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5G Green Mobile Communication Networks [1st ed.]
 978-981-13-6251-4;978-981-13-6252-1

Table of contents :
Front Matter ....Pages i-viii
Challenges of 5G Green Communication Networks (Xiaohu Ge, Wuxiong Zhang)....Pages 1-27
Energy Efficiency of 5G Wireless Communications (Xiaohu Ge, Wuxiong Zhang)....Pages 29-101
Energy Efficiency of Cellular Networks (Xiaohu Ge, Wuxiong Zhang)....Pages 103-184
Energy Efficiency of 5G Multimedia Communications (Xiaohu Ge, Wuxiong Zhang)....Pages 185-233
Wireless Resource Management for Green Communications (Xiaohu Ge, Wuxiong Zhang)....Pages 235-285
Energy Efficiency and Collaborative Optimization Theory of 5G Heterogeneous Wireless Multi Networks (Xiaohu Ge, Wuxiong Zhang)....Pages 287-325

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Xiaohu Ge · Wuxiong Zhang

5G Green Mobile Communication Networks

5G Green Mobile Communication Networks

Xiaohu Ge Wuxiong Zhang •

5G Green Mobile Communication Networks

123

Xiaohu Ge School of Electronic Information and Communications Huazhong University of Science and Technology Wuhan, Hubei, China

Wuxiong Zhang Shanghai Research Center for Wireless Communications Shanghai, China

ISBN 978-981-13-6251-4 https://doi.org/10.1007/978-981-13-6252-1

ISBN 978-981-13-6252-1

(eBook)

Jointly published with Publishing House of Electronics Industry, Beijing, China The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Publishing House of Electronics Industry. Library of Congress Control Number: 2019930643 © Publishing House of Electronics Industry, Beijing and Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1 Challenges of 5G Green Communication Networks . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Evolution of Green Communications . . . . . . . . . . . . . . . 1.2.1 Computation Power . . . . . . . . . . . . . . . . . . . . . . 1.2.2 New Issues Triggered by the Computation Power 1.3 Computation and Communication Power in 5G Wireless Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Power Consumption at BSs . . . . . . . . . . . . . . . . 1.3.2 Computation Power Model . . . . . . . . . . . . . . . . . 1.3.3 Evaluations of Computation Power . . . . . . . . . . . 1.3.4 Future Challenges . . . . . . . . . . . . . . . . . . . . . . . 1.4 New Issues in 5G Green Cellular Networks . . . . . . . . . . 1.4.1 Computation Capability Factor . . . . . . . . . . . . . . 1.4.2 Heat Dissipation Factor . . . . . . . . . . . . . . . . . . . 1.4.3 Maximum Receiving Rates for Smartphones . . . . 1.4.4 Simulation Results and Discussions . . . . . . . . . . 1.4.5 Future Challenges . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Energy Efficiency of 5G Wireless Communications . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Energy Efficient Hybrid Precoding Design . . . . . . . . . . . 2.2.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 2.2.4 Energy Efficient Hybrid Precoding Design . . . . . 2.2.5 Energy Efficient Optimization with the Minimum Number of RF Chains . . . . . . . . . . . . . . . . . . . .

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2.2.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Energy Efficient Optimization with RF Chains . . . . . . . . . . . 2.3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Hybrid Precoding Design for the Partially-Connected Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Energy Efficient Power Control Scheme . . . . . . . . . . . . . . . . 2.4.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Achievable Rate of MIMO PVT Random Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Green MIMO Random Cellular Networks . . . . . . . . . 2.4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Energy Efficiency of Cellular Networks . . . . . . . . . . . . . . . . . . . . 3.1 On the Energy-Efficient Deployment for Ultra-Dense Heterogeneous Networks with NLoS and LoS Transmissions . . 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Signal Propagation Model . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Network Transformation . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Performance Optimization and Tradeoff . . . . . . . . . . . . 3.1.7 Results and Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . 3.1.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Spatial Spectrum and Energy Efficiency of Random Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Models of PVT Random Cellular Networks . . . . . . . . . 3.2.4 Spatial Spectrum and Energy Efficiency of PVT Random Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular Networks: A Mean Field Game Approach . . . . . . . . . . . . . . . .

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3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 References

vii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . System Model . . . . . . . . . . . . . . . . . . . . . . Formulation of Energy Efficiency . . . . . . . . Network Energy Efficiency Optimization . . . Algorithm Design of Mean Field Game . . . Numerical Simulations of Mean Field Game Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

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4 Energy Efficiency of 5G Multimedia Communications . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Energy Efficiency Optimization for MIMO-OFDM Mobile Multimedia Communication Systems with QoS Constraints . . 4.2.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Energy Efficiency Modeling of MIMO-OFDM Mobile Multimedia Communication Systems . . . . . . . . . . . . . 4.2.4 Energy Efficiency Optimization of Mobile Multimedia Communication Systems . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Optimization Solution of Energy Efficiency . . . . . . . . . 4.2.6 Simulation Results and Performance Analysis . . . . . . . 4.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Multi-path Cooperative Communications Networks for Augmented and Virtual Reality Transmission . . . . . . . . . . . . . 4.3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Network Latency Model . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 AR/VR Multi-path Cooperative Transmissions . . . . . . 4.3.5 Service Effective Energy Optimization . . . . . . . . . . . . 4.3.6 Simulation Results and Performance Analysis . . . . . . . 4.3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Wireless Resource Management for Green Communications . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 User Traffic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Data Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Voice Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Downlink Average Rate and SINR Distribution in Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Bregman-Based Inexact Excessive Gap Method for Multiservice Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Regularized Lagrangian Dual Function and d-Excessive Gap Smoothing Technique . . . . . . . . . . . . . . . . . . . . . .

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5.4.3 Inexact Algorithm with Bregman Projection . . . . . . . . . 5.4.4 Multi-service Resource Allocation Across Heterogeneous Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Energy Efficiency and Collaborative Optimization Theory of 5G Heterogeneous Wireless Multi Networks . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Heterogeneous Wireless Multi-network Energy Efficiency Collaborative Optimization Architecture . . . . . . . . . . . . 6.1.2 Distributed Collaborative Architecture . . . . . . . . . . . . . . 6.1.3 Centralized Collaborative Architecture . . . . . . . . . . . . . 6.1.4 Hybrid Collaborative Architecture . . . . . . . . . . . . . . . . . 6.2 Power Reduction for Mobile Devices by Deploying Low-Power Base Stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Current Energy Efficiency Metrics . . . . . . . . . . . . . . . . 6.2.2 A Novel Energy Efficiency Metric Jointly Considered by Networks and Terminals . . . . . . . . . . . . . . . . . . . . . 6.2.3 Network Energy Efficiency Analysis in the Case of Macro and Micro Zone Coexistence . . . . . . . . . . . . . 6.3 Wireless Network Virtualization and Software Defined Wireless Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Wireless Virtualization . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Software Defined Wireless Network . . . . . . . . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Challenges of 5G Green Communication Networks

1.1 Introduction The deployment of a large number of small cells poses new challenges to energy efficiency, which has often been ignored in fifth generation (5G) cellular networks. While massive multiple-input multiple outputs (MIMO) will reduce the transmission power at the expense of higher computational cost, the question remains as to which computation or transmission power is more important in the energy efficiency of 5G small cell networks. Thus, the main objective in this chapter is to investigate the computation power based on the Landauer principle. Simulation results reveal that more than 50% of the energy is consumed by the computation power at 5G small cell base stations (BSs). Moreover, the computation power of 5G small cell BS can approach 800 W when the massive MIMO (e.g., 128 antennas) is deployed to transmit high volume traffic. This clearly indicates that computation power optimization can play a major role in the energy efficiency of small cell networks. Moreover, due to the higher wireless transmission rates in the fifth generation (5G) cellular networks, higher computation overhead is incurred in smartphones, which can cause the wireless transmission rates to be limited by the computation capability of wireless terminals. In this case, is there a maximum receiving rate for smartphones to maintain stable wireless communications in 5G cellular networks? The other objective of this chapter is to investigate the maximum receiving rate of smartphones and its influence on 5G cellular networks. Based on Landauer’s principle and the safe temperature bound on the smartphone surface, a maximum receiving rate of the smartphone is proposed for 5G cellular networks. Moreover, the impact of the maximum receiving rate of smartphones on the link adaptive transmission schemes has been investigated. Numerical analyses imply that the maximum receiving rate of smartphones cannot always catch up with the downlink rates of future 5G cellular networks. Therefore, the link adaptive transmission scheme for future 5G cellular networks has to take the maximum receiving rate of smartphones into account.

© Publishing House of Electronics Industry, Beijing and Springer Nature Singapore Pte Ltd. 2019 X. Ge and W. Zhang, 5G Green Mobile Communication Networks, https://doi.org/10.1007/978-981-13-6252-1_1

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1 Challenges of 5G Green Communication Networks

1.2 Evolution of Green Communications 1.2.1 Computation Power With the anticipated high traffic, small cell networks are emerging as an inevitable solution for 5G cellular networks [1]. In particular, the massive multiple input multiple-output (MIMO) and millimeter wave technologies are expected to be deployed towards improving the transmission rate and reduce the transmission power of 5G mobile communication systems [2]. On the other hand, more computation power will be required to process anticipated heavy traffic at small cell base stations (BSs). Under these conditions, a tradeoff between computation and transmission power needs to be thoroughly evaluated in order to achieve energy efficiency optimization for 5G small cell networks. This has been widely investigated in [3–6]. 5G Ultra-Dense Cellular Networks. Compared with transmission power, computation power was obviously smaller and usually fixed as a constant in a traditional energy efficiency evaluation of BSs [4]. As a consequence, the energy efficiency investigation of small cell networks has focused on the optimization of transmission power at BSs [5]. Furthermore, the BS sleeping scheme has been considered to improve energy efficiency where the radio frequency (RF) chains and transmitters of BSs are closed to save transmission power [6]. In addition, the computation power of small cell BSs has been improved by the volume and complexity of signal processing, which is weighted by massive MIMO and millimeter wave technologies [7]. When small cell BSs are ultra-densely deployed in 5G cellular networks [8], there exist scenarios in which the computation power of BSs will become larger than the transmission power of BSs despite lower power transmission requirements for small cell BSs. The transmission rate of 5G mobile communication systems is expected to reach to an average of 1 Gbps (20 Gbps at the peak rate) [2]. Hence, the huge traffic has to be handled at the base band units (BBUs) of small cell BSs and then the computation power of signal processing has to be accordingly improved at BBUs. Moreover, the cache communications and cloud computing network architecture will strengthen functions of signal processing and computing at small cell BSs. Nonetheless, the computation power of 5G small cell networks could be predicted to increase in the near future. All the above reasons trigger us to rethink the roles of computation and transmission power in 5G small cell networks.

1.2.2 New Issues Triggered by the Computation Power Currently, the peak rate of the fifth generation (5G) communication systems is expected to reach 20 gigabits per second (Gbps) [9]. Moreover, the future beyond 5G wireless communications is expected to reach 100 Gbps [10] for short-range

1.2 Evolution of Green Communications

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communication by utilizing the terahertz (THz) bands. Based on high transmission rates, some high-data-rate applications, such as virtual reality (VR) and augmented reality (AR), will be applied to wireless terminals, e.g., smartphones [11]. It is inevitable that smartphones will confront a dramatic data receiving and consume an enormous computation power in 5G and future 6G cellular networks. Moreover, the computation capability of smartphones is limited, making it a great challenge for smartphones to process massive wireless data in 5G cellular networks [1]. Therefore, new issues and limits triggered by smartphones are emerging for 5G cellular networks. Current and previous studies related to the capacities of cellular networks have a default assumption that the maximum receiving rate of smartphones always catches up with the downlink rate of base stations (BSs), irrespective of the downlink rate. In general, the maximum receiving rate of smartphones is the rate at which the data can be stably processed by chips in smartphones. Therefore, two factors restrict the maximum receiving rate of smartphones. One of the two factor is the computation capability of the baseband processor, and the other is heat dissipation. The chip in smartphones has integrated all components of computations and communications, such as the application processor (AP), storage unit and baseband processor (BP). When wireless communications run on smartphones, some critical processes related to receiving rates, such as digital signal processing and channel coding processes, are carried out by the BP. Thus, the maximum receiving rate of smartphones is limited by the computation capability of the BP. Moreover, the computation capability of the BP depends on the semiconductor technology. On the basis of utilizing the latest semiconductor technology to produce microchips, more transistors are being integrated into a microchip to increase the computation capability. Although the evolution of semiconductor technology has followed Moore’s law, Moore’s law has been invalidated by the verification of the Landauer limit [12] and the influence of thermal noise [13]. Additionally, silicon transistors will approach a projected scaling limit of 5-nanometer (nm) gate lengths [14, 15]. It is predicted that a chip based on 5 nm semiconductor technology will be produced around 2020, which will launch 5G cellular networks. Thus, the computation capability of the BP in smartphones will approach the limit for future 5G cellular networks. In addition to the computation capability limit, the maximum receiving rate is restricted by the heat dissipation of smartphones. Since wireless terminals have entered into the smart era, the development of smartphones is accompanied by overheating issues. In general, the major heat contributing to the overheating issues in smartphones is caused by chip computation. The relationship between heat generation and computation can be interpreted clearly by Landauer’s principle. Furthermore, the temperature of the smartphone surface has been limited to 45 [16]. When the temperature of a smartphone’s surface surpasses 45, it is necessary to diminish the heat generated by the chip. Consequently, smartphones must reduce the computation capability of the chip to decrease the heat generation. In some extreme conditions, the receiving rates of smartphones are cut down or stopped by a protection scheme. Therefore, the heat dissipation of smartphones restricts the maximum receiving rate of smartphones. Although the maximum receiving rate of smartphones is restricted by the computation capability and heat dissipation, detailed studies of basic models

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1 Challenges of 5G Green Communication Networks

used for evaluating the maximum receiving rate of smartphones are surprisingly rare in the available literature. Moreover, the impact of the computation capability and heat dissipation of smartphones on 5G cellular networks has not been investigated.

1.3 Computation and Communication Power in 5G Wireless Communication Systems Based on the Landauer’s principle, this section proposed a computation power model for 5G small cell networks. Considering that the massive MIMO and millimeter wave technologies are adopted at small cell BSs, the impact of the number of antennas and bandwidths on the computation power of 5G small cell networks is investigated. Simulation results indicate that the computation power will consume more than 50% of the energy at 5G small cell BSs. It is a surprising result for the energy efficiency optimization of 5G small cell networks.

1.3.1 Power Consumption at BSs To evaluate roles of computation and transmission power for BSs, the total BS power consumption needs to be analyzed in detail. Therefore, in this section 5G transmission technologies, such as massive MIMO and millimeter wave technologies will be incorporated for analyzing the power consumption of small cell BSs.

1.3.1.1

BS Power Consumption Types

Considering functions and architectures of BSs, the power consumption at BSs is typically classified into three types: transmission power, computation power and additional power which are described as follows. • The transmission power corresponds to the energy used by power amplifiers (PAs) and RF chains, which perform the wireless signals change, i.e., signal transforming between the base band signals and the wireless radio signals. Besides, the power consuming at feeders is included as a part of the transmission power. • The computation power represents the energy consumed at base band units (BBU’s) which includes digital single processing functions, management and control functions for BS’s and the communication functions among the core network and BS’s. All these operations are executed by software and realized at semiconductor chips. • The additional power represents the BS power, except for the transmission and computation power, e.g., the power consumed for maintaining the operation of BS’s. More specifically, the additional power includes the power lost at the

1.3 Computation and Communication Power in 5G Wireless Communication Systems

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exchange from the power grid to the main supply, at the exchange between different direct-current to direct-current (DC-DC) power supply, and the power consumed for active cooling at BS’s. The values of three types of consumed power are different depending on the types of BS. For example, unlike the macro cell BS the small cell BS normally does not have the active cooling system.

1.3.1.2

Total BS Power Consumption Model

The EARTH project has promoted energy efficiency optimization for wireless access networks and proposes a framework for the power consumption at BS’s [17]. Based on this energy efficiency framework, the BS is divided into seven parts (see Fig. 1.1): the antenna interface, the power amplifier, the RF chains, the BBU, the mains supply, and cooling and DC-DC. When the BS is equipped with NTRX antennas, the total BS PPA ·NTRX +PRF ·NTRX +PBB , where PPA is the power consumption Pin is calculated by Pin  (1−σ DC )(1−σMS )(1−σcool ) power of PA per antenna, PRF is the RF chain power per antenna, PBB is the power consumed at the BBU, σDC is the power loss rate of the DC-DC converter, σMS is the power loss rate of the alternating current supply, and σcool is the power loss rate of Pout , where Pout cooling. The power of PA per antenna is calculated by PPA  ηPA (1−σ feed ) is the transmission power at every antenna, ηPA is the exchange efficiency of PA, and feeder loss is configured as σfeed  −3 dB. For macro cell BS’s, the values of σDC , σMS and σcool are configured as 6, 7 and 9%, respectively [17]. For small cell BS’s, the values of σDC , σMS and σcool are configured as 8, 10 and 0%, respectively [17]. To simplify calculation, the RF chain power per antenna is usually fixed as different constants corresponding to different types of BS’s. Since PBB is obviously less than the power consumed at other parts of BS’s, the power consumed at the BBU is fixed as constant in a traditional BS power model [18].

1.3.2 Computation Power Model Because of the extensive traffic processing at 5G small cell BSs, the volume of data processing at 5G small cell BSs is evaluated by the operation per second at BBUs. Furthermore, Landauer’s principle is used to estimate the computation power consumed for data processing in this section. This section also studies the impact of massive MIMO and millimeter wave technologies on the computation power of 5G small cell BSs.

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1 Challenges of 5G Green Communication Networks

Fig. 1.1 Logistical architecture of eNodeB BS

1.3.2.1

Computation Power Types

In traditional macro cell BSs the power used at BBUs (BBU is the core unit of a BS) is small compared with the power consumed by PAs. With the recent advances of 5G of the massive MIMO and millimeter wave technologies, small cell BSs are replacing macro cell BSs to perform the function of wireless data transmission in 5G cellular networks. Moreover, the power consumed at BBUs is expected to gradually increase because of the massive traffic in 5G small cell BSs. Figure 1.1 is a typical logistical architecture of eNodeB BS, i.e., a macro cell BS in a cellular network. Without a loss of generality, the BBU of a macro cell BS includes four systems: the base band system, the control system, the transfer system, and the power system. The detailed functions of these systems in BBU are described as follows. • The functions of a base band system include signal filtering, fast Fourier transform/inverse fast Fourier transform (FFT/IFFT), modulation and demodulation, digital-pre-distortion (DPD) processing, signal detection, and wireless channel coding/decoding. Note that, the function of signal processing used for transmitters and receivers is performed by the BBU. • The control system takes charge of controlling and managing resource allocation at BS’s in order to provide control interface between the BS and other network units. Moreover, communication control protocols are run at the control system. The

1.3 Computation and Communication Power in 5G Wireless Communication Systems

7

control system also provides an interface of man-machine language (MML) for the local maintain terminal (LMT) to configure the resource allocation of BBU’s. • The transfer system connects with the mobility management entity/servinggateway (MME/S-GW) of the core network by the S1 interface (see Fig. 1.1). Moreover, the control and management information among BS’s are forwarded by the X2 interface of the transfer system in BBU’s. • The power system is responsible for power supply, cooling, and monitoring at BBU’s. For small cell BS’s, most functions are integrated into a few semiconductor chips and there is not a single power system. Therefore, the systems of BBU’s at small cell BS’s is simpler than the systems of BBU’s at macro cell BS’s.

1.3.2.2

Computation Power Model

Based on the four systems in the logistical architecture shown in Fig. 1.1, the main difficulty is how to calculate the computation power for every logistical system in BBUs. To achieve this, a BBU is partitioned into different parts based on the hardware architecture as shown in Fig. 1.2. These consist of DPD, Filter, CPRI, OFDM, FD, FEC, and CPU where DPD is the digital-pre-distortion processing part, filter is the hardware used for up/down signal sampling and filtering, CPRI is the common public radio interface part for connecting to the core network and RF chains by serial links, OFDM is the hardware used for FFT and orthogonal frequency-division multiplexing (OFDM)-specific signal processing, FD is the frequency-Domain processing part, which includes, symbol mapping/demapping and MIMO equalization, FEC is the forward error correction which includes the channel coding and decoding, and CPU is the BBU platform control processor. Based on Landauer’s principle, this section estimated the computation power of semiconductor chips using Giga operations per second (GOPS) and considering different semiconductor chip techniques. The computation power of BBU is summed up by the computation power of each hardware part, i.e., each semiconductor chip at BBU. Landauer’s principle was proposed in 1961 by Rolf Landauer who attempted to apply the thermodynamic theory to digital computers. Landauer’s principle elaborates the relationship between the information process and energy consumption from the viewpoint of a microscopic degree of freedom in statistical physics. This is based on a physical principle pertaining to the lower theoretical limit of energy consumption that corresponds to the computation. Bear in mind that the concept of entropy in information theory introduced by Claude Shannon is borrowed from the thermodynamic theory. Similarly, Landauer’s principle connects these two concepts of information and energy by using the thermodynamic theory and statistical physics. Therefore, in this section Landauer’s principle is first used to analyze the computation power consumption in 5G small cell networks. More specifically Landauer’s principle points out that any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by

8

1 Challenges of 5G Green Communication Networks

Fig. 1.2 Architecture of BBU

a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment [19]. In other words, erasing a bit information will consume more than kT ln(2) energy in a computing system, where k is the Boltzmann constant, i.e., 1.38 × 10−23 J/K, T is the kelvin temperature [12]. According to Landauer’s principle, the lower bound of computation power for a computing system can be obtained. Compared with the value of computation power at real semiconductor chips, there exists a difference of three orders of magnitude for the values of computation power derived by Landauer’s principle [20]. Moreover, the values of computation power are different when different semiconductor chip techniques are adopted at BBU’s. Under these conditions, the main difficulty is how to accurately calculate the computation power of small cell BS’s using the Landauer’s principle. To overcome the gap of computation power estimated by Landauer’s principle and real semiconductor chips, this section proposed a power coefficient ε is that can represent the level of the semiconduct chip technique in BBU’s. Moreover, the power coefficient ε is defined as the ratio of the active switching power of a transistor and the limit of Landauer’s principle. From Fig. 1.3, the power coefficient ε reflects the distance between semiconductor chip techniques and the limit of Landauer’s principle. Bear in mind that up till now the development of semiconductor chip techniques still follows Moore’s law. However, the international technology roadmap for semiconductors (ITRS) predicts that the development of semiconductor chip techniques will deviate from Moore’s law when the power coefficient approaches the limit of Landauer’s principle. For example, when nanomagnetic logic is used for chips, the computation power is expected to approach the limit of Landauer’s principle [21]. Considering the development of current chip techniques, this section focuses our attention on the computation power of semiconductor chips.

