetry. Indeed, we found the quality of coverage provided by the truncated cuboctahedron higher than that given by the truncated octahedron. Recall that Alam and Haas [12] found that the best polyhedral space-filler was the truncated octahedron. In this work, we found a much better polyhedral space-filler, which is great rhombicuboctahedron. As it can be seen from the above results, the great rhombicuboctahedron has the maximum quality of coverage compared to all other convex polyhedral space-fillers. Thus, we conclude that the great rhombicuboctahedron is the best convex polyhedral space-filler, which

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10 A Polyhedral Space Filler Tessellation-Based Approach …

will be used in our study of the connected k-coverage problem in spatial homogeneous wireless sensor networks. Lemma 10.2 below states this result. Lemma 10.2 (Optimal Convex Polyhedral) The great rhombicuboctahedron is the optimal convex polyhedral space-filler. Proof Based on all the results shown in Sects. 10.2.1–10.2.10, the great rhombicuboctahedron has the maximum quality of coverage among the set of all convex polyhedral space-fillers, denoted by 3DC P S F. That is: QoC gr (S) = max{QoCC P S F (S), C P S F ∈ 3DC P S F} Thus, the great rhombicuboctahedron is the optimal convex polyhedral spacefiller. ∎ Now, we derive the relationship between the sensing range and communication range of the sensors to ensure network connectivity. First, let us compute the distance between the centroid of a great rhombicuboctahedra and each of its faces, which could be a square, regular hexagon, or regular octagon. Lemma 10.3 computes these distances. Lemma 10.3 (Great Rhombicuboctahedra Distances) The distances between the centroid C of a great rhombicuboctahedra and its faces, denoted by γ (C, f ace), are computed as follows: ⎧ √ √ agr ⎪ 6 2 i f f ace = squar e ⎪ ⎨ 2 √ 11 + √ agr 9 + 6 2 i f f ace = hexagon γ (C, f ace) = 2 / √ ⎪ ⎪ ⎩ agr 10−√ 2 i f f ace = octagon 2

where agr = √

2 √ r 13+6 2

2− 2

is the edge length of the great rhombicuboctahedron.

Proof Let γ (C, squar e) the distance between the centroid C of a great rhombicuboctahedra and its square face. According to Fig. 10.7a, using Pythagore’s Theorem, we have: r 2 = γ 2 (C, squar e) + bs2 √ However, it is easy to check that bs = agr / 2. Now, given that agr = √ 2 √ r , we obtain: 13+6 2

γ (C, squar e) =

agr 2

/ √ 11 + 6 2

10.3 Solving the Connected Coverage Problem

327

Fig. 10.7 Distances γ (O, f ) for great rhombicuboctahedra

Let γ (C, hexagon) the distance between the centroid C of a great rhombicuboctahedra and its hexagonal face. According to Fig. 10.7b, using Pythagore’s Theorem, we have: r 2 = γ 2 (C, hexagon) + bh2 Given that bh = agr and agr = √

2 √ r, 13+6 2

we get:

agr γ (C, hexagon) = 2

/

√ 9+6 2

Let γ (C, octagon) the distance between the centroid C of a great rhombicuboctahedra and its octagonal face. According to Fig. 10.7c, using Pythagore’s Theorem, we have: r 2 = γ 2 (C, octagon) + bo2 √ √ √ √ ( ) Given that bo = agr /2 / sin π/8 = agr / 2 − 2 and agr = 2r/ 13 + 6 2, we get: / agr γ (C, octagon) = 2

√ 10 − 2 √ 2− 2

∎

Lemma 10.4 states the relationship that should exist between the sensors’ sensing range and communication range to guarantee network connectivity.

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10 A Polyhedral Space Filler Tessellation-Based Approach …

Lemma 10.4 (Great Rhombicuboctahedra-Based Network Connectivity) The ratio of the radius R of the communication sphere to the radius r of the sensing sphere of the sensors to achieve connected coverage of a spatial field is as follows: /

√ 11 + 6 2 R≥2 √ r = 1.90463961738r 13 + 6 2 Proof The distance between the centroids C1 and C2 of two adjacent great rhombicuboctahedra is the length of the line segment between these two centroids that is orthogonal (or perpendicular) to one of their shared face (i.e., square, regular hexagon, or regular octagon). Thus, the distance between the centroids C1 and C2 , denoted by γ (C1 , C2 ), is twice the distance between one of these two centroids and the shared face. Thus, two adjacent sensors can communicate with each other if the radius R of their communication spheres is at least twice the maximum of their three distances given above. That is, we have: R ≥ 2 × max{γ (C, squar e), γ (C, hexagon), γ (C, octagon)} In other words, R should satisfy the following inequality: /

√ 11 + 6 2 R≥2 √ r = 1.90463961738r 13 + 6 2

∎

10.3.1 Sensor Selection Algorithm Here, we show how to select sensors to cover a spatial field. First, we compute the number of sensors that are required to cover a spatial field, and the corresponding spatial sensor density. Then, we compute the minimum spatial sensor density for spatial coverage. 10.3.1.1

Spatial Sensor Density for Spatial Coverage

We geometrically divide a spatial field into nonoverlapping, congruent regions, each of which will be filled with the same polyhedral space-filler. Thus, we obtain a periodic space-filling grid with a fixed number of convex polyhedral space-fillers. To compute the minimum number of sensors, and, hence, the minimum spatial sensor density to cover a spatial field, we consider a deterministic sensor placement strategy in which the sensors are placed at the centroids of these nonoverlapping polyhedral space-fillers, thus, forming a periodic space-filling grid with a fixed number of sensors. Lemma 10.5 computes the number of sensors, denoted by n tot (C P S F), which are required to cover a spatial field with congruent polyhedral space-fillers. Lemma 10.6 computes the spatial sensor density of this periodic space-filling grid.

10.3 Solving the Connected Coverage Problem

329

Lemma 10.5 (Total Number of Sensors for Coverage) The total number of sensors that are needed to cover a spatial field is denoted by n tot (C P S F) and is computed as follows: n tot (C P S F) =

V (FoI ) V (C P S F)

where V (FoI ) stands for the volume of a spatial field, and V (C P S F) is the volume of a congruent polyhedral space-filler. Proof Given that the sensing range of the sensors is restricted to congruent convex polyhedral space-fillers (CPSF), the total number of sensors needed to cover a spatial field should be equal to the ratio of the volume of the spatial field to the volume of the convex polyhedral space-fillers. Thus, we have: n tot (C P S F) =

V (FoI ) V (C P S F)

∎

Lemma 10.6 (Spatial Sensor Density for Coverage) Let V (C P S F) be the volume of a congruent polyhedral space-filler. The spatial sensor density ρ(C P S F) of a periodic space filling grid, which uses these congruent space-fillers, is given by: ρ(C P S F) = 1/V (C P S F) Proof The spatial sensor density of a periodic space filling grid using congruent polyhedral space-fillers, denoted by ρ(C P S F), is defined as the number of sensors per unit volume. That is, ρ(C P S F) =

n tot (C P S F) V (FoI )

Substituting n tot (C P S F) for its value computed in Lemma 10.5 above, we obtain the desired result: ρ(C P S F) = 1/ V (C P S F)

∎

Now, Theorem 10.2 computes the minimum spatial sensor density to cover a spatial field. Theorem 10.2 (Minimum Spatial Sensor Density for Coverage) The minimum spatial sensor density, denoted by ρmin (r ), to achieve coverage of a spatial field is computed as: ρmin (r ) =

0.29782132732 r3

where r is the radius of the sensing sphere of the sensors.

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10 A Polyhedral Space Filler Tessellation-Based Approach …

Proof As per Lemma 10.2, the great rhombicuboctahedron is the optimal convex polyhedral space-filler that maximizes the quality of coverage metric. Also, the sensors are to be placed at the centroids of these nonoverlapping great rhombicuboctahedra. Thus, the minimum spatial sensor density ρmin (r ) to achieve coverage of a spatial field is computed as: 1 V (Gr eat Rhombicuboctahedr on) 1 = )3 ( √ 2 (22 + 14 2) √ r3 √

ρmin (r ) =

13+6 2

0.29782132732 = r3

10.3.1.2

∎

Random Sensor Selection

Here, we assume that every sensor has a unique ID that distinguishes it from all other sensors in the network. Our sensor selection method consists of two main steps. First, we compute the periodic space-filling grid. Second, based on this grid, we select the sensors to cover a spatial field. To allow all the sensors to participate in this coverage process, we generate this grid randomly and in every round. Given that the sink has infinite source of energy, we assume that this task is performed by the sink. Moreover, the sink is responsible for selecting the sensors in each round to cover a spatial field. Precisely, the sink divides a spatial field into nonoverlapping, tangential great rhombicuboctahedra. Then, it selects the sensors that are closer to the centroids of these great rhombicuboctahedra. However, some selected sensors’ locations may not coincide with those centroids. In this case, the sink should select other sensors to ensure full coverage of the spatial field. An appealing solution, which we leave as future work, is to use mobile sensors so they locate themselves at the centroids of those great rhombicuboctahedra. At the end of this sensor selection, the sink broadcasts its schedule for the next round in the network. A schedule SCHE has the following structure: SCHE = , where id j denotes the ID of a selected sensor, and m stands for the number of great rhombicuboctahedra forming the periodic space-filling grid. Due to the edge-effect problem, some great rhombicuboctahedra may not have their full shape (i.e., partial polyhedra). To achieve full coverage of the spatial field, the sink should select sensors for these partial great rhombicuboctahedra. Upon receiving a message SCHE, a sensor checks whether its ID is included. If so, it remains active and forwards SCHE to its one-hop neighbors. Otherwise, it simply forwards it to its one-hop neighbors and turns itself off. We assume that all the sensors are awake (or on) at the beginning of each round.

10.3 Solving the Connected Coverage Problem

331

10.3.2 Performance Evaluation In this section, we present simulation results of our great rhombicuboctahedron-based coverage in spatial wireless sensor networks. First, we describe our simulation environment. Then, we show the results of the performance evaluation of our proposed spatial coverage approach using a high-level simulator written in C.

10.3.2.1

Simulation Environment

We consider a cubic spatial field with side length 300 m. We assume that there are 1000 randomly deployed sensors in this cube. We suppose that the initial energy of each sensor is 70 J. We use the IEEE 802.11 distributed coordinated function with CSMA/CA as the underlying MAC protocol. Also, we consider a radio interference model given the pervasiveness of other 2.4 GHz radio sources. During their lifetime, the sensors spend energy in data sensing, data transmission, data reception, and control messages, which are necessary for our spatial coverage approach. We assume that the radii of the sensing and communication ranges of all the sensors are 30 m and 60 m, respectively. Moreover, all simulations are repeated 100 times and the results are averaged.

10.3.2.2

Simulation Results

Figure 10.8 considers the spatial sensor density to cover a spatial field versus the radius of the sensing sphere of the sensors. It shows a close match between our theoretical and simulation results. The difference is due to the edge-effect problem, which Fig. 10.8 Great rhombicuboctahedra— theoretical versus simulation

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10 A Polyhedral Space Filler Tessellation-Based Approach …

is briefly discussed in Sect. 10.3.1.2, as well as the locations of the sensors which may not necessarily coincide with the centroids of the great rhombicuboctahedra, which form the periodic space-filling grid generated by the sink. Figures 10.9 and 10.10 compare between Alam and Haas’ work [12], which proposed the truncated octahedron as the best space-filler, and ours in this chapter, which suggests the great rhombicuboctahedra as the optimal convex polyhedral space-filler. This comparison considers both the theoretical (Fig. 10.9) and simulation results (Fig. 10.10). Clearly, our proposed convex polyhedral space-filler (i.e., great rhombicuboctahedra) outperforms the one proposed by Alam and Haas [12] (i.e., truncated octahedron) for covering a spatial field. Figure 10.11 plots the remaining energy versus the time for our great rhombicuboctahedra-based spatial coverage approach, as compared to Fig. 10.9 Great rhombicuboctahedra versus truncated octahedron—theoretical

Fig. 10.10 Great rhombicuboctahedra versus truncated octahedron—simulation

10.4 Connected k-Coverage Problem

333

Fig. 10.11 Great rhombicuboctahedra versus truncated octahedron—energy = f (time)

the truncated octahedron-based Alam and Haas’ approach [12], where r = 25 m. As expected, Fig. 10.11 shows that our approach is more energy-efficient than the one proposed by Alam and Haas [12]. Indeed, our approach restricts the sensors’ sensing range to the great rhombicuboctahedra, while Alam and Haas [12] restricts the sensing range of the sensors to the truncated octahedron. Both of these convex polyhedra are space-fillers. However, the volume of the great rhombicuboctahedra is higher than that of the truncated octahedron. Therefore, the number of sensors needed by our approach is much less than that required by Alam and Haas’ method [12]. This extra number of sensors causes higher energy consumption. Also, the number of control messages exchanged among the sensors increases with their number. This increase of the number of sensors causes more energy. Thus, the energy consumption of our approach is much less than that of Alam and Haas’ method [12].

10.4 Connected k-Coverage Problem In this section, we focus on the problem of connected k-coverage of a spatial field using homogeneous wireless sensor networks, where all the sensors have the same sensing range and same communication range. As it can be seen from the above results, the great rhombicuboctahedron has the maximum quality of coverage compared to all other convex polyhedral space-fillers. Thus, we conclude that the great rhombicuboctahedron is the best space-filler, which will be used in our study of the connected k-coverage problem in spatial homogeneous wireless sensor networks. Recall that Alam and Haas [12] found that the best polyhedral space-filler was the truncated octahedron. In this work, we found a much better polyhedral space-filler, which is great rhombicuboctahedron. Indeed, we found that the quality of coverage

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10 A Polyhedral Space Filler Tessellation-Based Approach …

provided by the great rhombicuboctahedron is higher than the one given by the truncated octahedron.

10.4.1 Achieving Spatial k-Coverage As stated earlier, the great rhombicuboctahedron is the largest (i.e., with maximum volume) space-filler that can be inscribed in the sensing sphere of the sensors. Our analysis of the k-coverage problem uses this space-filler. Our goal is to investigate the placement of k sensors in the great rhombicuboctahedron so it is k-covered with exactly these k sensors. First, we divide a spatial field into tangential (or kissing), congruent great rhombicuboctahedra, thus, forming a grid. Next, we describe two classes of sensor placement strategies to k-cover a spatial field.

10.4.1.1

Centroid-Based Sensor Placement

One obvious strategy is to place k sensors at the centroid of each great rhombicuboctahedron of edge length agr of this grid, where agr is computed as follows: agr = √

2

√ r 13 + 6 2

with r being the radius of the sensors’ sensing sphere. Lemma 10.7 computes the spatial sensor density ρC (k, r ) associated with this type of sensor placement. Lemma 10.7 (Spatial Sensor Density for k-Coverage) The spatial sensor density ρC (k, r ) that is required to k-cover a spatial field and which corresponds to the placement of k sensors at the centroid of the great rhombicuboctahedron of edge length agr is given by: ρC (k, r ) = 0.29782132732

k r3

where r is the radius of the sensors’ sensing sphere. Proof The spatial sensor density ρC (k, r ) is the number of sensors per unit volume. Given that the placement of sensors at the centroid of the great rhombicuboctahedron of edge length agr ensures k-coverage of a spatial field, ρC (k, r ) is computed as follows:

10.4 Connected k-Coverage Problem

ρC (k, r ) =

335

k k ( )=( ( ) √ Vgr agr 22 + 14 2 √

)3 2

√ 13+6 2

= 0.29782132732

k r3

r3 ∎

We should note that it is not possible to place k sensors at the same point of a great rhombicuboctahedron. Indeed, the sensors are physical devices that cannot be reduced to just one point.

10.4.1.2

Plane-Based Sensor Placement

Here, we want to find a more practical yet energy-efficient solution to the sensor placement problem to ensure k-coverage of a spatial field. Our goal is to extend as much as possible the region where to deploy k sensors, while maintaining k-coverage of the great rhombicuboctahedron. Instead of placing sensors at the centroid C of each great rhombicuboctahedron of the grid, we place them on a planar area A that is inside each great rhombicuboctahedron and is tangential to the centroid C. To this end, we consider each planar area A as one of the three shapes corresponding to the square, hexagonal, and octagonal faces of the great rhombicuboctahedron. Case 1: Placing sensors on a tangential square of edge a gr . ( ) Let TS C, agr be a tangential square of edge length agr and ) center coincides ( whose a great rhombicuboctahedron G R C, a whose( edge length with the centroid C of gr ( ) ) is facing one of the square faces of G R C, agr . We is agr , such that TS C, agr ( ) ) ( place all k sensors on TS C, agr inside ( G R )C, agr However, with such a sensor placement strategy, some areas of G R C, agr are not guaranteed to be k-covered. To illustrate this problem, let us compute the ( ) farthest distance between a point p (possible location of a sensor)(on TS )C, agr and( a point ) q (possible location of an event) on a square face of G R C, agr facing TS C, agr and whose center is Os , as shown on Fig. 10.12. Using Pythagore’s Theorem, we have: δ 2 ( p, q) = δ 2 ( p, u) + δ 2 (u, q)

( ) Fig. 10.12 Tangential square TS C, agr

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10 A Polyhedral Space Filler Tessellation-Based Approach …

Also, we have the following equalities: 2 δ 2 (u, q) = δ 2 (u, v) + δ 2 (v, q) = 2agr

δ( p, u) = δ(C, Os ) ) ( Notice that q is one of the vertices of G R C, agr . Thus, we have: δ(C, q) = r . Let us compute δ(C, Os ). According to Fig. 10.12, using Pythagore’s Theorem, we have: δ 2 (C, q) = r 2 = δ 2 (C, Os ) + δ 2 (Os , q) √ However, we have: δ(Os , q) = δ(u, q)/2 = agr / 2, where agr = √

2

√ r 13 + 6 2

Therefore, we obtain: / δ(C, Os ) =

r2

−

2 agr

2

/ =

√ 11 + 6 2 √ r 13 + 6 2

We conclude that: √ δ( p, q) = δ 2 (C, Os ) + δ 2 (u, q) =

/

√ 19 + 6 2 √ r 13 + 6 2

That is, δ( p, q) = 1.1310441839r > r , where r is the radius of the(sensing) sphere of the sensors that contains the largest great rhombicuboctahedron G R C, agr . Thus, a sensor located at point p cannot ( sense) an event located at q. In other words, the great rhombicuboctahedron G R C, agr with edge (length a) gr cannot be k-covered with k sensors placed on the tangential square TS C, agr . Hence, it( is essential ) s s of a great rhombicuboctahedron G R C, agr that to determine the edge length agr k sensors when they are deployed on a tangential square is surely k-covered with ( ) TS C, agr . To achieve this, we require the farthest distance between those two points s p and q to be r . That is, δ( p, q) = r . Now, we need to compute the value of agr , which is given by: 2 s agr = √ √ rs 13 + 6 2

10.4 Connected k-Coverage Problem

337

with rs being the radius of the smallest( reduced ) sensing sphere ofsthe sensors in which s of edge length agr can be inscribed. the great rhombicuboctahedron G R C, agr s < agr . On the one hand, we have: Clearly, we have: agr δ 2 ( p, q) = δ 2 ( p, u) + δ 2 (u, q) = r 2 , where: ( s )2 δ 2 (u, q) = 2 agr On the other hand, we have: δ 2 (C, q) = (rs )2 = δ 2 (C, Os ) + δ 2 (Os , q) √ s with δ(Os , q) = agr / 2. We conclude that: / δ(C, Os ) =

(rs )2 −

( )2 s agr

Using δ 2 ( p, u)+δ 2 (u, q) = r 2 , where δ(u, q) = we get: rs = /

r

√ 19+6√2 13+6 2

2 √

s 2agr and δ( p, u) = δ(C, Os ),

= 0.884138758r

Also, we have: 2 2 s agr = √ √ r √ rs = √ 19 + 6 2 13 + 6 2 ( ) s The size of the deployment area, i.e., area of the square TS C, agr of edge length s agr , is given by: ( s ) ( s )2 = agr = 0.14553243772r 2 A agr (Now,s we ) can claim thats the placement of k sensors on the tangential square TS C, agr of edge length agr guarantees k-coverage of a great rhombicuboctahedron ) ( s s G R C, agr whose edge length is agr . Lemma 10.8 given below computes the spatial sensor density associated with such a sensor placement strategy. Lemma 10.8 (Spatial Sensor Density for k-Coverage) The spatial sensor density ρ S (k, r ) that is required to k-cover a spatial field( and which corresponds to the ) s s of edge length agr that placement of k sensors on a tangential square TS C, agr ) ( s s faces a square face of a great rhombicuboctahedron G R C, agr of edge length agr and centered at C is computed as follows:

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10 A Polyhedral Space Filler Tessellation-Based Approach …

ρ S (k, r ) = 0.43091787285

k r3

Proof field is k-covered through the placement of k sensors on each square ( As spatial ) s TS C, agr of edge length agr = √ 2 √ r that is tangential to the centroid of a ) 2 ( 19+6 s s great rhombicuboctahedron G R C, agr of the grid and whose edge length is agr . Thus, the spatial sensor density ρ S (k, r ), which is necessary to ensure k-coverage of a spatial field, is given by: ρ S (k, r ) =

k k ( ) = 0.43091787285 3 s r Vgr agr

√ )( s )3 ( s ) ( . where Vgr agr = 22 + 14 2 agr

∎

Case 2: Placing sensors on a tangential hexagon of edge a gr . ( ) Now, we place all k sensors on a tangential hexagon TH C, agr of edge length agr ) ( and centered at the centroid( C of a)great rhombicuboctahedron G R C,(agr whose ) edge length is agr . Also, TH C, agr is facing one hexagonal face of G R C, agr . As in the previous case, the latter is not fully k-covered with those k sensors. To show this problem, let us compute the farthest distance between a point p (or location of event a sensor) on TH and a point q (possible ( ) location) on the hexagonal face of this great rhombicuboctahedron G R C, agr and whose center is Oh , as shown on Fig. 10.13. As per Pythagore’s Theorem, we get: δ 2 ( p, q) = δ 2 ( p, u) + δ 2 (u, q) where δ(u, q) = 2agr and δ( p, u) = δ(C, Oh ). Let us compute δ(C, Oh ). Using Pythagore’s Theorem and Fig. 10.13, we have: δ 2 (C, q) = r 2 = δ 2 (C, Oh ) + δ 2 (Oh , q)

( ) Fig. 10.13 Tangential hexagon TH C, agr

10.4 Connected k-Coverage Problem

339

However, we have: δ(Oh , q) = agr , with agr = √ / 2 = δ(C, Oh ) = r 2 − agr

/

2 √ r. 13+6 2

Thus, we obtain:

√ 9+6 2 √ r 13 + 6 2

We conclude that: δ( p, q) =

√

/ δ 2 (C, Oh ) + δ 2 (u, q) =

√ 25 + 6 2 √ r 13 + 6 2

Thus, δ( p, q) = 1.24840774263r > r . Given that an event that occurs at location q cannot(be detected by a sensor located at point p, the great rhombicuboctahe) be k-covered with those k sensors placed on the tangential dron G R C,(agr cannot ) h of the great rhomhexagon TH C, agr . We want to compute the edge length agr ( ) h bicuboctahedron G R C, agr that can be k-covered with k sensors only. Likewise, the maximum distance δ( p, q) between those two farthest points p and q should be h is computed as follows: equal to r , i.e., δ( p, q) = r . Recall that agr h agr = √

2

√ rh 13 + 6 2

where rh stands for the radius of the smallest reduced sensing sphere of the sensors h can be inscribed. As per in which a great rhombicuboctahedron of edge length agr 2 2 2 Fig. 10.13, we have: δ ( p, q) = δ ( p, u) + δ (u, q) = r 2 , where: h δ(u, q) = 2agr

Following the same computational process as in the previous case, we have: / δ( p, u) = δ(C, Oh ) =

√ 9+6 2 √ rh 13 + 6 2

Finally, we obtain: rh = /

h agr = √

r

√ 25+6√2 13+6 2

= 0.80102034444r

2 √ rh = √ √ r 13 + 6 2 25 + 6 2 2

It is clear that the entire ) field, which is represented by a grid (of great ) ( spatial h h , is guaranteed to be k-covered if every G R C, agr rhombicuboctahedra G R C, agr

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10 A Polyhedral Space Filler Tessellation-Based Approach …

is k-covered, according to the above sensor placement method. Lemma 10.9 computes the spatial sensor density associated with such a sensor placement strategy. ( ) h The size of the deployment area, i.e., area of the regular hexagon TH C, agr of h edge length agr , is given by: √ ( h ) 3 3 ( h )2 agr = 0.31035441301r 2 A agr = 2 Lemma 10.9 (Spatial Sensor Density for k-Coverage) The spatial sensor density ρ H (k, r ) that is required to k-cover a spatial field and which corresponds to the h and whose center coincides placement of k sensors on a hexagon of edge length agr h with the centroid of the great rhombicuboctahedron of edge length agr is given by: ρ H (k, r ) = 0.57946225925

k r3

Proof field is k-covered through the placement of k sensors on each hexagon ( Ahspatial ) h TH C, agr of edge length agr = √ 2 √ r that is tangential to the centroid of a 25+6 ) 2 ( h h of the grid and whose edge length is agr . great rhombicuboctahedron G R C, agr Thus, the spatial sensor density ρ H (k, r ) that is necessary to ensure k-coverage of a spatial field is computed as: ρ H (k, r ) =

k k ( ) = 0.57946225925 3 h r Vgr agr

√ )( h )3 ( h) ( = 22 + 14 2 agr . where Vgr agr

∎

Case 3: Placing sensors on a tangential octagon of edge a gr . ( ) In this case, we place all k sensors on a tangential octagon ( TO )C, agr whose center is the centroid C of the great rhombicuboctahedron G R C, agr whose edge(length is ) agr . Such a sensor placement strategy does not guarantee(k-coverage ) of G R C, agr . Indeed, it is easy to check that there are regions in G R C, agr that are left not kcovered. Consider Fig. 10.14 and let us compute the distance between two(farthest) points from each other, namely point p (potential sensor position) ( on)TO C, agr and point q (potential event location) on a hexagonal face of G R C, agr and whose center is Oo , as shown on Fig. 10.14. As per Pythagore’s Theorem, we get: δ 2 ( p, q) = δ 2 ( p, u) + δ 2 (u, q) √ √ where δ(u, q) = 2δ(u, O0 ) = 2δ(O0 , q) = agr / sin(π/8) = 2agr / 2 − 2 and δ( p, u) = δ(C, Oo ). Let us compute δ(C, Oo ). According to Fig. 10.14, using Pythagore’s Theorem, we have:

10.4 Connected k-Coverage Problem

341

( ) Fig. 10.14 Tangential octagon TO C, agr

δ 2 (C, q) = r 2 = δ 2 (C, O0 ) + δ 2 (Oo , q) However, we have: / √ δ(Oo , q) = agr /2 sin(π/8) = agr / 2 − 2 where agr = √

2 √ r. 13+6 2

Thus, we obtain: /

δ(C, Oo ) =

2 agr r2 − √ = 2− 2

/

√ 2 √ r 14 − 2

10 −

We conclude that: √ δ( p, q) = δ 2 (C, Oo ) + δ 2 (u, q) =

/

26 −

√

2 √ r 14 − 2

It implies that δ( p, q) = 1.39766108518r > r . Given that there( is at least ) one point that is not k-covered, the entire great rhombicuboctahedron G R C, agr cannot be k-covered using such a sensor placement strategy. We want to find the edge length o agr of a great rhombicuboctahedron that can be k-covered with exactly k sensors. Similarly, it is necessary to have δ( p, q) = r . Now, we need to compute the value of o agr , where o agr = √

2

√ ro 13 + 6 2

with ro being the radius of the smallest reduced )sensing sphere ofo the sensors that ( o of edge length agr . First, we have: contains a great rhombicuboctahedron G R C, agr δ 2 ( p, q) = δ 2 ( p, u) + δ 2 (u, q) = r 2

342

10 A Polyhedral Space Filler Tessellation-Based Approach …

where: o δ(u, q) = 2agr

Furthermore, we have: δ 2 (C, q) = (ro )2 = δ 2 (C, O0 ) + δ 2 (Oo , q) √ √ h with δ(Oo , q) = agr / 2 − 2. We conclude that: [ | ( )2 | o agr ] 2 δ(C, Oo ) = (ro ) − √ 2− 2 Also, we have: δ( p, u) = δ(C, Oo ) h Using δ 2 ( p, u) + δ 2 (u, q) = r 2 , with δ(u, q) = 2agr and δ( p, u) = δ(C, Oo ), we get:

/ ro =

√ 2 √ r = 0.71548103514r 26 − 2

14 −

Also, we have: /

o agr

= √

√ 14 − 2 = 2 r √ r √ o 326 + 143 2 13 + 6 2 2

( ) o The size of the deployment area, i.e., area of the regular octagon TO C, agr of o , is given by: edge length agr ( √ )( o )2 ( o) = 2 1 + 2 agr A agr = 0.46017273141r 2 Lemma 10.10, which is given below, computes the spatial sensor density associated with such a sensor placement strategy. Lemma 10.10 (Spatial Sensor Density for k-Coverage) The spatial sensor density ρ O (k, r ) that is required to k-cover a spatial to the ( field ) and which corresponds o o of edge length agr and whose placement of k sensors on an octagon TO C, agr ) ( o of center coincides with the centroid of the great rhombicuboctahedron G R C, agr o is given by: edge length agr

10.4 Connected k-Coverage Problem

343

ρ O (k, r ) = 0.81313267830

k r3

Proof field is k-covered through the placement of k sensors on each octagon ( Aospatial ) o TO C, agr of edge length agr = √ 1 √ r that is tangential to the centroid of a great 14− 2

o . Thus, the spatial rhombicuboctahedron of the grid and whose edge length is agr sensor density ρ O (k, r ) that is necessary to ensure k-coverage of a spatial field is computed as:

ρ O (k, r ) =

k k ( ) = 0.81313267830 3 o r Vgr agr

√ )( o )3 ( o) ( = 22 + 14 2 agr . where Vgr agr

∎

Next, we derive the relationship that should exist between the sensing range and communication range of the sensors to ensure network connectivity.

10.4.2 Ensuring Spatial Connected k-Coverage We consider the above three sensor placement strategies. For each one of them, we compute the ratio of the communication range to the sensing range of the sensors.

10.4.2.1

( ( s Sensor Placement on Tangential Square TS C, a gr

Consider the configuration shown on Fig. 10.15. Let p1 and p2 be the locations of the two farthest sensors s1 (and s2 , respectively, ) ( from )each other, which are located on s s and TS C2 , agr centered at C1 and C2 , respectwo tangential squares TS C1 , agr ) ) ( ( s s and G R C2 , agr , tively, inside two adjacent great rhombicuboctahedra G R C1 , agr respectively. To maintain network connectivity, both sensors s1 and s2 should be able to communicate with each other. Lemma 10.11 states this result.

( ) s Fig. 10.15 Network connectivity based on TS C, agr

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10 A Polyhedral Space Filler Tessellation-Based Approach …

Lemma 10.11 (Relationship Between R and r for Network Connectivity) A spatial kcovered wireless sensor network, which is deployed using a tangential square-based sensor placement strategy, is connected if the ratio of the radius of the communication range of the sensors to the radius of their sensing range satisfy the following inequality: R ≥ 1.768277516 ≈ 1.768 r Proof As it can be seen from Fig. 10.15, sensors s1 and s2 can communicate with each other only if the radius of the sensors’ communication range satisfies the following inequality: R ≥ δ( p1 , p2 ) where, δ( p1 , p2 ) = δ( p1 , Os ) + δ(Os , p2 ) = 2δ( p1 , Os ). However, δ( p1 , Os ) = rs . Therefore, we have: R ≥ 2rs = 2 × 0.884138758r = 1.768277516r where r is the radius of the sensing sphere of the sensors.

10.4.2.2

∎

( ( h Sensor Placement on Tangential Hexagon TH C, a gr

Let us consider the configuration shown on Fig. 10.16, where sensors s1 and s2 are the farthest ones from each other, and located at points p1 and p2 , respectively. These ( twoh )points coincide ( )with the opposite corners of two tangential hexagons h TH C1 , agr and TH C2 , agr centered at C1 and C2 , respectively, inside two adjacent ) ) ( ( h h and G R C2 , agr , respectively. For this type great rhombicuboctahedra G R C1 , agr

( ) h Fig. 10.16 Network connectivity based on TH C, agr

10.4 Connected k-Coverage Problem

345

of sensor placement strategy, network connectivity is guaranteed when sensors s1 and s2 can communicate with each other. Lemma 10.12 states this result. Lemma 10.12 (Relationship Between R and r for Network Connectivity) A spatial k-covered wireless sensor network, which is deployed using a tangential hexagonbased sensor placement strategy, is connected if the ratio of the radius of the communication range of the sensors to the radius of their sensing range satisfy the following inequality: R ≥ 1.60204068888 ≈ 1.602 r Proof As it can be seen from Fig. 10.16, sensors s1 and s2 can communicate with each other only if the radius of the sensors’ communication range satisfies the following inequality: R ≥ δ( p1 , p2 ) where, δ( p1 , p2 ) = δ( p1 , Oh )+δ(Oh , p2 ) = 2δ( p1 , Oh ). However, δ( p1 , Oh ) = rh . Therefore, we have: R ≥ 2rh = 2 × 0.80102034444r = 1.60204068888r where r is the radius of the sensing sphere of the sensors.

10.4.2.3

∎

( ( o Sensor Placement on Tangential Octagon To C, a gr

Now, we consider the shown on Fig. 10.17 showing two tangential ( ) configuration ( ) o o and TO C2 , agr centered at C1 and C2 , respectively, inside octagons TO C1 , agr ) ) ( ( o o and G R C2 , agr , respectively. two adjacent great rhombicuboctahedra G R C1 , agr As it can be seen, sensors s1 and s2 , which are located at points p1 and p2 , respectively, are the farthest ones from each other. Network connectivity for this type of

( ) o Fig. 10.17 Network connectivity based on TO C, agr

346

10 A Polyhedral Space Filler Tessellation-Based Approach …

sensor placement strategy can be maintained provided that sensors s1 and s2 can communicate with each other. Lemma 10.13 states this result. Lemma 10.13 (Relationship Between R and r for Network Connectivity) A spatial k-covered wireless sensor network, which is deployed using a tangential octagonbased sensor placement strategy, is connected if the ratio of the radius of the communication range of the sensors to the radius of their sensing range satisfy the following inequality: R ≥ 1.43096207028 ≈ 1.431 r Proof As it can be seen from Fig. 10.17, sensors s1 and s2 can communicate with each other only if the radius of the sensors’ communication range satisfies the following inequality: R ≥ δ( p1 , p2 ) where, δ( p1 , p2 ) = δ( p1 , Oo )+δ(Oo , p2 ) = 2δ( p1 , Oo ). However, δ( p1 , Oo ) = r0 . Therefore, we have: R ≥ 2ro = 2 × 0.71548103514r = 1.43096207028r ∎

where r is the radius of the sensing sphere of the sensors.

10.4.3 Discussion Table 10.1 summarizes the results found in Sects. 10.4.1 and 10.4.2 with respect to the spatial sensor density, size of deployment area, and ratio of the radius of the communication range to the sensing range of the sensors. As it can be observed from Table 10.1, while the tangential square-based sensor placement has the minimum spatial sensor density to k-cover a spatial field, it has the minimum size of the deployment Table 10.1 Sensor placement strategy comparison

( ) s TS C, agr ( ) h TH C, agr ( ) o TO C, agr

ρ(k, r ) ×

r3 k

( ) s A agr r2

R r

0.43091787285

0.14553243772

1.768

0.57946225925

0.31035441301

1.602

0.81313267830

0.46017273141

1.431

10.5 Performance Evaluation

347

area inside a great rhombicuboctahedron and the highest ratio of the radius of the sensors’ communication range to the radius of their sensing range. In other words, on the one hand, this sensor placement strategy is the most energy-efficient one as it requires lesser number of sensors to achieve k-coverage of a spatial field. On the other hand, it requires the deployment of k more powerful sensors in a smaller area, which may yield some interference problem of higher magnitude compared to the other two sensor placement strategies. Therefore, there must be a trade-off among energy-efficiency, deployment area size, and sensor strength. It is essential to investigate this problem and find the best trade-off among these three metrics, which we leave as future work.

10.4.4 Sensor Selection Protocol In every round, the sink divides a spatial field randomly to form a grid of nonoverlapping, congruent great rhombicuboctahedra. To this end, the first great rhombicuboctahedron is generated randomly. Then, at the beginning of each round, the sink selects the sensors to cover a spatial field based on the above-mentioned sensor placement strategy being used. Also, we suppose that all the sensors are densely deployed on vertical planes that are perpendicular to the horizontal plane of a spatial field. Moreover, the distance between any pair of consecutive vertical planes is equal to the distance between the centroid of a great rhombicuboctahedron and one of its square, hexagonal, or octagonal faces, depending on the sensor placement strategy. Also, the first randomly generated great rhombicuboctahedron should have one of its faces on one of those vertical planes. Once the required sensors are selected, the sink broadcasts its schedule, which has the id’s of all selected sensors. When a sensor receives a schedule, it checks whether it has been selected to stay active. Otherwise, it turns itself off. In any case, it forwards the schedule to its one-hop neighbors. We suppose that all the sensors are awake (listening mode) at the beginning of each round.

10.5 Performance Evaluation In this section, we present our simulation setup. Then, we show the simulation results of our connected k-coverage approaches of a spatial field using a high-level simulator written in C.

10.5.1 Simulation Setup Our spatial field is a cube of side length 300 m, where 5000 sensors are randomly deployed as stated earlier. We assume that the radii of the sensors’ sensing and

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10 A Polyhedral Space Filler Tessellation-Based Approach …

communication ranges are 25 m and 50 m, respectively, unless stated otherwise. Moreover, all simulations are repeated 100 times and the results are averaged.

10.5.2 Simulation Results Figure 10.18 plots the spatial sensor density as a function of the radius of the sensors’ sensing range using the sensor placement strategy based on the tangential square, where k = 3. It shows a close match between the theoretical and simulation results. As it can be seen, the spatial sensor density ρ(k, r ) decreases proportionally to the radius r of the sensing sphere of the sensors. Figure 10.19 shows that the spatial sensor density increases linearly with the degree of coverage k, where r = 25 m. Also, it shows that the simulation results match the theoretical results. Figure 10.20 Fig. 10.18 ρ S (k, r ) versus r

Fig. 10.19 ρ S (k, r ) versus k

10.6 Related Work

349

Fig. 10.20 Theoretical results

Fig. 10.21 Simulation results

shows the theoretical spatial sensor density, while Fig. 10.21 plots the spatial sensor density from the simulation of each of those three sensor placement strategies, where r = 25 m. It is clear that the results obtained for the tangential square-based sensor placement method outperform the ones by the other two tangential hexagon and octagon-based strategies.

10.6 Related Work Alam and Haas [12] investigated the sensor placement problem in spatial wireless sensor networks to cover a spatial field based on the volumetric quotient of a few polyhedra. They found the truncated octahedron as the best space-filler. Bai et al.

350

10 A Polyhedral Space Filler Tessellation-Based Approach …

[71] suggested a set of patterns to cover a spatial field with a k-connected wireless sensor network, where k ≤ 4. Bail et al. [72] discussed a few optimal lattice patterns to cover a spatial field, while ensuring k-connectivity, where k = 6, 14. Andersen and Tirthapura [62] proposed a discretization approach for sensor deployment to cover a spatial field. Their approach transforms the coverage problem to a discrete optimization problem. But, it yields a coverage hole problem. Feng et al. [158] presented a coverage strategy by transforming points from spatial field to planar field using the manifold learning algorithm for the nonlinear dimensionality reduction, while maintaining the topology characteristic of the non-linear terrain. Huang et al. [208], [209] proposed a polynomial-time algorithm to solve the α-Ball-Coverage (α-BC) problem with a goal is to check α-coverage of a spatial field. Johnson and Qi [219] proposed an algorithm to optimize spatial sensor coverage. They used a photon mapping algorithm, which generates a photon map. Their approach performs virtual sensor placement optimization through queries to this generated photon map. Rao et al. [335] focused on computing the sensor density to cover a spatial field. Also, they proposed a method to discover redundant nodes and turn them off to achieve full coverage. Watfa and Commuri [404] suggested a self-healing algorithm to cover a spatial field by activating some sleeping sensors when sensors in the sensor cover fail due to their low energy. Zhang et al. [443] presented a set of patterns to cover a spatial field while guaranteeing k-connectivity, where there are at least k disjoint paths between any pair of sensors in the network with k ≤ 4. Liu and Ma [274] computed the sensor density to cover a mountainous region using stochastic sensor deployment, where sensors are deployed on spatial rolling surfaces. Pompili et al. [326] discussed a spatial communication architecture in underwater acoustic sensor networks, where the sensors float at different depths of the ocean to cover a spatial region. Ravelomanana [336] presented fundamental properties for connectivity and coverage, such as the sensing range that is required to guarantee certain degree of coverage of a region, the minimum and maximum network degrees for a given communication range, and the network hopdiameter. Aslam and Robertson [66] suggested a spatial distributed coverage algorithm, while maintaining network connectivity. Each sensor decides to become active based on “active probability”, which is computed randomly, and the announcement messages received from its neighbors. Cayirci et al. [105] provided a self-deployment scheme for spatial underwater wireless sensor networks, where the sensors adjust their depth to reduce the intersection with their neighboring nodes, thus, improving their coverage. Kumar et al. [241] proposed a method to select a subset of sensors to generate a 1-covered connected topology via the exchange of messages based on local information. Khalid and Durrani [229] discussed some connectivity properties, such as mean node degree and node isolation probability. They provided exact expressions for these two measures, where the sensors are distributed on a sphere or inside the volume of a ball. Watfa and Commuri [405] proposed a distributed algorithm for the selection of subset of sensors to cover a spatial field, where every node decides locally whether its sensing range is covered. Ammari [23] proposed an approach to k-cover a spatial field using Helly’s Theorem [85] and the Reuleaux tetrahedron [480]. He computed the corresponding sensor density, denoted by λ(k, r), and found that it is equal to λ(k, r ) = k/0.422r 3 ,

10.7 Conclusion

351

where r is the radius of the sensors’ sensing range and k ≥ 4. Also, Ammari and Das [44] studied the problem of 1-coverage (i.e., k = 1) in spatial wireless sensor networks using a probabilistic approach, and computed the critical sensor density above which a spatial field is almost surely covered. We should mention that this critical sensor density does not guarantee coverage of a spatial field as more sensors would be needed to fully cover it. Specifically, up to date, there is no exact solution to the problem of coverage of a spatial field using spheres. There are only a few interesting studies in the literature that attempted to solve the problem of coverage in spatial wireless sensor networks. To the best of our knowledge, Ammari’s work [23] is the only one in the existing literature that investigated the problem of determining the sensor density that is required to k-cover a spatial field based on the concept of Reuleaux tetrahedron. However, the proposed study considered k ≥ 4 and did not make use of the optimality ensured by space-fillers of a spatial field. In fact, the regular tetrahedron and Reuleaux tetrahedron are not space fillers.

10.7 Conclusion The problem of coverage of the space is an interesting one, and has similarity with the tiling problem in the same space. The latter is a fundamental yet challenging problem in a more concrete branch of mathematics, called elementary geometry, in the Euclidean space. It is well known that coverage of the Euclidean space of three dimensions is still an open and difficult problem in wireless sensor networks. In this chapter, we addressed the connected k-coverage problem in spatial wireless sensor networks using convex polyhedral space-fillers. We found that the great rhombicuboctahedra is the optimal polyhedron that maximize the coverage quality of the sensors. Precisely, our study shows that the great rhombicuboctahedron is the best convex polyhedral space-filler with respect to the coverage quality metric as it maximizes the sensors’ sensing range being utilized to cover a spatial field. In fact, it has higher coverage quality than the truncated octahedron proposed by Alam and Haas [12]. Based on our space-filler, we proposed several sensor placement strategies to ensure k-coverage of a spatial field based on the degree of coverage k, three of which are more practical and realistic. For each of these sensor placement methods, we computed the spatial sensor density for spatial k-coverage. Also, we computed the ratio of the sensors’ communication range to their sensing range to guarantee connected k-coverage of a spatial field. Moreover, we presented several simulations results of the above sensor placement strategies. While the polyhedral space-fillers studied in Sects. 10.3 and 10.4 involve only one type of polyhedron, there are other ones that are built using various types of polyhedra. Precisely, those polyhedral space-fillers could be obtained using more than one type of Archimedean polyhedra, or by combining Platonic and Archimedean polyhedra. We believe that this type of space covering perfectly models heterogeneous wireless sensor networks, where the sensors may not necessarily have the same communication and sensing capabilities. That is, their sensing ranges are spheres with different

352

10 A Polyhedral Space Filler Tessellation-Based Approach …

radii and their communication ranges are spheres with different radii. Therefore, these sensing spheres and communication spheres can be restricted to incongruent polyhedra. We can build polyhedral space-fillers using a mixture of Platonic and Archimedean polyhedra, or a combination of Archimedean polyhedra. • • • • • • • •

Regular Octahedra and Truncated Cubes Regular Octahedra and Tetrahedra with Ratio 1:2 Cubes and Truncated Rhombic Dodecahedra with Ratio 1:1 Truncated Octahedra, Cuboctahedra, and Truncated Tetrahedra Cubes and Rhombicuboctahedra Truncated Cubes and Octagonal Prisms Truncated Tetrahedra and Octahedra Truncated Tetrahedra and Hexagonal Prisms.

In reality, the sensors may not necessarily have the same characteristics in terms of their communication range and sensing range. Therefore, it is essential to extend our proposed convex polyhedral space-fillers based approaches, which are discussed in Sects. 10.3 and 10.4, to solve the problem of connected k-coverage in spatial heterogeneous wireless sensor networks [14]. The goal is to account for the more general problem of connected k-coverage, where k ≥ 1, where the sensors are not similar and may not have the same capabilities. For instance, it was shown that cubes can be combined with truncated rhombic dodecahedra in the ratio 1:1 to build a polyhedral space-filler [315]. The resulting network configuration is called octadecasil [315]. Given the above ratio, this type of space filling requires exactly two types of sensors, say types I and II. While type I sensors have their sensing range modeled by congruent cubes, type II sensors’ sensing range is represented by congruent truncated rhombic dodecahedra. Moreover, the number of type I sensors (whose sensing range is restricted to a cube) is equal to the number of type II sensors (whose sensing range is restricted to a truncated rhombic dodecahedron). Also, it was proved that regular octahedra and tetrahedra whose edges are equal can be packed together to fill a space provided they are combined in the ratio 1:2 [315].

Part V

Connectivity and Fault-Tolerance Measures of k-Covered Wireless Sensor Networks

Chapter 11

Planar Unconditional and Conditional Network Connectivity and Fault-Tolerance Measures for k-Covered Wireless Sensor Networks When wireless is perfectly applied, the whole earth will be converted into a huge brain, which in fact it is, all things being particles of a real and rhythmic whole. We shall be able to communicate with one another instantly, irrespective of distance. Not only this, but through television and telephony we shall see and hear one another as perfectly as though we were face to face, despite intervening distances of thousands of miles; and the instruments through which we shall be able to do his will be amazingly simple compared with our present telephone. A man will be able to carry one in his vest pocket. Nikola Tesla (1856–1943)

Overview This chapter gives our measures of unconditional (or traditional) and conditional connectivity and fault-tolerance of planar k-covered wireless sensor networks. The latter measures are more realistic than the former ones as they impose a restriction on the subsets of sensors that can fail at the same time. Precisely, conditional measures take into consideration the inherent properties of k-covered wireless sensor networks, such as high planar sensor density and sensor heterogeneity. In particular, the neighbor set of a sensor is defined as a forbidden faulty set, and hence cannot fail at the same time. This concept defines the new measure of connectivity, called conditional connectivity, which seems to be more realistic.

11.1 Introduction Connectivity, primarily a graph-theoretic concept, helps define the fault tolerance of wireless sensor networks in the sense that it enables the sensors to communicate with each other so their sensed data can reach the sink. On the other hand, sensing coverage, an intrinsic architectural feature of wireless sensor networks plays an important role in meeting application-specific requirements, for example, to reliably extract relevant data about a sensed field. A fundamental aspect in the design of wireless sensor networks is to keep them functional as long as possible. Because of scarce battery power (or energy), sensors may entirely deplete energy or have remaining energy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_11

355

356

11 Planar Unconditional and Conditional Network Connectivity …

below some threshold that is required for the sensors to function properly. Those sensors are called faulty as they cannot perform any monitoring task properly. A wireless sensor network is said to be functional if at any time there is at least one communication path between every pair of non-faulty sensors in the network. The existence of communication paths between pairs of sensors, however, is related to another fundamental property of wireless sensor networks, called vertex-connectivity (or simply connectivity). In general, sensing applications are required to be fault tolerant, where any pair of sensors is usually connected by multiple communication paths. Therefore, network functionality and hence network fault-tolerance strongly depends on connectivity. Another important issue in the design of wireless sensor networks is what is called sensing coverage, which is a good indicator of the quality of surveillance of a field of interest. Some sensing applications demand full coverage where every location in the field is covered by at least one sensor. Moreover, to cope with the problem of faulty sensors, duplicate coverage of the same region is desirable. Notice that sensing coverage and network connectivity are not totally orthogonal concepts. While sensing coverage depends on the sensing range, connectivity relates to the communication range of the sensors. Intuitively, sensing coverage becomes meaningless if the sensed data cannot be exchanged by the sensors so they reach a central gathering point, i.e., sink, for further analysis. Thus, for a network to function properly, sensing coverage and network connectivity should be maintained. In fact, it has been proven that connectivity strongly depends on coverage and hence considerable attention has been paid to establish tighter connection between them although only loose lower bound on connectivity of wireless sensor networks is known. It was proved that the connectivity of a homogeneous k-covered wireless sensor network is equal to k provided that the radius of the communication range of the sensors is at least double the radius of their sensing range [425]. In this chapter, we investigate connectivity based on the degree of sensing coverage by studying k-covered wireless sensor networks [45]. Although network connectivity can be used to measure the fault tolerance of smallscale networks, it is not appropriate for large-scale dense networks, such as k-covered wireless sensor networks. Traditional (or unconditional) connectivity has no restriction on the faulty sensor set and assumes that any subset of sensors can potentially fail at the same time, including all the neighbors of a given sensor. However, highly dense sensor networks, such as k-covered wireless sensor networks, can consist of thousands of sensors for which it is highly unlikely in this type of network that all the neighbors of a given sensor fail simultaneously. This is due to the following two reasons: • Assuming a uniform sensor distribution, the ratio of the size of the neighbor set of a given sensor to the total number √ of sensors in a planar field of area size A is given by π R 2 /A, where R (R ⋘ A) is the radius of the communication ranges of the sensors forming a homogeneous wireless sensor network. The probability of the failure of the entire neighbor set of a given sensor can be identified with this ratio and hence is very low. It is worth mentioning that there are two types of faults: random faults and arbitrary faults. Definitely, random fault can be dealt

11.1 Introduction

357

with much easier than arbitrary faults. In practice, the failure of sensors would come from the (uncontrollable) environment, and if a sensor is damaged due to a wild animal, it is very likely that the same animal will damage some neighboring sensors as well, but it is highly unlikely that all the neighboring sensors would be damaged by the same animal in highly densely deployed networks, such as k-covered wireless sensor networks. • In real-world scenarios, wireless sensor networks can be heterogeneous, where sensors have different sensing, processing, and communication capabilities, thus increasing the network reliability and lifetime [433]. Hence, the probability that an entire neighbor set of a sensor fail simultaneously in this type of heterogeneous network is very low. Next, we briefly describe the major tasks we want to accomplish in this chapter. Moreover, we briefly state how to achieve each one of them.

11.1.1 Major Tasks Clearly, the classical (or unconditional) connectivity may not reflect the actual fault tolerance of large-scale dense networks, such as k-covered wireless sensor networks, due to the above shortcomings. In this chapter, we focus on more realistic measures of network connectivity and fault tolerance of connected k-covered configurations in planar wireless sensor networks [45] through the accomplishment of some key tasks. Next, we specify these tasks and give our corresponding plan of actions. First, we want to compute traditional (or unconditional) measures of network connectivity and fault tolerance of planar connected k-covered wireless sensor networks. For this, we use the graph-theoretic concept of connectivity to compute the network connectivity of these networks. Then, we derive the corresponding network fault-tolerance. Second, we propose a new measure of fault tolerance for k-covered wireless sensor networks, called conditional fault-tolerance, to alleviate the problems, which are caused by the above shortcomings. To this end, we use a more general concept of connectivity, called conditional connectivity (or P-connectivity), which was originally introduced in [192] with respect to some property that is based on the concept of forbidden faulty sensor set that includes all the neighbors of a given sensor. Precisely, we exploit an instance of conditional connectivity, called restricted connectivity, which was proposed in [151]. Specifically, the restriction is on the faulty set [150, 418] (i.e., set of nodes that can fail). Restricted connectivity uses the concept of forbidden faulty set in which the entire neighbor set of a given node cannot be faulty at the same time.

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11 Planar Unconditional and Conditional Network Connectivity …

11.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 11.2 computes unconditional connectivity and fault-tolerance measures for planar k-covered wireless sensor networks while Sect. 11.3 computes their conditional connectivity and fault-tolerance measures. Section 11.4 reviews related on conditional connectivity and its variations. Section 11.5 concludes the chapter.

11.2 Unconditional Fault-Tolerance Measures In this chapter, we compute connectivity and fault-tolerance measures of homogeneous and heterogeneous planar k-covered wireless sensor networks. We consider a planar square field of area size A as shown in Fig. 11.1. We also assume that the size of the sensing and communication ranges of the sensors are much less than A.

11.2.1 Homogeneous Sensors We assume that the sink has the same communication range as the other sensors. In Chap. 5, we compute the planar sensor density required to achieve k-coverage of a planar field based on the Reuleaux triangle [481] model. Also, we prove that when a planar wireless sensor network is guaranteed to be connected when it is configured to provide k-coverage of a planar deployment field with k ≥ 3 and the radius of the Fig. 11.1 Sensor distribution in a planar square field of area A. The radius of the sensing ranges of the sensors is r while the radius of their communication ranges is R

11.2 Unconditional Fault-Tolerance Measures

359

communication ranges of the sensors is at least equal to the radius of their sensing ranges, i.e., R = r. To make this chapter self-contained, we recall Lemma 5.2, which is given earlier in Chap. 5. Lemma 5.2 (Guaranteed Planar k-Coveragee—Chap. 5) Let r be the radius of the sensing disks of the sensors and k ≥ 3. A planar field is k-covered if any Reuleaux triangle region of width r in the field contains at least k active sensors. ∎ From Lemma 5.2, we deduce that the planar sensor density required to guarantee k-coverage of a planar field is given by λ(r, k) =

2k √ (π − 3) r 2

(11.1)

Indeed, since we are interested in the number of neighbors of a sensor, we need to consider this lemma instead of Theorem 5.2, which characterizes minimum kcoverage, and Theorem 5.3, which gives a tight bound on the planar sensor density required for k-coverage, both of which are stated earlier in Chap. 5. Ammari and Das [51] proposed a sleep-wakeup scheduling protocol for full kcoverage of a planar field while using a minimum number of active sensors based on the above characterization. Theorem 11.1 below computes the connectivity of homogeneous k-covered wireless sensor networks and derives their fault tolerance. While the works in [425] compute connectivity based on boundary and interior sensors, our study considers the sink given its critical role in data collection and analysis. Precisely, we focus on the size of the connected component containing the sink. Indeed, it is more realistic to relate network connectivity measure to the sink. For instance, we may have a giant connected component of sensors that does not include the sink, and hence their data cannot reach the sink. Theorem 11.1 (Network Connectivity Measure) Let G be a communication graph of a homogeneous k-covered wireless sensor network deployed in a planar square field of area A. The connectivity of G is given by √ 2 π α2 k 2 R Ak √ √ ≤ κ(G) ≤ (π − 3) r 2 (π − 3)

(11.2)

where α = R/r and k ≥ 3. Its fault tolerance, η(G), is given by √ 2 π α2 k 2 R Ak −1 √ − 1 ≤ η(G) ≤ √ (π − 3) (π − 3) r 2

(11.3)

Proof To compute the network connectivity of homogeneous k-covered wireless sensor networks, we need to consider the following three cases, which depend on the sink, denoted by s0 and located at ξ0 . Since we assume that sensor failure is due to low battery power, we can use a powerful sink with infinite energy, thus eliminating the possibility of sink failure.

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11 Planar Unconditional and Conditional Network Connectivity …

Case 1—Isolated sink: This situation occurs when the disconnected network has at least two connected components, one of them being the trivial component containing the sink. Given the definition of network connectivity, the number of disconnected components should be equal to two. Notice that the optimum location of the sink in terms of energy-efficient data gathering from the available sensors is the center of the planar square field [281]. The sink can be isolated only when all its neighbors fail. Therefore, we compute the number of neighbors of the sink whose failure would disconnect the sink. Let n0 be the minimum number of sensor failures to isolate the sink s0 . Assuming that the sensors are deterministically deployed to achieve k-coverage of a planar field, the value of n0 is given by n 0 = λ(r, k) |D(ξ0 , R)|

(11.4)

where |D(ξ0 , R)| = π R 2 is the measure of the area of the communication disk D(ξ0 , R) of the sink s0 located at ξ0 . Hence, the network connectivity in this case is given by κ1 (G) = n 0 = λ(r, k) π R 2

(11.5)

Substituting Eq. (11.1) in Eq. (11.5) yields κ1 (G) =

2 π α2 k √ (π − 3)

(11.6)

where α = R/r and k ≥ 3. Notice that connectivity increase with the ratio α and the sensing coverage k. We observe that the network connectivity κ1 (G) is also higher than the sensing coverage k. Case 2—Non-trivial connected components: Similarly, the disconnected network has two connected component. We distinguish two particular network configurations that are worth of study. In the first one (Fig. 11.2), the connected component including the sink corresponds to its communication disk (whose area is π R 2 ) and is surrounded by a circular band that contains no sensors. Furthermore, the distance between any pair of sensors from these two components is at least R in order to disable any communication between the two connected components. Thus, the width of this empty band should be at least R. Hence, the area of the smallest empty circular band B(ξ0 , R) should be equal to |B(ξ0 , R)| = π (2R)2 − π R 2 = 3 π R 2

(11.7)

Thus, the minimum number n0 of sensor failures to isolate the connected component of the sink is given by

11.2 Unconditional Fault-Tolerance Measures

361

Fig. 11.2 First non-trivial connected component of the disconnected network

n 0 = λ(r, k) |B(ξ0 , R)|

(11.8)

Hence the network connectivity is equal to κ2 (G) = n 0 = 3 λ(r, k) π R 2

(11.9)

If we set α = R/r and substitute Eq. (11.1) in Eq. (11.9), we obtain κ2 (G) =

6 π α2 k √ (π − 3)

(11.10)

The second configuration of the disconnected network (Fig. 11.3) corresponds to the smallest connected component containing the sink if the planar field has to be divided into two regions such that none of them surrounds the other. Hence, the √ width of the empty rectangular band, denoted by B(R, A), which splits the planar field vertically should be equal to R. The minimum number n0 of sensor failures to isolate the sink is given by n 0 = λ(r, k) R

√

A

(11.11)

We find that the network connectivity is equal to κ3 (G) = n 0 = λ(r,k) R

√

A

Setting α = R/r and substituting Eq. (11.1) in Eq. (11.12) yields

(11.12)

362

11 Planar Unconditional and Conditional Network Connectivity …

Fig. 11.3 Second non-trivial connected component of the disconnected network

κ3 (G) =

√

Ak √ (π − 3) r 2 2R

Notice that κ3 (G) > κ2 (G) given the hypothesis R ⋘

(11.13) √

A.

Case 3—Largest connected component: One of the components of the disconnected network has only one sensor that is not the sink. This case is similar to the first case in that a single sensor becomes isolated when all of its neighbors fail. Applying the same reasoning leads to the same result found in Case 1. Thus, we have κ1 (G) ≤ κ(G) ≤ κ3 (G) and hence the network fault tolerance, η(G), is given by κ1 (G) − 1 ≤ η(G) ≤ κ3 (G) − 1. It is easy to check that κ(G) > k. However, it was proved in [425] that the connectivity of k-covered wireless sensor networks is equal to k provided that R ≥ 2r .

11.2.2 Heterogeneous Sensors In this section, we consider k-covered wireless sensor networks with heterogeneous sensors whose sensing and communications are not the same, i.e., both parameters are varying. In this case, it is easy to prove that the relationship R ≥ 2r cannot guarantee network connectivity even when the network is configured to provide sensing coverage. Figure 11.4 shows that the sensor s j can connect to the sensor si , but si cannot connect to s j . Lemma 11.1 establishes a necessary and sufficient condition for connectivity of heterogeneous k-covered wireless sensor networks.