1.3 Computation and Communication Power in 5G Wireless Communication Systems

9

Fig. 1.3 The power coefficient with respect to the development of chip techniques

Without a loss of generality, the power coefficient is configured as ε  103 when the 22 nm semiconductor technique is assumed to be adopted for chip manufacture in BBU’s. Moreover, the active switching power of a transistor is approximated by E FET ≈ εkT ln(2), which is used to calculate the power for operating 1 bit information at the semiconductor chip of BBU’s. In general, the data processing rate of semiconductor chips is represented by the instructions per second (IPS). Based on the definition of GOPS, in this section the 9 relationship between the IPS and the GOPS is expressed by I P S  G O P64S×10 when the logistical architecture of semiconductor chips is assumed to be 64 bit. According to the experimental results in [22], the information throughput of semiconductor 1 chips is denoted by ρ  ( I ωP S ) γ , where ω and γ are configured as 0.1 and 0.64, respectively. As a consequence, the computation power of different parts of a BBU is calculated by the product of the information throughput of semiconductor chips and the active switching power of transistors considering different values of GOPS at different parts of the BBU. Since different types of BS’s have different hardware components at BBU’s, it is difficulty to directly build a uniform model to evaluate the computation power of BBU’s in different types of BS’s. Therefore, it’s necessary to build a reference BS with typical parameters firstly. By comparing different types of BS’s with reference BS, we can derive the computation power of different BBU’s for different types of BS’s. Without a loss of generality, the system parameters are represented by i ∈ {BW, Ant, M, R, dt, d f }, where BW is the bandwidth parameter, Ant is the number of antennas parameter, M is the modulation coefficient parameter, R is the parameter of coding rate, dt is the parameter of time-domain duty-cycling, and df is the parameter of frequency-domain duty-cycling. To simplify symbols in this section, X iref is denoted as the reference BS. When the subscript i of X iref is replaced by different symbols, the new variable represents the corresponding system parameter ref is the bandwidth of the reference BS. Similarly, X ireal is in the reference BS, e.g., X BW denoted for a real BS. When the subscript i of X ireal is replaced by different symbols,

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1 Challenges of 5G Green Communication Networks

the new variable represents the corresponding system parameter in the real BS, e.g., real is the bandwidth of the real BS. X BW Considering different computation powers at different hardware parts of a BBU, the computation power of DPD, Filter, CRPI, OFDM, FD, FEC and CPU at reference ref ref ref ref ref ref ref , PFilter , PCRPI , POFDM , PFD , PFEC and PCPU , respectively. The BS are denoted by PDPD different hardware parts of a BBU depend on the different system parameters of BS. Si , i ∈ {BW, Ant, M, R, dt, d f } signifies the ratio of the different hardware parts of the BBU and the system parameters of the BS. When the relationship between the hardware part of BBU and the system parameter of BS is linear, the corresponding Si is configured as 1. If such a relationship is non-linear, the corresponding Si is set to 2. When the relationship between the hardware part of BBU and the system parameter of BS is independent, the corresponding Si is configured as 0. The detailed configuration parameters of Si are illustrated in Table 1.1. Based on measurement results from the reference BS, the computation power of the reference BBU can be obtained by ref ref ref ref ref ref ref ref PBB  PDPD + PFilter + PCPRI + POFDM + PFD + PFEC + PCPU .

To calculate the computation power of real BBU’s, the reference coefficient α is defined by α

  X real  Si i

X iref

i

 

real X BW ref X BW

 SBW 

real X Ant

 SAnt 

ref

X Ant

real XM ref XM

 SM 

X Rreal X Rref

 SR 

real X dt ref X dt

 Sdt 

real X df ref X df

 Sdf .

real ref Finally, the computation power of real BBU’s is calculated by PBB  α · PBB .

Table 1.1 Configuration parameters of the BBU BBU parameters

GOPS of macro

GOPS of small

DPD

160

0

Filter

400

250

1

0

0

1

1

0

CPRI/SERDES

720

0

1

1

1

1

1

1

OFDM

160

120

1

0

0

1

1

0

S BW

SM

SR

S Ant

Sdt

Sd f

1

0

0

1

1

0

FD (liner)

90

50

1

0

0

1

1

1

FD (non-liner)

30

15

1

0

0

2

1

1

FEC

140

130

1

1

1

1

1

1

CPU

400

40

0

0

0

1

0

0

1.3 Computation and Communication Power in 5G Wireless Communication Systems

11

1.3.3 Evaluations of Computation Power Considering that 5G small cell networks with massive MIMO’s and millimeter wave techniques have not yet been commercially deployed, it is difficult to compare our simulation results with real 5G small cell networks. To validate the performance of the proposed power consumption model, it’s indispensable to compare the results of the proposed model with those of the EARTH project [17], which measures the power consumption of macro cell and small cell BSs from real wireless networks. Without a loss of generality, the two wireless communication systems are configured with 10 MHz and 2 × 2 antennas at BSs and terminals. Based on the results from the EARTH project, the total power consumption and computation power of a macro cell BS are 321.6 and 29.68 W, respectively. Similarly, for our proposed power consumption model the total power consumption and the computation power of a macro cell BS are 317.84 and 24.78 W, respectively. In the case of a small cell BS for the EARTH project, the total power consumption and computation power of a small cell BS are 6.2 and 2.4 W, respectively. For the proposed power consumption model, the total power consumption and computation power are 7.22 and 3.6 W, respectively. Compared with the above, the results of the proposed power consumption model are in agreement with the results of real wireless networks. Therefore, our proposed power consumption model is shown to be capable of estimating the power consumption of 5G small cell networks. Without loss of generality, the system parameters of the reference BS are conref ref ref ref  20 MHz, X Ant  1, X M  6, X Rref  1, X dt  100% and figured as X BW ref X df  100%. Based on the configuration parameters of the BSs in Table 1.1, the computation power of BSs is simulated for 5G small cell networks. Since the massive MIMO and millimeter wave technologies are the core technologies for 5G mobile communication systems, in this section the impact of the number of antennas and bandwidths on macro cell BSs and small cell BSs are simulated in detail. Generally speaking, the PAs of macro cell BS and small cell BS are configured as 102.6 and 1.0 W. Figure 1.4 illustrates the computation power of BS with respect to the number of antennas and bandwidths. The default system parameters of real BSs are configured as follows: the bandwidth is 20 MHz, the modulation is 64quadrature amplitude modulation (QAM), the coding rate is 5/6, the time-domain duty-cycling is 100% and the frequency-domain duty-cycling is 100%. Figure 1.4a shows the computation power of BSs with respect to the number of antennas. Based on the results in Fig. 1.4a, the computation power of BSs quickly increases with the increase of the number of antennas. The reason is that the computation power consumed for frequency-domain processing is in proportion to the square of the number of antennas. Moreover, the computation power of macro cell BSs is always larger than the computation power of small cell BSs when the number of antennas is increased. When the number of antennas is equal to 128, i.e., adopting the massive MIMO technology, the computation power of macro cell BS is larger than 3000 W and the computation power of small cell BS is larger than 800 W.

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1 Challenges of 5G Green Communication Networks

(a)

(b) 4

10

3

10

2

10

1

10

0

10

1

4

Small cell BS

Small cell BS

Macro cell BS

Macro cell BS

Computation power of BSs ( W )

Computation power of BSs ( W )

10

2

4

8

16

32

Number of antennas

64

128

10

3

10

2

10

1

10

0

10

20

50

100

150

200

400

Bandwidths(MHz)

Fig. 1.4 Computation power of BS with respect to the number of antennas and bandwidths

In general, with the adaptation of millimeter wave techniques, 5G communication systems will be able to support large bandwidths (e.g., 400 MHz), or more precisely, high transmission rates. Consequently, this would require more processing at the BBU, hence further increasing the computation power at BSs. Therefore, in this section, the impact of a millimeter wave technique on the computation power of BSs is based on a wireless communication bandwidth. When the number of antennas is configured as 4, Fig. 1.4b depicts the computation power of BSs with respect to bandwidths. Based on the results in Fig. 1.4b, the computation power of BS increases with the increase of bandwidths. Moreover, the computation power of macro cell BSs is always larger than the computation power of small cell BSs when the bandwidth is increased. When the bandwidth is 400 MHz, i.e., adopting the millimeter wave technology, the computation power of macro cell BS is larger than 1000 W and the computation power of small cell BS is larger than 200 W. Based on results in Fig. 1.4, small cell BSs can save more computation power for BBUs than macro cell BSs in 5G mobile communication systems. To evaluate the role of computation power in the BS, the computation power ratio is defined by the computation power over the total power at a BS. Figure 1.5 illustrates the computation power ratio with respect to the number of antennas and bandwidths for small cell BSs and macro cell BSs. Figure 1.5a shows the computation power with respect to the number of antennas. Based on the results in Fig. 1.5a, the computation power ratio increases with the increased number of antennas. Moreover, the computation power ratio of small cell BSs is always larger than the computation power ratio of macro cell BSs. In addition, the computation power ratio of small cell BSs is obviously larger than 50%. Figure 1.5b depicts the computation power ratio with respect to bandwidths. Based on the results in Fig. 1.5b, the computation power

1.3 Computation and Communication Power in 5G Wireless Communication Systems

(a)

(b)

100 Small cell BS Macro cell BS

80 70 60 50 40 30

100

80 70 60 50 40 30

20

20

10

10

0

Small cell BS Macro cell BS

90

Computation power ratio %

Computation power ratio %

90

13

0 1

2

4

8

16

32

64

Number of antennas

128

10

20

50

100

150

200

400

Bandwidths(MHz)

Fig. 1.5 Computation power ratio with respect to the number of antennas and bandwidths

ratio increases with the increase of bandwidths. Moreover, the computation power of small cell BSs is always larger than the computation power of macro cell BSs. When millimeter wave technology is adopted, i.e., the bandwidth is larger than or equal to 20 MHz, and the computation power ratio of small cell BSs is obviously larger than 50%.

1.3.4 Future Challenges Based on the results in Figs. 1.4 and 1.5, the computation power will play a more important role than other power consumptions, including the transmission of power at 5G small cell BSs, no matter what the level of the absolute volume and the ratio for 5G small cell networks are. On the other hand, energy efficiency of 5G mobile communication systems is expected to improve 100–1000 times, compared with the energy efficiency of 4G mobile communication systems. However, most studies involving the energy efficiency of 5G cellular networks still focus on the transmission power optimization of BSs. To face the role of computation power in 5G small cell networks, some potential challenges are presented here. The first challenge is the impact of 5G network architectures on the computation power in 5G small cell networks. Based on the results in Fig. 1.5, the importance of computation power is improved for energy efficiency optimization of 5G small cell networks. One obvious reason is that the transmission power is reduced in 5G small cell networks that adopt the massive MIMO and millimeter wave technologies. With cloud/fog computing and cache communications emerging for 5G networks,

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1 Challenges of 5G Green Communication Networks

more and more data storage and computation will be performed at 5G small cell BSs. Therefore, it is possible to predict that computation power, no matter what the power consumption level of the absolute volume and the ratio will be shall further improve for 5G cellular networks. In this case, the energy efficiency optimization of 5G cellular networks will not only consider the transmission power but also the power consumed for data computation and storage at BSs. The second challenge is the optimization of computation power at BSs with massive MIMO and millimeter wave transmission technologies. Existing studies usually fix the value of computation power at BSs. Moreover, the impact of 5G wireless transmission technologies, such as the massive MIMO and millimeter wave technologies on the computation power, is ignored at BSs. Based on the results in Fig. 1.4, the massive MIMO and millimeter wave technologies have a greater impact on the computation power of 5G small cell BSs. Considering the role of computation power at 5G small cell BSs, it is inadvisable to ignore the impact of 5G transmission technologies on the computation power of 5G small cell BSs. When massive MIMO and millimeter wave technologies are adopted by 5G small cell BSs, a large number of antennas and bandwidths can be scheduled for resource optimization in 5G small cell networks. Determining how to schedule the number of antennas and bandwidths is an interesting question for the optimization of computation power at 5G small cell BSs. The third challenge is the tradeoff between computation power and transmission power in 5G networks. Based on the analysis in Sect. 1.2, the additional power of BSs depends on the computation and transmission powers of BSs. When the additional power of BSs is combined into the computation and transmission power of BSs, the energy efficiency of 5G networks can be calculated by the energy efficiency of computation and transmission powers at BSs. However, 5G transmission technologies have different effects on the energy efficiency of computation and transmission power of small cell BSs. In some specific scenarios, the effects on energy efficiency of computation and transmission powers are contradictory at 5G small cell BSs. Hence, the relationship between the computation and communication powers needs to be further investigated for 5G networks. Moreover, the tradeoff between computation and transmission power needs to be optimized for 5G small cell BSs. To face the above challenges in the energy efficiency optimization of 5G small cell networks, some potential research directions are summarized to solve these issues: • The new energy efficiency model of 5G small cell networks considering computation and transmission power needs to be investigated. Moreover, the softwaredefined networks (SDN) could be used to trade off computation and transmission powers at 5G small cell BS’s with cloud/fog computing functions. • To improve the energy efficiency of 5G small cell BS’s, joint optimization schemes and algorithms should be developed to save computation and transmission power at BBU’s and RF chains together. • Based on the simulation results in Fig. 1.4, lot of computation power of BBU’s has to be changed into heat and more cooling systems need to be designed to support computation functions at BBU’s. To save energy at BBU’s, the energy cycle should

1.3 Computation and Communication Power in 5G Wireless Communication Systems

15

be taken into account and some potential technologies are expected to change the heat from BBU’s into electrical energy based on the pyroelectric effect.

1.4 New Issues in 5G Green Cellular Networks This section analyzed the computation capability and heat dissipation of smartphones in detail. Then, a maximum receiving rate is derived for sustaining stable computations and communications at the smartphones. Moreover, the impact of the maximum receiving rate of smartphones on the link adaptive transmission scheme is analyzed.

1.4.1 Computation Capability Factor Before estimating the maximum receiving rate of smartphones, the computation capability of the BP needs to be analyzed. In this section, Landauer’s principle and the semiconductor technology limits will be incorporated to analyze the computation capability of the BP.

1.4.1.1

Computation Capability Analysis

The greatest heat generated by electronic products, such as laptops and smartphones, comes from the computation of the chips (Fig. 1.6). For the same semiconductor technology, the computation capability of different chips can be reflected by the heat generation rate. Moreover, the relationship between the computation capability and heat generation can be established by Landauer’s principle. Landauer’s principle indicates that any logically irreversible manipulations of information, such as the erasure of a bit or merging of two computation paths, must be accompanied by a corresponding entropy increase in the environment [23]. Based on thermodynamics, the entropy increase in the environment is in the form of heat. Moreover, the energy used to erase one bit of information has a lower bound, which is known as the Landauer limit: kT ln(2), where k is the Boltzmann constant, that is, 1.38 × 10−23 J/K, and T is the temperature in Kelvin. Furthermore, the Landauer limit has been proved as the lower bound of the transistor switching energy [12]. The switching energy of the transistors produced with the latest semiconductor technology still has a gap of two orders of magnitude to approach the Landauer limit. In this article, the gap between the transistor switching energy and Landauer limit is denoted as G S , where the subscript S represents the level of semiconductor technology. For instance, G 10 indicates the gap between the 10 nm semiconductor technology and Landauer limit. Based on Landauer’s principle and S nm semiconductor technology, more computations cause more heat generation at the chip, which consumes more computation

16

1 Challenges of 5G Green Communication Networks

Fig. 1.6 Computation and heat generation of electronic products

power. Accordingly, the computation capability can be indicated by the computation chip chip power consumption of the chip Pcomp . Pcomp mainly comes from three parts, PAP , PStorage and PBP , which are the power consumption of the AP, storage unit and BP, respectively. Among these three parts, the maximum receiving rate of smartphones is mainly limited by the computation capability of the BP. Before estimating the PBP by Landauer’s principle, it is necessary to consider the sum of computed or erased information in baseband processor CBP , CBP  K BP Rphone , where Rphone is the receiving rate of smartphones and K BP is the logic operations per bit of the algorithm in the BP that can be achieved, approximately 108 [24]. Therefore, the computation power consumption of the BP can be calculated by PBP  CBP F0 αG S kT ln 2, where F0 is the number of loading logic gates, known as the fanout, whose typical value is 3–4, and α is the activity factor whose typical range is 0.1–0.2 [24].

1.4.1.2

Semiconductor Technology Limits

The evolution of semiconductor technology has promoted the development of smart terminals, and Moore’s law has affected the semiconductor technology in the last fifty years. However, Moore’s law was invalidated by the verification of the Landauer limit and the effect of thermal noise [13, 24]. The Landauer limit shows that the switching energy of transistors has a lower bound [24]. Moreover, the thermal noise death of Moore’s law was proved from the viewpoint of communication in terms

1.4 New Issues in 5G Green Cellular Networks

17

of transmitting the internal signals of chips correctly with non-negligible effects of thermal noise [13]. As shown in Fig. 1.7, Moore’s law started to fail around 2015, and the silicon transistors approach the projected scaling limit of 5-nm gate lengths [14, 15]. Moreover, silicon transistors, with the limit of 5-nm gate lengths, still have a gap of two orders of magnitude between the transistor switching energy and the Landauer limit. This gap can be estimated as G 5 ≈ 454.2 [15]. Although 1-nm gate length transistors have been invented with molybdenum disulfide (MoS2 ) instead of silicon [14], applying the raw material MoS2 to chips will take a long time because it is difficult to integrate billions of transistors with new material into the chip. Therefore, the gap G S cannot be resolved until a long time after silicon transistors reach the projected scaling limit. When the gap G S cannot be diminished, the computation power consumption of the chip in smartphones increases with the growth of the data rates in 5G wireless communication systems. The peak rate is expected to reach 20 Gbps in 5G cellular networks and exceeds 100 Gbps for subsequent networks (Fig. 1.7). The higher wireless transmission rate implies that the computation power consumption will be increased in both smartphones and BSs. When the downlink rate is less than the Gbps level, the computation power consumption of chips is commonly treated as a small circuit power consumption or ignored. When the downlink rate in 5G cellular networks achieves 20 Gbps, the computation power consumption of the chips cannot be ignored or fixed as a constant.

Fig. 1.7 The gap between the transistor switching energy and Landauer limit versus wireless transmission rate

18

1 Challenges of 5G Green Communication Networks

1.4.2 Heat Dissipation Factor Analyzing the heat generation and transfer processes in smartphones is necessary to assess the restriction of heat dissipation on the maximum receiving rate of smartphones. In this section, the heat generation and transfer processes with respect to radio frequency (RF) chains and smartphone chips have been investigated.

1.4.2.1

Heat Generation and Transfer on RF Chains

In the downlink, smartphones passively receive the wireless signals from the base station. The received wireless signals cannot be directly processed by smartphones due to channel fading. Consequently, the smartphones utilize RF chains, including low noise amplifiers (LNAs), to amplify the received wireless signals. In this case, the major heat generated by RF chains comes from the LNAs. Without loss of generality, each antenna is configured with an RF chain at the smartphone. Therefore, the heat phone generation of the LNAs can be calculated by HLNA  NTRX PLNA (1 − η), where phone NTRX is the number of antennas, PLNA is the power of an LNA and η is the poweradded efficiency of the LNAs. Considering the electromagnetic compatibility in smartphones, RF chains are usually separated from the chip of smartphones. Hence, the ratio of the heat transferred from the LNAs to the chip can be configured as λ in this section. Based on Fourier’s law in the heat transfer theory, the value of λ can be estimated using the thermal conductivity of the printed circuit board (PCB) and the heat conduction distance between the LNAs and chip.

1.4.2.2

Heat Generation and Transfer on Chips

Based on Landauer’s principle, irreversible computations increase the entropy in the environment by generating heat. There is no other heat generation, such as friction of mechanical motion, during the computation of chips. Therefore, the heat generation chip power of the chip HThermal is equal to the computation power consumption of the chip chip Pcomp . To sustain stable computations and communications in smartphones, heat dissipation is required. When the heat has been generated by the computations of the chip, the heatsink in smartphones transfers the heat from the chip to the smartphone surface and other low-temperature components, e.g., the battery and PCB, with heat conduccomponents sur tion rates Q sur conduction and Q conduction , respectively. Moreover, the sum of Q conduction components chip and Q conduction can be approximated by the value of HThermal . Limited by the thickness of smartphones, the heat conduction distance from the chip to the surface of smartphones is only a few millimeters, which is shorter than the distance from the chip to other components. In this case, the value of Q sur conduction is larger than the

1.4 New Issues in 5G Green Cellular Networks components

19

value of Q conduction according to the heat transfer theory. Hence, most of the heat is transferred from the chip to the smartphone surface by the heatsink. Then, the heat accumulated on the smartphone surface is removed by free air convection at a rate Q convection . The size of smartphones is too small to use active cooling, such as forced convection caused by a fan, to dissipate the heat from smartphones. Therefore, free air convection, known as passive cooling technology, is the only technique to dissipate heat out of smartphones. Considering that the chip is the highest temperature component in smartphones (Fig. 1.8), the thermal design of smartphones mainly focuses on heat dissipation from the chip. Hence, the maximum heat dissipation power of the chip is configured as the components thermal design power of smartphones PTD , which is equal to the sum of Q conduction chip and Q convection . When the value of HThermal is larger than the value of PTD , heat accumulates on the surface of smartphones, and the temperature of the smartphone surface increases. To prevent a local high-temperature area on the smartphone surface, the thermal design of smartphones must turn smartphones into an isothermal body. However, a local high-temperature area on the smartphone surface always occurs near the chip (Fig. 1.8). When the temperature anywhere on the smartphone surface exceeds 45 ◦ C, a low-temperature burn is likely to occur on the users’ skin that touches the local high-temperature area on the smartphone surface [25]. Thus, the highest temperature of the smartphone surface is limited to 45 ◦ C [16], which can be regarded as a safe temperature bound Tsafe . Restricted by the passive cooling and safe temperature bound on the smartphone surface, the thermal design power of smartphones is a challenge to optimize,

Fig. 1.8 Structure and temperature distribution of smartphones

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1 Challenges of 5G Green Communication Networks

Table 1.2 Parameters of different chips Devices

Companies Chip products

Semiconductor technology (nm)

Power (W)

Package size (cm2 )

Heat density (W/cm2 )

Server

Intel

Xeon® Processor E7-8894 v4

14

165

23.63

6.98

Laptop

Core™ i7-7920HQ

14

45

11.76

3.83

Tablet

Core™ m3-7Y32 Processor

14

4.5

3.30

1.36

Qualcomm Snapdragon 835

10

3.6

0.72

5.00

Snapdragon 820

14

4.6

1.14

4.04

A10 Fusion

16

2.9

1.25

2.32

A9

16

4.3

1.05

4.10

A8

20

5.9

0.89

6.63

HiSilicon

Kirin 960

16

5.3

1.10

4.82

Samsung

Exynos 8895

10

2.9

1.06

2.74

Exynos 7420

14

5.5

0.78

7.05

Exynos 5433

20

6.1

1.13

5.40

Smartphone

Apple

especially considering the characteristics of the chip in smartphones, e.g., small size and high heat density. Although the power consumption of the chip has been scaled down by utilizing the latest semiconductor technology, the heat density of chips still has not been effectively reduced. The parameters of different chip products from various companies are illustrated in Table 1.2. The heat density of a full-load Qualcomm Snapdragon 835 used in smartphones is larger than that of the Intel® Core™ i7-7920HQ used in laptop computers. When smartphones have a high receiving rate, a large amount of heat is generated by the chip on a 1 cm2 chip surface area. Hence, HThermal has a large probability of exceeding PTD . 1.4.2.3

Heat Dissipation Analysis for Smartphones

Considering the heat transferred from the LNAs, the total heat of the chip is calculated chip as HTotal  HThermal +λHLNA . When HTotal is smaller than or equal to PTD , we assume that the surface temperature Tsur of smartphones is a constant equal to the ambient temperature Tenvir , i.e., 27 ◦ C or 300 K, to simplify the analysis. In this case, the heat dissipation power of smartphones is equal to HTotal . When HTotal is higher than PTD , the extra heat Q that cannot entirely be dissipated by smartphones causes a local high-temperature area on the smartphone surface. In this case, the heat dissipation power of smartphones is equal to PTD .

1.4 New Issues in 5G Green Cellular Networks

21

Most of the metal material used to produce the smartphone surface is 7075-T6 aluminum, which is a type of aluminum alloy and is used in the iPhone 7. The specific heat and density of 7075-T6 aluminum are C  870 J/kg K and ρ  3000 kg/m3 , respectively. Moreover, the smartphone surface in this article is assumed to adopt 7075-T6 aluminum. The local high-temperature area near the chip is assumed to be a rectangle whose area A is 1 cm2 , and the thickness of the smartphone surface D chip is 1 mm. Therefore, the relationship between Pcomp and Tsur can be established by chip the extra heat Q, Q  (Pcomp + λHLNA − PTD )t  CM(Tsur − Tenvir ), where M is the mass of the surface material with volume V  AD and t is the stable working duration of the chip. The extra heat Q raises the temperature of the smartphone surface from Tenvir to Tsafe when HTotal is greater than PTD . Then, the chips in smartphones have to decrease the computation capability to reduce the heat generation. Reducing the computation capability of chips usually means that the smartphones cannot work at the original receiving rates, and the worst case is to shut off wireless communications.

1.4.3 Maximum Receiving Rates for Smartphones To maintain stable computations and communications for smartphones, the value of HTotal must be less than or equal to PTD . Considering that the total proportion of chip PAP and PStorage is at least 64% in Pcomp [26, 27], the maximum value of β, which chip is the proportion of PBP in Pcomp , is 34%. Moreover, the maximum receiving rate of smartphones is limited by the computation capability of the BP and the thermal design power of smartphones. Thus, the maximum receiving rate of smartphones can TD −λHLNA ) . be expressed as Rmax  Kβ(P BP F0 αG S kT ln 2 In cellular networks, one important scheme that improves the spectral efficiency and maintains the reliability of communications is the link adaptive transmission scheme. The influence of Rmax on the link adaptive transmission scheme for 5G BSs is discussed below. The most important parameter of the link adaptive transmission scheme is the channel state information (CSI). Based on the CSI, BSs can estimate the wireless channel capacity, which is the maximum downlink rate of BSs. Therefore, BSs can adjust parameters, such as the transmission power and channel coding, to achieve the maximum downlink rate of BSs. In general, the CSI can be reflected in the signal-to-noise ratio (SNR). Considering the massive multiple-input multiple-output (MIMO) technology applied to 5G BSs, the maximum downlink rate of BSs for a smartphone Rdownlink depends on the phone SNR when the bandwidth BW and number of antennas of a smartphone NTRX are fixed. Based on the limits of Rmax and Rdownlink , the receiving rate of smartphones Rphone should take the smaller value between Rmax and Rdownlink in the link adaptive transmission scheme.

22

1 Challenges of 5G Green Communication Networks

1.4.4 Simulation Results and Discussions In this section, the average heat dissipation power of smartphones is taken as PTD  3 W [27]. The detailed simulation parameters are shown in Table 1.3. Without loss of generality, 5, 10 and 14-nm semiconductor technologies are used for the power analysis of the chip. Furthermore, the stable communication duration, i.e., the period that the receiving rate of smartphones can be held at a specific value, is analyzed for smartphones. Figure 1.9 illustrates the maximum receiving rate of smartphones as a function of chip β, i.e., the proportion of PBP in Pcomp , and the stable communication duration when the rate surpasses Rmax . In Fig. 1.9a, three types of semiconductor technologies, 5, 10 and 14-nm semiconductor technologies, have been adopted in the smartphone chip chip to estimate Rmax . Moreover, the value of Pcomp is equal to PTD in Fig. 1.9a. When β reaches the largest value, i.e., βmax  34 %, the values of Rmax for the 5, 10 and 14-nm semiconductor technologies are 9.74, 2.17 and 1.55 Gbps, respectively. Based on the results in Fig. 1.9a, the value of Rmax can be improved in the following two ways: applying the latest semiconductor technology to chips and reducing the proportions of PAP and PStorage . When the 5-nm semiconductor technology is applied to smartphones, Fig. 1.9b depicts the stable communication duration when the rate surpasses Rmax . In Fig. 1.9b, when the rate is 4 Gbps, the stable communication duration is 3 s for Rmax  2.9 Gbps, whose β value is 10%. Since 4 Gbps is larger than Rmax , whose value is 2.9 Gbps, the temperature of the smartphone surface reaches up to 45 ◦ C within a few seconds. In this case, smartphones must decrease the computation capability of the chip to reduce the heat generation, e.g., decrease the working frequency of the chip, to prevent low-temperature burns on the user’s skin. Thus, smartphones cannot sustain the original receiving rate and may even have to shut off wireless communications. Based on the results in Fig. 1.9b, the stable

Table 1.3 Simulation parameters

Parameters

Values

Number of antennas in BSs

256

Transmission power of BSs

5W

Number of antennas in smartphones

4

Noise power spectral density

−174 dBm/Hz

Carrier frequency

3.7 and 28 GHz

Cell radius

100 m

Bandwidth

20 and 500 MHz

Power-added efficiency of LNA

59%

Power of LNA

24.3 mW

Ratio of the heat transferred from the LNAs to the chip

30%

1.4 New Issues in 5G Green Cellular Networks

23

Fig. 1.9 a The maximum receiving rate of smartphones; b receiving rates of smartphones versus stable communication duration

communication duration depends on the value of Rmax . Furthermore, the value of Rmax depends on the value of β. To maintain reliable communications, the receiving rate of smartphones Rphone should be the smaller one of Rmax and Rdownlink in the link adaptive transmission scheme. Figure 1.10 illustrates the comparison of Rmax and Rdownlink with respect to different values of the SNR. In Fig. 1.10a, a 20 megahertz (MHz) bandwidth is used by 4G BSs, and 14-nm semiconductor technology is applied to the smartphone chip. When the SNR is less than 30 dB, the value of Rmax is larger than Rdownlink . Based on the results in Fig. 1.10a, Rphone takes the value of Rdownlink in the link adaptive transmission scheme for 4G BSs. Moreover, the redundancy of the receiving rates at smartphones in Fig. 1.10a, which is the gap between Rmax and Rdownlink , indicates that the baseband processor of smartphones has a redundant computation capability to support a higher receiving rate. In Fig. 1.10b, a 500 MHz millimeter wave bandwidth is used by 5G BSs [9], and 10-nm semiconductor technology is applied to the smartphone chip. When the SNR is less than or equal to 0.5 dB, Rphone takes the value of Rdownlink . When the SNR is larger than 0.5 dB, Rphone takes the value of Rmax . In Fig. 1.10c, a 500 MHz millimeter wave bandwidth is provided by 5G BSs, and 5-nm semiconductor technology is assumed to be applied to the smartphone chip. When the SNR is less than or equal to 14.6 dB, Rphone takes the value of Rdownlink . When the SNR is larger than 14.6 dB, Rphone takes the value of Rmax . Based on the results in Figs. 1.10b, c the value of Rmax is not always larger than Rdownlink , and the receiving rate of smartphones Rphone should be the smaller one of Rmax and Rdownlink in the link adaptive transmission scheme for 5G BSs. Furthermore, the redundancy of the downlink rates at BSs in Fig. 1.10b, c, which is the gap between Rdownlink and Rmax , indicates that the baseband processor of smartphones has no extra computation capability to make the maximum receiving rate of smartphones catch up with the

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1 Challenges of 5G Green Communication Networks

Fig. 1.10 Comparison of Rmax and Rdownlink with respect to different values of the SNR: a 20 MHz bandwidth and 14-nm semiconductor technology; b 500 MHz millimeter wave bandwidth and 10nm semiconductor technology; and c 500 MHz millimeter wave bandwidth and 5-nm semiconductor technology

maximum downlink rate of BSs. A way to reduce the redundancy of downlink rates at BSs is to improve the value of Rmax . Based on the results in Fig. 1.10, the maximum receiving rate of smartphones needs to be considered in the link adaptive transmission scheme for 5G BSs.

1.4.5 Future Challenges The maximum receiving rate of smartphones is not only restricted by the computation capability of the BP but also limited by the heat dissipation of smartphones. The computation capability of BP is an important challenge now that Moore’s law is no longer valid and given the projected scaling limit of 5-nm gate lengths for silicon transistors. Moreover, the heat dissipation of smartphones is restricted by the safe temperature bound on the smartphone surface and the thermal design power of smartphones. In consideration of the maximum receiving rate of smartphones, two potential challenges in 5G cellular networks are presented here. The first challenge is the impact of the maximum receiving rate of smartphones on the wireless transmission for 5G BSs, e.g., the link adaptive transmission scheme. In future studies related to the link adaptive transmission scheme for 5G, it is necessary to compare the values of the maximum receiving rate of smartphones and the maximum downlink rate of BSs. Although the maximum downlink rate of BSs can easily be estimated for 5G BSs from the CSI, the maximum receiving rate of smartphones is difficult to directly estimate. Hence, the impact of the maximum receiving rate of smartphones on the link adaptive transmission scheme for 5G BSs needs to be further investigated.

1.4 New Issues in 5G Green Cellular Networks

25

The second challenge is the trade-off between the communication and computation capabilities in 5G BSs. Based on the results in Fig. 1.10, redundancy of downlink rates at BSs occurs when the maximum receiving rate of smartphones is less than the maximum downlink rate of BSs. One way to reduce the redundancy of downlink rates at BSs is to apply mobile edge computing technology to offload the computation assignments of the AP and BP at smartphones. Offloading the computation assignments of the AP and BP can improve the maximum receiving rate of smartphones. Therefore, substantial computational resources will be allocated to 5G BSs. An ongoing problem is how to trade off the relationship between communication and computation capabilities in 5G BSs.

1.5 Conclusions Until recently, the computation power of BSs was ignored or just fixed as a small constant in the energy efficiency evaluation of cellular networks. In this chapter, the power consumption of BSs is analyzed for 5G small cell networks adopting massive MIMO and millimeter wave technologies. Considering the massive traffic in 5G small cell networks, the computation power of 5G small cell BSs is first estimated based on Landauer’s principle. Moreover, simulation results show that the computation power of BSs increases as the number of antennas and bandwidths increases. Compared with transmission power, computation power will play a more important role in the energy efficiency optimization of 5G small cell networks. Therefore, the energy efficiency optimization of 5G small cell networks should consider computation and transmission power together. How to converge computation and transmission technologies to optimize the energy efficiency of 5G networks is still an open issue. If this is accomplished, a different challenge would indeed emerge in the next round of the transmission and computation revolution. This chapter also considered the issue triggered by the computation power. The new issue is that the maximum receiving rate of smartphones, estimated in this chapter, cannot always catch up with the downlink rates of 5G BSs. Based on the maximum receiving rate of smartphones, the potential impacts on the link adaptive transmission scheme for 5G BSs have been highlighted. To mitigate these impacts, the maximum receiving rate of smartphones should be improved by the thermal design of future 5G smartphones, e.g., improving the heat conduction rate from the chip to other low-temperature components in smartphones. Additionally, mobile edge computing, one of the 5G technologies, can be applied to improve the maximum receiving rate of smartphones by offloading the computation assignments in the chips. In 5G and future 6G cellular networks, many potential impacts triggered by the maximum receiving rate of smartphones have not yet been investigated. An emerging challenge for industries and academic researchers is how to mitigate the limits of the maximum receiving rate at 5G and 6G smartphones.

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1 Challenges of 5G Green Communication Networks

References 1. Ge, X., S. Tu, G. Mao, C.X. Wang, and T. Han. 2016. 5G ultra-dense cellular networks. IEEE Wireless Communications 23 (1): 72–79. 2. Andrews, J.G., et al. 2014. What will 5G be? IEEE Journal on Selected Areas in Communications 32 (6): 1065–1082. 3. Yang, C., J. Li, and M. Guizani. 2016. Cooperation for spectral and energy efficiency in ultradense small cell networks. IEEE Wireless Communications 23 (1): 64–71. 4. Samarakoon, S., M. Bennis, W. Saad, M. Debbah, and M. Latva-aho. 2016. Ultra dense small cell networks: Turning density into energy efficiency. IEEE Journal on Selected Areas in Communications 34 (5): 1267–1280. 5. Kwon, B., S. Kim, H. Lee, and S. Lee. 2015. A downlink power control algorithm for long-term energy efficiency of small cell network (in English). Wireless Networks 21 (7): 2223–2236. 6. Liu, C., B. Natarajan, and H. Xia. 2016. Small cell base station sleep strategies for energy efficiency. IEEE Transactions on Vehicular Technology 65 (3): 1652–1661. 7. Choi, J. 2015. Energy efficiency of a heterogeneous network using millimeter-wave small-cell base stations. In 2015 IEEE 26th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), 293–297. 8. Björnson, E., L. Sanguinetti, and M. Kountouris. 2016. Deploying dense networks for maximal energy efficiency: Small cells meet massive MIMO. IEEE Journal on Selected Areas in Communications 34 (4): 832–847. 9. Xiao, M., et al. 2017. Millimeter wave communications for future mobile networks. IEEE Journal on Selected Areas in Communications 35 (9): 1909–1935. 10. Koenig, S., et al. 2013. Wireless sub-THz communication system with high data rate (in English). Nature Photonics 7 (12): 977–981. 11. Zhong, Y., T.Q.S. Quek, and X. Ge. 2017. Heterogeneous cellular networks with spatiotemporal traffic: Delay analysis and scheduling. IEEE Journal on Selected Areas in Communications 35 (6): 1373–1386. 12. Berut, A., A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz. 2012. Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483 (7388): 187–189. https://doi.org/10.1038/nature10872 (03/08/print 2012). 13. Izydorczyk, J. 2010. Three steps to the thermal noise death of Moore’s law. IEEE Transactions on Very Large Scale Integration (VLSI) Systems 18 (1): 161–165. 14. Desai, S.B., et al. 2016. MoS2 transistors with 1-nanometer gate lengths. Science 354 (6308): 99–102. 15. Qiu, C., Z. Zhang, M. Xiao, Y. Yang, D. Zhong, and L.-M. Peng. 2017. Scaling carbon nanotube complementary transistors to 5-nm gate lengths. Science 355 (6322): 271–276. 16. Chiriac, V., S. Molloy, J. Anderson, and K. Goodson. 2016. A figure of merit for mobile device thermal management. In 2016 15th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), 1393–1397. 17. Auer, G., et al. 2011. Energy efficiency analysis of the reference systems, areas of improvements and target breakdown. Tech. Rep. ICT-EARTH deliverable, Tech. Rep 2011. 18. Desset, C., et al. 2012. Flexible power modeling of LTE base stations. In Wireless Communications and Networking Conference (WCNC), 2012 IEEE, 2858–2862. 19. Landauer, R. 1961. Irreversibility and heat generation in the computing process. IBM Journal of Research and Development 5 (3): 183–191. 20. Cockshott, W.P., P. Cockshott, L.M. Mackenzie, and G. Michaelson. 2012. Computation and its limits. Oxford University Press. 21. Lambson, B., D. Carlton, and J. Bokor. 2011. Exploring the thermodynamic limits of computation in integrated systems: Magnetic memory, nanomagnetic logic, and the Landauer limit. Physical Review Letters 107 (1): 010604 (07/01/2011). 22. Zhirnov, V., R. Cavin, and L. Gammaitoni. 2014. Minimum energy of computing, fundamental considerations (ICT-energy-concepts towards zero—Power information and communication technology).

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23. Bennett, C.H. 2003. Notes on Landauer’s principle, reversible computation, and Maxwell’s Demon. Studies in History and Philosophy of Modern Physics 34B (3): 501–510. 24. Mammela, A., and A. Anttonen. 2017. Why will computing power need particular attention in future wireless devices? IEEE Circuits and Systems Magazine 17 (1): 12–26. 25. Moritz, A.R., and F.C. Henriques. 1947. Studies of thermal injury: II. The relative importance of time and surface temperature in the causation of cutaneous burns. The American Journal of Pathology 23 (5): 695–720. 26. Mohan, J., D. Purohith, M. Halpern, and V.C.V.J. Reddi. 2017. Storage on your smartphone uses more energy than you think. Screen 38: 37.0. 27. Ogawa, T., K. Ito, and K. Matsushima. 2013. Hardware platform supporting smartphones. Fujitsu Scientific & Technical Journal 49 (2): 231–237.

Chapter 2

Energy Efficiency of 5G Wireless Communications

2.1 Introduction The massive multi-input multi-output (MIMO) antennas and the millimeter wave communication technologies have been widely known as two key technologies for the fifth generation (5G) wireless communication systems [1–6]. Compared with the conventional MIMO antenna technology, massive MIMO can improve more than 10 times spectrum efficiency in wireless communication systems [7]. Moreover, the beamforming gain based on the massive MIMO antenna technology helps to overcome the path loss fading in millimeter wave channels. For MIMO communication systems with traditional radio frequency (RF) chains and baseband processing, one antenna corresponds to one RF chain [8, 9]. In this case, a large number of RF chains have to be employed for massive MIMO communication systems. These RF chains not only consume a large amount of energy in wireless transmission systems but also increase the cost of wireless communication systems [10]. Therefore, it is an important problem to find energy efficient solutions for 5G wireless communication systems with a large number of antennas and RF chains. Furthermore, it is impractical to perform a fully digital precoding solution, i.e., zero-forcing, for massive MIMO systems and millimeter wave technologies due to power consumption and space constraints in the analog front-end [11]. To reduce communication power consumption and the number of RF chains, a hybrid analogue/digital precoding solution is proposed as a viable approach for the deployment of massive MIMO systems with millimeter wave technology [12, 13]. Moreover, this technology is expected to have a major impact on promoting small cells as the main cellular architecture in 5G wireless networks. Bear in mind that, unlike traditional macro cells, the computation power of small cells equipped with massive MIMO systems can consume more than 40% of the total power [14, 15]. Therefore, the main objective in this chapter is to improve the energy efficiency by jointly optimizing the computation and communication power for multi-user massive MIMO systems.

© Publishing House of Electronics Industry, Beijing and Springer Nature Singapore Pte Ltd. 2019 X. Ge and W. Zhang, 5G Green Mobile Communication Networks, https://doi.org/10.1007/978-981-13-6252-1_2

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30

2 Energy Efficiency of 5G Wireless Communications

What’s more, in 5G mobile communication systems, the energy efficiency is proposed as one of the most important performance indicators. Considering that the 5G network will be a huge heterogeneous network, simple scenarios such as MIMO communication systems with finite interfering transmitters in a single cell are so simple that have no ability to accurately evaluate the energy efficiency of complex cellular networks. Moreover, studies of the impact of different power allocation schemes, which is the important influence factor in power consumption evaluation, on the energy efficiency of 5G cellular networks are surprisingly rare in the open literature.

2.2 Energy Efficient Hybrid Precoding Design With the massive MIMO antenna technology adopted for 5G wireless communication systems, a large number of RF chains have to be employed for RF circuits. However, a large number of RF chains not only increase the cost of RF circuits but also consume additional energy in 5G wireless communication systems. Energy and cost efficiency optimization solutions are investigated for 5G wireless communication systems with a large number of antennas and RF chains. An energy efficiency optimization problem is formulated for 5G wireless communication systems using massive MIMO antennas and millimeter wave technology. Considering the nonconcave feature of the objective function, a suboptimal iterative algorithm, i.e., the energy efficient hybrid precoding (EEHP) algorithm is developed for maximizing the energy efficiency of 5G wireless communication systems. To reduce the cost of RF circuits, the energy efficient hybrid precoding with the minimum number of RF chains (EEHP-MRFC) algorithm is also proposed. Moreover, the critical number of antennas searching (CNAS) and user equipment number optimization (UENO) algorithms are further developed to optimize the energy efficiency of 5G wireless communication systems by the number of transmit antennas and users.

2.2.1 Related Work To improve the performance of multiple antenna transmission systems, hybrid precoding technology combining digital baseband precoding with analog RF precoding was investigated in [9, 16–22]. Based on the joint design of RF chains and baseband processing, a soft antenna subset selection scheme was proposed for multiple antenna channels [9]. When a single data stream is transmitted, the soft antenna subset selection scheme is able to achieve the same signal-to-noise ratio (SNR) gain as a full-complexity scheme involving all antennas in MIMO wireless communication systems. By formulating the problem of millimeter wave precoder design as a sparsity-constrained signal recovery problem, algorithms were developed to approximate optimal unconstrained precoders and combiners in millimeter wave

2.2 Energy Efficient Hybrid Precoding Design

31

communication systems with large antenna arrays [16]. To maximize the sum rate of MIMO communication systems, a hybrid beamforming approach was designed indirectly by considering a weighted sum mean square error minimization problem incorporating the solution of digital beamforming systems [17]. Based on a low-complexity channel estimation algorithm, a hybrid precoding algorithm was proposed to achieve a near-optimal performance relative to the unconstrained digital solutions in the single user millimeter wave communication system [18]. To reduce the feedback overhead in millimeter wave MIMO communication systems, a low complexity hybrid analog/digital precoding scheme was developed for the downlink of the multi-user communication systems [19]. To maximize the minimum average data rate of users subject to a limited RF chain constraint and a phase-only constraint, a two stage precoding scheme was proposed to exploit the large spatial degree of freedom gain in massive MIMO systems with reduced channel state information (CSI) signaling overhead [20]. Based on phase-only constraints in the RF domain and a low-dimensional baseband zero-forcing (ZF) precoding, a low complexity hybrid precoding scheme was presented to approach the performance of the traditional baseband ZF precoding scheme [21]. Compared with the digital beamforming scheme, the hybrid beamforming scheme was shown to achieve the same performance with the minimum RF chains and phase shifters in multi-user massive MIMO communication systems [22]. In the above studies, hybrid precoding schemes have rarely been investigated to optimize the energy efficiency of massive MIMO communication systems. While in cellular communication systems, energy efficiency has been considered as a critical performance metric [23]. To improve the energy efficiency of MIMO communication systems, some digital precoding schemes have been studied in [24–26]. Based on static and fast-fading MIMO channels, an energy efficient precoding scheme was investigated when the terminals are equipped with multiple antennas [24]. Jointly considering the transmit power, power allocation among date streams and beamforming matrices, a power control and beamforming algorithm was developed for MIMO interference channels to maximize the energy efficiency of communication systems [25]. Transforming the energy efficiency of MIMO broadcast channel into a concave fractional program, an optimization approach with transmit covariance optimization and active transmit antenna selection was proposed to improve the energy efficiency of MIMO systems over broadcast channels [26]. With the massive MIMO concept emerging as a key technology in 5G communication systems, the energy efficiency of massive MIMO has been studied in several papers [10, 27–31]. Based on a new power consumption model for multi-user massive MIMO, closed-form expressions involving the number of antennas, number of active users and gross rate were derived for maximizing the energy efficiency of massive MIMO systems with ZF processing [10]. When linear precoding schemes are adopted at the base station (BS), it is proved that massive MIMO can improve the energy efficiency by three orders of magnitude [27]. Considering the transmit power and the circuit power in massive MIMO systems, a power consumption model has been proposed to help optimize the energy efficiency of multi-cell mobile communication systems by selecting the optimal number of active antennas [28]. When the sum spectrum efficiency was fixed, the impact of transceiver

32

2 Energy Efficiency of 5G Wireless Communications

power consumption on the energy efficiency of ZF detector was investigated for the uplinks of massive MIMO systems [29]. Considering the power cost by RF generation, baseband computing, and the circuits associated with each antenna, simulation results in [30] illustrated that massive MIMO macro cells outperforms Long Term Evolution (LTE) macro cells in both spectrum and energy efficiency. To maximize the energy efficiency of the massive MIMO orthogonal frequency division multiple access (OFDMA) systems, an energy efficient iterative algorithm was proposed by optimizing the power allocation, data rate, antenna number, and subcarrier allocation in [31]. However, in all the aforementioned studies, only the transmission rate and the hardware complexity of hybrid precoding systems were analyzed for MIMO or massive MIMO communication systems. Moreover, energy efficient solutions for the RF chains and the baseband processing of 5G wireless communication systems is surprisingly rare in the open literature. On the other hand, the energy and cost increase through using a large number of RF chains is an inevitable problem for 5G wireless communication systems. Motivated by the above gaps, in this section we propose energy and cost efficient optimization solutions for 5G wireless communication systems with a large number of antennas and RF chains.