11.2 Unconditional Fault-Tolerance Measures

363

Fig. 11.4 1-Coverage and Ri ≥ 2ri do not imply connectivity

Lemma 11.1 (Network Connectivity Condition for Heterogeneous Sensors) A heterogeneous k-covered wireless sensor networks is connected if for any sensor, the radius of its communication disk is at least equal to the sum of the radii of its own sensing disk and that of the most powerful sensor in terms of sensing capability, i.e., for all si ∈ S, Ri ≥ ri + rmax , where rmax = max{r j : s j ∈ S}. Proof Consider two sensors si and s j whose sensing disks are tangential to each other at point p (Fig. 11.4). Let Ri = ri + rmax and R j = r j + rmax . Thus, Ri ≥ ri + r j and R j ≥ ri + r j , thus implying |ξi − ξ j | ≤ min{Ri , R j }. Hence, si and s j are mutually connected. Thus, the underlying heterogeneous wireless sensor network is connected because it is k-covered. ∎ It is worth noting that achieving k-coverage depends on the least powerful sensors in terms of their sensing capability. Corollaries 11.1 and 11.2 for heterogeneous kcovered wireless sensor networks correspond to Lemmas 5.1 and 5.2, respectively, which are given earlier in Chap. 5. Corollary 11.1 (k-Covered Convex Area Width for Heterogeneous Sensors) If the width of a closed convex area A, ω(A), satisfies ω(A) ≤ rmin , where rmin = min{r j : s j ∈ S}, then A is guaranteed to be k-covered when k heterogeneous sensors are deployed in it, where k ≥ 3. Proof Our reasoning should be based on the least powerful sensors in terms of their sensing capability. In the worst case, when k least powerful sensors (all of them have

364

11 Planar Unconditional and Conditional Network Connectivity …

the smallest radius of their sensing disks) are deployed in the Reuleaux triangle of constant (maximum) width equal to rmin , denoted by RT (rmin ), then RT (rmin ) is ∎ guaranteed to be k-covered, where rmin = min{r j : s j ∈ S}. Corollary 11.2 (Guaranteed Planar k-Coverage Using Heterogeneous Sensors) The planar sensor density necessary to guarantee k-coverage of a planar field sensed by spatially distributed heterogeneous sensors is given by λ(rmin , k) =

2k √ 2 (π − 3) rmin

(11.14)

where rmin = min{r j : s j ∈ S} and k ≥ 3. Proof The proof is verbatim and stems from the fact that if k least powerful sensors (in terms of their sensing ranges) are able to k-cover a region A, then any subset of k sensors deployed in A will be able to do so. Using the AET model, we can √ r2 easily prove that the maximum size of A is Amax (rmin ) = (π − 3) min . Thus, 2 the required planar sensor density is λ(rmin , k) = Amaxk(rmin ) = (π−√2 k3)r 2 , where min rmin = min{r j : s j ∈ S}. ∎ While for homogeneous k-covered wireless sensor networks, “k-connected” is the proper parameter, we believe that “k-node disjoint paths to the sink” is the right argument for heterogeneous k-covered wireless sensor networks since differences in communication ranges induce unidirectional networks. Indeed, “k-strong connectivity” is an over-kill in the case of heterogeneous k-covered wireless sensor networks. From now on, connectivity of heterogeneous k-covered wireless sensor networks should be understood in the sense of “k-node disjoint paths to the sink”. Corollary 11.3 below computes connectivity of heterogeneous k-covered wireless sensor networks and their fault tolerance. Corollary 11.3 (Network Connectivity Measure Using Heterogeneous Sensors) Let G be a communication graph of a heterogeneous k-covered wireless sensor networks with k ≥ 3. The connectivity of heterogeneous k-covered wireless sensor networks (in the sense of k-node disjoint paths to the sink) is given by K 1 ≤ κ(G) ≤ K 2 where 2 k 2 π Rmin √ 2 (π − 3) rmin √ 2 Rmax A k K2 = √ 2 (π − 3) rmin

K1 =

(11.15)

11.2 Unconditional Fault-Tolerance Measures

365

rmin = min{r j : s j ∈ S} and Rmax = max{R j : s j ∈ S}. The fault tolerance, η(G), is given by K 1 − 1 ≤ η(G) ≤ K 2 − 1

(11.16)

Proof Similarly, we consider three cases depending on the types of components of the disconnected network that contain the sink. Case 1—Isolated sink: Although there are several cases for the sink to consider in terms of its communication capability, we limit our study to the following extreme two cases. Case 1.1: In this case, the sink is supposed to be the most powerful node in the network in terms of sensing, communication, computation, and storage capabilities. Hence, the radius of its communication disk is equal to Rmax . Indeed, in some sensing applications, the sink would receive all sensed data from the sensors in the network which has to be analyzed and processed for a better and more accurate decision making process. Thus, the minimum number n0 of sensor failures to isolate the sink is given by n 0 = λ(rmin , k) |D(ξ0 , Rmax )|

(11.17)

Substituting Eq. (11.14) in Eq. (11.17), we find that the network connectivity of heterogeneous k-covered wireless sensor networks is computed as κ1 (G) = n 0 =

2 k 2 π Rmax √ 2 (π − 3) rmin

(11.18)

where rmin = min{r j : s j ∈ S}, Rmax = max{R j : s j ∈ S}, and k ≥ 3. Case 1.2: In this case, the sink is supposed to be the least powerful node in the network and hence the radius of its communication disk is equal to Rmin . Indeed, in other sensing applications, in addition to traffic relaying, the sensors are also responsible for in-network data processing. In this case, the sink would receive only a small amount of data and hence does not have to possess high capabilities. Thus, the minimum number n0 ’ of sensor failures to isolate the sink is given by n '0 = λ(rmin , k) |D(ξ0 , Rmin )|

(11.17a)

Substituting Eq. (11.14) in Eq. (11.17a), we find that the network connectivity of heterogeneous k-covered wireless sensor networks is computed as κ1' (G) = n '0 =

2 k 2π Rmin √ 2 (π − 3)rmin

(11.18a)

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11 Planar Unconditional and Conditional Network Connectivity …

where rmin = min{r j : s j ∈ S}, Rmin = min{R j : s j ∈ S}, and k ≥ 3. Case 2—Non-trivial connected components: We consider the same two network configurations, which were studied in the case of homogeneous k-covered wireless sensor networks. Notice that the sensors located around the sink are heavily used in data forwarding and hence should be the most powerful ones in terms of sensing and communication capabilities. Otherwise, they will suffer severe energy depletion and die very quickly. Thus, the communication disk of the sink contains only powerful sensors. Hence, the width of the circular empty band surrounding the connected component containing the sink should be equal to Rmax . Thus, the area of this empty circular band B(ξ0 , Rmax ) should be equal to 2 2 |B(ξ0 , Rmax )| = π (2Rmax )2 − π Rmax = 3 π Rmax

(11.19)

The minimum number n1 of sensor failures to isolate the sink is given by n 1 = λ(rmin , k) |B(ξ0 , Rmax )|

(11.20)

If we substitute Eq. (11.14) in Eq. (11.20), we find that connectivity is equal to κ2 (G) = n 1 =

2 k 6 π Rmax √ 2 (π − 3) rmin

(11.21)

Likewise, for the second configuration of the disconnected network, the width of √ A) should be equal to Rmax and hence its area B(R, the empty rectangular band √ √ is equal to |B(R, A)| = Rmax A. Therefore, the minimum number n2 of sensor failures to isolate the sink is given by n 2 = λ(rmin , k)|B(R,

√

A)|

(11.22)

Thus, the network connectivity is given by √ 2 Rmax A k κ3 (G) = n 2 = √ 2 (π − 3) rmin

(11.23)

Case 3—Largest connected component: In this case, the single-node component may include the least powerful or the most power sensor in terms of its communication capability. Hence, the network connectivity will have lower and upper bounds depending on whether the isolated sensor is the least or most powerful sensor, respectively. The minimum number nlb of sensor failures to isolate a least powerful sensor is given by n lb = λ(rmin , k) |D(ξ0 , Rmin )|

(11.24)

11.3 Conditional Fault-Tolerance Measures

367

while the minimum number nub of sensor failures to isolate a most powerful sensor is given by n ub = λ(rmin , k) |D(ξ0 , Rmax )|

(11.25)

Let K lb and K ub be lower and upper bounds, respectively, on connectivity κ4 (G) of heterogeneous k-covered wireless sensor networks. In this case, it is easy to establish that κ4 (G) satisfies K lb (G) ≤ κ4 (G) ≤ K ub (G) where K lb (G) = n lb =

(11.26)

2 2 π Rmin k √ 2 (π − 3) rmin

K ub (G) = n ub =

2 k 2 π Rmax √ 2 (π − 3) rmin

Using the results of the above three cases, we find that connectivity of heterogeneous k-covered wireless sensor networks satisfies K 1 ≤ κ(G) ≤ K 2 and their fault 2 2 π Rmin k , tolerance is given by K 1 − 1 ≤ η(G) ≤ K 2 − 1, where K 1 = K lb (G) = (π−√3) r2 √

min

2 Rmax √ A2 k , r min = min{r j : s j ∈ S}, Rmin = min{R j : s j ∈ S}, and K 2 = κ3 (G) (π − 3) rmin Rmax = max{R j : s j ∈ S}. Similarly, it is easy to prove that κ(G) > k. ∎

11.3 Conditional Fault-Tolerance Measures Previous works on fault tolerance in wireless sensor networks assumed that any subset of sensors and, in particular, the neighbor set of any sensor, can potentially fail at the same time. As discussed earlier, this assumption is unrealistic for large-scale dense wireless sensor networks, such as k-covered wireless sensor networks, which are highly dense so they can achieve desirable levels of redundant coverage. Also, deploying heterogeneous sensors for real-world applications reduces the probability of simultaneous failure of the entire neighbor set of a sensor. In this chapter, we restrict a faulty sensor set to some subsets of sensors that do not include the forbidden faulty set. Next, we compute the conditional connectivity and conditional fault-tolerance of homogeneous and heterogeneous k-covered wireless sensor networks.

368

11 Planar Unconditional and Conditional Network Connectivity …

11.3.1 Homogeneous Sensors In this section, we consider homogeneous sensors that possess the same sensing range and the same communication range. Our results prove that k-covered wireless sensor networks can sustain a larger number of sensor failures under the restriction imposed on a faulty sensor set. Precisely, we show that the conditional connectivity of homogeneous k-covered wireless sensor networks is larger than k. Theorem 11.2 below computes the conditional connectivity and conditional fault-tolerance of homogeneous k-covered wireless sensor networks. Theorem 11.2 (Conditional Network Connectivity Measure) The conditional connectivity of homogeneous k-covered wireless sensor networks is given by κ(G : P) =

4 R (R + r ) k r2

(11.27)

where α = R/r and k ≥ 3. The conditional fault-tolerance of G, η(G : P), is given by η(G : P) = κ(G : P) − 1. Proof We consider two cases based on the type of component to which the sink belongs. Case 1—Smallest-size component including the sink: Under the assumption of forbidden faulty set, we assume the sink belongs to the smallest connected component that is disconnected from the rest of the network. Let ξ0 be the location of the sink s0 . By hypothesis, any location in the planar field is k-covered with k ≥ 3, and in particular the location ξ0 . Therefore, there must be a subset of sensors located at distance at most equal to r from ξ0 . Using the Reuleaux Triangle model, the Reuleaux triangle of width r + ε1 and centered at ξ0 , denoted by RT (ξ0 , r + ε1 ), where ε1 is an infinitesimal value, must be not empty; otherwise, the k-coverage property at ξ0 is not satisfied, and particularly the forbidden faulty set constraint is not met. Our goal is to compute the minimum number of sensors to fail in order to disconnect the sink under the forbidden set constraint. Notice that the smallest connected component including the sink requires ε1 = 0. The region RT (ξ0 , r ) is a guarantee that the sink will not be isolated by itself and hence the forbidden faulty sensor set constraint with respect to the sink is not violated. Indeed, only a subset of its neighbors fails and not all of them as in the case of classical connectivity (see Sect. 11.2.1, Case 1 of proof of Theorem 11.1). In this configuration, the majority of the sensors are not connected to the sink, and hence the network is dead. Now, to disconnect the sink together with its neighbors located in RT (ξ0 , r ), the annulus surrounding the region RT (ξ0 , r ) and centered at ξ0 should be empty and have a width equal to R + ε2 ; otherwise the network remains connected. Notice that the minimum number of sensor failure requires ε2 = 0. Thus, this empty annulus, denoted by A(ξ0 , R) (Fig. 11.5), will guarantee that the connected component in RT (ξ0 , r ) is disconnected. Notice that the width of the outmost Reuleaux triangle centered at ξ0 is equal to 2 R + r . Hence, the area of the annulus A(ξ0 , R) is given by

11.3 Conditional Fault-Tolerance Measures

369

Fig. 11.5 RT (ξ0 , r ) and A(ξ0 , R) regions

|A(ξ0 , R)| = |RT (ξ0 , 2 R + r )| − |RT (ξ0 , r )| = 2 (π −

√

3) R (R + r ) (11.28)

Thus, the conditional minimum number n(P) of sensor failures to disconnect the smallest component including the sink is computed as n(P) = λ(r, k) |A(ξ0 , R)|

(11.29)

Substituting Eqs. (11.1) and (11.28) in Eq. (11.29), we find that the conditional connectivity is computed as κ1 (G : P) = n(P) =

4 R (R + r ) k r2

(11.30)

where r and R are the radii of the sensing and communication disks of the sensors, respectively, and k is the degree of coverage of the planar field. It is easy to prove that the forbidden faulty sensor set constraint is satisfied for both the faulty and nonfaulty sensors. Any sensor inside the region RT (ξ0 , r ) still has non-faulty neighbors located in RT (ξ0 , r ). Also, any sensor outside the region RT (ξ0 , R + r ) has nonfaulty neighbors within RT (ξ0 , R+r ). Similarly, any faulty sensor within the annulus A(ξ0 , R) has non-faulty neighbors located in RT (ξ0 , r ) and outside RT (ξ0 , R + r ). Case 2—Largest connected component: We assume that the sensors located in the annulus A(ξi , R) as defined earlier fail. This case is similar to the previous one except that the sink belongs to the largest connected component of the disconnected network. Using the same reasoning as in Case 1, we obtain the same conditional network connectivity. Here again, to consider whether or not the resulting network is

370

11 Planar Unconditional and Conditional Network Connectivity …

connected or not depends on the type of coverage (full coverage or partial coverage) required by the sensing application. Thus, we have κ2 (G : P) = κ1 (G : P). From both cases 1 and 2, it follows that the conditional network connectivity is κ(G : P) = κ1 (G : P), and hence the conditional network fault-tolerance, η(G : P), is given by η(G : P) = κ(G : P) − 1. It is easy to check that κ(G : P) > κ(G) and hence κ(G : P) > k. By definition, the conditional network fault tolerance, denoted by η(G : P), as stated earlier in Chap. 2 (Sect. 2.2, Definition 2.58) is given ∎ by η(G : P) = κ(G : P) − 1.

11.3.2 Heterogeneous Sensors Computing the conditional connectivity of heterogeneous k-covered wireless sensor networks is not a straightforward generalization of the process used previously for homogeneous k-covered wireless sensor networks. We find that disconnecting the network while satisfying the forbidden faulty set constraint is a challenging problem. If, on the one hand, we choose the width of the annulus to be Rmax , then the sensors with communication range less than or equal to half of Rmax may be located in the annulus. Thus, the property P will be violated (Fig. 11.6) as the entire neighbor set of some sensors located within the annulus would fail at the same time. If, on the other hand, the width of the annulus is less than Rmax , then the non-faulty sensors of one connected component might be able to connect to the non-faulty sensors of the other connected component of the disconnected network. Hence, the obtained network is not disconnected (Fig. 11.7). As it can be seen, we cannot find an exact value of the conditional connectivity of heterogeneous k-covered wireless sensor Fig. 11.6 The forbidden fault set constraint is violated (neighbor set of si is within the circular band of width Rmax )

11.3 Conditional Fault-Tolerance Measures

371

Fig. 11.7 Connectivity is maintained (the radius of s j ’s communication disk is larger than Rmin )

networks in the absence of any deterministic sensor deployment strategy. Hence, we propose to compute lower and upper bounds on conditional connectivity based on particular sensor configurations in the annulus and around the annulus. While in the first scenario we assume that the annulus contains only least powerful sensors, the second scenario supposes that the annulus consists of most powerful sensors. Corollary 11.4 below computes the conditional connectivity and conditional faulttolerance of heterogeneous k-covered wireless sensor networks. Corollary 11.4 (Conditional Network Connectivity Measure Using Heterogeneous Sensors) The conditional connectivity of heterogeneous k-covered wireless sensor networks is given by κ1 (G : P) ≤ κ(G : P) ≤ κ2 (G : P) where κ1 (G : P) =

(11.31)

4 Rmin (Rmin +rmin ) k . 2 rmin

κ2 (G : P) =

4 Rmax (Rmax + rmax ) k 2 rmin

k ≥ 3 rmin = min{r j : s j ∈ S}, rmax = max{r j : s j ∈ S}, Rmin = min{R j : s j ∈ S}, and Rmax = max{R j : s j ∈ S}. The conditional fault tolerance of the network is given by κ1 (G : P) − 1 ≤ η(G : P) ≤ κ2 (G : P) − 1

(11.32)

372

11 Planar Unconditional and Conditional Network Connectivity …

Proof First assume that the annulus as well as the area surrounding it contains only least powerful sensors, and hence its width is equal to Rmin . Furthermore, in order to guarantee that the sink will not be isolated, which would violate the forbidden faulty sensor set constraint, the width of the Reuleaux triangle centered at the location ξ0 of the sink s0 should be equal to rmin . These two conditions yield a disconnected network that satisfies the forbidden faulty sensor set constraint. First assume that the annulus as well as the area surrounding it contains only least powerful sensors, and hence its width is equal to Rmin . The area of the annulus A(ξ0 , Rmin ) is given by |A(ξ0 , Rmin )| = |RT (ξ0 , 2 Rmin + rmin )| − |RT (ξ0 , rmin )| √ = 2(π − 3) Rmin (Rmin + rmin ) Hence, the conditional minimum number n(P) of sensor failures to disconnect the connected component including the sink from the rest of the network is computed as n 1 (P) = λ(rmin , k)|A(ξ0 , Rmin )| Thus, the lower bound on conditional connectivity is computed as κ1 (G : P) = n 1 (P) =

4 Rmin (Rmin + rmin ) k 2 rmin

(11.33)

where rmin = min{r j : s j ∈ S} and Rmin = min{R j : s j ∈ S}. To compute the upper bound on network connectivity, we assume that the sensors inside the annulus and around it are the most powerful ones. The analysis is similar to the previous one except that we just replace rmin by rmax and Rmin by Rmax in the denominator of the first part of Eq. (11.7) in order to disconnect the network while meeting the forbidden faulty set constraint. We find that the network connectivity is given by κ2 (G : P) =

4 Rmax (Rmax + rmax ) k 2 rmin

(11.34)

where rmax = max{r j : s j ∈ S}, Rmin = min{R j : s j ∈ S}, and Rmax = max{R j : s j ∈ S}. Thus, the conditional network connectivity of heterogeneous k-covered wireless sensor networks with respect to the forbidden faulty set constraint P satisfies κ1 (G : P) ≤ κ(G : P) ≤ κ2 (G : P) and their conditional network fault-tolerance is given by κ1 (G : P) − 1 ≤ η(G : P) ≤ κ2 (G : P) − 1

∎

11.5 Conclusion

373

Note that there is neither a polynomial-time algorithm for computing κ(G : P) for a general graph nor any tight upper bound for κ(G : P). However, our characterization of k-coverage based on the intersection of k sensing disks and the Reuleaux triangle make it possible to compute the corresponding minimum planar sensor density. This helps us derive conditional connectivity and conditional fault-tolerance of k-covered wireless sensor networks.

11.4 Related Work Existing works on coverage and connectivity in wireless sensor networks assumed the concept of traditional connectivity. Our proposed approach, however, considers both concepts of traditional and conditional connectivity. The latter is based on the concept of forbidden faulty sensor set that includes subsets of sensors that cannot be faulty at the same time. Furthermore, our measures of connectivity and fault tolerance for k-covered wireless sensor networks take into consideration the morphology of wireless sensor networks, where the sink is the most crucial node. In this section, we describe existing work that exploited the concept of conditional connectivity. The concept of conditional connectivity [192] has been investigated in several research works. Esfahanian [150] presented a new fault-tolerance analysis for the n-cube networks based on the concept of forbidden faulty set. Latifi et al. [250] introduced a new measure of conditional connectivity for the n-dimensional cube, where every node is required to have at least g good neighbors. Wu and Guo [418] computed the fault tolerance of the m-ary n-dimensional hypercubes using forbidden faulty sets. Also, Chen et al. [114] proposed a probabilistic approach for computing the fault tolerance of hypercube network using forbidden faulty sets. Malde and Oellermann [287] introduced the notion of F-connectivity as the smallest number of vertices of G whose removal produces a trivial graph or a disconnected graph with each component a subgraph of F, where F is an induced subgraph of G. Oellermann [313] proposed the P-connectivity of a graph G with respect to hereditary properties, where every induced subgraph F of a graph G having property P also has property P. Ammari and Das [54] proposed measures of conditional fault-tolerance of k-covered wireless sensor networks but considered connectivity from graph theory perspective, which is quite different from connectivity with the sink. Indeed, network connectivity is not necessarily a condition for the network to operate whereas connectivity to the sink is. Thus, more realistic measures of fault tolerance should be defined with respect to the sink.

11.5 Conclusion One important issue in the design of wireless sensor networks is fault tolerance. Precisely, the network should remain functionally connected in spite of some sensor

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11 Planar Unconditional and Conditional Network Connectivity …

failures. In the context of graph theory, connectivity is an appropriate measure of fault tolerance. In this chapter, we investigate coverage, connectivity, and fault tolerance in k-covered wireless sensor networks [45]. In order to compute their connectivity and fault tolerance, we find that it is necessary to compute the minimum planar sensor density to guarantee k-coverage of a convex planar field. For this purpose, we characterize k-coverage based on the intersection of sensing disks of k sensors and the Reuleaux Triangle model. Our measures take into account the size of the connected component that includes the sink. Indeed, the sink is a very critical node in data collection and decision making. Regardless of the degree of coverage k provided by the network, we prove that connectivity is always higher than k. We derive fault tolerance of k-covered wireless sensor networks based on connectivity. Traditional (or unconditional) connectivity assumes that any subset of sensors can be potentially faulty at the same time. For large-scale dense wireless sensor networks, such as kcovered wireless sensor networks, this assumption is not realistic. To alleviate this problem, we propose a new measure of fault-tolerance for k-covered wireless sensor networks based on the concepts of conditional connectivity and forbidden faulty sensor set, where the entire neighbor set of a given sensor cannot fail at the same time. We prove that k-covered wireless sensor networks can sustain a large number of sensor failures provided that the faulty sensor set does not include a forbidden faulty sensor set. We believe that our results can be used in the design of wireless sensor networks with prescribed degrees of coverage, connectivity, and fault tolerance. In the next chapter, i.e., Chap. 12, we provide network and fault-tolerance measures for spatial (or three-dimensional) homogeneous and heterogeneous k-covered wireless sensor networks using the same approach for their counterpart in planar deployment fields.

Chapter 12

Spatial Unconditional and Conditional Network Connectivity and Fault-Tolerance Measures for k-Covered Wireless Sensor Networks When the wireless transmission of power is made commercial, transport and transmission will be revolutionized. Already motion pictures have been transmitted by wireless over a short distance. Later the distance will be illimitable, and by later I mean only a few years hence. Pictures are transmitted over wires—they were telegraphed successfully through the point system thirty years ago. When wireless transmission of power becomes general, these methods will be as crude as is the steam locomotive compared with the electric train. Nikola Tesla (1856–1943)

Overview This chapter computes the unconditional (or traditional) connectivity and fault-tolerance of spatial (or three-dimensional) homogeneous and heterogeneous k-covered wireless sensor networks. The proposed measures are based on the Reuleaux tetrahedron model, which is used to characterize k-coverage of a spatial field as discussed earlier in Chap. 7. This choice is to make this computation problem more tractable. Moreover, based on the concepts of conditional connectivity and forbidden faulty sensor set, which cannot include all the neighbors of a sensor, this chapter proposes conditional network connectivity and fault-tolerance measures for the above networks. It proves that spatial k-covered wireless sensor networks can sustain a large number of sensor failures. Precisely, it shows that these networks have connectivity higher than their degree of coverage k. Furthermore, some widely used assumptions in coverage and connectivity in wireless sensor networks, such as sensor homogeneity and unit sensing and communication model, are relaxed so as to promote the practicality of the results of this chapter in real-world scenarios.

12.1 Introduction In the literature, most of the work on wireless sensor networks dealt with planar (or two-dimensional) settings, where sensors are deployed on a planar field. However, there exist applications that cannot be effectively modeled in the plane. For instance, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_12

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underwater applications [8, 9] require the design in the space. Oceanographic data collection, pollution monitoring, offshore exploration, disaster prevention, and assisted navigation are typical applications of underwater sensor networks [8]. Indeed, Pompili et al. [328] proposed different deployment strategies for planar and spatial communication architectures in underwater acoustic sensor networks, where sensors are anchored at the bottom of the ocean for the planar design and float at different depths of the ocean to cover the entire space. As discussed by Soro and Heinzelman [363], both coverage of space and energy-efficient data routing for a telepresence application require a spatial design. In order to cope with the problem of faulty sensors due to low battery power and also to ensure high data accuracy, it is essential that redundant coverage (in particular k-coverage) of the same region be achieved. As mentioned earlier, the major function of wireless sensor networks is to monitor a phenomenon in a field of interest and report the sensed data to the sink for further analysis and processing. However, sensing coverage may not be sufficient if the sensed data cannot reach the sink. This is mainly due to the absence of any communication paths between the source sensors (or data generators) and the sink. Thus, network connectivity must be guaranteed to allow interactions between them all the time. In particular, a wireless sensor network is said to be fault tolerant if it remains functionally connected in spite of the occurrence of sensor failures. Therefore, each source sensor has to be connected to the sink through multiple communication paths. This is why we focus on connectivity when coverage is provided. Hence, a wireless sensor network will be able to function properly only if both coverage and connectivity are maintained simultaneously. This implies that all source sensors and the sink should belong to the same connected component during the network operation. Data accuracy depends on the size of the connected component that contains the sink. It reaches the highest value when the sink belongs to the largest connected component of the network. Thus, high-quality coverage requires all source sensors be connected to the sink. That is why we focus on the sink to compute the connectivity of spatial k-covered wireless sensor networks [41]. In other words, connectivity of wireless sensor networks should be defined so as to take into consideration the inherent structure of these networks. Indeed, sensors have neither the same role nor the same impact on the network performance. Thus, measuring the connectivity of wireless sensor networks should account for their specific morphology, where the sink is the most critical node in the network. Hence, we compute the connectivity of spatial k-covered wireless sensor networks based on the size of the connected component that includes the sink. Traditional (or unconditional) connectivity is a graph-theoretic concept that does not take into account the type of network being modeled by a graph. In fact, it is defined under the assumption that any subset of nodes can potentially fail at the same time. Also, this definition of connectivity does not exclude the case where all the neighbors of a given node may fail. While such definition of network connectivity may be reasonable to measure the fault tolerance of small-scale networks, it is not effective for large-scale dense networks, such as highly dense spatial k-covered wireless sensor networks. Indeed, in such networks with thousands of sensors, it is highly

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unlikely that all the neighbors of a given sensor fail at the same time. This is due to two important factors, namely high-density deployment and sensor heterogeneity. Assuming a uniform sensor distribution, the ratio of the size of the neighbor set of a given sensor to the total number of sensors in a wireless sensor network covering a spatial field of volume V is 4 π R 3 /3 V , where R ⋘ V 1/3 is the radius of the communication range of the sensors. The probability of failure of all the neighbors of a given sensor could be identified with this ratio and hence is very low. Moreover, in real-world sensing applications [246], wireless sensor networks may have heterogeneous sensors with unequal energy levels and different sensing, processing, and communication capabilities, thus increasing the network reliability and lifetime [433]. This also implies that the probability that the entire neighbor set of a sensor fail at the same time is very low. Next, we briefly present the major tasks we want to accomplish in this chapter. Moreover, we discuss how to achieve each one of them.

12.1.1 Major Tasks To account for the above-mentioned specificities of spatial k-covered wireless sensor networks, we use a more general concept of connectivity, called conditional connectivity [192] with respect to some property P, defined as follows. The conditional connectivity of a connected graph G is the smallest number of nodes of G whose removal disconnects G into components each of which has property P. Another generalization of connectivity, called restricted connectivity [150] is based on the concept of forbidden faulty set, where all neighbors of a node cannot simultaneously fail. In this chapter, we focus on more realistic measures of network connectivity and fault tolerance of connected k-covered configurations in planar wireless sensor networks through the accomplishment of some key tasks. Next, we specify these tasks and give our corresponding plan of actions. First, we want to compute traditional (or unconditional) measures of network connectivity and fault tolerance of planar connected k-covered wireless sensor networks. For this, we use the graph-theoretic concept of connectivity to compute the network connectivity of these networks. Then, we derive the corresponding network fault-tolerance. Second, we propose a new measure of fault tolerance for k-covered wireless sensor networks, called conditional fault-tolerance, to alleviate the problems, which are caused by the above shortcomings. To this end, we use a more general concept of connectivity, called conditional connectivity (or P-connectivity), which was originally introduced in [192] with respect to some property that is based on the concept of forbidden faulty sensor set that includes all the neighbors of a given sensor. Precisely, we exploit an instance of conditional connectivity, called restricted connectivity, which was proposed in [151]. Specifically, the restriction is on the faulty set (i.e., set

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of snodes that can fail). Restricted connectivity uses the concept of forbidden faulty set in which the entire neighbor set of a given node cannot be faulty at the same time.

12.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 12.2 computes unconditional connectivity for spatial homogeneous and heterogeneous k-covered wireless sensor networks. Section 12.3 computes the conditional counterparts for the above networks. Section 12.4 discusses the results obtained in the previous sections. Section 12.5 shows how to relax the unit sphere model and account for a convex model for sensing. Section 12.6 discusses the results for underwater sensor networks. Section 12.7 provides a review of existing approaches on coverage and connectivity in spatial wireless sensor networks. Finally, Sect. 12.8 concludes the chapter.

12.2 Unconditional Connectivity In this section, we compute measures of unconditional (or traditional) connectivity for homogeneous and heterogeneous spatial k-covered wireless sensor networks, where any subset of sensors can fail. First, recall the following result in Lemma 12.1 below, which is stated earlier in Chap. 7. It specifies the condition under which a spatial field is guaranteed to be k-covered with at least k sensors. Lemma 12.1 (Guaranteed Spatial k-Coverage) A spatial field is guaranteed to be k-covered if any Reuleaux tetrahedron region of side r0 in the field contains at least ∎ k sensors, where r 0 = r/1.066 and k ≥ 4. Our analysis in this section is based on the preliminary value of the spatial sensor density stated in Theorem 12.1 below [41], which is also computed earlier in Chap. 7 in Sect. 7.3.3. This value is based on the Reuleaux tetrahedron model. Theorem 12.1 (Spatial Sensor Density for k-Coverage): Let r be the radius of the sensing spheres of sensors and k ≥ 4. The spatial sensor density required to k-cover a spatial (or three-dimensional) field is computed as λ(r, k) = where r 0 = r/1.066.

k 0.422 r03

(12.1) ∎

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12.2.1 Homogeneous Sensors Theorem 12.2 deals with homogeneous wireless sensor networks. Theorem 12.2 (Network Connectivity Measure) Let G be a communication graph of a homogeneous spatial k-covered wireless sensor network deployed in a cubic field of volume V. The connectivity of G is given by κ1 (G) ≤ κ(G) ≤ κ3 (G)

(12.2)

where κ1 (G) = 12.024 α 3 k κ3 (G) =

R V2/3 k 0.422 r03

r 0 = r/1.066, α = R/r, and k ≥ 4. Proof The optimum position of the sink in terms of energy efficient data gathering is the center of the cubic field [281]. Let ξ0 be the location of the sink s0 . We consider the following three cases depending on the size of the connected component that includes the sink. Also, given that sensor failure is due to low battery power, we assume that the sink has infinite source of energy, thus excluding the possibility of a faulty sink. Case 1: Single-node connected component. In this case, there are at least two components, one of them being the single-node component containing the sink. Finding the minimum number of nodes to disconnect the network requires that the disconnected network have only two components. In order to isolate the sink, all its neighbors must fail. Hence, at least the communication sphere of the sink should contain no sensor but the sink. Let N be a random variable that counts the number of sensor failures to isolate the sink s0 . Given that sensors are randomly and uniformly deployed in a volume V with spatial density λ(r, k) per unit volume, where R ⋘ V 1/3 , the expected number of neighbors of the sink is given by E [N] = λ(r, k) ∥ B(ξ0 , R) ∥

(12.2a)

where ∥ B(ξ0 , R) ∥ = 4 π R 3 /3 is the measure of the volume of the communication sphere B(ξ0 , R) of the sink s0 located at ξ0 . Thus, the expected minimum number of sensor failures to isolate s0 is equal to E [N]. Substituting Eq. (12.1) in Eq. (12.2a), we find that the network connectivity is given by

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κ1 (G) = E [N] = 12.024 α 3 k

(12.2b)

where α = R/r . Figure 12.1 plots the function κ1 (G) in Eq. (12.2b) by varying α = R/r for several fixed values of k, while Fig. 12.2 plots the function κ1 (G) by varying k for several fixed values of α = R/r. Clearly, κ1 (G) increases with the ratio α and the degree of coverage k. More importantly, κ1 (G) is much higher than k. Case 2: Non-trivial connected components. As in the case of planar sensor deployment, which is discussed earlier in Chap. 11, two configurations of the disconnected network are of particular interest where the two connected components of the disconnected network are separated by a vacant region (or gap). Furthermore, any pair of sensors, one from each component, are separated by a distance at least equal to R in order to prohibit any communication between the two components. In the first Fig. 12.1 Plot of the function κ1 (G) (fix k and vary α = R/r)

Fig. 12.2 Plot of the function κ1 (G) (fix α = R/r and vary k)

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configuration as shown in Fig. 12.3a, the component including the sink is reduced to its communication sphere. Thus, the volume of the vacant region, denoted by gap(ξ0 , R), and surrounding the component of the sink, is given by Fig. 12.3 Planar projection of non-trivial connected components of the disconnected network (case 2: non-trivial connected components)

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∥ gap(ξ0 , R) ∥ = 4π (2R)3 /3 − 4π R 3 /3 = 9.333 π R 3

(12.2c)

Thus, the expected minimum number of sensor failures to isolate the component of the sink is given by E [N] = λ(r, k) ∥ gap(ξ0 , R) ∥

(12.2d)

Substituting Eq. (12.1) in Eq. (12.2d), we find that the network connectivity is equal to κ2 (G) = E [N] = 84.164 α 3 k

(12.2e)

where α = R/r. In the second configuration as shown in Fig. 12.3b, the original network is split into two components such that the vacant region forms a cuboid, denoted by cub(R), and whose sides are R, V 1/3 , and V 1/3 . Now, this configuration corresponds to the smallest connected component containing the sink if the field has to be divided into two regions such that none of them surrounds the other. Thus, the expected minimum number of sensor failures to isolate the connected component containing the sink is given by E [N] = λ ∥ cub(R) ∥

(12.2f)

∥ cub(R) ∥ = R V 2/3

(12.2g)

where

Substituting Eqs. (12.1) and (12.2g) in Eq. (12.2f), it follows that network connectivity is equal to κ3 (G) = E [N] =

R V2/3 k 0.422 r03

(12.2h)

It is easy to check that κ3 (G) > κ2 (G) since R ⋘ V 1/3 . Case 3: Largest connected component. This configuration is totally opposite to the one given in Case 1 and has only one isolated sensor. Since we are interested in k-coverage of the entire spatial field, such a network is considered as disconnected. The network connectivity is the same as in Case 1. Thus, κ1 (G) ≤ κ(G) ≤ κ3 (G)

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It is easy to check that κ(G) > k. ∎ All these bounds and those to be derived in next sections are based on our fundamental result stated in Theorem 12.1 related to the minimum spatial sensor density for full k-coverage of a spatial field. Thus, these are lower bounds, and hence tight.

12.2.2 Heterogeneous Sensors Achieving k-coverage of a spatial field by heterogeneous sensors would depend on the least powerful ones in terms of their sensing capabilities. The following results are derived from those given earlier in Chap. 7 as well as those stated above in this chapter. We present them without any proofs. Lemma 12.2 (Breadth for Guaranteed k-Coverage) If the breadth of a spatial convex region C is at most equal to the minimum radius rmin of the sensing spheres of sensors, then C is guaranteed to be k-covered, where k ≥ 4 and rmin = min{r j /1.066 : s j ∈ ∎ S}. From Lemma 12.2, it follows that connectivity between sensors located in the Reuleaux tetrahedron of side rmin requires that Rmin ≥ rmin . Lemma 12.3 (Spatial Sensor Density for k-Coverage) Let rmin be the minimum radius of the sensing spheres of sensors and k ≥ 4. The minimum spatial sensor density needed for k-coverage of a spatial field by heterogeneous wireless sensor networks is given by λ(rmin , k) =

k 3 0.422 rmin

where rmin = min{r j /1.066 : s j ∈ S}.

(12.3) ∎

Lemma 12.4 computes the connectivity measures for heterogeneous spatial kcovered wireless sensor networks. Lemma 12.4 (Network Connectivity Measure Using Heterogeneous Sensors) Let G be a communication graph of a heterogeneous spatial k-covered wireless sensor network with Rmin ≥ rmin and k ≥ 4. The connectivity of the graph G is given by κ4 (G) ≤ κ(G) ≤ κ3 (G) where κ3 (G) =

Rmax V2/3 k 3 0.422 rmin

(12.4)

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κ4 (G) = 12.024 α23 k α2 = Rmin /rmin , k ≥ 4, rmin = min{r j /1.066 : s j ∈ S}, Rmin = min{R j : s j ∈ S}, and Rmax = max{R j : s j ∈ S}. ∎

12.2.3 Boundary Effects Network connectivity is defined as the minimum number of sensors whose failure (or removal) disconnects the network. Given the cubic geometry of a spatial field we consider, sensors located close to the border of the field are affected by the boundary effects. Indeed, the communication ranges of these sensors cover areas outside of the deployment area, and hence have fewer number of communication neighbors compared to all other sensors (especially the ones whose communication ranges lay entirely inside the field). However, the proposed network connectivity measures consider the sink as the most critical node in the network and whose isolation would definitely kill the entire network. Because the optimum position of the sink in terms of energy efficient data gathering is the center of the cubic field [281], the boundary effects do not exist at all, and thus our derived bounds on the connectivity are correct. Even when our approach for computing network connectivity measures considers all sensors as critical (see Sect. 12.6 for detailed discussion), the boundary effects do not have any impact on the derived bounds on network connectivity. Indeed, we are interested in the minimum number of sensor failures to disconnect the network. As far as sensor deployment to achieve spatial k-coverage is concerned, we should mention that the boundary effects have an impact on the performance of the network. By Lemma 12.1 stated above and also in Chap. 7, a spatial field is guaranteed to be fully k-covered if each Reuleaux tetrahedron region in the field contains at least k sensors. However, it is impossible to randomly decompose a spatial field into an integer number of Reuleaux tetrahedron regions because of the boundary of the field. Indeed, most of the Reuleaux tetrahedron regions close to the border of a spatial field do not entirely lay inside the deployment area. Therefore, more than necessary sensors would be needed to achieve k-coverage of these Reuleaux tetrahedron regions on the border of the field. Simulation results reported in Chap. 7 show that the spatial sensor density necessary to fully k-cover a cubic field is a bit higher than the bound given in Eq. (12.1) at the beginning of Sect. 12.2, mainly due to the boundary effects. Next, we introduce new measures of connectivity of spatial k-covered wireless sensor networks by placing a specific constraint on a subset of sensors that would fail.

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12.3 Conditional Connectivity In this section, we compute measures of conditional connectivity for homogeneous and heterogeneous spatial k-covered wireless sensor networks. These measures are based on the concepts of conditional connectivity [192] and forbidden faulty set [150]. As discussed in the next sections, our results prove that spatial k-covered wireless sensor networks can sustain a larger number of sensor failures under the restriction imposed on the faulty sensor set. Similarly to our discussion in Chap. 7, it is easy to show that the traditional connectivity, which does not impose any restriction on the faulty sensor set, is not a useful metric for spatial k-covered wireless sensor networks, which are highly dense networks and denser than their planar counterparts.

12.3.1 Homogeneous Sensors Theorem 12.3 computes the conditional connectivity of homogeneous spatial kcovered wireless sensor networks. Theorem 12.3 (Conditional Network Connectivity Measure) The conditional connectivity of a homogeneous spatial k-covered wireless sensor network, where k ≥ 4, is given by κ(G : P) =

((r0 + 2 R)3 −r03 ) k r03

(12.5)

where r 0 = r/1.066. Proof We consider the following two cases based on the type of connected component that contains the sink. Case 1: Smallest connected component. According to our conditional connectivity model, no sensor can be isolated and hence no trivial component can be part of the disconnected network. Under the assumption of forbidden faulty sensor set, the smallest connected component that is disconnected from the rest of the network and contains the sink can be determined as follows. In order to achieve k-coverage of the cubic field, every location must be k-covered, including the location ξ0 of the sink s0 . Otherwise, the k-coverage property will not be satisfied. Therefore, the smallest connected component that includes the sink consists of k sensors deployed in the Reuleaux tetrahedron of side r 0 and centered at ξ0 . In order to disconnect the sink under the forbidden faulty sensor set constraint, the Reuleaux tetrahedron should be surrounded by an empty annulus of width equal to R as shown in Fig. 12.4, i.e., sensors located in the annulus have failed. The Reuleaux tetrahedron together with

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Fig. 12.4 Two nested concentric Reuleaux tetrahedra

this annulus forms a larger Reuleaux tetrahedron of side r0 + 2R. The volume of the annulus, denoted by A(ξ0 , R), is equal to ∥ A (ξ0 , R) ∥ = 0.422 (r0 + 2 R)3 − 0.422 r03 Therefore, the expected conditional minimum number of sensor failures to disconnect the smallest component including the sink can be computed as E [N : P] = λ(r, k) ∥ A(ξ0 , R) ∥

(12.5a)

Substituting Eq. (12.1) in Eq. (12.5a), we find that the conditional connectivity is given by κ1 (G : P) = E [N : P] =

((r0 + 2 R)3 −r03 ) k r03

(12.5b)

Figure 12.5 shows the planar projection of an annulus of width R surrounding a Reuleaux tetrahedron of side r . It is easy to check that the forbidden faulty set constraint is satisfied for both the faulty sensors (located inside the annulus and which have failed) and non-faulty sensors (located outside the annulus). Indeed, any sensor in the inner Reuleaux tetrahedron still has non-faulty neighbors in the inner Reuleaux tetrahedron. Besides, any sensor outside the outer Reuleaux tetrahedron still has non-faulty neighbors outside the outer Reuleaux tetrahedron. Also, any faulty sensor within the annulus A(R) has non-faulty neighbors located in the inner Reuleaux tetrahedron and outside the outer Reuleaux tetrahedron. Case 2: Largest connected component. This case is similar to the previous one. However, the sink belongs to the largest connected component. Hence, the disconnected network consists of two components: the one including the sink and a second

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Fig. 12.5 Planar projection of an annulus of width R surrounding a Reuleaux tetrahedron of side r

component associated with sensors located in a Reuleaux tetrahedron of side r. Using the same analysis as in Case 1, we obtain the same conditional network connectivity: κ2 (G : P) = κ1 (G : P)

(12.5c)

From both cases 1 and 2, the conditional connectivity of homogeneous spatial k-covered wireless sensor networks is computed as κ(G : P) = κ1 (G : P)

∎

12.3.2 Heterogeneous Sensors We observe that computing the conditional connectivity of heterogeneous spatial k-covered wireless sensor networks is not a straightforward generalization of the approach used previously for homogeneous spatial k-covered wireless sensor networks. If we use the same process as previously, we either violate the forbidden faulty sensor set constraint or maintain network connectivity. Precisely, if the width of the annulus containing the faulty sensors is Rmax as shown in Fig. 12.6a, then sensors whose communication radii are less than or equal to Rmax /2, may be located in the annulus. Thus, the entire neighbor set of this type of sensors would fail at the same time and hence the property P would be violated. Now, if the width of the annulus containing the faulty sensors is less than Rmax as shown in Fig. 12.6b, then the non-faulty sensors of one connected component will be able to communicate with the non-faulty sensors of the other connected component of the disconnected network. Hence, the network is still connected. In this case, it is impossible to find an exact value of conditional connectivity for heterogeneous spatial k-covered wireless sensor networks. In the following, we compute their lower and upper bounds based on the types of sensors in and around the annulus.

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Fig. 12.6 Planar projection: a forbidden fault sensor set constraint violated and b connectivity maintained

Lemma 12.5 (Conditional Network Connectivity Measure Using Heterogeneous Sensors) The conditional connectivity of the heterogeneous spatial k-covered wireless sensor networks is given by κ1 (G : FP ) ≤ κ(G : P) ≤ κ2 (G : FP )

(12.6)

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where κ1 (G : P) = κ2 (G : P) =

() ) )3 0 0 3 rmin k + 2Rmin − rmin 0 rmin

3

3 )k ((rmiax + 2 Rmax )3 −rmax 3 rmin

k ≥ 4, rmin = min{r j /1.066 : s j ∈ S}, rmax = max{r j /1.066 : s j ∈ S}, Rmin = min{R j : s j ∈ S}, and Rmax = max{R j : s j ∈ S}. Proof As above, we consider the following two cases depending on the size of the connected component that includes the sink. Case 1: Smallest connected component. In order to compute a lower bound on the conditional connectivity of heterogeneous spatial k-covered wireless sensor networks, we consider the Reuleaux tetrahedron centered at location ξ0 of the sink s0 , which will be disconnected from the network. First, we assume that the annulus containing the faulty sensors as well as the volume surrounding it contains only least powerful sensors, and hence the width of this annulus is equal to Rmin . Also, to guarantee that the sink will not be isolated, which would violate the forbidden faulty sensor set constraint, the Reuleaux tetrahedron centered at ξ0 should have a side equal to rmin . These two conditions help disconnect the network while satisfying the forbidden faulty sensor set constraint. The volume of the annulus A(ξ0 , Rmin ) is given by 3 ∥ A (ξ0 , Rmin ) ∥ = 0.422 (rmin + 2 Rmin )3 − 0.422 rmin

(12.6a)

Hence, the expected conditional minimum number of sensor failures to disconnect the connected component including the sink (or the inner Reuleaux tetrahedron) from the rest of the network is computed as E[N : P] = λ(rmin , k)∥ A(ξ0 , Rmin ) ∥

(12.6b)

where λ(rmin , k) = 0.422k r 3 and rmin = min{r j /1.066 : s j ∈ S}. Thus, the min conditional network connectivity is given by κ1 (G : P) = E[N : P] =

3 )k ((rmin + 2 Rmin )3 −rmin 3 rmin

(12.6c)

where k ≥ 4 and Rmin = min{R j : s j ∈ S}. In order to compute an upper bound on the conditional connectivity of heterogeneous spatial k-covered wireless sensor networks, we assume that sensors inside the

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annulus are the most powerful ones. Thus, the side of the inner Reuleaux tetrahedron should be equal to rmax , while the width of the annulus surrounding it should be equal to Rmax . It is easy to check that this setup will disconnect the network, while satisfying the forbidden faulty set constraint. The upper bound on the conditional connectivity is given by κ2 (G : P) = E[N : P] = λ(rmin , k)∥ A(ξ0 , Rmax ) ∥ =

3 )k ((rmax + 2 Rmax )3 −rmax 3 rmin

(12.6d)

where λ(rmin , k) = 0.422k r 3 , k ≥ 4, rmin = min{r j /1.066 : s j ∈ S}, rmax = min max{r j /1.066 : s j ∈ S}, and Rmax = max{R j : s j ∈ S}. Case 2: Largest connected component. In this case, the sink belongs to the largest connected component of the disconnected network. Hence, the previous analysis applies to any sensor in the network instead of the sink. Thus, the conditional connectivity of heterogeneous spatial k-covered wireless sensor networks satisfies. κ1 (G, P) ≤ κ(G, P) ≤ κ2 (G, P) ∎ Next, we relax the assumptions used in our previous analysis to enhance the practicality of our results.

12.4 Discussion The analysis of the minimum spatial sensor density necessary for k-coverage of a spatial field and network connectivity of spatial k-covered wireless sensor networks is based on the unit sphere model, where the sensing and communication ranges of sensors are modeled by spheres. In other words, sensors are supposed to be typically isotropic. Although this assumption is the basis for most of the protocols for coverage and connectivity in wireless sensor networks, it may not hold universally and thus may not be valid in practice. In Sect. 12.5, we show how to relax this assumption in order to promote the applicability of our results to real-world spatial wireless sensor network scenarios, and summarize our results for the convex model, where the sensing and communications ranges of sensors are convex but not necessarily spherical. Moreover, we assume that our results for network connectivity hold for a degree of sensing coverage k, where k ≥ 4. In this section, we show how to relax the latter assumption. Then, we discuss a sensor placement strategy for full k-coverage of a spatial field.

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12.4.1 Relaxing the Assumption of k ≥ 4 The analysis of k-coverage and connectivity for spatial k-covered wireless sensor networks are valid for all k ≥ 4. Since the breadth of the Reuleaux tetrahedron is equal to r (or rmin ), our results can also be used for k ≤ 3. In other words, k-coverage of the cube can be met by deploying k sensors in the Reuleaux tetrahedron, where k ≤ 3. However, the network would be denser than necessary (especially for k = 1) and the coverage degree would be higher than that dictated by the application.

12.4.2 Sensor Placement Strategy Notice that under the assumption of spherical model, it is impossible to achieve a degree of coverage exactly equal to k in all the locations of the cube. Therefore, a sensor placement strategy to achieve k-coverage should be devised in such a way that every location in the cube is k ' -covered, where k ' is very close to k. This placement strategy should benefit from the geometry of the Reuleaux tetrahedron. The sensor placement problem can be transformed into a problem of covering a cube with overlapping sets of congruent Reuleaux tetrahedra. An optimal covering consists to use a minimum number of Reuleaux tetrahedra by minimizing the overlap volume between them. More precisely, two adjacent Reuleaux tetrahedra overlap such that the faces of their corresponding regular tetrahedra are entirely coinciding with each other. Thus, the curved edges of the Reuleaux tetrahedra should overlap so the same subset of sensors deployed on these curved edges could participate in covering the space associated with both Reuleaux tetrahedra at the same time, thus minimizing the total number of sensors required to k-cover the cube. As in the case of planar deployment, the sensors located in a spatial lens, which corresponds to the overlap volume of two adjacent Reuleaux tetrahedra of side length r 0 , are at distance r 0 from any point in their volumes, and hence participate in k-covering both tetrahedra. The design of duty-cycling protocols to k-cover a spatial field with a minimum number of sensors should select sensors based on this observation.

12.4.3 Sink-Independent Connectivity Measures Although in centralized algorithms the concept of sink is well defined, it is likely that distributed algorithms, such as the consensus-based algorithm, will be implemented for wireless sensor networks. In this case, the concept of (fixed) sink would lose value. It would be interesting to revise the definition of connectivity to take this concept into account. We suggest that all the nodes be considered as peer-to-peer. Thus, we define connectivity with respect to all sensors in the network. Given the geometry of the spatial deployment field that we consider (cube), the boundary sensors, i.e.,

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Fig. 12.7 Eight boundary sensors located on the corners of a cube

sensors located at the boundary of the cube, have small neighbor sets. In particular, the sensors located at the eight corners of the cube sb1 , …, sb8 as shown in Fig. 12.7 are highly likely to have the smallest neighbor set. Our measures of connectivity will be based on one of these boundary sensors to find the minimum number of sensors of its neighbor set that should fail in order to disconnect the network. It is easy to check that compared to the sink, the size of the neighbor set of a boundary sensor is equal to a quarter of that of the neighbor set of the sink. Thus, the connectivity measures computed in the previous sections with respect to the sink remain the same for a boundary sensor but are weighted by a coefficient equal to 1/4. For more details, the interested reader is referred to Sect. 12.6.

12.4.4 Spatial Sensing Applications In the case of wireless sensor networks deployed on the trees of different heights in a forest, the sensors could be seen almost everywhere in the space. For wireless sensor networks deployed in buildings with multiple floors, sensors are placed on the ground and/or the wall, but the networks seldom contain sensors floating in the middle of the room. The first example show that our proposed spatial model is valid and can be applied to choose the spatial sensor density in practical problems. The second example, however, shows the limited validity of our model due to the restriction imposed on the placement of sensors inside buildings or rooms.

12.5 Relaxing the Unit Sphere Model: Convex Model

393

12.5 Relaxing the Unit Sphere Model: Convex Model The assumption of spherical sensing and communication ranges of the sensors may not hold in real-world wireless sensor network platforms. It has been observed in [451] that the communication range of MICA motes [204] is asymmetric and depends on the environments. It is also found in [456] that the communication range of radios is highly probabilistic and irregular. In this chapter, for problem tractability, we consider a convex model, where the sensing and communication ranges of sensors are convex but not necessarily spherical. First, we define the notion of largest enclosed sphere of a spatial convex region C as a sphere that lies entirely inside C and whose diameter is equal to the minimum distance between any pair of points on the boundary of the region C.

12.5.1 Homogeneous Sensors We consider homogeneous sensors that have the same convex sensing ranges and communication ranges. The following results are also derived from those stated earlier in Chap. 7. Their proofs are similar to their counterparts in Chap. 7 by using the largest enclosed sphere instead of the sensing sphere. Lemma 12.6 (Breadth of k-Covered Region Using Convex Model) If k ≥ 4 homogeneous sensors are deployed in a spatial convex region C, then the region C is k-covered if its breadth does not exceed rled , the radius of the largest enclosed sphere of the sensing range. ∎ Lemma 12.7 (Spatial Sensor Density for k-Coverage Using Convex Model) The minimum spatial sensor density required to guarantee k-coverage of a spatial field is given by λ(rled , k) =

k 0 0.422 rled

3

(12.7)

where rled stands for the radius of the largest enclosed sphere of the sensing range, 0 rled = rled /1.066, and k ≥ 4. ∎ Now, we discuss how those results can be derived using a convex model, where the sensing and communication ranges of the sensors may not necessarily be spherical.

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12 Spatial Unconditional and Conditional Network Connectivity …

12.5.2 Heterogeneous Sensors We state the following results, which are also derived from those stated earlier in Chap. 7. Their proofs are similar to their counterparts in Chap. 7 by using the largest enclosed sphere instead of the sensing sphere. Lemma 12.8 (Breadth of k-Covered Region Using Heterogeneous Sensors) A spatial convex region C is guaranteed to be k-covered when exactly k heterogemin , where neous sensors are deployed in it, if the breadth of C does not exceed rled min rled = min{rled /1.066 : s j ∈ S} and k ≥ 4. ∎ Lemma 12.9 (Spatial Sensor Density for k-Coverage Using Heterogeneous Sensors) The minimum spatial sensor density required to k-cover a spatial field using heterogeneous sensors is given by min λ(rled , k) =

k min 2 0.422 (rled )

min where rled = min{rled /1.066 : s j ∈ S} and k ≥ 4.

(12.8)

∎

The measures of network connectivity can be derived using the same approach as in the previous sections. Thus, the assumption of unit sphere model for sensing and communication ranges of sensors can be relaxed with the aid of largest enclosed sphere of their sensing range.

12.6 Underwater Sensor Networks The results in the previous sections are only applicable to the connectivity of sink node. Although the connectivity of sink is critical, in some scenarios, such as underwater wireless sensor networks [8, 9], any sensor may be critical due to the high cost, for instance. In the following, we extend our network connectivity measures for spatial kcovered wireless sensor networks to the case where any sensor in the network is critical. Specifically, we consider a boundary sensor, i.e., a sensor located at one corner of the cubic field. Such a boundary sensor has the minimum number of communication neighbors given that all sensors are located within the deployment region—the cube, and hence the actual communication range of a boundary sensor is only a quarter of its communication sphere. In [425], a boundary sensor is considered to compute the connectivity of planar k-covered wireless sensor networks. Theorem 12.4 summarizes the connectivity measures with respect to a boundary sensor for homogeneous spatial k-covered wireless sensor networks. The case of heterogeneous wireless sensor networks and the case of sensors with convex sensing

12.7 Related Work

395

and communication ranges can be treated similar to the previous sections. Thus, we omit the proof of Theorem 12.4. Theorem 12.4 (Network Connectivity Measures with respect to Boundary Sensors) Let G be a communication graph of a homogeneous spatial k-covered wireless sensor network deployed in a cubic field, where the radii of the sensing and communication spheres of sensors are r and R, respectively. The connectivity of G is computed as κ(G) = 3.02 α 3 k

(12.9)

whereas the conditional connectivity of G is given by κ(G : P) =

3.02 ((r0 + R)3 −r03 ) k r03

where r 0 = r/1.066, α = R/r, and k ≥ 4.

(12.10) ∎

12.7 Related Work Coverage and connectivity in spatial wireless sensor networks have gained relatively less attention in the literature. A placement strategy based on Voronoi tessellation of the space is proposed in [12], which creates truncated octahedral cells. Several fundamental characteristics of randomly deployed spatial wireless sensor networks for connectivity and coverage are investigated in [336], which compute the required sensing range to guarantee certain degree of coverage of a region, the minimum and maximum network degrees for a given communication range as well as the hop-diameter of the network. Related to the work in this chapter is the novel result discussed in [480], which proved that the breadth of the Reuleaux tetrahedron is not constant. This shows that the properties of the plane cannot be directly extended to the space (i.e., the Euclidean three-dimensional space). Indeed, the Reuleaux triangle [481] (counterpart of Reuleaux tetrahedron in the plane) has a constant width. This helps us provide correct measures of connectivity and fault-tolerance of spatial wireless sensor networks based on an accurate characterization of k-coverage of spatial fields. Note that the Reuleuax tetrahedron is the symmetric intersection of four congruent spheres such that each sphere passes through the centers of the other three spheres. However, the Reuleaux triangle [481] corresponds to the symmetric intersection of three congruent disks such that each disk passes through the centers of the other two disks. Therefore, its constant width is equal to the radius of these disks. The work discussed in this chapter can be viewed as an extension of [12] by considering k-coverage in spatial wireless sensor networks. Moreover, existing work on coverage and connectivity in wireless sensor networks assumed the notion

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12 Spatial Unconditional and Conditional Network Connectivity …

of traditional connectivity, while the work in this chapter considers a more realistic measure, namely conditional connectivity [192], which is based on the concept of forbidden faulty set [150].

12.8 Conclusion In this chapter, we investigate coverage and connectivity in spatial k-covered wireless sensor networks with emerging applications, such as underwater acoustic sensor networks that require spatial design. We propose the Reuleaux tetrahedron model to guarantee k-coverage of a spatial field. Based on the geometric properties of Reuleaux tetrahedron, we derive the spatial sensor density for guaranteeing k-coverage of a spatial space. We also compute the connectivity of homogeneous and heterogeneous spatial k-covered wireless sensor networks. Our results on connectivity take into consideration an inherent characteristic of wireless sensor networks in that the sink has a critical role in terms of data processing and decision making, compared to the rest of the network. Therefore, we compute the connectivity of spatial k-covered wireless sensor networks based on the size of the connected component that includes the sink. We conclude that the connectivity of spatial k-covered wireless sensor networks is much higher than the degree of sensing coverage k provided by the network. The traditional connectivity metric, however, is defined in an abstract way and does not consider the inherent properties of wireless sensor networks because it assumes that any subset of nodes can fail at the same time. This assumption is not valid for heterogeneous spatial k-covered wireless sensor networks. To compensate for these shortcomings, we propose more realistic measures of connectivity based on the concept of forbidden faulty set. We find that spatial k-covered wireless sensor networks can sustain a large number of sensor failures provided that the neighbor set of a sensor cannot fail at the same time. That is, we show that the traditional connectivity metric, which is used to capture network fault-tolerance, underestimates the resilience of spatial k-covered wireless sensor networks. Particularly, based on the minimum spatial sensor density, we prove that the connectivity of spatial k-covered wireless sensor networks is much higher than the degree k of sensing coverage provided by the network. We believe that our results have practical significance for sensor network designers to develop spatial applications with prescribed degrees of coverage and connectivity. These connectivity measures can be exploited in the design of fault-tolerant topology control protocols for spatial k-covered wireless sensor networks.

Part VI

Geographic Data Forwarding, Gathering, and Delivery in Wireless Sensor Networks

Chapter 13

A Planar Checkpoints-Based Approach for Geographic Forwarding on Always-on Sensors

Present wireless receiving apparatus will be scrapped for much simpler machines; static and all forms of interference will be eliminated, so that innumerable transmitters and receivers may be operated without interference. It is more than probable that the household’s daily newspaper will be printed ‘wirelessly’ in the home during the night. Domestic management—the problems of heat, light and household mechanics—will be freed from all labor through beneficent wireless power. Nikola Tesla (1856–1943)

Overview This chapter addresses the problem of energy-efficient data forwarding in wireless sensor networks. It presents our solution to this problem for always-on wireless sensor networks, where the sensors are active all the time. Specifically, it investigates both of the short-range and long-range data forwarding schemes on always-on sensors and provide several theoretical and simulation results. It shows that short-range data forwarding scheme is more appropriate than long-range data forwarding scheme to save energy of the sensors and promote the longevity of wireless sensor networks with scarce energy resources. Our proposed solution is based on the geometric properties of Delaunay triangulation.

13.1 Introduction In contrast to traditional wireless networks, energy efficiency is a critical determinant to extend the lifetime of wireless sensor networks. Data forwarding is an essential function in wireless sensor networks. As discussed earlier, the sensors communicate with each other via wireless, multi-hop links and have limited battery power (or energy), which is the most crucial resource. With these challenges in mind, energy efficiency is the primary constraint that should be met in the design and implementation of protocols for wireless sensor networks, which will be used by the sensors in their sensing, communication, and processing tasks. Because energy is the most crucial resource for the sensors to perform efficiently and correctly, the design of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_13

399

400

13 A Planar Checkpoints-Based Approach for Geographic …

energy-efficient data forwarding protocols for wireless sensor networks has been receiving much attention to extend the network lifetime. Next, we give a brief description of the major tasks we want to accomplish in this chapter. Then, we briefly state how to achieve each one of them.

13.1.1 Major Tasks It is well-known that geographic forwarding is an energy-efficient and practical scheme for wireless sensor networks in the sense that the sensors are not required to maintain global and detailed information on the topology of the entire network. The sensors need only maintain local knowledge on their one-hop communication neighbors with respect to their geographic location information in order to progress data toward their final destinations, i.e., to forward data to a central gathering point, i.e., the sink, for further analysis and processing. In this chapter, we focus on an ongoing debate: Should short-range or long-range data forwarding be opted for in multi-hop wireless networks, such as wireless sensor networks? This chapter supports the short-range data forwarding strategy for wireless sensor networks, where energy should be given the highest priority [42]. This type of data forwarding strategy enables protocol design to efficiently use the limited energy of the sensors by minimizing their average energy consumption in forwarding data originated from sources to a single sink. The goal is to prolong the operational lifetime of the whole network. This short-range data forwarding process requires accomplishing some important tasks. Next, we describe these tasks and provide our corresponding plan of actions. First, we want to design an energy-efficient data forwarding protocol for wireless sensor networks so they remain operational as long as possible. To this end, our proposed protocol, called Weighted Localized Delaunay Triangulation-based data forwarding (WLDT), uses 1-lookahead scheme to guarantee data delivery to the sink. The protocol WLDT aims at minimizing the average energy consumption of the sensors during data forwarding towards the sink. It exploits the geometric properties of the Delaunay triangulation [67] to build an energy-efficient path between the source and the sink as a sequence of sub-paths whose endpoints are called checkpoints, in order to forward data over short range. These checkpoints are selected based on their location in the planar field and their remaining energy. A sub-path between a pair of checkpoints consists of a series of forwarders, which are the endpoints of short Delaunay edges. These forwarders are selected based on their location and remaining energy to forward the data between their checkpoints. Second, we want to extract useful theoretical results to assess these short-range forwarding paths. For this, we compute a lower bound on the energy consumption in short-range data forwarding and the corresponding optimum number of forwarders between a source and the sink. This bound should help extend the battery lifetime of the sensors, thus prolonging the operational network lifetime.

13.2 The WLDT Protocol

401

13.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 13.2 describes our WLDT protocol. Section 13.3 analyzes WLDT while Sect. 13.4 discusses both short-range and long-range data forwarding schemes and present several theoretical results. Section 13.5 addresses a few reliability issues of WLDT. Section 13.6 reviews related work. Section 13.7 concludes the chapter.

13.2 The WLDT Protocol In this section, we propose an energy-efficient data forwarding protocol for wireless sensor networks so they remain operational as long as possible. Our protocol, Weighted Localized Delaunay Triangulation-based data forwarding (WLDT), uses 1-lookahead scheme to guarantee data delivery to the sink. WLDT aims to minimize the average energy consumption of the sensors during data forwarding towards the sink. It exploits the geometric properties of the Delaunay triangulation [67] to build an energy-efficient path between the source and the sink as a sequence of sub-paths whose endpoints are called checkpoints. These checkpoints are selected based on their location in the planar field and their remaining energy. A sub-path between a pair of checkpoints consists of a series of forwarders, which are the endpoints of short Delaunay edges and are selected based on their location and remaining energy to forward the data between their checkpoints. Next, we describe WLDT in details.

13.2.1 Long-Range Versus Short-Range Forwarding Let S = {s0 , . . . , sn−1 } be a set of n sensor nodes and sm the single sink connected by wireless links over a wireless sensor network. Lemma 13.1 states that, under some specific condition, data forwarding through a short-range forwarding scheme is more energy efficient than that using a long-range forwarding scheme. We assume that the sensors si and s j are one-hop neighbors of each other. Note that the “one-hop neighbor” relationship is symmetric. In other words, we assume that all the sensors are homogeneous, i.e., have the same physical capabilities and, in particular, their transmission range. Lemma 13.1 (Short-Range vs. Long-Range Forwarding Total Energy Consumption) The total energy consumption for forwarding one data packet from the sensor si to the sensor s j along a short-range path is smaller than the energy spent along a long-range path between them if there is a sensor sk ∈ N S(si ) such that

402

13 A Planar Checkpoints-Based Approach for Geographic …

/

(

δ(sk , sk, p )
Eelec ε E 1−hop (si , sk ) + E 1−hop (sk , s j ) < E 1−hop (si , s j )

∎

Corollary 13.1 recommends two-hop data forwarding between any pair of sensors si to s j only if they are separated by a distance δ(si , s j ) such that δ(si , sk )δ(sk , s j ) >

E elec ε

where sk is a forwarder that lies on the line segment [si , s j ].

13.2.2 A Two-Step Data Forwarding Protocol We propose a data forwarding protocol (Fig. 13.2), which benefits from the energy gain introduced by short-range data forwarding as stated in Lemmas 13.1 and 13.2 and Corollary 13.1, and uses the geometric properties of the Delaunay triangulation (DT). Since we are interested in the area between the sending and the receiving sensors, the neighbor set N S(si ) of the sensor si will contain only the sensors located between si and the sink sm . Figure 13.3 shows a localized DT (Chap. 2, Sect. 2.2, Definition 2.17) as well as a set of candidate checkpoints of s0 (Chap. 2, Sect. 2.2, Definition 2.19). The WLDT protocol is composed of two steps: checkpoint selection and checkpoint-based short-range data forwarding.

13.2.2.1

Checkpoint Selection

We consider a scenario where the source s0 wishes to forward its sensed data to the sink sm . The goal of this step is to identify a subset of candidate checkpoints, CC P(s0 , sm ), of N S(s0 ) that are closest to the sink sm . First, the source s0 constructs its localized Delaunay triangulation, L DT (s0 ), as shown in Fig. 13.3. Intuitively, only the sensors in CC P(s0 , sm ) will be able to get the sensed data out of the transmission range of s0 . Figure 13.3 shows the L DT (s0 ) and the subset CC P(s0 , sm ) that includes s p1 , s p2 , s p3 , s p4 , and s p5 , which are adjacent to the sink sm in the L DT (s0 ). From the subset of candidate checkpoints, CC P(s0 , sm ), the source s0 selects the checkpoint sensor s p such that C E p = max{C E i = ci ei : si ∈ CC P(s0 , sm )} where

(13.1)

13.2 The WLDT Protocol Fig. 13.2 The WLDT protocol

405

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13 A Planar Checkpoints-Based Approach for Geographic …

Fig. 13.3 Localized DT and candidate checkpoints

ci =

δ(s0 , sm ) δ(s0 , si ) + δ(si , sm )

(13.2)

Er em (si ) s j ∈CC P(s0 ,sm ) E r em (s j )

(13.3)

and ei = Σ

As it can be seen, the weight ci measures the degree of closeness of si to the shortest path [s0 , sm ], while the term ei is the percentage of the remaining energy of the sensor si with respect to the total remaining of the subset of sensors CC P(s0 , sm ). Note that ci attains its maximum (ci = 1) when si lies on [s0 , sm ]. Intuitively, the source s0 and the sink sm are the first and last checkpoints, respectively.

13.2.2.2

Checkpoint-Based Short-Range Forwarding

The objective is to forward the sensed data to the checkpoint s p that was selected in the previous step. First, the source s0 assigns weights to each of the Delaunay edges adjacent to it as follows: if s j is an adjacent node to s0 in the L DT (s0 ), then the weight placed on the edge (s0 , s j ) is C E D j (s0 ) = c j e j d j , where cj =

δ(s0 , s p ) δ(s0 , s j ) + δ(s j , s p )

ej = Σ

Er em (s j ) sk ∈Ad j (s0 ) E r em (sk )

13.2 The WLDT Protocol

407

and dj = Σ

1/δ(s0 , s j ) sk ∈Ad j (s0 ) 1/δ(s0 , sk )

where Ad j (s0 ) denotes the subset of sensors adjacent to s0 in the L DT (s0 ). As it can be observed, the term d j measures the degree of closeness of the sensor s j to the source s0 . This means that s0 favors closer sensors so it transmits its sensed data over short distances and hence saves its energy. Then, the source s0 selects its next forwarder using 1 − lookahead scheme described as follows: • The source s0 sorts the list of Delaunay edges adjacent to it, sorted-list, based on their weights in decreasing order. • The source s0 considers the node, sh , with the highest weight and examines its adjacent neighbors in the L DT (s0 ). If sh has at least one path that originates from one of its adjacent sensors and leads to the checkpoint s p using only positive progress (condition COND), the node sh is selected as the next forwarder. Otherwise, the source s0 repeatedly picks the next node in sorted-list and checks if it satisfies COND. The WLDT protocol is said to be 1 − lookahead because any forwarding sensor uses the information about the adjacent of its adjacent sensors so that it can make appropriate forwarding decision. At the end of this phase, the source sensor identifies its appropriate forwarder and forwards the sensed data to it. By Lemma 13.1, the source s0 prefers to forward its data to its corresponding checkpoint (located at the perimeter of its transmission range) through a series of forwarders, hence favoring short-range over long-range data forwarding. This will enable all the sensors deplete their energy slowly, thus extending the network lifetime. From the function C E D j , it is clear that when the sensor s0 selects a forwarder sk , it takes into consideration three metrics: remaining energy of the sensor sk , position of the sensor sk with respect to the shortest path [s0 , s p ], and the Euclidean distance, δ(s0 , sk ), between the sensors sk and s j ; recall that E tr (s0 , sk ) ∝ δ(s0 , sk ). This means that WLDT attempts to build energy-efficient sub-paths between the source and its checkpoint or between any pair of consecutive checkpoints, which include a series of forwarders linked by short Delaunay edges. Then, the source s0 fills in two fields in the data packet, namely Checkpoint, which contains the checkpoint s p , and Forwarder, which contains the forwarder sh , and forwards the data to sh . When a node sk receives the sensed data, it will examine both fields to check whether it is a checkpoint or a forwarder. If sk is a checkpoint, it will act like the source s0 by running steps 1 and 2. Otherwise, it will run only step 2. The pseudo-code of WLDT is given in Fig. 13.2.

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13 A Planar Checkpoints-Based Approach for Geographic …

13.2.3 Illustrative Example Figure 13.4 shows a path marked by arrows between the source s0 and its checkpoint s p2 . Data is forwarded by s1 , s2 , s3 , and s4 along the corresponding Delaunay edges before it reaches s p2 . In order to identify its checkpoint, s0 constructs L DT (s0 ). Then, it computes its forwarder s1 , sets up fields Checkpoint = s p2 and For war der = s1 , and forwards its data to s1 , which in turn forwards it to s2 . The same forwarding process repeats until s p2 receives the data. When s p2 gets the data, it will act as s0 in order to forward the data to its next checkpoint. The entire process of determining checkpoints and series of forwarders between any pair of consecutive checkpoints repeats until the sink sm receives the data. At each forwarding step, the fields Checkpoint and For war der are updated accordingly.

13.3 Analysis of WLDT In this section, we compute a lower bound on the energy consumption in short-range data forwarding and the corresponding optimum number of forwarders between a source and the sink. This bound helps extend the battery lifetime of the sensors, thus prolonging the operational network lifetime. Specifically, we prove that any checkpoint on the data forwarding path between the source s0 and the sink sm is reachable (Lemma 13.3), and hence the sink itself (Corollary 13.2). Next, we approximate the length of any edge in the transmission graph G = (S, T ) (Lemma 13.4) and deduce the length of any Delaunay edge in L DT (si ) (Lemma 13.5). Then, we analyze the energy consumption in short-range data forwarding (Theorems 13.1 and 13.2) and compare WLDT with BVGF and GPSR, which implement a long-range data forwarding scheme (Theorems 13.3 and 13.5). Finally, we show that checkpoints have a positive impact on data forwarding in terms of energy savings (Theorem 13.5). Lemma 13.3 (Checkpoint Reachability) Any checkpoint between the source s0 and the sink sm is reachable.

Fig. 13.4 Data forwarding path between s0 and s p2

13.3 Analysis of WLDT

409

Proof Let us prove that the checkpoint of the source s0 , say s p , is reachable. Assume that the current forwarder (the sensor that currently holds the sensed data) is si . If the sensor si has a Delaunay edge (si , s p ) in L DT (s0 ), then si could either directly forward the data to s p (long-range) or forward the data to s p through a series of forwarders (short-range) depending on the length of the Delaunay edge (si , s p ) compared to the length of other Delaunay edges adjacent to si . The WLDT protocol uses a 1−lookahead scheme, which helps the sensors choose appropriate forwarders. Therefore, si selects its forwarder s j only if there is at least one path from s j to s p along the Delaunay edges, where the x-coordinate of any forwarder along this path is less than the x-coordinate of s p . This means that the checkpoint s p will be reached using only positive progress from any forwarder between the source s0 and the sink sm . Using the same argument as above, we can prove that every checkpoint between s0 and sm is reachable. ∎ The sink sm is also a checkpoint, and hence should be reachable. Corollary 13.2 below follows directly from Lemma 13.3. Corollary 13.2 (Sensed Data Reachability) Any sensed data originated from the ∎ source s0 is guaranteed to reach the sink sm , assuming no packet loss. The total energy consumption in data forwarding depends on the distance between the sending and receiving nodes. Lemma 13.4 approximates the minimum length of any edge between any pair of sensors in the transmission graph G = (S, T ). Lemma 13.4 (Minimum Edge Length) The minimum edge length in the transmission graph G = (S, T ) can be approximated by ( dmin =

E elec ε

)1/α (13.4)

Proof We need to compute the minimum transmission distance used by a sensor in data transmission. According to the energy consumption model [195] discussed earlier in Chap. 2 in Sect. 2.5, Er x = κ E elec and E t x (d) = κ E elec + κ ε d α , where d is the transmission distance used by a sender. By Assumption 4, Er x represents the minimum energy which could be spent by a given sensor. It is always the case that when a sensor transmits, it always consumes more than when it receives. However, we can approximate the second term in the formula of E tx (d), which corresponds to the energy consumed by the amplifier transmitter component, with the energy consumed by the receiver in data reception. Furthermore, if we consider the values of E elec and ε given earlier in Chap. 2 in Sect. 2.5, we observe that E elec is much higher than ε. Thus, there must be certain value of the distance d such that εd α = E elec . Hence, the minimum value of κ ε d α in E t x can be approximated by Er x . That is, κ ε d α = κ E elec . Thus, the minimum transmission distance that can be used by a sensor in data transmission is equal to dmin = (E elec /ε)1/α

∎

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13 A Planar Checkpoints-Based Approach for Geographic …

It is worth noting that the minimum transmission distance dmin is totally different from the minimum distance between neighboring sensors. While the former is quantified by using energy parameters, the latter is a geometric parameter that can be quantified based on the planar sensor density as one possible parameter. In fact, the minimum distance between any pair of neighboring sensors can be very small if the planar sensor density if very high. Lemma 13.5 computes the length of any Delaunay edge in a localized DT, L DT (si ), based on the result of Lemma 13.4. Lemma 13.5 (Delaunay Edge Length) The length of any Delaunay edge between two sensors s j and sk , denoted by (s j , sk ), in a localized DT, L DT (si ), satisfies dmin ≤ (s j , sk ) ≤ r Proof Let G(si ) be the sub-graph of the transmission graph G = (S, T ) induced by N S(si ) ∪ {sm }, and L DT (si ) the localized DT of si . First, both G(si ) and L DT (si ) have the same vertex set N S(si ) ∪ {sm }. By definition of G(si ), an edge t j,k = (s j , sk ) if and only if δ(s j , sk ) ≤ r. On the one hand, if δ(s j , sk ) = dmin , then the corresponding Voronoi diagram should have two Voronoi cells V C(s j ) and V C(sk ) with a common Voronoi edge. By definition of the DT, there must be a Delaunay edge connecting s j and sk whose length is (s j , sk ) = δ(s j , sk ). Thus, (s j , sk ) = dmin . On the other hand, no edge between two sensors exists unless they are within the transmission range of each other. Therefore, the maximum length of any Delaunay edge cannot exceed r. Hence, (s j , sk ) ≤ r. Both results yield dmin ≤ (s j , sk ) ≤ r

∎

Theorem 13.1 computes the minimum energy consumption and the corresponding optimum number of forwarders required for forwarding one data packet from the source s0 to the sink sm . Theorem 13.1 (Minimum Energy Consumption and Optimum Number of Forwarders) The lower bound on the energy consumption required to forward one sensed data packet from the source s0 to the sink sm along [s0 , sm ] is given by δ(s0 , sm ) dmin α × [(α − 1)E elec + 21/α (α − 1)1/α−1 εdmin ]

E min (s0 , sm ) = 2α−1 κ

(13.5)

and the corresponding optimum number of forwarders is ( m opt (s0 , sm ) =

α−1 2

)1/α

δ(s0 , sm ) dmin

(13.6)

13.3 Analysis of WLDT

411

Proof Assume that there are m forwarders including s0 (s0 , s1 , ..., sm−1 ) between the source s0 and the sink sm . The total energy E tot (s0 , sm ) spent in forwarding one sensed data packet from the source s0 to the sink sm is computed as Σ

E tot (s0 , sm ) = E tot (s0 , s1 ) + E tot (sm ) +

E tot (si )

i=1..m−1

where E tot (s0 , s1 ) = κ E elec +κ ε δ α (s0 , s1 ), E tot (sm ) = κ E elec , and E tot (si , si+1 ) = 2 κ E elec + κ ε δ α (si , si+1 ) (Chap. 2, Sect. 2.5). Thus, Σ

E tot (s0 , sm ) =

2 κ E elec + κ ε δ α (si , si+1 )

i=0..m−1

In general, the distances between all pairs of consecutive forwarders are not equal. Indeed, E tot (s0 , sm ) reaches its minimum when all distances δ(si , si+1 ) are the same. In other words, δ(si , si+1 ) = δ(s0m,sm ) . Thus, E tot (s0 , sm ) ≥ E(m), where Σ

(

δ(s0 , sm ) E(m) = 2 κ E elec + κ ε m i=0..m−1

)α

Thus, E(m) = 2 m κ E elec + κ ε

δ α (s0 , sm ) m α−1 ∗

(13.7)

) = 0. It is easy to check that The function E(m) reaches its minimum when ∂ ∂E(m m∗ ∂ 2 E(m ∗ ) > 0, for all α ≥ 2. That is, E(m) is strictly convex as shown in Figs. 13.5 ∂ 2 m∗ ∂ E(m ∗ ) ∗ and 13.6. Hence, m in ∂m ∗ = 0 corresponds to the minimum of E(m). Thus, the

Fig. 13.5 Energy function E(m) for α = 2

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13 A Planar Checkpoints-Based Approach for Geographic …

Fig. 13.6 Energy function E(m) for α = 3

optimum number of forwarders and checkpoints is given by ∗

(

m opt (s0 , sm ) = m =

α−1 2

)1/α

δ(s0 , sm ) dmin

and hence the minimum energy consumption required in data forwarding between the source s0 and the sink sm is δ(s0 , sm ) dmin α × [(α − 1)E elec + 21/α (α − 1)1/α−1 εdmin ]

E min (s0 , sm ) = 2α−1 κ

Theorem 13.1 shows that we make a good approximation of the transmission distance dmin between any pair of sensors in the transmission graph G = (S, T ). Given that 2 ≤ α ≤ 4, we have ( 0.707 ≤

α−1 2

)1/α ≤ 1.107

Thus, using the above result and Eq. (13.6), we obtain m opt (s0 , sm ) ≈

δ(s0 , sm ) dmin

Thus, we get E min (s0 , sm ) ≈ κ

δ(s0 , sm ) α (2 E elec + ε dmin ) dmin

13.3 Analysis of WLDT

413

This means that the energy consumption reaches its minimum when the distance between any pair of consecutive forwarders is dmin . The numerical values of all the constants are as follows: κ = 216, E elec = 50 nJ/bit, ε f s = 10 pJ/bit/m2 , and εmp = 0.0013 pJ/bit/m2 . Figures 13.5 and 13.6 show the plot of E(m) [see Eq. (13.7)], where dmin = 70.71 m for α = 2 and dmin = 156.68 m for α = 3 [see Eq. (13.4)]. According to Theorem 13.1, the optimum number of forwarders [see Eq. (13.6)] is m opt (s0 , sm ) ≈ 50 for α = 2 and m opt (s0 , sm ) ≈ 23 for α = 3 given that δ(s0 , sm ) = 3500 m. We find that the numerical results are consistent with our theoretical results since 50 × 70.71 m ≈ 3500 m and 23 × 156.68 m ≈ 3500 m. Theorem 13.2 computes the expected minimum and maximum energy required by the WLDT protocol in data forwarding from the source s0 to the sink sm . Theorem 13.2 (Expected Minimum and Maximum Energy Consumption) The expected minimum and maximum energy spent in forwarding one data packet from the source s0 to the sink sm are respectively given by min α min E exp (s0 , sm ) = (2 κ E elec + κ ε dmin ) × (Nexp (s0 , sm ) + 1) max α max E exp (s0 , sm ) = (2 κ E elec + κ ε dmin ) × (Nexp (s0 , sm ) + 1)

(13.8) (13.9)

where • dmin : the expected minimum length of a Delaunay edge. • r: the expected distance between any pair of consecutive checkpoints (r is the transmission range ] sensors). [ of the δ(s ,s ) min 0 m • Nexp (s0 , sm ) = − 1: the expected minimum number of forwarders and dmin checkpoints that[lie on [s0] , sm ].[ ] r max 0 ,sm ) • Nexp × δ(s − 1: the expected maximum number of (s0 , sm ) = P(dmin ,θ0 ) P(r,θ ) forwarders and checkpoints that lie on the data forwarding path between s0 and sm . √ • P(r, θ ) = δ(s0 , sm ) − r 2 + δ 2 (s0 , sm ) − 2 r δ(s0 , sm ) cos θ : the progress made towards the sink sm ./ 2 • P(dmin , θ0 ) = r − dmin + r 2 − 2 d min r cos θ0 : the progress made towards a checkpoint. Proof The expected minimum energy consumption in data forwarding occurs when all forwarders and checkpoints between the source s0 and the sink sm lie on [s0 , sm ]. Because the checkpoints are located at the perimeter of the transmission range of the sensors, the expected distance between any pair of consecutive checkpoints is r. Thus, the[ expected ] minimum number of checkpoints between the source s0 and the sink sm is δ(s0r,sm ) −1. Similarly, the expected minimum number of forwarders between any [ ] r − 1, where the expected distance between pair of consecutive checkpoints is dmin any pair of consecutive forwarders is dmin . Recall that our protocol prefers forwarding

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13 A Planar Checkpoints-Based Approach for Geographic …

the sensed data via a series of forwarders linked by short Delaunay edges in L DT (si ). Thus, the expected minimum total number of forwarders and checkpoints is given by [ min Nexp (s0 , sm )

=

r dmin

]

[ ×

] ] [ δ(s0 , sm ) δ(s0 , sm ) −1 −1= r dmin

α α Because E tot (s0 ) = κ E elec + κ ε dmin , E tot (s j ) = 2κ E elec + κεdmin (s j is a forwarder or a checkpoint) and E tot (sm ) = κ E elec , the expected minimum energy consumption is min α E exp (s0 , sm ) = [E tot (s0 ) + (2 κ E elec + κ ε dmin ) min × Nexp (s0 , sm )] + E tot (sm )

Thus, min α min E exp (s0 , sm ) = (2κ E elec + κεdmin ) × (Nexp (s0 , sm ) + 1)

Now, we consider data forwarding paths that do not coincide with the shortest path [s0 , sm ]. Let P(r, θ ) be the progress made towards the sink at each forwarding action, where θ = ∠s p , s0 , sm is the expected maximum angle between segments [s p , s0 ] and [s √ p , sm ] (Fig. 13.7). By definition, P(r, θ ) = δ(s0 , sm )−δ(s p , sm ), where δ(s p , sm ) = r 2 + δ 2 (s0 , sm ) − 2 r δ(s0 , sm ) cos θ and r is the expected distance between two consecutive [ ] checkpoints. Thus, the expected maximum number of checkpoints is, δ(s0 ,sm ) − 1. Likewise, the expected maximum number of forwarders between P(r,θ ) [ ] r − 1, where the expected distance any pair of consecutive checkpoints is P(dmin ,θ0 ) between any pair of consecutive forwarders is dmin . Therefore, the expected maximum number of forwarders and checkpoints between the source s0 and the sink sm is Fig. 13.7 Progress made towards s p and sm

13.3 Analysis of WLDT

415

min (s , s ) for Fig. 13.8 E exp 0 m α=2

[ max Nexp (s0 , sm ) =

r P(dmin , θ0 )

]

[ ×

] δ(s0 , sm ) −1 P(r, θ )

Thus, the expected maximum energy consumption for forwarding one sensed data packet from the source s0 to the sink sm is given by max α max E exp (s0 , sm ) = (2 κ E elec + κ ε dmin ) × (Nexp (s0 , sm ) + 1)

∎

min Figure 13.8 shows the plot of the function E exp (s0 , sm ) [see Eq. (13.8)] for different values of the distance δ(s0 , sm ) between the source s0 and the sink sm . As it can be seen, the minimum energy consumption grows proportionally to the distance separating max (s0 , sm ) [see the source and the sink. Figure 13.9 shows the plot of the function E exp Eq. (13.9)] for α = 2 and different values of r and θ while keeping δ(s0 , sm ) constant. We observe that when the angle θ increases, the length of the data forwarding path between the source and the sink increases. As a result, more energy consumption will be introduced. In other words, any deviation from the shortest path between the source and the sink will increase the number of forwarders and hence yields additional cost in terms of energy consumption. We also observe that small and large values of r yield more energy consumption. Because the distance between any pair of checkpoints is r , the number of checkpoints is determined by the value of r . Thus, there are an optimum number of checkpoints that need to be used to optimize the deviation from the shortest path and hence leads to max (s0 , sm ). According to Fig. 13.9, for θ = π/7, the optimum a minimum value of E exp value or r is ropt = 400 m, implying that the optimum number of checkpoints is [3500/400] = 9.

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13 A Planar Checkpoints-Based Approach for Geographic …

max (s , s ) for Fig. 13.9 E exp 0 m α=2

13.4 Short-Range Versus Long-Range In this section, we compare the WLDT protocol with the BVGF [424] and GPSR [224] protocols. We show that WLDT outperforms BVGF and GPSR even when BVGF considers its optimal path in terms of network dilation or number of hops between the source s0 and the sink sm . Precisely, we compute the energy gain percentage in data forwarding from a source to the sink in comparison with the BVGF [424] and GPSR [224] protocols, which we slightly update so they account for energy in the selection of a next forwarder. Note that the BVGF and GPSR protocols forward data via long range. Moreover, we prove that the presence of checkpoints in WLDT introduces an energy gain in comparison with a similar protocol, called WLDT w/c (or WLDT without checkpoints), which forwards sensed data via short range but does not use checkpoints.

13.4.1 Energy Gain The BVGF and GPSR protocols follow a greedy forwarding approach, where the data is forwarded to the sensor with the closest distance to the sink, thus enabling longrange data forwarding. The WLDT protocol, however, enables short-range transmission of the data between a source and its checkpoint or between any pair of consecutive checkpoints. Theorem 13.3 computes the energy gain percentage of our protocol compared to the BVGF protocol. Theorem 13.3 (Energy Gain Percentage of WLDT Compared to BVGF): The energy gain percentage of our protocol compared to the BVGF protocol along the shortest

13.4 Short-Range Versus Long-Range

417

path [s0 , sm ] between the source s0 and the sink sm is given by. 1−1/α

E G P(s0 , sm ) = 1 −

3 ε1/α E elec r 2 E elec + ε r α

(13.10)

where r is the radius of the transmission range of the sensors. Proof The total energy consumption required by BVGF [424] to forward a data packet from the source s0 to the sink sm is E BV G F (s0 , sm ) = (2 κ E elec + κ ε r α )

δ(s0 , sm ) r

In fact, The BVGF protocol performs δ(s0r,sm ) forwardings of the data packet along [s0 , sm ], where two consecutive forwarders are separated by a distance equal to r . Likewise, the total energy consumption required by the WLDT protocol is α E W L DT (s0 , sm ) = (2 κ E elec + κ ε dmin )

δ(s0 , sm ) dmin

,sm ) Indeed, the WLDT protocol requires δ(sd0min forwardings of the sensed data along [s0 , sm ]. The energy gain percentage of the WLDT protocol compared to the BVGF protocol is 1−1/α

E G P(s0 , sm ) = 1 −

3 ε1/α E elec r E W L DT (s0 , sm ) =1− E BV G F (s0 , sm ) 2 E elec + ε r α

∎

Theorem 13.4 computes the energy gain percentage of WLDT compared to the GPSR using the non-direct data forwarding path between s0 and sm . Theorem 13.4 (Energy Gain Percentage of WLDT Compared to GPSR) The energy gain percentage of our protocol compared to the GPSR protocol along the nonshortest path between the source s0 and the sink sm is given by E G P(s0 , sm ) = 1 −

r 3 E elec × α 2 E elec + ε r P(dmin , θ0 )

(13.11)

/ 2 where P(dmin , θ0 ) = r − dmin + r 2 − 2 dmin r cos θ0 is the progress made towards the sink along any segment between any pair of consecutive forwarders (checkpoint), θ0 = ∠si+1 , si , sm is the angle between the segments [si , si+1 ] and [si , sm ], si and si+1 are two consecutive forwarders, and r is the radius of the transmission range of sensors.

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13 A Planar Checkpoints-Based Approach for Geographic …

Proof We assume that the distance between any pair of consecutive forwarders in GPSR and checkpoints in WLDT is equal to r . Thus, the total energy consumption of GPSR is given by E G P S R (s0 , sm ) = (2 κ E elec + κ ε r α )

since the sensed data will be forwarded energy consumption of WLDT is

δ(s0 ,sm ) P(r,θ )

δ(s0 , sm ) P(r, θ )

times. On the other hand, the total

α E W L DT (s0 , sm ) = (2 κ E elec + κ ε dmin )

r P(dmin , θ0 )

×

δ(s0 , sm ) P(r, θ )

where P(r, θ ) = δ(s0 , sm ) −

√ r 2 + δ 2 (s0 , sm ) − 2r δ(s0 , sm ) cos θ

and P(dmin , θ0 ) = r −

/

2 dmin + r 2 − 2dmin r cos θ0

In fact, there are δ(s0 , sm )/P(r, θ ) checkpoints between s0 and sm and r/P(dmin , θ0 ) forwarders between any pair of consecutive checkpoints. Thus, E W L DT (s0 , sm ) E G P S R (s0 , sm ) 3 r E elec =1− (2 E elec + ε r α ) P(dmin , θ0 )

E G P(s0 , sm ) = 1 −

∎

Figure 13.10 shows the impact of the radius of the transmission range of sensors, r , and the angle θ0 on E G P(s0 , sm ) [see Eq. (13.11)], assuming α = 2. We observe that the maximum energy gain percentage is obtained when every series of forwarders lie on the segment between their corresponding checkpoints, i.e., θ0 = 0. When θ0 = 0, the BVGF and GPSR protocols are similar in the sense that both of them forward the data to the closest sensors to the sink on the shortest path [s0 , sm ]. We find that the energy gain percentage of WLDT compared to BVGF and GPSR is about 55%. As θ0 increases, the length of the path between two consecutive checkpoints increases and hence more energy will be spent to forward the sensed data towards the next checkpoint. Notice that when r increases, GPSR would consume more energy as E G P S R (s0 , sm ) ∝ r , while WLDT transmits data over short Delaunay edges. Thus, E G P(s0 , sm ) increases with r as shown in Fig. 13.10. Thus, WLDT achieves significant energy savings for higher values of r compared to the GPSR protocol.

13.4 Short-Range Versus Long-Range

419

Fig. 13.10 Impact of r and θ0 on E G P(s0 , sm )

Fig. 13.11 Impact of α on E G P(s0 , sm )

Figure 13.11 shows the impact of r and α on E G P(s0 , sm ), assuming θ0 = 0. As r and α increase, E G P(s0 , sm ) increases and reaches 100% for α = 3. Moreover, for α = 3, the energy gain percentage is more than 100%.

13.4.2 Controlled Short-Range Data Forwarding As mentioned earlier, the presence of checkpoints helps build short data forwarding paths between the source s0 and the sink sm by reducing their deviation from the shortest path [s0 , sm ]. Theorem 13.5 states that the use of checkpoints yields short data forwarding paths and hence significant energy savings.

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13 A Planar Checkpoints-Based Approach for Geographic …

Theorem 13.5 (Energy Gain Percentage of WLDT Due to Checkpoints) The energy gain percentage of our protocol due to the presence of checkpoints in forwarding one sensed data packet from the source s0 to the sink sm is given by E G P(s0 , sm ) = 1 −

r P(dmin , θ + θ0 ) P(r, θ )P(dmin , θ0 )

(13.12)

where P(dmin , θ0 ) = R − P(dmin , θ + θ0 ) = δ(s0 , sm ) −

/

2 dmin + R 2 − 2 R dmin cos θ0

/ 2 δ 2 (s0 , sm ) + dmin − 2 dmin δ(s0 , sm ) cos(θ + θ0 )

P(r, θ ) = δ(s0 , sm ) −

√ δ 2 (s0 , sm ) + r 2 − 2 r δ(s0 , sm ) cos θ

and φ is an infinitesimal angle in radian. Proof In order to measure the energy gain introduced by checkpoints, let us consider a similar protocol to WLDT, which forwards the sensed data through short distances but does not use checkpoints. For the sake of clarity of the notation, let us call this protocol N oC P (No Checkpoint). The progress made towards the sink using the N oC P scheme is given by P(dmin , θ + θ0 ) = δ(s0 , sm ) −

/ 2 δ 2 (s0 , sm ) + dmin − 2 dmin δ(s0 , sm ) cos(θ + θ0 )

where θ = ∠si+1 , si , sm , si and si+1 are two consecutive forwarders, and dmin is the distance between any pair of consecutive forwarders. Figure 13.7 illustrates this scenario. Therefore, the number of forwarders between the source s0 and the sink sm δ(s0 ,sm ) . Thus, the total energy consumption required is given by N F(N oC P) = P(d min ,θ ) by the N oC P protocol for forwarding one sensed data packet from the source s0 to the sink sm is α E N oC P (s0 , sm ) = (2 κ E elec + κ ε dmin ) × N F(N oC P)

On the other hand, our protocol requires that the sensed data be forwarded through checkpoints. These checkpoints lie on or closely to the shortest path [s0 , sm ]. Thus, the progress made towards the sink using the WLDT protocol is given by P(r, θ ) = δ(s0 , sm ) −

√ δ 2 (s0 , sm ) + r 2 − 2 r δ(s0 , sm ) cos θ

13.4 Short-Range Versus Long-Range

421

where φ < θ and r is the distance between any pair of consecutive checkpoints. In order to compare the WLDT protocol to the N oC P protocol, the progress made towards any checkpoint is P(dmin , θ0 ), which is given by P(dmin , θ0 ) = R −

/

2 dmin + R 2 − 2 R dmin cos θ0

Thus, the total number of forwarders and checkpoints is N=

δ(s0 , sm ) r × P(r, θ ) P(dmin , θ0 )

0 ,sm ) checkpoints between the source s0 and the sink sm and In fact, there are δ(s P(r,θ ) r forwarders between any pair of consecutive checkpoints. Therefore, the P(dmin ,θ0 ) total energy consumption for forwarding one sensed data packet from the source s0 to the sink sm is given by

α E W L DT (s0 , sm ) = (2 κ E elec + κ ε dmin )× N

Thus, the energy gain percentage of WLDT compared to N oC P is given by E G P(s0 , sm ) = 1 −

r P(dmin , θ + θ0 ) E W L DT (s0 , sm ) =1− E N oC P (s0 , sm ) P(r, θ )P(dmin , θ0 )

∎

Figures 13.12 and 13.13 show the impact of r and δ(s0 , sm ) on E G P(s0 , sm ) [see Eq. (13.12)], respectively, where θ = π/3 and θ0 = π/7. As it can be seen, regardless of the value of the radius R of the communication range of the sensors, there is always an energy gain percentage between 0.842 and 0.837 when we use checkpoints. This result shows the benefits of using checkpoints, which shorten the data forwarding paths between sources and the sink and hence yields significant energy savings. This Fig. 13.12 Impact of r on E G P(s0 , sm ) δ(s0 , sm )

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13 A Planar Checkpoints-Based Approach for Geographic …

Fig. 13.13 Impact of δ(s0 , sm ) on E G P(s0 , sm )

will extend the network lifetime. However, this gain is inversely proportional to R. We observe that the energy gain percentage using checkpoints reaches its maximum, which is about 0.842, for r = 250 m and its minimum, which is about 0.837, for r = 450 m. We should mention that the performance of the NoCP protocol does not depend on R at all while that of WLDT does. Indeed, any increase in the value of R will reduce the number of checkpoints between the source sensor and the sink. But, it will lengthen the data forwarding path between any pair of consecutive checkpoints. The presence of more checkpoints guarantees short forwarding paths between consecutive checkpoints, and hence less energy consumptions of the sensors.

13.5 Discussion In this section, we briefly discuss sensor reliability issues, which are left for future work. When the sensors fail, WLDT may also fail to forward the data to the sink. 1

What would WLDT do when the checkpoints fail? We need to consider the following two cases: Case 1. A checkpoint goes down before forwarding the sensed data to its next forwarder: Any checkpoint s p is required to send back an acknowledgment message to its previous forwarder sk from which it has received the sensed data packet. In other words, as soon as the checkpoint s p forwards the sensed data towards the sink, it replies back with a message saying that the data has been successfully forwarded to its next forwarder. Therefore, when the forwarder sk does not hear from its checkpoint s p before certain time-out, it will understand

13.6 Related Work

423

that s p went down. In this case, the forwarder sk will have to act like a checkpoint. Thus, the sensor sk will run phase 1 of the WLDT protocol to identify a new checkpoint and forward the sensed data to it through a series of forwarders.

2

Case 2. A checkpoint goes down before having the sensed data forwarded to it: If a forwarder, say sk , learns that its checkpoint s p has disappeared, it will ignore it and find another checkpoint. Then, it forwards the sensed data towards this new checkpoint through a series of forwarders. What if a forwarder fails? If a forwarder disappears before forwarding the sensed data to its next forwarder or checkpoint, the sensed data will be lost. To solve this problem, an acknowledgment with a time-out could be used to check whether a forwarder has successfully forwarded the sensed data towards the sink. Therefore, any forwarder has to send back an acknowledgment message to inform the previous forwarder (or checkpoint) that the sensed data packet has been sent out to the next forwarder. Otherwise, the previous forwarder will have to select another sensor as forwarder so the sink can receive the sensed data.

13.6 Related Work In this section, we discuss a sample of energy-aware data forwarding protocols for wireless sensor networks as well as those protocols using the notion of proxies in data forwarding in wireless networks. Xing et al. [424] proposed a greedy geographic routing protocol, called Bounded Voronoi Greedy Forwarding (BVGF), which allows sensing-covered networks to achieve a lower routing path length compared to other existing protocols. The nodes eligible to act as the next hops are the ones whose Voronoi regions are traversed by the segment line joining the source and the destination. The BVGF protocol chooses as the next hop the neighbor that has the shortest Euclidean distance to the destination among all eligible neighbors. This protocol does not help the sensors deplete their battery power uniformly. Each sensor has, indeed, only one next hop to forward its data to the sink. Thus, any data forwarding path between a source sensor and the sink will always have the same chain of next hops, thus suffering from battery power depletion. Bose and Morin [87] proposed a Voronoi routing for Delaunay triangulations that moves data along the nodes whose Voronoi regions intersect the straight line between the sender and the receiver. The major problem of this algorithm is that it requires the construction of the Voronoi diagram and the Delaunay triangulation of all the wireless nodes. This strategy is very expensive in distributed environments, such as sensor networks. Also, this protocol considers the same path between the source and destination, and hence would deplete the battery power of the sensors quickly. Karp and Kung [224] proposed a Greedy Perimeter Stateless Routing (GPSR) protocol for mobile wireless ad hoc networks. GPSR forwards data packets through long distances and hence consumes much energy. Our protocol, however, forwards sensed data through short Delaunay edges and hence achieves significant energy savings. Li et al. [257] studied compass routing [238],

424

13 A Planar Checkpoints-Based Approach for Geographic …

random compass routing [238], greedy routing [88], and most forwarding routing [368] on different graphs. Lindsey et al. [266] presented a scheme, called PEGASIS (Power-Efficient Gathering in Sensor Information Systems), where each node can receive from and send to close neighbors. Choi and Das [120] proposed an applicative indirect routing (AIR) protocol for ad hoc wireless networks using proxy candidates, which are defined as the neighbors that are shared by the sender and the receiver. A survey of routing and data dissemination algorithms and protocols in wireless sensor networks can be found in [128].

13.7 Conclusion In this chapter, we propose a data forwarding protocol for wireless sensor networks, which helps sensors save their energy by forwarding the sensed data towards the sink over short distances [42]. In addition, preference is always given to the sensors with high remaining energy and whose locations lie on or closely to the shortest path between the source and the sink. The objective of our protocol is to prolong the network lifetime by controlling sensed data transmission. Specifically, the proposed protocol builds a data forwarding path between the source and the sink as a sequence of sub-paths, each of which is composed of a series of forwarders between two endpoints, called checkpoints. These checkpoints are selected based on their remaining energy and closeness to the sink based on the geometric properties of Delaunay triangulation. The data should be forwarded to the sink through these checkpoints. To demonstrate the effectiveness of the WLDT protocol, we present theoretical results supported by extensive numerical results. We compute lower bound on the energy cost of forwarding one sensed data packet from a source to the sink as well as the corresponding optimum number of forwarders between them. We prove that the proposed protocol yields significant energy savings. Also, we compare WLDT with BVGF and GPSR, which enable data forwarding through long distances (i.e., forward data via long range), and find that WLDT achieves an energy gain percentage in the order of 55% for the free space model and close to 100% for the multi-path model. Moreover, we prove that the checkpoints help shorten data forwarding paths between the sources and the sink. Note that we slightly update BVGF and GPSR so they account for energy in the selection of a next forwarder. We prove that the presence of checkpoints in WLDT introduces an energy gain. Indeed, we find that WLDT yields an energy gain percentage compared to a protocol similar to WLDT, which favors short-range data forwarding but does not use checkpoints. These energy gain percentages are in the order of 84.2% for R = 250 m and 83.7% for R = 450 m. These significant energy savings will definitely increase the network lifetime. There is an ongoing debate on short-range versus long-range data forwarding in multi-hop wireless sensor networks. We believe that the primary concern of wireless sensor networks is energy savings to prolong the network lifetime. The WLDT protocol supports the short-range strategy to achieve this goal.

Chapter 14

A Planar Energy-Delay Trade-off Based Approach for Geographic Forwarding on Always-on Sensors

If we want to avert an impending calamity and a state of things which may transform this globe into an inferno, we should push the development of flying machines and wireless transmission of energy without an instant’s delay and with all the power and resources of the nation. Nikola Tesla (1856–1943).

Overview This chapter proposes a data forwarding protocol for wireless sensor networks that trades off between energy and delay. Specifically, this protocol helps achieve minimum energy consumption while ensuring uniform battery power depletion of the sensors and meeting the required delay constraints in the sense that data gathering points must receive the sensed data within a specified time bound. Given that these are conflicting goals, this trade-off is formulated as a multi-objective optimization problem whose solution is an input to our data forwarding scheme.

14.1 Introduction Data forwarding, which is an essential component and critical determinant of the effectiveness of wireless sensor networks, where source sensors (or simply sources) send their sensed data possibly through multi-hop wireless links to the sink. The longevity of multi-hop wireless sensor networks requires that the load of data forwarding be balanced among all the sensors so they deplete their battery power uniformly. Indeed, battery power (or energy) is the most crucial resource in wireless sensor networks, especially when battery recharging or replenishing is impossible. Thus, sensors should use energy-efficient data forwarding protocols that guarantee uniform energy depletion of the sensors. This will keep the sensors operating for longer periods of time, thus extending the network lifetime. However, some sensing applications should satisfy strict source-to-sink delay (or simply delay) constraints in the sense that data gathering points, also known as sinks, must receive the sensed data originated from source sensors within a specified time bound. Therefore, ensuring the longevity of wireless sensor networks becomes a challenging issue, especially © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_14

425

426

14 A Planar Energy-Delay Trade-off Based Approach for Geographic …

for sensing applications with strict delay constraints [289, 290] that must be satisfied at the sink so it can make decisions in a timely fashion regarding the collected data. Hence, appropriate data forwarding protocols should be designed to achieve minimum energy consumption while ensuring uniform battery power depletion of the sensors and meeting the required delay constraints, thus leading to a multi-objective optimization problem. Next, we briefly introduce the major tasks we want to accomplish in this chapter. In addition, we discuss how to achieve each one of them.

14.1.1 Major Tasks Given that minimum energy consumption, minimum delay, and uniform energy depletion are conflicting goals, which have to be dealt with simultaneously, finding a trade-off between them is necessary. Indeed, minimizing energy consumption requires transmitting the sensed data over short distances; recall that the energy (E mt ) spent in data transmission over a physical distance d between a pair of transmitting and receiving points, is proportional to d, i.e., E mt ∝ d α , with 2 ≤ α ≤ 4 being the path-loss exponent. Minimizing delay, however, requires minimizing the number of intermediate forwarders between a source and the sink. This goal could be achieved by maximizing the distance between any pair of consecutive forwarders. Furthermore, a reduced search space of candidate forwarders yields an unbalanced distribution of the data forwarding load among the sensors, thus causing a nonuniform depletion of their available energy. Indeed, the candidate forwarders located in a small search space would suffer heavy depletion of their energy as they will be frequently selected as forwarders. In contrast, a large search space ensures a more balanced data forwarding load among the sensors and hence helps achieve uniform energy depletion of the sensors. Haenggi [187] and Haenggi and Puccinelli [188], however, took an extreme position by arguing that long-hop routing is a very competitive strategy compared to short-hop routing, thus sacrificing the very scarce energy resource of the sensors. Haenggi [187] provided twelve reasons explaining the advantages of long-range over short-range forwarding. We believe that a more balanced approach should be used to account for delay and energy uniformity [28]. In this chapter, we focus on finding the best trade-off between energy and delay in order to satisfy the needs of various sensing applications. Recall that in Chap. 13, we opt for short-range over long-range data forwarding given that energy is the only metric to be considered. However, if we have to account for delay, we should use both of short-range and long-range data forwarding schemes by accomplishing some specific tasks. Next, we present these tasks and provide our corresponding plan of actions. First, we want to meet the needs of a variety of sensing applications. To this end, we slice the communication range of the sensors into concentric circular bands in order to trade-off the above-mentioned three conflicting goals, namely minimum energy consumption, minimum delay, and uniform energy depletion. Then, we classify these

14.2 A Slicing Approach

427

concentric circular bands with a goal to satisfy those sensing applications’ particular requirements of in terms of energy consumption, delay, and energy depletion. Second, we focus on finding the best trade-off among these three conflicting goals. For this purpose, we formulate this trade-off as a multi-objective optimization problem. Then, we solve it using a method called, weighted scale-uniformunit sum (WES) [226]. Based on the solution to this multi-objective optimization problem, we propose a data forwarding protocol, called TED, which accommodates various sensing applications based on their needs as stated earlier. Compared to other methods, such as multi-objective optimization genetic algorithm (MOGA) [160], the WES method is more flexible in that it helps solve an optimization problem with several weighted objective functions. Precisely, each weight reflects the relative importance of the corresponding objective function. Also, these weights are carefully selected in order to account for these three objective functions, which possess different units and orders of magnitude. The optimum solution (or best trade-off of the above conflicting goals) to this multi-objective optimization problem depends on these weights. Third, we want to generalize this study so as to make the proposed TED protocol more practical and more effective. We proceed by relaxing several widely adopted assumptions in the design of wireless sensor networks. Fourth, we want to assess the performance of the proposed protocol TED. To this end, we conduct extensive simulations of TED and compare it with existing ones. We believe that this simulation study is an important step that would help gain more insight into TED before its implementation on a sensor test-bed.

14.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 14.2 introduces the concepts of slicing the communication range of the sensors and proxy forwarders. It also characterizes the uniform energy depletion of the sensors. Section 14.3 presents an approach to trade-off between energy consumption, delay, and energy depletion in data forwarding based on the needs of the sensing applications. Then, it discusses our proposed data forwarding protocol, which trades off energy with delay, and shows how to relax the assumptions stated in Sect. 14.4. Section 14.5 evaluates the performance of our protocol. Section 14.6 reviews sample data forwarding protocols for wireless sensor networks. Section 14.7 concludes the chapter.

14.2 A Slicing Approach This section shows how to slice the communication range of the sensors into concentric circular bands (CCBs). It also characterizes the uniform battery power depletion of the sensors.

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14.2.1 Slicing of Communication Range The idea of slicing the communication range of the sensors stems from the simple fact that any sensor has higher preference to some of its neighbors than to others. This notion of preference becomes apparent in the next-forwarder selection process when a sensor has to decide to which sensor it wishes to forward its data so it reaches the sink while meeting some energy and/or delay constraints. The slicing approach is based on an approximation of a minimum transmission distance dmin in data transmission. As will be seen, the communication range slicing approach helps a sensor classify its neighbors depending on which of the above-mentioned metrics it wishes to optimize. From Lemma 13.4, which is stated earlier in Chap. 13 (Sect. 13.3), the minimum transmission distance that can be used in data transmission can be approximated by / )1/α ( dmin = E elec ε

(14.1)

In order to achieve a better balance between minimum energy consumption, minimum delay, and uniform energy depletion, we ] to slice the commu[ propose / nication range C D(ζi , R) of a sensor si into n ccb = R dmin CCBs, each of which is centered at si and has a width of dmin . As will be seen, a set of CCBs can be divided

Fig. 14.1 Slicing of the communication range of sensors

14.2 A Slicing Approach

429

into three categories (Fig. 14.1). The inner CCBs favor minimizing energy consumption over minimizing delay and uniform energy depletion; the middle CCBs give the same degree of interest to the three performance metrics; the outer CCBs favor minimizing delay and uniform energy depletion over minimizing energy consumption of the sensors.

14.2.2 Selection of Candidate Proxy Forwarders We consider a set of sensors and a single sink sm , all in a planar field. Next, we define the notion of candidate proxy forwarder. Definition 14.1 (Candidate Proxy Forwarder Set): From a neighbor set N S(si ) of a sensor si , we define C P F(si , sm , k, β) as a subset of sensors, called candidate proxy forwarder set of si , which belong to the kth CCB and located within a zone ∎ determined by a wedge with an angle β centered at si (Fig. 14.2). The size of C P F(si , sm , k, β) depends on the values of β and k, where 1 ≤ k ≤ n ccb and 0 < β < π. We design our trade-off protocol between energy and delay in such a way that when a source selects a specific concentric circular band, say k, from which it will designate a sensor as a forwarder, any future proxy forwarder will have to use the same value of k. We use this design decision for the sake of ease of the analysis of our proposed protocol based on this trade-off. Also, this decision would help us find some theoretical results in terms of upper bounds on some specific

Fig. 14.2 Impact of k on the size of the subset C P F(si , sm , k, β)

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14 A Planar Energy-Delay Trade-off Based Approach for Geographic …

metrics. We further relax this decision by allowing the sensors to select their own values of k before forwarding data originated from a source to the sink.

14.2.3 Uniform Energy Depletion Characterization Now, we propose to characterize uniform energy depletion. First, we define this notion with respect to our slicing approach. Definition 14.2 (Uniform Energy Depletion Achievability): We say that uniform energy depletion is achieved if each sensor guarantees that a large number of its neighbors located in a given CCB are equally likely to act as candidate proxy forwarders, and hence participate in forwarding the sensed data to the sink. ∎ As it can be seen from Fig. 14.2, the size of the subset C P F(si , sm , k, β) increases with k. On the one hand, high values of k yield a large subset of sensors to participate in the selection process of a proxy forwarder. This leads to a uniform depletion of the battery power of sensors. On the other hand, when the size of C P F(si , sm , k, β) is small, the same sensors will be frequently selected as proxy forwarders and hence their remaining energy drains faster compared to the other sensors in the network, leading to a non-uniform energy depletion. Thus, uniform energy depletion of sensors can be achieved by maximizing the size of C P F(si , sm , k, β), i.e., maximizing k. In other words, the uniform energy depletion can be characterized by a large size of subset C P F(si , sm , k, β), implying high values of k. It is worth noting that higher values of k mean that the proxy forwarders would be selected from CCBs that are closer to the boundary of the communication range of the sensors. This will introduce an additional computational cost due to the selection of a proxy forwarder from the subset C P F(si , sm , k, β), which is large, compared to choosing a proxy forwarder from a CCB that is far away from the boundary of the communication range of the sensor, and hence a small set of candidate proxy forwarder is considered. Thus, the origin of this “additional computational cost” is due to the size of the set of candidate proxy forwarder. Indeed, any sensor would need to select a candidate proxy forwarder that has the highest remaining energy as it is stated in Sect. 14.3.3.3. This step would need the sensor to sort its neighboring sensors based on their remaining energy. However, the cost of this sorting step depends on the size of the set of candidate proxy forwarders. Clearly, choosing proxy forwarders from CCBs closer to the boundary of the communication ranges of the sensors would require higher sorting cost compared to choosing from CCBs farther from the boundary of the communication ranges of the sensors.

14.3 Trading-off Energy with Delay

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14.3 Trading-off Energy with Delay In this section, we discuss our approach that trades off between minimum energy consumption, minimum delay, and uniform energy depletion in data forwarding in wireless sensor networks. Precisely, we propose a data forwarding protocol that trades off between these goals by slicing the communication range of sensors into concentric circular bands. In particular, we benefit from an approach that we call weighted scale-uniform-unit sum and which will be used by source sensors to solve this multi-objective optimization problem. Our proposed data forwarding protocol, called TED (short for Trade-off Energy with Delay), makes use of a solution to a multi-objective optimization problem to find a “best” trade-off between minimum energy consumption, minimum delay, and uniform sensors’ battery power depletion. We present several numerical results to show the effectiveness of TED. Then, we relax several assumptions to enhance the practicality of TED. We evaluate TED performance through simulations and find that it is near optimal with respect to the energy × delay metric. This simulation study seems to be an essential step to gain more insight into TED before implementing it on a sensor testbed. We also compute upper bounds on these three metrics and the optimal values of k corresponding to their optimum trade-offs.

14.3.1 Simple Analytical Bounds 14.3.1.1

Data Forwarding Along Shortest Paths

Lemma 14.1 computes upper bounds on the expected number of candidate proxy forwarders, energy consumption, and delay. We omit its proof since it is verbatim. Lemma 14.1 (Upper Bounds on Expected Number of Candidate Proxy Forwarders, Energy Consumption, and Delay Along Shortest Path): Let λ be the planar sensor density (i.e., the number of sensors per unit area) and c = qd +td. The expected total number of candidate proxy forwarders, energy consumption, and delay associated with the kth CCB in forwarding a data packet from a source s0 to the sink sm along the shortest path [s0 , sm ] are computed as λβ(2k − 1) δ(s0 , sm )dmin 2k ) ( 2 E elec α−1 α−1 +εk dmin δ(s0 , sm ) E exp (s0 , sm , k) = a k dmin |C P Fexp (s0 , sm , k, β)| =

Dexp (s0 , sm , k) =

c δ(s0 , sm ) k dmin

(14.2) (14.3) (14.4)

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respectively, and their respective upper bounds are given by | | |C P Fexp (s0 , sm , β)| ≤ λ β (2 R − dmin ) δ(s0 , sm )dmin 2R ) ( 2 E elec α−1 δ(s0 , sm ) + εR E exp (s0 , sm ) ≤ a R Dexp (s0 , sm ) ≤

c δ(s0 , sm ) dmin

2 Proof The area of the kth CCB is π (2 k − 1) dmin , where 1 ≤ k ≤ n ccb . Thus, the 2 λ β (2 k−1) dmin Given that the expected total size of the subset C P F(si , sm , k, β) is 2 0 ,sm ) number of forwarding of the sensed data is N f = δ(s , the expected total number k dmin of candidate proxy forwarders between a source s0 and the sink sm is given by N f −1 Σ| | | | |C P Fexp (s0 , sm , k, β)| = |C P Fexp (si , sm , k, β)| i=0

λβ(2k − 1) δ(s0 , sm )dmin = 2k Because |C P Fexp (s0 , sm , k, β)| is concave, its upper bound corresponds to the largest value of k, i.e., k = n ccb . That is, | | |C P Fexp (s0 , sm , β)| ≤ λβ(2 R − dmin )δ(s0 , sm )dmin 2R Also, the expected delay is given by Dexp (s0 , sm , k) =

c δ(s0 , sm ) k dmin

It is easy to check that an upper bound on delay corresponds to k = 1 since Dexp (s0 , sm , k) is inversely proportional to k. Thus, Dexp (s0 , sm ) ≤

c δ(s0 , sm ) dmin

The expected total energy consumption along the shortest path between the source s0 and the sink sm is a(2 E elec + ε(k dmin )α )δ(s0 , sm ) k dmin ) ( 2 E elec α−1 δ(s0 , sm ) + ε k α−1 dmin =a k dmin

E exp (s0 , sm , k) =

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433

To find an upper bound on the energy consumption, we solve ∂ E exp (s0 , sm , k) =0 ∂k As 2 ≤ α ≤ 4, the unique solution to this equation is given by k∗ = ∂2 E

2 (α − 1)1/α

(s ,s ,k ∗ )

exp 0 m ≥ 0, the function E exp (s0 , sm , k) reaches its lower bound at Since ∂ 2 k∗ ∗ k = k . Thus, we have ) ( E elec α−1 δ(s0 , sm ) + ε(2 dmin ) E exp (s0 , sm ) ≥ a dmin

Afterwards, E exp (s0 , sm , k) starts growing proportionally to k. Hence, an upper bound on the energy consumption corresponds to the last CCB, i.e., k = n ccb . Thus, the distance between s0 and its proxy forwarder is equal to dmax = R and hence the minimum number of data forwarding is N f (si , sm ) =

δ(si , sm ) R

Thus, we obtain ) 2E elec α−1 δ(s0 , sm ). + εR E exp (s0 , sm ) ≤ a R (

14.3.1.2

Data Forwarding Along Non-direct Paths

In general, shortest paths between the senders and the sink are not always available given that our data forwarding protocol is based on the remaining energy of sensors. Precisely, the sensors that lie on the shortest path between a sender and the sink could be used as proxy forwarders a few times before their remaining energy get smaller than that of other sensors. The following analysis focuses on non-direct paths between sources and the sink based on the angle θ = ∠si+1 , si , sm (Fig. 14.3). In Lemma 14.2, we assume that the progress that is made toward the sink, which is denoted by ψ(k, θ ), is constant. This assumption is to simplify the analysis so the formulas that we obtain for the expected total energy consumption, delay, and number of candidate proxy forwarders, would look like the ones given in Eqs. (14.5), (14.6), and (14.7). Otherwise, we would obtain three summations that depend on the value of the angle, θ. Note that the maximum value of θ is β/2, i.e., θ could be any value in the interval

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Fig. 14.3 Non-shortest path between s0 and sm

[0, β/2], where the value of β determines the size of the candidate proxy forwarder set as shown in Fig. 14.2. Thus, in practice, the value of θ could be chosen between 0 and β/2. Lemma 14.2 is a generalization of Lemma 14.1. Lemma 14.2 (Upper Bounds on Expected Number of Candidate Proxy Forwarders, Energy Consumption, and Delay Along Non-direct Path): Let ψ(k, θ ) be a constant progress that is made toward the sink sm at each forwarding action, i.e., θ is constant (Fig. 14.3). The expected total energy consumption, delay, and number of candidate proxy forwarders considered when a data packet is disseminated along a non-direct path from a source s0 to the sink sm with respect to the kth CCB are given by E exp (s0 , sm , k, θ ) =

) ( α δ(s0 , sm ) a 2E elec + εk α dmin ψ(k, θ )

(14.5)

cδ(s0 , sm ) ψ(k, θ )

(14.6)

Dexp (s0 , sm , k, θ ) =

2 | | |C P Fexp (s0 , sm , k, θ )| = λβ(2k − 1)dmin δ(s0 , sm ) 2ψ(k, θ )

(14.7)

p

respectively, where 1 ≤ k ≤ n ccb and ψ(k, θ ) = δ(s0 , s1 ) = k dmin cos θ. p

Proof By Pythagorean theorem, we have ψ(k, θ ) = δ(s0 , s1 ) = k dmin cos θ, where p s1 is the orthogonal projection of s1 on [s0 , sm ] and θmax = β/2. Also, we have N f = δ(s0 , sm )/ψ(k, θ ). Thus, we obtain E exp (s0 , sm , k, θ ) =

) ( α δ(s0 , sm ) a 2E elec + εk α dmin ψ(k, θ )

Similarly, the expected delay is given by

14.3 Trading-off Energy with Delay

Dexp (s0 , sm , k, θ ) =

435

cδ(s0 , sm ) ψ(k, θ )

Using the same reasoning as in Lemma 14.1, we obtain (Figs. 14.4, 14.5 and 14.6)

Fig. 14.4 Impact of CCB id (k) and angle θ on the energy consumption for α = 2

Fig. 14.5 Impact of CCB id (k) and angle θ on the energy consumption for α = 3

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Fig. 14.6 Impact of CCB id (k) and angle θ on the energy consumption for α = 4 2 | | |C P Fexp (s0 , sm , k, θ )| = λβ(2k − 1)dmin δ(s0 , sm ) 2ψ(k, θ )

14.3.1.3

Numerical Results

We consider the values of the parameters defined in Table 14.1. Figure 14.4 shows the impact of k, θ, and α on E exp (s0 , sm , k, θ ) (Eq. 14.5). Notice that the energy consumption increases with k and reaches its minimum at k ∗ = 2 as computed earlier. This result can be proved as follows. Let E exp (s0 , sm , k) = a [A1 (k) + A2 (k)] δ(s0 , sm ), α−1 elec and A2 (k) = ε k α−1 dmin . Thus, A2 (k) becomes a dominant where A1 (k) = 2k Edmin factor (i.e., A2 (k) ≥ A1 (k)) for some k that makes E exp (s0 , sm , k) grow proportionally to k. In fact, A2 (k) ≥ A1 (k) ⇒ k α ≥ 2, implying k ≥ 2 as 2 ≤ α ≤ 4. Moreover, more energy consumption would be incurred as we deviate from the shortest path [s0 , sm ] between a source s0 to the sink sm , i.e., as θ increases (Figs. 14.7, 14.8 and 14.9). Table 14.1 Parameters setting

δ(s0 , sm )

R

3500 m

350 m

c 0.001

λ 0.001

α=2

α=3

α=4

dmin = 70.71 m nccb = 5

dmin = 156.68 m nccb = 3

dmin = 44.29 m nccb = 8

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437

Figure 14.5 shows that Dexp (s0 , sm , k, θ ) (Eq. 14.6) decreases with k since fewer proxy forwarders would be needed. Also, α has an impact on the delay. Indeed, Dexp (s0 , sm , k, θ ) is inversely proportional to dmin which in turn depends on α. Similarly, more delay would be incurred as we deviate from [s0 , sm ].

14.3.2 Multi-objective Optimization Approach In this section, we discuss a weighted scale-uniform-unit sum (WES) approach [226] to solve our multi-objective optimization problem for trading off between the abovementioned three metrics, namely minimum energy consumption, minimum delay, and uniform energy depletion.

14.3.2.1

Overview of the WES Approach

Assume we want to minimize a multi-objective function (Figs. 14.10, 14.11 and 14.12) F(x) = (F1 (x), ..., Fn (x))T where Fi (x) is an objective function, for 1 ≤ i ≤ n. WES is a simple approach that introduces a weighting coefficient wi ci for each Fi (x), where wi is a weight selected by a network designer to reflect the relative importance of Fi (x) and ci is a coefficient that not only scales Fi (x) but also helps produce a one-dimensional function F(x). A survey on similar approaches for solving multi-objective optimization problems can be found in [291]. Using WES, a multi-objective optimization problem can be formulated as follows: Σn Minimi ze F(x) = wi ci Fi (x). i=1

Subject to x = X. where ci =

μ , Fimax

Fimax = max{Fi (x): ∀x ∈ X }, wi ≥ 0,

Σn i=1

wi = 1,

{ } μ = max Fimax : 1 ≤ i ≤ n ,

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and X is a set of admissible solutions. It is assumed that Fimax /= 0 : 1 ≤ i ≤ n. 14.3.2.2

Solving the Trade-off Problem Using WES

Notice that the function C P Fexp (s0 , sm , k) (Eq. 14.2) is concave for any θ and in particular for θ = 0. In fact, | | ∂ 2 |C P Fexp (s0 , sm , k)| < 0, for 1 ≤ k ≤ n ccb . ∂2 k Thus, we consider its opposite function given by | | P Fexp (s0 , sm , k) = −|C P Fexp (s0 , sm , k)|

(14.8)

which is a convex function (Fig. 14.6), so we can formulate our unconstrained multi-objective optimization problem using the WES approach. Now, uniform energy depletion requires minimizing P Fexp (s0 , sm , k). Notice that E exp (s0 , sm , k), Dexp (s0 , sm , k), and P Fexp (s0 , sm , k) (Eqs. 14.3, 14.4, and 14.8, respectively) are conflicting objective functions. Minimizing E exp (s0 , sm , k) requires minimizing k, whereas minimizing P Fexp (s0 , sm , k) and Dexp (s0 , sm , k) requires maximizing k. Moreover, E exp (s0 , sm , k), Dexp (s0 , sm , k), and P Fexp (s0 , sm , k) are strictly convex [177] on the interval [1 . . . n ccb ] given that ∂ 2 E exp (s0 , sm , k) > 0, ∂ 2k ∂ 2 Dexp (s0 , sm , k) > 0, ∂ 2k and

∂ 2 P Fexp (s0 , sm , k) > 0, for 1 ≤ k ≤ n ccb . ∂2 k

In addition, because the feasible set {1, 2, ..., n ccb } is a convex set, the WES approach yields correct solutions [109]. Let M(k) = (E exp (s0 , sm , k), Dexp (s0 , sm , k), P Fexp (s0 , sm , k))T be our multiobjective function, which we wish to minimize. E exp (s0 , sm , k) reaches its maximum value ) ( 2E elec α−1 max δ(s0 , sm ) at k = n cbb for α ∈ {2, 4} + εR E exp = a R and

14.3 Trading-off Energy with Delay

( max E exp =a

439

) 2E elec α−1 δ(s0 , sm ) at k = 1 for α3 + εdmin dmin

Also, Dexp (s0 , sm , k) and P Fexp (s0 , sm , k) attain their maximum max Dexp =

c δ(s0 , sm ) dmin

and max P Fexp =

λπ dmin δ(s0 , sm ) , 6

respectively, at k = 1. Using the WES approach, where the weights w1 , w2 , and w3 indicate the relative importance of E exp (s0 , sm , k), Dexp (s0 , sm , k), and P Fexp (s0 , sm , k), respectively, our unconstrained multi-objective optimization problem can be written as follows: (MO) Minimi ze M(k) Subject to 1 ≤ k ≤ n cbb where ⎧ max P F (s , s , k) max D E exp E exp exp 0 m exp (s0 , sm , k) ⎪ ⎪ w1 E exp (s0 , sm , k) + w2 + w3 ⎪ ⎪ max max ⎪ Dexp P Fexp ⎪ ⎪ ⎪ ⎪ ⎪ max = max{E max , D max , P F max } ⎪ i f E exp ⎪ exp exp exp ⎪ ⎪ ⎪ ⎪ max E max P F (s , s , k) ⎪ Dexp D ⎪ exp (s0 , sm , k) exp 0 m exp ⎪ ⎨ w1 + w2 Dexp (s0 , sm , k) + w3 max max E P Fexp exp M(k) = ⎪ ⎪ max = max{E max , D max , P F max } ⎪ ⎪ i f Dexp ⎪ exp exp exp ⎪ ⎪ ⎪ ⎪ max E max D ⎪ P Fexp P F exp (s0 , sm , k) ⎪ exp exp (s0 , sm , k) ⎪ ⎪ w1 + w2 + w3 P Fexp (s0 , sm , k) ⎪ max max ⎪ E D ⎪ exp exp ⎪ ⎪ ⎪ ⎩ max = max{E max , D max , P F max } i f P Fexp exp exp exp

max 0 ≤ w1 , w2 , w3 ≤ 1 with w1 + w2 + w3 = 1, E exp = max{E exp (s0 , sm , k) : max max = 1 ≤ k ≤ n ccb }, Dexp = max{Dexp (s0 , sm , k) : 1 ≤ k ≤ n ccb }, and P Fexp max{P Fexp (s0 , sm , k) : 1 ≤ k ≤ n ccb }. Let us now study the non-linear multi-objective function M(k), which depends max max max , Dexp , and P Fexp of their corresponding objective on the maximum values E exp functions. Hence, we consider the following three cases depending on the values max max max E exp , Dexp , and P Fexp . max max max max Case 1: E exp = max{E exp , Dexp , P Fexp }. Let k1∗ be a solution to (MO).

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14 A Planar Energy-Delay Trade-off Based Approach for Geographic … ∂ M(k1∗ ) ∂ k1∗

= 0 ⇒ k1∗ =

/ α

2 E elec α (α−1) ε dmin

+

max w2 c E exp α max w1 (α−1) a ε dmin Dexp

+

max w3 π λ E exp

α−2 max 6(α−1) w1 a ε dmin P Fexp

∂ 2 M(k ∗ )

Notice that ∂ 2 k ∗1 > 0. Thus, k1∗ corresponds to the minimum of M(k1∗ ). Further1 more, varying the weights w1 , w2 , and w3 from 0 to 1, where w1 + w2 + w3 = 1, generates the corresponding minimum solutions of M(k1∗ ). max max max max Case 2: Dexp = max{E exp , Dexp , P Fexp }. ∗ Let k2 be a solution to (MO). ∂ M(k2∗ ) ∂ k2∗

[ | | α = 0 ⇒ k2∗ = ]

max max w2 c E exp w3 π λ E exp 2 E elec α + w (α − 1) a ε d α D max + α−2 max (α − 1) ε dmin 1 6(α − 1) w1 a ε dmin P Fexp min exp

max max max max Case 3: P Fexp = max{E exp , Dexp , P Fexp }. ∗ Let k3 be a solution to (MO). ∂ M(k3∗ ) ∂ k3∗

[

| max max | w2 c E exp w3 π λ E exp 2 E elec α = 0 ⇒ k3∗ = ] α + w (α − 1) a ε d α D max + α−2 P F max (α − 1) ε dmin 1 6(α − 1) w1 a ε dmin min exp exp

Notice that WES generates a unique, optimum solution (k1∗ = k2∗ = k3∗ ) to the multi-objective optimization problem (MO) regardless of the outcome of the comparmax max max , Dexp , and P Fexp . Although there are other ison between the maximum values E exp methods for solving multi-objective optimization problems, such as multi-objective optimization genetic algorithm (MOGA) [160], WES fits well our purpose. Indeed, the TED protocol is designed for sensing applications with different requirements in terms of energy and delay. That is, real-world applications could give priority to one or more metrics rather than others and vice-versa. The flexibility of WES enables a network designer to emphasize the importance of a specific objective function over the others by choosing a suitable weight vector. Particularly, the different objective functions could be given the same preference by using equal weights. Moreover, we obtain a Pareto-optimum solution [109].