2.2.2 System Model Considering the impact of massive MIMO antennas on the RF chains and the baseband processing, the energy efficiency of 5G wireless communication systems has to be rethought. In the following, we describe the system configuration of 5G wireless communication systems, including massive MIMO antennas, RF chains and the baseband processing. Moreover, the energy efficiency optimization problem of 5G wireless communication systems is formulated. Without loss of generality, a single cell scenario is illustrated in Fig. 2.1, where a BS and K user equipments (UEs) are located in the 5G wireless communication system. The BS is assumed to be equipped with NTx antennas. And based on the massive MIMO configuration, we assume NTx  100 in our system [7, 27]. There are K active UEs, each with a single antenna, that are associated with the BS. Moreover, the transmission system of BS is equipped with NRF RF chains. One baseband data stream is assumed to be associated with one UE in 5G wireless communication systems. The studies in this section focus on the downlinks of 5G wireless communication systems. The received signal at the k-th UE is expressed as yk  hkH BRF BBB x + wk ,

(2.1)

where x  [x1 , . . . , xk , . . . , x K ] H is the signal vector transmitted from the BS to K UEs, where the {xk }, k  1, . . . , K , are assumed to be independently and identically

2.2 Energy Efficient Hybrid Precoding Design

33 UE1

UE 2

RF Chain

K

Β BB

N RF

Β RF

N Tx UE k

RF Chain

Baseband Precoding

RF Precoding

UE K

Fig. 2.1 System model

distributed (i.i.d.) Gaussian random variables with zero mean and variance of 1; BBB ∈ C NRF ×K is the baseband precoding matrix, where the kth column of BBB is denoted as bBB,k which is the baseband precoding vector associated with the kth UE; BRF ∈ C NTx ×NRF is the RF precoding matrix which is performed by NRF RF chains; wk is the noise received by the kth UE. Moreover, all noises received by UEs are denoted as i.i.d. Gaussian random variables with zero mean and variance of 1. The vector hkH is the downlink channel vector between the BS and the kth UE. The downlink channel matrix between the BS and K UEs is denoted as H H  [h1 , . . . , hk , . . . , h K ] H . The power consumed to transmit signals for the kth UE is expressed as  2  2 Pk  BRF bBB,k xk   BRF bBB,k  .

(2.2)

Note that the consumed power for transmitting signals in (2.2) is expended for a given bandwidth W , which is set to be 20 MHz in this section [10]. The millimeter wave communication technology is adopted for 5G wireless communication systems. Considering the propagation characteristic of millimeter waves in wireless communications, a geometry-based stochastic modeling (GBSM) is used to express the millimeter wave channel as follows [16, 17, 32, 33]  hk 

Nray NTx βk  ρki u(ψi , ϑi ), Nray i1

(2.3)

34

2 Energy Efficiency of 5G Wireless Communications γ

where Nray is the number of the multipath between the BS and K UEs. βk  ζ /lk is the large scale fading coefficient over the wireless link between the BS and the kth UE. ζ is the lognormal random variable with the zero mean and the variance of 9.2 dB. lk is the distance between the BS and the kth UE. γ is the path loss exponent. ρki is the complex gain of the ith multipath over the kth UE link, which denotes the small-scale fading in wireless channels and is governed by a complex Gaussian distribution. Moreover, ρki is i.i.d. for different values of k(k  1, . . . , K ) and i(i  1, . . . , Nray ). ψi and ϑi are the azimuth and the elevation angle of the ith multipath at the BS antenna array, respectively. u(ψi , ϑi ) is the response vector of BS antenna array with the azimuth ψi and the elevation angle ϑi . Without loss of generality, the BS antenna array is assumed as the uniform planar antenna array. Therefore, the response vector of BS antenna array with the azimuth ψi and the elevation angle ϑi is expressed as [34] u(ψi , ϑi )  √

1  2π 1, . . . , e j λ d(m sin(ψi ) sin(ϑi )+n cos(ϑi )) NTx

, . . . , e j(NTx −1) λ d((M−1) sin(ψi ) sin(ϑi )+(N −1) cos(ϑi )) 2π

T

,

(2.4)

where d is the distance between adjacent antennas, λ is the carrier wave length, M and N are the row and column number of the BS antenna array, respectively. m is denoted as the mth antenna in the row of the BS antenna array, 1  m < M; n is denoted as the nth antenna in the column of the BS antenna array, 1  n < N .

2.2.3 Problem Formulation Based on the system model in Fig. 2.1, the link spectrum efficiency of the kth UE is expressed as  H H BRF hk hkH BRF bBB,k bBB,k Rk  log2 1 +  K . (2.5) H H H 2 i1,ik hk BRF bBB,i bBB,i BRF hk + σn Note that from (2.5) we get the instantaneous spectrum efficiency. We assume that the BS transmitter has perfect CSI, i.e. channel vectors hk , k  1, . . . , K are known at the BS. This assumption is widely adopted for the investigation of precoding problems in massive MIMO systems and millimeter wave transmission systems [16, 17, 27, 28]. In practical wireless communication systems, the CSI can be obtained through uplink channel estimation then applied to downlink precoding based on the channel reciprocity in the time division duplex (TDD) mode [7, 35]. Moreover, the millimeter wave multipath channel estimation utilizing compressed channel sensing was investigated in [18] and [36]. Furthermore, considering all the UEs, the sum spectrum efficiency is expressed by

2.2 Energy Efficient Hybrid Precoding Design

Rsum 

35 K 

Rk .

(2.6)

k1

The total BS power is expressed as Ptotal

K  1  BRF bBB,k 2 + NRF PRF + PC ,  α k1

(2.7)

 K  BRF bBB,k 2 is where α is the efficiency of the power amplifier, and the term k1 the power consumed to transmit signals for K UEs over the given bandwidth, PRF is the power consumed at every RF chain which is comprised by converters, mixers, filters, phase shifters, etc. Considering the number of antennas is fixed in a wireless communication system, the number of phase shifters is also fixed since each phase shifter is associated with one antenna. PC is the power consumed for site-cooling, baseband processing and synchronization in the BS. To simplify the derivation, PC is fixed as a constant. We focus on how to maximize the BS energy efficiency (bits per Joule) by optimizing the baseband precoding matrix BBB , the RF precoding matrix BRF and the number of RF chains NRF . This optimization problem is formed by

W Rsum opt opt opt NRF , BRF , BBB  arg max η  Ptotal NRF ,BRF , BBB 2 1 s.t. [BRF ]i, j  NTx Rk  k , k  1, . . . , K K    BRF bBB,k 2  Pmax ,

(2.8)

k1

where W is the transmission bandwidth, k is the minimum spectrum efficiency required by UEk , and Pmax is the maximum transmit power required for wireless downlinks. In general, the RF precoding is performed by phase shifters, which can change the signal phase but cannot change the signal amplitude. Therefore, the amplitude of RF precoding matrix is fixed as a constant, which is added as a 2 constraint for (2.8), i.e., [BRF ]i, j  N1Tx . Furthermore, there exists a minimum data rate required by each UE in practical applications. The minimum data rate is obtained by multiplying the corresponding minimum spectrum efficiency by the bandwidth. Since the bandwidth is fixed, an equivalent minimum spectrum efficiency constraint is used to satisfy the minimum data rate requirement, which is expressed as  K  BRF bBB,k 2  Pmax is the maximum Rk  k , k  1, . . . , K in (2.8). And k1 transmit power constraint.

36

2 Energy Efficiency of 5G Wireless Communications

2.2.4 Energy Efficient Hybrid Precoding Design To maximize the energy efficiency in (2.8), an EEHP algorithm is developed to jointly optimize the baseband precoding matrix, the RF precoding matrix and the number of RF chains in the following.

2.2.4.1

Upper Bound of Energy Efficiency

Based on definitions of the baseband precoding matrix and the RF precoding matrix, the size of BRF ∈ C NTx ×NRF and BBB ∈ C NRF ×K is related with the number of RF chains NRF . To simplify the derivation, we first fix the number of RF chains. Based on the optimization objective function in the Sect. 2.2.3, (2.8) is a non-concave function with regard to BRF and BBB . In general, there does not exist an analytical solution for such non-concave functions. To tackle this problem, B  BRF BBB is configured as a digital precoding matrix B ∈ C NTx ×K , B  [b1 , . . . , bk , . . ., b K ] whose size does not depend on the number of RF chains NRF . Furthermore, the amplitude of B is free 2 from the constraint [BRF ]i, j  N1Tx . Substituting the digital precoding matrix B 2 into (2.8) and neglecting the constraint [BRF ]i, j  N1Tx , the optimization problem in the Sect. 2.2.3 is transformed as opt

B

 arg max η¯  B

1 α

W

K

K k1

Rk

bk  + NRF PRF + PC 2

k1

s.t. R k  k , k  1, . . . , K K 

bk 2  Pmax ,

(2.9)

k1

where  R k  log2 1 +  K

hkH bk bkH hk

i1,ik

hkH bi biH hk + σn2

.

(2.10)

Based on (2.9), the energy efficient is maximized by optimizing the digital precoding matrix B, where B is only constrained by the maximum transmit power Pmax and the minimum spectrum efficiency k . Assume that the maximum energy efficiency in (2.8) is denoted as ηmax . Similarly, assume that the maximum energy efficiency in (2.9) is denoted as η¯ max . Compared (2.8) with (2.9), two optimization problems have the same objective function. But (2.9) has less constraints than that of (2.8). Therefore, the solution of (2.9), i.e., the maximum energy efficiency of (2.9) is larger than or equal to the solution of (2.8), i.e., the maximum energy efficiency of (2.8). Therefore, the maximum energy efficiency ηmax is upper bounded by the maximum energy efficiency η¯ max , i.e., ηmax  η¯ max .

2.2 Energy Efficient Hybrid Precoding Design

2.2.4.2

37

Energy Efficiency Local Optimization

Considering (2.9) is a non-concave function, it is difficult to find a global optimization solution for this optimization problem. A η¯ opt with the   local optimization solution opt opt opt optimal digital precoding matrix Bopt  b1 , . . . , bk , . . . , b K is first derived for the energy efficiency optimization in (2.9). Denoting the energy efficiency and spectrum efficiency in (2.11) as functions of ¯ k ) and R k (bk ). The gradient of η(b bk , i.e., η(b ¯ k ) with respect to bk is derived by  ∂ η(b ¯ k) 2  2 k − k bk , ∂bk P

(2.11)

with Phk hkH

k   K

H H 2 j1 hk b j b j hk + σn

K k 

i1 R i I NTx + P

α ln 2

K  i1,i k



,

hiH bi biH hi (δi )2 + δi hiH bi biH hi ⎞

(2.12) · hi hiH ,

⎛ K 1 ⎝1  bk 2 + NRF PRF + PC ⎠, P W α

(2.13) (2.14)

k1

δi 

K  j1, j i

2 hiH b j b H j hi + σn ,

(2.15)

where δi is the sum of the received interference power and the noise power for the UE UEi , i  1, . . . , K . R i is obtained from (2.10) by replacing k with i. ¯ k)  0 is applied, the local optimization When the zero-gradient condition ∂ η(b ∂bk solution for the UE UEk , k  1, . . . , K is derived as k bk  k bk .

(2.16)

To obtain the optimal digital precoding matrix Bopt of the local optimization solution, an iterative algorithm is developed and called the energy efficiency hybrid precoding-A (EEHP-A) algorithm. Algorithm 1: Energy Efficient Hybrid Precoding-A (EEHP-A) Algorithm. Begin: (0) (1) Assuming B(0) to be the initial digital precoding matrix, (0) k and k are calculated by (2.11) for the UE UEk , k  1, . . . , K ; (2) At the nth, n  1, 2, . . ., iteration step, the iterative step length μ(n) k is searched within [0,1] for all K UEs

38

2 Energy Efficiency of 5G Wireless Communications

μ(n) k  arg max η μ(n) k ∈[0,1]

    −1 (n−1) (n−1) (n−1)  b I NTx + μ(n)  − I NTx k k k k

    −1 (n) (n−1) (n−1) (n−1) k s.t. R k I NTx + μk k − I NTx bk  k 2   K  −1    (n−1)   I N + μ(n) (n−1) (n−1) − I N bk Tx k k k  Tx   Pmax ; k1

(2.17) (n) (3) Based on μ(n) k , the digital precoding matrix bk is calculated at the nth, n  1, 2, . . . iteration step    −1 (n) (n−1) (n−1)  b(n−1) b(n)  I + μ  − I ; (2.18) NTx NTx k k k k k (n) (n) (4) Based on b(n) k and (2.11), k and k are updated; (5) Return to step 2 and keep iterating till b(n) k converges. end Begin

Based on the EEHP-A algorithm, the bk is obtained by the converged b(n) k . After opt obtaining bk , k  1, . . . , K for all K UEs, the local optimization solution η¯ opt is achieved. To ensure the convergence of EEHP-A algorithm, the corresponding proof is given as follows. −1  (n−1) b(n−1) , (2.17) and (2.18) are rewritten as Proof Assuming X  (n−1) k k k  

(n) (n) b(n−1) μ(n) k  arg max η μk X + 1 − μk k opt



μ(n) k ∈[0,1]

s.t. R k b(n) k



 k

K     (n) 2 bk   Pmax ,

(2.19)

k1

 −1

(n) (n−1) (n−1) (n−1) (n) b(n)  b(n−1)  μ  b + 1 − μ k k k k k k k

(n) (n) (n−1)  μk X + 1 − μk bk .

(2.20)

For the UE UEk , k  1, . . . , K , (n−1) based on (2.13) is a Hermitian symmetric k positive matrix. Hence, (n−1) can be denoted as (n−1)  ZZ H , where Z is a k k symmetric positive definite matrix. Furthermore, the following result is derived as

2.2 Energy Efficient Hybrid Precoding Design ⎤H

(n−1)

∂ η¯ bk ⎣ ⎦ X − b(n−1) k (n−1) ∂bk    2  (n−1)  H (n−1) (n−1) (n−1) −1 (n−1) (n−1)  2 bk k × k − k k − 1 bk P    2  (n−1)  H (n−1) (n−1) −1 (n−1) (n−1) (n−1) (n−1) k bk k  2 bk k − 2k + k P  



2 (n−1) H (n−1) (n−1) H (n−1) (n−1) (n−1) Z−1 k bk  2 bk − k × Z−1 k − k P  0.

39



(2.21)





(1) (n)  η ¯ b  · · ·  η ¯ b Based on (2.21) and the proposition in [37], η¯ b(0) k k k ¯ k ) is are non-decreasing sequences for the UE UEk , k  1, . . . , K . Moreover, η(b upper-bounded according to the proposition in [25]. It is easily known that the upperbounded non-decreasing sequence approaches to a convergence value. Therefore, η(b ¯ k ) in EEHP-A is proved to converge.

2.2.4.3

Hybrid Precoding Matrices Optimization

Based on (2.9) and the EEHP-A algorithm, the local optimization solution η¯ opt is obtained. When the hybrid precoding matrices BRF BBB approach the optimal digital 2 precoding matrix Bopt with the constraint [BRF ]i, j  N1Tx , the energy efficiency η will approaches the local optimization solution η¯ opt . Therefore, the optimal hybrid opt opt precoding matrices BRF and BBB can be solved by minimizing the Euclidean distance opt between BRF BBB and B [16, 17, 38]

  opt opt BRF , BBB  arg min Bopt − BRF BBB  F BRF , BBB

2 1 s.t. [BRF ]i, j  . NTx

(2.22)

2 Considering the non-convex constraint [BRF ]i, j  N1Tx [39], it is not tractable to analytically solve the optimization problem in (2.22). Based on the millimeter of BS antenna array steering matrix U  

wave channel in (2.3), the entries u(ψ1 , ϑ1 ), . . . , u(ψi , ϑi ), . . . , u ψ Nray , ϑ Nray ∈ C NTx ×Nray are constant-amplitude which can be implemented by phase shifters in BS RF circuits. Meanwhile, as pointed out in [16], the columns vectors of steering matrix U are independent from each other in millimeter wave channels. Moreover, U ∈ C NTx ×Nray and BRF ∈ C NTx ×NRF have the same row numbers. To simplify the engineering application, NRF column vectors are selected from U to form the column vectors of BRF . The detailed vector selection method is described in the EEHP-B algorithm. Furthermore, the baseband precoding matrix BBB is optimized by approaching BRF BBB to Bopt . As a consequence, the optimization problem in (2.22) is transformed as follows

40

2 Energy Efficiency of 5G Wireless Communications

   opt   opt   B BBB  arg min  − U B BB   F BBB       H  s.t.  diag BBB BBB   NRF 0

    2  opt 2 U B BB   B  ,  

(2.23)



where BBB ∈ C Nray ×K is a digital precoding matrix constrained by          2    H    NRF , and BBB has NRF non-zero rows; U BBB   Bopt 2 diag BBB B BB     0 is the transmit power constraint. As a result, the energy efficient hybrid precoding-B (EEHP-B) algorithm is developed for the optimization problem in (2.18) as follow. Algorithm 2: Energy Efficient Hybrid Precoding-B (EEHP-B) Algorithm. Begin: (1) Preset BRF as an NTx × NRF empty matrix, and set Btemp  Bopt ; (2) For i  1 : 1 : NRF   U H Btemp ;

 v  arg max  H v,v ; v1,...,Nray 

BRF  BRF [U]:, v ;  H −1 H opt BRF BRF B ; BBB,temp  BRF Bopt − BRF BBB,temp  ; Btemp   opt B − BRF BBB,temp  F

End For opt (3) BRF is solved by the step 2. The optimal baseband precoding matrix is   B opt calculated by BBB  Bopt  F  opt BB,temp  . BRF BBB,temp 

end Begin

F

Since the EEHP-B algorithm has the specified cycle number, i.e., NRF , the EEHPB algorithm is guaranteed to converge. opt opt When the optimal hybrid precoding matrices BRF and BBB are submitted into (2.8), an optimal energy efficiency ηopt can be obtained. Based on the EEHP-B algorithm, the value of ηopt approaches to the value of η¯ opt , i.e., ηopt  η¯ opt . Considering η¯ opt is a locally optimal solution for the energy efficiency optimization in (2.9), ηopt is also a locally optimal solution for the energy efficiency optimization in (2.8).

2.2 Energy Efficient Hybrid Precoding Design

2.2.4.4

41

Number of RF Chains Optimization

Based on EEHP-A and EEHP-B algorithms, a local optimization solution ηopt with opt opt optimized hybrid precoding matrices BRF and BBB is available for the energy efficiency optimization in (2.8). However, the number of RF chains is fixed for the solution ηopt . To maximize the energy efficiency, the number of RF chains is further optimized based on the locally optimal solution ηopt . By analyzing (2.9) and (2.22), the number of RF chains NRF is related not only to opt opt the size of the optimized hybrid precoding matrices BRF and BBB but also to the entries opt opt of BRF and BBB . Therefore, it is difficulty to derive an analytical solution for the optimal number of RF chains. However, the number of RF chains NRF is an positive integer and is limited in the specific range [K , NTx ]. In this case, we can utilize the ergodic searching method to find the optimal number of RF chains maximizing the energy efficiency in (2.8). Therefore, the EEHP algorithm is developed to achieve the global optimization solution for the energy efficiency optimization in (2.8). Algorithm 3: Energy Efficient Hybrid Precoding (EEHP) Algorithm. Begin: (1) For NRF  K : 1 : NTx (search all the possible values of NRF from K to NTx ) For a certain value of NRF , calculate Bopt (NRF ) according to the EEHP-A algorithm; opt opt Based on Bopt (NRF ) and NRF , calculate BRF (NRF ) and BBB (NRF ) according to the EEHP-B algorithm; opt opt Calculate ηopt (NRF ) with BRF (NRF ) and BBB (NRF ); End For opt (2) Find the optimal number of RF chains NRF maximizing the energy efficiency;

opt opt (3) Configure the global optimal hybrid precoding matrices as BRF NRF

opt opt and BBB NRF . end Begin opt

Based on the EEHP algorithm, the global maximum energy efficiency ηglobal is opt achieved by , the RF

configuring the number of RF chains NRF

chain precoding matrix opt

opt

BRF NRF

opt

opt

opt

and the baseband precoding matrix BBB NRF . Considering ηglobal 

η¯ opt  η¯ max , an suboptimal energy efficiency optimization solution is found by the EEHP algorithm in this section. Furthermore, according to the computational complexity of matrix calculation and iterative algorithms in [40] and [41], the computational complexity of the proposed

42

2 Energy Efficiency of 5G Wireless Communications

algorithm is explained as follows: The complexity of Algorithm 1 is calculated as  3 floating point operations (flops); the complexity of Algorithm 2 O(NTx K ) + O NTx  2 3 2 K + NRF + NRF NTx + NTx NRF K flops; combining Algorithm is calculated as O NTx 1 and 2, the complexity of the EEHP algorithm, i.e. Algorithm 3 is calculated as   2

3 2 3 flops. K + NRF + NRF NTx +NTx NRF K + NTx O (NTx − K ) NTx

2.2.5 Energy Efficient Optimization with the Minimum Number of RF Chains In general, the cost of RF chain is very high in wireless communication systems. To reduce the cost of massive MIMO systems, an energy efficient solution with the minimum number of RF chains is investigated in the following.