14.3.2.3

Numerical Results

In Fig. 14.7, we plot M(k) with respect to different weight vectors (w1 , w2 , w3 ), where w1 + w2 + w3 = 1, while varying the path-loss exponent 2 ≤ α ≤ 4. As expected, the minimum of M(k) depends on the weights assigned by a network designer to the individual objective functions. When α = 2, the minimum of M(k) is obtained at k = 2 for high values of w1 (w1 ∈ {0.8, 0.7, 0.6}), meaning that the network designer wishes to minimize the energy consumption of sensors, which could be achieved for k = 2 as was proved theoretically. However, when w1 has comparable values to those of w2 and w3 ((w1 , w2 , w3 ) ∈ {(0.5, 0.3, 0.2), (0.4, 0.4, 0.2)}), M(k) reaches its minimum at k = 3 as the network designer wants to achieve some

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441

Fig. 14.7 Impact of CCB id (k) and angle θ on the delay for α = 2

balance between the three objective functions. When α = 3, the best trade-off corresponds to k = 2 for ((w1 , w2 , w3 ) ∈ {(0.8, 0.1, 0.1), (0.7, 0.2, 0.1)}), which favor minimizing energy consumption over minimizing delay and guaranteeing uniform energy depletion, and k = 3 for lower values of w1 . However, when α = 3, M(k) attains its minimum at k = 3 and k = 4, depending on the weights w1 , w2 , and w3 . It is clear that as w1 decreases, the value of k corresponding to the minimum of M(k) increases, meaning that outer CCBs are more preferred (Figs. 14.13, 14.14 and 14.15). Figures 14.16, 14.17, and 14.18 plot M(k) favoring minimizing delay over the other two metrics using appropriate values of w1 , w2 , and w3 . When α = 2, the minimum of M(k) is obtained at the outmost CCB, i.e., k = 5. Similarly, when α = 3, the best trade-off occurs at k = 3. We observe from Figs. 14.16 and 14.17 that the value of k increases as we increase the value of w2 . Recall that the minimum delay is reached for the outer CCBs. For α = 4, the minimum of M(k) is reached at k = 2. The plots of M(k) in Figs. 14.19, 14.20, and 14.21 use the weights w3 > w1 ≥ w2 , which favor uniform battery power depletion over the other two metrics. As it can be observed from Figs. 14.19, 14.20, and 14.21, the minimum of M(k) occurs at k = 5 for α = 2. Also, when α = 3, M(k) attains its minimum at k = 3 (outmost CCB). However, when α = 4, the minimum of M(k) is reached at 2 ≤ k ≤ 3. Figures 14.22, 14.23, and 14.24 show that the best trade-off between the three objective functions with the same weight corresponds to k = 5 for α = 2, k = 3 for α = 3, and k = 2 for α = 4. It is worth noting that minimizing delay and guaranteeing uniform energy depletion are not conflicting metrics since both of them require maximizing k. Thus, the weights w2 and w3 can be viewed as a combined weight against the weight w1 .

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Fig. 14.8 Impact of CCB id (k) and angle θ on the delay for α = 3

Fig. 14.9 Impact of CCB id (k) and angle θ on the delay for α = 4

14.3.3 TED Detailed Description The TED protocol given in Fig. 14.25 has three phases: communication range slicing, concentric circular band selection, and proxy forwarder selection.

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Fig. 14.10 Impact of CCB id, k, and angle θ on P Fexp (s0 , sm , k) for α = 2

Fig. 14.11 Impact of CCB id, k, and angle θ on P Fexp (s0 , sm , k) for α = 3

14.3.3.1

Communication Range Slicing

This phase is run by each sensor only once at the beginning of the sensing task. Each sensor creates a table with n ccb entries, each including a subset of neighbors located

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Fig. 14.12 Impact of CCB id, k, and angle θ on P Fexp (s0 , sm , k) for α = 4

Fig. 14.13 Trade-off between the three metrics with (w1 > w2 , w3 ) for α = 2

in the corresponding CCB. Sensors located at the boundaries of two consecutive CCBs are assigned to the inner one.

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Fig. 14.14 Trade-off between the three metrics with (w1 > w2 , w3 ) for α = 3

Fig. 14.15 Trade-off between the three metrics with (w1 > w2 , w3 ) for α = 4

14.3.3.2

Concentric Circular Band Selection

When a source s0 wishes to disseminate its data to the sink sm , it computes the id of the CCB to be used. This id is denoted by k, where 1 ≤ k ≤ n ccb . The value of k corresponds to the optimum solution of the multi-objective optimization problem

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Fig. 14.16 Trade-off between the three metrics with (w2 > w1 , w3 ) for α = 2

Fig. 14.17 Trade-off between the three metrics with (w2 > w1 , w3 ) for α = 3

defined earlier. It is the responsibility of a source to choose the appropriate weights associated with each of the three metrics. The selection of the values of these weights is guided by the requirements of the underlying sensing application in terms of energy savings and delay. In other words, a source may wish to favor one of the metrics over the others or find a “best” trade-off between them. Once a source s0 has selected the

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Fig. 14.18 Trade-off between the three metrics with (w2 > w1 , w3 ) for α = 4

Fig. 14.19 Trade-off between the three metrics with (w3 > w1 , w2 ) for α = 2

corresponding weights, it solves the multi-objective optimization problem to find the value of k that meets the application needs.

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Fig. 14.20 Trade-off between the three metrics with (w3 > w1 , w2 ) for α = 3

Fig. 14.21 Trade-off between the three metrics with (w3 > w1 , w2 ) for α = 4

14.3.3.3

Proxy Forwarder Selection

A source s0 selects its proxy forwarder s P F1 among the subset C P F(s0 , sm , k), such that

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Fig. 14.22 Trade-off between the three metrics with (w1 = w2 = w3 = 1/3) for α = 2

Fig. 14.23 Trade-off between the three metrics with (w1 = w2 = w3 = 1/3) for α = 3

( ) } { Er em (s P F1 ) = max Er em s j : s j ∈ C P F(s0 , sm , k, θ ) where Er em (s j ) is the remaining energy of sensor s j . In other words, the protocol takes into consideration the remaining energy of sensors and prefers the ones with high

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Fig. 14.24 Trade-off between the three metrics with (w1 = w2 = w3 = 1/3) for α = 4

available energy, thus avoiding sensors with little remaining energy whose selection would possibly lead to network disconnections. When a proxy forwarder receives sensed data, it runs the same protocol to select its next proxy forwarder. This process continues until the sink receives data originated from a source.

14.3.3.4

Is k Fixed for All Proxy Forwarders or Not?

Recall that WES generates a unique, optimum solution (k = k1∗ = k2∗ = k3∗ ) to the multi-objective optimization problem (MO). Furthermore, the value of k depends only on the weighting coefficients w1 , w2 , and w3 , which are associated with each of the three metrics, i.e., minimum energy consumption, minimum delay, and uniform energy depletion. The question that we should address is whether the same value of k computed by a source is used by all proxy forwarders. In order to answer this question, we consider the following two scenarios for computing the value of k.. In the first one, the TED protocol requires all proxy forwarders on the data forwarding path between s0 and sm to use the same value of k to identify their proxy forwarders. In other words, the formulation of the multi-objective optimization problem and its solving is done only once by a source s0 and any proxy forwarder should use the value of k found by s0 . In the second scenario, however, each of the senders (i.e., sources and proxy forwarders) computes its own value of k based on its own values of w1 , w2 , and w3 . The second scenario seems more practical for the following reason. As we get closer to the sink, sensors could get more active in data forwarding toward the sink. In particular, sensors located around the sink act as relay of all data originated

14.3 Trading-off Energy with Delay

Algorithm: TED Begin // Actions executed by a source s 0 Phase 1: Slice the communication range CD(ζ 0 , R) of s 0 1. Slice CD(ζ 0 , R) into nccb CCBs Phase 2: Select an appropriate CCB using k 2. Select the appropriate weights 0 ≤ w1, w2 , w3 ≤ 1 such that w1 + w2 + w3 = 1 to solve the multi-objective optimization problem: (ΜΟ ) Minimize M (k ) Subject to 1 ≤ k ≤ nccb 3. Choose a CCB id, k , which is a solution to (ΜΟ ) 4. If sink s m ∈ NNS ( s 0 ) and sm ∈ k 'th with k ' ≤ k Then Begin 5. Forward the sensed data directly to the sink s m 6. Break; End 7. Else Begin Phase 3: Select a proxy forwarder from k th CCB 8. Identify a subset candidate proxy forwarders CPF ( s 0 , s m , k ) from the k th CCB 8. If this subset is empty Then Begin 9. Randomly pick the closest qth non-empty lower/higher CCB 10. k = q; End 11. Determine the first proxy forwarder sPF1 such that Erem ( sPF1 ) = max{Erem ( s j ) : s j ∈ CPF ( s0 , sm , k ,θ )}

12. Forward the sensed data packet to sPF1 // Actions executed by any proxy forwarder 13. While (sensed data has not reached sm ) Do Begin 14. If s m ∈ NNS ( s PFi ) and sm ∈ k 'th CCB with k ' ≤ k Then Begin 15. Forward the sensed data directly to s m 16. Break; End 17. Else Replace s0 with sPFi and run steps 5-15 End End End Fig. 14.25 The TED data forwarding protocol

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from all sources. Specifically, the data traffic model would have an impact on the energy consumption. In fact, if each sensor is required to transmit data to the sink, sensors nearer the sink would consume more energy than all other sensors in the network. Thus, for these sensors, it would be more important for them to minimize their energy consumption as much as possible by sending/forwarding data to the sink through short distances. This implies that the value of w1 is higher than those of w2 and w3 . Therefore, in our design and simulation of the TED protocol, we consider the second scenario in which the value of k is not the same for a source and all subsequent proxy forwarders when forwarding sensed data toward the sink.

14.4 Relaxation of Several Key Assumptions The assumptions that we make about wireless sensor networks, namely sensor homogeneity (Assumption 14.1), communication disk model (Assumption 14.2), highly dense wireless sensor networks (Assumption 14.3), energy consumption dominated by energy spent in data transmission and reception (Assumption 14.4), and alwayson sensor model (Assumption 14.5), form the basis for most of the existing data forwarding and topology control protocols for wireless sensor networks. In this section, we discuss how to relax these assumptions and assess their impact on the TED protocol in order to promote its applicability to real-world sensing applications.

14.4.1 Relaxing the Sensor Homogeneity Model In real-world scenario, our assumption (Assumption 14.1) may not be true. Indeed, sensors may have different capabilities, particularly in terms of communication ranges. Several studies [137, 433] showed that wireless sensor networks with heterogeneous sensors possessing unequal energy levels and different sensing, processing, and communication capabilities, have increased reliability and lifetime. In the second scenario (i.e., each sensor computes its own value of k), there is no change to TED. For the first scenario (i.e., same value of k for all source and proxy forwarders), however, it may happen that the number of CCBs of some proxy forwarder si , say n ccb (si ), is less than k (i.e., solution to the multi-objective optimization problem (M O)). In this case, the sensor si selects its proxy forwarder from its k'th CCB, where k' = n ccb (si ) < k. Given that k' < k, the data forwarding process will consume less energy than expected. Also, given that the proxy forwarder si selects its last CCB, a minimum delay and a uniform energy depletion will be ensured. Thus, we obtain the best trade-off with respect to the three goals.

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14.4.2 Relaxing the Communication Disk Model The circular radio model (Assumption 14.2) is used to simplify the analysis and may be unrealistic. Indeed, empirical studies have found that the communication range of radios is highly probabilistic and irregular [414, 451, 456]. For tractability, we consider convex communication ranges that are not necessarily disks. Using the notion of largest enclosed disk, which is introduced earlier in Chap. 2 (Sect. 2.2, Definition 2.10), each sensor would be able to run the TED protocol by slicing the largest enclosed disk of its communication range. As in Sect. 14.4.1, there is no change to TED protocol in the second scenario. For the first scenario, however, the number n ccb (si ) of CCBs of a sensor si may be less than the value of k computed by a source. Similarly, sensor si would consider its last CCB from which it would selects its proxy forwarder. Using the same argument as in Sect. 14.4.1, we can see that we obtain the best trade-off between all the three individual objective functions.

14.4.3 Relaxing the Dense Network Model One of the main characteristics of wireless sensor networks is their high density compared to ad hoc wireless networks [10, 11]. Thus, our assumption (Assumption 14.3) is realistic. Now, what would happen if TED is used by a sparse wireless sensor network? In this case, a sensor may not find any sensor located in the selected kth CCB. Although some proxy forwarder can be located in another CCC, say the ith CCB, where i /= k, and hence selected to participate in the forwarding process of the sensed data, it may not be always possible to find the best trade-off required by the underlying sensing application. Thus, the TED protocol would be able to trade-off energy with delay with respect to the specific needs of a sensing application only when the network has some density such that each CCB has some sensors that would be selected to act as proxy forwarders. This is, indeed, a valid assumption since wireless sensor networks are densely deployed in general to provide high-quality monitoring.

14.4.4 Relaxing the Energy Consumption Model The slicing approach of the communication range of sensors depends on the minimum transmission distance dmin . Recall that dmin is computed based on our assumption (Assumption 14.4) which considers only the energy spent in data transmission and reception. Nevertheless, sensors spend energy in computation and sensing. However, both types of energy are negligible compared to the one spent in communication. Also, neither the energy spent in computation nor the one spent in sensing is quantifiable. Hence, if either type of energy has some known formula, the value of dmin

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would be modified accordingly. Precisely, this value would be smaller than the one that is computed earlier in Chap. 13 (Sect. 13.3, Lemma 13.4). Hence, the number of CCBs would be higher, which would have an impact on TED performance.

14.4.5 Relaxing the Always-on Sensors Model Most of the existing geographic routing and data forwarding protocols assume that the radios of all sensors are turned on all the time (Assumption 14.5). In particular, all sensors in the network are awake during the forwarding activity. However, in real-world scenarios, sensors switch between on and off states in order to save their limited energy. It is not even practical to keep a sensor awake all the time while it is in reality active for only some short periods of time. Indeed, keeping sensors always on may cause failures that have severe impact on the performance of the wireless sensor network. Precisely, the network could be partitioned into at least two noncommunicating sub-networks and hence the existence of the whole network may become meaningless. Therefore, it is important to duty-cycle sensors so that they deplete their energy resources uniformly and slowly [401, 425, 447]. Unfortunately, duty-cycling may create a problem for routing the current message to the next hop while it is asleep. The challenge is how to duty-cycle sensors running TED while guaranteeing good routing performance [309, 458, 459]. The handling of highly dynamic wireless sensor networks that experience time-varying connectivity due to sensor duty-cycling is left as one of our future work.

14.5 Simulation Results Our approach to trading-off minimum energy consumption, minimum delay, and uniform energy depletion is unique in several ways as described below, and hence it is impossible to make a fair quantitative comparison between TED and other existing approaches, such as the ones given in [267, 268, 299, 432, 458, 459] and reviewed in Sect. 14.6. First, TED allows a network designer to optimize the above-mentioned three metrics according to the specific needs of the underlying sensing application by using weights that specify the interest in each of these three metrics. Second, TED works at the network layer and does not assume any sleep-wakeup scheduling protocol, where sensors are duty-cycled (i.e., turned on or off ) to save energy. TED applies to always-on and many-to-one wireless sensor networks, where sensors are always on to collect data about a specific phenomenon in a target planar field and send them to a single sink for further analysis and processing. Third, TED does not assume any aggregation of sensed data originated from sources toward the sink. In other words, all data should be received by the sink without undergoing any fusion or aggregation at any intermediate sensor. On the other hand, PEGASIS (PowerEfficient Gathering in Sensor Information Systems) [267, 268] is a simple and elegant

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data aggregation protocol in that it forms a chain among sensors so that each sensor receives from and transmits to a close neighbor and only one designated sensor sends the combined sensed data to the sink in each round. Precisely, all sensors take turns to directly transmit the combined data to the sink. PBBF (Probability-Based Broadcast Forwarding) [299] works at the MAC layer and assume a sleep-wakeup mechanism for all sensors. Indeed, PBBF benefits from the redundancy in broadcast communication and forwards packets using a probability-based approach with two parameters that need to be used by a sleep-wakeup scheduling protocol. The first parameter is a probability p that a sensor rebroadcasts a packet in the current active time while not all its neighbors are guaranteed to be awake to receive the broadcast. The second parameter is a probability q that a sensor would remain awake after the active time when it would be asleep. Furthermore, the goal of PBBF applies to many-to-many wireless sensor networks and ensures that a sensor receives at least one copy of each broadcast packet with high probability, while reducing the latency due to sleeping. Also, PTW (Pipelined Tone Wakeup) [432] helps sensors forward their data to their final destinations based on a wakeup scheme that helps to achieve a balance between energy saving and end-to-end delay. Also, GeRaF assumes a sleepwakeup scheduling protocol, where each sensor keeps forwarding data until at least one of its neighboring sensors is awake and able to receive them.

14.5.1 Simulation Settings In this section, we present the performance results of the TED protocol for the freespace model (α = 2) based on simulation programs written in the C programming language. We consider a planar square deployment field of side length is equal to 1000 m where sensors are randomly and uniformly distributed. Furthermore, we assume that every sensor continuously generates constant bit rate (CBR) data of 1024 bits/s (i.e., 4 data packets of size 256 bits per second). Moreover, the radius of the communication range of the sensors is equal to 250 m. Also, we assume that the total number of deployed sensors in the default case is 1000, which corresponds to a planar sensor density equal to λ = 0.001 sensor per m2 . In addition, we assume that all sensors have the same amount of initial energy that is equal to 1 J.

14.5.2 Impact of Selection Space Size In this experiment, we consider one metric, namely number of communication rounds, and show the impact of the size of the selection space of proxy forwarders on TED. We assume that all sensors use the same value of the angle β, which together with a value of k define the size of the space of the kth CCB from which proxy forwarders are selected. As it can be seen from Fig. 14.26, as the value of β increases, the number of communication rounds increases too. The existence of more communication rounds

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Fig. 14.26 Uniform energy depletion

means that there are more communication paths between sensors and the sink so that data originated from the former would reach the latter. In the absence of a balance of the load of data forwarding among all sensors, some sensors would be more heavily used as proxy forwarders than others, and hence die quickly. Thus, holes (i.e., void regions) in the network would appear and the network would consequently disconnect, thus prohibiting sensed data from reaching the sink. This situation appears if the space from which sensors select their proxy forwarders is small. Hence, the same neighbors would be selected frequently to forward data toward the sink, thus depleting their energy faster than others. Thus, uniform energy would guarantee that most of sensors would evenly participate in data forwarding to the sink and deplete their energy as slowly and uniformly as possible. However, we should mention that the energy sink-hole problem [259, 264, 299, 316, 415, 416] cannot be avoided while using a static sink.

14.5.3 Using the Energy × Delay Metric In this experiment, we consider another metric, namely energy × delay, which was introduced by Lindsey et al. [267] to evaluate the performance of their PEGASIS protocol. Since it is impossible to compare our TED protocol with any existing approach, we consider two instances of TED, namely short-range forwarding (SRF) and long-range forwarding (LRF). Using SRF, sensors forward data over short distances. With LRF, sensors forward data over long distances. SRF performs the

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Fig. 14.27 Impact of planar sensor density

best in terms of energy consumption but yields a highest delay, and hence provides us with lower bound on energy. LRF performs the best in terms of delay but consumes the maximum energy, and hence provides us with lower bound on delay. On the other hand, TED helps find a balance between energy with delay. Thus, we expect that TED performs like SRF when sensors care more about energy consumption (w1 ⋙ w2 , w3 ), and performs like LRF when sensors care more about delay and uniform battery depletion (w2 , w3 ⋙ w1 ). Given that our goal is to find a trade-off between energy and delay, we consider the energy × delay metric to compare TED against SRF and LRF. Figure 14.27 shows that LRF has the lowest energy × delay whereas SRF yields the highest energy × delay. As expected, Fig. 14.27 shows that energy × delay for the three schemes increases with the planar sensor density. Figure 14.28 shows that energy × delay depends on the location of the sink. It is worth mentioning that the optimum location of the sink in terms of energy-efficient data gathering corresponds to the center of the planar deployment field [281]. Figure 14.28 shows that the best performance of the three protocols is obtained only when the sink is located at the center of the planar deployment field. Thus, the obtained results are conforming to the claim given in [281]. Figure 14.29 shows the impact of sensed data packet size on energy × delay. Recall that the energy a sensor spends in data transmission (E mt ) depends on the size of the sensed data forwarded to the sink. As it can be observed, TED has energy × delay close to the one produced by LRF. Given the orders of magnitude of energy and delay, we can claim that energy ×delay reaches its minimum for the smallest value of delay. Our intuition matches the results of the analysis of the function E exp (s0 , sm , k) × Dexp (s0 , sm , k). Indeed, it is easy to

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Fig. 14.28 Impact of location of the sink

Fig. 14.29 Impact of data packet size

check that this function reaches its minimum at k = 5, which corresponds to LRF as it always selects proxy forwarders from the last CCB. Thus, LRF is optimal with respect to the energy × delay metric, and hence TED is near optimal. Although LRF has the best energy × delay, it is not suitable for energy-constrained wireless

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sensor networks. SRF, on the other hand, is not appropriate for time-critical sensing applications. As it can be seen in Fig. 14.30, SRF yields better network lifetime than TED. However, as shown in Fig. 14.31, we find that TED outperforms LRF in terms of network lifetime. This is due to the nature of forwarding schemes of SRF and LRF.

Fig. 14.30 TED compared to SRF

Fig. 14.31 TED compared to LRF

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Compared to SRF and LRF, the TED protocol is more balanced in terms of energy and delay, and hence is a best candidate for wireless sensor networks, where both energy savings and delay are (equally) important. Also, TED offers more flexibility to wireless sensor network designers to meet the specific needs of sensing applications in terms of energy and delay.

14.5.4 Impact of Variability of k In this experiment, we focus on the variability of k whose value of k may not be the same for all the proxy forwarders. We consider two cases: In the first case, the source sensor computes the value of k and each proxy forwarder involved in the forwarding data toward the sink has to use the same value of k. In the second case, computing the value of k is done locally. In fact, each proxy forwarder computes its proper value of k to identify the CCB from which it would select its next proxy forwarder. Figure 14.32 shows the simulation results for both cases. Although both results are very similar, we find that the variability of k yields better performance. It is worth mentioning that the sensors nearer the static sink represent hot-spot traffic points in the sense that they are heavily used in the process of data forwarding to the sink. This situation creates a problem known as the energy sink-hole problem, which could possibly isolate the sink, thus disconnecting the network. These sensors care more about minimizing their energy consumption by forwarding the sensed data to the sink over short distances although it may be within their communication range.

Fig. 14.32 Impact of variability of k

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Thus, their selection scheme enables them to extend their individual lifetime, and consequently, the network lifetime.

14.5.5 Impact of Sensor Heterogeneity In this experiment, we evaluate the performance of our TED protocol in the presence of heterogeneous sensors. We assume that the sensors may not necessarily have the same amount of initial energy and the same radii of their communication ranges. The initial energy of the sensors is randomly picked from the interval [1 J, 2 J]. Similarly, the radii of their communication ranges are randomly selected from the interval [250 m, 350 m]. Given that the sensors may not have the same radii of their communication range, they would not have the same number of CCBs when they slice their communication ranges. Thus, a value of k that is initially computed by the source sensor may exceed the total number of CCBs a proxy forwarder would have in its communication range. In that case, a proxy forwarder would simply choose the last CCB from which it would select its next proxy forwarder. Therefore, the value of k would not be the same for all the proxy forwarders, i.e., k is dynamic. It is as if each sensor would have its proper value of k. Figure 14.33 shows the results for homogeneous sensors and heterogeneous sensors network setup. Not surprisingly, we find that the performance of TED using heterogeneous sensors outperforms that of the network with homogeneous sensors. Here, we show that sensor heterogeneity helps extend the lifetime of the individual sensors, thus prolonging the network lifetime.

Fig. 14.33 Impact of sensor heterogeneity

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Indeed, Yarvis et al. [433] presented several analytical, simulations, and real testbed results showing the potential benefit and impact of energy and link heterogeneity on sensor nets, where all the sensors report their sensed data to a single static sink.

14.6 Related Work This section reviews a sample of data forwarding protocols for wireless sensor networks that trade off energy with other metrics, such as delay and robustness. Yang and Vaidya [432] proposed a wakeup scheme, called Pipelined Tone Wakeup (PTW), which achieves a balance between energy saving and end-to-end delay. The PTW scheme is based on an asynchronous wakeup pipeline that overlaps the wakeup procedures with the packet transmissions. It uses wakeup tones which allow a large value of duty cycle ratio without causing a large wakeup delay at each hop. Miller et al. [299] studied the trade-off between energy, latency and reliability. They presented a Probability-Based Broadcast Forwarding (PBBF) scheme which minimizes energy usage and optimizes latency and reliability. Zorzi and Rao [458] proposed a transmission technique for wireless sensor networks called geographic random forwarding (GeRaF), where relay nodes are decided only after the transmission has started. The packet duplication problem is solved using a scheme for contention among receivers. Zorzi and Rao [459] also gave a detailed description of a MAC protocol and an evaluation of the latency and energy performance. Bandyppadhyay and Coyle [75] proposed a transmission scheduling scheme using a collision-free protocol for gathering sensor data. They also studied many trade-offs between energy usage, sensor density, temporal and spatial sampling rates. Sohrabi et al. [357] proposed a sequential assignment routing (SAR) protocol which is used by sensors to select a path among multiple ones to the sink node. The SAR protocol selects a path based on the energy resources and the priority level of a packet. Lindsey et al. [266] presented a scheme, called PEGASIS (Power-Efficient Gathering in Sensor Information Systems), where each node can receive from and send to close neighbors. The data gathered by nodes in each round has to be collected and transmitted to the base station by only one designated node in order to reduce energy consumption and extend the network lifetime. PEGASIS, which outperforms LEACH protocol [195], considered energy × delay as the optimization metric per round of data gathering in wireless sensor networks [267, 268]. Krishnamachari et al. [239] showed by well-selected examples that when robustness and energy efficiency are the main concern, single-path routing outperforms multipath routing under the assumption of perfectly reliable source and destination sensors. Choi and Das [119] proposed a data gathering scheme which trades off coverage and data reporting latency while

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enhancing energy conservation. Hynh and Hong [212] proposed an energy*delayaware routing protocol for wireless sensor networks using cluster-based and chainbased approaches. Each communication round consists of a cluster and chain formation phase and a data transmission phase. The construction of the network configuration, which is defined by the cluster of sensors and their cluster-heads, is accomplished by a base station. Soltan et al. [359] proposed a mobility-aware multi-hop routing scheme for a hierarchical wireless sensor network using mobile relays with a goal to optimize the network lifetime, delay, and local storage size. They proposed to solve an optimization problem to maximize the network lifetime under local storage, delay, and maintenance cost constraint.

14.7 Conclusion In this chapter, we study a communication range slicing-based approach to tradeoff between conflicting objectives of sensing applications. Our approach aims to slice the communication range of the sensors into concentric circular bands and classify them with a goal to satisfy specific requirements of sensing application in terms of energy consumption, delay, and energy depletion. In this chapter, we formulate the trade-off between these three conflicting goals as a multi-objective optimization problem which is solved using a weighted scale-uniform-unit sum (WES) approach [226]. Specifically, we propose a data forwarding protocol for wireless sensor networks, which exploits a solution to a multi-objective optimization problem to find an optimum trade-off between three conflicting goals, namely minimum energy consumption, minimum delay, and uniform energy depletion. To account for the uniform energy depletion of the sensors, we propose an approach to characterize it based on the size of the subset of candidate forwarders of the data toward the sink. Though there are other efficient methods, such as multi-objective optimization genetic algorithm (MOGA) [160], the WES approach offers more flexibility to a network designer to find solutions that satisfy some associated priorities of the objective functions. Theoretical results show that an optimum trade-off between the three abovementioned goals exists. Moreover, it depends on the weighting coefficients, which are introduced to reflect the relative importance of the individual objective functions and to address the problem of their different units and order of magnitude. For tractability, the communication ranges of the sensors are supposed to be circular and have the same radius. Then, we discuss ways of relaxing these assumptions to enhance the practicality of TED. Also, we evaluate the performance of TED through extensive simulations and compare it with existing ones. We find that the performance of TED is near optimal with respect to the energy × delay metric. This simulation study seems to be an essential step to gain more insight into TED before implementing it on a sensor testbed. To the best of our knowledge, although the design of energy-efficient data forwarding protocols for wireless sensor

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networks has received much attention, there is no previous work that jointly considered minimum energy consumption, minimum delay, and uniform energy depletion to find the best trade-off between them.

Chapter 15

A Planar Approach for Solving the Energy Sink-Hole Problem with Always-on Sensors

“To gain your energy from the right sources means you gain more energy from less; reducing the retarding factors means you have less friction that wastes your energy; focusing your motive power towards the direction of the collective human movement means you use others’ energy to boost your energy. Gain more energy, waste less energy, spend the energy so next time you’ll need less to achieve more. Increase human mass, reduce retarding force, and increase the force accelerating the human mass. Follow this process with reason. Gain more, waste less, spend efficiently, learn.” Nikola Tesla (1856–1943).

Overview This chapter investigates the energy sink-hole problem, which is inherent to static always-on wireless sensor networks, where the sensors located around a static sink act as relays to the sink on behalf of all other sensors, thus suffering from severe energy depletion. Indeed, some analysis and simulation, which are applied to static, uniformly distributed, always-on wireless sensor networks with constant data reporting to the sink, show that uniform energy depletion of all the sensors cannot be achieved when the sensors use their nominal communication range in their data transmission to the sink. This chapter presents a theoretical analysis of the energy sink-hole problem. It shows that this problem can be solved provided that the sensors adjust their communication ranges. However, this solution imposes a severe restriction on the size of a planar deployment field. To overcome this limitation, we propose a sensor deployment strategy based on energy heterogeneity with a goal that all the sensors deplete their energy at the same time. Also, it proposes an energy-efficient protocol to solve the energy sink-hole problem using sensor mobility and our newly introduced concept, called energy-aware Voronoi diagram. Our simulations show that EVEN outperforms existing greedy geographical data forwarding protocols and has similar performance of an existing data collection protocol using a joint mobility and routing strategy. In general, the EVEN protocol helps extend significantly the network lifetime.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_15

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15.1 Introduction One way to extend the lifetime of a wireless sensor network is through load balancing so that all sensors deplete their energy slowly and uniformly during their monitoring activity. Particularly, the behavior of the sink has an impact on the network lifetime. Indeed, static always-on wireless sensor network (i.e., the radios of the sensors are turned on all the time) are much affected by the energy sink-hole problem, where sensors located around a sink suffer from severe battery power depletion problem. Indeed, the sensors close to the sink act as relays to the sink on behalf of all other sensors, and hence deplete their battery power more quickly, thus leading to possible disconnection of the network and disruption of the sensed data from reaching the sink. It was proved that it is impossible to guarantee uniform energy depletion of all the sensors in static, uniformly distributed, always-on wireless sensor network with constant data reporting to the sink when sensors use their maximum communication range to transmit sensed data to the sink [259, 260, 264, 316, 415, 416]. Next, we describe the major tasks we want to accomplish in this chapter. Furthermore, we give a brief discussion on how to achieve each one of them.

15.1.1 Major Tasks The deployment of static sink and sensors in real-world applications is very common, and hence efficient solutions should be provided to tackle the energy sink-hole problem, which is inherent to static wireless sensor networks. We believe that the network lifetime depends on three key design metrics, namely type of data forwarding (long range vs. short range), type of sensors (homogeneous vs. heterogeneous), and type of sink (static vs. mobile). This motivates us to account for these three design metrics in order to solve the energy sink-hole problem. First, we consider the transmission distance that distinguishes between short-range and long-range forwarding. Second, we consider sensor heterogeneity when deploying sensors for its ability to improve the reliability of the network and extend its lifetime [137, 433]. Third, when sensors have the same initial energy, we consider sink mobility for its ability to evenly distribute the data forwarding load among all the sensors to extend the network lifetime [397]. In this chapter, we investigate the energy sink-hole problem in static always-on wireless sensor networks. Our proposed solutions to this problem are based on the above-mentioned metrics [49] and are guided by the accomplishment of well-defined tasks. Next, we briefly describe these tasks and present our corresponding plan of actions. First, we want to find out how a uniform energy depletion of all sensors in the network can be achieved and to what extent it can be guaranteed so as to avoid the energy sink-hole problem in wireless sensor networks. To this end, we consider static, uniformly distributed always-on wireless sensor networks with constant data

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reporting to a static sink. We prove that the energy sink-hole problem can be solved if the sensors are capable of adjusting their communication ranges during their data transmission or forwarding to the sink. More specifically, we divide a planar circular field of interest into concentric circular bands whose width is equal to the radius of the nominal communication range of sensors. Then, we focus on solving two problems. In the first one, called perfect uniform energy depletion, we check whether the network has the ability to ensure uniform energy depletion of the sensors in all the bands. In the second problem, called partial uniform energy depletion, we determine a subset of bands, where the sensors can deplete their energy uniformly. However, our study shows that these solutions have some limitations. Precisely, they impose restriction on the size of the planar field of interest with regard to the number of its bands. To overcome these shortcomings, we benefit from the deployment of heterogeneous sensors with respect to their energy reserves. Our goal is to guarantee that all the sensors in all the bands can deplete their energy uniformly. We conduct extensive simulations to check whether this type of sensor deployment approach would be able to help the sensors in all the bands deplete their initial energy at the same time. Second, we want to tackle the energy sink-hole problem in homogeneous wireless sensor networks and see how it can be addressed. For this purpose, we propose to use an original concept of Voronoi diagram, called energy aware Voronoi diagram, for uniformly distributed always-on wireless sensor networks. Then, we propose a protocol, called energy aware Voronoi diagram-based data forwarding (EVEN), for geographic data forwarding. A sensor computes locally this type of Voronoi diagram based on the virtual locations of its neighbors. This protocol is based on a mobile sink using the random waypoint mobility model [218]. Notice that this mobility helps the sink change its neighbors over time, which allows different subsets of sensors to act as relays to the sink. In addition, this concept of energy aware Voronoi diagram allows a sensor to select its neighbors, which have the highest remaining energy, to serve as data forwarders. While the philosophy of Haenggi and Puccinelli [188] supports the long-hop routing over the short-hop routing, we believe that there is a need for a more balanced approach in order to ensure a fair load distribution among all the sensors. Third, we want to assess the performance of the proposed protocol (EVEN) using a combination of the energy-aware Voronoi diagram and sink mobility. For that, we conduct extensive simulations of EVEN to check whether EVEN can help extend the network lifetime. In particular, we compare EVEN with a similar data forwarding protocol using static sink. Moreover, we compare EVEN with an existing data collection protocol, which uses joint mobility and routing [281].

15.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 15.2 analyzes the energy sink-hole problem and proposes a restricted solution [49]. Section 15.3 exploits

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energy heterogeneity to solve the energy sink-hole problem [49]. Section 15.4 makes use of the sink mobility and our new proposed concept of energy aware Voronoi diagram [49] to solve the energy sink-hole problem for homogeneous wireless sensor networks. Section 15.5 reviews related work. Finally, Sect. 15.6 concludes the chapter.

15.2 Energy Sink-Hole Problem Analysis We consider always-on wireless sensor networks, where the sensors constantly report their sensed data to a single static sink. Hence, the sensor cannot be turned off while monitoring a physical phenomenon. We assume that the sensors are static and uniformly distributed in a planar circular field of radius R with planar sensor density λ (Fig. 15.1). First, we discuss a base protocol, where the network has a static sink and uses a short-path routing protocol [165]. The static sink is supposed to be located at its optimum position in terms of energy efficient data gathering, which corresponds to the center of the planar field [281]. We show that the sensors around the sink have higher energy consumption than all other sensors.

15.2.1 Base Protocol Average Energy Consumption The model that we use to compute the maximum average energy consumption of sensors is similar to the model in [164]. The average energy consumption of a node located in an area of size A2 that forwards traffic for other nodes located in another area of size A1 is proportional to A1 + A2 / A2 . Our model focuses on the nodes within a distance L ≤ σ ≤ R from the sink, where R is the radius of the nominal communication range of sensors and L 1 bands such that R = k R.

15.2.2 Nominal Communication Range–Based Data Forwarding Under the above-mentioned planar field slicing method, the energy consumption rate of sensors depends on which band they belong to as well as the communication range used in reporting their data to the sink. We assume that sensors are homogeneous and use their nominal communication range. Lemma 15.1 proves that sensors located in the kth (i.e., outmost) band consume less energy than all other sensors. Lemma 15.1 (Energy Consumption of Outmost Band) Assume a uniform sensor distribution with planar density λ and sensors constantly transmit their data to the sink using their nominal communication range of radius R. Sensors located in the kth band have longer lifetime than all sensors in the network. Proof Any sensor located in the kth band has only to report its own data to the sink. Indeed, given that the communication range of sensors coincides with the width of the bands, no one of the sensors in the kth band can participate in forwarding data to the sink on behalf of others. Using the energy model presented earlier in Chap. 2 in Sect. 2.5, the energy consumption rate per sensor in the kth band is computed as

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E R(k) = E Rt x (R) = (ε R α + E elec )b Let E init be the initial energy of a given sensor. The average lifetime of a given init sensor in the kth band is equal to EER(k) . To the contrary, all sensors located in any other band forward data on behalf of others. Precisely, a sensor in the ith band forwards data originated from sensors in the jth band, where i < j ≤ k. The number of sensors in the ith band is equal to ) ( N (i) = λ π i 2 − (i − 1)2 R 2 = λ π (2i − 1)R 2 Let si be an arbitrary sensor located in the ith band. Thus, under uniform sensor distribution and constant data reporting, the average number of messages forwarded by si per unit of time is given by ( ) λ π R 2 k 2 − (i − 1)2 N (l) k 2 − (i − 1)2 = = N (i ) λ π R 2 (2 i − 1) 2i − 1

Σk M(i ) =

l=i

Hence, the average energy consumption rate of a sensor in the ith band is given by E R(i ) = (M(i ) − 1)(E Rtx (R) + E Rrx ) + E Rtx (R) Using the energy consumption model, which is given in Chap. 2 in Sect. 2.5, and the value of M(i ), the above equation leads to k 2 − (i − 1)2 E R(i ) = b ε Rα + 2i − 1

(

) 2k 2 − 2i 2 + 1 b E elec 2i − 1

(15.1)

init Hence, the average lifetime of the sensors in the ith band is EER(i) , where i < k. E init E init It is easy to check that E R(i ) < E R(k) , meaning that the lifetime of the sensors in the kth band is longer than that of the sensors in all other bands. ∎

Lemma 15.2, which follows from Lemma 15.1, states that uniform energy depletion cannot be guaranteed under the assumption of constant data reporting by sensors using their nominal transmission range. Thus, all the sensors do not have same lifetime. Lemma 15.2 (Energy Consumption of Sensors in Different Bands) Assume a uniform sensor distribution with planar sensor density λ. Also, suppose that sensors are always on and constantly report their sensed data to the sink using their nominal communication range of radius R. It is impossible for a given pair of sensors in two different bands to have the same energy consumption rate in their lifetime. ∎ Next, we investigate the case where sensors may use their adjustable communication range to transmit or forward data to the sink.

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15.2.3 Adjustable Communication Range-Based Data Forwarding Given that the sensor distribution is uniform and that sensors constantly report their data to the sink, the transmission distance remains the key parameter to check whether it is possible to guarantee uniform energy depletion of sensors. We consider the following two problems.

15.2.3.1

Perfect Uniform Energy Depletion

In the case of perfect uniform energy depletion, sensors in all bands consume energy at the same rate. Precisely, we want to compute the number k of bands of a planar field such that the sensors located in the first and kth bands have the same lifetime. Let N = λ π R2 be the total number of sensors forming the network. Now, consider two arbitrary sensors s1 and sk that belong to the first and kth bands, respectively. Given that sensors are uniformly distributed in the planar field, the average number of sensors in the first and the remaining (k −1) bands, denoted by N (1) and N (2 → k), respectively, are equal to N (1) = λ π R 2 ) ( ( ) N (2 → k) = λ π R2 − R 2 = λ π k 2 − 1 R 2 Moreover, the sensors in the first band and, in particular, sensor s1 acts as forwarder of the data coming from all other bands. Thus, the average number of messages forwarded by s1 (including its own message) per unit of time is M(1) =

λ π R2 N = = k2 N (1) λ π R2

Out of these k 2 messages, (k 2 − 1) were sent by sensors located in the (k − 1) remaining bands. To simplify the analysis, we assume that the sensor s1 uses the transmission distance d1 . Hence, the average rate of energy consumption of s1 is given by ⎛

2 kΣ −1

E R(d1 , 1) = ⎝

⎞ E Rt x (d1 ) + E Rr x ⎠ + E Rt x (d1 )

l=1

( ) ( ) = k 2 ε d1α + 2 k 2 − 1 E elec b

(15.2)

On the other hand, the average rate of energy consumption of the sensor sk is computed as

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) ( E R(Dk , k) = E Rt x (Dk ) = ε Dkα + E elec b where Dk is the transmission distance used by sk . The metric of energy depletion uniformity requires that the average rate of energy consumption of all sensors is the same. Hence, equating the above two equations, i.e., E R(d1 , 1) = E R(Dk , k), yield Dkα − k 2 d1α −

) 2E elec ( 2 k −1 =0 ε

(15.3)

where 0 < Dk ≤ R, 0 < d1 ≤ R, 2 ≤ α ≤ 4, and k > 1. Notice that under uniform sensor distribution and constant data reporting, it is possible to guarantee uniform energy consumption of sensors located in the first and kth bands if the transmission distances d and Dk satisfy Eq. (15.3). Thus, given that sensors in the kth band do not forward data on behalf of others, their transmission distance Dk should be larger than that used by sensors in the lower (k − 1) bands. Achieving the goal of energy depletion uniformity requires that sensors in the lower (k − 1) bands adjust their transmission distances according to Dk . Particularly, the transmission distance d1 for sensors in the first band given by d1 =

1

(

k 2/α

Dkα

) ) 1/α 2E elec ( 2 k −1 − ε

(15.4)

Lemma 15.3 [50, 58, 59] approximates the minimum transmission distance dmin a given sensor can use for transmitting its own data or forwarding data on behalf of others to the sink. Lemma 15.3 (Minimum Transmission Distance) The minimum transmission distance used by a sensor when it sends/forwards data to the sink, can be approximated by ( dmin =

E elec ε

) 1/α ∎

From Lemma 15.3, it follows that a physical solution to Eq. (15.3) exists if and only if 1 k 2/α

(

Dkα −

( ) ) ) 1/α E elec 1/α 2E elec ( 2 k −1 ≥ ε ε

The above inequality implies that guaranteeing uniform energy depletion of all sensors is possible if and only if the number k of bands of the planar field satisfies / k≤

ε 2 Dkα + 3 E elec 3

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Table 15.1 Values of E elec and ε depending on α.

E elec

ε

α=2

50n J/bit

10 p J/bit/m 2

2 r s . • All mobile proxy sinks are assumed to be heterogeneous. Moreover, they are more powerful than those static and mobile sensors with respect to all of their capabilities, including sensing range, communication range, and battery power.

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Indeed, these mobile proxy sinks will be responsible to ensure k-coverage of any region of interest in the planar field. They are expected to communicate with the static and mobile sensors to k-cover the region. Also, they are responsible for delivering the gathered sensed data to the static sink, thus, acting as data Mules [347]. It is well known that compared to processing and sensing, communication is the major source of energy consumption. This is the reason why mobile proxy sinks are more powerful than all the static and mobile sensors. • All mobile proxy sinks as well as all static and mobile sensors are assumed to be fully cooperative. They are willing to k-cover any region of interest when they are requested to participate in accomplishing any mission. • Sensor mobility is purposeful. That is, all mobile sensors move toward a region of interest to be k-covered only when they are instructed by special sensors, called leader mobile proxy sinks, which will be defined later.

17.3.3 Achieving Mobile k-Coverage In this section, we propose a distributed protocol to k-cover a region of interest while reducing the energy consumption due to sensor communication and mobility. In [60], we suggest a pseudo-random deployment approach for densely deployed sensors, which are distributed in different layers in a planar circular field with respect to the sink based on the strength of their sensing range, communication range, and initial energy. We propose centralized and distributed k-coverage protocols. This multi-tier sensor deployment architecture suffers from a few shortcomings. First, all the sensors including the sink, are static, thus suffering from the energy sink-hole problem [181, 316], where all the sensors located around the sink are heavily used in forwarding data to the sink. This makes them deplete their energy very quickly as they act as relays to the sink on behalf of all other sensors in the network. Second, this type of pseudo-random deployment strategy is restrictive in the sense that the sensors should be placed in those layers with respect to their resources in terms of their initial energy supplies and sensing and communication capabilities. Third, we assume a densely deployed wireless sensor networks to k-cover a planar field. Thus, our proposed pseudo-random deployment approach [60] may not be suitable for sparse wireless sensor networks. In particular, the k-coverage requirement of a sensing application may not be satisfied due to the lack of sensors to k-cover an entire planar field. Figure 17.3 shows this protocol. Our k-coverage approach consists of two main phases, namely mobile proxy selection and mobile sensor selection. In the first selection phase, which is described in Sect. 17.3.3.1, the goal is to identify a mobile proxy sink that will act as a leader to control the accomplishment of a mission. This selection phase takes into account the size of the neighborhood of mobile proxy sinks, called local planar density, which is defined below (Definition 17.1). This local planar density is based on the number of mobile sensors in the communication range of a mobile proxy sink. In addition, this phase depends on the location of a mobile proxy sink with respect to the target

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Fig. 17.3 Mobile k-coverage algorithm

location of the region of interest to be k-covered. Mobile proxy sinks that are closer to the region and have a local planar density that matches the total number of sensors needed to k-cover a region, are preferred than others. In the second selection phase, which is described in Sect. 17.3.3.2, a leader mobile proxy sink, which is selected in the first phase, selects the mobile sensors that will participate in k-covering the region. This selection phase depends on the remaining energy of mobile sensors and their locations with respect to the region. This helps ensure that those mobile sensors have enough energy to k-cover the region, and also use them efficiently to minimize the energy consumption due to sensor mobility and communication. Our ultimate goal is to extend the lifetime of individual sensors so as to prolong the entire network lifetime. Next, we describe our k-coverage approach in details. First, we specify how to identify a mobile proxy sink to take the lead on a mission and accomplish it. Second,

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we discuss how mobile sensors are selected by a mobile proxy sink so they move to a region of interest and participate to its k-coverage.

17.3.3.1

Mobile Proxy Sink Selection

Without loss of generality, we assume that each region of interest is a square area with center (x 0 , y0 ) and side length a. Also, each mobile proxy sink has a unique id and keeps track of the number of mobile sensors located within its communication range. Moreover, each mobile proxy sink is aware of the locations of the static sensors in the planar deployment field. In particular, every mobile proxy sink knows all the static sensors that are located in a given region of interest. First, we define the notion of local planar density of a mobile proxy sink, which will be used in our selection algorithm of mobile proxy sinks. Definition 17.1 (Local Planar Density) The local planar density of a mobile proxy sink si , denoted by LD(si ), is the number of mobile sensors located in the ∎ communication range of si . Definition 17.2 (Mobile proxy sink density) A mobile proxy sink si is said to be denser than a mobile proxy sink sj if si ’s local density is greater than that of sj , i.e., ∎ LD(si ) > LD(sj ). Based on Theorem 5.3 stated earlier in Chap. 5 in Sect. 5.2.2 and which computes the planar sensor density to achieve k-coverage of a planar field, Theorem 17.1 stated below computes the total number of sensors needed to k-cover a region of interest [39]. Theorem 17.1 (Required number of sensors) Let k ≥ 3 and assume that a region of interest is sliced into overlapping Reuleaux triangles of width r. The total number of sensors, denoted by n(r, k, I 0 ), which is required to guarantee k-coverage of this region is given by n(r, k, I0 ) =

0.8141 k × I0 r2

where I 0 is the area of the region of interest.

∎

Another problem similar to k-coverage is k-fold coverage [318]. Likewise, Pach [318], Pach and Toth [320], Aloupis et al. [13], and Gibson et al. [170] discussed the problem of k-fold coverage of the plane from a discrete geometry point of view. In particular, whereas Theorem 17.1, which is stated above, gives a result for how many sensors are required to ensure k-coverage of any region of interest in a planar field, the work of Pach [318], Pach and Toth [320], Aloupis et al. [13], and Gibson et al. [170], gives results for the number of disjoint k-covers that can be constructed from a family of polygons that k-cover the plane. For more details on the k-fold coverage problem, the interested reader is referred to [318].

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Given its main role described earlier, the static sink specifies the region of interest to be k-covered by determining its center (x 0 , y0 ) and side length a. Then, it broadcasts a QUERY packet whose structure is: QUERY = . The selection of a mobile proxy sink, which will be responsible to carry out this mission, is determined in a distributed manner via negotiation among all mobile proxy sinks. Next, we provide a detailed description of the underlying mobile proxy sink selection algorithm. When a mobile proxy sink si receives the above-mentioned QUERY packet from the static sink, it executes the following four steps, whenever possible, for the selection of a mobile proxy sink that will be responsible for accomplishing the mission: • Candidacy Step: Using its local planar density LD(si ) (Definition 17.1 above) and the number of static sensors NSS located in the region of interest, si checks whether k-coverage of this region can be achieved based on Theorem 17.1 above. Specifically, si determines whether it is a candidate mobile proxy sink so it can perform the mission successfully if the next inequality is satisfied: L D(si ) + N SS ≥ n(r, k) where n(r, k) stands for the total number of sensors that are needed to k-cover a region of interest. Also, r denotes the radius of the sensing range of static sensors, in case at least one static sensor would be selected by mobile proxy sink si to kcover a region of interest. Otherwise, r is the radius of the sensing range of mobile sensors. As indicated in Section 17.3.2, all the static sensors are heterogeneous, and so are all mobile sensors, although the latter are more powerful than the former. – If the above inequality holds, si is considered as a candidate mobile proxy sink for accomplishing this mission. It sends a short VOLUNTEER packet including all of its id, local planar density (Definition 17.1 above), and current location. This packet should be received by all the mobile proxy sinks within t candidate-mps-found time so that they know that there is at least one candidate mobile proxy sink that has been found to carry out the mission. Then, si waits for some time to listen to other VOLUNTEER packets. – Otherwise, si ignores the QUERY packet that it has received from the static sink as it is unable to accomplish the requested mission. Indeed, the sum of the number of mobile sensors located in the transmission range of the mobile proxy sink, i.e., LD(si ), and the number of static sensors already located in the region of interest, i.e., NSS, is not enough (or less than the total number of sensors n(r, k) that is needed to k-cover the region), i.e., LD(si ) + NSS < n(r, k). • Volunteering Step: When a mobile proxy sink si receives a VOLUNTEER packet from another mobile proxy sink sj , it checks whether it is a better candidate than

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sj . Hence, si focuses on the energy consumption due to mobility, which depends on the distance being traveled by a sensor as stated in our energy consumption model presented earlier in Chap. 2 in Sect. 2.5. Given that si is a mobile proxy sink that is potentially responsible for accomplishing the underlying mission, si estimates the average energy consumption per sensor caused by mobility as the energy consumption caused by the movement of si toward the center (x 0 , y0 ) of the region of interest. Thus, si acts as follows: – If si is closer to the center (x 0 ,y0 ) of the region than sj , then si will maintain its candidacy. Given that sj already received a VOLUNTEER packet from si . In this case, sj will simply give up as at least si is a better candidate than it for playing the role of leader mobile proxy sink. – Otherwise, si will simply give up as its selection would cause higher energy consumption due to sensor mobility to the region of interest. • Leading Step: After t leader-mps-selected time, a mobile proxy sink (known as leader mobile proxy sink) should have been selected to take the lead on accomplishing a mission. When a mobile proxy sink finds out that it is a leader mobile proxy sink, it sends out a short LEADER packet, including its id and the QUERY packet originated from the sink. • Merge Step: If after t candidate-mps-found time, no VOLUNTEER packet has reached any of the mobile proxy sinks, it means that no mobile proxy sink has enough local planar density (Definition 17.1 above) to ensure k-coverage of the region of interest. In this case, each mobile proxy sink si advertises a short HELLO packet including its id, location, and local planar density. When si receives a HELLO packet, it compares its local planar density to those of all other mobile proxy sinks. Then, si chooses to merge its local planar density with the mobile proxy sink (sj ) that has the smallest local planar density so that the sum of the local planar densities of si and sj is enough k-cover the region. If si find more than one mobile proxy sink with the smallest local planar density, it breaks the tie by using the closest one to it. In case there is still tie, si chooses to merge with the one with the smallest id. To reduce energy consumption due to control overhead, only si would reply if its local planar density is less than that of sj . More precisely, sj gives up and si becomes a candidate mobile proxy sink. In this case, si is said to be a merger mobile proxy sink. Thus, if si is selected as a candidate or leader mobile proxy sink, the local planar density of sj will join that of si . At the end of this merge operation, each candidate mobile proxy sink broadcasts its VOLUNTEER packet to initiate the Volunteering Step. Notice that packets, such as QUERY and VOLUNTEER, have to reach all mobile proxy sinks, which are randomly deployed in the planar field of interest. Thus, the static sink and mobile proxy sinks have to broadcast those packets throughout the whole network. To this end, given the sparseness of sensor deployment, the static and mobile sensors participate in this broadcast process by forwarding those packets whenever they receive them so other mobile proxy sinks in the network receive them

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539

too. This forwarding activity will incur a routing overhead, which will be assessed in Sect. 17.6.2. As it can be seen from the description of the Merge Step, a given mobile proxy sink may merge with more than one mobile proxy sink to ensure k-coverage of a region of interest. In fact, the end of the Merge Step always initiates the Volunteering Step. This enables the selection process to keep running until finding a suitable leader mobile proxy sink that is responsible for supervising the accomplishment of the current mission. Moreover, during the merge operation, which is specified in the Merge Step, a mobile proxy sink would move together with its own mobile sensors to the target location. It is true that at the beginning, some mobile sensors could be “orphans”. That is, they are not located in the communication range of any mobile proxy sink. This is due to the fact that all mobile sensors and mobile proxy sinks are randomly deployed. However, since the region of interest changes dynamically, mobile proxy sinks may be located anywhere in the planar deployment field due to their mobility toward the region to be k-covered. This allows every mobile sensor not to stay “orphan” and be associated with one mobile proxy sink. It is worth mentioning that there are several missions to accomplish during the network operation. Nevertheless, our k-coverage approach considers only one mission at a time. Therefore, there is always only one leader mobile proxy sink that is responsible for selecting mobile sensors to move toward a region of interest and k-cover it. The problem of having multiple missions, say n, to be dealt with simultaneously would need the selection of n leader mobile proxy sinks. Each of these leader mobile proxy sinks will control the accomplishment of one of those n missions. Clearly, this scenario will raise several major issues. For instance, since leader mobile proxy sinks independently choose mobile sensors to achieve k-coverage, it is possible that two or more leader mobile proxy sinks choose one same mobile sensor. This would create a conflict that needs to be addressed appropriately. All the issues regarding this problem will be investigated and solved as part of our future work of this study.

17.3.3.2

Mobile Sensor Selection

The main goal of the design of our k-coverage approach is to reduce the energy consumption of the sensors while ensuring k-coverage of any region of interest. Our k-coverage approach attempts to achieve this design goal by reducing the energy consumption due to several sources, such as data transmission, data reception, sensing, and sensor mobility. Knowing the number of static sensors located in the region, denoted by NSS, a leader mobile proxy sink computes the necessary number of mobile sensors, denoted by NMS, to move toward the region and k-cover it. To this end, the leader mobile proxy sink randomly decomposes the region into overlapping Reuleaux triangles (or slices) of radius r, where r is the radius of the sensing range of the static sensors (r = r s ) that have been selected to participate in k-covering the region. In case there are only mobile sensors that have been selected to k-cover the region, r is the radius of the sensing range of those mobile sensors (r = r m ).

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17 A Planar Approach for Joint k-Coverage and Data Collection …

Fig. 17.4 Extreme points A and B of a lens

Recall that mobile sensors are more powerful than static sensors. Then, based on the distribution of lenses in the slicing grid as well as the distribution and the number of static sensors in the region of interest, the leader mobile proxy sink performs the next actions: • Computes the two extreme points A and B of each lens as shown in Fig. 17.4. This pair of points will be used by the selected mobile sensors to position themselves within a lens when they move to it. This will ensure full k-coverage of the corresponding adjacent slices in the region by exactly k sensors as stated earlier in Theorem 5.2 in Chap. 5 in Sect. 5.2.2 [39]. • Computes the number of mobile sensors, NMS, which should move to the region to be k-covered. This should account for the presence of NSS static sensors in the region. Recall that NMS should verify the equality: N M S = n(r, k) − N SS • Selects NMS mobile sensors with the highest remaining energy. There is a threshold for the remaining energy below which a mobile sensor cannot be selected to move to the region of interest. Thus, selected mobile sensors should have enough energy to k-cover the region. • Computes the parameters of its mobility according to the corresponding mobility model. Then, it communicates them to the selected mobile sensors whenever applicable. We distinguish the following three cases, each of which corresponds to one of the three mobility models being considered in this study. – Case 1—Random waypoint mobility model: As described earlier, there is no coordination among mobile sensors with regard to their mobility trajectories in this type of mobility model. In particular, the leader mobile proxy sink does not have any control on the mobile sensors’ movements. – Case 2—Reference point group mobility model: The leader mobile proxy sink determines its speed vref (t), maximum speed max_speed, and direction deviation θ ref (t), and communicates these parameters to the mobile sensors that are selected to move to the region of interest. These mobile sensors use those

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541

parameters to define their movement behavior according to the specification of the reference point group mobility model as described earlier in Chap. 2 in Sect. 2.8. – Case 3—Manhattan mobility model: The static sink computes the street map with its horizontal and vertical lanes, and sends it to all the mobile sensors in the network. This step is done only once at the beginning of the sensor deployment. Given that a region of interest for a given mission is dynamic (i.e., the center (x 0 , y0 ) and side length a of the square region of interest change from one mission to another), a street map needs to be computed carefully in order to ensure k-coverage of the region of interest. To this end, the distance Δ between two streets should not exceed the width of a slice (i.e., Reuleaux triangle) in the dynamic slicing grid of the region of interest. Recall that the leader mobile proxy sink is in charge of randomly decomposing a region of interest into Reuleaux triangles. Precisely, this distance Δ is set to ω/3, where ω stands for the width of a slice. Also, each street has three lanes, each of which has a width equal to ω/3. Thus, the width of a street is equal to ω. • Assigns the selected mobile sensors to the lenses, where they should be located, and announces this assignment. To this end, the leader mobile proxy sink sends a short SELECT packet including the parameters of its mobility as stated above, if applicable, followed by the id’s of the selected mobile sensors and the pairs of the extreme points of the lenses. Similarly, we distinguish three cases for the three mobility models: Random waypoint mobility model, reference point group mobility model, and Manhattan mobility model. – Case 1—Random waypoint mobility model: A SELECT message has NMS triplets, each of which has the structure , where id i is the id of mobile sensor si and Aj and Bj are the two extreme points of lens j in the region. Sensor si can be located anywhere in the lens when it moves to the region to be k-covered. Each selected mobile sensor chooses its constant mobility speed randomly between 0 and vmax , where vmax is the maximum speed. Then, it selects its target waypoint within the slice whose extreme points are specified in the SELECT packet. Also, it sets its pause time to 0. Finally, it moves to its selected target waypoint using the mobility speed it has selected. – Case 2—Reference point group mobility model: A SELECT message has one triplet and NMS triplets, each of which has the structure as specified in Case 1. Similarly, sensor si can be located anywhere in the lens when it moves to the region to be k-covered. Each selected mobile sensor derives its speed and motion direction from those of their reference point given in the above-mentioned triplet as stated earlier in Chap. 2 in Sect. 2.8.2. – Case 3—Manhattan mobility model: A SELECT message has NMS triplets, each of which has the structure as specified in the previous two cases. Similarly, sensor si can be located anywhere in the lens when it moves to the region of interest. In this case, each mobile sensor picks the closest lane

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to its target lens specified in the SELECT message. Then, it moves along that lane until it reaches that target lens.

17.4 Data Gathering Algorithms In this section, we address the problem of data gathering in sparsely deployed wireless sensor networks using heterogeneous sensors while considering sensor mobility. Precisely, we propose two algorithms for data gathering in k-covered heterogeneous and sparse wireless sensor networks based on sensor mobility: Direct data gathering and chain-based data gathering. The main component in data gathering is routing, which specifies how data is forwarded from one sensor to another until it reaches its final destination, i.e., the sink. A good routing protocol in data gathering should balance the energy consumption among the sensors, thus, prolonging the lifetime of the network. The use of sensor mobility is motivated by the following two scenarios. First, as stated earlier, the sink is supposed to be static. Thus, all the sensors surrounding it may suffer from a severe battery power depletion problem if they were the only ones that would forward the data on behalf of all other sensors in the network. As discussed earlier in Sect. 17.3.3 above, this problem is known as the energy sinkhole problem [181, 316]. To alleviate this problem, we propose to exploit sensor mobility in order to avoid that all the sensors located around the sink are heavily used in forwarding data to it. More precisely, in addition to those sensors nearer the sink, sensor mobility enables other sensors to participate in the data forwarding process to the static sink. Second, given that the sensors are sparsely deployed, there must be some support to help establish communication paths between source sensors and the sink. Mobile sensors can be used to build energy-efficient data forwarding paths so data originated from source sensors can be received by the sink. First, we describe each of those data gathering algorithms along with their routing component. Then, we discuss their advantages and disadvantages. In Sect. 17.6 below, we evaluate their performance.

17.4.1 Direct Data Gathering In this algorithm, each leader mobile proxy sink is solely responsible for delivering all collected data of a mission to the static sink. That is, once a leader mobile proxy sink collects data from the static and mobile sensors that accomplished a mission in a given region of interest, it moves to the center of the planar field to deliver the data to the static sink. As it can be seen, on the one hand, this approach does not introduce any routing overhead between a leader mobile proxy sink and any other sensor in the network. Thus, the only source of energy consumption is due to the mobility of a leader mobile

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543

proxy sink and the direct transmission of its collected data to the static sink. In order to reduce energy expenditure during data transmission, a leader mobile proxy sink would send its data to the static sink over a short distance. To reduce the amount of energy dissipated due to its mobility, a leader mobile proxy sink needs to travel as close to the shortest path between it and the static sink as possible. On the other hand, this data gathering approach incurs more delay. Indeed, if forwarded through intermediate sensors, the data would travel at a speed faster than that of a leader mobile proxy sink. Thus, this approach is not suitable for time-critical applications. The advantages of this direct data gathering scheme are described as follows: • Extended network lifetime: Thanks to its mobility, a leader mobile proxy sink transmits its collected data to the static sink through short distances, which yields the consumption of a little amount of energy. This helps extend the lifetime of mobile proxy sinks, thus prolonging the operational network lifetime. • Alleviated data forwarding load: Using this direct data gathering scheme, no other sensor is needed to act as a relay node between a leader mobile proxy sensor and the static sink. Therefore, all the static sensors and mobile sensors save their energy significantly, which helps them prolong their lifetime. Given that this is a zero-routing based data gathering protocol, the static sensors nearer the static sink do not suffer at all the energy-hole problem. This allows those sensors save their energy, which enables them to extend their lifetime. Also, it helps maintain the whole network operational for a longer period of time. • Improved data delivery ratio: Given that the mobile proxy sinks are very powerful compared to all other sensors, the likelihood of their failure is very low. All other sensors, however, may fail due to low battery power. Thus, avoiding these sensors improves the ratio of successful data transmission to the static sink. The disadvantages of this scheme are summarized as follows: – High end-to-end delay: The mobility trajectory of a leader mobile proxy sink may not coincide with the shortest path between it and the static sink. Also, there may be obstacles in the planar field, and hence, leader mobile proxy sink mobility is not straight. This will incur more delay. – Potential data loss: A leader mobile proxy sink failure is crucial to the sensed data lifetime. Indeed, once data is lost, it cannot reach the static sink as there is no way to regenerate it.

17.4.2 Chain-Based Data Gathering In this algorithm, all mobile proxy sinks are involved in the data gathering process. Specifically, once a leader mobile proxy sink has been identified, all other mobile proxy sinks form a forwarding chain of relay nodes to forward data to the static sink. Thus, there is a routing overhead that should be minimized. Moreover, the routing load should be evenly distributed among the involved sensors in order to balance their energy consumption and, thus, extend the network lifetime. To minimize

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the energy consumption due to multi-hop data forwarding, the mobile proxy sinks forming the chain should forward the data toward the sink over short distances. Furthermore, to achieve balanced energy consumption among them, they need to keep the same distance between any pair of consecutive mobile proxy sinks. Thus, our goal is to ensure that the data is routed (or forwarded) toward the sink using the same minimum distance between any pair of consecutive forwarders on the data forwarding chain. This routing strategy helps minimize and balance the total energy consumption among all the involved mobile proxy sinks. Assume there are n mobile proxy sinks. The total energy consumption required to forward a data packet from a leader mobile proxy sink to the static sink is proportional to the sum of the distances, i.e., Σ E tot ∝ diα , for all 1 ≤ i ≤ n where d i is the distance between two adjacent mobile proxy sinks si and si+1 , 1 ≤ i ≤ n − 1, and d n is the distance between the static sink and mobile proxy sink sn . It is easy to prove that E tot is minimum only when all those distances d i are the same, i.e., d1 = d2 = . . . = di = . . . = dn , for all 1 ≤ i ≤ n To simplify the analysis and without loss of generality, assume that there are three mobile proxy sinks, say mps1 , mps2 , and mps3 . Also, assume α = 2 and that the distance between mps1 and mps2 , and that between mps2 and mps3 , are respectively equal to d 1 and d 2 , where d 1 + d 2 = d. We can prove that f (d 1 , d 2 ) = d 1 2 + d 2 2 is minimum when d 1 = d 2 = d/2. Lemma 17.1 stated below proves this result. Then, Corollary 17.1 shown below gives a generalization of Lemma 17.1 to a function f , where f has n variables d 1 , d 2 , …, d n . Lemma 17.1 (Squared Distance Sum Function Minimum) Let d 1 and d 2 be two real numbers, where d 1 + d 2 = d. The function f (d 1 , d 2 ) = d 1 2 + d 2 2 + c reaches its minimum at d 1 = d 2 = d/2, where c is a constant value. Proof Given that d 1 + d 2 = d, we have d 2 = d – d 1 . Hence, f (d 1 , d 2 ) = d 1 2 + d 2 2 + c = g(d 1 ) = d 1 2 + (d – d 1 )2 + c = 2d 1 2 – 2d d 1 + d 2 + c. It is clear that g(d 1 ) 1) = 4d1 − 2d = 0. That is, d 1 = d/2, which implies reaches its minimum when ∂g(d ∂d1 ∎ that d 2 = d – d 1 = d/2. Corollary 17.1 (Generalization of Lemma 17.1) Consider a forwarding chain between a leader mobile proxy sink and a static sink that has n forwarders (i.e., n − 1 mobile proxy sinks and a leader mobile proxy sink). Assume that the Euclidean distance between the leader mobile proxy sink Σ and the static sink is equal to d. The total energy consumption, denoted by E tot = i=1..n diα + c, which is required to forward a data packet from the leader mobile proxy sink to the static sink, reaches ∎ its minimum when d 1 = d 2 = … = d n = d/n.

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Theorem 17.2 stated below is similar to Lemma 15.3, which is given in Chap. 15, and can be used to compute the minimum distance between any pair of consecutive mobile proxy sinks in the above-mentioned forwarding chain. Theorem 17.2 (Minimum transmission distance) The minimum transmission distance, denoted by d 0 , between any pair of consecutive mobile proxy sinks in the forwarding chain between a leader mobile proxy sink and the static sink is given by: ( d0 =

E elec ε

)1/α ∎

The above result shows that there exists a minimum transmission distance d 0 between any pair of consecutive mobile proxy sinks in the forwarding chain that leads to minimum transmission energy consumption. It is worth mentioning that communication between any pair of consecutive mobile proxy sinks can take place only when d 0 ≤ Rmin , where Rmin stands for the minimum radius of the communication range of mobile proxy sinks. Now, let us compute the optimum number of mobile proxy sinks, denoted by nopt , forming the forwarding chain between a leader mobile proxy sink and the static sink. Theorem 17.3 stated below computes nopt . Theorem 17.3 (Optimum number of mobile proxy sinks) The optimum number of mobile proxy sinks forming a forwarding chain between a leader mobile proxy sink and the static sink separated by a distance d is given by ( n opt =

α−1 2

)1/α

d d0

where d 0 is the minimum transmission distance between any pair of consecutive mobile proxy sinks in the forwarding chain. Proof Assume that there are n forwarders in the forwarding chain between a leader mobile proxy sink s0 and the static sink sn . Let (s0 , s1 , …, sn−1 ) be this forwarding chain. According to the energy consumption model described earlier in Chap. 2 in Sect. 2.5, the total energy consumption when a mobile proxy sink receives an m-bits message and transmits it over a distance d is given by E rx-tx (d) = m × (εd α + 2E elec ). Thus, the total energy E tot (s0 , sn ) spent in forwarding a sensed data packet from a leader mobile proxy sink s0 to the static sink sn is computed as follows: E tot (s0 , sn ) = E tot (s0 , s1 ) + E tot (sn ) +

Σ

E tot (si )

i=1..n−1

where E tot (s0 , s1 ) = m × (ε δ α (s0 ,s1 ) + E elec ) is the transmission energy consumed by a leader mobile proxy sink s0 to transmit its m-bits message to the first mobile proxy

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sink s1 in the forwarding chain, E tot (sn ) = mE elec is the reception energy consumed by the static sink sn to receive an m-bits message from the last mobile proxy sink sn-1 in the forwarding chain, and E tot (si ) = m × (ε δ α (si ,si+1 ) + 2E elec ) is the receptiontransmission energy consumption consumed by a mobile proxy sink si to receive m-bits message from a mobile proxy sink si-1 and transmit it to the next mobile proxy sink si+1 in the forwarding chain, for 1 ≤ i ≤ n − 1, with δ(si , s j ) being the Euclidean distance between sensors si and sj . Thus, with a little algebra, we have: Σ

E tot (s0 , sn ) =

m × ( ε δ α (si , si+1 ) + 2 E elec )

i=0..n−1

In general, the distances between all pairs of consecutive mobile proxy sinks in the forwarding chain between a leader mobile proxy sink and the static sink are not necessarily equal. However, the total energy consumption E tot (s0 , sn ) reaches its minimum when all distances δ(si , si+1 ) between adjacent sensors in the chain are equal, i.e., δ(si , si+1 ) = δ(s0n,sn ) , with n being the number of mobile proxy sinks, including the leader mobile proxy sink, forming the forwarding chain between the latter and the static sink. Thus, E tot (s0 ,sn ) ≥ E(n), where E(n) =

) ) ( ( δ(s0 , sn ) α + 2 E elec m× ε n i=0..n−1 Σ

Therefore, E(n) can be rewritten as: ) ( α δ (s0 , sn ) E(n) = m × ε + 2 n E elec n α−1 ∗

) The minimum of the function E(n) can be found using ∂ E(n = 0. Notice that ∂n ∗ ∂ 2 E(n ∗ ) > 0, for all α ≥ 2, thus, showing that E(n) is strictly convex. This can be seen ∂ 2 n∗ in Fig. 17.5 for the case α = 2, where m = 256, E elec = 50 nJ/bit, εfs = 10 pJ/bit/m2 , ∗ ) d = 3500 m, and nopt ≈ 50. Hence, the value of n ∗ in ∂ E(n = 0 corresponds to the ∂n ∗ minimum of E(n). Thus, the optimum number of mobile proxy sinks is given by

∗

(

n opt = n =

α−1 2

)1/α

d d0

∎

Theorem 17.2 given above shows that we make a good approximation of the minimum transmission distance d 0 between any pair of consecutive mobile proxy sinks in the forwarding chain between a leader mobile proxy sink and the static sink. Indeed, since 2 ≤ α ≤ 4, we have ( 0.707 ≤

α−1 2

)1/α ≤ 1.107

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Fig. 17.5 Total energy consumption E(n)

Thus, using the above result, we obtain n opt ≈

d d0

This means that the total energy consumption E(n) that is required to forward an m-bits message from a leader mobile proxy sink to the static sink reaches its minimum when the distance between any pair of consecutive mobile proxy sinks in the forwarding chain connecting the leader mobile proxy sink to the static sink is equal to d 0 as computed in Theorem 17.2 shown above. Moreover, the corresponding optimum number of mobile proxy sinks in this forwarding chain is equal to nopt as stated in Theorem 17.3 and approximated above. The sensed data will be forwarded through the forwarding chain only until it reaches the static sink. We assume that there is a sufficient number of mobile proxy sinks to form a forwarding chain between the static sink and a leader mobile proxy sink when it is located at the farthest point in a planar square field, which is one of its corners. Furthermore, based on Theorem 17.3 given above, the leader mobile proxy sink computes the necessary number of mobile proxy sinks to form the forwarding chain along with their respective physical locations. In other words, only a subset of mobile proxy sinks and not necessarily all of them would move to build this forwarding chain. This would further reduce the total energy consumption. Then, it broadcasts a short packet, denoted by Fwd_Chain = into the network, where dest_mps states that Fwd_Chain packet is destined to mobile proxy sinks only, nopt is the number of mobile proxy sinks forming the forwarding chain, and (x i , yi ) is the physical location of the ith mobile proxy sink in the forwarding chain, with 1 ≤ i ≤ nopt . Only mobile proxy sinks would consider a Fwd_Chain packet. Otherwise, it would be discarded by any other type of sensor. When a mobile proxy sink mpsj receives a Fwd_Chain packet,

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it runs the following procedure to check whether it can act as a forwarder in the forwarding chain: • It picks the closest destination location (x i , yi ) in the forwarding chain to its current location (x j , yj ). First, it advertises a short Willing_Fwd packet with following structure: Willing_Fwd = , where E rem (i) stands for the remaining energy of mobile proxy sink mpsj . Then, it waits for some time t wait-fwd before it makes its final decision. • If it does not receive any Willing_Fwd packet from any other mobile proxy sink before t wait-fwd expires, mpsj decides to act as a forwarder and moves to the destination location (x i , yi ). Otherwise, if mpsj receives a Willing_Fwd packet from another mobile proxy sink mpsl , it discards its candidacy if mpsl is closer to (x i , yi ) than mpsj . Now, if both of the mobile proxy sinks mpsj and mpsl are equidistant from the location (x i , yi ), the one with the highest remaining energy would be selected as forwarder in the underlying forwarding chain. The benefits of this chain-based data gathering scheme are described as follows: • Low delay: These mobile proxy sinks forming the forwarding chain are powerful. Thus, they can forward the data to the static sink through the sensors on the chain only. Hence, the data will reach the static sink in a timely manner. • Guaranteed data delivery: The failure of any mobile proxy sink in the planar field will not affect the forwarding chain. The latter will be reorganized by the rest of the mobile proxy sinks when others fail. Therefore, the data originated from a leader mobile proxy sink is guaranteed to be delivered to the sink. The shortcomings of this data gathering scheme are stated as follows: • Chain formation energy overhead: There is extra energy consumption due to the forwarding chain formation and maintenance. These mobile proxy sinks need to coordinate their motion to build an efficient forwarding chain. To this end, the distance between any pair of consecutive mobile proxy sinks in the chain is d 0 . • Chain maintenance energy overhead: When mobile proxy sinks fail due to low battery power, the chain needs to be reorganized. This requires more energy expenditure. In fact, those mobile proxy sinks forming the forwarding chain should adjust their positions to forward sensed data over the same distance d 0 .

17.5 Impact of Sensor Heterogeneity The deployment of heterogeneous sensors with regard to their energy supply, sensing range, and communication range, may cause a few problems. But, it results in several desired advantages and characteristics in comparison with homogeneous sensors. More importantly, this heterogeneity reflects the actual features of realworld networked sensing systems. Next, we discuss those problems as well as the resulting advantages.

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549

First, it is obvious that the use of sensors with different energy supply potentially increases the network lifetime based on the following arguments: Assuming that the sensors may fail due to low battery power, the powerful sensors remain functional for a longer time interval. This helps delay the occurrence of sensor failures and, thus, keeps the entire network operational for an extended period of time. In addition, those sensors equipped with higher battery supply are highly likely to be selected more often than those sensors with lower battery power to participate to the network operation by accomplishing a given task. As a result, the latter sensors are avoided, which reduces the chances of their failures. Also, this copes with the energy-hole problem, thus, overcoming the potential coverage-hole and connectivity-hole problems. Second, in Sect. 17.2 above, we discuss the impact of the differences in the sensing range of the sensors on the problem of k-coverage. More precisely, we describe the two problems, namely under k-coverage and over k-coverage that may arise when the sensors do not have the same radius of their sensing range. In other words, it is not practical to design a centralized k-coverage protocol that would be run under the control of a centralized entity, such as the static sink. However, the deployment of powerful sensors in terms of their sensing capability has some advantages. Indeed, the existence of powerful sensors helps improve coverage efficiency. In fact, fewer powerful sensors are able to achieve k-coverage of a region of interest compared to using homogeneous and less powerful sensors. This definitely leads to significant amount of energy savings, which in turn helps extend the network lifetime. Third, the differences in the communication range of the sensors may create a potential problem regarding their interactions. Let si and sj be two sensors whose radii of communication range are Ri and Rj , respectively. Assume that Ri > Rj . This scenario may create asymmetry in the communication between si and sj . Indeed, we may have a configuration in which sj is within the communication range of si , whereas si is outside the communication range of sj . While si can send messages to sj , the latter cannot transmit any message to si . This is due to the asymmetric communications links between the two sensors. In addition to sensing coverage, network connectivity is an essential concept that allows the sensors to communicate with each other so the data collected by the sensors can reach a central gathering point, such as the static sink, for further analysis and processing. As stated earlier in Sect. 17.4.2 above, the minimum transmission distance d 0 and the minimum radius Rmin of the communication range of the mobile proxy sinks should satisfy the inequality when d 0 ≤ Rmin , in order to enable communication between any pair of consecutive mobile proxy sinks forming the forwarding chain. Also, the presence of powerful sensors with regard to their communication capability helps shorten the length of the communication paths between the sensors. This yields a minimum cost to establish network connectivity by minimizing the number of active sensors. Consequently, a large amount of energy is saved, which contributes to the network lifetime prolongation. As it can be noticed from the above discussion, the common advantage of sensor heterogeneity at the sensing range, communication range, and energy supply levels, is

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the extension of the whole network operational lifetime. We argue that a better prolongation of the network lifetime can be obtained if those three levels of sensor heterogeneity are combined together in the same framework. Our proposed k-coverage approach considers groups of sensors that have intra-group and inter-group heterogeneity. More precisely, our k-coverage approach considers three heterogeneous groups of sensors. Moreover, the sensors in each group are heterogeneous. In the next section, we consider those three sensor heterogeneity levels. Our goal is to assess their impact on our k-coverage approach.

17.6 Performance Evaluation In this section, we evaluate the performance of our proposed framework for kcoverage and data collection in heterogeneous and sparsely deployed wireless sensor networks. First, we specify the simulation setup. Then, we present the simulation results of our framework based on several performance metrics and using a high-level simulator written in C.

17.6.1 Simulation Setup Our framework works independently of the shapes of the planar deployment field and region of interest to be k-covered. In particular, it can be applied to square and nonsquare shapes. Without loss of generality, we consider a planar square deployment field of side length 700 m and a square region of interest of side length 300 m. We assume that the total number of sensors is 1000, where 400 of them are mobile (i.e., there are 600 static sensors). Among those 400 mobile sensors, there are 100 mobile proxy sinks. We assume that the initial energy of each static sensor is randomly selected from the interval [60 J, 70 J]. Moreover, the initial energy of each mobile sensor is randomly selected from the interval [80 J, 90 J], while that of each mobile proxy sink is randomly selected from the interval [100 J, 110 J]. Recall that mobile sensors are more powerful than static sensors with regard to all of their capabilities. In addition, we use the energy model specified in Chap. 2 in Sect. 2.5. The sensor energy consumption in idle and sleep modes are 0.012 J and 0.0003 J, respectively [435]. Also, the energy cost for a sensor to move one unit distance, emove , is randomly selected between 0.008 J/m and 0.012 J/m [398]. In a long-term running, mobile sensors powered by batteries have to be recharged somewhere. We leave this issue as our future work, where an appropriate recharge scheme is added to our k-coverage approach. The communication energy cost is based on the energy model defined in Chap. 2 in Sect. 2.5, which consider the energy due to data transmission and reception [195]. Moreover, we assume that the size of data packets is 256 bits and that every sensor continuously generates constant bit rate data of 1024 bits/second, i.e., 4 data packets per second. Also, the radii of the sensing and communication ranges of all

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551

static sensors are randomly selected from the intervals [20 m, 30 m] and [40 m, 50 m], respectively, while those of all mobile sensors are randomly selected from the intervals [40 m, 50 m] and [60 m, 70 m], respectively. Also, the radii of the sensing and communication ranges of all mobile proxy sinks are randomly selected from the intervals [60 m, 70 m] and [80 m, 90 m], respectively. We use the IEEE 802.11 distributed coordinated function with CSMA/CA as the underlying MAC protocol. Also, we consider a radio interference model given the pervasiveness of other 2.4 GHz radio sources. We consider various sources of energy consumption, including data sensing, data transmission, data reception, sensor mobility, and control messages, which are exchanged among all the involved sensors for the proper working of our framework. All simulations are repeated 100 times and the results are averaged. Table 17.1 summarizes all the simulation settings. Table 17.1 Simulation settings

Parameters

Values

Total number of sensors

1000

Number of static sensors

600

Number of mobile sensors

300

Number of mobile proxy sinks

100

Initial energy of a static sensor

Random value in [60 J, 70 J]

Initial energy of a mobile sensor

Random value in [80J, 90 J]

Initial energy of a mobile proxy sink

Random value in [100 J, 110 J]

Sensor energy consumption in idle mode

0.012 J

Sensor energy consumption in sleep mode

0.0003 J

Sensor mobility energy per meter (emove )

Random value in [0.008 J/m, 0.012 J/m]

Static sensor sensing range

Random value in [20 m, 30 m]

Mobile sensor sensing range

Random value in [40 m, 50 m]

Mobile proxy sink sensing range

Random value in [60 m, 7 0 m]

Static sensor communication range

Random value in [40 m, 50 m]

Mobile sensor communication range

Random value in [60 m, 70 m]

Mobile proxy sink communication range

Random value in [80 m, 90 m]

Packet size

256 bits

Constant bit rate

1024 s/second

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17.6.2 Simulation Results In the first experiments given in Sect. 17.6.2.1, we consider the reference point group mobility model. However, all the experiments discussed in Sect. 17.6.2.2 use various mobility models, namely the reference point group mobility model, Manhattan mobility model, and the random waypoint mobility model.

17.6.2.1

Simulations Using Reference Point Group Mobility Model

In these experiments, we use an implementation of the reference point group mobility model [168], where each mobile sensor is supposed to move at a constant speed of 1 m/s until it reaches its target destination in the region of interest to be k-covered. As discussed earlier, there are some mobile sensors, called mobile proxy sinks, which would be selected as reference points (or group leaders) to control the accomplishment of any given mission in the planar deployment field. Recall that our framework mainly includes two essential components, namely mobile k-coverage and data gathering on top of k-covered network configurations. In order to fairly compare our mobile k-coverage protocol with Wang and Tseng’s mobile k-coverage approach [398], we consider only homogeneous sensors given that Wang and Tseng’s approach [398] used only sensors having the same capabilities. In other words, we assume that all the static and mobile sensors in our four-tier architecture are the same with respect to their main characteristics, namely sensing range, communication range, and initial energy, whose values are set to 20 m, 40 m, and 60 J, respectively. Figure 17.6 shows that our mobile k-coverage protocol requires less number of active sensors than the Competition protocol [398] to ensure k-coverage of the region of interest. In fact, our characterization of k-coverage, which is based on Helly’s Theorem [85], helps find a tight bound on the number of sensors that is necessary to Fig. 17.6 Our proposed k-coverage approach versus competition (number of sensors)

17.6 Performance Evaluation

553

Fig. 17.7 Our proposed k-coverage approach versus competition (total mobility energy)

ensure k-coverage of the region. This yields significant energy savings, thus, enabling several inactive sensors to be efficiently used in other subsequent missions. Hence, our mobile k-coverage protocol prolongs the lifetime of the whole network more than the Competition protocol [398] so it accomplishes more missions in any region of interest in the planar deployment field. As shown in Fig. 17.7, the energy consumption due to mobility, which is required for the correct operation of k-SCHEMES, is less than the one needed for the Competition protocol [398]. This is due in part to the coordination between all mobile proxy sinks to select one of them as a leader to accomplish the mission. Indeed, the mobile proxy sink that causes the least amount of mobility energy consumption is selected as a leader mobile proxy sink. In addition, this shows the effect of the reference point group mobility model. In fact, all the selected mobile sensors form a group that follows the movement trajectory of its reference point, i.e., leader mobile proxy sink. As expected and in accordance with the results found in [433], sensor heterogeneity helps reduce the total energy consumption, thus extending the operational network lifetime. Figure 17.8 shows that our k-SCHEMES protocol using heterogeneous sensors (Heterogeneous k-SCHEMES) outperforms k-SCHEMES using homogeneous sensors (Homogeneous k-SCHEMES) as indicated in our four-tier architecture. Indeed, the mobile sensors are more powerful than the static ones, and particularly, in terms of their sensing range. Hence, the number of heterogeneous static and mobile sensors to k-cover a region of interest is less than that of homogeneous sensors to achieve the same goal. Thus, a smaller number of mobile sensors have to move to the region to be k-covered, thus, causing less mobility energy consumption. Consequently, the whole network would save significant amount of energy by extending the lifetime of the individual sensors. Figure 17.9 shows that chain-based data gathering protocol consumes less energy than direct data gathering protocol. Using the former protocol, mobile proxy sinks should communicate with each other to move to the appropriate positions in the

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Fig. 17.8 Homogeneous versus heterogeneous sensor deployment

Fig. 17.9 Direct versus chain-based data gathering

planar field. This helps them form a chain in a way that they are equidistant from each other. First, mobile proxy sinks need to coordinate between themselves to select only the necessary number of nodes forming the forwarding chain, which is given by nopt . Second, based on Theorem 17.2 shown above, all of the selected mobile proxy sinks should move toward their exact locations in the forwarding chain between the leader mobile proxy sink and the static sink. As discussed earlier, both of nopt and the target locations of the selected mobile proxy sinks are computed by the leader mobile proxy sink. Although the construction of this forwarding chain incurs certain additional energy consumption (or energy overhead) compared to the direct data gathering protocol, we find that, in general, the chain-based data gathering protocol is more energy efficient than the direct data gathering protocol. This shows that transmitting data over short distances yields significant energy savings, thus, proving the effectiveness of our chain-based data gathering protocol.

17.6 Performance Evaluation

17.6.2.2

555

Simulations Using Group Mobility Models and Random Waypoint Mobility Model

In these experiments, we investigate the impact of mobility on the performance of our proposed framework. More specifically, we vary the mobility speed of the mobile sensors and mobile proxy sensors from 1 m/s to 5 m/s in order to assess the impact of purposeful mobility on our framework, where the degree of coverage k required for any mission is set to 3 (i.e., k = 3). Moreover, we consider four quantitative metrics, namely data delivery ratio, delay, overhead, and energy consumed, to evaluate the performance of all the mobility models and their impact on our framework. The third metric, namely overhead, includes all the messages that are needed for data routing as well as any communication that is necessary among the sensors in the network for accomplishing a given mission. Furthermore, we assume that all mobile proxy sinks use the direct data gathering scheme, which is described in Sect. 17.4.1 above, as the underlying data routing protocol to collect data from source sensors and deliver it to the static sink. Figure 17.10 shows that the reference point group mobility model (RPGM) has the highest data delivery ratio compared to all other mobility models, namely Manhattan mobility model (MMM) and random waypoint mobility model (RWP). As it can be seen, the random waypoint mobility model has the lowest data delivery ratio among all the mobility models. Also, as the mobility speed of mobile sensors and mobile proxy sinks increases, the data delivery ratio decreases for all mobility models. The group mobility nature of the reference point group mobility model as well as the collaboration among mobile sensors and leader mobile proxy sink helps gather more data and deliver them to the static sink. The Manhattan mobility model gives better data delivery ratio than the random waypoint mobility model. This is due to the randomness in the movement of mobile sensors using the random waypoint mobility model. With the latter, maintaining continuous connectivity among mobile sensors Fig. 17.10 Data delivery ratio versus mobility speed

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Fig. 17.11 Average delay versus mobility speed

is very challenging due to the dynamic nature of their communication links. Both of the group mobility models outperform the random waypoint mobility model with respect to the data delivery ratio. With regard to the average delay, RPGM gives the lowest value as it can be seen in Fig. 17.11, while the RWP mobility model provides the highest average delay. Here also, the Manhattan mobility model exhibits better performance than the RWP mobility model. Notice that the average delay increases as the mobility speed of mobile sensors increases. This is true for only RPGM and Manhattan mobility model, whereas the average delay incurred by the RWP mobility model is almost constant regardless of the mobility speed. It is clear that group mobility models are more suitable than random mobility models for this type of application, where coordination and collaboration among nodes are required. Figure 17.12 shows that the random waypoint mobility model requires more routing overhead compared to both group mobility models. However, when the speed increases, the difference between the performance of both group mobility models and the random mobility model is getting smaller. In general, as the mobility of mobile sensors increases, regardless of the mobility model being considered, there is more need for mobile sensors to exchange control messages to establish communication paths among themselves. This type of routing overhead is necessary for the correct working of our mobile k-coverage and data gathering protocols in this dynamically changing network topology whose connectivity is time-varying due to high mobility of mobile sensors. According to Fig. 17.13, the random waypoint mobility model has expectedly higher energy consumption compared to both group mobility models. This is mainly due to the higher routing overhead it has as discussed earlier in Fig. 17.12. It is trivial that moving in groups yields lower energy consumption, which is required for sensor mobility and their communication. As it can be observed, all of these experiments

17.7 Related Work

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Fig. 17.12 Routing overhead versus mobility speed

Fig. 17.13 Energy consumed versus mobility speed

prove that group mobility models are more suitable than random mobility models for our framework. Figure 17.14 shows how the total remaining energy of the whole network varies with time. This helps quantify the network lifetime under the above network setting.

17.7 Related Work In this section, we describe a sample of approaches to achieve coverage in mobile wireless sensor networks. Recently, significant efforts have been focused on studying coverage using mobile wireless sensor networks. Sensor mobility has been recognized as an efficient way to

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Fig. 17.14 Remaining energy versus time

guarantee better sensor deployment. Yang and Cardei [428] dealt with the Movementassisted Sensor Positioning (MSP) problem with a goal to increase the network lifetime. First, they proposed a solution to address the energy-hole problem caused around the sink by computing the desired non-uniform planar sensor density in the monitored area. Then, they proposed a centralized algorithm to relocate mobile sensors while satisfying the planar density requirement with minimum cost. Wu and Yang [420] proposed a method, called Scan-based Movement-Assisted sensoR deploymenT (SMART), to achieve a balance state by balancing the workload of the sensors while avoiding the communication-hole problem in wireless sensor networks. SMART uses clustering, where a planar rectangular deployment field is partitioned into planar mesh, which also partitioned into one-dimensional arrays by rows and columns, and each square area is assigned a cluster-head. These rows and columns are scanned to determine the overload and under-load in clusters so the load is shifted from overloaded clusters to under-loaded clusters in order to achieve a balance state. Wu and Yang [419] took a step further and proposed an optimal but centralized approach based on the Hungarian method to minimize the total moving distance. They also proposed non-optimal distributed solutions for the same purpose based on the scan-based approach in [420]. Rao and Kesidis [334] investigated mobility in mission-oriented wireless sensor networks, where a sensor moves to a location so it can perform any task or all the tasks in a better fashion, and hence the notion of purposeful mobility. Using this type of controlled mobility, Cao et al. [99] proposed techniques for mobility assisted sensing and routing while considering the computation complexity, network connectivity, the energy consumption due to both mobility and communication, and the network lifetime. Wang et al. [393] addressed the problem of how to meet sensing coverage requirements using mobile sensors. They proposed a Grid-Quorum solution that locate the closest redundant sensors with low message complexity and relocate them in a timely, efficient and

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559

balanced way using cascaded movement. Du and Lin [138] proposed an approach to improve the performance of wireless sensor networks in terms of coverage, connectivity, and routing by introducing a few mobile sensors in addition to the static ones, which constitute the majority of the sensors in the network. Moreover, they proposed several schemes for effective sensor mobility that helps achieve the abovementioned goals. Wang et al. [392] proposed a proxy-based approach that allows the sensors to move directly to their target locations and not a zig-zag way with a goal to provide satisfactory coverage. In harsh environments, such as battlefields, sensor mobility helps ensure the required coverage, where mobile sensors can reach areas that cannot be reached by static sensors. Wang et al. [391] proposed a Voronoi diagram-based approach to detect coverage holes and three approaches, namely VEC (VECtor-based), VOR (VORnoi-based) and Minmax, each of which enables the sensors to move from densely deployed areas to sparsely deployed areas using less time, movement distance, and message complexity. Heo and Varshney [197] proposed distributed, energy-efficient deployment algorithms that employ a synergetic combination of cluster-structuring and a peer-to-peer deployment scheme. Also, they proposed an energy-efficient Voronoi diagram-based deployment algorithm. Wang et al. [394] proposed solutions to two related deployment problems in wireless sensor networks, namely sensor placement and sensor dispatch, in the presence of obstacles. For the first problem, they proposed a solution that considers arbitraryshaped obstacles as well as arbitrary relationship between the communication and sensing ranges of the sensors. For the second problem, they proposed centralized and distributed approaches. While the centralized one exploits the results for sensor placement and converts the dispatch problem to the maximum-weight maximum-matching problem whose objective function is minimizing the total energy consumption due to mobility or maximizing the average remaining energy of the sensors after movement, the distributed solution enables the sensors to compute their moving directions autonomously. Also, Wang and Tseng [398] generalized their solutions to the sensor selection and dispatch problems [394] by considering multi-level coverage without obstacles. To the best of our knowledge, this is the only work on mobile k-coverage, which is compared to our k-coverage approach discussed in this chapter.

17.8 Conclusion In real-world sensor deployment, it may not be always possible to cover the entire monitored planar field due to sensor sparseness. It is necessary to account for such sparse wireless sensor networks by providing the required level of coverage, where events can occur in any region in the planar field. In this chapter, we consider sparsely deployed wireless sensor networks along with two essential dimensions, namely sensors mobility and heterogeneity. More precisely, we propose an energy-efficient four-tier architecture to provide k-coverage in this type of network. This layered architecture includes one static sink (or central gathering point), mobile proxy sinks,

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mobile sensors, and static sensors. Moreover, all of those sensors have different capabilities with regard to their sensing and communication capabilities as well as their power supplies. We propose a k-coverage protocol that exploits sensor mobility to achieve k-coverage of any region of interest. We find that our on-demand k-coverage protocol outperforms an existing mobile k-coverage protocol, called Competition [398], with regard to the network lifetime and the number of active sensors to ensure k-coverage. On top of k-covered network configurations, we propose two data gathering protocols that use mobile proxy sinks to deliver the collected sensed data to the static sink. The first one, called direct data gathering protocol, uses only one mobile proxy sink that moves toward the static sink and delivers data to it. However, the second one, called chain-based data gathering protocol, uses the concept of forwarding chain to deliver data to the static sink through multi-hop communication paths that include several mobile proxy sinks. For energy-efficient data forwarding, we compute the minimum transmission distance between any pair of consecutive mobile proxy sinks in the forwarding chain as well as the corresponding optimum number of mobile proxy sinks forming this chain. Simulation results show that the chain-based data gathering protocol outperforms the direct data gathering protocol in terms of energy consumption. In particular, this result shows the benefits of using a short-range based data forwarding scheme to route sensed data toward the static sink via the mobile proxy sinks forming the forwarding, where each pair of consecutive mobile proxy sinks are equidistant from each other. Furthermore, we find that our k-coverage approach has better performances for heterogeneous rather than for homogeneous wireless sensor networks. Also, we consider a variety of group and random waypoint mobility models to assess their impact on our proposed k-coverage approach. We find that group mobility models, such as reference point group mobility model and Manhattan mobility model, are more convenient than random mobility models, such as random waypoint mobility model, given the collaborative nature of wireless sensor networks. More specifically, the reference point group mobility model exhibits the best performance compared to Manhattan mobility model and random waypoint mobility model. In addition, Manhattan mobility model has better performance in comparison with the random waypoint mobility model, with respect to various metrics, namely data delivery ratio, delay, routing overhead, and consumed energy.

Chapter 18

Planar Approaches for Joint k-Coverage and Data Collection Using Heterogeneous Duty-Cycled Sensors

Our first endeavors are purely instinctive prompting of an imagination vivid and undisciplined. As we grow older reason asserts itself and we become more and more systematic and designing. But those early impulses, though not immediately productive, are of the greatest moment and may shape our very destinies. Indeed, I feel now that had I understood and cultivated instead of suppressing them, I would have added substantial value to my bequest to the world. But not until I had attained manhood did I realize that I was an inventor. Nikola Tesla (1856–1943)

Overview This chapter focuses on the problem of joint k-coverage and data collection in planar heterogeneous wireless sensor networks. It is well known that in static wireless sensor networks with constant data reporting, the sensors nearer the sink are responsible for forwarding data to it on behalf of all other sensors in the network. Those sensors suffer a severe batter power depletion problem, also known as the energy sink-hole problem. In this chapter, we study the above problem in dutycycled connected k-covered wireless sensor networks. In order to cope with this problem, this chapter proposes a few solutions based on mobile proxy sinks, which are responsible for collecting the sensed data from source sensors and dropping them off at a static sink. It provides a formal analysis of the performance of joint mobility and routing in this type of network. These are fundamental results for the design of duty-cycled connected k-covered wireless sensor networks. More precisely, it investigates the best mobility strategy of mobile proxy sinks to minimize the total energy consumption in data collection. To this end, it discusses two types of three-tier architectures, which have static source sensors, static sinks, and mobile proxy sinks. For the first architecture, a planar circular field is divided into a set of concentric circular bands of constant width. A few joint mobility and routing schemes are proposed for data collection based on the number of static sinks and mobile proxy sinks. This chapter provides a thorough analytical model of these schemes. However, for the second architecture, a planar circular field is divided into a set of concentric circular bands of varying widths. Also, it discusses a data collection protocol for this type of architecture. In addition, it shows the performance evaluation by simulation of the proposed architectures and the associated protocols for data collection. The results © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_18

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show significant improvement of the network lifetime compared to a solution without mobile proxy sinks.

18.1 Introduction The connected k-coverage problem in wireless sensor networks has been studied extensively in the literature [69, 182, 210]. Unfortunately, most existing work focused on homogeneous wireless sensor networks, where all the sensors have the same capabilities with regards to their storage, computational power, sensing range, communication range, and initial energy, to name a few. Also, the problem of k-coveragepreserving sensor duty-cycling in homogeneous wireless sensor networks has gained considerable attention [39, 398, 425, 447, 460]. However, this type of homogeneous sensors poses a severe restriction on the design of real-world sensor net applications given that all the sensors are required to have the same above-mentioned capabilities. In general, this assumption is unrealistic as the sensors may not necessarily have the same capabilities even when they are built by the same company. In other words, sensors may be heterogeneous. It is worth mentioning that the problem of selecting a minimum subset of sensors to k-cover a planar field using homogeneous sensors is NP-hard [182]. With heterogeneous sensors, this problem is also NP-hard. Intuitively, the NP-hardness in the homogeneous case leads to the NP-hardness in the heterogeneous case. Thus, we propose approximation algorithms to solve the joint k-coverage and data collection problem in heterogeneous wireless sensor networks with mobile sinks. Our work in this chapter is motivated by the following observations. First, in realworld applications, sensor nets have heterogeneous sensors which do not necessarily have the same capabilities (i.e., sensing range, communication range, energy, storage, computation, etc.). These networks have a potential to increase the network lifetime and reliability without causing significant increase in their cost [433]. For instance, Intel deployed two types of sensors (line-powered and battery-power sensors) in the design of a pilot application of sensor nets in order to monitor the health of mechanical equipment in its fabrication plants [433]. While line-powered sensors can be attached to pumps and motors in the fabrication plant, battery-power sensors can be used to reduce installation cost and complexity. Indeed, Yarvis et al. [433] presented several analytical, and testbed results showing the potential benefit and impact of energy and heterogeneity on sensor nets, where all the sensors report their data to a sink. Second, the use of static sink potentially yields the problem of energy sink-hole, where the sensors around the sink are continuously involved in forwarding data on behalf of all other sensors to the sink. This causes a severe problem of depletion of their battery power, thus, isolating the sink. To remedy this problem, we use mobile sinks for data collection. Next, we present the major tasks we want to accomplish in this chapter. In addition, we briefly discuss how to achieve each one of them.

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18.1.1 Major Tasks In this chapter, we consider heterogeneous sensors, which are deployed in a planar field of interest. These sensors may differ by their sensing range, communication range, and/or energy reserve. Our focus is on the problem of joint k-coverage and data collection in heterogeneous duty-cycled wireless sensor networks using one or multiple mobile sinks. More specifically, we attempt to address the energy sink-hole problem, which may occur during data collection in connected k-covered wireless sensor networks. Our study of this problem is motivated by the existence of a wide range of real-world sensing applications that deploy static sensors and sinks. It is necessary that this problem be addressed and energy-efficient solution be provided for the network operational effectiveness and lifetime elongation. We believe that the latter depend on two design decisions, namely whether the sensors are homogeneous or heterogeneous, and whether they are mobile or static. To solve the energy sinkhole problem, we deploy heterogeneous sensors that have different initial energy. This helps extend the network reliability and lifetime [298, 433]. Also, we use mobile sinks in order to achieve balanced load among all the sensors in the network. We expect that the joint use of sensor heterogeneity and sink mobility be able to efficiently address the problem of energy sink-hole in duty-cycled connected k-covered wireless sensor networks. This combination of sensor heterogeneity and sink mobility provides a more realistic and accurate view of the network design for a variety of sensing applications. Next, we present these tasks and briefly describe our corresponding plan of actions. First, we want to design a sensor deployment architecture and specify all of its participating components so as to avoid the energy sink-hole problem in duty-cycled connected k-covered wireless sensor networks. Hence, we introduce a general framework that jointly considers k-coverage and data collection in wireless sensor networks using one or multiple mobile sinks. This framework accounts for heterogeneous sensors, which may not have the same sensing range, communication range, and/or initial energy. Specifically, we propose two types of three-tier architectures, which forms the basis for addressing the energy sink-hole problem in duty-cycled connected k-covered wireless sensor networks. They differ by the number of deployed static and mobile sinks for their joint k-coverage and data collection task. Notice that for the k-coverage component, we exploit our previous results [39] that are discussed earlier in Chap. 5 and which are based on Helly’s Theorem [85] and the geometric properties of the Reuleaux triangle [481]. Precisely, we suggest a pseudo-random sensor deployment approach, where the sensors are deployed in a planar circular field according to their strengths with regard to their sensing range, communication range, and initial energy. Then, we propose energy-efficient centralized and distributed protocols for generating energy-efficient k-coverage configurations in heterogeneous wireless sensor networks based on this pseudo-random deployment approach. Second, we need to find out how the mobile proxy sinks would move in order to ensure uniform energy consumption of all the sensors and cope with the problem

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of energy sink-hole in duty-cycled connected k-covered wireless sensor networks. To achieve this goal, we investigate the optimal mobility strategy of the sink during data collection with a goal to minimize the average total energy consumption due to both of data transmission and reception between the source sensors and mobile proxy sinks as well as sink mobility in a planar circular sensor field. To this end, we divide the planar field into concentric circular bands either with the same width for the first three-tier architecture, or with different widths for the second three-tier architecture. Also, we derive a closed-form solution for the optimal sink mobility. To the best of our knowledge, this chapter provides the first analysis of the best mobility trajectory of the sink during data collection in heterogeneous k-covered wireless sensor networks. Third, we want to determine how the sensed data would be collected from source sensors by mobile sinks and dropped off at the static sinks so as to minimize the average total energy consumption. First, we use our proposed three-tier architectures for data collection using joint mobility and routing on top of energy-efficient connected k-coverage configurations. We use the above analysis of the optimal sink mobility to design data collection protocols using mobile proxy sinks on top of our multi-tier sensor deployment architecture. For the first three-tier architecture, we propose four data collection schemes depending on the numbers of static sinks and mobile proxy sinks in a planar circular field. For the second three-tier architecture, we discuss two data collection protocols. The first protocol is based on an adaptive hybrid forwarding scheme, where the sensors could adaptively choose between short-range and long-range forwarding based on their location with respect to the sink. Moreover, as it is hybrid, the sensors take advantage of deterministic and opportunistic forwarding by specifying in their sensed data packets m active candidate next forwarders from their communication neighbor set, where m > 1. The second one uses a mobile sink to collect the sensed data from all the sensors in the network. For each data collection scheme, we give the placement of static sinks and mobile proxy sinks, and show how the latter collect data from source sensors and deliver them to the former. Furthermore, we provide a theoretical analysis of each of those data collection protocols. Fourth, we want to assess the performance of the proposed two three-tier architectures along with their joint k-coverage and data collection protocols, and corroborate it with simulation results. To this end, we corroborate our analysis with simulation results to assess the performance of both frameworks. In addition, we provide a comparison study among the proposed protocols for k-coverage and data collection in this chapter. Furthermore, we compare our proposed schemes against other solutions that do not use mobile proxy sinks. Our goal is to show the utility of sink mobility and its impact on the joint protocol performance.

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18.1.2 Chapter Organization The remainder of this chapter is organized as follows. Section 18.2 presents a basic two-tier architecture, which suffers the energy sink-hole problem in k-covered wireless sensor networks using static homogeneous and duty-cycled sensors, and provides an analysis of its energy consumption. Section 18.3 discusses a three-tier architecture, which helps cope with the energy sink-hole problem, along with four protocols for joint k-coverage and data collection through the deployment of heterogeneous and duty-cycled sensors. This type of architecture is associated with a planar circular deployment field that is sliced into concentric circular bands of constant width. Section 18.4 focuses on a three-tier architecture to solve the energy sink-hole problem and gives a data collection protocol based on this architecture. The latter is associated with a planar circular deployment field that is sliced into concentric circular bands of varying widths. Finally, Sect. 18.5 concludes the chapter.

18.2 Basic Two-Tier Architecture The challenge is to select a minimum subset of sensors to remain active in order to k-cover a planar field and ensure connectivity between all active sensors. In this type of duty-cycled wireless sensor networks, connectivity is time-varying due to the fact that sensors may be on or off. Also, some sensors may deplete their energy and die. Thus, routing on duty-cycled sensors is challenging as those ones that are selected as next forwarders may not be on or have depleted all of their energy when data reached them. Sensors’ power depletion may have a serious problem that affects the network performance. In particular, this problem gets aggravated in the case of static wireless sensor networks when some specific sensors deplete their energy. Indeed, the sensors nearer the sink are very critical to this problem as they act as the points of contact between the sink and the rest of the sensors in the network. It is easy to check that the sensors around the sink severely suffer from a battery power depletion problem. Those sensors are responsible for forwarding data on behalf of all other sensors in the network to the static sink. Thus, the death of those sensors may yield a coverage hole around the static sink, which prohibits the data from reaching it. This phenomenon is known as the energy sink-hole problem.

18.2.1 Impact of the Energy Sink-Hole Problem It is well known that homogeneous wireless sensor networks using only static source sensors and static sinks suffer from the energy sink-hole problem [259, 316]. Given that k-covered wireless sensor networks are dense in nature, the energy sink-hole

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problem may have negative impact on those sensing applications that require kcoverage of their sensor deployment fields. Precisely, those applications will suffer from some or all of the following problems: • The area surrounding the static sink may not be k-covered as those sensors near the sensors may have depleted their energy. Therefore, the level of k-coverage required by the sensing application may not be achieved in the whole field. • The entire network may be split into at least two disconnected sub-nets. In particular, the static sink may be disconnected from the rest of the network. It is worth noting that connectivity to the sink is the true metric that should be considered when considering network connectivity in wireless sensor networks. In fact, if the sink is not reachable by the sensors, it is as if the network does not have any existence since no sensed data can reach the static sink.

18.2.2 Energy Consumption Analysis In our study of the energy sink-hole problem in duty-cycled connected k-covered wireless sensor networks, we exploit previous results on connected k-coverage in wireless sensor networks and a randomized distributed protocol [39] that are based on Helly’s Theorem [85], which is a fundamental result of the theory of convexity. In this section, we study the basic architecture that has only one static sink, zero mobile proxy sink, and densely deployed source sensors to k-cover a planar field. As stated earlier, this architecture suffers from the energy sink-hole problem. We assume that the source sensors report their data constantly to the sink using a shortpath routing protocol [165]. We show that the source sensors around the sink consume higher energy compared to all other source sensors in the network. In order to compute the maximum average energy consumption of the source sensors, we use a model that is similar to the one in [164]. Precisely, the average energy consumption of a node located in an area of size A2 that forwards traffic for other nodes located in another area of size A1 is proportional to ( A1 + A2 )/A2 . We focus on the source sensors within a distance l ≤ σ ≤ R from the sink, where R stands for the radius of the communication range of the source sensors and l ⋘ ⋙ π ⇌ R is an infinitesimal value, as shown in Fig. 18.1. More specifically, we consider a circle Cσ of radius σ around the static sink is 1 . It is clear that Cσ includes the source sensors that are actively forwarders data on behalf of all other source sensors to is 1 . 2 The number ) sensors inside Cσ is λπ σ , while that of source sensors outside ( 2of source 2 Cσ is λπ D − σ , where D is the radius of the planar circular deployment field C D and λ is the planar sensor density to k-cover the planar field. Let η be the radius of an infinitesimal circular region A whose area is d A = ηdηdθ , where 0 ≤ η ≤ σ and 0 ≤ θ ≤ 2π . The average distance davg between a source sensor in Cσ and the static sink is 1 is computed as follows:

18.2 Basic Two-Tier Architecture

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Fig. 18.1 Average energy consumption analysis

davg

1 = πσ2

∫ Cσ

1 ηd A = πσ2

∫σ

∫2π η dη

dθ =

2

0

2σ 3

0

The energy consumption per source sensor in Cσ is given by ) ( E(Cσ ) = E(Cσ , is 1 ) + E Cσ , is 1 where E(Cσ , is 1 ) is the average energy consumption ( per )source sensor in Cσ to directly send its data to the static sinkis 1 , and E Cσ , is 1 is the average energy consumption per source sensor in Cσ to forward a subset of data packets originated from source sensors in Cσ = C D − Cσ to the sink. Using the energy consumption model defined earlier in Chap. 2 in Sect. 2.5, we obtain 1 E(Cσ , is 1 ) = λπ σ 2

∫ E t x (η)λd A = E elec + Cσ

(

( ) π D2 − σ 2 E Cσ , is 1 = πσ2

)( 2E elec + ε

(

2ε α σ α+2

2σ 3

)α )

Thus, the energy consumption per source sensor using the base protocol is given by

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Fig. 18.2 Plot of E(Cσ )

( E(Cσ ) = ε

2 + α+2

( )α ) ( 2 ) 2 2D σα + − 1 E elec 3 σ2

Figure 18.2 shows that E(Cσ ) increases significantly as we approach the center of the planar sensor field (i.e., location of the sink). It is clear that all sensed data will be forwarded by sensors whose distance from the sink is at most equal to R = 300 m.

18.3 Three-Tier Architecture with Constant Band Width 18.3.1 Proposed Architecture As stated earlier in Sect. 18.2, in static wireless sensor networks with constant data reporting, the sensors nearer the sink are responsible for forwarding data to it on behalf of all other sensors in the network. Those sensors suffer from a severe batter power depletion problem, also known as the energy sink-hole problem. In this section, we study the above problem in duty-cycled connected k-covered wireless sensor networks. In order to change the neighbors of a sink over time, our solutions suggest the use of mobile proxy sinks that collect data from source sensors and drop them off at a static sink. Our proposed three-tier architecture has static source sensors, static sinks, and mobile proxy sinks. First, we provide the first formal analysis of the performance of joint mobility and routing in this type of network. Precisely, we investigate the best mobility strategy of mobile proxy sinks to minimize the total energy consumption for data collection. Second, we propose joint mobility and routing schemes based on the number of static sinks and mobile proxy sinks. We provide a thorough analytical model for our schemes. Finally, we evaluate their

18.3 Three-Tier Architecture with Constant Band Width

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Fig. 18.3 Concentric circular bands with constant width

performance by simulation. Our results show their significant improvement of the network lifetime compared to a solution without mobile proxy sinks.

18.3.2 Joint Mobility and Routing In this section, we propose four different approaches for data collection to address the problem of energy sink-hole problem in connected k-covered wireless sensor networks. We assume that there is at least one static sink, where the collected data is stored. Also, we suppose that any static sink has an infinite source of energy [433]. Specifically, these approaches are based on the number of static sinks and that of mobile proxy sinks. First, we give a decomposition of a planar deployment field. Then, we investigate the optimum mobility of a mobile proxy sink to minimize the average total energy consumption of the source sensors.

18.3.2.1

Planar Sensor Field Deployment Decomposition

We divide a planar circular field of interest of diameter D into n concentric circular bands of width R as shown in Fig. 18.3, where n = D/R with R being the radius of the sensors’ communication range. Our proposed data collection schemes are based on this decomposition. Particularly, the placement of the static sinks and the mobility of mobile proxy as well as their numbers sinks depend on this planar deployment field decomposition.

18.3.2.2

Optimum Proxy Sink Mobility Trajectory

To ensure energy-efficient data collection, we need to specify the mobility trajectory of the mobile proxy sink and how data is being collected from source sensors. The mobile proxy sink will have circular trajectories interrupted by short linear

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moves. First, we determine the optimum mobility trajectory of the mobile proxy sink that yields the minimum energy consumption when collecting data. Theorem 18.1 states this result under the energy consumption model specified earlier in Chap. 2 in Sect. 2.5. Theorem 18.1 (Optimum Proxy Sink Mobility Trajectory) Let ρ = (i − 1)R. The optimum proxy sink mobility trajectory inside a band bi of width R that yields the minimum energy consumption during data collection, corresponds to a circle of radius σ given by σ =

1 √ 2 4ρ + 2R 2 + 4Rρ 2

Proof Consider Fig. 18.4. Assume that a mobile proxy sink mps1 moves inside band bi on the perimeter of a circle of radius ρ + x denoted by C(O, ρ + x), where 0 ≤ x ≤ R. We distinguish two groups of source sensors in band bi . The first group is in band bi,1 of interior and exterior radii ρ and ρ + x, respectively. The second group belongs to band bi,2 of interior and exterior radii ρ + x and ρ + R, respectively. That is, band bi is split into bands bi,1 and bi,2 . Let N 1 and N 2 be the numbers of source sensors in bands bi,1 and bi,2 , respectively. Based on the planar sensor density λ stated earlier in Chap. 5 (Theorem 5.3, Sect. 5.2.2) to ensure k-coverage of a planar field, Ni,1 and Ni,2 can be computed as follows: ) ( Ni,1 = λAi,1 = λπ (ρ + x)2 − ρ 2 ) ( Ni,2 = λAi,2 = λπ (ρ + R)2 − (ρ + x)2 where Ai,1 and Ai,2 stand for the areas of bands bi,1 and bi,2 , respectively, and ρ = (i − 1)R. Fig. 18.4 Sink trajectory mobility

18.3 Three-Tier Architecture with Constant Band Width Table 18.1 Values of xopt depend on bi

571

bi

xopt

b1

0.707R

b5

0.527R

b100

0.50125R

To minimize the average energy consumption due to data transmission and reception, it is necessary that both subsets of sensors in bands bi,1 and bi,2 consume the same amount of energy. Given that the energy consumption depends on the number of source sensors transmitting their data, a balanced energy consumption between bands bi,1 and bi,2 is achieved when Ni,1 = Ni,2 . Hence, we have the equality: ) ( ) ( λπ (ρ + x)2 − ρ 2 = λπ (ρ + R)2 − (ρ + x)2 Thus, the optimum value of x, denoted by xopt , which yields the minimum total energy consumption by all source sensors in both bands bi,1 and bi,2 , is given by xopt = −ρ +

1 √ 2 4ρ + 2R 2 + 4ρ R 2

Thus, the radius σ = ρ + xopt of the mobility circle C(O, σ ) of mobile proxy sink mps1 is given by σ = ρ + xopt =

1 √ 2 4ρ + 2R 2 + 4ρ R 2

∎

Notice that xopt depends on the band being visited by mps1 . As i increases, bi,1 and bi,2 tend to have the same area, and, thus, the same number of source sensors. xopt tends to decrease and reach a value close to R2 . Table 18.1 gives the values of xopt for a few bands, while Fig. 18.5 plots xopt as a function of band bi . Next, we discuss each of our data collection approaches in details and compute their average total energy consumption.

18.3.3 Architecture 1: 1 Static Sink—1 Mobile Proxy Sink There are various scenarios for a mobile proxy sink to collect data. Next, we discuss two specific ones. Scenario 1 ( Single band-based data collection ): In this scenario, the mobile proxy sink mps1 moves inside each band to gather data from the sensors located inside that band. More precisely, mps1 starts its movement at one location ls in band bi , moves on the perimeter of a circle of radius ρ + xopt inside bi , and returns to l s . Once mps1 revisits ls , it goes to the next band by moving through a linear trajectory (see Fig. 18.6). We assume that mps1 moves at a constant speed between 0 and vmax ,

572

18 Planar Approaches for Joint k-Coverage and Data Collection …

Fig. 18.5 xopt versus bi

Fig. 18.6 Mobile proxy sink circular trajectory

and stops at some locations to collect data. The mobile proxy sink mps1 follows the same trajectory pattern until it visits all bands b1 , …, bn and collects data from their respective source sensors. Let us compute the average transmission distance used by all the source sensors. As shown in Fig. 18.7, each source sensor si sends its data to the mobile proxy sink mps1 only when the segment connecting the respective locations of si and mps1 , say ξi and ξ j , is orthogonal to the tangential passing by ξ j located at the perimeter of the ) ( mobility circle C O, ρ + xopt . In fact, this scenario corresponds to the minimum transmission distance si can use to minimize its energy consumption when sending its data. While source sensors in bi,1 transmit their data to mps1 over distances ranging between 0 and xopt , those in bi,2 send their data to mps1 over distances ranging between 0 and R − xopt . Lemma 18.1 computes the average distance between source sensors in band bi,1 and mps1 .

18.3 Three-Tier Architecture with Constant Band Width

573

Fig. 18.7 Optimal sink mobility

Lemma 18.1 (Average Distance 1) The average Euclidean distance, davg,i,1 , between a source sensor in band bi,1 whose width is xopt , and the mobile proxy sink mps1 is computed as davg,i,1 =

1 xopt 3

Proof Consider band bi,1 whose width is equal[ to xopt]. If we unfold bi,1 , we get a diagram that can be modeled by the segment 0, xopt , where each source( sensor) is located at position (0, x) whereas mps1 is positioned at location x j = 0, xopt with x ≤ xopt , as shown in Fig. 18.8. All the locations of the source sensors form a uniform distribution. We pick a source sensor si randomly located at xi = (0, x). The average Euclidean distance between xi and x j is given by ∫

∫ davg,i,1 =

| | | x i − x j |d x i d x j

[0,xopt ] [0,xopt ] ∫ ∫

d xi d x j

[0,xopt ] [0,xopt ] ∫

∫xopt ∫x j

∫ d xi d x j =

[0,xopt ] [0,xopt ]

Fig. 18.8 Mobile proxy sink position mapping

d xi d x j = 0

0

1 xopt 2 2

574

18 Planar Approaches for Joint k-Coverage and Data Collection …

| | ) ( Given that x j = 0, xopt , xi = (0, x), and x ≤ xopt , we deduce that |xi − x j | = x j − xi . Thus, we have ∫

∫

| | |xi − x j |d xi d x j

[0,xopt ] [0,xopt ] ∫xopt ∫x j = 0

(

) 1 x j − xi d xi d x j = xopt 3 6

0

( ) We conclude that davg,i,1 xi , x j = 13 xopt .

∎

Corollary 18.1 states the average distance between a source sensor si in band bi,2 and mps1 , using Lemma 18.1. Corollary 18.1 (Average Distance 2) The average Euclidean distance, davg,i,2 , between a source sensor in band bi,2 whose width is R − xopt , and the mobile proxy sink mps1 is given by davg,i,2 =

) 1( R − xopt 3

∎

Let E avg,i,1 (x) be the average energy consumption that is due to the transmission of data by all the source sensors located inside band bi,1 and its reception by the mobile proxy sink mps1 . Knowing the number of source sensors N1 in band bi,1 and the average distance davg,i,1 , the energy E avg,i,1 (x) is given by ( ) ( ) E E tr x davg,i,1 ((avg,i,1 xopt) = N1)( ) 2 α = λ κ π ρ + xopt − ρ 2 εdavg,i,1 + 2E elec )( ( ) )α ( 2 + 2ρxopt ε xopt /3 + 2E elec = λ κ π xopt Likewise, we denote by E avg,i,2 (x) the average energy consumption, which results from the transmission of data by all the source sensors located inside band bi,2 and its reception by the mobile sink mps1 . Based on the number of source sensors N2 in band bi,2 and the average distance davg,i,2 , the energy E avg,i,2 (x) is computed as follows: ( ) ( ) E avg,i,2 xopt ( = N2 E tr x ( davg,i,2 ) )( ) 2 α = λ κ π (ρ + R)2 − ρ + xopt εdavg,i,2 + 2E elec ( )2 ) ( = λ κ π (ρ + R)2 − ρ + xopt ) )α ) ( (( ε R − xopt /3 + 2E elec Thus, the total average energy consumption, E avg,i (x), due to data transmission by all source sensors located inside band bi and its reception by mps1 , is given by:

18.3 Three-Tier Architecture with Constant Band Width

575

( ) ( ) ( ) E avg,i xopt = E avg,i,1 xopt + E avg,i,2 xopt ) ( Now, the energy spent by mps1 when moving on the circle C O, ρ + xopt located inside band bi is given by ( ) ) ( E mob,C,i xopt = emove 2π ρ + xopt Also, the energy spent by mps1 to move up to the next band bi+1 using a rectilinear trajectory of length R is E mob,i,i+1 (R) = emove R Moreover, after collecting all data from all source sensors in all bands, mps1 should deliver ( it to ) the static sink located in band b1 at the center of the planar field. Let E mob,n,1 xopt be the energy consumed by mps1 to move from band bn to band b1 , where it drops off all collected data to the sink. It is clear that mps1 has to traverse n − 1 bands (bn-1 , …, b1 ), each of which has width equal to R. The distance that mps1 has to travel while in band bn is equal to xopt . Hence, we have ( ) ( ) E mob,n,1 xopt = emove (n − 1)R + xopt The total energy consumption is due to the mobility of mps1 and data transmission by source sensors in band bi(to mps ) 1 , and its reception by mps1 . Thus, the total average tot xopt , per round of data collection from all n bands b1 , energy consumption, E avg …,bn , is given by tot E avg

(

)

(

)

xopt = E mob,n,1 xopt +

i=n−1 Σ

E mob,i,i+1 (R)

i=1

+

i=n Σ

( ) ( ) E avg,i xopt + E mob,C,i xopt

i=1

Scenario 2 (Adjacent bands-based data collection): We assume that mps1 moves on the perimeters of the bands. During its mobility on the perimeter of band bi , mps1 collects data from the source sensors located in the two adjacent bands bi and bi+1 . Then, it takes a linear trajectory to move up to band bi+2 in order to collect data from source sensors in bands bi+2 and bi+3 . As it can be seen, in every round of movement for data collection, mps1 gathers data from a new pair of adjacent bands. The mobile proxy sink mps1 repeats the same movement strategy until it collect data from source sensors in all bands b1 , …,bn . Corollary 18.2 (Average Distance 3) The average Euclidean distance, davg , between a source sensor located in band bi whose width is R and the mobile proxy sink mps1 moving on the outer circle of band bi is computed as follows:

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18 Planar Approaches for Joint k-Coverage and Data Collection …

davg =

1 R 3

∎

Let us compute the total energy consumption due to data collection from the source sensors in all bands b1 , …, bn as well as the proxy sink mobility. We distinguish two cases depending on whether the number of bands n is odd or even. Case 1: n is even (n = 2 m). The mobile ( ) proxy sink mps1 collects data from a pair of adjacent bands. Let E avg,i davg be the average energy consumption for data collection from all the source sensors located in band bi . We have: ( ) ( ) E avg,i davg = Ni E tr x davg Thus, the total average energy consumption, E avg (R), which is due to the transmission of data by the source sensors located in all bands b1 , …, bn and its reception by mps1 , is given by: E avg (R) =

i=n Σ

( ) E avg,i davg

i=1

=

i=2m Σ

( ) Ni E tr x davg

i=1

=

i=n Σ

) ( )( α λ κ π R 2 + 2ρ R εdavg + 2E elec

i=1

( ) where Ni = λAi = λπ (ρ + R)2 − ρ 2 . Now, the mobile proxy sink mps1 moves on the outer circles of the bands b1 , b3 , b5 , …,b2m-1 . That is, mps1 moves on the circles C(O, (2i − 1)R), where i = 1, 2, 3, ..., m. Thus, the energy spent by mps1 during its mobility on the circle C(O, (2i − 1)R) is given by E mob,C,i (R) = emove 2π (2i − 1)R On the other hand, mps1 should move until band bn-1 to collect data from both bands bn-1 and bn . Thus, the energy consumed by mps1 to move from b1 to bn-1 , denoted by E mob,1,n−1 (R), is given by E mob,1,n−1 (R) = emove (2m − 1)R = emove (n − 1)R Let E mob,n−1,1 (R) be the energy consumed by mps1 to move from band bn-1 to band b1 , where it drops off all collected data to the sink. It is clear that mps1 has to traverse n − 1 bands (bn−1 , …, b1 ), each of which has width equal to R. Thus,

18.3 Three-Tier Architecture with Constant Band Width

577

E mob,n−1,1 (R) is given by E mob,n−1,1 (R) = emove (n − 1)R tot Thus, the total energy spent per round of data collection, denoted by E avg (R), is given by tot E avg (R) = E mob,1,n−1 (R) + E mob,n−1,1 (R)

+

i=n Σ

m ( ) Σ E avg,i davg + E mob,C,i (R)

i=1

i=1

Case 2: n is odd (n = 2 m + 1) Now, the mobile proxy sink mps1 collects data from a pair of adjacent bands during each move. However, in the last move, mps1 collects data ( from ) only one band, i.e., bn . Thus, the total average energy consumption, E avg davg , which is due to the transmission of data by all source sensors located inside all bands b1 , …, bn and its reception by the mobile sink mps1 , is given by: i=n ( ) Σ ( ) E avg davg = E avg,i davg i=1

=

i=2m+1 Σ

( ) Ni E tr x davg

i=1

=

i=n Σ

) ( )( α λ κ π R 2 + 2ρ R εdavg + 2E elec

i=1

Now, mps1 moves on the outer circles of the bands b1 , b3 , b5 , …, b2m−1 . That is, mps1 moves on the circles C(O, (2i − 1)R), where i = 1, 2, 3, ..., m. Thus, the energy spent by mps1 during its mobility on the circle C(O, (2i − 1)R) is given by E mob,C,i (R) = emove 2π (2i − 1)R Notice that mps1 should move until band bn to collect data from this band only. Thus, the energy consumed by mps1 to move from band b1 to band bn , denoted by E mob,1,n (R), is given by E mob,1,n (R) = emove (2m + 1)R = emove n R Let E mob,n,1 (R) be the energy consumed by mps1 to move from band bn-1 to band b1 , where it drops off all collected data to the sink. It is clear that mps1 has to traverse n bands (bn , …, b1 ), each of which has width equal to R. Thus, we have

578

18 Planar Approaches for Joint k-Coverage and Data Collection …

E mob,n,1 (R) = emove n R tot Therefore, the total energy spent for data collection, denoted by E avg (R), is computed as follows: tot E avg (R) = E mob,1,n (R) + E mob,n,1 (R)

+

i=n Σ

Σ ( ) m+1 E avg,i davg + E mob,C,i (R)

i=1

i=1

18.3.4 Architecture 2: 1 Static Sink—N Mobile Proxy Sinks Assuming there are n bands, our architecture requires n mobile proxy sinks, namely mps1 , mps2 , mps3 , …, mpsn . As stated in Theorem 18.1, each mobile proxy sink mpsi should be moving inside its band bi along a circular trajectory of radius Rin + xopt , where is Rin is the inner radius of band bi and 0 < xopt < Rout − Rin with Rout being the outer radius of band bi , where Rin = ρ = (i − 1)R and Rout = i R. Thus, mpsi moves on the perimeter of a circle of radius (i − 1)R + xopt . Given that the bands do not have the same inner and outer radii, it is easy to check that the mobile proxy sinks located in the upper (or outer) bands would consume more energy compared to their counterparts that are located in the lower (or inner) bands. It is clear that mobile proxy sinks in the outer bands would receive more data than those located in the inner bands, thus, yielding more energy consumption due to data reception. For instance, the number of source sensors in bands 1 and 2 are respectively given by N1 = λA1 = λπ R 2 ) ( N2 = λA2 = λπ (2R)2 − R 2 = 3λπ R 2 with N2 > N1 . Also, the energy spent by a mobile proxy sink due to its mobility depends on the traveled distance. As it can be seen, the mobility trajectory of mobile proxy sinks in the outer bands is longer than that of mobile proxy sinks in the inner bands, thus, causing more energy consumption due to mobility. For instance, the lengths of the trajectories of the mobile proxy sinks mps1 and mps2 that are moving inside their respective bands b1 and b2 , are respectively given by L 1 = 2π xopt ( ) L 2 = 2π xopt + R

18.3 Three-Tier Architecture with Constant Band Width

579

Fig. 18.9 Mobile proxy sink cone

with L 2 > L 1 . Therefore, to achieve load balancing between all n mobile proxy sinks in terms of their energy consumption, which is due to data reception and their mobility for data collection, we divide the entire planar circular field into n cones centered at the field center. Each cone has an angle equal to 2π/n, as shown in Fig. 18.9. Now, each mobile proxy sink collects data from only one cone. Assuming a uniform sensor deployment, all mobile proxy sinks collect data from the same size set of source sensors. Also, these mobile proxy sinks have the same mobility trajectory as stated below. Thus, this cone-based decomposition strategy of the planar field ensures load balancing among all of the mobile proxy sinks in their data collection task. Initially, each mobile proxy sink mpsi is positioned on the perimeter of the circle that divides bn into bn,1 and bn,2 at one of the two endpoints, say epci , of its respective cone, as shown on Fig. 18.9. For instance, in the first round, mobile proxy sink mpsi follows a top-down movement from the outmost band bn until the inmost band b1 . Precisely, it moves along the arc (epci , vn ) inside band bn , then segment [vn , vn−1 ] whose length is R, then arc (vn−1 , un−1 ) inside band bn−1 , etc. This mobility pattern repeats until mobile proxy sink reaches the first band b1 , where it delivers all collected data to the sink sm . Thus, the trajectory mobility of mobile proxy sink mpsi is a sequence of arcs and segments, (epci , vn ), [vn , vn−1 ], (vn−1 , un−1 ), [un−1 , un−2 ], (un−2 , vn−2 ), [vn−2 , vn−3 ], (vn−3 , un−3 ), …, as shown in Fig. 18.9. After dropping off the data to the static sink is1 in this first round, the mobile proxy sink mpsi moves back to its original position epci at a constant speed vmax , where it pauses for some time t pause . It is clear that the sum of time spent by mpsi to reach epci and t pause are needed for the sources sensors to obtain data that will be picked up in the next round of data collection. Let us compute the total energy consumption due to data transmission by all the source sensors in a cone and their reception by the corresponding mobile proxy sink. at the planar circular field. Consider the portion of band bi , Let ∇ j be a cone ( centered ) denoted by bi ∇ j , which is included in the cone ∇ j(that) is being managed by ( mobile ) ∇ . As it can be seen in Fig. 18.10, b is divided into b proxy sink mps j i j i,1 ( ) ( ) ( ) (∇ j ) and bi,2 ∇ j . Let Ni,1 ∇ j and Ni,2 ∇ j be the number of source sensors in bi,1 ∇ j and