2.2.5.1

Energy Efficiency Hybrid Precoding with the Minimum Number of RF Chains

Based on the function of RF chains and the hybrid precoding scheme, the number of RF chains is larger than or equal to the number of baseband data streams in massive MIMO communication systems [9, 42]. In this section the number of baseband data streams is assumed to be equal to the number of active UEs. Without loss of min  K , where K generality, the minimum number of RF chains is configured as NRF is the number of active UEs in Fig. 2.1. When the minimum number of RF chains is configured, the RF precoding matrix BRF has a size of NTx × K , which is exactly the size of the conjugate transpose of the downlink channel matrix. Therefore, the entry of RF precoding matrix BRF is directly configured as [21] [BRF ]i, j  √

1 e jθi, j , NTx

(2.24)

where [BRF ]i, j denotes the (i, j)th entry of the RF precoding matrix BRF , and θi, j is the phase of the (i, j)th entry of the conjugate transpose of the downlink channel matrix. When the downlink channel matrix H H and the RF chain precoding matrix are combined together, an equivalent downlink channel matrix for all K UEs is given by H

H H H H Heq  H H BRF  BRF h1 , . . . , BRF hk , . . . , BRF hK H

 h1,eq , . . . , hk,eq , . . . , h K ,eq ∈ C K ×NRF .

(2.25)

Substitute (2.25) into (2.8), the energy efficiency optimization problem is transformed as

2.2 Energy Efficient Hybrid Precoding Design opt ! BBB  arg max η˜  BBB

s.t.

43

1 α

K

! k1 Rk,eq 2 K    + K PRF + PC k1 BRF bBB,k W

!k,eq  k , k  1, . . . , K R

K K     BRF bBB,k 2  Pk  Pmax , k1

(2.26)

k1

where  ! Rk,eq  log2 1 +  K



H H bBB,k bBB,k hk,eq hk,eq

i1,ik

H H hk,eq bBB,i bBB,i hk,eq + σn2

.

(2.27)

Similar to the optimization problem in (2.8), the energy efficiency η˜ in (2.25) is opt maximized by the optimal baseband precoding matrix ! BBB . Considering the objective function in (2.26) is non-concave, it is intractable to solve for the global optimum of η. ˜ Therefore, a local optimum solution is developed for (2.26) as follow.  of the baseband precoding vector of the kth UE, i.e.  Given that η˜ is a function η˜ bBB,k , the gradient of η˜ bBB,k is derived by   ∂η bBB,k 2 ! !  2  (2.28) k − k bBB,k , ! ∂bBB,k P with !k    K

H Phk,eq hk,eq

, (2.29) H H 2 hk,eq bBB, j bBB, j hk,eq + σn ⎛ ⎞ K K H H  ˜ b b h h i,eq BB,i BB,i i,eq ⎜ H H ⎟ ! k  i1 Ri,eq BRF BRF + P hi,eq hi,eq  ⎝ 2 ⎠, α ln 2 H H ˜δi + δ˜i hi,eq i1,ik bBB,i bBB,i hi,eq j1

! 1 P W



δ˜i 



K  1  BRF BBB,i 2 + K PRF + PC , α i1 K 

H H 2 hi,eq bBB, j bBB, j hi,eq + σn ,

(2.30) (2.31)

(2.32)

j1, ji

!k,eq where δ˜i is the sum of the received interference and noise power for the ith UE, R is obtained by replacing k with i in (2.27). Replacing the results of (2.11) by the results opt of (2.28), the baseband precoding matrix ! BBB is solved by the EEHP-A algorithm. opt Substituting the baseband precoding matrix ! BBB into the EEHP algorithm, a local optimum of η˜ is solved. To differentiate this approach from the EEHP algorithm,

44

2 Energy Efficiency of 5G Wireless Communications

Fig. 2.2 Energy efficiency of the EEHP-MRFC algorithm

this algorithm is denoted as the energy efficient hybrid precoding with the minimum number of RF chains (EEHP-MRFC) algorithm. To analyze the performance of EEHP-MRFC algorithm, numerical simulation results are illustrated in Fig. 2.2. In generally, the optimization result of EEHPMRFC algorithm depends on the initial values of the baseband precoding matrix, i.e. (0) B(0) BB . Without loss of generality, BBB can be configured to have equal entries as [25] $ B(0) BB



Pmax K ×K 1 , K

(2.33)

where 1 K ×K denotes the K × K matrix whose entries are equal to 1, the coefficient % Pmax is due to the maximum transmit power constraint in the BS. The detailed K simulation parameters are list in Table 2.1. Based on the results in Fig. 2.2, the EEHP-MRFC algorithm converges after a limited iteration number. Besides, the convergence rate decreases the number of UEs increases. Moreover, the converged value of the energy efficiency also decreases with the increase of the number of UEs. This result indicates that the energy efficiency of massive MIMO system is inversely proportional to the number of active UEs.

2.2 Energy Efficient Hybrid Precoding Design

45

Table 2.1 Simulation parameters of EEHP algorithm [10, 42–44] Parameter

Value

Maximum transmit power Pmax

33 dBm

Minimum spectrum efficiency for each UE k

3 bit/s/Hz

Power consumed by each RF chain PRF

48 mW

Power consumed by other parts of the BS PC

20 W

The number of BS antennas NTx

200

Power amplifier efficiency α

0.38

Noise power spectral density

– 174 dBm/Hz

Carrier frequency

28 GHz

Bandwidth

20 MHz

Cell radius

200 m

Minimum distance between the UE and the BS

10 m

Path loss exponent γ

4.6

The number of multi-paths Nray

30

Azimuth ψi and elevation angle ϑi

Uniformly distributed within [0, 2π ]

2.2.5.2

Energy Efficiency Optimization Considering the Numbers of Antennas and UEs

In practical engineering applications, the optimal number of transmit antennas and the optimal number of UEs can be performed by the resource management and the user schedule schemes for maximizing the energy efficiency of 5G wireless communication systems. However, the EEHP algorithm cannot directly derive the analytical number of transmit antennas and UEs for maximizing the energy efficiency of 5G wireless communication systems. Therefore, we try to derive the optimal number of transmit antennas and UEs for maximizing the energy efficiency of 5G massive MIMO communication systems. To simplify the derivation, a specified scenario with rich scattering and multi-paths propagation in a millimeter wave wireless channels is considered for the ergodic capacity calculation in this section. Based on measurement results of millimeter wave channels in [33, 45, 46], the Rayleigh fading model can be adopted to describe the considered millimeter wave wireless channels. Assume that the transmitter, i.e., the BS has the perfect CSI. When the minimum number of RF chains is configured and the ZF precoding is adopted in the system model [8], based on (2.24) and (2.25), the baseband precoding matrix is given by  H Heq BBB  Heq Heq

−1

D,

(2.34)

where D is a K × K diagonal matrix, which aims to normalize BBB . Assume that Pout is the total BS downlink transmit power consumed by K active UEs. To simplify the derivation, the equal power allocation scheme is assumed to be adopted for all K

46

2 Energy Efficiency of 5G Wireless Communications

active UEs. Substituting (2.34) into (2.5), the normalized link capacity, i.e. the link spectral efficiency of UEk is expressed as [27] ⎛ ⎞ ⎜ Rk,ZF  log2 ⎝1 +

Pout  H K Heq Heq

−1



⎟ ⎠.

(2.35)

k,k

Based on Jensen inequality, the upper-bound of the ergodic link capacity of UEk is derived by ⎛ ⎡ ⎞⎤  Pout ⎜ ⎢ ⎟⎥  ⎠⎦, E Rk,ZF  R k,ZF  log2 ⎣1 + E⎝  (2.36) −1 H K Heq Heq &

k,k

where E(·) is the expectation operation taken over the Rayleigh fading channel H H within Heq  BRF H. Diagonal and off-diagonal entries of Heq are given by NT x 

1  [H]i,k , Heq k,k  hkH bRF,k  √ NTx i1

Heq

 j,k

 h Hj bRF,k  √

NT x 1  [H]i, j e jθi,k , NTx i1

(2.37)

(2.38)

where bRF,k is the kth column of BRF . With Rayleigh fading channel considered and based on results in [21], the of Heq is governed by a normal

√diagonal entry 

π NTx π , 1 − 4 and the off-diagonal entry of Heq is distribution, i.e., Heq k,k ∼ N 2 

governed by a standard normal distribution, i.e., Heq j,k ∼ CN (0, 1). Considering massive MIMO antennas are equipped at the BS, without loss of generality, the number of BS antennas is assumed to be larger than or equal to 100, i.e. NTx  100. In this case the expected value of diagonal entries is much larger than the expected value of off-diagonal entries in Heq . Therefore, the expected value of the off-diagonal entries can be set to zero in Heq , and Heq is approximated as a diagonal matrix [21]. Based on (2.36), the upper-bound of the ergodic link capacity of UEk is approximated as 

2  R k,ZF ≈ log2 1 + E Pk Heq k,k     NTx π 2 − π + 4 . (2.39)  log2 1 + E Pk 4 &

When the ergodic link capacity is replaced by the upper-bound of the ergodic link capacity, the upper-bound of the BS energy efficiency is derived by

2.2 Energy Efficient Hybrid Precoding Design

ηˆ ZF 

K log2 1 + 1 α

47 Pout K



NTx π 2 −π+4 4



Pout + K (PRF + PBB ) + PC

,

(2.40)

where PBB is the power consumed by the baseband processing for the baseband data stream, PC is the fixed BS power consumption without the power consumed for the downlink transmit, the RF chains and the baseband processing. For the upper-bound of the BS energy efficiency in (2.40), the following proposition is given. Proposition Considering the impact of the number of UEs and transmit antennas on the BS energy efficiency, a function G(K , N T x ) is formed as (2.41) )  Pout (PRF + PBB ) NTx π 2 − π + 4 4 G(K , N T x )  P + 4PC ln 2 α out * ) *   Pout NTx π 2 − π + 4 Pout NTx π 2 − π + 4 + − 1+ 4K ln 2 4K ) *  Pout NTx π 2 − π + 4 × log2 1 + . (2.41) 4K Cri . The critical When G(1, 100) ≥ 0, there exists a critical number of antennas NTx Cri number of antennas NTx is the smallest integer which is larger than or equal to the Cri , there exists an optimal root of the equation G(1, NTx )  0. When NTx ≥ NTx Max opt . K opt number of UEs K ≥ 1 which maximizes the BS energy efficiency as ηˆ ZF is solved by the integer closest to the root of the equation G(K , NTx )  0. When Cri Max , the maximum BS energy efficiency ηˆ ZF is achieved with the number NTx < NTx of UEs K  1. Moreover, the BS energy efficiency ηˆ ZF decreases with the increase of the number of UEs K. When G(1, 100) < 0, there exists an optimal number of UEs K opt which maxiMax . When the number of antennas NTx is given, mizes the BS energy efficiency as ηˆ ZF opt the optimal number of UEs K is solved by the integer closest to the root of the equation G(K , NTx )  0.

Proof To simplify the derivation, some variables are defined as follows: z  K1 ,

2 z ∈ ( 0, 1] , a  Pout NTx π 4−π+4 , b  α1 Pout + PC and c  (PRF + PBB ). The derivative of ηˆ ZF with respect to z is given by

log2 (1+az) d c+zb dηˆ ZF  dz dz a(c+zb) − b log2 (1 + az) (1+az) ln 2  (c + zb)2

48

2 Energy Efficiency of 5G Wireless Communications

 

ac b ln 2

+

a z ln 2 1 b (c

− (1 + az) log2 (1 + az) + zb)2 (1 + az)

.

When the numerator of (2.42) is defined by

ac a f (z, a)  + z − (1 + az) log2 (1 + az), b ln 2 ln 2

(2.42)

(2.43)

f (z,a) ηˆ ZF (2.42) is simply rewritten as ddz  1 (c+zb) . 2 (1+az) b  ∂ f (z,a) Since ∂z  lna2 − a log(1 + az) + lna2 < 0, f (z, a) monotonously decreases with increase of z ∈ ( 0, 1] . Moreover, the limitation of f (z, a) is derived by

lim f (z, a) 

z→0+

ac ac > 0. lim+ f (z, a)  > 0. z→0 b ln 2 b ln 2

(2.44)

If f (1, a)  0, then f (z, a)  0, z ∈ ( 0, 1] . Furthermore, we have the result  0. Therefore, ηˆ ZF monotonously increases with increase of z ∈ ( 0, 1] . In other words, ηˆ ZF monotonously decreases with the increase of K. In this case, the BS energy efficiency ηˆ ZF is maximized by K  1. If f (1, a) < 0, there must exist an optimal value z opt , z opt ∈ ( 0, 1] , which ensures f z opt , a  0. The integer closest to 1/z opt is the optimal number of UEs K opt which maximizes the BS energy efficiency ηˆ ZF . The differential of f (1, a) with respect to a is derived by dηˆ ZF dz

∂ f (1, a) c  − log2 (1 + a). ∂a b ln 2

(2.45)

(1,a) decreases with increase of a. When the number of transmit (2.45) implies that ∂ f ∂a antenna is configured as N Tx  1, i.e. a is minimized as amin , the corresponding (1,a) < 0 considering the practical value range of Pout , differential result is ∂ f ∂a aamin

(1,a) PC , PRF and PBB [10, 30, 47, 48]. Therefore, we have the result ∂ f ∂a < 0 for all available values of a. As a consequence, f (1, a) monotonously decreases with increase of a ∈ [amin , ∞).

2 Substitute z  K1 , z ∈ ( 0, 1] , a  Pout NT x π 4−π+4 , b  α1 Pout + PC and c  (PRF + PBB ) into (2.43), the function G(K , N T x ) is transformed by f (z, a). opt which When f (1, min ) < 0, i.e., G(1, 100) < 0, there exists an optimal value z  aopt ensures f z , a  0 and maximizes the BS energy efficiency ηˆ ZF . The integer closest to 1/z opt is the corresponding optimal number of UEs K opt . When f (1, amin )  0, i.e. G(1, 100)  0, there exist a critical number of Cri Cri which ensures f (1, a)| NTx NTxCri  0. When NTx  NTx , there exists antennas NTx  opt opt an optimal value z which ensures f z , a  0 and maximizes the BS energy efficiency ηˆ ZF . As a consequence, the integer closest to 1/z opt is the corresponding Cri , ηˆ ZF monotonously decreases with optimal number of UEs K opt . When NTx < NTx

2.2 Energy Efficient Hybrid Precoding Design

49

the increase of K. In this case, the BS energy efficiency ηˆ ZF is maximized by K  1. The proposition is proved. Note that in the above proposition and the corresponding proof, the minimum number of antennas is set as 100 considering the massive MIMO scenario. Based on the results in [21], the approximation in (2.39) is guaranteed to hold when the number of antennas is larger than or equal to 100. Based on the proof, G(1, NTx ) is a monotonous decreasing function with respect to NTx . Utilizing the bisection method, the critical number of antennas searching (CNAS) algorithm is developed to solve Cri . the critical number of antennas NTx Algorithm 4: Critical Number of Antennas Searching (CNAS) Algorithm. Begin: Low  100 and the (1) Preset the initial lower bound of searching range NTx High initial upper bound of searching range NTx  1000; High

(2) Adjust the upper bound of the range until G 1, NTx equal to 0.

High >0 While G 1, NTx High

Do NTx

is less than or

High

 NTx × 2;

End

Temp High Low (3) Preset the median value as NTx  NTx + NTx /2.

Temp (4) Keep searching until the median value satisfies G 1, NTx  0.

Temp While G 1, NTx ! 0

Temp If G 1, NTx >0 Temp

Low NTx  NTx ; Else High Temp NTx  NTx ; End

Temp High Low + NTx /2; NTx  NTx

End (5) Configure the critical number of antennas+ as the,smallest integer not less Temp Cri .  NTx than the final median value, i.e., NTx end Begin

50

2 Energy Efficiency of 5G Wireless Communications

Cri When the critical number of antennas NTx is obtained by the CNAS algorithm, the opt optimal number of UEs K is solved by G(K , NTx )  0 which maximizes the BS Max . Considering G(K , NTx ) monotonously decreases with the energy efficiency ηˆ ZF increase of the number of UEs K, the UE number optimization (UENO) algorithm is developed to solve the optimal number of UEs K opt by the bisection method.

Algorithm 5: The UE Number Optimization (UENO) Algorithm. Begin: (1) Preset the initial lower bound of the searching range K Low  1 and the initial upper bound of the searching range K High   40; (2) Adjust the upper bound of the range until G K High , NTx is less than or equal to 0;  While G K High , NTx > 0 Do K High  K High × 2; End  (3) Preset the median value as K Temp  K Low + K High  /2 (4) Keep searching until the median value satisfies G K Temp , NTx  0;  While G K Temp , NTx !  0  If G K Temp , NTx > 0 K Low  K Temp Else K High  K Temp ; End  K Temp  K Low + K High /2; End (5) Configure the optimal number - of UEs as. the integer closest to the final median value, i.e., K opt  K Temp − 1/2 . end Begin Since the above CNAS and UENO algorithm are based on the bisection method, the computational complexity of the CNAS and UENO algorithms are calculated as



  High Low and O log2 K High − K Low , respectively [49]. O log2 NTx − NTx

2.2.6 Simulation Results Energy efficiency optimization solutions with respect to the number of RF chains, transmit antennas and active UEs are simulated in the following. Without loss of

2.2 Energy Efficient Hybrid Precoding Design

51

generality, the number of active UEs is configured as 10. Other default parameters are listed in Table 2.1. The EEHP algorithm is proposed in Sect. 2.2.4 to maximize the BS energy efficiency. To tradeoff the energy and cost efficiency of the BS RF circuits, the EEHPMRFC algorithm is developed in Sect. 2.2.4. To analyze the proposed EEHP and EEHP-MRFC algorithms, the energy efficient digital precoding (EEDP) algorithm and sparse precoding algorithm are simulated for performance comparisons. In the EEDP algorithm, the energy efficiency of 5G wireless communication systems is opt obtained by substituting the optimal digital precoding vectors bk derived from the EEHP-A algorithm into (2.9). The sparse precoding algorithm in [16] is also simulated, which implements a spectrum efficiency maximization hybrid precoding scheme. The two proposed algorithms along with the EEDP algorithm, the sparse precoding algorithm and the traditional ZF precoding algorithm are compared in Fig. 2.3. In order to better compare the energy efficiency performance of the algorithms, the total transmit power for all the UEs is used as the comparison standard [24, 50]. For each algorithm, the energy efficiency first increases with the increasing total transmit power. When the total transmit power exceeds a given threshold, the energy efficiency starts to decrease. The variation tendencies of the curves coincide with the previously published results [24, 50]. When the total transmit power is fixed, the EEDP algorithm has the highest energy efficiency which is consistent with the analysis result in Sect. 2.2.4. The ZF precoding algorithm has the lowest energy efficiency because it uses the same number of RF chains as antennas, which leads to the highest power consumption for the RF chains. Besides, the sparse precoding algorithm performs more poorly than the EEHP algorithm but outperforms the EEHP-MRFC algorithm in terms of energy efficiency for 5G wireless communication systems. Although the energy efficiency of the EEHP-MRFC algorithm is not the best result, this algorithm is valuable for reducing the transmitter cost and the design complexity by minimizing the number of RF chains. In practical applications, the selection of the EEHP algorithm or the EEHP-MRFC algorithm depends on the desired tradeoff between the energy and cost efficiency for 5G wireless communication systems. Figure 2.4 illustrates the BS energy efficiency with respect to the number of transmit antennas considering different numbers of multipath components. The sparse precoding and EEDP algorithms are also simulated to compare with the two proposed EEHP and EEHP-MRFC algorithms. It can be seen that the BS energy efficiency increases with increasing numbers of the transmit antennas. Meanwhile, increasing the number of multipath components yields the highest BS energy efficiency when the number of transmit antennas is fixed. Decreasing energy efficiency results are yielded by the algorithms in the order EEDP, EEHP, sparse precoding and finally the EEHP-MRFC algorithm, respectively. The BS energy efficiency with respect to the number of RF chains is shown in Fig. 2.5. As for the EEDP and EEHP algorithms, the BS energy efficiency first increases then decreases with increasing numbers of the RF chains. For the sparse precoding and EEHP-MRFC algorithms, the BS energy efficiency always decreases when increasing the number of RF chains. When the number of multi-paths is fixed

52

2 Energy Efficiency of 5G Wireless Communications

Fig. 2.3 Energy efficiency with respect to the total transmit power

Fig. 2.4 Energy efficiency with respect to the number of transmit antennas considering different numbers of multipaths

2.2 Energy Efficient Hybrid Precoding Design

53

Fig. 2.5 Energy efficiency with respect to the number of RF chains considering different numbers of multi-paths

as 30 and the number of RF chains is less than or equal to 26, the energy efficiency of the EEHP-MRFC algorithm is larger than that of the sparse precoding algorithm. Further, when the number of RF chains is larger than 26, the energy efficiency of the EEHP-MRFC algorithm is always less than that of the sparse precoding algorithm. In Fig. 2.6, the spectrum efficiency as a function of the number of RF chains and the number of multi-paths is illustrated. It can be seen that sparse precoding algorithm has the highest spectrum efficiency. As for EEDP and EEHP algorithms, the spectrum efficiency increases with increasing the numbers of the RF chains. Considering that the number of RF chains always equals the number of UEs in the EEHP-MRFC algorithm, the spectrum efficiency of the EEHP-MRFC algorithm decreases with the increase of the number of RF chains. When the number of the RF chains is fixed, decreasing spectrum efficiency results are yielded by the algorithms in the order sparse precoding, EEDP, EEHP, and finally the EEHP-MRFC algorithm, respectively. The impact of the number of transmit antennas and the number of UEs on the BS energy efficiency is investigated in Sect. 2.2.5.2. Based on the CNAS and UENO algorithms, numerical simulations are shown in Fig. 2.7. Without loss of generality, the number of transmit antennas are configured as 100, 150 and 200, respectively. The power consumed by other parts of the BS is configured as in Fig. 2.7. The BS energy efficiency first increases then decreases with increasing of the number of UEs.

54

2 Energy Efficiency of 5G Wireless Communications

Fig. 2.6 Spectral efficiency with respect to the number of RF chains considering different numbers of multi-paths

The maximums of the BS energy efficiency correspond to the optimal numbers of UEs 35, 50 and 55, respectively, which is consistent with the optimal number of UEs obtained from the proposition.

2.2.7 Conclusions The BS energy efficiency considering the energy consumption of RF chains and baseband processing is formulated as an optimization problem for 5G wireless communication systems. Considering the non-concave feature of the objective function, an available suboptimal solution is proposed by the EEHP algorithm. To tradeoff the energy and cost efficiency in RF chain circuits, the EEHP-MRFC algorithm is developed for 5G wireless communication systems. Based on the CNAS and UENO algorithms, the energy efficiency of 5G wireless communication systems can be maximized by optimizing the number of UEs and BS antennas, which is easily employed in the user scheduling and resource management schemes. Compared with the maximum energy efficiency of conventional ZF precoding algorithm, numerical results indicate that the maximum energy efficiency of the proposed EEHP and EEHP-MRFC algorithms are improved by 220% and 171%, respectively. Moreover, the difference between the EEHP algorithm and the EEHP-MRFC algorithm

2.2 Energy Efficient Hybrid Precoding Design

55

Fig. 2.7 Energy efficiency with respect to the number of UEs considering ZF baseband precoding and the minimum number of RF chains

is illustrated by numerical simulation results. Furthermore, the results provide some available suboptimal energy efficiency solutions and insights into the energy and cost efficiency of RF chain circuits for 5G wireless communication systems.