580

18 Planar Approaches for Joint k-Coverage and Data Collection …

Fig. 18.10 Mobile proxy sink band

( ) bi,2 ∇ j , respectively. Then, we have: ) ( ) 1 (( )2 Ni,1 ∇ j = λπ ρ + xopt − ρ 2 n ) 1 ( 2 = λπ xopt + 2ρxopt n ( ) 1 ( )2 ) ( Ni,2 ∇ j = λπ (ρ + R)2 − ρ + xopt n ) ) ( 1 ( 2 = λπ R 2 + 2ρ R − xopt − xopt n The energy spent due to) collect the data from ( to)data transmission and reception ( all source sensors in bi ∇ j is denoted by E avg,i,∇ j R, xopt , n and given by: ( ( ) ( ) ( ) ( ) ) E avg,i,∇ j R, xopt , n = Ni,1 ∇ j E tr x davg,i,1 + Ni,2 ∇ j E tr x davg,i,2 λκπ 2 α ((xopt = + 2ρxopt )(εdavg,i,1 + 2E elec ) n 2 α )(εdavg,i,2 + 2E elec )) + (R 2 + 2ρ(R − xopt ) − xopt ( ) Thus, the total transmission and reception energy, denoted by E avg,∇ j R, xopt , n , ( ) which is needed to collect the data from all portions of bands bi ∇ j , i = 1, . . . , n, in the cone ∇ j is given by: n ( ( ) Σ ) E avg,∇ j R, xopt , n = E avg,i,∇ j R, xopt , n i=1

Now, let us compute the energy spent by a mobile proxy sink in one round of data collection due to its mobility. As discussed earlier, the mobility trajectory of a

18.3 Three-Tier Architecture with Constant Band Width

581

mobile proxy sink is a sequence of alternated segments and arcs. Each segment has length equal to R except the last segment whose length is xopt . On the other hand, the length of an arc of a circle of radius ϕ and subtending an angle θ is equal to ϕθ . In our case, we have θ = 2π and ϕ varies between xopt (i.e., radius of the bottom n arc located in the inmost band—band b1 ) and (n − 1)R + xopt (i.e., radius of the top bn(). The arc located in the outmost band—band ( ) ) length(of the ) arc(u i , vi ), denoted by (u i , vi ), which separates bi ∇ j into bi,1 ∇ j and bi,2 ∇ j is equal to (u i , vi ) =

) 2π ( ) 2π ( ρ + xopt (i − 1)R + xopt = n n

Thus, the length of the mobility trajectory of a mobile proxy sink from band bn ( ) to b1 is denoted by L mob,n,1,∇ j R, xopt , n and computed as i=n ( ) 2π Σ L mob,n,1,∇ j R, xopt , n = (n − 1)R + xopt + ρ + xopt n i=1

( ) The corresponding mobility energy, denoted by E mob,n,1,∇ j R, xopt , n , is given by ( ( ) ) E mob,n,1,∇ j R, xopt , n = emove L mob,n,1,∇ j R, xopt , n ( ) i=n 2π Σ = emove (n − 1)R + xopt + ρ + xopt n i=1 Also, the mobile proxy sink mpsj has to move up to its initial location epci to start a new data collection round. The energy spend in this movement is denoted by E mob,1,n,∇ j (R) and given by ( ( ) ) E mob,1,n,∇ j R, xopt , n = emove (n − 1)R + xopt Hence, the energy consumed by a mobile proxy sink during its mobility in its cone, denoted by E mob,∇ j (R, xmin , n), is given by (

)

(

E mob,∇ j R, xopt , n = emove 2(n − 1)R + 2xopt

i=n 2π Σ + ρ + xopt n i=1

)

The ( total energy consumption in the data collection, denoted by ) tot R, x E avg,∇ , n , which is due to the transmission of data by all the source opt j sensors and its reception by mobile proxy sink mpsj , is given by ( ( ) ) tot R, xopt , n = E mob,∇ j R, xopt , n + E avg,∇ j (R) E avg,∇ j

582

18 Planar Approaches for Joint k-Coverage and Data Collection …

( = emove 2(n − 1)R + 2xopt +

n Σ

i=n 2π Σ ρ + xopt + n i=1

)

( ) E avg,i,∇ j R, xopt , n

i=1 tot We conclude that the average total energy consumption, denoted by E avg , by all n mobile proxy sinks in one round of data collection is equal to

( ) tot tot R, xopt , n E avg = n E avg,∇ j ( ( ) )) ( = n E mob,∇ j R, xopt , n + E avg,∇ j R, xopt , n

18.3.5 Architecture 3: N Static Sinks—1 Mobile Proxy Sink Assuming there are n bands, our architecture requires n static sinks, namely is1 , …, isn . Each static sink is in charge of one band and will gather the data from its respective source sensors through the single mobile proxy sink mps1 . All static sinks are positioned on a segment line originating from the center of the planar circular sensor deployment field and ending at one of the points on its perimeter. The distance between any pair of adjacent static sinks is equal to R, while the first static sink is1 is located at a distance equal to xopt from the center O of the planar circular field. In other words, every static sink is located at a distance equal to xopt from the inner circle of its corresponding band. In the first round of data collection, mps1 moves from the inmost band b1 toward the outmost band bn . Specifically, mps1 moves on the perimeter of a circle of radius xopt inside band b1 to collect data and deliver it to the static sink is1 . Then, it moves toward band b2 along a segment of length R and follows a circular trajectory on the perimeter of a circle of radius R + xopt inside band b2 to collect data from source sensors inside b2 and drop it off to the static sink is2 . This pattern repeats until collecting data from all remaining bands b3 , …,bn . After collecting data from band bn and delivering it to the static sink isn , the mobile proxy sink mps1 pauses for some time t pause before starting its second round of data collection. This time t pause is required for all source sensors to get data. During the second round of data collection, mps1 moves from the outmost band bn toward the inmost band b1 using the reverse path while collecting data from the source sensors in their respective bands. This mobility trajectory repeats for the network lifetime. Let E avg,i,1 (x) be the average energy consumption that is due to the transmission of data by all the source sensors located inside band bi,1 and its reception by the mobile proxy sink mps1 . E avg,i,1 (x) is given by ( ) ( ) E avg,i,1 xopt = N1 E tr x davg,i,1

18.3 Three-Tier Architecture with Constant Band Width

583

)( (( )2 ) α ρ + xopt − ρ 2 εdavg,i,1 + 2E elec ( 2 )( ( ) )α = λ κ π xopt + 2ρxopt ε xopt /3 + 2E elec

= λκ π

Likewise, we denote by E avg,i,2 (x) the average energy consumption, which results from the transmission of data by all the source sensors located inside band bi,2 and its reception by the mobile proxy sink mps1 . E avg,i,2 (x) is computed as follows: ( ) ( ) E avg,i,2 xopt = N2 E tr x davg,i,2 ( )2 )( α ) ( = λ κ π (ρ + R)2 − ρ + xopt εdavg,i,2 + 2E elec ( ))( (( ) )α ) ( 2 = λ κ π R 2 − xopt + 2ρ R − xopt ε R − xopt /3 + 2E elec The total average energy consumption, denoted by E avg (R, xopt ), which is due to the transmission of data by all the source sensors located inside all bands b1 , …, bn and its reception by the mps1 , is given by: n ( ) Σ ( ) ( ) E avg R, xopt = E avg,i,1 xopt + E avg,i,2 xopt i=1

Recall that the mobility trajectory of mps1 is a sequence of alternated circles and segments. Assuming the mobile proxy sink mps1 is originally at the center of the planar field, the length of the first segment is xopt , while the length of any subsequent segment is R. Also, the radius of the first mobility circle along which mps1 moves is xopt , whereas the difference of the radii of two consecutive mobility circles is equal to R. To summarize, when it reaches band bi , mps1 moves on a circle of radius xopt + (i − 1)R. Thus, the energy spent by mps1 during its mobility to collect data from all active source sensors is given by: (

)

(

E mob R, xopt , n = emove xopt + (n − 1)R + 2π

i=n Σ

) xopt + (i − 1)R

i=1

( ) tot xopt , which is necessary for the data Thus, the total energy consumption, E avg collection process, is given by ( ) ( ) tot xopt = E avg (R) + E mob R, xopt , n E avg

584

18 Planar Approaches for Joint k-Coverage and Data Collection …

18.3.6 Architecture 4: N Static Sinks – N Mobile Proxy Sinks Having only one static sink would incur significant delay and yield considerable energy consumption due to proxy sink mobility to reach the static sink and deliver data to it. Assuming there are n bands, our architecture includes n static sinks and n mobile proxy sinks. Each band has one static sink and one mobile proxy sink. Each mobile proxy sink moves inside its band at a constant speed between 0 and vmax along a circular trajectory to collect data from its source sensors. Then, it moves toward its static sink to deliver all gathered data to it. To summarize, mobile proxy sink mpsi moves on the perimeter of a circle of radius (i − 1)R + xopt inside band bi . Each static sink isi is positioned deterministically inside one band. Specifically, is1 is located at the perimeter of a circle of radius xopt inside band b1 at location (xopt , 0) while is2 is positioned at the perimeter of a circle whose radius is xopt + R inside band b2 at location (0, xopt + R). Then, is3 is positioned at the perimeter of a circle of radius xopt + 2R inside band b3 at location (−xopt − 2R, 0) whereas is4 is located at the perimeter of a circle with radius xopt + 3R inside band b4 at location (0, −xmin −3R). Then, is5 is positioned at the perimeter of a circle of radius xopt +4R inside band b5 at location (xopt + 4R, 0), etc. To summarize, the static sink isi is located at the perimeter of a circle of radius xopt + (i − 1)R in band bi at location (x, y), where 1 ≤ i ≤ n and x and y are computed as follows: (x, y) = (xmin + (i − 1)R, 0)

if i mod 4 = 1

(x, y) = (0, xmin + (i − 1)R)

if i mod 4 = 2

(x, y) = (−xmin − (i − 1)R, 0)

if i mod 4 = 3

(x, y) = (0, −xmin − (i − 1)R)

if i mod 4 = 0

It is worth mentioning that we propose such a placement of static sinks to avoid interference and collisions due to simultaneous transmission of data by source sensors to their respective mobile proxy sinks. Let E avg,i,1 (x) be the average energy consumption that is due to the transmission of data by all the source sensors in band bi,1 and its reception by mobile proxy sink mpsi . This energy E avg,i,1 (x) is computed as: ( ) ( ) E avg,i,1 xopt = Ni,1 E tr x davg,i,1 (( )( )2 ) α = λ κ π ρ + xopt − ρ 2 εdavg,i,1 + 2E elec ( 2 )( ( ) )α = λ κ π xopt + 2ρxopt ε xopt /3 + 2E elec

18.3 Three-Tier Architecture with Constant Band Width

585

Let E avg,i,2 (x) be the average energy consumption that results from the transmission of data by all the source sensors located inside band bi,2 and its reception by the mobile sink mpsi . This energy E avg,i,2 (x) is given by ( ) ( ) E avg,i,2 (xopt = Ni,2 E tr x davg,i,2 ) )2 ( = λ κ π (ρ + R)2 − ρ + xopt ) )α ) ( (( ε (R − xopt /3 +(2E elec )) 2 + 2ρ R − xopt = λ κ π R 2 − xopt ( (( ) )α ) ε R − xopt /3 + 2E elec The total average energy consumption, E avg (x), due to the data transmission by all the source sensors in band bi and its reception by mpsi , is given by: ( ) ( ) ( ) E avg,i xopt = E avg,i,1 xopt + E avg,i,2 xopt Also, the energy spent by mpsi during its mobility along the perimeter of a circle of radius xopt + (i − 1)R inside band bi is given by ( ) E mob,i (R, xmin , n) = emove 2π xopt + (i − 1)R ( ) tot xopt , which is spent by all the Thus, the average total energy, denoted by E avg source sensors and mobile proxy sinks in the data collection process is ( ) Σi=n ( ) tot E avg E avg,i xopt + E mob,i (R, xmin , n) xopt = i=1

18.3.7 Performance Evaluation In this section, we present the simulation results of our data collection protocols using a high-level simulator written in C language. First, we specify our simulation environment. Then, we discuss our simulation results.

18.3.7.1

Simulation Environment

We consider a planar circular deployment field of radius D = 1000 m. We assume that the radii of the sensing and communication ranges of all the sensors (including both of the source sensors and mobile proxy sinks) are r = 50 m and R = 100 m, respectively. Thus, the number of concentric circular bands is n = D/R = 10. Moreover, we consider the free-space model, where α = 2. In addition, according to the energy model given in [435], the energy consumption in transmission, reception, idle, and

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sleep modes are 60 mW, 12 mW, 12 mW, and 0.03 mW, respectively [435]. We suppose that the initial energy of each source sensor is 60 J, while that of each mobile proxy sink is 100 J. To the best of our knowledge, there is no experimental result that has reported in the literature regarding the energy emove that is spent per unit of distance due to sensor mobility. In the absence of such an empirical value, we assign two different values to emove to assess the impact of mobility. We assume that the moving speed of each proxy sink is constant and equal to 1 m/s. All simulations are repeated 100 times and the results are averaged.

18.3.7.2

Simulation Results

We use the distributed connected k-coverage protocol, called DIRACCk , which is proposed for homogeneous wireless sensor networks, where all the source sensors and the single static sink are static [39]. Here, we focus on the energy consumption due to the data collection process (data transmission, data reception, and mobility). In all the experiments related to Figs. 18.11, 18.12, 18.13, 18.14, and 18.15, we set the value of emove to E Elec , i.e., emove = E Elec = 50 × 10−9 J . Let our data collection protocols described earlier in Sects. 18.3.3, 18.3.4, 18.3.5, and 18.3.6, be called 1–1 protocol, 1–n protocol, n–1 protocol, and n–n protocol, respectively. Figure 18.11 shows the results of the first scenarios of the 1–1 protocol, while Fig. 18.12 shows those of the second scenario. We find a match between theoretical and simulation results. Also, the 1–1 protocol in Scenario 1 gives better results than in Scenario 2. This is due to the different transmission distances used by the source sensors. In Scenario 1, the average transmission distances are davg,i,1 = xopt /3 (in band ) ( bi,1 ) and davg,i,2 = R − x opt /3 (in band bi,2 ), while in Scenario 2, it is equal to davg,i,1 = R/3. Fig. 18.11 Protocol 1–1 (Scenario 1)

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Fig. 18.12 Protocol 1–1 (Scenario 2)

Fig. 18.13 Protocol 1–n

Figure 18.13 shows the theoretical and simulation results of the 1–n protocol. Likewise, there is a match between both results. In Fig. 18.14, we plot both of the theoretical and simulation results of the n–1 protocol. Similarly, we notice that there is a match between both results. Figure 18.15 shows the theoretical and simulation results of the n–n protocol, with a good match between them. Comparing Fig. 18.13 and Fig. 18.14, it is easy to see that the n–1 protocol outperforms the 1–n protocol. While they both have the same energy consumption due to data transmission and reception, the 1–n protocol incurs more mobility given that is has n mobile proxy sinks. Indeed, each of the n mobile proxy sinks has to move towards the center of the planar field to deliver the collected data to the single static sink. Thus, the 1–n protocol is expected to consume more energy than the n–1 protocol. We find that both of the n–1 protocol and n–n protocol have comparable energy consumption. We

588

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Fig. 18.14 Protocol n–1

Fig. 18.15 Protocol n–n

expect the n–n protocol to consume the least amount of energy for data collection. Indeed, compared to the n–1 protocol, there is no rectilinear movement in the n–n protocol as every mobile proxy sink is in charge of only one band. Also, there is no need for any mobile proxy sink in the n–n protocol to move out of its band as there is a static sink in its corresponding band to deliver data to. Thus, the energy savings of the n–n protocol is due to the absence of rectilinear mobility of mobile proxy sinks. Now, we consider a different value of emove in the experiments related to Figs. 18.16 and 18.17, where emove = 0.0008 J. Figure 18.16 shows the energy consumption during data collection and delivery of the 1–1 protocol, while Fig. 18.17 shows that of the of the 1–n protocol. It is clear that the former outperforms the latter. In fact, the 1–n protocol requires more mobility given that all n mobile proxy sinks have to move toward the center of the planar field to deliver the collected data to

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Fig. 18.16 Protocol 1–1 (emove = 0.0008 J)

Fig. 18.17 Protocol 1–n (emove = 0.0008 J)

the static sink. Thus, the selection of the best protocol with regard to the number of mobile proxy sinks to solve the energy sink-hole problem should depend on the mobility cost.

18.4 Three-Tier Architecture with Varying Band Widths In general, wireless sensor networks may have various types of sensors, thus, increasing the network reliability and lifetime [433]. In this section, we focus on heterogeneous k-covered wireless sensor networks to generate energy-efficient kcoverage configurations. To this end, we exploit our previous results on k-coverage

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for homogeneous sensor nets [39] based on Helly’s Theorem [85]. We extend those results to account for the case of heterogeneous sensors deployment.

18.4.1 Proposed Architecture We consider the same multi-tier (or hierarchical) architecture of heterogeneous sensors, which is discussed earlier in Chapter 6 and shown in Fig. 6.2 (Sect. 6.2.2), along with two data collection protocols [25]. While the first protocol is based on an adaptive hybrid forwarding scheme, the second one uses a mobile sink to collect the sensed data from all the sensors in the network. To this end, we investigate the optimal mobility strategy of the sink in order to minimize the average total energy consumption due to both of data communication and sink mobility in a planar circular sensor field. As presented in Chap. 6 (Sect. 6.2.2), we divide the planar field into concentric circular bands with different widths, and derive a closed-form solution for the optimal sink mobility. Moreover, we consider the same deployment strategy (Chap. 6, Sect. 6.2.2), where the sensors are pseudo-randomly deployed in the planar field, in order to exploit the benefits of sensor heterogeneity. Next, we propose a few data collection schemes. In the first set of data collection schemes, we assume all the sensors are static. In the second set, we consider proxy sink mobility and investigate its impact.

18.4.2 Static Data Collection Schemes In this section, we propose three data collection protocols on top of connected kcoverage configurations in wireless sensor networks. While the first two protocols use a deterministic approach, the third one is based on an opportunistic approach, which copes with the shortcomings of the former method. Specifically, our opportunistic forwarding approach will combine the use of both deterministic forwarding schemes along with other parameters to be specified shortly. Furthermore, we assume that all the sensors and the sink do not move (i.e., static). We assume that the sink is located at the center of the planar circular field. Indeed, Luo and Hubaux [281] proved that the center is the optimum position for a sink in terms of energy-efficient data collection. Moreover, each sensor node is aware of the location information of each of its neighboring sensor nodes, as well as the location of the sink. This type of information can be gathered by each sensor node at the beginning of deployment of the sensor nodes. All the sensors located at the different bands are supposed to forward their sensed data to the single static sink using multi-hop communication path through other intermediate sensors. In this type of deterministic geographic forwarding approach [224], a sensor node chooses its next forwarder deterministically and based on some metrics, namely distance to the

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source sensor and remaining energy. More precisely, we consider two deterministic schemes, which we discuss below in details.

18.4.2.1

Scheme 1: Short-Range Data Forwarding

First, each sensor node si prefers to select one of its neighboring sensors that is active and has the highest remaining energy (HRE) as its next forwarder. Obviously, this forwarder is located in the area between si and the sink so the sensed data is forwarded progressively toward the sink (i.e., phenomenon known as positive progress). This design decision ensures that all the neighboring sensors of a sensor node si are equally likely to participate in the forwarding process. In addition, it leads to load balancing in the entire network. Second, sensor node si chooses one of its active closest neighboring sensors (CNS) to act as next forwarder. This design decision is motivated by the fact that the energy consumption of a sensor node due to data transmission is proportional to the distance a message would travel from a sender to a receiver. Hence, sensor node si aims at minimizing its energy consumption, thus, extending its lifetime. Third, sensor node si prefers a neighboring sensor that lies as close as possible to the shortest path (SP) between si and the sink. This latter choice ensures that all selected forwarders will not deviate much from the shortest path between sensor node si and the sink, thus, reducing the delay incurred by a message to reach the sink. To account for these three attribute, namely highest energy reserve, closeness, and shortest path, we propose a metric, which is denoted by HRE_CNS_SP(sj ) and defined as: ( ) H R E_C N S_S P s j [ ( ) ( )] [ ) ( ( ( ))] = Er em s j /δ si , s j × δ(si , sink)/ δ si , s j + δ s j , sink where sj is a neighboring sensor of si (i.e., sj ∈ NS(si )), E rem (sk ) stands for the remaining energy of sensor node sk , and δ(su , sv ) stands for the Euclidean distance between sensor nodes su and sv . It is easy to check that ) ( ( ( )) δ(si , sink)/ δ si , s j + δ s j , sink ≤ 1 It is clear that the maximum value 1 is reached when sj lies on the shortest path (or line segment) between si and the sink. In summary, sensor node si selects its neighboring sensor snf as its next forwarder such that HRE_CNS_SP(snf ) is the highest value among all the neighboring sensors sj of sensor node si . In other words, {[ ( ) ( )] ( ) H R E_C N S_S P sn f = max Er em s j /δ si , s j [ ) ( ( ( ))] } × δ(si , sink)/ δ si , s j + δ s j , sink | s j ∈ N S(si )

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Notice that HRE_CNS_SP increases inversely proportionally to δ(si , sj ). That is, the closest sensor sj to si will lead to maximize HRE_CNS_SP as the transmission distance of sensor si gets smaller. This feature is highly desirable for lowpowered sensors. Indeed, this is the fundamental driving force behind short-range data forwarding. Any selected next forwarder will apply the same algorithm to forward sensed data to the sink.

18.4.2.2

Scheme 2: Long-Range Data Forwarding

This forwarding scheme is similar to the above-mentioned one except that sensor node si prefers one of its farthest neighboring sensors (FNS) to act as next forwarder. To account for these three attributes, namely highest energy reserve, fartherness, and shortest path, we introduce a metric, which is denoted by HRE_FNS_SP(sj ) and defined as: H R E_F N S_S P(s j ) [ )] [ ( ( ))] ( ) ( ) ( = Er em s j /δ s j , sink × δ(si , sink)/ δ si , s j + δ s j , sink Thus, sensor node si selects its neighboring sensor snf as its next forwarder such that HRE_FNS_SP(snf ) is the highest value among all the neighboring sensors sj of sensor node si . In other words, {[ ( ) ( ) ( ) )] [ ( ( H R E_F N S_S P sn f = max Er em s j /δ s j , sink × δ(si , sink)/ δ si , s j } +δ(s j sink))]|s j ∈ N S(si ) Now, one can see that HRE_FNS_SP increases inversely proportionally to δ(sj , sink). That is, the closest sensor sj to the sink (i.e., the farthest sensor from si ) will help maximize HRE_FNS_SP as the transmission distance of sensor si gets larger, thus, minimizing the number of hops (or intermediate relay nodes) between the source sensors and the sink. This is a highly desirable feature for time-critical sensing applications that have hard deadlines. Indeed, this is the fundamental driving force of long-range data forwarding. While Scheme 1 is a short-range data forwarding protocol, Scheme 2 is a long-range data forwarding protocol. Compared to Scheme 1, one can easily notice that Scheme 2 yields higher energy consumption by each sensor. However, it progresses the sensed data faster toward the sink, thus, incurring lower delay. Clearly, Scheme 2 is more suitable for sensing applications where delay is a very critical attribute to the effectiveness of the underlying wireless sensor networks.

18.4.2.3

Major Problems with Deterministic Schemes

Here, we show that both of the deterministic approaches, namely Schemes 1 and 2, suffer from two major problems due to sensor heterogeneity and duty-cycling. Then,

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we describe our third data forwarding scheme, which is opportunistic, in order to cope with these two problems. As it can be noticed, the first problem is that the two above schemes do not benefit from the heterogeneity of the sensors in their respective bands. Indeed, regardless of the bands they belong to, all sensors use the same forwarding scheme: Either short-range data forwarding (Scheme 1) or long-range data forwarding (Scheme 2). It is important to design an adaptive data collection scheme in which data forwarding depends on our layered architecture, where sensors in different bands have different features. To account for sensor heterogeneity, we propose a hybrid forwarding scheme which benefits from Scheme 1 and Scheme 2. Precisely, we require that the sensors in the outer bands forward sensed data over short distances, which helps them save their energy. In fact, those sensors have lower energy and, thus, should have lower forwarding load using short-range data forwarding. Also, the sensors in the inner bands send data over long distances. Indeed, those sensors have higher energy, and, thus, should have higher forwarding load using long-range data forwarding. This helps bypass the sensors nearer the sink, thus, saving their energy. The second problem is that both of Scheme 1 and Scheme 2 use deterministic forwarding where a sensor chooses a next best forwarder based on those three metrics and forwards data to it. As it can be seen, the next forwarder is determined a priori. Notice that the sensors that are selected to k-cover the planar circular field are the only ones that will be responsible for forwarding data to the sink on behalf of others. Moreover, those selected sensors will be active for one round. Some of them (if not all of them) may not be selected in the next round to participate in the k-coverage of the planar circular field, and, thus, remain off (or inactive). Thus, a sensor that has been selected to act as next forwarder for a current round may not remain active in the next one. Thus, because of duty-cycling, sensors holding data to be forwarded to the sink and using deterministic forwarding are not totally certain that their currently awake sensing neighbors would remain awake after data is being forwarded. Clearly, duty-cycling introduces uncertainty at the sender side when selecting a next best forwarder. To cope with this problem, we require the sensors to use opportunistic forwarding [83] where a next best forwarder is decided on-the-fly and after the data is sent. However, with opportunistic forwarding, several active sensors may hear the transmitted data, thus creating high contention at the receiver side to select a next best forwarder. Thus, it is important to find a trade-off between uncertainty due to duty-cycling with deterministic forwarding, and contention due to opportunistic forwarding. Next, we describe our hybrid forwarding approach in details.

18.4.2.4

Scheme 3: Adaptive Hybrid Forwarding

Our proposed forwarding scheme, as described in Fig. 18.18, is adaptive in the sense it exploits the abovementioned design decision, where sensors located away from the sink in the outer bands use short-range data forwarding (or Scheme 1), while sensors belonging to the inner bands use long-range data forwarding (or Scheme 2). Without loss of generality, all sensors in bands 1, 2, . . . , [n/2] − 1 use Scheme 2, while all

594 Fig. 18.18 Scheme 3 (joint k-coverage and adaptive hybrid forwarding protocol)

18 Planar Approaches for Joint k-Coverage and Data Collection …

ALGORITHM : Scheme 3 Procedure Forward_Data Begin 1. sf forwards sensed data to the sink sm using Scheme 1 if sf’s band’s id is greater than or equal to ⎡n/2⎤; otherwise, sf uses Scheme 2 (n being the number of bands in the planar circular deployment field) 2. sf sends an ACK to the previous forwarder and m – 1 candidate forwarders End Begin /* This code section is run by a sender or current forwarder si */ 1. Sort all potential forwarders in a descending order of their remaining energy and break their tie using their closeness to the sink sm 2. Select the first three candidate forwarders, store them in a sensed data packet, and broadcast it using Scheme 1 if si’s band’s id is greater than or equal to ⎡n/2⎤; otherwise, si uses Scheme 2 /* This code section is run by a candidate next forwarder scf */ 3. Exchange messages between the m candidate next forwarders 4. If scf has the highest remaining energy Then 5. scf calls Forward_Data End

sensors in bands [n/2], [n/2] + 1, . . . , n use Scheme 1, where n is the number of bands forming the planar circular field and which can be computed using Eqs. (6.1) and (6.2), which are given earlier in Chap. 6 (Sect. 6.2.2). Also, it is hybrid as it takes advantages of deterministic and opportunistic forwarding approaches in order to achieve good data forwarding performance in terms of data delivery ratio, delay, and control overhead. Precisely, a sensor node si specifies in its sensed data packet the id’s of m active candidate next forwarders (scf ) from its communication neighboring set, where m > 1. Upon receiving this data packet, the m designated sensors exchange short messages including the current value of their remaining energy, provided they are active. The one with the highest value is officially selected as the next forwarder (sf ) of the received data packet. Intuitively, one of the m candidate next forwarders that is active and has the largest remaining energy is selected as the next forwarder. After forwarding this data packet, sf sends an acknowledgment message to the sensor node si and the m – 1 other candidate next forwarders letting them know that the data packet has been forwarded. It is worth noting that these m candidate next forwarders are located in one of the lenses of the sensing disks of sensor node si . Thus, they are able to communicate with each other. To minimize the energy consumption due to the communication between these κ candidate next forwarders, we simply set m = 3, which seems to be a reasonable value. Thus, our adaptive hybrid forwarding scheme

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Fig. 18.19 Width of mobility band

(Scheme 3), which is shown in Fig. 18.19, requires little communication between those m candidate next forwarders.

18.4.3 Mobile Data Collection In this section, we study sink mobility during data collection in order to save the energy of the sensors, thus, extending the whole network lifetime. Based on this study, we propose a data collection protocol using a single mobile sink.

18.4.3.1

Optimal Sink Mobility

Here, we investigate the mobility of the sink within each band in order to collect data from the sensors. Our goal is to minimize the total energy consumption due to data transmission, data reception, and sink mobility. As it can be seen, the shortest mobility Σ trajectory of the sink within a band bj of width wj is a circle of radius ρ = i=1... j−1 wi + x as shown in Fig. 18.19. This circular trajectory splits band bj into two sub-bands, namely bj1 and bj2 . Thus, the mobile sink s0 collects data from the sensors located in bands bj1 and bj2 of widths x and wj – x, respectively. To minimize the energy consumption due to data transmission, each sensor si submits its data directly to the sink s0 only when the line segment connecting the respective locations of si and s0 , say ξi and ξ j , is orthogonal to the tangential passing by ξ j located at the perimeter of the mobility circle of s0 . As discussed earlier in Sect. 18.3.3 (Lemma 18.1), the average transmission distance between sensor si in band bj1 and the mobile sink s0 is computed as follows: davg, j1 =

1 x 3

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Notice that sensors in bj1 transmit their data to mps1 over distances ranging between 0 and xopt , those in bi,2 send their data to mps1 over distances ranging between 0 and R − xopt . Based on the above lemma, it is clear that the average transmission distance between a sensor si in band bj2 and the mobile sink s0 , denoted by davg, j2 , can be computed as follows: davg, j2 =

1 (R − x) 3

Next, we compute the optimum value of x, denoted by x opt , which helps minimize the average total energy consumption due to data transmission and reception as well as sink mobility. Theorem 18.2 computes the value of x opt , which we refer to as the width of mobility band. Theorem 18.2 (Mobility Band Width) The width of mobility band that corresponds to the optimum mobility trajectory of the mobile sink s0 inside band b j is given by xopt, j =

√ −B + Δ 2A

with u, v, and Δ being defined as follows: 2λπ κε R j 3 ⎞2 ⎛ ⎞2 ⎞ ⎛⎛ Σ Σ 2λπ κε ⎝⎝ Σ ⎠ wt +⎝ wt ⎠ + 4∈ R j wt − R 2j ⎠ B= 9 t=1.. j t=1.. j−1 t=1.. j−1 A=

C = 2π emove Δ = B 2 − 4 AC where wj is the width of band bj and Rj is the radius of the communication range of the sensors located in band bj . Proof Let us compute the average total energy consumption for data transmission, data reception, and sink mobility. Assume that emove is the average energy spent when the mobile sink moves a unit of distance. Thus, the energy spent during sink mobility is computed as. ( Σ E avg−mob−bj (x) = 2π emove x +

t=1.. j−1

wt

)

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where wt is the width of band bt , for t = 1.. j – 1. Given that the sensors are uniformly distributed in each band, the number of sensors nj1 and nj2 in sub-bands bj1 and bj2 , respectively, are given by ⎛⎛

Σ

n j1 = λπ ⎝⎝x +

⎞2

⎛

wt ⎠ − ⎝

t=1... j−1

⎛⎛ n j2 = λπ ⎝⎝

Σ

⎞2

wt ⎠ ⎠

t=1... j−1

⎛

Σ

wt ⎠ − ⎝ x +

t=1... j

⎞2 ⎞

Σ

⎞2 ⎞ wt ⎠ ⎠

t=1... j−1

Σ Σ Σ where σ1 = t=1.. j−1 wt , σ2 = x + t=1.. j−1 wt , and σ3 = t=1.. j wt are the inner radius of sub-band bj1 , outer radius of sub-band bj1 (also, inner radius of sub-band bj2 ), and outer radius of sub-band bj2 , respectively. Using Lemma 18.1 and Corollary 18.1 stated above, the total energy spent in data transmission and reception is given by: E avg−tr x−bj (x) = n j1 E avg−tr x−bj1 (x) + n j2 E avg−tr x−bj2 (x) where: ( ( ) ) ) ( x 2 2 + 2E_elec E avg−tr x−bj1 (x) = κ εdavg, j1 + 2E elec = κ ε 3 ) ( 2 E avg−tr x−bj2 (x) = κ εdavg, j2 + 2E elec ) ( ( ) R−x 2 + 2E_elec =κ ε 3 Therefore, we have: E avg−tot−bj (x) = E avg−mob−bj (x) + E avg−tr x−bj (x) To compute the optimum value of x, x opt , in order to minimize the average total energy consumption, we solve the following derivative: ∂ E avg−tot−bj (x) =0 ∂x With a little algebra, we obtain the following unique closed-form solution: xopt, j

√ −B + Δ = 2A

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with A, B, and Δ being defined as follows: 2λπ κε R j 3 ⎞2 ⎛ ⎞2 ⎞ ⎛⎛ Σ Σ 2λπ κε ⎝⎝ Σ B= wt ⎠ + ⎝ wt ⎠ + 4∈ R j wt − R j 2 ⎠ 9 t=1... j t=1... j−1 t=1... j−1 A=

C = 2π emove Δ = B 2 − 4 AC

18.4.3.2

∎

Scheme 4: Sink Mobility-Based Data Collection

Recall that there is only one mobile sink, denoted by ms1 , for data gathering. As mentioned earlier, the mobile sink ms1 adopts a circular trajectory in its mobility in each band for data collection from all the sensors inside the underlying band. More precisely, when gathering data from band bj , the mobile sink ms1 moves inside bj on a circle of radius Ψ j , which is computed as follows: Ψj =

Σ

Rt + xopt, j

t=1... j−1

with Rt being the radius of the communication range of the sensors located inside band bt . Then, ms1 follows a linear trajectory to move to the next band bj+1 , where it will have a similar circular trajectory on a circle of radius Ψ j+1 . This alternation of circular movement and linear movement of the mobile sink repeats until it visit the last band bn . Then, it follows a linear movement toward the center of the planar field of interest until it reaches the circle of radius xopt,1 (as defined earlier in Theorem 18.2), which is located in the first band b1 . The mobile sink ms1 uses the same trajectories in the next round as in the previous one to collect data from each of the visited bands, b1 , b2 , …, bn . Figure 18.20 illustrates this mixed movement of circular trajectory and linear trajectory of the mobile sink during any round of data gathering with respect to two consecutive bands, namely bj and bj+1 . We assume that during any round of data gathering, the single mobile sink ms1 has a constant mobility speed between v, and stops at some locations on its circular trajectories to collect data.

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Fig. 18.20 Linear and circular trajectories of sink mobility

18.4.4 Performance Evaluation In this section, we present a variety of simulation results of our k-coverage and data collection protocols for various network setups using a high-level simulator written in C. First, we specify our simulation environment. Then, we discuss our simulation-based findings with respect to several metrics and scenarios, such as the number of active sensors as a function of the degree of coverage k; homogeneous kcoverage compared to heterogeneous k-coverage; random sensor deployment against pseudo-random sensor deployment; centralized k-coverage compared to distributed k-coverage; delay as a function of the coverage degree k; data delivery ratio depending on the coverage degree k; and adaptive hybrid forwarding using a static sink compared to mobile sink-based data collection.

18.4.4.1

Simulation Setup

We consider a planar circular field of radius D = 1000 m. We use the energy model in [435], where the sensor energy consumption in transmission, reception, idle, and sleep modes are 60 mW, 12 mW, 12 mW, and 0.03 mW, respectively. Following [447], the energy required for a sensor to stay idle for 1 s is equivalent to one unit of energy. Given that the sensors are heterogeneous, we assume that the initial energy of each sensor is between 60 and 80 J. Also, we assume that the mobility speed of the sink is v = 1 m/s during any round of data gathering. All simulations are repeated 200 times and the results are averaged.

600

18.4.4.2

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Simulation Results

Now, we use our proposed data gathering protocol using adaptive hybrid forwarding (Scheme 4). Figure 18.21 shows that PR-Het-DCCk incurs less delay than R-HomDCCk as heterogeneous sensors tend to transmit data over long distances, thus, reducing the number of forwarders between source sensors and the mobile sink ms1 . This results in time saving due to less processing of the data before it reaches the sink. Also, as shown in Fig. 18.22, PR-Het-DCCk has better data delivery ratio than R-Hom-DCCk . This higher delay caused by R-Hom-DCCk may yield data loss since the next forwarder may not be active in the next round. We compare between our proposed data gathering protocol using optimal sink mobility (Scheme 4) as stated in Sect. 18.4.3.2 with our adaptive hybrid data Fig. 18.21 Comparing PR-Het-DCCk and R-Hom-DCCk (delay)

Fig. 18.22 Comparing PR-Het-DCCk and R-Hom-DCCk (data delivery ratio)

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forwarding protocol (Scheme 3), which is discussed in Sect. 18.4.2.4. We consider our PR-Het-DCCk k-coverage protocol. We find that Scheme 4 outperforms Scheme 3 in terms of average delay (Fig. 18.23) and average data delivery ratio (Fig. 18.24). Clearly, Scheme 4 provides more guarantee for delivering data to the mobile sink given its pattern of mobility. Also, Scheme 4 less delay than Scheme 3 for the above reason. This shows the positive impact of optimal sink mobility on the global system performance. Fig. 18.23 Comparing data gathering protocols (delay)

Fig. 18.24 Comparing data gathering protocols (data delivery ratio)

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18.5 Conclusion In this chapter, we discuss two types of three-tier architectures for joint k-coverage and data collection in planar heterogeneous duty-cycled connected k-covered wireless sensor networks. More specifically, these types of architectures form the basis to address the energy sink-hole problem, which is inherent in static wireless sensor networks. The first type of three-tier architecture uses heterogeneous sensors that differ only by their initial energy reserves. The first layer has a set of static source sensors that have the same capabilities and are densely and randomly deployed to achieve k-coverage of a planar circular field. The second layer has one or more static sinks (i.e., central gathering points) that have infinite sources of energy. These sinks are positioned at specific locations in the planar circular field. The third layer contains one or multiple mobile proxy sinks. The latter have the same capabilities, but are more powerful than the source sensors in terms of their initial energy. Their sole task is to collect data from source sensors and drop them off at the static sinks. We use our architecture for data collection using joint mobility and routing on top of energy-efficient connected k-coverage configurations. We propose four data collection schemes depending on the numbers of static sinks and mobile proxy sinks. First, we divide a planar circular field into concentric circular bands, which have the same width. Then, we determine the best mobility strategy of a mobile proxy sink that minimizes the average total energy consumption due to data and control packet communication, as well as sink mobility. Third, we investigate sink mobility in a planar circular sensor field, and computed the mobility trajectory of the sink. Second, for each data collection scheme, we give the placement of static sinks and mobile proxy sinks, and show how the latter collect data from source sensors and deliver them to the former. Third, we provide a rigorous analysis of the performance of each scheme and corroborate it with simulation results. We show that our proposed schemes significantly improve the network lifetime compared to a solution without mobile proxy sinks. To the best of our knowledge, this is the first work that studies the energy sink-hole problem in duty-cycled connected k-covered wireless sensor networks and provides a thorough analysis of joint mobility and routing schemes to cope with it. The second type of three-tier architecture, which is different from the first one, has static source sensors, one static sink, and one mobile proxy sink. We focus on joint k-coverage and data gathering in heterogeneous wireless sensor networks using two different strategies for data collection. The first data collection protocol uses adaptive hybrid forwarding, which runs using one static sink and multiple source sensors. The second data collection protocol is different from the first one in the sense that it also uses one mobile proxy sink. We exploit the optimal sink mobility in a planar circular sensor field that is found earlier for this single mobile proxy sink. First, we propose centralized and distributed connected k-coverage protocols for this type of three-tier architecture, where a planar deployment field is divided into concentric circular bands with different widths. Each band has different type of sensors from all other bands with respect to their initial energy reserve, sensing

18.5 Conclusion

603

range, and communication range. However, each band contains the same type of sensors with regard to the above characteristics. Second, we present an adaptive hybrid forwarding-based data collection protocol, where the sink is static. Our data forwarding protocol using a hybrid scheme benefits from the advantages of deterministic and opportunistic forwarding schemes. We derive a closed-form solution that characterizes the optimal mobility trajectory of the sink. In addition, we suggest a data collection protocol based on this optimal mobility of the sink. The simulation results, not surprisingly and conform to previous results on the impact of heterogeneity on the design of wireless sensor networks [433], show that the proposed kcoverage protocols for heterogeneous sensor networks outperform their counterparts for homogeneous wireless sensor networks. Also, we find that our data collection protocol using our proposed optimal sink mobility is more efficient in terms of both average delay and data delivery ratio compared to our adaptive hybrid geographic forwarding-based data collection protocol.

Part VIII

Connected k-Barrier Coverage in Wireless Sensor Networks

Chapter 19

A Planar Approach for Physical Security Using Connected k-Barrier Coverage

As regards the security of a country against foreign invasion, it is interesting to note that it depends only on the relative, and not the absolute, number of the individuals or magnitude of the forces, and that, if every country should reduce the war-force in the same ratio, the security would remain unaltered. An international agreement with the object of reducing to a minimum the war-force which, in view of the present still imperfect education of the masses, is absolutely indispensable, would, therefore, seem to be the first rational step to take toward diminishing the force retarding human movement. Nikola Tesla (1856–1943)

Overview This chapter investigate the problem of physical security using planar wireless sensor networks, where a critical space is surrounded by a barrier (or a belt of sensors) preventing any intruder’s attempt to cross it and/or access it. Precisely, this chapter focuses on the connected k-barrier coverage problem, where any path crossing this belt intersects with the sensing range of at least k sensors, while all the sensors form a connected network, where k ≥ 1. It considers two planar deterministic sensor deployment strategies: Square lattice and hexagonal lattice wireless sensor networks. For each one of them, it computes the number of intruder’s abstract paths along a k-barrier covered sensor belt region. Then, it provides a polynomial representation of all these abstract paths. Also, it calculates the number of sensors, which is required to achieve k-barrier coverage of a planar belt region. In addition, it computes the length of weakly k-barrier covered path crossing a k-barrier covered sensor belt region. Moreover, it computes the observability of intruder’s abstract path. Furthermore, it extends these results to account for random intruder’s moves across a k-barrier covered sensor belt region. Finally, it shows several simulation results to corroborate the proposed analysis.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_19

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19.1 Introduction For the safety and/or proper operation of critical environments, it is important to provide physical security. In fact, when a critical space undergoes an unauthorized access, this yields a fundamental physical breach, which can be classified as a physical security problem. For example, when people’s offices and/or labs are not occupied, their equipment is vulnerable to theft and damage as it may be stolen by someone. Therefore, it is essential to cope with the physical security problem of facilities and equipment. Notice that a physical security problem could lead to a cybersecurity problem, where the data or information either on paper or in an electronic device is in danger. Physical security is an essential component that enables the detection and prevention of any intrusion. To address the physical security problem, a variety of solutions have been proposed. For instance, from the earliest medieval castles to the middle ages, moats were used as a defensive strategy against an attacking army. Generally, several traditional physical solutions, such as guards, barricades, and fences, to name a few, have been used to protect a facility. However, each of these safeguards provides a single layer of physical security. Because physical security is not just a barb wire fence that should be placed around a facility, advanced technologies with a detection capability, such as video surveillance systems (e.g., cameras), can be used for tracking any unauthorized access to a protected area. In this chapter, we use a layered protection system by leveraging lattice wireless sensor networks, which provide several layers of protection, where each one of them includes several sensors. Our choice of this type of protection strategy is motivated by the fact that when an intruder attempts to traverse one layer of sensors, there is another layer that should be capable of further detecting and preventing such an intrusion activity. Because there exists more than one layer, it is highly likely that an intruder would be detected, which helps avoid any malicious acts. Indeed, it is almost impossible that an intruder penetrates all layers without being caught. The strength of the physical security that is provided would depend on the number of layers forming the protection system, denoted by k. In other words, the higher value of k is, the stronger the physical security of the underlying system is. Next, we present the major tasks we want to accomplish in this chapter. In addition, we briefly discuss how to achieve each one of them.

19.1.1 Major Tasks This chapter addresses the protection of a facility perimeter (e.g., international border), which requires building a barrier in order to prevent any intruder’s attempt to cross this perimeter and access any critical region. Specifically, we focus on the physical security problem in stealthy lattice wireless sensor networks, where a critical area is surrounded by a belt of sensors (i.e., barrier). Specifically, we propose a

19.1 Introduction

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theoretical framework to analyze the connected k-barrier coverage problem [20, 21], where every path crossing this belt of sensors intersects with the sensing range of at least k sensors and all the sensors are connected to each other, where k ≥ 1. In other words, a path is said to be k-barrier covered if some (i.e., at least one) or all of its points are covered by at least k sensors, which are deployed in a barrier. That is, those points on the path intersect with at least k sensors. We propose to analyze the k-barrier coverage problem using a tiling-based approach, where the sensing disks of the sensors are simply tangential to each other. Next, we introduce these tasks and briefly discuss our corresponding plan of actions. First, we want to determine how the sensors are deterministically placed in a belt so that every crossing path of this belt is k-barrier covered, i.e., every path intersects with at least k sensors. To this end, we propose to analyze the k-barrier coverage problem from a tiling perspective, where the sensors’ sensing disks are kissing each other. More precisely, we propose two lattice-based deterministic deployment approaches, where a belt of sensors surrounds a critical area. While the first approach yields a square lattice, the second one produces a hexagonal lattice. In each of these two lattices, any path crossing this belt is guaranteed to be k-barrier covered (or intersecting with at least k sensors). Second, we need to compute the minimum number of sensors required to achieve k-barrier coverage of a belt region. First, we determine whether the above deterministic sensor deployment strategies are the same, and whether they require the same number of sensors to achieve k-barrier coverage. That is, we investigate both approaches to check whether the two square and hexagonal lattices are isomorphic (i.e., have the same structure). Also, we compute the number of sensors necessary to ensure k-barrier coverage of a senor belt for each lattice. Third, we need to determine when connectivity is insured provided that k-barrier coverage is satisfied. We determine the relationship that should exist between the communication range and sensing range of the sensors for each lattice so all the sensors forming a k-barrier covered belt are connected. That is, we calculate the ratio of the communication range to the sensing range of the sensors so a belt is connected k-barrier covered. Fourth, we want to compute the number of possible (or abstract) paths of an intruder crossing a k-barrier covered sensor belt. Also, we should find out how these abstract paths are represented. Thus, we study all possible paths taken by an intruder when accessing a critical area. We introduce the concept of intruder’s abstract path as a sequence of k progressive moves, including left-oblique, right-oblique, and vertical line-segments only. Also, we compute the number of these paths that cross a k-barrier covered sensor belt based on the value of k. In addition, we propose a more compact form to represent all the intruder’s abstract paths. Fifth, we aim at calculating the minimum size of the intersection set among all possible intersection sets between an intruder’s abstract path and k-barrier covered sensor belt. For this purpose, we introduce the concept of intruder’s abstract path observability as the minimum cardinality of the intersection set among all possible intersection sets between an intruder’s abstract path and a k-barrier covered sensor belt. We compute its value for each lattice.

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Sixth, we want study all the intruder’s paths, including those random ones, which may include non-progressive moves by the intruder to access a critical area. Therefore, we generalize the above results to random intruder’s motion across a k-barrier covered sensor belt. To do so, we redefine an abstract path as a sequence of lefthorizontal (a non-progressive move), right-horizontal (a non-progressive move), left-oblique, right-oblique, and/or vertical line-segments. Seventh, we want to analyze those paths originating from one location and targeting another location. Hence, we extend our study of the problem of k-barrier coverage in stealthy lattice wireless sensor networks to account for intruder’s movement from one source location to a destination location. We discuss both of progressive Manhattan and random paths across a k-barrier covered sensor belt in square lattice wireless sensor networks. Eighth, we want to assess the performance of our k-barrier coverage sensor deployment. Thus, we corroborate our analysis of square and hexagonal lattice wireless sensor networks with several simulation results.

19.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 19.2 investigates the kbarrier coverage problem for two deterministic sensor deployment strategies yielding square and hexagonal lattice wireless sensor networks. Section 19.3 generalizes the above discussion to account for random movement of intruders. Section 19.4 discusses the intruder’s movement from one source location to a destination location. Section 19.5 briefly presents other possible generalizations of the proposed study in this chapter. Section 19.6 provides simulation results of our proposed study. Section 19.7 reviews existing related approaches. Section 19.8 concludes the chapter.

19.2 Tiling-Based k-Barrier Coverage In this section, we analyze the k-barrier coverage problem from a tiling perspective. In other words, we tile a sensor belt region so it is k-barrier covered, while there is no overlap between the sensors’ sensing disks. We consider two deterministic sensor deployment strategies, which yield square (Sect. 19.2.3) and hexagonal (Sect. 19.2.4) lattice wireless sensor networks, respectively.

19.2.1 Intruder’s Abstract Path Counting First, we state the assumptions made to study the k-barrier coverage problem in wireless sensor networks.

19.2 Tiling-Based k-Barrier Coverage

611

Assumption 19.1 (Belt Region and Sensor Deployment) All the sensors are deterministically deployed in a rectangular belt region of width w and length l, using square lattice based sensor deployment approach (Chap. 2, Sect. 2.2, Fig. 2.9a) or hexagonal lattice-based sensor deployment approach (Chap. 2, Sect. 2.2, Fig. 2.9b), ∎ where k ≥ 1 and r stands for the radius of the sensing disk of the sensors. Assumption 19.2 (Stealthy Sensors) All the deployed sensors in a belt region are stealthy. ∎ Assumption 19.3 (Intruder Detection) Any intruder moving along a k-barrier covered path when walking through a belt region to cross a border or access a protected area, will surely be detected by at least k sensors, where k ≥ 1 is a natural number. ∎ Here, we exploit the concept of structural k-node line, which is defined earlier in Chap. 2 (Sect. 2.2, Definition 2.40). We present some theoretical results regarding the number and height of structural k-node lines. Theorem 19.1 below computes the number of structural k-node lines for square lattice wireless sensor networks. Theorem 19.1 (k-Node Lines Number for Square Lattice Wireless Sensor Networks) The number of structural k-node lines for square lattice wireless sensor networks is 3k−1 , where k ≥ 1 is a natural number. Proof We can proceed using a proof by mathematical induction on k. Let P(k) be the following statement: P(k) : “There are 3k−1 different structural k-node lines, where k ≥ 1 is a natural number.” Basis step: Let us prove that P(k) is true for k = 1. This is trivial. In fact, there is only one structural 1-node line (30 = 1). Inductive step: We assume that P(m) is true, i.e., there are 3m−1 . different structural m-node lines, where m ≥ 1. We want to prove that P(m + 1) is true. That is, the number oftructural (m + 1)-node lines is 3m . We start from those 3m−1 . different structural m-node linedd the (m + 1)th node to each one of them. Let q be the only leaf node of each of these structural m-node lines. There are three possibilities to add that (m + 1)th node to each of these structural m-node lines so as to produce structural (m + 1)-node lines. In fact, the newly added (m + 1)th node can be a left child, direct child, or right child of q. Therefore, the total number of produced structural (m + 1)-node lines is 3 × 3m−1 = 3m , thus, proving P(m + 1) is true. Thus, the statement P(k) is true, for all k ≥ 1. Indeed, we have the inference rule: (P(1) ∧ (∀m ≥ 1, P(m) → P(m + 1))) → P(k), ∀k ≥ 1 This concludes our proof.

∎

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Fig. 19.1 Crossing intersection point of two kissing sensing disks

Notice that for hexagonal lattice wireless sensor networks, any node, except the leaf node, has only one left node or one right node. That is, there is no vertical node. Indeed, there is one special case when an intruder moves from the sensing disk of a sensor node to another area through the intersection point of two kissing sensing disks of two neighboring sensors. In this situation, we choose one of these two nodes (i.e., either left node or right node). Figure 19.1 illustrates this special case. Theorem 19.2 computes the number of structural k-node lines for hexagonal lattice wireless sensor networks. Theorem 19.2 (k-Node Lines Number for Hexagonal Lattice Wireless Sensor Networks) The number of structural k-node lines for hexagonal lattice wireless sensor networks is 2k−1 , where k ≥ 1 is a natural number. Proof Same proof method as the one for Theorem 19.1, except that a node can have only one left child or right child (i.e., there is no direct child). ∎ Lemma 19.1 calculates the height of a structural k-node line. Lemma 19.1 (Structural k-Node Line Height) The height of a structural k-node line is k − 1, where k ≥ 1. Proof It is easy to show this result using a mathematical induction proof on k. Let Q(k) be the following statement: Q(k) : “A structural k-node line has a height equal to k − 1, where k ≥ 1”

Basis step: Let us prove that Q(k) rue for k = 1. This structural 1-node line has height equal to 0. Thus, Q(1) true. Inductive step: We assume that P(m) is true, i.e., the height of structural m-node line is m − 1, where m ≥ 1. We want to prove that Q(m + 1) is true. That is, the height of a structural (m + 1)-node line is m. We add the (m + 1)th. node to a structural m-node line whose height is m − 1, and attach it to its leaf node as left, vertical, or right child. We obtain a structural (m + 1)-node line whose height is equal to that of the structural m-node line augmented by 1, i.e., (m − 1) + 1 = m. Thus, Q(m + 1) is true. We have the following inference rule, which proves that Q(k) is true for any k ≥ 1. (Q(1) ∧ (∀m ≥ 1, Q(m) → Q(m + 1))) → Q(k), ∀k ≥ 1

19.2 Tiling-Based k-Barrier Coverage

This concludes our proof.

613

∎

Assumption 19.4 (Fast Sensor Belt Region Crossing) An intruder attempts to cross a sensor belt region using a shortest path. ∎ Intuitively, an intruder always aims at crossing a sensor belt region (or barrier) as fast as possible so they can have access to a protected area without being detected. Hence, their movement trajectory is a sequence of progressive moves (or linesegments), where each line-segment is left-oblique, right-oblique, or vertical (i.e., orthogonal to the sensor belt region). That is, an intruder would not make any horizontal move along a sensor belt region, thus, eliminating horizontal line-segment from their movement trajectory. Thus, an intruder’s path can be viewed as a random sequence of left-oblique, right-oblique, and vertical line-segments. Given that all the sensing disks are tangential to each other for both square and hexagonal lattices, the movement trajectory of an intruder can be considered as a sequence of transitions from one sensing disk to another. In other words, all the intruder’s movement trajectory, which allow them to move from the sensing disk of one sensor to that of another sensor, can be abstracted (or summarized) by only one left-oblique, right-oblique, or vertical line-segment. This depends on whether this move is from one sensing disk to another one located at its left, right, or below it, as shown in Fig. 19.2. That is, an intruder’s path can be represented by an abstract path, which is defined earlier in Chap. 2 (Sect. 2.2, Definition 2.39) and is denoted by I A P = (N , E), where the node set N represents the set of sensing disks, and the edge set E stands for transitions between the sensors’ sensing disks. Moreover, any intruder’s abstract path has exactly k nodes and k − 1 edges, i.e., |N | = k and |E| = k − 1. Consequently, we conclude that the number of intruder’s abstract paths along a k-barrier covered sensor belt region S B R w,l corresponds to the number of structural k-node lines as computed in Theorems 19.1 and 19.2. Corollary 19.1 below states this result. Example As shown in Fig. 19.2, the intruder’s abstract path corresponding to their movement trajectory and denoted by I A P = (N , E) can be defined by the following two sets: • N = {s1 , s2 , s3 , s4 , s5 , s6 , s7 } • E = {(s1 , s2 ), (s2 , s3 ), (s3 , s4 ), (s4 , s5 ), (s5 , s6 ), (s6 , s7 )}. where the first node is s1 and last node is s7 , from top to bottom. Corollary 19.1 (Intruder’s Abstract Path Cardinality) The total number of intruder’s abstract paths with k vertices and k − 1 edges along a k-barrier covered sensor belt region is 3k−1 for square lattice, and 2k−1 for hexagonal lattice, where k ≥ 1 is a natural number. ∎ Although the number of paths crossing a sensor belt region is infinite, those paths can be represented by those 3k−1 and 2k−1 intruder’s abstract paths for square lattice wireless sensor networks and hexagonal lattice wireless sensor networks,

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Fig. 19.2 a Intr mement trajectory, b abstract path

respectively. That is, for square lattice wireless sensor networks, all the possible intruder’s paths to cross the sensor belt region can be classified into 3k−1 abstract paths. Notice that those 3k−1 intruder’s abstract paths can be classified into three families. The first family has only one path, which consists of only vertical linesegments, whereas the second and third ones have the same number of abstract paths. Moreover, the abstract paths in the second family are symmetric to their counterparts in the third one. Table 19.1 shows those abstract paths in the three families for k = 2 and k = 3. However, for hexagonal lattice wireless sensor networks, those 2k−1 intruder’s abstract paths can be classified into two families, which have the same number of abstract paths. Furthermore, those paths are symmetric to each other. Table 19.2 shows those abstract paths in those families for k = 2 and k = 3.

19.2 Tiling-Based k-Barrier Coverage Table 19.1 All intruder’s abstract paths for square lattice wireless sensor networks

615 k=2

k=3

1st path family

2nd path family

3rd path family

Table 19.2 All intruder’s abstract paths for hexagonal lattice wireless sensor networks

k=2

k=3

1st path family

2nd path family

19.2.2 Intruder’s Abstract Path Analysis As stated earlier, there are three types of line-segments in an intruder’s abstract path, namely left-oblique, right-oblique, and vertical line-segments. Let x L O , x R O , and x V be three variables denoting those three types of line-segments, respectively. Theorem 19.3 provides a polynomial representation of all intruder’s abstract paths for square lattice wireless sensor networks. Theorem 19.3 (Intruder’s Abstract Path Representation for Square Lattice Wireless Sensor Networks) All the 3k−1 possible intruder’s abstract paths that have exactly k vertices and (k − 1) edges can be represented by the following polynomial: (x L O + x R O + x V )

k−1

=

(

(

Σ k L O +k R O +k V =k−1

k−1 kL O ,

)

=

k−1 k L O , k R O , kV

) x Lk LOO x Rk ROO x VkV

(k − 1)! k L O !k R O !k V !

where k ≥ 1 is a natural number, x Lk LOO x Rk ROO x VkV is an intruder’s abstract path k R O . right-oblique, and k V vertical line-segments, and that has k L O left-oblique, ( ) k−1 is the corresponding total number of such a path. k L O , k R O , kV Proof As per Lemma 19.1, any intruder’s abstract path has a height equal to k − 1. That is, it has exactly (k − 1) levels. Each level contains exactly one type of linesegment, i.e., left-oblique, right-oblique, or vertical. Let us assimilate a level to a

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box, and left-oblique (type 1), right-oblique (type 2), and vertical (type 3) linesegments to three different types of objects. Therefore, we have (k − 1) boxes and three types of objects. Precisely, we have (k − 1) objects of each type, and each of those k − 1 boxes can hold exactly one instance of any type of object. We can select k L O . objects of type 1, k R O objects of type 2, and k V objects of type 3 such that 0 ≤ k L O , k R O , k V ≤ k − 1 and k L O + k R O + k V = k − 1 Basically, we are counting the number of possible permutations of((k − 1) objects)subject to the above two k−1 computes the number of conditions. The multinomial coefficient k L O , k R O , kV distinct ways to permute a multiset of (k − 1) objects. We used the term “multiset” because we allow the use of the same object many times. The factor x Lk LOO x Rk ROO x VkV correspond to a given multiset with k L O objects of type 1, k R O objects of type of a multiset corresponds to an 2, and k V objects of type 3.(Thus, a permutation ) k−1 computes all possible permutation of intruder’s abstract path, and k L O , k R O , kV a given multiset, thus, generating all possible intruder’s abstract paths having exactly k L O left-oblique, k R O right-oblique, and k V vertical line-segments. The summation symbol is used to account for all possible permutations of all possible multisets, thus, producing all possible intruder’s abstract paths by varying the variables k L O , k V , and k R O , subject to the following conditions: 0 ≤ k L O , k R O , k V ≤ k − 1 and k L O + k R O + k V = k − 1, where k ≥ 1.

∎

Table 19.3 shows a few examples of the above results for various values of k. abstract path is reduced to one node. For For instance, for k = (1, the intruder’s ) k = 2, there are three i.e., 32−1 intruder’s abstract paths, each of which has only one edge that could be left-oblique, right-oblique, or vertical. For k (= 3,) there are ( ) 2 nine i.e., 33−1 intruder’s abstract paths. For instance, ( 2 )the first path x L O has only two left-oblique line-segments, ( )the second path x R O has two right-oblique linesegments, and the third path x V2 has two vertical line-segments. Also, there are two paths (2 × x L O × x R O ), each of which has one left-oblique and one right-oblique line-segments; two paths (2 × x L O × x V ), each of which has one left-oblique and one vertical line-segments; and two paths (2 × x R O × x V ), each of which has one Table 19.3 Polynomial representation of all intruder’s abstract paths k (x L O + x R O + x V )k−1 1 1 2 x L O + x R O + xV 3 x L2 O + x 2R O + x V2 + 2 × x L O × x R O + 2 × x L O × x V + 2 × x R O × x V 4 x L3 O + x 3R O + x V3 + 3x L2 O x R O + 3x L2 O x V + 3x L O x 2R O + 3x L O x V2 + 3x 2R O x V + 3x R O x V2 + 6x L O x R O x V

19.2 Tiling-Based k-Barrier Coverage

617

right-oblique and one vertical line-segments. We use the same interpretation for k = 4. Theorem 19.4 gives a polynomial representation of all intruder’s abstract paths for hexagonal lattice wireless sensor networks. Theorem 19.4 (Intruder’s Abstract Path Representation for Hexagonal Lattice Wireless Sensor Networks) All the 2k−1 possible intruder’s abstract paths that have exactly k vertices and (k − 1) edges can be represented by the following polynomial: (x L O + x R O )

k−1

(

(

Σ

=

k L O +k R O =k−1

k−1 kL O , k R O

)

=

) k−1 x kL O x kRO kL O , k R O L O R O

(k − 1)! k L O !k R O !

where k ≥ 1, x Lk LOO x Rk ROO is an intruder’s abstract path that has k L O left-oblique ( ) k−1 is the corresponding total and k R O right-oblique line-segments, and kL O , k R O number of such a path. Proof We follow the same proof as the one for Theorem 19.3. But, there is no vertical line-segment. ∎ Let Ω SL I A P (k L O , k R O , k V , k) denote the total number of possible intruder’s abstract paths x Lk LOO x Rk ROO x VkV for square lattice wireless sensor networks, each of which contains k L O left-oblique line-segments, k L O right-oblique line-segments, and k V vertical linesegments, where 0 ≤ k L O , k R O , k V ≤ k − 1 and k L O + k R O + k V = k − 1. From Theorem 19.3, we have the following equality: Ω SL I A P (k L O , k R O , k V , k)

(

Σ

=

k L O +k R O +k V =k−1

Σ

=

k L O +k R O +k V

k−1 k L O , k R O , kV

)

(k − 1)! k !k !k ! =k−1 L O R O V

From Theorem 19.4, the total number Ω IHALP (k L O , k R O , k V , k). of intruder’s abstract paths x Lk LOO x Rk ROO for hexagonal lattice wireless sensor networks, each of which contains k L O left-oblique line-segments and k L O right-oblique line-segments, where 0 ≤ k L O , k R O ≤ k − 1 and k L O + k R O = k − 1, is computed as follows: Ω IHALP (k L O , k R O , k)

=

Σ k L O +k R O =k−1

(

k−1 kL O , k R O

) =

Σ k L O +k R O

(k − 1)! k !k ! =k−1 L O R O

There are two optimal deterministic sensor deployment approaches in a planar field, namely square and hexagonal lattice-based sensor deployment, over a k-barrier

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Fig. 19.3 k-barrier covered sensor belt region S B Rw,l for square (left) and hexagonal (right) sensor deployment

covered sensor belt region S B R w,l , with w being its width, and l its length. Indeed, there are two types of lattice patterns in the literature, namely square lattice and hexagonal lattice. As per Assumption 19.1, the sensors are deployed according to each of these two sensor deployment strategies. Figure 19.3 shows the configuration for each of these two types of lattice patterns. Next, we study the k-barrier coverage problem in both lattices.