2.3 Energy Efficient Optimization with RF Chains 2.3.1 Related Work Generally, there exist two types of hybrid precoding solutions in RF systems: the fully-connected structure and the partially-connected structure [11]. In the former, all the antennas are connected to each RF chain by phase shifters where multiplexing between RF chains and antennas can be achieved for massive MIMO systems [21, 51–57]. For instance, a hybrid precoding solution with fully-connected structures using orthogonal matching pursuit that utilizes the structure of millimeter wave channels was proposed in [51]. Based on the simulation results, it was observed that the proposed algorithm can approach the theoretical limit of spectral efficiency. As a trade-off between performance and complexity, four precoding hybrid algorithms were investigated in [52] for a single user massive MIMO system. An algorithm based on iteratively updating phases of phase shifters in the RF precoder was proposed in

56

2 Energy Efficiency of 5G Wireless Communications

[53]. The proposed approach aims at minimizing the weighted sum of squared residuals between the optimal full-baseband design and the hybrid design. To guarantee that precoding can converge to a locally optimal solution, a hybrid precoding algorithm was developed in [53]. In addition, a hybrid precoding algorithm based on an adaptive channel estimation was developed in [54]. The proposed algorithm aims at relaxing hardware constraints on the analogue only beamforming and achieving spectral efficiency of fully digital solutions [54]. Considering multi-user massive MIMO scenarios, a hybrid precoding scheme that approaches spectral efficiency of a traditional baseband ZF precoding scheme was proposed in [21]. Furthermore, to harvest a large array gain through phase-only RF precoding, a hybrid block diagonalization (BD) scheme capable of approaching the capacity of the traditional BD processing method in massive MIMO systems was investigated in [55]. When the number of RF chains is less than twice the number of data streams, the authors in [56] developed a heuristic algorithm to solve the problem of spectral efficiency maximization for transmission scenarios, such as a point-to-point massive MIMO system and a multi-user multiple-input-single-output (MISO) system. Based on the fully-connected structure, [57] developed a hybrid precoding scheme to optimize the energy efficiency of multi-user massive MIMO systems. Although the fully-connected structure of a hybrid precoding solution can approach the theoretical limit of spectral efficiency for fully digital precoding systems, the partially-connected hybrid precoding approach (i.e., every RF chain is connected to a limited number of antennas) is more attractive for practical implementation due to low complexity and cost [11, 58–64]. A comparison between fullyconnected and partially-connected structures of hybrid precoding for massive MIMO systems with millimeter wave technology was performed in [11], which indicates that the partially- connected structure of a hybrid precoding solution can offer a potential advantage of balancing cost and performance for massive MIMO systems. Furthermore, based on a prototype system, the advantages of a hybrid beamforming scheme in 5G cellular networks with a partially-connected structure were demonstrated [58]. In [59] a multi-beam transmission diversity scheme was proposed for single stream and single user case in massive MIMO systems with partially-connected structures [59]. To improve the transmission rate, a hybrid precoding scheme for partiallyconnected structures capable of adaptively adjusting the number of data streams was developed by [60]. This approach is based on the rank of an equivalent baseband MIMO channel matrix and the received SNR. Treating the hybrid precoder design as a matrix factorization problem, [61] proposed the effective alternating minimization algorithms that can be used to optimize the transmission rate of massive MIMO systems with partially-connected structures. Considering the issue of power consumption in massive MIMO systems, energy efficiency optimization of a hybrid precoding solution with a partially-connected structure was studied in [62–64]. For instance, for multi-user massive MIMO systems with millimeter wave technology, it was shown that the partially-connected structure can outperform the fully- connected structure in terms of both spectral efficiency and energy efficiency [62]. Considering a single user massive MIMO system, the baseband and RF precoding matrices were optimized to improve energy efficiency of massive MIMO systems [63]. Based on

2.3 Energy Efficient Optimization with RF Chains

57

the successive interference cancellation (SIC)-based hybrid precoding method, the authors in [64] had shown that energy efficiency of a single user massive MIMO system can be improved with low complexity. It is partially-connected structure that attracts practical implementation. However, researches on it are rare, especially on energy efficiency optimization. Moreover, all the aforementioned studies which optimize energy efficiency for partially- connected structures use simple precoding optimization methods, such as optimizing baseband and RF precoding independently. Although the ratio of computation power to total power has shown improvement in massive MIMO systems, detailed investigation of the computation power model used for massive MIMO systems has received little attention in the open literature. In fact, these investigations simply treat energy consumption of massive MIMO systems solely as communication power [62–64].

2.3.2 System Model Although the fully-connected structure of massive MIMO RF systems can easily approach the spectral efficiency limit for multi-user massive MIMO systems, the cost and complexity of massive MIMO RF systems are becoming a major issue for their future deployment. To reduce cost and simplify complexity of massive MIMO RF systems, the partially-connected structure, where each RF chain corresponds to multiple phase shifters and antennas as shown in Fig. 2.8, is a promising solution for industrial applications. In this section, we jointly optimize the computation and communication power of massive MIMO RF systems to reduce the cost and complexity of multi-user massive MIMO systems with the partially-connected structure.

2.3.2.1

Wireless Transmission Model

A multi-user massive MIMO communication system with the partially-connected or fully-connected structures is illustrated in Fig. 2.8. As can be observed the transmitter in the massive MIMO communication system includes: a baseband unit with K input data streams, NRF RF chains, and NT ≥ 100 antennas. Considering the partiallyconnected structure, one RF chain is connected with NNRFT phase shifters and NNRFT antennas in such a way that antennas connected to each RF chain do not overlap. The receivers are configured as K active UEs, each with a single antenna. In this work we focus on the downlink of multi-user massive MIMO communication systems. The received signal at the kth UE is expressed by yk  hkH BRF BBB x + wk .

(2.46)

58

2 Energy Efficiency of 5G Wireless Communications

Data 1 Data 2 . . .

UE

RF Chain 1

Baseband Precoding

. . .

1

UE

N RF

RF Precoding

. . .

2

. . .

RF Chain N RF

Data K

UE

K

+ +

N RF

NT

Partially-connected structure

N RF

NT

Fully-connected structure

Fig. 2.8 Multi-user massive MIMO communication systems

In the above, x  [x1 , . . . , xk , . . . , x K ] H is the signal vector transmitted from the transmitter to K UEs, where xk is assumed to be independently and identically distributed i.i.d. Gaussian random variables with zero mean and a variance of 1, hk is the downlink channel vector between the BS and the kth UE, and wk is the noise received by the kth UE. Moreover, all noise samples in the UE are i.i.d. Gaussian random variables with zero mean and variance of σn2 . The BBB ∈ C NRF ×K is the baseband precoding matrix, where the kth column of BBB is denoted as bBB,k which is the baseband precoding vector for the kth UE. The BRF ∈ C NT ×NRF is the RF precoding matrix, which is realized by NT phase shifters. For the partiallyconnected structure, every RF chain is equipped with an antenna sub-array as shown in Fig. 2.8. In this 0 matrix BRF is a block diagonal matrix, / case, the RF precoding i.e., BRF  diag m1 , . . . , mi , . . . , m NRF , where mi is the ith block matrix which corresponds to the precoding matrix between the ith RF chain and the connected NNRFT antennas, mi is a NNRFT × 1 complex vector and the amplitude of vector element is fixed as 1. When the bandwidth of the kth UE is configured as W, considering interference caused by sidelobe beam, the available rate of the kth UE is expressed by

2.3 Energy Efficient Optimization with RF Chains



59

H H hkH BRF bBB,k bBB,k BRF hk

Rk  W log2 1 +  K

i1,ik

H H hkH BRF bBB,i bBB,i BRF hk + σn2

,

(2.47)

where superscript H is the conjugate transposition operation on the matrix. When the transmissions of all UEs are considered, the available sum rate of multiuser massive MIMO communication system is expressed by Rsum 

K 

Rk .

(2.48)

k1

To support massive wireless traffic in 5G wireless communication systems, millimeter wave technology is adopted for multi-user massive MIMO communication systems. Based on the propagation characteristic of millimeter wave in wireless communications, a geometry-based stochastic model (GBSM) is used to describe the millimeter wave channel of multi-user massive MIMO communication systems [32, 65, 66]  hk 

Nray NT εk  ρki u(θi , ϑi ), Nray i1

(2.49)

where Nray is the number of the multi-paths between the transmitter and K UEs, εk is the path loss between the transmitter and the kth UE, ρki is the complex gain of the kth UE over the ith multi-path, θi and ϑi are the azimuth and elevation angle of the ith multi-path over the antenna array at the transmitter, respectively. The u(θi , ϑi ) is the response vector of transmitter antenna array with the azimuth θi and elevation angle ϑi . By assuming a uniform planar antenna array for the sake of simplicity, the response vector of transmitter antenna array with the azimuth θi and elevation angle ϑi is expressed as [10] 2π 1 d(msin(θi )sin(ϑi ) + n cos(ϑi )), u(θi , ϑi )  √ [1, . . . , exp j λ NT 2π . . . , exp j(NT − 1) d((M − 1)sin(θi ) sin(ϑi ) + (N − 1) cos(ϑi ))]T , (2.50) λ where d is the distance between adjacent antennas, λ is the carrier wavelength, M and N are the number of rows and columns of the transmitter antenna array, respectively. m and n represent the mth and nth antenna corresponding to the transmitter antenna array (0 ≤ m < M, 0 ≤ n < N ), and T is the transposition operation over the vector.

60

2.3.2.2

2 Energy Efficiency of 5G Wireless Communications

Power Model

Since the massive traffic data needs to be computed at the baseband unit and RF transmission systems, the computation power cannot be ignored for multi-user massive MIMO communication systems. Based on results in [67–69], we express the total power at the transmitter as Ptotal  Pcommun + PCMPT + Pfix ,

(2.51)

where Pcommun is the communication power, PCMPT is the computation power and Pfix is the fixed power at the transmitter of multi-user massive MIMO communication systems. In general, the fixed power Pfix includes the cooling power, losses incurred by direct-current to direct-current (DC–DC) power supply and the mains power supply. The communication power of multi-user massive MIMO communication systems is consumed by the power amplifiers (PAs) and RF chains, which is extended by Pcommun  PPA + PRF ,

(2.52)

where PPA is the power consumed by PAs and is calculated by PPA 

K  1  BRF bBB,k 2 , F α k1

(2.53)

where α is the efficiency factor of PAs, F represent the Frobenius-norm. The power consumed at RF chains is expressed by [70] PRF  NRF Pone_RF ,

(2.54)

where Pone_RF is the power consumed of an RF chain. Substitute (2.53) and (2.54) into (2.52), the communication power of multi-user massive MIMO communication systems is expressed by Pcommun 

K  1  BRF bBB,k 2 + NRF Pone_RF . α k1

(2.55)

The computation power of multi-user massive MIMO communication systems is consumed by wireless channel estimation, channel coding, linear processing at the baseband units and RF transmission systems, and processing to derive the precoding matrix, which is expressed by [10] PCMPT  PCE + PCD + PLP + Pcomplex ,

(2.56)

2.3 Energy Efficient Optimization with RF Chains

61

where PCE is the power consumed by wireless channel estimation, PCD is the power consumed by channel coding and PLP is the power consumed by linear processing at baseband units and RF transmission systems, Pcomplex is the power consumed by our proposed algorithm to generate the precoding matrix. To avoid explicit estimation of the channel, channel estimation is done by beam training. For simplicity, the estimation power is obtained as a product of the number of subcarriers (N) times the number of paths of each subcarrier (Nray ), times the estimated power of a subcarrier in a single path. For the latter

2 term, the channel  is assumed logκ N 1 2 2 (κ −1) logκ N −2 to be frequency-flat and can be expressed as κ γ¯ s1 δ G BS (s) [54], where κ is the number of BS precoding vectors used in each training stage. N is the number of discrete points taken from the angle of departure (AOD) quantization, γ¯ is the average channel SNR, δ is the probability of estimation error. G BS (s)  NT Cs2 is the beamforming gain at stage s, Cs is a normalization constant. So the channel estimation power for OFDM systems is derived as PCE

  log κ N  2 (κ 2 − 1) logκ N 1 −2 .  K Nray κ BS (s) γ¯ δ G s1 2

(2.57)

Without loss of generality, the power of channel coding is assumed to be proportional to the available sum rate of a multi-user massive MIMO communication system [10]. Therefore, the power of channel coding is expressed by PCD  PCOD

K 

Rk ,

(2.58)

k1

where PCOD is the efficiency factor of channel coding, i.e., measured in Watt per bit per second. We assume that the power of linear processing in multi-user massive MIMO communication systems is limited to the power consumed for precoding at both baseband units and RF transmission systems. Under these conditions, the power of linear processing can be extended as PLP  PLP_BB + PLP_RF ,

(2.59)

where PLP_BB is the power consumed for the precoding at the baseband units and PLP_RF is the power consumed for the precoding within the RF transmission systems. Regardless of the CSI or precoding algorithm, the former, which is caused by the product of the signal vector times precoding matrix, can be expressed as: , where ϕ is the number of floating-point computations in one basePLP_BB  ϕνLPCD TR band precoding operation, νPCD is the number of baseband precodings per second, and L TR is the computation efficiency of the transmitter. One baseband precoding operation is assumed to handle K symbols. Moreover, one symbol is configured to

62

2 Energy Efficiency of 5G Wireless Communications

contain  bits. To satisfy the available sum rate Rsum at the transmitter, then the number of baseband precoding operations per second is expressed as νPCD  RKsum . At the baseband unit, ϕ can be calculated by ϕ  2NRF K [29]. Based on (2.47), the power of linear processing is calculated by PLP_BB  

2

K k1

Rk NRF

L T R K 2NRF  L T R



H H BRF hk hkH BRF bBB,k bBB,k

W log2 1 +  K

i1,ik

k1

H H hkH BRF bBB,i bBB,i BRF hk + σn2

. (2.60)

Since the precoding of RF transmission system is performed by phase shifters, its power consumption is calculated by PLP_RF  Nshifter Pshifter ,

(2.61)

where Nshifter is the number of phase shifters and Pshifter is the power of a phase shifter. Substitute (2.60) and (2.61) into (2.59), the power of linear processing is calculated by PLP 

2

K k1

Rk NRF

L TR

+ NT Pshifter .

(2.62)

The power to run the precoding algorithm can be calculated by Pcomplex 

Ccmplx , L TR

(2.63)

where  denotes the complexity of the proposed algorithm, L TR denotes the transmitter efficiency and Ccmplx denotes a constant factor. Substitute (2.57), (2.59), (2.62) and (2.63) into (2.56), and the computation power of multi-user massive MIMO communication systems is derived by PCMPT  PCOD

K  k1

+

Rk +

2

K k1

Rk NRF

L TR

+ NT Pshifter

  log κ N  Ccmplx 2 (κ 2 − 1) logκ N 1 −2 . + K Nray κ 2 BS L TR γ¯ δ G (s) s1

(2.64)

Furthermore, substituting (2.64) and (2.55) into (2.51), the total power of the transmitter is given by

2.3 Energy Efficient Optimization with RF Chains

Ptotal 

63

K  1  BRF bBB,k 2 + NRF Pone_RF α k1

  log κ N  2 (κ 2 − 1) logκ N 1 −2 BS (s) γ¯ δ G s1 K K  Ccmplx 2 k1 Rk NRF + PCOD Rk + + NT Pshifter + + Pfix . L TR L TR k1

+ K Nray κ 2

(2.65)

2.3.3 Problem Formulation 2.3.3.1

Energy Efficiency

Considering computation and communication power consumption, next we focus on optimizing the energy efficiency of multi-user massive MIMO communication systems for partially-connected structures by optimizing the hybrid precoding matrices of baseband and RF systems. This optimization problem is formed by Rsum , Ptotal BBB 0 /  diag m1 , . . . , m NRF ,

maximize

BRF

∈C NT ×NRF ,

s.t. BRF

∈C NRF ×K

ηEE 

BRF BBB 2F ≤ Pmax ,

(2.66)

where ηEE is the energy efficiency, and Pmax is the maximum transmission power. BRF BBB 2F ≤ Pmax is the maximum transmission power constraint. Since RF precoding is performed by phase shifters, only the signal phases change. For the partiallyconnected structure of RF transmission systems, the element amplitude of complex vector: mi is fixed as 1. We should point out that when (2.48) and (2.65) are substituted into (2.66), the optimization problem of energy efficiency is a non-concave optimization problem.

2.3.3.2

Cost Efficiency

Energy efficiency is an important indicator for service providers. For telecommunication equipment providers, the cost efficiency is another important indicator impacting their design strategies. To evaluate the benefits of the partially-connected structure in RF transmission systems, the cost efficiency of the multi-user massive MIMO communication systems is defined by ηcost 

Rsum , Ctotal

(2.67)

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2 Energy Efficiency of 5G Wireless Communications

where Ctotal is the total cost, which is comprised of power consumption cost: Cpower and the hardware cost: Chardware in communication systems. Without loss of generality, the total cost is calculated by Ctotal  Chardware + Cpower ,

(2.68)

Cpower  βpower Ptotal ,

(2.69)

Chardware  βT NT + βshifter Nshifter + βRF NRF + βBB ,

(2.70)

where βpower is the power rate, βT is the cost coefficient per antenna, βT is the cost coefficient per phase shifter, βRF is the cost coefficient per RF chain and βBB is the cost efficient per baseband unit.

2.3.4 Hybrid Precoding Design for the Partially-Connected Structure Taking into consideration the complexity and non-concave properties of the optimization problem in (2.66), it is difficult to directly solve the baseband and RF precoding matrices. Therefore, we first derive the upper bound of the energy efficiency and then propose a suboptimal solution with joint optimized baseband and RF precoding matrices that can approach the upper bound.

2.3.4.1

Upper Bound of Energy Efficiency

To derive the upper bound of energy efficiency, the constraints of energy efficiency optimization in (2.66) are relaxed. Moreover, to simplify derivations, the product of the baseband precoding matrix BBB and the RF precoding matrix BRF is replaced by the fully-digital precoding matrix: B ∈ C NT ×K , i.e., B  BRF BBB , where the kth column of B is denoted as bk which is the baseband precoding vector for the kth UE. Theorem 1 (Upper bound of energy efficiency) When the joint precoding matrix B is a stationary matrix and the value of B satisfies the following result: φk−1 ψk bk − bk  0, k  1, 2 . . . K ,

(2.71)

with   K K  hiH bi biH hi 2 ¯ 2 ¯ H · hi hi Ri I NTx + φk  PW α i1 ln 2 (δi )2 + δi hiH bi biH hi i1,ik

(2.72)

2.3 Energy Efficient Optimization with RF Chains

65



  K  hiH bi biH hi 4 H ¯ ⎣ ψk  P · · hi hi W ln 2 i1,ik (δi )2 + δi hiH bi biH hi * K    W hk hkH 2 2NRF ¯ + · P + R +  i COD ln 2 Kj1 hkH b j b Hj hk + σn2 L TR i1 ) W hk hkH 2 · K ln 2 j1 hkH b j b Hj hk + σn2  ⎤ K  h Hj b j b Hj h j 2 W − · h j h Hj ⎦  ln 2 j1, jk δ j 2 + δ j h H b j b H h j j j δi 

K 

hiH b j b Hj hi + σn2 ,

(2.73)

(2.74)

j1, ji

the upper bound of energy efficiency is achieved for multi-user massive MIMO communication systems. Proof When the product of the baseband precoding matrix BBB and the RF precoding matrix BRF is replaced by the joint precoding matrix B ∈ C NT ×K , a relaxed optimization problem is formulated as follows; maximize η¯ EE  B∈C NT ×K

Rsum Ptotal

,

(2.75)

with Rsum 

K  k1

 W log2 1 +  K

hkH bk bkH hk

i1,ik

hkH bi biH hk + σn2

,

(2.76)

1 B2 + NT Pshifter + NRF Pone_RF α   log κ N  2 (κ 2 − 1) logκ N 1 −2 + K Nray κ 2 BS γ¯ δ G (s) s1  K  hkH bk bkH hk + PCOD W log2 1 +  K H H 2 i1,ik hk bi bi hk + σn k1  K Ccmplx hkH bk bkH hk 2NRF  W log2 1 +  K + Pfix . + + H H 2 L TR k1 L TR i1,ik hk bi bi hk + σn

Ptotal 

(2.77)

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2 Energy Efficiency of 5G Wireless Communications

Based on differential calculus, the solution of the partial derivative of η¯ EE is zero if the value of η¯ EE is an extremum. Therefore, the partial derivative of η¯ EE is expressed as

 ∇ η¯ EE (B)  (η¯ EE )b1 (B), (η¯ EE )b2 (B), . . . , (η¯ EE )b K (B) ,

(2.78)

with (η¯ EE )b K (B) 

ψk bk − φk bk P

2

, k  1, 2 . . . K .

(2.79)

Let ∇ η¯ EE (B)  0, then (2.71), (2.72), (2.73) and (2.74) can be derived. When the value of B satisfies (2.71), the value of η¯ EE is an extremum point. Denoting superscript (i) as the ith iteration, since φk(i) is a Hermitian symmetric positive matrix, φk(i) can be extended as φk(i)  ZZ H , where Z is a symmetric positive  −1 definite matrix. If X  φk(i) ψk(i) b(i) k , we have the following result:  H

(η¯ EE )b(i) (B(i) ) X − b(i)  k k H

2  (i)  H −1 (i) (i) (i) −1 (i) b Z Z b(i) ψ − ψ ψ − ψ k k k k k k . P¯ 2

(2.80)

Since the right expression of (2.80) can be formulated as A T A, the left expression of (2.80) is a Hermitian symmetric positive semidefinite matrix. Hence, ≥ 0 is satisfied for all values of b(i) (η¯ EE )b(i) (B(i) ) H X − b(i) k k . By starting from k

any b(i) ¯ EE (B(i) ) is a non-decreasing function. When a fullyk and moving to X, η digital precoding matrix is configured as a stationary point, the result of [25] proves that the energy efficiency optimization function of MIMO systems can be converted to an upper bound. When the fully-digital precoding matrix b is assumed as a stationary point (based on the result of [32]), the upper bounds of η¯ EE (B(i) ) can be computed. Consequently, η¯ EE (B(i) ) is a convergent function and the upper bound of energy efficiency is achieved for multi-user massive MIMO communication systems. Algorithm 6 is developed to obtain the optimized fully-digital precoding matrix Bopt . Algorithm 6: Upper Bound of Hybrid Precoding Design. Input: K , NT , NRF Output: Bopt i  0, initialize B(0) with random complex value repeat compute φk(i) , ψk(i) , k  1 . . . K based on (2.72), (2.73) (2.74)

2.3 Energy Efficient Optimization with RF Chains

67

for μ  0 :  : 1 // μ is the step length,  is the step length interval. for k  1:K

temp _ b(k

i +1) ( μ )

−1

i i i (μ ) i (μ ) = μ ⎡⎣φk( ) ⎤⎦ ⎡⎣ψ k( )b(k ) ⎤⎦ + (1 − μ )b(k )

end for k end for μ (i+1)(μ) as column to form matrix temp_B(i+1)(μ) use temp_bk  the kth (i+1)(μ) and let B(i+1)  temp_B(i+1)(μ) find the highest η¯ EE temp_B i i +1 until   (i+1) B − B(i) F ≤ ε1 , // ε1 is the stopping criterion Bopt  B(i+1)

2.3.4.2

Hybrid Precoding Matrix Design

When the product of hybrid precoding matrices BRF BBB approaches to the optimized fully-digital precoding matrix Bopt , the energy efficiency ηEE will approach the upper bound of energy efficiency in multi-user massive MIMO communication systems. opt opt Therefore, the optimized baseband and RF precoding matrices, i.e., BBB and BRF can be solved by minimizing the Euclidean distance between BRF BBB and Bopt [52, 62, 66], which is formulated by  opt  B − BRF BBB  F BBB 0 /  diag m1 , . . . , m NRF ,

minimize

BRF

∈C NT ×NRF ,

s.t. BRF

∈C NRF ×K

BRF BBB 2F ≤ Pmax .

(2.81)

To solve the optimized baseband and RF precoding matrices, an alternating minimization method is adopted in this section [61, 71, 72]. Based on the principle of alternating minimization and without loss of generality, we first fix the RF precoding matrix BRF and derive a solution of baseband matrix BBB . In this case, (2.81) is transferred as   minimize Bopt − BRF BBB F

BBB ∈C NRF ×K

s.t. BRF BBB 2F ≤ Pmax .

(2.82)

Based on the result in [61], (2.82) is a nonconvex quadratically constrained opt quadratic program (QCQP). Let xc  vec(BBB ), bc  vec(Bopt ) and ζc  I K ⊗BRF , opt where xc , bc and ζc are complex vectors, and vec() denotes vectorization. To transfer (2.82) into a real QCQP, let t 2  1 and

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2 Energy Efficiency of 5G Wireless Communications



⎤ real(xc ) x  ⎣ imag(xc ) ⎦, t  opt  real(bc ) , bopt  opt imag(bc )   real(ζc ) ζ  . imag(ζc )

(2.83)

(2.84) (2.85)

As a consequence, (2.82) becomes a real QCQP as follows minimize x H Tx 2 s.t. x(n)  1,

x H 1 x ≤

Pmax NRF , NT

(2.86)

with  T

 ζ H ζ −ζ H bopt , −bopt H ζ bopt H bopt

n  2K NRF + 1,

(2.87) (2.88)

where x(n) denotes the nth value of x. Considering x H Tx  Tr(xx H ) and let X  xx H , (2.86) is simplified as minimize Tr(TX) n×n X∈R

s.t. Tr( 2 X)  1, Pmax NRF , Tr( 1 X) ≤ NT X ≥ 0, rank(X)  1,

(2.89)

with 

 In−1 0 , 0 0   0n−1 0

2  . 0 1

1 

(2.90) (2.91)

Except for the constraint condition rank(X)  1, the objective function and other constraint conditions in (2.89) are convex. To obtain an approximate solution of

2.3 Energy Efficient Optimization with RF Chains

69

(2.89), we first relax the constraint condition rank(X)  1 [73]. Thus, (2.89) is transferred into a semidefinite relaxation program (SDR) as follows minimize Tr(TX) n×n X∈R

s.t. Tr( 2 X)  1, Pmax NRF , Tr( 1 X) ≤ NT X ≥ 0.

(2.92)

When a standard convex algorithm, such as the interior point algorithm [74] is performed for (2.92), an optimized solution Xopt is solved. Since the constraint condition: rank(X)  1 is removed in (2.92), the solution of (2.92), i.e., Tr(TXopt ) is the lower bound of (2.91). If Xopt does not satisfy the constraint condition: rank(X)  1, Xopt can not be decomposed as the product of two vectors, i.e., xx H . Consequently, the baseband matrix: BBB cannot be solved from Xopt . To solve this issue, the approximate method as Xopt  in  [74, 75] is adopted to obtain the vector x. Firstly, Xopt is decomposed  H opt U U , where every column of U is the eigenvector of X . is a diagonal matrix 1 and its diagonal entries are the eigenvalues of Xopt . Secondly, let x  U 2 v, where v is a random vector and each element is a complex circularly symmetric uncorrelated Gaussian random variable with zero mean and variance of 1. Therefore, a series of approximate solutions with E[xx H ]  Xopt are obtained. To simplify the calculation, we select the random vector x which satisfies BRF BBB 2F ≤ Pmax . Based on the selected vector x, an approximate solution of (2.82), i.e., the corresponding baseband precoding matrix, is obtained. Since the RF precoding matrix only changes the signal phase but not the signal amplitude, the power constraint, i.e., BRF BBB 2F ≤ Pmax can be ignored in our calculations. In this case, (2.81) is transferred as   minimize Bopt − BRF BBB F 0 / s.t. BRF  diag m1 , . . . , m NRF .

(2.93)

BRF BBB can be regarded as the ith column of BRF phasing on the ith row of BBB and then the sum of all results, i.e.,

BRF BBB

⎤ ⎡ NRF 

 bBB(1,:)  bRF(:,1) , . . . , bRF(:,NRF ) ⎣ . . . ⎦  bRF(:,i) bBB(i,:) . i1 bBB(NRF ,:)

(2.94)

Moreover, we assume that the value range of the element in the ith column of BRF to be continuous. Thus, the following result is derived

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2 Energy Efficiency of 5G Wireless Communications

  minimize Bopt − BRF BBB F ⇔ phase(Bopt )  phase(BRF BBB ),

BRF ∈C NT ×NRF

(2.95)

where phase(Bopt ) is the operation to get phase of each element of matrix Bopt . So 2 1 NRF opt H , (2.96) phase(BRF(i, j) )  phase(B(i,:) BBB( j,:) ), 1 ≤ i ≤ NT , j  i · NT where  denotes rounding up to an integer. Based on (2.82) and (2.93), an iteration algorithm of the alternating minimization method is developed by Algorithm 7. Algorithm 7: Baseband and RF Precoding Matrices. Input: Bopt opt opt Output: BBB , BRF i  0, initialize B(i) RF with random phases repeat i i +1 (i) fix B(i−1) RF , calculate BBB (i) fix B(i) BB , calculate BRF until B opt − B RF (i ) B BB( i ) < ε 2 // ε 2 is the stopping criterion F opt

opt

(i) BBB  B(i) BB , BRF  BRF

Considering the multi-user massive MIMO communication system with partiallyconnected structure, an optimized hybrid precoding algorithm to approach the upper bound of the energy efficiency is developed by the oPtimal Hybrid precOding with computation and commuNication powEr (PHONE) algorithm. Algorithm 8: oPtimized Hybrid precOding with computation and commuNication powEr (PHONE). Input: K , NT , NRF opt opt Output: BBB , BRF compute Bopt based on Algorithm 6 opt opt compute BBB and BRF based on Algorithm 7 Based on the computational complexity of matrix calculation and the iterative algorithms in [73] and [76], the computation complexity of Algorithm 6 is estimated as O(K 2 + N T3 x ) floating point operations (flops); the computation complexity of Algorithm 7 is estimated as O(K 3.5 ) flops; Combining Algorithms 6 and 7, the

2.3 Energy Efficient Optimization with RF Chains

71

computational complexity of the PHONE algorithm, i.e.  is estimated as O(N T3 x + K 3.5 ) flops.

2.3.5 Simulation Results Energy efficiency optimization solutions with respect to the number of transmission antennas, RF chains, and users are simulated for multi-user massive MIMO communication systems in this section. Without loss of generality, the number of active UEs is configured as 5 and the number of RF chains as 5. Other default values of simulation parameters are listed in Table 2.2. To analyze the proposed PHONE algorithm with a partially-connected structure, the orthogonal matching pursuit (OMP) algorithm [51] with fully-connected and partially-connected structures are simulated for performance comparisons of multi-user massive MIMO communication systems. Figure 2.9 illustrates the computation and communication power with respect to the number of transmission antennas for fully-connected and partially-connected structures. In this figure, the “Fully-connected structure” corresponds to the OMP algorithm which is based on the spatial sparse precoding algorithm [51], and “Partially-connected structure” represents the proposed PHONE algorithm. Figure 2.9a, b show that the communication power practically remains flat, whereas the computation power increases. Comparing the results in Fig. 2.9a, b, the increment of computation power with fully-connected structure is larger than the increment of computation power with partially-connected structure. Moreover, Fig. 2.9c, d

Table 2.2 Default values of simulation parameters Parameters

Value

Parameters

Value

τ

1

W

200 kHz

L TR

12.8 × 109

Pone_RF

12,900 mWatt

α

0.38

βT

188

Nray

20

βshifter

1800

Pnoise

−174 dBm/Hz

βRF

7800

Pfix

1W

βBB

6800

Pmax

33 dBm

βpower

0.9

PCOD

0.1 × 10−9 W/bit/s

Ccmplx

1

Pshifter

88 mW

κ

2

N

64

δ

0.1

K

5

flops/Watt

72

2 Energy Efficiency of 5G Wireless Communications

(a)

(b)

(c)

(d)

Fig. 2.9 Computation and communication power with respect to the number of transmission antennas considering fully-connected and partially-connected structures

indicate that the proposed PHONE algorithm with the partially-connected structure outperforms the OMP algorithm with the fully-connected structure due to the saving of communication and computation power in multi-user massive MIMO communication systems. Figure 2.10a depicts the energy efficiency with respect to the number of RF chains. As can be observed, the energy efficiency of the PHONE and OMP algorithms decreases with the increase of the number of RF chains. Moreover, the energy efficiency of the proposed PHONE algorithm is larger compared to the OMP algorithm. Figure 2.10b illustrates the power saving ratio with respect to the number of RF chains compared with the OMP algorithm with fully-connected and partiallyconnected structures in massive MIMO systems. Power saving is defined by the difference of unit rate power consumption. The results in Fig. 2.10b show that the power saving ratio of the PHONE algorithm increases with a greater number of RF chains. For instance, when the number of RF chains is 14, the power saving ratio of massive MIMO system is 75.59% and 38.38% compared to the OMP algorithm adopting partially-connected and fully-connected structures, respectively. Figure 2.11 describes the energy efficiency with respect to the number of transmission antennas. When the computation power is considered for massive MIMO systems, the energy efficiency of the PHONE and OMP algorithms decreases with increasing the number of transmission antennas. When the number of transmit antennas is fixed, the energy efficiency of the PHONE algorithm is larger than the energy efficiency of OMP algorithms with fully-connected and partially-connected structures in massive MIMO systems.

10

12

14

Number of RF chains

0

8

1

20

30

40

50

60

70

80

90

10

6

PHONE OMP with fully-connected structure OMP with partially-connected structure

100

2

3

4

5

6

7

8

9

10

11

(a)

Power saving ratio (%)

(b)

6

10

Number of RF chains

8

12

OMP with fully-connected structure OMP with partially-connected structure

14

Fig. 2.10 a Comparing the PHONE and OMP algorithms in terms of energy efficiency with respect to the number of RF chains, b power saving ratio with respect to the number of RF chains

Energy efficiency (Mbits/J)

12

2.3 Energy Efficient Optimization with RF Chains 73

74

2 Energy Efficiency of 5G Wireless Communications

22 PHONE OMP with fully-connected structure OMP with partially-connected structure

20

Energy efficiency (Mbits/J)

18 16 14 12 10 8 6 4 20

40

60

100 80 120 140 Number of antennas

160

180

200

Fig. 2.11 Comparing the PHONE and OMP algorithms in terms of energy efficiency with respect to the number of antennas

Figure 2.12 shows the energy efficiency with respect to the number of users. When the computation power is considered for massive MIMO systems, the energy efficiency of the PHONE and OMP algorithms decreases with increasing the number of users. When the number of users is fixed, the energy efficiency of the PHONE algorithm is larger than the energy efficiency of OMP algorithms with fully-connected and partially-connected structures in massive MIMO systems. The spectral efficiency with respect to the number of RF chains is analyzed in Fig. 2.13. When the computation power is considered for massive MIMO systems, the spectral efficiency of the PHONE algorithm with partially-connected structure and the OMP algorithm with fully-connected structure increases with the increasing of the number of RF chains. However, the spectral efficiency of the OMP algorithm with partially-connected structure decreases with increasing the number of RF chains. When the number of RF chains is fixed, the spectral efficiency of the PHONE algorithm is less than the spectral efficiency of OMP algorithm with fully-connected structure. When the number of RF chains is less than or equal to 6, the spectral efficiency of the PHONE algorithm is less than the spectral efficiency of OMP algorithm with partially-connected structure. When the number of RF chains is larger than 6, the spectral efficiency of the PHONE algorithm is larger than the spectral efficiency of OMP algorithm with partially-connected structure.

2.3 Energy Efficient Optimization with RF Chains

75

15

Energy efficiency (Mbits/J)

PHONE OMP with fully-connected structure OMP with partially-connected structure

10

5

0 2

3

2.5

3.5 Number of users

4.5

4

5

Fig. 2.12 Comparing the PHONE and OMP algorithms in terms of energy efficiency with respect to the number of users 10

Spectral efficiency (bits/s/Hz)

9

PHONE OMP with fully-connected structure OMP with partially-connected structure

8

7

6

5

4

3 5

6

7

8

9 10 11 Number of RF chains

12

13

14

Fig. 2.13 Comparing the PHONE and OMP algorithms in terms of spectral efficiency with respect to the number of RF chains

76

2 Energy Efficiency of 5G Wireless Communications -5

1.4

x 10

1.2

PHONE OMP with fully-connected structure OMP with partially-connected structure

Cost efficency

1

0.8

0.6

0.4

0.2

0 5

6

7

8

9 10 11 Number of RF chains

12

13

14

Fig. 2.14 Comparing the PHONE and OMP algorithms in terms of cost efficiency with respect to the number of RF chains

The cost efficiency of multi-user massive MIMO communication systems, with respect to the number of RF chains, is compared in Fig. 2.14. As can be observed, the cost efficiency of the PHONE algorithm improves with the increasing number of transmission antennas; whereas with OMP, it decreases with the increasing number of transmission antennas. When the number of RF chains is fixed, the cost efficiency of the PHONE algorithm outperforms the OMP algorithms with fully and partially connected structures. For instance, compared with the fully and partially connected structures of OMP algorithms, the maximum cost efficiency is improved by 6.78 and 17.97 times, respectively. The cost efficiency with respect to the number of transmission antennas is compared in Fig. 2.15. The cost efficiency of multi-user massive MIMO communication systems decreases when the number of transmission antennas for PHONE and OMP algorithms increases. When the number of transmission antennas is fixed, the cost efficiency of the PHONE algorithm is larger than the OMP algorithms with fullyconnected and partially-connected structures.

2.3.6 Conclusions With massive traffic processing, the computation power is emerging as an important part of the energy consumption for 5G massive MIMO communication systems. When computation power is considered, the results of this section reveal that

2.3 Energy Efficient Optimization with RF Chains 1

x 10

77

-4

PHONE OMP with fully-connected structure OMP with partially-connected structure

0.9 0.8

Cost efficency

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20

40

60

80

100 120 140 Number of antennas

160

180

200

Fig. 2.15 Comparing the PHONE and OMP algorithms in terms of cost efficiency with respect to the number of antennas

the energy efficiency of massive MIMO systems decreases when the number of antennas and RF chains increases, which is different than conventional energy efficiency analysis of massive MIMO systems, i.e., only communication power is considered. Faced with this challenge, an optimized solution of energy efficiency that considers communication and computation power is proposed for multi-user massive MIMO communication systems with partially-connected structures. First, an upper bound on energy efficiency of multi-user massive MIMO communication systems is derived. Secondly, the optimized baseband and RF precoding matrices are derived to approach the upper bound on energy efficiency of massive MIMO systems with partially-connected structures. Furthermore, a PHONE algorithm is developed to optimize the performance of multi-user massive MIMO communication system with partially-connected structures. Numerical results indicate that the proposed PHONE algorithm outperforms the OMP algorithm in energy and cost efficiency and the maximum power saving is achieved by 76.59% and 38.38% for multi-user massive MIMO communication systems with partially connected and fully-connected structures, respectively. In future work, we plan to explore the trade-off between the energy and cost efficiency for multi-user massive MIMO communication systems with partially connected structures.

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2 Energy Efficiency of 5G Wireless Communications

2.4 Energy Efficient Power Control Scheme 2.4.1 Related Work With the development of wireless transmission technologies, multi-input multioutput (MIMO) antenna technology is widely used to improve the capacity of wireless communication systems. Moreover, numerous energy efficiency models have been investigated for MIMO communication systems in [25, 77–84]. To maximize the energy efficiency of MIMO communication systems over time varying channels, the impact of line-of-sight, out-of-cell interferers and the antenna correlation was discussed for downlink channels in [77]. An optimal power control algorithm was proposed for the generalized energy-efficiency proportional fair metric in a multiuser MIMO communication system [78]. A tight upper bound of the energy efficiency with a spectrum efficiency constraint was derived for a virtual-MIMO communication system which has one destination and one relay using the compress-and-forward (CF) cooperation scheme [79]. Based on the proposed energy efficiency upper bound, the optimal power and bandwidth allocation have been derived for maximizing the energy efficiency of MIMO communication systems. In [80], an energy efficient adaptive transmission scheme was proposed for MIMO beamforming communication systems with orthogonal space-time block coding (OSTBC) with imperfect channel state information (CSI) at transmitters. An algorithm that jointly considering the transmit power, power allocation among streams and beamforming matrices was developed to maximize the energy efficiency of MIMO communication systems with interference channels [25]. Due to the trade-off between the traffic rate and the hardware power consumption, an antenna selection algorithm was developed in MIMO communication systems [81]. By jointly choosing the transmission power and precoding vector among codebooks, a radio resource optimization scheme was proposed to improve the spectrum and energy efficiency of MIMO communication systems with user fairness constraints [82]. Assuming that channel state information is known to the transmitters, an optimal power control scheme was proposed for maximizing the energy efficiency of a base station (BS) using multiple antennas [83]. Considering distributed transmitter systems employing a zero-forcing based multiuser MIMO precoding, a heuristic power control method was proposed to improve the energy efficiency of MIMO communication systems under constraints on the peruser target rate and the per-antenna instantaneous transmit power [84]. However, the above studies concerning the energy efficiency of MIMO communication systems have been limited to finite numbers of interfering transmitters. Many studies indicated that improving the energy efficiency of cellular networks is a critical problem for the future of the telecommunication industry [42, 85–92]. The purpose of traditional cellular wireless communications always is higher throughput for the user and higher capacity for the service provider, regardless of energy efficiency. Davaslioglu et. al. discussed the specific reasons for inefficiency and potential improvement in the physical layer as well as in higher layers of the communication protocol of cellular networks [85]. Hasan et. al. presented a review of methods of

2.4 Energy Efficient Power Control Scheme

79

improving the energy efficiency of cellular networks, and explored some related topics and challenges, moreover suggested some techniques to make green cellular networks possible [86]. A novel user cooperation scheme termed inter-network cooperation was investigated to improve uplink emission energy efficiency of cellular networks with the help of a short-range communication network [87]. Three typical multi-cell cooperation scenarios, i.e., the energy efficiency coordinated multi-point transmissions, the traffic-intensity-aware and the energy-aware multi-cell cooperation were also discussed for reducing the energy consumption of cellular networks in [88]. The downlink performance evaluation of small cell networks including capacity and energy efficiency was investigated in [89], where BSs and users are modeled as two independent spatial Poisson point processes. In related work on a MIMO cellular network with one single macrocell base station (MBS) and multiple femtocell access points, an opportunistic interference alignment scheme was proposed for reducing the intra and inter tier interference and the energy consumption [90]. Through the deployment of sleeping strategies and small cells, the success probability and energy efficiency were improved for homogeneous macrocell single tier wireless networks and heterogeneous multiple tiers wireless networks in [91]. Using the signal-to-interference-and-noise ratio (SINR) as the function of the user’s location, an analytical model was proposed for calculating the spectrum and energy efficiency of cellular networks with orthogonal frequency division multiplexing access (OFDMA) [92]. Based on single antenna transmission systems, the energy efficiency of random cellular networks with the statistical analysis of traffic load and power consumption was also evaluated in [42]. However, in future 5G mobile communication systems, the energy efficiency is proposed as one of the most important performance indicators. Considering that the 5G network will be a huge multi-layer heterogeneous network simple scenarios such as MIMO communication systems considering finite interfering transmitters in one single cell are so simple that have no ability to accurately evaluate the energy efficiency of complex cellular networks. Moreover, studies of the impact of different power allocation schemes, which is the important influence factor in power consumption evaluation, on the energy efficiency of MIMO random cellular networks are surprisingly rare in the open literature.