19.2.3 Square Lattice-Based Sensor Deployment Consider a row of sensors in a square lattice. Notice that the difference between the x-coordinates of the centers of two adjacent sensors is 2r . That is, the x-coordinate increases by 2r , while the y-coordinate remains the same as we move from one sensor to another in the same row. Likewise, the difference between the y-coordinates of the centers of two adjacent sensors located on the column is 2r . In other words, the y-coordinate increases by 2r , while the x-coordinate does not change as we move from one sensor to another in the same column. Theorem 19.5 computes the total number of deployed sensors to achieve k-barrier coverage in square lattice wireless sensor networks. Theorem 19.5 (Sensor Cardinality for Square Lattice Deployment) The number of sensors deployed over a k-barrier covered sensor belt region S B R w,l according to a square lattice-based sensor deployment, denoted by n SL , is computed as n SL = αk where k ≥ 1 is a natural number, α = [l/2r ], and k = [w/2r ]. Proof Assume that the sensors are deployed over a k-barrier covered sensor belt region S B Rw,l according to a square lattice-based sensor deployment. Given that S B Rw,l has a width w and a length l, there are k = [w/2r ] rows of sensors, each of which has α = [l/2r ]. sensors (i.e., α. is the number of columns). Therefore,

19.2 Tiling-Based k-Barrier Coverage

619

the total number of sensors deployed over S B Rw,l is n SL = [w/2r ][l/2r ], where k ≥ 1. ∎ As stated in Theorem 19.5 above, there are w/2r . rows and l/2r columns of sensors in a square lattice. A more detailed description of the locations of the sensors is given below. • The first row (from bottom to top) includes sensors located at (r, r ), (3r, r ), (5r, r ), …, ((2 j + 1)r, r ), …, ([l/2r ]r, r ). • The second row includes sensors located at (r, 3r ), (3r, 3r ), (5r, 3r )., …,((2 j + 1)r, 3r ), 2026,([l/2r ]r, 3r ) • the third row includes sensors located at (r, 5r ), (3r, 5r ), (5r, 5r ), ((2 j + 1)r, 5r ), ([l/2r ]r, 5r ) • The ith row (from bottom to top) includes sensors located at (r, (2i − 1)r ), (3r, (2i − 1)r ), (5r, (2i − 1)r ), …,((2 j + 1)r, (2i − 1)r ), …, ([l/2r ]r, (2i − 1)r ) • The last, (kth. or w/2r th), row includes sensors located at locations whose x–y coordinates are (r, (2k − 1)r ), (3r, (2k − 1)r ), (5r, (2k − 1)r ), …, ((2 j + 1)r, (2k − 1)r ), …, ([l/2r ]r, (2k − 1)r )

19.2.4 Hexagonal Lattice-Based Sensor Deployment In this case, the difference between the x-coordinates of the centers of two adjacent sensors located in the same row is 2r . However, the difference between the y-coordinates of the centers of two sensors located in two adjacent rows is Δ. As shown in Fig. 19.4, the latter can be computed as follows: Δ2 + r 2 = (2r )2 ⇒ Δ =

√ 3r

Theorem 19.6 computes the total number of deployed sensors to achieve k-barrier coverage in hexagonal lattice wireless sensor networks.

Fig. 19.4 Difference Δ, and type 1 and type 2 rows

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Theorem 19.6 (Sensor Cardinality for Hexagonal Lattice Deployment) The number of sensors deployed over a k-barrier covered sensor belt region S B Rw,l according to a hexagonal lattice-based sensor deployment, denoted by n H L , is given by [

([ ] ] ) l l + k2 +1 n H L = k1 2r 2r ] [ w k1 = √ 2 3r ] [ w k2 = √ 2 3r where k = k1 + k2 ≥ 1 is a natural number. Proof As shown in Fig. 19.4, k-barrier covered sensor belt region S B Rw,l has two types of rows, namely Type 1 and Type 2. The sensors are deployed deterministically using a top-down approach. That is, we start with forming the first row of Type 1, then the second row of Type 2, then the third row of Type 1, etc. In other words, S B Rw,l . is built as an alternation of rows of Type 1 and Type 2. Let k1 and k2 be the numbers of rows of Type 1 and Type 2, respectively. It is clear that k1 ≥ k2 since the starting row is of Type 1. More precisely, we have k1 = k2 or k1 = k2 + 1. Given that type of hexagonal tiling of S B Rw,l , any pair of consecutive rows of Type 1 and Type 2, respectively, have a width, denoted by ωT 1,T 2 , which is less than twice the diameter of the sensors’ sensing disk of radius 2r (i.e., ωT 1,T 2 < 2 × 2r = 4r ). Let ωT 1,T 2 = 2r + β as shown in Fig. 19.4. As it can be seen, we have β = 2(d1 + d2 ), where d1 = r × sin θ = r sin 30◦ =

r . 2

Also, we have: ( √ ) d = 2r − 2D = 2 − 3 r Using Pythagorean’s Theorem, we have: / d2 =

d2

( √ ) √ ( )2 2 3−3 3d d = = − r 2 2 2

Thus, we have: √ Consequently, the width ωT 1,T 2 = 2r + β = 2 3r . The numbers of sensors deployed in any row of Type 1 and Type 2 are, respectively, given by:

19.2 Tiling-Based k-Barrier Coverage

621

] l 2r [ ] l +1 = 2r [

nT 1 = nT 2

If the width w of S B Rw,l is a multiple of the width ωT 1,T 2 of two consecutive rows of Type 1 and type 2, respectively, i.e., w = a × ωT 1,T 2 ., we necessarily have k1 = k2 . Otherwise, we get k1 = k2 + 1. Thus, the numbers of rows of Type 1 and Type 2, denoted by k1 . and k2 , respectively, are computed as: k1 =

w

ωT 1,T 2 w k2 = ωT 1,T 2

w = √ 2 3r w = √ 2 3r

Therefore, the total number of sensors deployed over S B Rw,l according to a hexagonal lattice-based sensor deployment, denoted by n H L , is given by [ n H L = k1

l 2r

([

] + k2

l 2r

]

) +1

∎

√ Consequently, there are w/ 3r rows in a hexagonal lattice. • The first row (from bottom to top) includes sensors located at (r, r ), (3r, r ), (5r, r ),…, ((2 j + 1)r, r ), . . . , ([l/2r ]r, r ) ( ( ) ) ( (√ ) ) √ • The second row includes sensors located at 0, 3 + 1 r , 2r, 3+1 r , (√ ( (√ ( (√ ( ) ) ) ) ) ) 4r, 3 + 1 r , …, 2 jr, 3 + 1 r , …, ([l/2r ] + 1)r, 3+1 r . ( ( √ ) ) ( ( √ ) ) • The third row includes sensors located at r, 2 3 + 1 r , 3r, 2 3 + 1 r , ( ( ( √ ( ( √ ) ) ) ) ) ) ( √ 5r, 2 3 + 1 r , …, (2 j + 1)r, 2 3 + 1 r , …, [l/2r ]r, 2 3 + 1 r ( ( √ ) ) ( ( √ ) ) • The fourth row includes sensors located at 0, 3 3 + 1 r , 2r, 3 3 + 1 r , ( √ ( ) ) √ √ (4r, (3 3 + 1)r ), . . . , (2 jr, (3 3 + 1)r ), . . ., ([l/2r ] + 1)r, 3 3 + 1 r • The row) )( (om (bottom to top)) includes ( ( (2i − 1)th ) ( ( sensors √located) )at √ √ r, (2i − 2) 3 + 1 r 3r, (2i − 2) 3 + 1 r , 5r, (2i − 2) 3 + 1 r , ( ( ( ( ) ) ) ) √ √ …, (2 j + 1)r, (2i − 2) 3 + 1 r , …, [l/2r ]r, (2i − 2) 3 + 1 r • The to top) ) includes located) )at ( ( ( (2i )th √row (from ) ) ( bottom ) ( (sensors √ √ 0, (2i − 1) 3 + 1 r , 2r, (2i − 1) 3 + 1 r , 4r, (2i − 1) 3 + 1 r , ( ( ( ( ) ) ) ) √ √ …, 2 jr, (2i − 1) 3 + 1 r , …, ([l/2r ] + 1)r, (2i − 1) 3 + 1 r • The sensor deployment in the last row depends on whether k is odd or even. • Case 1: k is odd

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• The includes located ) )at ( ( last, √(kth ) or) (w/2r(th), row ( ) ) ( sensors √ √ r, (k − 1) 3 + 1 r , 3r, (k − 1) 3 + 1 r , 5r, (k − 1) 3 + 1 r , ( ( ( ( ) ) ) ) √ √ (2 j + 1)r, (k − 1) 3 + 1 r …, …, [l/2r ]r, (k − 1) 3 + 1 r . • Case 2: k is even • The (from top) includes located) )at ( ( to √ ( ( ( kth row ) ) bottom ) ) ( sensors √ √ 0, (k − 1) 3 + 1 r ., 2r, (k − 1) 3 + 1 r , 4r, (k − 1) 3 + 1 r , ( ( ( ( ) ) ) ) √ √ …, 2 jr, (k − 1) 3 + 1 r , …, ([l/2r ] + 1)r, (k − 1) 3 + 1 r

19.2.5 Square Lattice Versus Hexagonal Lattice Next, we study the k-barrier coverage problem for each of these two sensor deployment strategies. Corollary 19.2 shows that square lattice and hexagonal lattice are not isomorphic. Corollary 19.2 (Lattice Isomorphism) A square lattice wireless sensor network and a hexagonal lattice wireless sensor network deployed over a k-barrier covered sensor belt region S B R w,l are not isomorphic. Proof There is at least one sensor in a hexagonal lattice wireless sensor network that does not have any corresponding sensor in a square lattice wireless sensor network. In fact, the two lattice wireless sensor networks have unequal number of sensors, i.e., a hexagonal wireless sensor network has more sensors than a square lattice deployed ∎ over S B R w,l . Theorem 19.7 shows that hexagonal lattice wireless sensor networks are denser than square lattice wireless sensor networks. Theorem 19.7 (Hexagonal Lattice vs. Square Lattice-Based Sensor Deployment) A hexagonal lattice-based sensor deployment over a k-barrier covered sensor belt region S B R w,l is denser than its counterpart using a square lattice. We have the following relationship between n H L and n SL : nHL

) ( 1 1 = √ 2 + l n SL 3 2r

Proof Without loss of generality, let us assume that the numbers ]of rows of Type 1 [ w (i.e., k1 ) and Type 2 (i.e., k2 ) are equal. That is, k1 = k2 = 2√3r . We obtain: [ nHL =

w √ 2 3r

]( [ ] ) l 2 +1 2r

19.2 Tiling-Based k-Barrier Coverage

623

Fig. 19.5 Voronoi diagram for a square lattice of 100 sites

Therefore, we get: [

nHL n SL nHL

]( [ ] ) ) ( 2 2rl + 1 1 1 [ w ][ l ] = = √ 2+ [ l ] ⇒ 3 2r 2r 2r ) ( 1 1 = √ 2 + [ l ] n SL 3 2r w √ 2 3r

∎

It is well known that Voronoi diagram is one of the fundamental constructs that is defined by a discrete set of points. As stated in [67], the Voronoi diagram associated with a set of points in the plane divides the plane based on the nearest-neighbor rule, where every point is associated with the closest region of the plane to it. In our case, we compute the Voronoi diagram [67] of a set of points that correspond to the locations of the sensors, which are positioned according to square and hexagonal sensor deployment strategies. Figures 19.5 and 19.6 show the Voronoi diagram corresponding to a square lattice and hexagonal lattice of 100 points, respectively. All the Voronoi regions are identical for each type of lattice. A Voronoi region is a square for a square lattice, while it is a hexagon for a hexagonal lattice. In this comparison between square lattice and hexagonal lattice wireless sensor networks, our analysis focuses on their weakly k-barrier covered paths, i.e., worstcase scenario for intruder detection. Hence, we want to compute the observability metric in order to compare between these two types of lattice wireless sensor networks. Indeed, observability defines the worst-case behavior of S B R w,l in terms of its detection capability. By definition of this geometric structure, a path from the top to the bottom through Voronoi diagram, which is as far as possible from any point, is a sequence of Voronoi edges. This path is a weakly k-barrier covered path crossing S B R w,l . As it can be seen, there is a large number of such a path for both lattice-based sensor deployment strategies. However, for each one of them, there are

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Fig. 19.6 Voronoi diagram for a hexagonal lattice of 100 sites

a few shortest weakly k-barrier covered paths crossing S B R w,l . For a square lattice, this path is a sequence of vertical Voronoi edges, and the length of this path is equal to the number of sites forming a column of the lattice. Each Voronoi edge has a length that is equal to the diameter of the sensors’ sensing disks, i.e., 2r . However, for a hexagonal lattice, a shortest weakly k-barrier covered path consists of alternating sequence of vertical, left-oblique, and right-oblique edges. Theorem 19.8 computes the length of a shortest weakly k-barrier covered path for square and hexagonal lattice wireless sensor networks. Theorem 19.8 (Shortest Weakly k-Barrier Covered Path) The lengths of the shortest weakly k-barrier covered paths crossing a k-barrier covered sensor belt region S B R w,l are kes = w and 4k1 er h = 4w/3 for square lattice and hexagonal lattices, respectively, where es and er h are the edge lengths of the smallest square and regular hexagon, respectively, which are inscribed in a circle of radius r (i.e., radius of sensing disk). Proof For a square lattice, a Voronoi region is a square whose edge length es is equal to the diameter of the sensors’ sensing disks, i.e., es = 2r . Indeed, this Voronoi region corresponds to the smallest square containing a disk of diameter equal to 2r , as shown in Fig. 19.5. For a hexagonal lattice, a Voronoi region is the smallest regular hexagon that includes a disk of diameter equal to 2r . Figure 19.6 shows a disk of radius r inscribed in a regular hexagon. The edge length er h of this smallest regular hexagon can be computed as follows: er h 2 1 tan θ = tan 30◦ = √ = 2 ⇒ er h = √ r. r 3 3

First, let us consider a square lattice wireless sensor network. In this case, a weakly k-barrier covered path crossing a-barrier covered sensor belt region S B R w,l has to follow only the Voronoi edges for a square lattice. As shown in the Voronoi diagram associated with a square lattice, the shortest weakly k-barrier covered path along

19.2 Tiling-Based k-Barrier Coverage

625

S B Rw,l has to traverse all k = [w/2r ] rows, each of which has a width equal to the edge length es of the smallest square including a circle of radius r . Thus, the length of this shortest path, denoted by l SL , is given by: l SL = kes =

w 2r = w 2r

Now, for a hexagonal lattice wireless sensor network, without loss of generality, assume [ that√the]number of rows of Type 1 is equal to that of Type 2, i.e., k1 = k2 = w/2 3r . In this case, as it can be seen from the Voronoi diagram associated with a hexagonal lattice, the shortest weakly k-barrier covered path along S B R w,l includes a pair of vertical Voronoi edge from a row of Type 1 and right-oblique Voronoi edge shared between a row of Type 1 and its neighboring one of Type 2, and a pair of vertical Voronoi edge and left-oblique Voronoi edge shared between a row of Type 1 and its neighboring one of Type 2. While those vertical Voronoi edges belong alternatively to two parallel lines, the oblique Voronoi edges form an alternating sequence of right-oblique and left-oblique edges. Precisely, this path has 2k1 vertical, 2k2 /2 right-oblique, and 2k2 /2 left-oblique Voronoi edges. Thus, the length of this shortest path, denoted by l H L , is computed as: l H L = (2k1 + 2k2 /2 + 2k2 /2)er h = 4k1 er h w 2 4w >w =4 √ √ r = 3 2 3r 3

∎

Theorem 19.9 computes the intruder’s abstract path observability for square and hexagonal lattice wireless sensor networks. Theorem 19.9 (Intruder’s Abstract Path Observability) The intruder’s abstract path observability, denoted by O I A P , along a k-barrier covered sensor belt region [ √S B]Rw,l is equal to [w/2r ] for a square lattice wireless sensor network, and 2 w/ 3r for a hexagonal lattice wireless sensor network, where w is the width of S B Rw,l , and r stands for the radius of the sensors’ sensing disk. Proof Given the defition of intruder’s abstract path observability, which is given earlier in Chap. 2 (Sect. 2.2., Definition 2.44), we consider the weakly k-barrier covered path crossing a k-barrier covered sensor belt region S B R w,l . That is, observability defines the worst-case behavior of S B R w,l in terms of its detection capability. It computes the minimum number of times an intruder would be detected as they cross S B R w,l . Indeed, an intruder would be detected only Θ(k) times when they cross S B R w,l through a weakly k-barrier covered path. Because all the sensing disks are touching each other, regardless of whether we deal with a square lattice wireless sensor network or hexagonal lattice wireless sensor network, the intersection set between this weakly k-barrier covered path and S B R w,l coincides with the set of its vertical, left-oblique, and/or right-oblique Voronoi edges. Thus, for

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a square lattice, we get O I A P = k = w/2r , √ and for a hexagonal lattice, we have √ ∎ O I A P = 2k1 + 2k2 /2 + 2k2 /2 = 4k1 = 4w/2 3r = 2w/ 3r .

19.2.6 Discussion We can claim that a hexagonal lattice-based sensor deployment is better than its counterpart using square lattice over a k-barrier covered sensor belt region S B R w,l . Indeed, a hexagonal lattice has more advantageous features compared to a square lattice. • First, the sensing disks are distributed over S B R w,l more tightly for a hexagonal lattice than a square lattice, thus, allowing a better communication among the sensors. For instance, in a square lattice wireless sensor network, each sensor’s sensing disks touches exactly four other sensing disks. Assuming that the radius of the sensors’ communication range is twice their sensing range, i.e., R = 2r , any sensor would be able to communicate with only four neighboring sensors although it has eight neighboring ones. Indeed, as shown in Fig. 2.9a (Chap. 2, Sect. 2.2), each sensor is at √distance 2r away from four of his eight neighboring sensors, and at distance 2 2r away from the remaining four neighboring ones. However, for a hexagonal lattice, each sensor is at distance 2r away from each of its six neighboring sensors, as shown in Fig. 2.9b (Chap. 2, Sect. 2.2). This kind of uniformity helps the sensors in a hexagonal lattice exchange more useful data for intruder tracking compared with their counterpart in a square lattice. • Second, the length of the weakly k-barrier covered path over S B R w,l for a hexagonal lattice is longer than its counterpart for a square lattice. Hence, an intruder would take more time to cross the sensor belt region S B R w,l , thus, increasing their detection by the sensors. • Third, the observability for a hexagonal lattice is higher than that for a square lattice. This helps increase both the quality of detection and tracking.

19.3 Generalization Assumption 19.5 (Random Sensor Belt Region Crossing) An intruder moves randomly across a sensor belt region. ∎ In general, an intruder may cross a sensor belt region randomly, thus, following a path including a random sequence of line-segments that are not necessarily progressive. Precisely, the trajectory of an intruder’s movement could include moves between the sensing disks of sensors located at the same level of the sensor belt region although those moves may be through left-oblique or right-oblique line-segments. It is worth noting that an abstract path through a hexagonal lattice does not include

19.3 Generalization

627

Fig. 19.7 Intruder’s movement trajectory and associated abstract path for square and hexagonal lattices

any vertical line-segment. Indeed, when an intruder passes vertically through the (unique) intersection point of two kissing sensing disks, we represent that intruder’s move by either a left-oblique or a right-oblique line-segment. Figure 19.7 shows such intruder’s moves and the corresponding abstract path to both square and hexagonal lattices. Consequently, an intruder’s abstract path may include left-horizontal, righthorizontal, left-oblique, right-oblique, and/or vertical line-segments (this last type of line-segment is for square lattice only). Likewise, an intruder’s abstract path can be modeled by a graph denoted by I A P = (N , E), where the node set N represents the set of sensing disks, and the edge set E stands for transitions between the sensors’ sensing disks, with |N | ≥ k and |E| ≥ k − 1. Theorem 19.10 below computes the lengths of the shortest and longest intruder’s paths, respectively. Theorem 19.10 (Shortest and Longest Intruder’s Abstract Paths) The length of the shortest intruder’s abstract path along a k-barrier covered sensor belt region S B Rw is k − 1. That is, it has k nodes and (k − 1) edges, including only left-oblique, right-oblique, and/or vertical line-segments (for square lattice only). The length of the longest intruder’s abstract path along a k-barrier covered sensor belt region S B Rw,l is αk − 1. That is, it has αk nodes and αk − 1 edges, including exactly (k − 1) vertical line-segments (for square lattice only), and (α − 1)k left-horizontal and right-horizontal line-segments, where k ≥ 1 is a natural number, w = 2kr , and l = 2αr . Proof Since the sensor belt region S B Rw,l has k rows of sensors, there are at least (k − 1) transitions between these k rows in order to cross S B Rw,l . That is, the shortest path crossing S B Rw,l has (k − 1) edges, as shown in Fig. 19.8. Thus, the length of the shortest intruder’s abstract path is(k − 1). Moreover, this path does not include any left-horizontal or right-horizontal line-segments. Otherwise, it would have more than (k − 1) edges. It can have only left-oblique, right-oblique, and/or

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(a) Square lattice

(b) Hexagonal lattice

Fig. 19.8 Shortest intruder’s abstract path for both lattices

vertical edges (or line-segments). Indeed, only these types of edges help the intruder to move from one row to another. The longest intruder’s abstract path includes all nodes in S B Rw,l . Although all the intruder’s moves are progressive, they could traverse all the sensors’ sensing disks in S B Rw,l . That is, the intruder has to visit the sensing disks of all the sensors deployed in S B Rw,l . Given that each row has α nodes, there a. left-horizontal or right-horizontal edges between them. Since S B Rw,l . has k rows, there are (α − 1)k left-horizontal and right-horizontal edges. Precisely, if k is even, the numbers of left-horizontal and right-horizontal edges, denoted by E L H and E R H , respectively, are the same, and are equal to (α − 1)k/2. If k. is odd, the difference between E L H . and E R H . is α − 1. That is, we have |E L H − E R H | = α − 1. More specifically, if the starting edge of this longest path is left-horizontal, we have E L H − E R H = α − 1. Otherwise (i.e., the starting edge is right-horizontal), we have E R H − E L H = α − 1. In the case of square lattice, for an intruder to move from one row to its next one, there is one vertical edge. For all the transitions between consecutives rows in S B Rw,l , there should be (k − 1) vertical edges. In the case of hexagonal lattice, there should be (k − 1) left-oblique and right-oblique edges to allow transitions between consecutives rows in S B Rw,l . Hence, the total number of edges that are needed to cross the entire S B Rw,l is (α − 1)k + k − 1 = αk − 1. Thus, ∎ the length of the longest intruder’s abstract path is αk − 1 in both lattices. Lemma 19.2 states the main characteristic of all intruder’s abstract paths. Lemma 19.2 (Intruder’s Abstract Path Characteristic) Every path across a k-barrier covered sensor belt region S B Rw,l is characterized by the presence of (k − 1) leftoblique, right-oblique, and/or vertical edges, where k ≥ 1 is a natural number. Proof The only edges that enable an intruder’s abstract path to cross k-barrier covered sensor belt region S B Rw,l are left-oblique, right-oblique, and vertical edges. Given that S B Rw,l has k rows of sensors, there are exactly (k − 1) transitions between consecutive rows. Thus, any path crossing S B Rw,l has (k − 1) left-oblique, rightoblique, and/or vertical edges. ∎ Theorem 19.11 characterizes the structure of all random intruder’s abstract paths. Theorem 19.11 (Random Intruder’s Abstract Path Structure) Let μ IRand A P be the length of a random intruder’s abstract path I A PRand = (E Rand , N Rand ) across a k-barrier

19.3 Generalization

629

covered sensor belt region S B Rw,l . If μ IRand A P = k − 1, the edge set E Rand includes only left-oblique, right-oblique, and/or vertical edges. If k ≤ μ IRand A P ≤ αk − 1, the − − 1) left-horizontal and/or right-horizontal edge set E Rand has exactly μ IRand (k AP edges, where k ≥ 1 is a natural number, w = 2kr , and l = 2αr . Proof From Lemma 19.2, if μ IRand A P = k − 1, all the edges should be left-oblique, right-oblique, and/or vertical. Given that we have only five types of edges (i.e., left-oblique, right-oblique, vertical, left-horizontal and right-horizontal edges), if k ≤ μ IRand A P ≤ αk − 1, by Lemma 19.2, there are exactly (k − 1) left-oblique, right-oblique, and/or vertical edges, and all other μ IRand A P − (k − 1) edges should be left-horizontal and/or right-horizontal. ∎ Theorem 19.12 computes the total number of random intruder’s abstract paths based on Assumption 19.5 and the above analysis. Theorem 19.12 (Random Intruder’s Abstract Path Cardinality) Let Ω IRand A P (k L O , k R O , k V , k L H , k R H , k) denote the total number of all intruder’s abstract paths x Lk LOO x Rk ROO x VkV x Lk LHH x Rk RHH across a k-barrier covered sensor belt region S B Rw,l , under Assumption 19.5 stated earlier. Each of these paths has k L O left-oblique line-segments, k L O . right-oblique line-segments, k V vertical line-segments (for square lattice only), k L H left-horizontal line-segments, and/or k R H right-horizontal line-segments, where 0 ≤ k L O , k R O , k V ≤ k − 1, k L O + k R O + k V = k − 1, Fast Rand k L H + k R H = μ IRand A P − μ I A P = μ I A P − (k − 1), k ≥ 1 is a natural number, w = 2kr , and l = 2αr . We have this result: k−1 Ω IRand A P (k L O , k R O , k V , k L H , k R H , k) = 3 ( )) αk−1 Σ Σ( μ IRand A P + k L O , k R O , kV , k L H , k R H Rand μ I A P =k

def

C1 ,C2

def

where C1 = k L O + k R O + k V = k − 1 and C2 = k L H + k R H = μ IRand A P − (k − 1). Proof We demonstrate in Theorem 19.1 that the total number of intruder’s abstract paths, which have only left-oblique, right-oblique, and/or vertical linesegments, is 3k−1 . The length of each of these paths is k − 1. The first part of k−1 Ω IRand , accounts for those paths having A P (k L O , k R O , k V , k L H , k R H , k), namely 3 Rand such a length. However, the send part of Ω I A P (k L O , k R O , k V , k L H , k R H , k), namely ( ( )) Σ Σαk−1 μ IRand AP , accounts for those paths whose C1 ,C2 μ IRand A P =k k L O , k R O , kV , k L H , k R H length μ IRand A P is larger than k − 1. Each of such paths should contain k − 1 leftoblique, right-oblique, and/or vertical line-segments (conditionC1 ), and the other remaining μ IRand A P − (k − 1) edges have to be left-horizontal and/or right-horizontal line-segments (conditionC2 ). The inner summation of this second part is similar to the one given in Theorem 19.1. It computes the number of intruder’s abstract paths whose length is μ IRand A P and each of which has k L O . left-oblique line-segments,

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k L O right-oblique line-segments, and/or k V . vertical line-segments, and k L H . lefthorizontal line-segments and/or k R H right-horizontal line-segments, subject to those two conditions, i.e., C1 and C2 . The outer summation of this second part accounts for all paths whose length varies between k and αk − 1, which corresponds to the longest intruder’s abstract path. ∎

19.4 Source-to-Destination Path Analysis In this section, we discuss the intruder’s abstract paths from one source location to a destination location along a k-barrier covered sensor belt region S B R w,l in square lattice wireless sensor networks, where k ≥ 1 is a natural number, w = 2kr , and l = 2αr . We consider two different intruder’s movements. The first one uses only right-horizontal and vertical moves, while the second movement involves righthorizontal, vertical, and/or right-oblique moves. That is, we assume that all these moves are progressive. Because of their structure, we call the paths associated with the first movement Manhattan paths. Next, we characterize all of these paths associated with these two types of movements, and compute their numbers. Our study accounts for both square and rectangular S B R w,l .

19.4.1 Square k-barrier Covered Sensor Belt First, we consider a square S B Rw,l , denoted by SS B Rw,l , where w = l = 2kr (i.e., α = k). Also, we assume that all these moves are progressive. That is, any intruder’s abstract path includes only right-horizontal, vertical, and right-oblique edges. Moreover, we focus on intruder’s movements that cross SS B Rw,l , where an intruder moves from the top corner of SS B Rw,l to its bottom one. These paths are called source-to-destination paths. In fact, there are four groups of intruder’s abstract source-to-destination paths: • Group 1: All line-segments are located in the top part of SS B Rw,l . That is, these line-segments are either on the diagonal or above it, where at least one linesegment is not on the diagonal. Some of these paths are Manhattan paths, while any other path has at least one right-oblique line-segment and at most (k − 2) right-oblique line-segments (see Fig. 19.9a). • Group 2: All line-segments are located in the bottom part of SS B Rw,l . In other words, these line-segments are either on the diagonal or below it, where at least one line-segment is not on the diagonal. Similar to Group 1, this group includes Manhattan paths as well as other paths, each of which has at least one right-oblique line-segment and at most (k − 2) right-oblique line-segments (see Fig. 19.9b).

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Fig. 19.9 Intruder’s abstract source-to-destination paths of Group 1 (a) and Group 2 (b) for square lattice

Fig. 19.10 Shortest (a) and longest (b) intruder’s abstract source-to-destination paths for square lattice

• Group 3: All line-segments are right-oblique and are located on the diagonal of SS B Rw,l . This is the shortest intruder’s abstract source-to-destination path, which is discussed in Lemma 19.3 below (see Fig. 19.10a). • Group 4: Some of the line-segments are located in the top part of SS B Rw,l , whereas some others belong to its bottom part. That is, any path in this group has one of the two structures: For the first one, an intruder’s abstract source-todestination path consists of a sub-path in the top part of a square SS B Rw,l followed by a vertical line segment, which is followed by a sub-path in the bottom part of SS B Rw,l (see Fig. 19.11a). For the second structure, an intruder’s abstract sourceto-destination path is composed of a sub-path in the bottom part of SS B R w,l followed by a horizontal line segment, which is followed by a sub-path in the top part of SS B R w,l (see Fig. 19.11b). Notice that some of the paths in this group are Manhattan paths.

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Fig. 19.11 Intruder’s abstract source-to-destination paths of Group 4 structure 1 (a) and structure 2 (b) for square lattice

It is worth noting that the intruder’s abstract source-to-destination paths across SS B R w,l in Group 1 are symmetric to those in Group 2, whereas the paths in Group 4 having the first structure are symmetric to those ones in the same group and possessing the second structure. Thus, Group 1 and Group 2 have the same number of paths. Also, the number of paths in Group 4 with the first structure is identical to that in the same group having the second structure. Therefore, our focus will be on the intruder’s abstract source-to-destination path in Group 1 as well as those in Group 4 and having the first structure. As we mention earlier, there are Manhattan paths in Group 4, which will be computed later. Lemma 19.3 compute the length of intruder’s abstract source-to-destination paths. Lemma 19.3 (Intruder’s Abstract Source-to-Destination Path Length) The length B ρ SL−S I AS D P (k) of an intruder’s abstract source-to-destination path across SS B Rw,l satisfies the following inequality: B k − 1 ≤ ρ SL−S I AS D P (k) ≤ 2(k − 1)

Proof Let us first compute the shortest intruder’s abstract source-to-destination paths across a square k-barrier covered sensor belt region, SS B Rw,l . It is clear that the shortest path lies on the diagonal of SS B Rw,l (see Fig. 19.10a). Given that this path goes through k sensing disks, there are (k − 1) right-oblique linekRO segments (i.e., k R O =( k − 1). This ) diagonal path is represented as x R O . Thus, the SL−S B minimum length min ρ I AS D P (k) of intruder’s abstract source-to-destination path ( ( SL−S B ) ) B is min ρ SL−S I AS D P (k) = k − 1. Now, let us compute max ρ I AS D P (k) , which is the maximum length of an intruder’s abstract source-to-destination paths across SS B Rw,l . The longest path corresponds to intruder’s movements using only righthorizontal and vertical line-segments (see Fig. 19.10b), which yield progressive Manhattan paths. More specifically, each Manhattan path across SS B Rw,l includes exactly k V = (k − 1) vertical line-segments and k R H = (k − 1) right-horizontal line-segments. This Manhattan path has the following structure: x VkV x Rk RHH . Its length

19.4 Source-to-Destination Path Analysis

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is computed as:) k V + k R H = (k − 1) + (k − 1) = 2(k − 1). Thus, we have: ( B max ρ SL−S ∎ = 2(k − 1). (k) I AS D P These progressive source-to-destination paths can be either Manhattan or random paths. Next, we study each of these two classes of paths. Then, we give some illustrative examples.

19.4.1.1

Progressive Manhattan Paths

In this section, we focus on progressive Manhattan paths through a square k-barrier covered sensor belt region, SS B R w,l . We study three categories of these paths: • Paths on or above the diagonal of SS B R w,l . • Paths on or below the diagonal of SS B R w,l . • Paths crossing the diagonal of SS B R w,l . We characterize the paths in each of these three categories, and compute their numbers in square lattice wireless sensor networks.

Above-Diagonal Manhattan Paths The intruder attempts to avoid any Manhattan path that falls below the diagonal of SS B R w,l . They may think that it would be safer not to follow any path that crosses the diagonal of SS B R w,l . Lemma 19.4 computes the total number of Manhattan paths on or above this diagonal. Lemma 19.4 (Intruder’s Abstract Above-Diagonal Progressive Manhattan Path Cardinality) The total number of progressive Manhattan paths on or above the B diagonal of SS B Rw,l , denoted by M ISL−S AS D P (k), is given by: (k−2) B M ISL−S AS D P (k) = C 2(k−2) =

(2k − 4)! (k − 2)!2

Proof It is easy to check that the first edge in any progressive Manhattan path on or above the diagonal of SS B Rw,l is right-horizontal, whereas the last one is vertical. As we state earlier in Lemma 19.3, there are exactly k V = (k − 1) vertical edges and k R H = (k − 1) right-horizontal edges in any Manhattan path across SS B Rw,l , and, particularly, those ones on or above the diagonal of SS B Rw,l . In total, there are 2(k − 1) edges (or moves). To compute the total number of these progressive Manhattan paths, we model all these moves with a window that has 2(k − 1) boxes, where the first box is occupied by a right-horizontal edge and the last one contains a vertical edge. Now, we are left with k 'R H = (k − 2) right-horizontal edges and k V' = (k − 2) vertical edges, which need to be placed in the 2(k − 1) − 2 = 2(k − 2) remaining boxes. The total number of progressive Manhattan paths on or above the

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Fig. 19.12 Above-diagonal Manhattan path (a) and its symmetric below-diagonal Manhattan path (b) for square lattice SL−S B diagonal of S B Rw,l , denoted by M AD M P (k), corresponds to the number of ways (or combinations) of placing (k − 2) right-horizontal edges (or (k − 2) vertical edges) in those 2(k − 2) remaining boxes. Thus, we have: (k−2) SL−S B M AD M P (k) = C 2(k−2) =

(2k − 4)! (k − 2)!2

where the degree of k-barrier coverage is k ≥ 2.

∎

Below-Diagonal Manhattan Paths It is worth noting that all progressive Manhattan paths on or above the diagonal of SS B R w,l are symmetric to those on or below the diagonal of SS B R w,l . Therefore, one can easily see that the first edge in any Manhattan path on or below the diagonal of SS B R w,l is vertical, while the last one is right-horizontal. Figure 19.12 illustrates this symmetry through a couple of paths. Based on Lemma 19.4 and this symmetry, Corollary 19.3 computes the number of paths progressive Manhattan paths on or below the diagonal of SS B R w,l . Corollary 19.3 (Intruder’s Abstract Below-Diagonal Progressive Manhattan Path Cardinality) The total number of progressive Manhattan paths on or below the B diagonal of SS B Rw,l , denoted by M BSL−S D M P (k), is given by: (k−2) B M BSL−S D M P (k) = C 2(k−2) =

under the assumption that k ≥ 2.

(2k − 4)! (k − 2)!2 ∎

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Crossing-Diagonal Manhattan Paths Some of the vertical and right-horizontal line-segments are located in the top part of the square SS B R w,l , whereas some others belong to its bottom part. That is, any path in this group has one of the two structures: For the first one, a progressive Manhattan path consists of a sub-path in the top part of SS B R w,l followed by a vertical line segment, which is followed by a sub-path in the bottom part of SS B R w,l (see Fig. 19.11a). For the second structure, a progressive Manhattan path is composed of a sub-path in the bottom part of SS B R w,l followed by a horizontal line segment, which is followed by a sub-path in the top part of SS B Rw,l (see Fig. 19.11b). Lemma 19.5 compute the number of progressive Manhattan paths that cross the diagonal of a square k-barrier covered sensor belt region SS B Rw,l . Lemma 19.5 (Intruder’s Abstract Crossing-Diagonal Progressive Manhattan Path Cardinality) The total number of progressive Manhattan paths crossing the diagonal B of SS B Rw,l , denoted by MCSL−S D M P (k), is given by: B MCSL−S D M P (k) =

(2k − 4)! (2k − 2)! −2 (k − 1)!2 (k − 2)!2

B Proof Let us first compute the total number M PSL−S M P (k) of progressive Manhattan paths across SS B Rw,l . As mentioned earlier, each of these paths has exactly (k − 1) vertical line-segments and (k − 1) right-horizontal line-segments. Using the same B window-based model described above, M PSL−S M P (k) corresponds to the number of ways of placing (k − 1) right-horizontal edges (or (k − 1) vertical edges) in a window that has 2(k − 1) boxes. Thus, we have: (k−1) B M PSL−S M P .C 2(k−1) =

(2k − 2)! (k − 1)!2

However, the total number of progressive Manhattan paths across SS B Rw,l is equal to the sum of all those paths on, above, below, and crossing the diagonal of SS B Rw,l . Thus, we have the following: B SL−S B SL−S B SL−S B M PSL−S M P (k) = M AD M P (k) + M B D M P (k) + MC D M P (k)

This implies that B SL−S B SL−S B SL−S B MCSL−S D M P (k) = M P M P (k) − M AD M P (k) − M B D M P (k)

= where k ≥ 2.

(2k − 2)! (2k − 4)! −2 (k − 1)!2 (k − 2)!2 ∎

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19.4.1.2

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Progressive Random Paths

Here, we consider all the progressive paths across a square k-barrier covered sensor belt region, SS B R w,l . More precisely, a progressive path through SS B R w,l may contain vertical, right-horizontal, and/or right-oblique line-segments. In this case, the window-based model should be applied to progressive random paths carefully. This is due to the fact that there are three types of line-segments that cannot be combined in any order. It depends on the path length, which varies between (k − 1) and 2(k − 1), as stated in Lemma 19.3. Also, in any progressive path across SS B R w,l , the number of vertical line-segments is always equal to that of right-horizontal line-segments. Moreover, the number of right-oblique line-segments is zero in any Manhattan path, whereas it varies between 1 and (k − 1) in any other path. First, we characterize progressive random paths across SS B R w,l . Then, we compute their number. Theorem 19.13 provides a characterization of progressive random paths across SS B R w,l . Theorem 19.13 (Intruder’s Abstract Progressive Random Path Structure) Any progressive random path across a square k-barrier covered sensor belt region, S B R w,l , has k V vertical line-segments, k R H right-horizontal line-segments, and/or k R O right-oblique line-segments, thus, possessing the following structure: x VkV x Rk RHH x Rk ROO where 0 ≤ k V , k R H , k R O ≤ k − 1, k V = k − 1 − m, k R H = k − 1 − m, k R O = m, 0 ≤ m ≤ k − 1, and k ≥ 1. Proof Let us proceed using a proof by mathematical induction on m. Let P(m) be the following statement: P(m) : “There are k V vertical line-segments and k R H right-horizontal line-segments in any progressive random path that has k R O right-oblique line- segments, where k V = k R H = k − 1 − m, k R O = m, m ≥ 0, and k ≥ 1 is a natural number.”

Basis step: Let us prove that P(m) is true for m = 0. This is true for any Manhattan path (i.e., path having no right-oblique line-segment). In fact, any Manhattan path has (k − 1) vertical line-segments and (k − 1) right-horizontal line-segments. Thus, P(0) is true. Inductive step: We assume that P(n) is true, i.e., there are (k − 1 − n) vertical line-segments and (k − 1 − n) right-horizontal line-segments in in any progressive random path that has n right-oblique line-segments, where n ≥ 0. We want to prove that P(n + 1) is true. That is, there are (k − 2 − n) vertical line-segments and (k − 2 − n) right-horizontal line-segments in in any progressive random path having (n + 1) right-oblique line- segments. Given a progressive random path that has n right-oblique line-segments, we replace a consecutive pair of one vertical linesegment followed by one right-horizontal line-segment (or one right-horizontal linesegment followed by one vertical line-segment) with one right-oblique line-segment

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Fig. 19.13 Substituting a pair of right-horizontal and vertical line-segment (a) with one rightoblique line-segment (b)

(see Fig. 19.13). We obtain a progressive random path with (k − 1 − n − 1) = (k − 2 − n) vertical line-segments, (k − 1 − n − 1) = (k − 2 − n) right-horizontal line-segments, and (n + 1) right-oblique line- segments. This proves that P(n + 1) is true. Therefore, the statement P(m) is true, for all m ≥ 0. Indeed, we have the inference rule: (P(0) ∧ (∀n ≥ 1, P(n) → P(n + 1))) → P(m), ∀m ≥ 0 ∎

This concludes our proof.

Theorem 19.14 computes the total number of intruder’s abstract progressive random paths (or intruder’s abstract source-to-destination paths) across SS B R w,l for square lattices. Theorem 19.14 (Intruder’s Abstract Progressive Random Path Cardinality) Let B ϕ SL−S I A P R P (k, m) denote the total number of all intruder’s abstract progressive random paths x VkV x Rk RHH x Rk ROO across a square k-barrier covered sensor belt region, SS B Rw,l . Each of these paths has k V vertical line-segments, k R H right-horizontal linesegments, and/or k R O right-oblique line-segments, where 0 ≤ k V , k R H , k R O ≤ k −1, k V = k R H = k − 1 − m, k R O = m, m ≥ 0, and k ≥ 1 is a natural number, and w = l = 2kr . We have this result: ( k−2 ) Σ SL−S B k−1 k−1−m k−1−m ϕ I A P R P (k, m) = C2k−2 + C2k−2−m × C2k−3−m + 1 m=1 B k−1 Proof The first term in ϕ SL−S I A P R P (k, m), namely C 2k−2 , corresponds to the total SL−S B number M P M P (k) of Manhattan paths as computed in the proof of Lemma 19.5, while the last value (i.e., 1) is the unique diagonal path, which has exactly (k − 1) right-oblique line-segments. Now, we need to compute the total number of paths

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that have at least one right-oblique line-segment and at most (k − 2) right-oblique line-segments. To this end, we use the same window-based model described above. In this case, the size of this window is not fixed. Specifically, this size depends on the length of the path, which in turn depends on the number of its right-oblique line-segments. Indeed, the size of this window decreases by one as the length of the path decreases by one through adding a new right-oblique line-segment. In fact, a right-oblique line-segment replaces a consecutive pair of a right-horizontal linesegment and a vertical line-segment (or a vertical line-segment followed by a rightB horizontal line-segment). Let) us focus on the second term in M PSL−S M P (k), namely ( Σk−2 k−1−m k−1−m m=1 C 2k−2−m × C 2k−3−m . This term corresponds to all the progressive random paths, each of which has at least one right-oblique line-segment and at most (k − 2) right-oblique line-segments. These two values show on the summation symbol. Each of these paths has (k − 1 − m) vertical line-segments, (k − 1 − m) right-horizontal line-segments, and m right-oblique line-segments, where 1 ≤ m ≤ k − 2. We consider a window of size (2k − 2 − m), which is the length of any of these paths k−1−m different ways to place (i.e., 2(k − 1 − m) + m = 2k − 2 − m). There are C2k−2−m (k − 1 − m) right-horizontal line-segments in those (2k − 2 − m) boxes. For each k−1−m different ways to arrange (k − 1 − m) vertical lineof these ways, there are C2k−3−m segments in the remaining (k − 1) boxes, i.e., (2k − 2 − m) − (k − 1 − m) = k − 1. k−1−m In other word, C2k−3−m corresponds to the number of ways to distribute (k − 1 − m) identical vertical line-segments among (k − 1) distinct boxes. More precisely, k−1−m C2k−3−m is derived as follows: k−1−m k−1−m C2k−3−m = C(k−1)+(k−1−m)−1 =

This concludes our proof.

19.4.1.3

(2k − 3 − m)! (k − 1 − m)!(k − 2)! ∎

Illustrative Examples

For a square k-barrier covered sensor belt region, SS B Rw,l , we consider different values of k, and give the entire set of all the source-to-destination paths. This set includes all intruder’s abstract progressive random paths. Example 1: k = 2

19.4 Source-to-Destination Path Analysis

Group 1

Group 2

Group 3

Example 2: k = 3 Group 1

Group 2

Group 3

Group 4

Example 3: k = 4

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Group 1

Group 2

Group 3

19.4 Source-to-Destination Path Analysis

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Group 4

19.4.2 Rectangular k-barrier Covered Sensor Belt Considering a rectangular k-barrier covered sensor belt region, S B R w,l , denoted by RS B R w,l , for square lattice wireless sensor networks, there is no diagonal path across RS B R w,l . That is, any progressive non-Manhattan path across RS B R w,l has at least one vertical or right-horizontal line-segment. In this section, first, we extend our previous results in Sect. 19.4.1 to account for a rectangular k-barrier covered sensor belt region, RS B R w,l , which is a more general case. Recall that the width and length of RS B R w,l are respectively given by w = 2kr and l = 2αr , where k /= α. Then, we provide some illustrative examples.

19.4.2.1

Progressive Path Analysis

Lemma 19.6 compute the length of intruder’s abstract source-to-destination paths across RS B R w,l .

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Lemma 19.6 (Intruder’s Abstract Source-to-Destination Path Length) The length B ρ SL−R I AS D P (k) of an intruder’s abstract source-to-destination path across a rectangular k-barrier covered sensor belt region, RS B Rw,l satisfies the following inequality: ( ) B max ρ SL−R I AS D P (k, α) = k + α − 2 ( ) B min ρ SL−R I AS D P (k, α) =

⎧

k − 1 if k > α α − 1 if α > k

where k ≥ 1 and α ≥ 1. Proof As stated earlier, the longest progressive source-to-destination path across RS B Rw,l corresponds to one of the Manhattan paths through RS B Rw,l , which has k V = (k − 1) vertical line-segments and k R H = (α − 1). right-horizontal linesegments. Thus, the maximum path length across RS B Rw,l is given by: ( ) B max ρ SL−R I AS D P (k, α) = (k − 1) + (α − 1) = k + α − 2 The shortest progressive source-to-destination path across RS B Rw,l includes the maximum number of right-oblique line–segments. Given that the width w = 2kr and length l = 2αr of RS B Rw,l are not equal, we consider the following two cases: Case 1: k > α A path across RS B Rw,l can have at most (α − 1) right-oblique line-segments. Without loss of generality, assume that all these (α − 1) right-oblique line-segments are consecutive (i.e., adjacent to each other). The last right-oblique will end in the last column (i.e., αth column) of RS B Rw,l ., as shown in Fig. 19.14a. Notice that the projection of those (α − 1) right-oblique line-segments are equivalent to (α − 1) vertical line-segments starting from the top right most corner of that last column in Fig. 19.14 Shortest paths or k > α (a) and α > k (b) for square lattice

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RS B Rw,l . Thus, the length of this underlying path is (k − 1). Thus, we have: ( ) B min ρ SL−R I AS D P (k, α) = k − 1. Case 2: α > k This case is symmetric to therevious one. That is, a path across RS B Rw,l has at most (k − 1) right-oblique le-segments. Likewise, without loss of generality, assume we have a sequence (k − 1). right-oblique line-segments. The last right-oblique will end in the last row (i.e., kth row) of RS B Rw,l , as shown in Fig. 19.14b. Now, the projection of those right-oblique line-segments are equivalent to (k − 1) right-horizontal line segments starting from the bottom left most corner of that last row in RS B Rw,l . Thus, the length of this underlying path is (α − 1). Thus, we obtain the following result: ( ) B min ρ SL−R I AS D P (k, α) = α − 1 ∎

is concludes our proof.

Lemma 19.7 computes the total number of intruder’s abstract progressive random paths across RS B Rw,l for square lattice wireless sensor networks. Lemma 19.7 (Intruder’s Abstract Progressive Random Path Cardinality) Let B ϕ SL−R I A P R P (k, α, m) denote the total number of all intruder’s abstract progressive random paths x VkV x Rk RHH x Rk ROO across a rectangular k-barrier covered sensor belt region, SS B Rw,l . Each of these paths has k V vertical line-segments, k R H right-horizontal line-segments, and/or k R O right-oblique line-segments, where 0 ≤ k R H ≤ α − 1, 0 ≤ k V ≤ k − 1, k V = k − 1 − m, k R H = α − 1 − m, k R O = m, 0 ≤ m ≤ min(k − 1, α − 1), α ≥ 1, k ≥ 1, w = 2kr , l = 2αr , and k /= α. We have this result: B k−1 ϕ SL−R I A P R P (k, α, m) = C k+α−2 (min(k−1,α−1) ) Σ k−1−m k−1−m + Ck+α−2−m × Ck+α−3−m m=1

Proof The total number of Manhattan paths across RS B Rw,l is denoted by B M PSL−R M P (k, α) and computed as follows: B k−1 M PSL−R M P (k, α) = C k+α−2

Indeed, each Manhattan path has (k − 1) vertical line-segments and (α − 1) B right-horizontal line-segments. We compute M PSL−R M P (k, α) using the same argument as in the proof of Theorem 19.14 above. That is, using the above-mentioned B window-based model, M PSL−R M P (k, α) corresponds to the number of ways to place

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Fig. 19.15 Separating diagonal for k > α (a) and α > k (b) for square lattice

(k − 1) vertical line-segments into (k + α − 2) boxes. This is given by the first B k−1 term in ϕ SL−R I A P R P (k, α, m), namely C k+α−2 . Likewise and as discussed in the proof of Theorem 19.14, the second term accounts for all the progressive random paths that have at least one right-oblique line-segment and at most min(k − 1, α − 1) right-oblique line-segments. ∎

19.4.2.2

Illustrative Examples

For a rectangular k-barrier covered sensor belt region, RS B R w,l , we consider different values of k and α, and provide the whole set of all the source-to-destination paths. This set contains all intruder’s abstract progressive random paths. Also, we have the following conventions to define each of the above-mentioned four groups of progressive paths in lattice wireless sensor networks: • Given the rectangular shape of a k-barrier covered sensor belt region, there is no separating diagonal. Therefore, the diagonal in the square formed by min(k, α) rows and min(k, α) columns is used to distinguish those four groups of progressive paths defined earlier in Sect. 19.4.1. Figure 19.15 shows this convention. • When k > α, any path that includes all line-segments on the separating diagonal would belong to Group 1 of above-diagonal paths. • if α > k, any path that has all line-segments on the separating diagonal would belong to Group 2 of below-diagonal paths. These conventions do not have any impact on the total number of progressive paths using a rectangular k-barrier covered sensor belt region, RS B R w,l . They simply help classify the generated path and assign them to their respective groups. This is mainly due to the absence of a clear separating diagonal because of the shape of the k-barrier covered sensor belt region. We need to define one based on the dimensions of this sensor belt region, namely its width k and length α, in order to build those four groups discussed above. Example 1: k = 4, α = 3

19.4 Source-to-Destination Path Analysis

Group 1

Group 2

Group 4

Example 2: k = 3, α = 4

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Group 1

Group 2

Group 4

19.5 Other Possible Generalizations Our proposed study of the problem of k-barrier coverage in wireless sensor networks considers three particular models: • Model 1: Deterministic sensor deployment using lattice-based sensor strategies, namely square and hexagonal stealthy lattices. • Model 2: Unit disk model, where both of the sensing and communication ranges of the sensors are modeled as disks of radii r and R, respectively. • Model 3: Homogeneous model, where all the deployed sensors have the same sensing range and the same communication range. Regarding the first model above (i.e., Model 1) While both of these lattice-based sensor deployment approaches are useful and needed for some physical areas to be monitored, a more general approach considering random sensor deployment would be more desirable. That is, given that sensors are randomly deployed around a physical space to be secured, the main question is:

19.5 Other Possible Generalizations

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How can k-barrier coverage be achieved? Unless sensors are densely deployed, it may not be possible to ensure k-barrier coverage using random sensor deployment. Even in the presence of high sensor density, some regions around this physical space may be left uncovered due to this randomness. We believe that the most efficient solution to achieve k-barrier coverage is to provide sensors with a mobility capability. This dimension (i.e., sensor mobility) would enable sensors to move, if needed, to construct k layers of sensors around a critical space that needs to be monitor. Because of their limited capabilities, and, particularly, their battery power, sensors mobility should be purposeful. This helps save the sensors’ energy, thus, extending the network lifetime. In other words, sensors would need to collaborate between themselves to have enough number of sensors to be relocated at specific locations around the critical space so that every intruder’s path crossing a sensor belt region is k-barrier covered. We plan to study further this mobility dimension in our future work. For our second model (i.e., Model 2), we plan to relax it to consider a more general model to represent the sensing and communication ranges of the sensors using a more realistic way. In fact, these ranges may be irregular and do not follow the unit-disk model, where an event is either sensed or not. A stochastic model for both the sensing and communication ranges would be more practical. We extend our results to account for this irregularity as part of our second future work. As far as our third model (i.e., Model 3) is concerned, a homogeneous model for sensing and communication may not be always the only scenario. In fact, sensors many not necessarily have the same sensing and communication ranges. That is, some sensors could be more powerful than others in terms of these two capabilities, in addition to their energy (or power) reserves. Deployed more powerful sensors may operate at more than one layer of a k-barrier sensor belt region, which would reduce the total number of sensors for k-barrier coverage. As our third future work, we focus on using a heterogeneous model to account for deploying sensors that may not have the same capabilities with regard to their battery power as well as their sensing and communication ranges. The above three possible generalizations will have an impact on our current results to solve the problem of k-barrier coverage using stealthy and homogeneous sensors whose sensing and communication ranges are represented by disks. In particular, we study the possibility of extracting square and hexagonal lattices when sensors are randomly deployed in a k-barrier covered sensor belt region. To this end, we exploit sensor mobility to generate these lattices. This mobility support introduces an overhead due to sensor communication, which in turn incurs more energy consumption due to sensor mobility and their interactions with each other, which are necessary for the proper operation of the network.

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19.6 Performance Evaluation In this section, we specify the simulation setup. Then, we present some simulation results using a high-level simulator written in C.

19.6.1 Simulation Setup We consider a rectangular belt of the following dimensions: Its width w takes on its values in the set {50, 60, 70, 80, 90, 100 m}, and its length l = 3000 m. We assume that the sensors are densely and randomly deployed in this rectangular belt, and that their sensing and communication ranges are equal to 5 m and 10 m, respectively. The number of deployed sensors is set to 5000. Notice that the number of required sensors for deterministic sensor deployment based on square lattice (see Theorem 19.5) and hexagonal lattice (see Theorem 19.6) for w = 100 m are respectively given by n SL = 3000 < 5000, and n hl = 3305 < 5000. This shows that the sensors ardensely dloyed. In fact, the number of required sensors is much smaller for the other smaller values of w, i.e., {50, 60, 70, 80, 90 m}. Also, we assume that an intruder moves across the rectangular belt from top to bottom. That is, if (x(t), y(t)). is the position of an intruder at time t, (x(t + 1), y(t + 1)). is its position at time t + 1 such that the following conditions are met: 0 ≤ x(t), x(t + 1) ≤ l = 3000 0 ≤ y(t + 1) ≤ y(t) ≤ w There is no relationship between the x-coordinates of the intruder’s location at times t (i.e., x(t)) and t + 1 (i.e., x(t + 1)). However, the y-coordinates of the intruder’s location should satisfy y(t + 1) ≤ y(t) so an intruder makes progressive movement as they cross the rectangular sensor belt.

19.6.2 Simulation Results Figures 19.16 and 19.18 are for square lattice-based analysis, while Figs. 19.17 and 19.19 are for hexagonal lattice-based analysis. For random sensor deployment, we want to select the sensors whose locations correspond or are close to their counterparts in square and hexagonal lattices, respectively, so the rectangular sensor belt is kcovered. Figures 19.16 and 19.17 show that the number of selected sensors to generate square lattice and hexagonal lattice, respectively, is higher than the number of sensors needed for square lattice-based and hexagonal lattice-based deterministic sensor deployment, respectively. This is mainly due to randomness. It is not always possible to extract a square lattice or hexagonal lattice from randomly deployed sensors.

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Fig. 19.16 Number of selected sensors for square lattice

Also, the number of sensors needed to generate a square lattice is smaller than its counterpart to produce a hexagonal lattice. Figures 19.18 and 19.19 show that rate of intruder detection associated with the generated hexagonal lattice is higher than its counterpart corresponding to the produced square lattice. Indeed, a denser sensor deployment improves the percentage of detection of intruder while crossing the kcovered sensor belt region. Moreover, as w increases, k increases (i.e., k is linearly proportional to w), the rate of intruder detection increases. As discussed earlier in Sect. 19.1, the presence of multiple layers (i.e., layered protection system) makes the likelihood of an intruder being caught (or detected) high.

19.7 Related Work In this section, we describe a sample of approaches that dealt with the barrier coverage problem in wireless sensor networks. Kumar et al. [242] proposed the first study of the k-barrier coverage problem in wireless sensor networks. They established an optimal deployment pattern for achieving k-barrier coverage was established, developed efficient global algorithms for checking k-barrier coverage of a given region. First, they investigated the weak barrier-coverage with high probability, where all intruders are guaranteed to be detected when crossing a barrier of stealthy sensors. Then, they discussed the strong barrier-coverage with high probability, which ensures the detection of all intruders

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Fig. 19.17 Number of selected sensors for hexagonal lattice

Fig. 19.18 Intruder detection rate for square lattice

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Fig. 19.19 Intruder detection rate for hexagonal lattice

when crossing a barrier of non-stealthy sensors. In addition, they showed the nonexistence of localized algorithms for testing the existence of global barrier coverage. To address this limitation, Chen et al. [116] proposed localized algorithms so sensors can locally determine the existence of local barrier coverage. Moreover, Kumar et al. [245] proposed optimal polynomial-time algorithms to solve the sleep-wakeup problem for the barrier coverage model using sensors with equal and unequal lifetimes. Liu et al. [271] presented an efficient distributed algorithm for the construction of sensor barriers on long strip areas of irregular shape. Also, they presented results related to intruder detection depending on the relationship between the width and length of a rectangular area. Yang and Qiao [431] focused on the barrier information coverage problem, which aims at reducing the number of active sensors to cover a barrier through exploiting collaborations and information fusion among neighboring sensors. Chen et al. [115] introduced the concept of local barrier coverage, which allows sensors to locally determine whether a given sensor deployment can provide global barrier coverage, where all movements with trajectory confined to a slice of the belt of region of deployment are guaranteed to be detected. Saipulla et al. [341] considered the problem of using mobile sensors with limited mobility to efficiently improve barrier coverage. They provided a sensor mobility scheme that maximizes the number of barriers with minimum sensor moving distances. Yang et al. [430] studied the minimum-energy cost k-barrier coverage problem in wireless sensor networks, where each sensor has several sensing power levels. They modeled

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this problem as a minimum cost flow problem, and used Lagrangian relaxation technique to solve it. Wang et al. [395] focused on the problem of efficient use of mobile sensors to achieve k-barrier coverage. First, given a number of deployed stationary sensors, they determined the number of mobile sensors that are required to form k-barrier coverage. Then, given the deployment of stationary and mobile sensors, they computed the maximum number of formed barriers. Megerian et al. [294], [295] used the concept of Voronoi diagram [67] to study the problems of worst and best-case coverage in wireless sensor networks. He et al. [194] investigated the barrier coverage problem in wireless sensor networks for line-based and curvebased sensor deployment. They identified the characteristics for optimal curve-based deployment. Saipulla et al. [342] studied the barrier coverage of a line-based sensor deployment and used mobile sensors to improve barrier coverage. They devised an efficient algorithm for mobile sensor relocation on the deployed line to improve barrier coverage by filling gaps and balancing energy consumption among mobile sensors. He et al. [193] generalized their previous work [194] to take into consideration a heterogeneous sensing model. Kim et al. [233] proposed three remedies for the scheduling algorithms developed by Kumar et al. [244], which achieved optimal lifetime via the identification of a collection of disjoint subsets of sensors, each of which provide barrier-coverage over the area. Kim et al. [232] proposed four approaches for constructing reinforced barriers, which sense any intruder movement and detect any penetration. Zhang et al. [450] formulated the k-barrier coverage problem as a constrained optimization problem and introduce the energy constraint of sensor node to prolong the lifetime of the k-barrier coverage. Also, proposed hybrid particle swarm optimization and gravitational search algorithm for adjusting the velocity updating by integrating the ability to exploit in particle swarm optimization to enhance the global search capability and introduce agent boundary mutation strategy to increase population diversity and search accuracy. Luo and Zou [283] proposed a strong k-barrier coverage algorithm, which is a local-barrier constructing algorithm that can detect any intruder crossing the k-barrier with a full probability. Also, it can differentiate between legal intruder and illegal intruder. For more details on various coverage problems in wireless sensor networks, the reader is referred to a comprehensive survey [390].

19.8 Conclusion Providing perimeter intrusion detection to a critical area is a requirement to achieve physical security objective. Among others, an international border is one instance of the above-mentioned physical security problem. We find that wireless sensor networks can be deployed to build a belt of sensors (or barrier) around a protected area to detect and prevent any intruder crossing it. In this chapter, we attempt to solve the problem of k-barrier coverage, where every path crossing this barrier intersects with at least k sensors. First, we consider two deterministic lattice-based sensor deployment strategies: Square lattice and hexagonal lattice. Second, we represent the

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walk of an intruder across this barrier by an abstract path, which includes progressive moves, each of which having the form of left-oblique, right-oblique, or vertical line segment. Then, we derive several theoretical results, such as the total number of intruder’s abstract paths along with their polynomial representation, the total number of sensors, and the intruder’s abstract path observability for square and hexagonal lattices-based k-barrier coverage. We propose a generalization of these results by considering a random intruder’s trajectory across a k-barrier covered sensor belt, which is characterized by a sequence of left-horizontal, right-horizontal, left-oblique, right-oblique, and/or vertical (for square lattice only) line segments.

Chapter 20

A Spatial Approach for Physical Security Through Connected k-Barrier Coverage

General disarmament being for the present entirely out of question, a proportionate reduction might be recommended. The safety of any country and of the world’s commerce depending not on the absolute, but relative amount of war material, this would be evidently the first reasonable step to take towards universal economy and peace. But it would be a hopeless task to establish an equitable basis of adjustment. Population, naval strength, force of army, commercial importance, water-power, or any other natural resource, actual or prospective, are equally unsatisfactory standards to consider. Nikola Tesla (1856–1943)

Overview This chapter investigates physical security, which is essential to safeguarding critical areas (e.g., high-risk chemical facilities and plants’ physical assets) using machinery and facilities, such as alarms, locked gates, and cameras. To cope with the problem of accessing a critical space, which is a physical breach, it is important to build a barrier that prevents any intruder’s attempt to cross it and access its facilities. Precisely, this chapter focuses on the physical security problem in spatial stealthy lattice wireless sensor networks using a spatial sensor belt around a critical space. Specifically, we suggest a theoretical framework to investigate the spatial kbarrier coverage problem, where any path that crosses this sensor belt intersects with the sensing range of at least k sensors. We propose to study the problem of spatial k-barrier coverage from a tiling viewpoint, where the sensing spheres of the sensors are touching (or kissing) each other. We analyze various spatial deterministic sensor deployment methods: Simple cubic, body centered cubic, face centered cubic, and hexagonal close-packed lattice wireless sensor networks. First, using the concept of the unit cell covered volume ratio, we prove that none of these spatial lattices guarantee k-barrier coverage. Second, to remedy this problem, we consider polyhedral space-fillers, which meet the k-barrier coverage property, and, particularly, the great rhombicuboctahedron. Then, we introduce the concept of intruder’s abstract paths along a spatial k-barrier covered sensor belt region, and compute their number. Also, we propose a polynomial representation of all abstract paths. In addition, we compute the number of sensors deployed over a spatial k-barrier covered sensor belt region

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_20

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using various polyhedral space-fillers. Third, we corroborate our analysis with both numerical and simulation results.

20.1 Introduction Physical security of facilities and equipment becomes a necessity for their safety and/or operation. Any unauthorized access to them constitutes a fundamental physical breach, causing a physical security problem. Various solutions were proposed to cope with this problem. From the earliest medieval castles to the middle ages, moats were used as a defensive strategy against an attacking army. Also, physical safeguards, such as guards, barricades, and fences, are used to protect a facility by providing a single layer of physical security. More sophisticated video surveillance systems, such as cameras, can be used to track any unauthorized access to a protected area. In this chapter, we focus on spatial lattice wireless sensor networks that are capable of providing multiple layers of protection, where each layer consists of several sensors. The use of this type of layered protection system assures that if an intruder succeeds to traverse one layer of sensors without being caught, one of the other layers of sensors would be able to further detect and prevent such an intrusion attempt. In other words, because there are several layers, it is highly likely that an intruder would be caught, thus, helping avoid any undesirable consequences. The strength and efficacy of the physical security of the underlying system increases with the number k of its layers. Our study focuses on the protection of the perimeter of a facility, such as an international border. Thus, it is essential to build a barrier that prevents any intruder’s attempt to cross it and access a critical area. In this chapter, we address the problem of physical security in stealthy lattice wireless sensor networks through the use of a spatial belt of sensors (i.e., barrier) surrounding a critical area. Specifically, we propose a theoretical framework to analyze the problem of k-barrier coverage, where every path crossing this spatial belt of sensors intersects with the sensing range of at least k sensors, where k ≥ 1. That is, a path is k-barrier covered if some (i.e., at least one) or all of its points are covered by (or intersect with) at least k sensors, which are deployed in a barrier. Next, we specify the major tasks we aim at accomplishing in this chapter. Moreover, we briefly describe how each one of them can be achieved.

20.1.1 Major Tasks In this chapter, we want to solve the physical security problem through k-barrier coverage using spatial wireless sensor networks. Note that the problem of k-barrier

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coverage in spatial wireless sensor networks has not been studied extensively like its counterpart in planar wireless sensor networks. In this chapter, we shed more light on this problem and its various tasks in a more realistic environment in order to resolve it. Next, we introduce these tasks and briefly discuss our corresponding plan of actions. First, we want to place the sensors deterministically in a spatial belt region so that every path crossing this belt is k-barrier covered, i.e., every path intersects with at least k sensors. Thus, we investigate the k-barrier coverage problem from a tiling perspective, where the sensors’ sensing spheres are touching (or kissing) each other. We propose various deterministic deployment methods of a spatial belt of sensors around a critical area leading to various lattices, including simple cubic, body centered cubic, face centered cubic, and hexagonal close-packed lattices. We prove that none of these lattices ensures k-barrier coverage. Second, we want to compute the minimum number of sensors to achieve k-barrier coverage of a spatial belt region. We use a spatial polyhedral space filler, namely the great rhombicuboctahedron, which assimilates the sensors’ sensing range in order to achieve k-barrier coverage of a spatial belt. Then, we compute the corresponding number of sensors. Third, we want to determine how to ensure network connectivity. To this end, we compute the ratio of the communication range to the sensing range of the sensors so a spatial k-barrier covered belt is connected. Fourth, we investigate the number of all possible paths (or abstract paths) of an intruder crossing a spatial k-barrier covered belt region. We exploit the concept of abstract path, which is introduced earlier in Chap. 19, and account for all intruder’s sequences of k progressive moves in their trajectories through the great rhombicuboctahedron. Then, we compute the number of these paths crossing a spatial k-barrier covered sensor belt as a function of k. Fifth, we want to represent all the intruder’s abstract paths crossing a spatial connected k-barrier covered belt region. We provide a polynomial representation of all intruder’s abstract paths while taking into consideration the great rhombicuboctahedron. Sixth, we want to assess the performance of the great rhombicuboctahedron to ensure connected k-barrier coverage of a spatial belt region. Thus, we corroborate our analysis with various numerical and simulation results.

20.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 20.2 studies the k-barrier coverage problem for the above lattices. Section 20.3 focuses on the great rhombicuboctahedron to solve the k-barrier coverage problem in spatial wireless sensor networks. Section 20.4 provides numerical and simulation results of our proposed investigation. Section 20.5 reviews sample related work. Section 20.6 concludes the chapter.

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20.2 Spatial k-Barrier Coverage Problem Analysis In this section, we analyze the k-barrier coverage problem from a tiling perspective. In other words, we attempt to tile a spatial sensor belt region so it is k-barrier covered, while there is no overlap between the sensors’ sensing disks. We consider various deterministic sensor deployment strategies leading to the following spatial lattice wireless sensor networks: • • • •

Simple cubic lattice (Fig. 20.1a) Body centered cubic (BCC) lattice (Fig. 20.1b) Face centered cubic (FCC) lattice (Fig. 20.1c) Hexagonal close-packed (HCP) lattice (Fig. 20.1d).

Now, we want to check whether each of these lattice wireless sensor networks can guarantee k-barrier coverage using a tiling-based approach. That is, we have to found out whether any path crossing each of these lattice wireless sensor networks intersects with the sensing ranges of at least k sensors. First, we exploit the notion of unit cell covered volume ratio, which is introduced earlier in Chap. 2 (Sect. 2.2, Definition 2.46). We use it to assess the quality of tiling to solve the problem of k-barrier coverage. First, we present our assumptions, which are used in our study of the physical security problem in spatial wireless sensor networks defines some key terms, and states the assumptions made to analyze the k-barrier coverage problem in spatial wireless sensor networks. Assumption 20.1 (spatial Belt Region and Sensor Deployment) All the sensors are deterministically deployed in a rectangular parallelepiped belt region of width w, length l, and height h using simple cubic, body-centered cubic, face-centered cubic, hexagonal close packed, or polyhedral space-filler lattice-based sensor deployment approach. ∎ Assumption 20.2 (Stealthy Sensors) All the deployed sensors in a rectangular parallelepiped belt region are stealthy. ∎ Assumption 20.3 (Intruder Detection) Any intruder moving along a k-barrier covered path when walking through a belt region to cross a border or access a ∎ protected area, will surely be detected by at least k sensors, where k ≥ 1. Assumption 20.4 (Fast Sensor Belt Region Crossing) An intruder attempts to cross a sensor belt region as fast as possible. ∎

20.2.1 Simple Cubic Lattice In this sensor deployment strategy, each pair of sensing disks on any edge of any square face of a simple cubic lattice quite touch one another. Also, all the sensing

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Fig. 20.1 a Simple cubic lattice, b body centered cubic (BCC) lattice, c face centered cubic (FCC) lattice, and d hexagonal close-packed (HCP) lattice

disks are homogeneous (i.e., having same radius) and are placed on the vertices of this cubic lattice such that each of its vertices coincides with the center of exactly one sensing disk. As it can be seen from Fig. 20.2, a sensor placement strategy yielding a simple cubic lattice cannot guarantee k-barrier coverage. This is due to the holes (or gaps) caused by placing four spheres tangential to each other to form a square face of a cube. In fact, any vertical path that is perpendicular to any sequence of holes does not intersect with any sensing sphere at all. Theorem 20.1 states that a simple

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Fig. 20.2 Unit cell of a simple cubic lattice

cubic lattice-based sensor deployment strategy does not meet the k-barrier coverage property. Theorem 20.1 (Simple Cubic Lattice) The simple cubic lattice cannot be deployed to successfully k-barrier cover a spatial sensor belt region. Proof Let us compute the unit cell covered volume ratio ϑ as defined earlier in Chap. 2 (Sect. 2.2, Definition 2.46). Notice that the unit cell is a cube whose edge length is a = 2r and volume is V (a) = a 3 = 8r 3 . Moreover, only one eighth of a sensing sphere at each of the eight vertices of a cube that covers (or occupies) the cube. That is, we use exactly 8 × 18 Vs (r ) = Vs (r ) to cover the cube, where Vs (r ) is the volume of a sensing sphere with radius r . We obtain: ϑ=

π Vs (r ) = = 0.52 V (a) 6

This implies that the cube is not totally covered (or occupied) by the sensing spheres. Given that there are holes (or gaps, uncovered regions) at every square face, there exists at least one path through a vertical (or horizontal) stack of adjacent cubes that cannot be k-barrier covered. Thus, a simple cubic lattice cannot be used to k-barrier cover a spatial sensor belt region. ∎ Next, our analysis of the k-barrier coverage problem focuses on BC, FCC, and HCP lattice wireless sensor networks. Specifically, we consider other sensor placement strategies based on BCC, FCC, or HCP lattice, and check whether they ensure kbarrier coverage. That is, we need to find out whether any path crossing any of these lattices intersect with at least one sensing sphere at any given layer.

20.2.2 Body Centered Cubic (BCC) Lattice To remedy the violation of the k-barrier coverage property caused by those holes in a cubic lattice wireless senor network, we place a sensor in the center of every cube of

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the lattice. This leads to a body centered cubic (BCC) lattice wireless senor network (Fig. 20.3). Here, we distinguish two different topologies based on the placement of the eight sensing spheres with respect to each other. Case 1: Pairwise touching adjacent sensing spheres Here, the eight sensing spheres of any cube have to overlap with this additional sensing sphere in the middle of this cube. Given that we do not allow any overlap among the sensing spheres, the one in the center of each cube cannot be similar to the rest of the other sensing spheres. Therefore, we obtain a BCC lattice of heterogeneous sensors (Fig. 20.4). Theorem 20.2 states that this BCC lattice cannot k-barrier cover a spatial sensor belt region. Theorem 20.2 (Heterogeneous Body Centered Cubic Lattice) A BCC lattice of heterogeneous sensors with pairwise touching adjacent sensing spheres cannot be deployed to k-barrier cover a spatial sensor belt region. Fig. 20.3 Body centered cubic lattice

Fig. 20.4 Unit cell of a body centered cubic lattice (Case 2)

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Proof Let us compute the radius rcBCC of this central sensing sphere. Under the above conditions, the edge length of any cube is 2r , where r stands for the radius of the sensing disks of the sensors placed on the vertices of cubes of the lattice. Also, the diagonal through the body of a cube has a length d BCC that can be computed as follows: AH = d BCC = r + 2rcBCC + r = 2r + 2rcBCC Using√Pythagore’s Theorem, we have: AE 2 + E H 2 = AH 2 , where AE = 2r , E H = 2 2r (diagonal through a face of the cube), and AH = d BCC = 2r + 2rcBCC . Thus, we obtain: ) (√ rcBCC = 3−1 r 2r , it is clear that the above condition is met. Also, this means that any pair of central sensing spheres in two adjacent cubes are not touching each other. This type of topology yields successive gaps along the verticals of the four sides of every cube of the lattice. Therefore, this structure cannot provide k-barrier coverage. ∎

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20.2.3 Face Centered Cubic (FCC) Lattice Another attempt to remedy to the k-barrier coverage property violation problem is to place a sensing sphere at the center of each of the eight square faces of a cube. This yields a face centered cubic (FCC) lattice wireless sensor network. First, we consider the following two cases as in Sect. 20.2.2. Case 1: Pairwise touching adjacent sensing spheres We require that any pair of sensing spheres on the edge of any square face are tangential to each other. Also, the additional central sensing sphere must be tangential to the pair of sensing spheres located on the diagonal of any square face. Theorem 20.4 states that this FCC lattice cannot k-barrier cover a spatial sensor belt region. Theorem 20.4 (Touching Face Centered Cubic Lattice) The FCC lattice with pairwise touching adjacent sensing spheres cannot be deployed to successfully k-barrier cover a spatial sensor belt region. Proof Let us compute the radius rcFCC of this central sensing sphere. The diagonal through the square face of a cube has a length d FCC equal to d FCC = r + 2rcFCC + r = 2r + 2rcFCC We have: C A2 + AB 2 = C B 2 , where C A = AB = 2r (cube edge length) and C B = d FCC = 2r + 2rcFCC . We get: rcFCC =

(√ ) 2−1 r

Clearly, rcFCC < rcBCC < r . That is, the central sensor has to be less powerful than the other ones in terms of its sensing range. Also, no pair of sensing spheres located on any pair of parallel square faces can touch each other. Notice that the placement of an additional sensing sphere at the center of a square face of a cube leaves four small gaps. Each of these gaps creates a vertical (and horizontal) path through a vertical (and, also horizontal) stack of adjacent cubes. Let us compute the unit cell covered volume ( ratio ) ϑ. A unit cell is a cube whose edge length is a FCC = 2r and volume is V a FCC = a FCC3 = 8r 3 . Also, only one eighth of a sensing sphere at each of the eight vertices of a cube, and six halves of the sensing sphere of radius rcFCC (on the) six faces that cover ( the ) cube. That is, we use exactly 8 × 18 Vs (r ) + 6 × 21 Vs rcFCC = Vs (r ) + 3Vs rcFCC to cover the cube. We get: ( ) Vs (r ) + Vs rcBCC ( ) ϑ= = 0.64 V a BCC Thus, it violates the k-barrier coverage property.

∎

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Next, we focus on another variation of FCC lattice based on hexagonal tiling, and study the k-barrier coverage problem. Case 2: Pairwise non-touching adjacent sensing spheres Here, we assume that all the additional sensing spheres located at the centers of all six square faces of the cube are similar to all other sensing spheres on the eight vertices of the cube. We place each of these additional sensing spheres as tightly as possible to those four spheres on the four vertices of a square face of the cube. In this case, only the three sensing spheres on the diagonal of each square face touch each other. Any four sensing spheres located on the four vertices of any square face are separate and do not intersect at all. Theorem 20.5 states that this FCC lattice cannot k-barrier cover a spatial belt. Theorem 20.5 (Non-Touching Face Centered Cubic Lattice) A FCC lattice with pairwise non-touching adjacent sensing spheres cannot k-barrier cover a spatial sensor belt region. Proof Let us compute the edge length a FCC of a unit cell (i.e., cube). Intuitively, all the three sensors on the diagonal of every square face are connected to each other as shown on Fig. 20.6. That is, the diagonal through each square face of a cube has a length d FCC equal to 4r , i.e., d FCC = 4r . Using Pythagore’s Theorem, we have: B A2 + AC 2 = BC 2 , where B A = AC = a FCC , BC = d FCC = 4r . Thus, we obtain: √ a FCC = 2 2r Given that a FCC > 2r , it is clear that any pair of sensing spheres on the edge of any square face of a cube of the lattice do not intersect. It is possible to move between cubes without touching any sphere. That is, we can find a path through a vertical stack of cubes without intersecting with any sensing sphere. Let us compute the unit cube whose √cell covered volume (ratio ϑ. ) A unit cell is a √ edge length is a FCC = 2 2r and volume is V a FCC = a FCC3 = 16 2r 3 . Also, only one eighth of a sphere at each of the eight vertices of a cube, and six halves of

Fig. 20.6 Unit cell of a face centered cubic lattice (case 2)

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the sensing sphere of radius r on the six faces that cover the cube. That is, we use exactly 8 × 18 Vs (r ) + 6 × 21 Vs (r ) = Vs (r ) + 3Vs (r ) = 4Vs (r ) to cover the cube. We have: ϑ=

4Vs (r ) ) = 0.74 ( V a FCC

Given that the unit cell covered volume ratio ϑ is less than 1 (i.e., ϑ = 0.74), it is clear that there are several empty regions in the cube (and, in the FCC lattice). Even though all of those trigonal and octahedral holes seem to be covered by second layer and third layer, respectively, those empty regions are not covered by any sensing sphere. Although those empty regions are tiny, they may be traversed by an infinite number of paths without touching any sensing sphere in the FCC lattice. Thus, this type of lattice structure fails to satisfy the k-barrier coverage property. ∎ We should point out that face centered cubic (FCC) lattice and cubic close-packed (CCP) lattice (Fig. 20.7) are two different names for the same lattice structure. First, let us show how to build a CCP lattice using a hexagonal tiling-based approach. We start with a hexagonal layer of sensing spheres (blue color), which yields holes. These gaps are trigonal holes (Fig. 20.8) that are formed between any three adjacent sensing spheres in this first layer. We add a second hexagonal layer of sensing spheres (gray color) to cover those trigonal holes. However, this second layer cannot cover all the holes. Indeed, there are octahedral holes (Fig. 20.8), which are holes from the Fig. 20.7 CCP lattice formation

Fig. 20.8 Trigonal and octahedral holes

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Fig. 20.9 Formation of a unit cell of a face centered cubic lattice

second layer that fall directly over holes in the first layer. The process of building these two first layers is identical to the one used in the generation of hexagonal closepacked (HCP) lattice, which is discussed below. Now, we add a third layer of sensing spheres (pink color) to cover all these octahedral holes. As it can be seen, all these three layers are different from each other as shown in Fig. 20.7. If we repeat this pattern several times (i.e., stacking these three layers indefinitely, following Plan A, Plan B, Plan C, Plan A, etc.), and rotate this structure, it will reveal its cubic structure that is similar to FCC structure. As discussed above, this structure contains hexagonally packed layers, and does not show cubic. In fact, we can generate a unit cell of a face centered cubic lattice as follows. We consider a hexagonal packing of six spheres forming an equilateral triangle. Then, we place another sensing sphere on top, which yields a triangular pyramid with seven sensing spheres. We use the same process to build another triangular pyramid. Now, we place the two pyramids with 14 sensing spheres together facing in opposite directions as shown in Fig. 20.9. Notice that if we rotate the obtained structure, we obtain a cube, where the centers of eight of the spheres lie at the centers of the eight vertices of the cube, and the centers of the remaining six spheres coincide with the centers of the six faces of the cube.

20.2.4 Hexagonal Close-Packed (HCP) Lattice We build an HCP lattice as follows: First, we start with a hexagonal layer of sensing spheres (blue color, or Plan A), which yields holes. These gaps are trigonal holes that are formed between any three adjacent sensing spheres in this first layer. We add a second hexagonal layer of sensing spheres (gray color, or Plan B) to cover some of

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Fig. 20.10 HCP lattice formation

those trigonal holes. Then, we repeat this same process using an alternation of Plan A and Plan B (i.e., Plan A, Plan B, Plan A, Plan B, etc.) to build subsequent layers as shown in Fig. 20.10. Theorem 20.6 states that this HCP lattice cannot k-barrier cover a spatial sensor belt region. Theorem 20.6 (Hexagonal Close-Packed Lattice) The HCP lattice cannot be successfully deployed to k-barrier cover a spatial sensor belt region. Proof As it can be seen from Fig. 20.10, half of the trigonal holes are not covered by the second layer (i.e., Plan B). Hence, there is a path through a stack of each of these trigonal holes that does not intersect with any sensing sphere. Let us compute the unit cell covered volume ratio ϑ. Notice that the unit cell of a hexagonal close-packed lattice (Fig. 20.11) is a hexagonal prism. It is clear that the side length a of the base of the prism is a = 2r . Also, the√height h of the prism is twice √ the distance between two adjacent layers, which is 2 2/3r . Thus, we have h = 4 2/3r . The volume of this hexagonal prism is given by: √ √ V (hexagonal prism) = (3/2) 3a 2 h = 24 2r 3 Furthermore, only one sixth of a sensing spheres at each of the six vertices of both the bottom and top bases of the prism, three complete sensing spheres inside the prism, and one half of a sensing sphere at the center of each of the bottom and

20.3 Polyhedral Space-Filling Lattice

669

Fig. 20.11 Formation of a unit cell of a hexagonal close-packed lattice

top bases of the prism that cover (or occupy) the hexagonal prism. That is, we use exactly 12 × 16 Vs (r ) + 3 × Vs (r ) + 2 × 21 Vs (r ) = 6 × Vs (r ) to cover the hexagonal prism, where Vs (r ) is the volume of a sensing sphere with radius r . We obtain: ϑ=

6 × Vs (r ) = 0.74 V (hexagonal prism)

Therefore, HCP lattice cannot assure k-barrier coverage.

∎

We find that FCC–Case 2 (or CCP) and HCP have the same unit cell covered volume ratio ϑ. Indeed, we show earlier that CCP lattice can be built using a hexagonal tiling-based method. Result: We conclude that with the concept of coverage, it is impossible to ensure k-barrier coverage using simple cubic, HCP, BCC, or FCC lattice wireless sensor networks. This is mainly due to the presence of holes by the process of tiling using sensing spheres. Next, we turn our attention to the concept of polyhedron, which is a spatial solid whose faces join at their edges. Specifically, we focus on convex polyhedra that are space-fillers. The latter are capable of filling (or covering) a spatial space through replication of its congruent copies without any overlap of the interiors of or any gap between any pair of adjacent copies.

20.3 Polyhedral Space-Filling Lattice In this section, we investigate the k-barrier coverage problem using spatial space fillers (or spatial space filling polyhedra) [15, 17]. These are convex polyhedra that tile a spatial space without any overlap or any gap. In fact, the sphere is not a spatial space filler given that a lattice of spheres leaves gaps regardless of how efficient this lattice is. As discussed above, even the most efficient sphere packing lattice (i.e., FCC lattice) yields holes and cannot ensure k-barrier coverage. First, we describe our method for sensor placement to guarantee k-barrier coverage. Then, we provide a thorough analysis of intruder’s paths.

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Fig. 20.12 The great rhombicuboctahedron

First, we identify a space filler that can be used to tile a spatial space without overlaps and gaps. In his work on connected coverage [23] and connected k-coverage [22], Ammari studied several space fillers, namely the cube [125], regular right hexagonal prism [125], truncated octahedron [292], great rhombicuboctahedron [292], rhombic dodecahedron [292], elongated dodecahedron [292], rhombic triacontahedron [292], Sommerville’s largest tetrahedron [345], and Goldberg’s equilateral octahedron [173]. However, the great rhombicuboctahedron (also rhombitruncated cuboctahedron or truncated cuboctahedron) [292] is the most efficient one (Fig. 20.12). Indeed, it has the maximum quality of coverage compared to all other spatial convex polyhedral space-fillers [22, 23]. That is, the great rhombicuboctahedron is the largest spatial space filler that can be inscribed in a sphere. Its structure consists of three different types of faces. It has 26 faces, including 12 squares, 8 hexagons, and 6 octagons (Fig. 20.12). Without loss of generality, assume that any intruder’s path trajectory will be top-down. In order to achieve k-barrier coverage, we use the above spatial space filler, i.e., the great rhombicuboctahedron, to tile a spatial sensor belt region of width wk , height h k , and length lk . More precisely, we build a wall (or rectangular parallelepiped) that has k layers, each of which has a width w = wk , a length l = lk , and a height h = hkk . This implies that in this rectangular parallelepiped, all the k planar, horizontal layers are adjacent to each other and there is no emptiness between them as shown in Fig. 20.7. Moreover, all of these k layers are the same, i.e., we build a layer and duplicate it k times. This wall can be used to protect any country’s border from any intrusion. Given that each layer consists of a set of sensors whose sensing range is assimilated to our selected spatial space filler, i.e., the great rhombicuboctahedron, any intruder will always be within the sensing range of at least one sensor. In fact, if an intruder is located or moving on any of the

20.3 Polyhedral Space-Filling Lattice

671

edges or faces of the great rhombicuboctahedron, they could be within the sensing range of at least two or more sensors. In other words, any intruder who would cross this wall, would be continuously detected and tracked by at least k sensors. As it can be seen from Figs. 20.12, 20.13, and 20.14, the great rhombicuboctahedron is one of the 13 Archimedean solids that are distinguished by their high

Fig. 20.13 Unfolding the great rhombicuboctahedron

Fig. 20.14 Orthogonal projections of the great rhombicuboctahedron

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20 A Spatial Approach for Physical Security …

symmetry. Without loss of generality, we build a layer such that the first great rhombicuboctahedron, which is referred to as the reference great rhombicuboctahedron, rests on one of its octagonal faces. In this way, the great rhombicuboctahedron presents five (5) levels and has its center located on the middle level: • The one in the middle has eight (8) vertical, planar faces. Specifically, this middle level has four (4) octagonal faces and four (4) square faces in alternation. • The two levels above the middle one are symmetric to their counterpart ones below the middle level. These two levels (respectively, their counterpart ones) are as follows: The first one immediately above (respectively, below) the middle one has eight (8) oblique, planar faces: Four (4) hexagonal faces and four (4) square faces in alternation. The top (respectively, bottom) level has only one (1) octagonal face.

20.3.1 Intruder’s Path Analysis Now, we present some theoretical results based on the use of the great rhombicuboctahedron. Let us consider Fig. 20.12, and focus on a special version of this space filler, called reference great rhombicuboctahedron, denoted by G Rr e f . It is located at the center of the middle level. It has nine types of neighboring nodes: • Eight nodes located at the level immediately above the middle level (one back right diagonal node, one back left diagonal node, one front right diagonal node, and one front left diagonal node, which are connected to G Rr e f through segments orthogonal to its four square faces on the middle level; one right oblique node, one left oblique node, one back oblique node, and one front oblique node, which are connected to G Rr e f through segments orthogonal to its four hexagonal faces on the middle level). • One vertical node, which is connected to G Rr e f through the segment orthogonal to its octagonal face on the top level. It worth noting that G Rr e f has the same nine types of neighboring nodes at the levels below the middle one. However, the great rhombicuboctahedron presents a perfect symmetry in the sense that its top half is similar to its bottom half (Fig. 20.14). That is, a horizontal plan passing through the center of the great rhombicuboctahedron splits it into two equal halves. Also, a vertical plan passing through the center of the great rhombicuboctahedron splits it into two equal halves (Fig. 20.14). Thus, all these nine types of neighboring nodes at the levels below the middle one are similar to their counterpart ones located above the middle level. More specifically, these types are as follows: • Eight nodes located at the level immediately below the middle level:

20.3 Polyhedral Space-Filling Lattice

673

First set of four nodes: One back right diagonal node, one back left diagonal node, one front right diagonal node, and one front left diagonal node, which are connected to G Rr e f through segments orthogonal to its four square faces on the middle level. Second set of four nodes: One right oblique node, one left oblique node, one back oblique node, and one front oblique node, which are connected to G Rr e f through segments orthogonal to its four hexagonal faces on the middle level). • One vertical node, which is connected to G Rr e f through the segment orthogonal to its octagonal face on the bottom level. We should mention that G Rr e f has eight horizontal nodes, which are connected to it through segments orthogonal to its four square faces and its four octagonal faces on the middle level. However, these eight nodes are not included in any structural k-node line. Theorem 20.7 computes the number of structural k-node lines for space-filling great rhombicuboctahedron lattice wireless sensor networks. Theorem 20.7 (Great Rhombicuboctahedron Lattice) The number of structural knode lines for great rhombicuboctahedron lattice wireless sensor networks is 9k−1 , where k ≥ 1 is a natural number. Proof We can proceed using a proof by mathematical induction on k. Let P(k) be the following statement: P(k) : “There are 9k−1 different structural k-node lines, where k ≥ 1 is a natural number.”

Basis step: Let us prove that P(k) is true for k = 1. This is trivial. In fact, there is only one structural 1-node line (90 = 1). Inductive step: We assume that P(m) is true, i.e., there are 9m−1 different structural m-node lines, where m ≥ 1. We want to prove that P(m + 1) is true. That is, the number of structural (m + 1)-node lines is 9m . We start from those 9m−1 different structural m-node lines, and add the (m + 1)th node to each one of them. Let q be the only leaf node of each of these structural m-node lines. There are nine (9) possibilities to add that (m + 1)th node to each of these structural m-node lines so as to produce structural (m + 1)-node lines. In fact, the newly added (m + 1)th node can be one of those nine types of node. Therefore, the total number of produced structural (m + 1)-node lines is 9 × 9m−1 = 9m , thus, proving P(m + 1) is true. Thus, the statement P(k) is true, for all k ≥ 1. Indeed, we have the inference rule: (P(1) ∧ (∀m ≥ 1, P(m) → P(m + 1))) → P(k), ∀k ≥ 1 Lemma 20.1 calculates the height of a structural k-node line. Lemma 20.1 (Structural k-Node Line Height) The height of a structural k-node line is k − 1, where k ≥ 1. Proof It is easy to show this result using a mathematical induction proof on k. Let Q(k) be the following statement:

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Q(k) : “A structural k-node line has a height equal to k − 1, where k ≥ 1”

Basis step: Let us prove that Q(k) is true for k = 1. This structural 1-node line has height equal to 0. Thus, Q(1) is true. Inductive step: We assume that P(m) is true, i.e., the height of structural m-node line is m −1, where m ≥ 1. We want to prove that Q(m + 1) is true. That is, the height of a structural (m + 1)-node line is m. We add the (m + 1)th node to a structural m-node line whose height is m − 1, and attach it to its leaf node as left, vertical, or right child. We obtain a structural (m + 1)-node line whose height is equal to that of the structural m-node line augmented by 1, i.e., (m − 1) + 1 = m. Thus, Q(m + 1) is true. We have the following inference rule, which proves that Q(k) is true for any k ≥ 1. (Q(1) ∧ (∀m ≥ 1, Q(m) → Q(m + 1))) → Q(k), ∀k ≥ 1

∎

Usually, in order to access a protected area without being detected, an intruder attempts to cross a spatial sensor belt region (or barrier) using a shortest path. Consequently, the latter consists only of a random sequence of progressive line-segments (i.e., back right diagonal, back left diagonal, front right diagonal, front left diagonal, right oblique, left oblique, back oblique, front oblique, or vertical). Intuitively, any horizontal move through a spatial sensor belt region should be excluded from any intruder’s movement trajectory. Because all the great rhombicuboctahedra tile a spatial sensor belt region (i.e., no gaps or overlaps), an intruder’s movement trajectory can be viewed as a sequence of transitions from one sensing sphere to another. We represent this trajectory by a spatial abstract path, which is described as follows: An intruder’s spatial abstract path has only some or all of the nine types of the above progressive line-segments representing the trajectory of an intruder’s movement. It includes all the great rhombicuboctahedra traversed by the intruder.

It is worth mentioning that an intruder’s spatial abstract path, I A P = (N , E), has k nodes and k − 1 edges (i.e., |N | = k and |E| = k − 1). Consequently, we conclude that the number of intruder’s abstract paths along a k-barrier covered sensor belt region S B Rw,l corresponds to the number of structural k-node lines as computed in Theorem 20.7. Corollary 20.1 states this result. Corollary 20.1 (Intruder’s Abstract Path Cardinality): The total number of intruder’s abstract paths with k nodes and k − 1 edges along a strongly k-barrier covered spatial sensor belt region is 9k−1 for the great rhombicuboctahedra lattice, ∎ where k ≥ 1. In reality, the total number of possible intruder’s paths crossing a spatial sensor belt region is infinite. However, owing to the concept of intruder’s abstract path, it is more adequate to represent that infinite set of paths by a finite set that has only 9k−1 paths that are worth of study. Given the symmetry presented by the great rhombicuboctahedron, those intruder’s abstract paths are symmetric to each other. Indeed, the following pairs of line-segments exhibit a symmetry:

20.3 Polyhedral Space-Filling Lattice

675

• Back right diagonal (brd) and back left diagonal (bld) line-segments (i.e., right symmetric to left). • Front right diagonal (frd) and front left diagonal (fld) line-segments (i.e., right symmetric to left). • Right oblique (ro) and left oblique (lo) line-segments (i.e., right symmetric to left). • Back oblique (bo) and front oblique (fo) line-segments (i.e., back symmetric to front). Trivially, vertical (v) line-segments are symmetric to themselves. Moreover, all of those 9k−1 intruder’s abstract paths can be classified into three categories. The first one (Category 1) has only one path that has only vertical line-segments, while the second (Category 2) and third (Category 3) categories have the same number of abstract paths. Furthermore, the abstract paths in the second category are symmetric to their counterparts in the third one. We represent a path by a set of its consecutive edges. Assume that k = 3. Thus, there are 92 = 81 intruder’s abstract paths, each of which has two edges. More precisely, each of the above three categories has the following set of abstract paths. • Category 1 (Only one path with two vertical line-segments): ⟨v, v⟩ • Category 2 (40 paths): ⟨v, brd⟩ , ⟨v, frd⟩ , ⟨v, ro⟩ , ⟨v, bo⟩ , ⟨brd, v⟩ , ⟨brd, brd⟩ , ⟨brd, bld⟩ , ⟨brd, frd⟩ , ⟨brd, fld⟩ , ⟨brd, ro⟩ , ⟨brd, lo⟩ , ⟨brd, bo⟩ , ⟨brd, fo⟩ , ⟨frd, v⟩ , ⟨frd, brd⟩ , ⟨frd, bld⟩ , ⟨frd, frd⟩ , ⟨frd, fld⟩ , ⟨frd, ro⟩ , ⟨frd, lo⟩ , ⟨frd, bo⟩ , ⟨frd, fo⟩ , ⟨ro, v⟩ , ⟨ro, brd⟩ , ⟨ro, bld⟩ , ⟨ro, frd⟩ , ⟨ro, fld⟩ , ⟨ro, ro⟩ , ⟨ro, lo⟩ , ⟨ro, bo⟩ , ⟨ro, fo⟩ , ⟨bo, v⟩ , ⟨bo, brd⟩ , ⟨bo, bld⟩ , ⟨bo, frd⟩ , ⟨bo, fld⟩ , ⟨bo, ro⟩ , ⟨bo, lo⟩ , ⟨bo, bo⟩ , ⟨bo, fo⟩ . • Category 3 (40 paths): ⟨v, bld⟩ , ⟨v, fld⟩ , ⟨v, lo⟩ , ⟨v, fo⟩ , ⟨bld, v⟩ , ⟨bld, brd⟩ , ⟨bld, bld⟩ , ⟨bld, frd⟩ , ⟨bld, fld⟩ , ⟨bld, ro⟩ , ⟨bld, lo⟩ , ⟨bld, bo⟩ , ⟨bld, fo⟩ , ⟨fld, v⟩ , ⟨fld, brd⟩ , ⟨fld, bld⟩ , ⟨fld, frd⟩ , ⟨fld, fld⟩ , ⟨fld, ro⟩ , ⟨fld, lo⟩ , ⟨fld, bo⟩ , ⟨fld, fo⟩ , ⟨lo, v⟩ , ⟨lo, brd⟩ , ⟨lo, bld⟩ , ⟨lo, frd⟩ , ⟨lo, fld⟩ , ⟨lo, ro⟩ , ⟨lo, lo⟩ , ⟨lo, bo⟩ , ⟨lo, fo⟩ , ⟨fo, v⟩ , ⟨fo, brd⟩ , ⟨fo, bld⟩ , ⟨fo, frd⟩ , ⟨fo, fld⟩ , ⟨fo, ro⟩ , ⟨fo, lo⟩ , ⟨fo, bo⟩ , ⟨fo, fo⟩ .

20.3.2 Intruder’s Path Representation and Counting Recall that there are nine types of line-segments in an intruder’s abstract path: Back right diagonal (brd), back left diagonal (bld), front right diagonal (frd), front left diagonal (fld), right oblique (ro), left oblique (lo), back oblique (bo), front oblique (fo), and vertical (v). Let xbr d , xbld , x f r d , x f ld , xr o , xlo , xbo , x f o , and xv be nine variables denoting those nine types of line-segments, respectively. Theorem 20.8 gives a polynomial representation of all intruder’s abstract paths for great rhombicuboctahedra lattice wireless sensor networks. Theorem 20.8 (Intruder’s Abstract Path Representation for Great Rhombicuboctahedra Lattice Wireless Sensor Networks) There is a polynomial representation of those 9k−1 possible intruder’s abstract paths, each of which has k nodes and (k − 1) edges:

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(

xbr d + xbld + x f r d + x f ld + xr o + xlo + xbo + x f o + xv ( ) k−1 kbr d , kbld , k f r d , k f ld , kr o , klo , kbo , k f o , kv Σ = kbr d kbld kr d k xbr d xbld x f r d x f ld f ld κ=k−1

)k−1

kf o

klo kbo xrkor o xlo xbo x f o xvkv

where ) ( k−1 • = kbr d , kbld , k f r d , k f ld , kr o , klo , kbo , k f o , kv (k − 1)! kbr d !kbld !k f r d !k f ld !kr o !klo !kbo !k f o !kv ! κ = kbr d + kbld + k f r d + k f ld + kr o + klo + kbo + k f o + kv • k ≥ 1 is a natural number kbr d kbld k f r d k f ld kr o klo kbo k f o kv • xbr d x bld x f r d x f ld xr o xlo x bo x f o x v is an intruder’s abstract path that has kbr d back right diagonal, kbld back left diagonal, k f r d front right diagonal, k f ld front left diagonal, kr o right oblique, klo left oblique, kbo back oblique, k f o front oblique, and ( kv vertical line-segments ) k−1 • designates the total number of such kbr d , kbld , k f r d , k f ld , kr o , klo , kbo , k f o , kv an abstract path. Proof Based on Lemma 20.1, an intruder’s abstract path consists of (k − 1) levels. Also, each level has exactly one line-segment of one of the above nine types, i.e., back right diagonal, back left diagonal, front right diagonal, front left diagonal, right oblique, left oblique, back oblique, front oblique, and vertical. A level can be viewed as a box, and those nine types of line-segments as nine different types of objects. In other words, we have (k − 1) boxes and nine types of objects. More specifically, there are (k − 1) objects of any of those nine types, and each of those (k − 1) boxes can contain one of those objects. Out of those (k − 1) objects, there can be kbr d back right diagonal, kbld back left diagonal, k f r d front right diagonal, k f ld front left diagonal, kr o right oblique, klo left oblique, kbo back oblique, k f o front oblique, and kv vertical line-segments, subject to the two following conditions: • Cond 1: 0 ≤ kbr d , kbld , k f r d , k f ld , kr o , klo , kbo , k f o , kv ≤ k − 1 • Cond 2: kbr d + kbld + k f r d + k f ld + kr o + klo + kbo + k f o + kv = k − 1 Our problem reduces to counting the total number of possible permutations of (k − 1) objects such that the(above-mentioned two conditions are met. ) k−1 Notice that the multinomial coefficient kbr d , kbld , k f r d , k f ld , kr o , klo , kbo , k f o , kv

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677

stands for the total possible number of different ways to permute a given kbr d kbld k f r d k f ld kr o klo kbo k f o kv multiset xbr d x bld x f r d x f ld xr o xlo x bo x f o x v of (k − 1) objects. We call it “multiset” as the same object (i.e., line-segment or move) can be selected several times. Therefore, an intruder’s abstract path ( ) is a permutation of a multiset, and k−1 computes all possible permutations of kbr d , kbld , k f r d , k f ld , kr o , klo , kbo , k f o , kv the underlying multiset. This helps generate all possible intruder’s abstract paths that contain kbr d back right diagonal, kbld back left diagonal, k f r d front right diagonal, k f ld front left diagonal, kr o right oblique, klo left oblique, kbo back oblique, k f o front oblique, and kv vertical line-segments. In order to consider all possible permutations of all possible multisets, and produce all possible intruder’s abstract paths, we use the summation symbol and vary the integer variables kbr d , kbld , k f r d , k f ld , kr o , klo , kbo , k f o , and kv , subject to the two conditions Cond 1 and Cond 2. ∎ Example 2 Let us consider Table 20.1. It shows a polynomial representation of some intruder’s abstract paths depending on the values of k. For k = 1, the intruder’s abstract path reduces to just one node. For k = 2, there are exactly nine (i.e., 92−1 = 9) intruder’s abstract paths, each of which contains only one type of edge (i.e., back right diagonal, back left diagonal, front right diagonal, front left diagonal, right oblique, left oblique, back oblique, front oblique, or vertical) that could be left-oblique, rightoblique, or vertical. For k = 3, there are 81 (i.e., 93−1 = 81) intruder’s abstract paths, each of which has exactly two edges. Nine of those paths have two edges of the same 2 2 2 type (e.g., xbr d , x bld , . . . x v ). All other paths have two edges of different types. Each of the nine types of edges is associated with one of the remaining eight types of edges. That is, there are 72 (i.e., 9 × 8) paths of this kind. For instance, to be more accurate, the polynomial 2 × xbr d × xbld represents, indeed, two paths: • Path ⟨brd, bld⟩ with one back right diagonal line-segment followed by one back left diagonal line-segment. It is represented by the polynomial xbr d × xbld . • Path ⟨bld, brd⟩ with one back left diagonal line-segment followed by one back right diagonal line-segment. It is represented by the polynomial xbld × xbr d . Without loss of generality, we assume that the length of the wall (or rectangular parallelepiped) is given by lk = nwG R , where wG R stands for the great rhombicuboctahedron’s width, and n ⋙ 1. For example, for border security, the value of is very large as we need a long wall. Theorem 20.9 computes the spatial sensor density Table 20.1 Polynomial representation of all intruder’s abstract paths )k−1 ( k xbr d + xbld + x f r d + x f ld + xr o + xlo + xbo + x f o + xv 1

1

2

xbr d + xbld + x f r d + x f ld + xr o + xlo + xbo + x f o + xv

3

2 + x2 + x2 2 2 2 2 2 2 xbr d bld f r d + x f ld + xr o + xlo + x bo + x f o + x v + 2 × x br d × x bld + . . . + 2 × xbr d × xv + 2 × xbld × x f r d + . . . + 2 × x f o × xv

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to achieve strong k-barrier coverage in great rhombicuboctahedron lattice wireless sensor networks, and the corresponding total number of deployed sensors. Theorem 20.9 (Spatial Sensor Density for the Great Rhombicuboctahedron Lattice Deployment) The spatial sensor density ρ(r) for strong k-barrier coverage in the great rhombicuboctahedron lattice wireless sensor networks, and the corresponding total number of deployed sensors ℵ(r ) are respectively given by: 0.298 r3 ℵ(r ) = 1.785nk 2 ρ(r ) =

where r is the radius of the sensing sphere of the sensors, n ⋙ 1 is a real number, and k ≥ 1 is a natural number, with k being the total number of layers forming the wall. Proof Let wk = kwG R , h k = kh G R , and lk = nwG R be, respectively, the width, height, and length of the wall (or rectangular parallelepiped), with wG R and h G R being, respectively, the great rhombicuboctahedron’s width and height. Also, let aG R be the great rhombicuboctahedron’s edge length, which is defined as follows [479]: aG R = √

2

√ r 13 + 6 2

where r is the radius of the sensing sphere of the sensors. The great rhombicuboctahedron’s volume is computed as follows [479]: )3 ( ( √ ) 3 √ ) 2 r3 = 22 + 14 2 aG R = 22 + 14 2 √ √ 13 + 6 2 (

VG R

By definition, the spatial sensor density ρ(r ) is the number of sensors per unit volume. Given that we place one sensor at the centroid of the great rhombicuboctahedron, the spatial sensor density ρ(r ) that is required for k-barrier coverage can be computed as follows: ρ(r ) =

0.298 1 = VG R r3

According to [479], the distance between the great rhombicuboctahedron’s center and centroid of the square faces is given by: dG R,s f

√ 3+ 2 = aG R 2

20.4 Performance Evaluation

679

Also, distance between the great rhombicuboctahedron’s center and centroid of the octagonal faces is computed as: dG R,o f

√ 1+2 2 aG R = 2

Given that our great rhombicuboctahedron -based tiling approach considers that the great rhombicuboctahedron reposes on its octagonal face, the great rhombicuboctahedron’s width is wG R = 2dG R,s f , while its height is h G R = 2dG R,o f . Thus, the width, height, and length of the wall (or rectangular parallelepiped) can be computed as follows: ( √ ) 2 3+ 2 wk = kwG R = √ √ kr 13 + 6 2 ( √ ) 2 1+2 2 h k = kh G R = √ √ kr 13 + 6 2 ( √ ) 2 3+ 2 lk = nwG R = √ √ nr 13 + 6 2 Let ℵ(r ) be the total number of deployed sensors to achieve k-barrier coverage. Thus, we have: ℵ(r ) = ρ(r ) × VS B R = ρ(r )wk h k lk where VS B R = wk h k lk is the volume of the sensor belt region (also called wall or rectangular parallelepiped). We obtain: ( √ )2 ( √ ) 3+ 2 1+2 2 ℵ(r ) = nk 2 = 1.785nk 2 √ 22 + 14 2 with n ⋙ 1 being a real number and a natural number.

∎

20.4 Performance Evaluation In this section, we specify the simulation setup. Then, we present a few numerical results as well as some simulation results using a high-level simulator written in C.

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20.4.1 Simulation Setup We consider a rectangular parallelepiped belt region of width w, length l, and height h whose values are defined as follows: Its width w takes on its values in the set {50, 60, 70, 80, 90, 100 m}, its height takes on its values in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 m}, and its length l = 3000 m. We assume that the sensors are densely and randomly deployed in this rectangular parallelepiped belt, and that their sensing and communication ranges are equal to 5 m and 10 m, respectively. The number of deployed sensors is set to 3000.

20.4.2 Numerical Versus Simulation Results Figure 20.15 shows the spatial sensor density as a function of the radius of the sensing spheres of the sensors. The former is inversely proportional to the latter. The spatial sensor density decreases with the radius of the sensors’ sensing spheres. In fact, as we increase the sensing range of the sensors, it is possible to cover more space, thus, requiring a smaller number of sensors. As it can be seen from this figure, there is a close-to-perfect match between the numerical results and simulation results. The slight difference is due to edge effect problem, where only portions of the great rhombicuboctahedron cover the spatial belt region. Figure 20.16 compares between two spatial space fillers, namely the great rhombicuboctahedron and the

Fig. 20.15 The great rhombicuboctahedron’s performance

20.5 Related Work

681

Fig. 20.16 The great rhombicuboctahedron versus the truncated octahedron

truncated octahedron [292]. It shows that the great rhombicuboctahedron is a better space-filling polyhedron than the truncated octahedron.

20.5 Related Work While the k-barrier coverage problem in planar wireless sensor networks has been studied extensively [115, 116, 193, 194, 232, 233, 242, 244, 245, 271, 341, 342, 395, 430, 431, 283, 450] as discusses earlier in Chap. 19, there is a little work done in spatial wireless sensor networks [17]. Barr et al. [76] computed a spatial stealth distance to measure how far a submarine can travel in a sensor network before being detected by a sensor. Also, they proved the absence of strong barrier coverage in a large spatial fixed emplacement field. In addition, they presented an energy conserving approach to constructing a strong spatial barrier using mobile nodes so that intruding submarines cannot pass through without being detected. Shen et al. [352] proposed a scheme to transform the spatial weak k-barrier coverage problem into planar complete k-coverage problem, based on which they suggested an algorithm for spatial weak k-barrier decision problem. They devised a Hungarian Method-based sensor assignment algorithm to construct weak k-barrier coverage while minimizing the total movement distance of all sensors in underwater wireless sensor networks. Shen et al. [353] proposed a fully distributed deployment algorithm for constructing maximum-level underwater strong k-barrier coverage with available

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mobile sensors in spatial underwater environment. Si et al. [355] proposed a resolution criterion based on a spatial sensing model of a camera sensor for capturing the intruder’s face. They used this resolution criterion to study the barrier coverage of a feasible deployment strategy in camera sensor networks. The interested reader is referred to Wu et al. [417] for a review on barrier coverage in wireless sensor networks, and Tao and Wu [376] for a survey of barrier coverage in directional wireless sensor networks.

20.6 Conclusion Spatial wireless sensor networks are able to model more accurately several real-world physical security scenarios. They can be deployed to provide an intrusion detection system by building a barrier whose goal to detect and prevent any intruder crossing or accessing a protected area, such as an international border. This barrier is a spatial belt of sensors that helps achieve physical security. In this chapter, we study the problem of spatial k-barrier coverage, where every path crossing this spatial belt region intersects with at least k sensors. We investigate four deterministic latticebased sensor deployment methods: Simple cubic, body centered cubic, face centered cubic, and hexagonal close-packed lattices. We prove that none of these lattices can guarantee spatial k-barrier coverage. Then, we consider a spatial space-filling polyhedron, great rhombicuboctahedron, to solve the problem of spatial k-barrier coverage. First, we introduce the concept of abstract path to represent an intruder’s walk across a spatial sensor belt region (i.e., barrier), which includes only progressive moves. Second, we compute the total number of an intruder’s abstract paths and accompany them with a polynomial representation. Third, we compute the total number of sensors to ensure k-barrier coverage using a great rhombicuboctahedronbased lattice.

Part IX

Applications of Wireless Sensor Networks and Concluding Remarks

Chapter 21

An Overview of Sensing Hardware, Standards, Operating Systems, Software Development, and Applications and Systems Whatever the future may bring, the universal application of these great principles is fully assured, though it may be long in coming. Nikola Tesla (1856–1943).

Overview This chapter provides a description of sensing hardware and software along with a classification of wireless sensor networks applications and systems based on their application domains. Precisely, it describes the main hardware and software components that are essential for the design and development of this type of network. Also, it states the challenges that can be encountered during the software development for these networks along with their solutions. Moreover, it focuses on five application domains, namely healthcare, agriculture, environmental, industry, and military. For each one of them, it gives an overview of a variety of applications and systems of wireless sensor networks.

21.1 Introduction The last two decades witnessed the emergence of a new networking technology, called wireless sensor networks. The latter has attracted the attention of several researchers from both academia and industry. A huge number of research papers and software and hardware products have been generated. In fact, recent advances in wireless communication, miniaturization, and low-cost, low-power circuit design, have led to the design and development of tiny communication devices, called sensor nodes. The latter has enabled the design of cost-effective, large-scale wireless sensor networks. These types of Networks are becoming increasingly important in our lives. Though we may not realize it, they are already heavily used in most industries and by the military [323, 329]. Basically, a wireless sensor network consists of a collection of sensor nodes that collaborate to successfully accomplish their monitoring task by communicating with each other via multi-hop, wireless links. The main difference © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. M. Ammari, Theory and Practice of Wireless Sensor Networks: Cover, Sense, and Inform, Studies in Systems, Decision and Control 214, https://doi.org/10.1007/978-3-031-07823-1_21

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between traditional computer networks, particularly, ad-hoc networks, and wireless sensor networks resides in the size and density of the devices forming the underlying network. On the one hand, a traditional node computer has large size and very powerful computation, storage, communication, and power supply resources, while the sensor node is a tiny device with very limited features, especially, its energy supply. On the other hand, sensor nodes have an extra feature, which is their sensing capability, and are, in general, densely deployed. It is worth mentioning that there is really no limit to the styles and applications of wireless sensor networks. Before giving an overview of these applications, we examine the typical hardware and software components that make up wireless sensor networks [329]. Regardless of the underlying application of wireless sensor networks, it is important to analyze the requirements that are needed for a sensor node (or simply node) hardware. Precisely, a sensor node consists of a mote and one or more sensors. From now on, we use the terms “sensor node” and “node” interchangeably unless stated otherwise. Node hardware includes five broad components, namely microcontroller, transceiver, external memory, power supply, and sensors. As it can be seen, a mote has computation, communication, and storage capabilities along with battery power (or energy), while a sensor has the sensing capability and may have several sensing modalities, such as light, sound, temperature, vibration, motion, etc. Typically, the motes have some sort of sensors attached on them and will relay information about their environment back to a central gathering node, called base station (or sink), which can process the data or send them to a remote server [95, 166, 329].

21.1.1 Major Tasks In this chapter, we propose a taxonomy of applications of wireless sensor networks based on their application domains (or categories). Specifically, we give an overview of a variety of applications of wireless sensor networks for each category. Also, we describe the main hardware and software components, which are needed for the design and development of this type of network. We should notice that this chapter complements another existing survey of applications and hardware components of wireless sensor networks [203].

21.1.2 Chapter Organization The remainder of this chapter is organized as follows: Sect. 21.2 describes the hardware components of a wireless sensor network. Section 21.3 reviews the software components for the design and development of wireless sensor networks. Section 21.4 discusses the issues that could be raised while programming the sensors. Section 21.5 gives an overview of a variety of applications of wireless sensor

21.2 Sensing Hardware

687

networks. Section 21.6 focuses on future applications of wireless sensor networks. Section 21.7 concludes the chapter.

21.2 Sensing Hardware In this section, we give an overview of a sample of existing hardware elements that are indispensable for the design and implementation of wireless sensor networks.

21.2.1 Mote Hardware Depending on the manufacturer and model of sensor nodes, the attributes of the components of a mote can vary significantly. Thus, it is important to understand what exactly an application might require from the motes. Next, we give a brief description of the hardware specifications of various popular mote families. The Berkeley-style Mica family of motes is distributed by Crossbow Technology [467]. They support the TinyOS operating system and are programmed in network embedded C. All of these motes had Atmel AVR micro-controllers that operate on a modified Harvard architecture with RISC. The first mote in the family, the Mica, used the ATmega 103 4 MHz 8-bit processor with 4 K bytes of EEPROM memory [462]. The ATmega103 provides throughput of about 6MIPS at 6 MHz and has 32 general purpose registers directly connected to the controllers ALU. When fully active, the controller consumes approximately 16.5 mW of power. During idle periods, it consumes approximately 30 µ-W. The ATmega103 provides the Mica with a good measure of power for a relatively low energy cost [199]. The Mica uses the TR 1000 as its radio transceiver. This transceiver provides developers with access to signal strengths and background noise levels while also providing easy software-based start-up and shutdown. The TR 1000 operates within a frequency of 916.30–916.70 MHz with a data transfer rate of 2.4 Kbps. The TR 1000 is ideal for wireless sensor applications where small size, low power consumption and low cost are necessary [470]. Figure 21.1 shows the block diagram for Mica mote [471]. The Mica2 expands on its predecessor with the addition of a more powerful processor and radio as well as more flash memory. The ATmega 128L is the microcontroller used in the Mica2. It has a throughput of 16MIPS at 16 MHz along with 128 K bytes of in-system programmable flash. The AVR micro controller provides 32 registers similar to the ATmega 103. The Mica 2 uses Texas Instruments CC1000 as its radio transceiver. The CC1000 provides data rates of about 12 Kbps at a full duty cycle. It operates between the frequencies of 300–1000 MHz and utilizes hardwarebased Manchester encoding. It also provides selectable power and duty cycle states for more precise power management [462, 472].

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Fig. 21.1 Block diagram for Mica Mote [471]

21.2.2 Sensor Technology The sensors are an integral part of any wireless sensor network whose main goal is monitoring a physical environment. Depending on what the network is being used for, motes can support one or many different sensors. These sensors actually sample the data from the environment. The mote will then typically send the data that it gathers back to a master node, i.e., a sink, for the data to be stored and analyzed. There is virtually no limit to what type of factors a sensor board can pick up. Typically, the sensor board is a separate hardware attachment to the mote. This allows a user to re-purpose motes eliminating the need to buy different types of motes for different applications. Although there are numerous types of sensors, in this section, we describe a sample of common sensors and their potential use. Also, we discuss some issues that come up when sampling data from these sensors.

21.2.2.1

Problems

One of the major concerns in wireless sensor networks is power conservation. As wireless sensor networks are often randomly deployed and rely on the forwarding capability of the nodes to forward data on behalf of other nodes in the network toward their final destinations, some actively forwarding nodes would run out of battery power very quickly. As a result, these nodes die and the network becomes weaker until, at a certain point, it becomes disconnected and may no longer be able to operate correctly. Since constantly replacing the batteries in individual nodes is usually not a viable solution, all activities of the sensors from communication methods to data gathering are focused on conservation of power [237]. As we may imagine, taking power conservation only into account can cause some problems when trying to collect real-time data from the sensors. On the one hand, if the sensors are sampled too often, this would waste valuable battery power. On the other hand, if

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the sensor is not sampled enough, however, the data collected could be outdated and inaccurate. In certain systems, such as networks monitoring the health of a patient or the detection of hazardous chemicals or conditions, this could even be fatal. There are many methods that have been developed to solve these issues and provide enough data while still conserving power in the networked motes. One such method is developing algorithms to switch nodes on and off when necessary to provide continuous coverage while still preserving battery life in the individual nodes [321]. Ideally, all the sensors in the network should be kept at relativity the same level of power to make sure no blind spots appear in the network, which could prevent data from being collected and isolate parts of the network by destroying routes for data to reach the sink. A common method for preventing this is activating nodes on a basis of need. The time between sensor samplings can also be decreased in sections of the network where there is no noticeable change in data. If these sensors begin to detect activity, they can increase the sampling rate to provide more constant coverage of the area. One example of this is in a security network, detecting the location of an intruder in the system. If no movement is detected in that area, the nodes will bring down their sampling rate to conserve power. However, if they get notification from another node that an intruder is moving into their coverage area, or if they detect the intruder themselves, they will bring up their sampling rate to a level where they can accurately track the intruder’s movement through their area.

21.2.2.2

Types of Sensors

There is virtually no limit to the types of sensors that can be installed on a wireless mote. If a manufacturer can make the sensor small and versatile enough to perform the necessary tasks, it can be part of a sensor network. Depending on the function or industry using them, sensor boards can provide a variety of different sensing modalities. One sensor board can even contain multiple sensors with different sensing modalities, such as temperature, light, sound, vibration, etc. This reduces the number of motes that are needed in the network overall. Personal Sensors: Personal sensor boards can be built to measure vital signs or changes in the wearer’s body chemistry. As we can imagine, these types of sensors can revolutionize the healthcare industry. Sensors can be tuned to measure everything from O2 saturation in a patient’s blood to blood pressure. They can even detect irregular heart rhythms [302]. Location: Using GPS technology, wireless sensors can also be used to track movements of people and animals. This is especially useful for the military, giving them the ability to accurately track all of their troops on a battlefield or while performing special operations behind enemy lines [95]. Also, tracking locations is useful to environmental researchers, which allows them to track the movement and migration patterns of animals [356]. The ability of sensor boards to determine their global location using GPS is also useful in networks where the sensors are randomly deployed. If the sensors are, for example, dropped from a passing plane, the users of the network

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would need to know exactly where every mote was located in order to make effective use of the data that they collect. Although the adding GPS to sensors may not be completely useful in small scale wireless sensor networks (as the GPS can typically only place each mote within about a 3-m radius), these sensors are defiantly useful in many types of dynamic wireless sensor networks [369]. Integrity Monitoring: One type of sensor that is particularly widely used for industrial applications is a component integrity sensor. Before the advent of wireless sensor networks, machinery had to be monitored by wired sensors. These sensors often could not be placed where they were most needed because moving parts inside the particular component would prevent running wires through the location. These sensors have revolutionized the way industrial assets are monitored. In many cases, they allow the operators to anticipate mechanical failures well in advance, thus, saving time and preventing any permanent damage to the mechanism [7]. This family of sensors makes use of several sensing modalities, such as temperature, vibration, and humidity, to successfully accomplish their task. Hazard Detection: Detecting workplace hazards is very important in many industries. It has the potential to save several lives by providing advance notice to any sort of danger, such as a gas leak or imminent collapse of a mining shaft [7]. These types of sensors are not only used for protecting workers but also for product protection. In industries, such as food processing and pharmaceuticals, the products often need to be kept in specific conditions to ensure the quality and safety for the consumer. These sensors can constantly monitor the temperature and humidity of the product throughout the entire production, storage and transportation of the items to ensure that they are safe to use when they reach the shelves [329]. Also, the military makes use of hazard detecting sensors, which can provide advance warning to chemical attacks, radiation and even hazardous weather conditions to come, when deployed in the field. Whether used by private industry or the military, sensors can be vital in preserving life and preventing harm from coming to the population [143]. Environmental Monitoring: One of the great benefits that wireless sensor networks provide is that they can be placed in hazardous locations and take measurements that are inconvenient or even dangerous for researchers to take [203]. This, coupled with the fact that they can stay in place for long periods of time and provide constant data on the location at which they are deployed, make wireless sensor networks ideal for environmental monitoring. This can range from analyzing data on certain ecosystems to detect changes for biological and environmental research, to monitoring volcanoes in an effort to provide early warning to an eruption [375]. Researchers often have to sample data from locations that are awkward to access, such as the polar ice caps and the inside of a volcano. The use of motes having temperature and movement sensors can make this task much easier and provide more comprehensive data. Placing the motes along fault lines in the earth’s crust could also allow scientists to observe more subtle changes in the pressure and positioning of the area, thus, improving earthquake detection.

21.2 Sensing Hardware

21.2.2.3

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Real Sensor Boards

The MTS400CC sensor board, which is manufactured my MEMSIC, is an example of a multipurpose sensor, which allows a mote to sample several different types of data from the surrounding environment. According to the manufacturer, the MTS400CC sensor board can be used in several different fields, such as environmental monitoring, agriculture, preservation of art and artifacts, and location mapping [466]. This one board contains the hardware to sense acceleration, barometric pressure, ambient light, temperature and humidity. Also, the manufacturer provides an additional attachment for GPS data to be acquired. Figure 21.2a shows MTS400CC sensor board. The STB80 sensor board, which is manufactured by EasySen, contains six separate sensor types. It can be used to detect both visual and infrared light, sound, temperature, strength and direction of magnetic fields, and acceleration [464]. This sensor board could be used for a variety of applications, such as intruder detection (infrared, sound, light) or even in industrial machines to monitor performance (temperature, magnetism). Figure 21.2b shows STB80 sensor board. As we can tell from the above examples, sensor boards are versatile and their uses are only limited to the imaginations of the users. Later on, in this chapter, we discuss more in depth how these many different sensor types can be used to benefit a whole range of fields. First, we need to get a better understanding of how the hardware and software operate. Tables 21.1, 21.2, 21.3, and 21.4 provide a summary of the main characteristics of several well-known motes, such as TelosB, MICA2, IRIS, and MICAZ, respectively. For more information, the reader is referred to [482]. Also, a comparison of several motes can be found in [94].

Fig. 21.2 a MTS400CC [466], b STB80 [464] sensor boards

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Table 21.1 TelosB mote

Specifications

TPR2420CA

Remarks

Module Processor performance

16-bit RISC

Program flash memory

48 K bytes

Measurement serial flash

1024 K bytes

RAM

10 K bytes

Configuration EEPROM

16 K bytes

Serial communications

UART

0–3 V transmission levels

Analog to digital converter

12 bit ADC

8 channel, 0–3 V input

Digital to analog converter

12 bit DAC

2 ports

Current draw

1.8 mA

Active mode

5.1 µA

Sleep mode

Frequency band

2400–2483.5 MHz

ISM band

Transmit (TX) data rate

250 kbps

RF transceiver

RF power

−24–0 dBm

Receive sensitivity

−90 dBm (min), − 94 dBm (typ)

Outdoor range

75–100 m

Inverted-F antenna

Indoor range

20–30 m

Inverted-F antenna

Current draw

23 mA

Receive mode

21 µA

Idle mode

1 µA

Sleep mode

Visible light sensor range

320–730 nm

Hamamatsu S1087

Visible to IR sensor range

320–1100 nm

Hamamatsu S1087-01

Humidity sensor range

0–100% RH

Sensirion SHT11

Resolution

0.03% RH

Accuracy

±3.5% RH

Sensors

Absolute RH (continued)

21.2 Sensing Hardware Table 21.1 (continued)

693 Specifications

TPR2420CA

Remarks

Temperature sensor range

−40–123.8 °C

Sensirion SHT11

Resolution

0.01 °C

Accuracy

±0.5 °C

@25 °C

Electromechanical Battery

2X AA batteries

Attached pack

User interface

USB

v1.1 or higher

Size (in)

2.55 × 1.24 × 0.24

Weight (oz)

0.8

21.2.3 Gateways One key component of wireless sensor networks is the gateway. The gateway provides the link between a wireless sensor network and the outside world. Its primary functions are to collect the data from the network, pre-process it, if needed, and send commands to the motes. Depending on the application, the gateway can operate in many different fashions. It could simply forward the data to an external server or it could perform some data processing before sending [117]. In this section, we discuss the function of gateways in wireless sensor networks and provide some examples of how these gateways are actually used.

21.2.3.1

Gateway Functionality

Since one of the major goals of wireless sensor networks is longevity and power consumption, it is important that the data collected by the sensor nodes be analyzed by other nodes. In fact, the data is typically stored on some external server or monitoring station and is analyzed there. However, the gateway may perform some data processing. If the gateway can selectively send data or send the aggregated data, such as average, the load on the external server can be greatly decreased [117]. The benefit would be significant when many networks are contacting the same server. Besides just receiving data from the network, gateways are also used to send commands from the server or user to nodes. They can even analyze the field and coordinate which nodes should remain active and which are not needed at the present time [333]. In addition, certain gateways are used to connect the nodes to a computer, allowing for software to be loaded and updated [463]. This can be done either by directly connecting the node to the gateway, or even by updating the software on nodes that have already been deployed.

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Table 21.2 MICA2 mote

Processor/radio board

MPR400CB

Remarks

Processor performance Program flash memory

128 K bytes

Measurement (serial) flash

512 K bytes

Configuration EEPROM

4 K bytes

Serial communications

UART

0–3 V transmission levels

Analog to digital converter

10 bit ADC

8 channel, 0–3 V input

Current draw

>100,000 measurements

8 mA

Active mode

300 m

Indoor range

>50 m

Current draw

16 mA

Receive mode

10 mA

TX, −17 dBm

13 mA

TX, −3 dBm

17 mA

TX, 3 dBm

Battery

2X AA batteries

Attached pack

External power

2.7–3.3 V

Molex connector provided Red, green, yellow

Electromechanical

User interface

3 LDs

Size (in)

2.25 × 1.25 × 0.25

Weight (oz)

0.7

Expansion connector

51-pin

All major I/O signals

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Table 21.4 MICAZ mote

Processor/radio board

MPR2400CA

Remarks

Processor performance Program flash memory

128 K bytes

Measurement (serial) flash

512 K bytes

Configuration EEPROM

4 K bytes

Serial communications

UART

0–3 V transmission levels

Analog to digital converter

10 bit ADC

8 channel, 0–3 V input

Current draw

>100,000 measurements

8 mA

Active mode