2.4.2 System Model Assume that in the infinite plane R2 BSs and mobile stations (MSs) are deployed randomly, of which the locations are approximated to be two independent Poisson point processes [93] with intensities λ M and λ B , which are expressed as / 0  B  {y Bi , i  0, 1, 2, . . .},  M  x M j , j  0, 1, 2, . . . ,

(2.97)

80

2 Energy Efficiency of 5G Wireless Communications

Fig. 2.16 Illustration of PVT random cellular networks

where y Bi and x M j are two-dimensional location coordinates of the ith BS B Si and the jth MS M S j , respectively. Assume that MSs communicate with the closest BS for suffering the minimum path loss in the process of radio propagation. All other BSs in the infinite plane R2 are interfering BSs. The OFDMA scheme is adopted for wireless transmission to avoid the intra-interference in the cell. Thus, we can split the plane R2 into a number of irregular polygons approximately expressing coverage areas of different cells through the Delaunay Ttiangulation method [94]. The illustration of stochastic and irregular topology in Fig. 2.16 is so-called Poisson Voronoi Tessellation (PVT) random network, where each cell is identified by Ci (i  0, 1, 2, . . .). According to Palm theory, one of most important features of PVT random cellular networks is that geometric characteristics of all cells coincide with each other, such that can be viewed as coinciding with a typical PVT cell C0 [95]. Thus, analytical results for a typical PVT cell C0 can reveal properties of the whole PVT random cellular network. Assume that each BS is integrated with Nt transmission antennas and each MS is equipped with Nr receive antennas. In this chapter, our study is focused on the downlinks of cellular wireless communication systems. Without loss of generality, the signal received at an MS M S0 in the typical PVT cell C0 is expressed as y0  H00 x0 +

∞  i1

Hi0 xi + n0

2.4 Energy Efficient Power Control Scheme &

 H00 V0 0 c0 +

81 ∞ 

&

Hi0 Vi i ci + n0 ,

(2.98)

i1

where c0 is Nt dimension desired transmitted symbol satisfying c0H c0  1 from the BS B S0 , ci (i  1, 2, . . .) is the interfering transmitted symbol H satisfying √ ci√ci  13 from the interfering BS B Si (i  1, 2, . . .), i  diag Pi1 , Pi2 , . . . , Pi Nt (i  0, 1, 2, . . .) is Nt × Nt transmit power allo Nt cation vector satisfying j1 Pi j  PT i . The scalars PT 0 and PT i (i  1, 2, . . .) denote the transmission power at the desired BS B S0 and the interfering BS B S0 respectively. Afterwards, the transmitted symbol vector is precoded by an Nt × Nt matrix Vi as xi  Vi i ci (i  0, 1, 2, · · ·); H00 is the Nr × Nt channel matrix between the MS M S0 and the desired BS B S0 , Hi0 (i  1, 2, 3 . . .) is the Nr × Nt channel matrix between the MS M S0 and the interfering BS B Si , the element h 0,k,n (k  1, 2, . . . , Nr ; n  1, 2, . . . , Nt ) of channel matrix H00 and the element h i,k,n (i  1, 2, . . . ; k  1, 2, . . . , Nr ; n  1, 2, . . . , Nt ) of channel matrix Hi0 are independently and identically distributed i.i.d.; n0 is the Nr dimension additive white Gaussian noise (AWGN) vector in the wireless channel, the noise power is equal to N0 . Due to the infinite sum of interferers in (2.98), it is reasonable to assume that the system model of MIMO PVT random cellular networks is an interference limited scenario. Our analysis can also approximate fading correlation scenarios by performing moment matching to simplify to a single Gamma distribution [96]. 4

4

2.4.3 Achievable Rate of MIMO PVT Random Cellular Networks 2.4.3.1

Interference Model

The received signals including the interference signals are assumed to be propagated though independent wireless channels [97]. The shadowing effect is assumed to be follow a log-normal distribution, to which a Gamma fading distribution is an alternative approximately [98] for simplifying calculation. The multi-path fading is assumed to be follow a Nakagami-m distribution which spans via the m parameter the widest range of the amount of fading (from 0 to 2) among all the multi-path distributions [99]. In this case, the wireless channel gain from the nth transmission antenna at the interfering BS B Si to the kth receive antenna at the MS M S0 is expressed as   h i,k,n 2  1 wi,k,n z i,k,n 2 , σ Ri

(2.99)

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2 Energy Efficiency of 5G Wireless Communications

where Ri is the Euclidean distance between the MS M S0 and the interfering BS B Si , σ is the path loss coefficient in radio propagation, wi,k,n is a random variable governed by Gamma distribution, and z i,k,n is a random variable governed by Nakagami-m distribution. Considering that the OFDMA scheme and a relevant interference cancellation scheme [100] are used for intra-cell signals, there is no significant co-channel interference from within one PVT cell [101]. Therefore, the co-channel interference is assumed to be transmitted from all BSs in the infinite plane except for the BS in the typical PVT cell. Assume that the active interfering BSs set is modeled an independent thinning process on the BS Poisson point process, which still form a Poisson point process with intensity λInf [42], and generally satisfies 0  λInf  λ B . Therefore, the interference power aggregated at the MS M S0 is expressed as [102] ∞ Nr ∞   Ii,k   Ii Ii PI  ,  σ σ  Ri Ri Riσ k1 i1 i1 i∈

(2.100)

inf

with Ii 

Nr  

Ii,k ,

(2.101)

k1

where Ii,k is the received interference of the kth antenna of the MS M S0 from the ith BS regardless of the pass loss fading. Considering that every antenna of the MS M S0 will receive multiple interference streams transmitted from Nt antennas of interfering BSs, Ii represents the interference power received by Nr antennas at the MS M S0 , which is further expressed as [102] Ii  PT i

Nt Nr  

Ti,k,n ,

(2.102)

2 Ti,k,n  wi,k,n z i,k,n .

(2.103)

k1 n1

with

Assume that the average power of interference terms that are transmitted from the nth antenna of interfering BS B Si and received at the kth antenna of MS M S0 are approximately equal in the statistical meaning [104]. Thus, the Gamma fading over all sub-channels is simplified as wi,k,n (i  1, 2, . . . ; k  1, 2, . . . , Nr ; n  1, 2, . . . , Nt )  wi . Furthermore, the interference power received by Nr antennas of the MS M S0 is derived by Ii  PT i Hi ,

(2.104)

2.4 Energy Efficient Power Control Scheme

83

with Hi  wi

Nt Nr   z i,k,n 2 .

(2.105)

k1 n1

A channel that experiences the product of both Gamma fading and Nakagami fading follows a K G distribution. Therefore, the probability density function (PDF) of Hi is derived by [98, 105, 106]  $ Nt Nr m+λ  2 2 mλ Nt Nr m+λ−2 mλy  y 2 K λ−Nt Nr m 2 f Hi (y)  (y > 0, i  1, 2, 3 . . .),

(Nt Nr m) (λ)  (2.106) with 

3

2

2 (λ + 1)/λ λ  1/ e(σd B /8.686) − 1 ,

(2.107)

where (·) is a Gamma function, m is a Nakagami shaping factor, K λ−m (·) is the modified Bessel function of the second kind with order λ − m and σd B is the variance of shadowing effect values. Based on Eq. (2.100) and the Campbell theory in [94], the characteristic function of the interference power aggregated at the MS M S0 can be written as   ¨

/ jω PI 0 jωy σ R  PI  E e f I (y)dy Rd R  exp −2π λinf 1−e   5 

ω  1 − φI Rd R , (2.108)  exp −2π λinf Rσ r where f I (y) and φ I (ω) are the PDF and the characteristic function of the total interference power Ii received at the MS M S0 , respectively; E{·} is the expectation operation. Based on the result in [102], the characteristic function  PI represents an alpha stable random process, which can be simply denoted as PI ∼ Stable(α  2/σ, β  1, δ, μ  0), where α and δ are the stability parameter and the scale parameter, respectively. Based on the alpha stable characteristic function expression,  PI can be re-written as 

π α   , (2.109)  PI  exp −δ|ω|α 1 − jβsign(ω) tan 2 with π (2 − α) cos δ  λinf 1−α

 πα 2

/ 0 / 0 E PTαi E Hiα

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2 Energy Efficiency of 5G Wireless Communications

(α  1, i  1, 2, 3 . . .),

(2.110)

/ 0 where E PTαi is the moment of / the0receiving power raised to the power α at the MS M S0 . Based on Eq. (2.106), E Hiα is expressed as / 0 E Hiα 

5∞ 0

 $ Nt Nr m+λ  2 2 mλ Nt Nr m+λ−2 mλy α  y 2 K λ−Nt Nr m 2 y dy,

(Nt Nr m) (λ) 

(2.111)

From the table of integrals in [106], Eq. (2.111) can be written in closed form as / 0 E Hiα 



mλ 

−α

(λ + α) (Nt Nr m + α) (i  0, 1, 2, 3 . . .),

(Nt Nr m) (λ)

(2.112)

Substituting Eq. (2.112) into (2.110), the PDF of the interference power aggregated at the MS M S0 can be written as 1 f PI (y)  2π

2.4.3.2

5+∞  PI ( jω) exp(−2π jωy)dω.

(2.113)

−∞

Achievable Rate Model

Based on the proposed interference model of MIMO random cellular networks in (2.113), the achievable rate at the MS is derived in this section. We assume the network is interference rather than noise limited, due to the infinite sum of interferers in (2.98) [103]. Therefore, the received signal-to-interference ratio (SIR) at the MS M S0 in the typical PVT cell is expressed as 4

H

4

c H  H V H H H00 V0 0 c0 S I R0  0 0 0 00 , PI

(2.114)

where H is the conjugate transpose operation. Furthermore, the achievable rate at the MS M S0 is expressed as )

* H H H00 V0 0 c0 c0H 0H V0 H00 R0  BW log[1 + S I R0 ]  BW log 1 + , PI 4

4

(2.115)

where BW is the bandwidth allocated for the MS M S0 . Transmitters are assumed to obtain the CSI from receivers without delay via uplink feedback channels. Moreover, the MIMO channel is divided into a number of parallel single-input single-output (SISO) channels by the single value decomposition (SVD) method. In this case, the channel matrix H00 in (2.98) can be decomposed as

2.4 Energy Efficient Power Control Scheme

85

H H00  U00 D00 V00 ,

(2.116)

√ √ √ where D00  diag λ1 , λ2 , . . . , λ L is the L × L diagonal matrix, λ1 ≤ λ2 ≤ H H00 , and L  rank(H00 ) is the rank of · · · ≤ λ L , are eigenvalues of the matrix H00 H00 , U00 is the Nr × L unitary matrix, V00 is the L × Nt unitary matrix. Furthermore, assuming that the full matrix V00  V0 , the achievable rate at the MS M S0 is expressed as ) ) * * c0H 0H D200 0 c0 0H D200 0 R0  BW log 1 +  BW log 1 + . (2.117) PI PI 4

2.4.4 Green MIMO Random Cellular Networks Considering that the MS required traffic load will influence the BS transmission power, the energy efficiency of MIMO PVT cellular networks will be related with the traffic rates in MSs. In this section, we discuss this relationship in more detail. Two classical power control schemes are discussed with the proposed model, numerical results show inherent relationships among the energy efficiency, traffic load, and the prevailing channel environment conditions.

2.4.4.1

Basic Energy Efficiency Model

We define the energy efficiency of MIMO PVT cellular networks as the average ratio of traffic load over total power consumption [112] at a BS in a typical PVT cell based on Palm theory [94] / 0  E C0 nat r (P) , (2.118)  EE  P(P)s E{PB S } Joule where C0 is the traffic rate in a typical PVT cell C0 , PB S is the total power consumed at a BS in a typical PVT cell C0 . The total BS power consumption includes both fixed power and dynamic power consumption terms [113]. According to [107], the total power consumption PB S is written following PB S 

PC0 _r eal + Nt Pdyn + Psta , η

(2.119)

where PC0 _r eal is the total BS radio frequency transmission power for all transmit antennas, η is the average efficiency of the BS power amplifiers, Nt is the number of active BS antennas, Pdyn is the RF circuit power for an antenna and Psta is the fixed power consumption in a BS. Moreover, there is a maximal BS transmission power

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2 Energy Efficiency of 5G Wireless Communications

Pmax in practical. In this case, when the required BS transmission power exceeds Pmax the corresponding transmission traffic will / be0 interrupted. Therefore, the average transmission traffic rate is calculated by E C0 FPC0 (Pmax ), where FPC0 (Pmax ) is the probability that the BS transmission power less than Pmax . Furthermore, the energy efficiency of the MIMO PVT random cellular networks is expressed as   / 0 / 0 E C0 FPC0 PC0 _max E C0 FPC0 PC0 _max 0 EE   1 / , (2.120) E{PB S } E PC0 _r eal + N T Pdyn + Psta η 0 / where E PC0 _r eal is the average BS actual transmission power in the typical PVT cell. Many empirical measurement results have demonstrated that the traffic load in both wired and wireless networks, including cellular networks, is self-similar and bursty. Considering the infinite variance characteristic of self-similar distributions, Pareto distributions with infinite variance are proposed for modeling the self-similar traffic in wireless networks [114]. Therefore, the traffic rate ρ(x Mi ) at the MS M Si is assumed to be a Pareto distribution with infinite variance. Traffic rates of all MSs are assumed to be i.i.d. Then, the PDF of traffic rate is expressed by f ρ (x) 

θρminθ , x ≥ ρmin > 0, x θ+1

(2.121)

where θ ∈ (1, 2] is a shape parameter, also known as the tail index. ρmin is minimum possible value of traffic rate that is needed to meet the MS’s quality of service (QoS) requirements. Furthermore, the average traffic rate at a MS is expressed as E{ρ} 

θρmin . θ −1

(2.122)

Based on the results in [97], the average traffic rate for all MSs in a typical PVT cell θρmin is denoted as E{ C0 }  λλBM(θ−1) . As a consequence, the energy efficiency of MIMO PVT random cellular networks is derived by EE 

2.4.4.2

λ M θρmin F λ B (θ−1) PC0



PC0 _max 0 / . 1 E PC0 _r eal + N T Pdyn + Psta η

(2.123)

Energy Efficiency of Average Power Allocation Scheme

The average power allocation scheme is a simple antenna power control scheme which has been widely used for practical MIMO wireless communications systems. The maximum ratio transmission/maximum ratio combining (MRT/MRC) methods [115] are assumed to be adopted in MIMO PVT random cellular networks, the

2.4 Energy Efficient Power Control Scheme

87

achievable rate with an average power allocation scheme satisfying 0  NP0t I (Nt ) at the MS in the typical PVT cell is derived as ) * 0H D200 0 R0  BW log 1 + PI ) ) * * P0 P0 H λ (H00 H00 ) λ Nt max Nt rank(H00 )  BW log 1 +  BW log 1 + PI PI ⎡ 2 ⎤ ) * P0 P0  Nr  Nt 2 H  h 00 0,k,n F k1 n1 N N ⎦  BW log 1 + t  BW log⎣1 + t PI PI ⎡ 2 ⎤ ) P0  Nr  Nt P0 H0 * 1 k1 n1 R0σ w0 z 0,k,n Nt Nt Rσ0 ⎣ ⎦ ≈ BW log 1 +  BW log 1 + , (2.124) PI PI H H where λmax

(H00 H00 ) is the maximum eigenvalue operation for the matrix H00 H00 . Let τ  NP0t RHσ0 /PI , the relationship between achievable rate and traffic rate of M S0 0 is regarded as

BW log2 (1 + τ )  ρ(x M0 ).

(2.125)

Based on the PDF of traffic rate, the PDF of τ is derived as f τ (z) 

−θ  θρminθ BW log2 (1 + z) ln 2 · (1 + z)

−θ−1 

z > z0  2ρmin /BW − 1 .

(2.126)

Consider that the MS communicates with the closest BS in a PVT cell. Furthermore, the probability of the Euclidean distance between an MS and the ith near BS can be expressed as  i−1 λB π R2 2 e−λ B π R , Pr(i − 1 BSs in a circle area with radius R)  (i − 1)!

(2.127)

where Pr(·) is the probability operation. Thus, the PDF of the distance R is derived by f R0 (R) 

d Pr{R0 > R} d Pr{R0  R} − d R dR   2 0 λ B π R 2 ) e−λ B π R ( d

−

0!

dR

 2π λ B Re−πλ B R . 2

(2.128)

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2 Energy Efficiency of 5G Wireless Communications

where Pr{R0 > R} is the probability that there is no BS in the circular area with the center x M0 and the radius. Considering the path loss effect on the distance R, the corresponding PDF is derived as f R0σ (R) 

1 2 −1 2/σ R σ · 2π λ B e−πλ B R . σ

(2.129)

Furthermore, the downlink transmission power P0 between the BS B S0 and the MS M S0 is expressed as P0  PI ·

Nt · R0σ · τ . H0

(2.130)

The characteristic function of P0 is derived by 5 φ P0 (ω)  φ PI (ωx) f Nt R0σ τ (x)d x H0

x

5 ¨  x

¨ 

y,z

  xy yφ PI (ωx) f H0 (y) d xd ydz f τ (z) f R0σ z Nt z Nt π λB 2 σ

y,z

G(ω)z y

−2 σ

+ π λB

f H0 (y) f τ (z)dydz,

(2.131)

with  π , G(ω)  δ|ω|2/σ 1 − j · sign(ω) · tan σ

(2.132)

2  Nt  Nr where f H0 (y) is the PDF of the channel variable H0  w0 n1 k1 z 0,k,n . In this section, the channel variable Hi is i.i.d. for all channels. Based on (2.106), the function f H0 (y) is expressed by  $ Nt Nr m+λ  2 2 mλ Nt Nr m+λ−2 mλy  y 2 K λ−Nt Nr m 2 f H0 (y)  (y > 0).

(Nt Nr m) (λ) 

(2.133)

Assume that BS transmission power is dynamically adjusted to meet the required traffic rates for all MSs in the typical PVT cell. The required BS transmission power for all MSs is expressed by de f

PC0 



Ps · 1{x Ms ∈ C0 },

(2.134)

where Ps is the consumed power transmitted from the BS B S0 to the MS M Ss in the typical cell, 1{. . .} is an indicator function for gathering together all MSs belong to the same typical PVT cell. Assume that Ps is a series of i.i.d. random variables, of

2.4 Energy Efficient Power Control Scheme

89

which the PDF and the characteristic function of which are denoted as f P ( p) and φ P (ω), respectively. Based on the Campbell theory, the characteristic function of required BS transmission power PC0 in the typical PVT cell is derived as ⎡ ⎤ ¨  jωp e − 1 f P ( p)1{x ∈ C0 }2π λ M xdpd x ⎦ φ PC0 (ω)  exp⎣ x, p



 exp⎣−2π λ M

5∞

⎤ (1 − φ P (ω))(1{x ∈ C0 })xd x ⎦

0

⎡  exp⎣−2π λ M

5∞

⎤ (1 − φ P (ω))e−πλ B x xd x ⎦ 2

0

  λM   exp − 1 − φ P0 (ω) . λB

(2.135)

And f PC0 (x) is the PDF of the required BS transmission power, which can be calculated by applying the inverse Fourier operation to φ PC0 (ω). Considering the limit of BS transmission power Pmax , the energy efficiency of MIMO random cellular networks with the average power allocation scheme is derived by

EE 

2.4.4.3

1 η

6 PC0 _max 0

λ M θρmin λ B (θ−1)

·

6 Pmax 

0

f PC0 (x)d x

2

x f PC0 (x)d x + N T Pdyn + Psta ·

6 Pmax 0

f PC0 (x)d x

.

(2.136)

Energy Efficiency of MIMO Cellular Networks Using Water-Filling Power Allocation Scheme

When perfect CSI is assumed to be available at both transmitters and receivers in wireless communication systems, the water-filling power allocation scheme is used for improve the capacity of wireless communication systems [108]. The downlink capacity over wireless channels between the BS B S0 and the MS M S0 is expressed as  0H D200 0 C  max log det I Nt + PI subject to

Nt 

Pl  PT 0 ,

(2.137)

l1

where N0 is the noise power at the transmitters. The objective function in (2.137) is jointly concave in the powers and this optimization problem can be solved by

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2 Energy Efficiency of 5G Wireless Communications

Lagrangian methods [109], the optimal transmission power for the lth sub-channel is given by   L N0 PT 0 1  N0 N0 +  − , Pl  ν − λl + L L l1 λl λl

(2.138)

where ν is the water-filling threshold in water-filling power allocation scheme. Based on (2.138) and (2.117), the achievable rate with the water-filling power scheme at the MS M S0 is derived as R0  BW  BW  BW

 BW

  2 0 0H D00 log 1 + PI    H D 2 0 log det I Nt + 0 00 PI   L  Pl λl log 1 + PI l1

⎛ L PT 0 L + L1 l1  L log⎝1 + PI l1

N0 λl



N0 λl

⎞ λl ⎠.

(2.139)

Assume that the traffic rate is satisfied by the achievable rate at the MS M S0 , thus the corresponding balance equation is expressed by

⎞ ⎛ N0 PT 0 N0 1 L L λl + −  l1 λl L L λl ⎠  ρ(x M0 ). log⎝1 + (2.140) BW P I l1 When the maximal BS transmission power limit Pmax is considered, based on (2.140) the Monte-Carlo is configured to iteratively solve the transmit 0 / simulation power PT 0 . Then, E PC0 _r eal and FPC0 (Pmax ) can be averaged and statistically 0 / computed from the simulation results. Substituting the values for E PC0 _r eal and FPC0 (Pmax ) into (2.120), the energy efficiency of MIMO PVT cellular networks with the water-filling power allocation scheme can be obtained as EE 

/

1 E η

λ M θρmin λ B (θ−1)



FPC0 (Pmax )

2

0  . PC0 _r eal + N T Pdyn + Psta · FPC0 (Pmax )

(2.141)

Based on (2.140), the CDF and PDF of the required BS transmission power with the water-filling power allocation scheme are analyzed as follows. Unless otherwise specified, the key parameters are set as σd B  6, σ  4, m  1, Nt  8, Nr  4,

2.4 Energy Efficient Power Control Scheme

91

Fig. 2.17 The CDF of the required BS transmission power with water-filling power allocation scheme

Fig. 2.18 The PDF of the required BS transmission power with water-filling power allocation scheme

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2 Energy Efficiency of 5G Wireless Communications

  2/σ  10−2 W, λ B  Pmax  40 Watt (W), the moment of receiving power E PT i  1/ π · 8002 m−2 , λ M /λ B  30, λinf  0.9λ B , θ  1.8, ρmin  2.5 bits/s/Hz. Figure 2.17 reveals the CDF of the required BS transmission power with waterfilling power allocation scheme considering different path loss coefficients σ . Figure 2.17 indicates that the CDF curve shifts to the left with the increasing value of σ , i.e., the required BS transmission power with the water-filling power allocation scheme is decreased when the value of σ is increased in MIMO PVT random cellular networks. Figure 2.18 evaluates the impact of intensity ratio of MSs to BSs λ M /λ B on the required BS transmission power with the water-filling power allocation scheme. Figure 2.18 shows that the probability mass shifts to the right when increasing the value of λ M /λ B , i.e., the required BS transmission power with water-filling power allocation scheme is increased when the value of λ M /λ B is higher.

2.4.5 Simulation Results The effect of two power allocation schemes on the proposed energy efficiency model of MIMO random cellular networks is investigated in detail following. In the following simulations, the Monte-Carlo simulation method is adopted for performance analysis. Moreover, the total BS transmission power including the required BS transmission power and RF circuit power, the BS fixed power is investigated in this section. Default system parameters are configured as: the average efficiency of power amplifier is η  0.38, the RF circuit power for an antenna Pdyn  83 W and the fixed power consumption in a BS is Psta  45.5 W [26, 110]. Figure 2.19 illustrates the energy efficiency of MIMO random cellular networks versus the number of transmitting antennas Nt and the intensity ratio of MSs to BSs λ M /λ B , where “WF” denotes the water-filling power allocation scheme and “AV” represents the average power allocation scheme. First, Fig. 2.19 shows that the energy efficiency curve of MIMO random cellular networks shrinks down when increasing the value of Nt . Base on the result of (2.123), the total BS power consumption increases with the increase of the number of transmit antennas, while the average traffic rate remains unchanged. Hence, the energy efficiency of PVT random cellular networks decreases with the increase of Nt . Second, we force on one of curves and analyze the energy efficiency for both power allocation schemes with impact of λ M /λ B . Simulation results indicate that the water-filling/average power allocation schemes can achieve the maximum energy efficiency for MIMO random cellular networks. When the intensity ratio of MSs to BSs is low, indicating a few MSs in a typical PVT cell, the increase of the intensity ratio of MSs to BSs conduces to a moderate increase in total BS power consumption including mainly fixed BS power consumption and a small portion of dynamic BS power consumption. In this case, the energy efficiency of PVT cellular networks is increased. However, when the intensity ratio of MSs to BSs in a PVT typical cell exceeds a given threshold, a high aggregate

2.4 Energy Efficient Power Control Scheme

93

Fig. 2.19 Energy efficiency of MIMO random cellular networks versus λ M /λ B and Nt

traffic load resulted from a large number of MSs will significantly increase the total BS power consumption including mainly dynamic BS power consumption and a small portion of fixed BS power consumption. In this case, the energy efficiency of PVT cellular networks is decreased. Moreover, the energy efficiency of the waterfilling power allocation scheme is always larger than for the energy efficiency of the average power allocation scheme in MIMO random cellular networks. Figure 2.20 reveals the impact of the path loss coefficient σ and λ M /λ B on the energy efficiency of MIMO random cellular networks. The energy efficiency curve lifts up as the value of λ M /λ B increases. Again the energy efficiency of the waterfilling power allocation scheme is always larger than for average power allocation scheme under diffierent values of σ. Figure 2.21 evaluates the effect of the minimum traffic rate ρmin and the tail index θ on the energy efficiency of MIMO random cellular networks with the two power allocation schemes. There always exists a maximum energy efficiency of MIMO random cellular networks considering different system parameters. However, numerical results indicate that the energy efficiency of the water-filling power allocation scheme is always larger than for the energy efficiency of average power allocation scheme in MIMO random cellular networks. Finally, the effect of the interfering BS intensity λinf on the energy efficiency with different power allocation schemes is investigated in Fig. 2.22. When the values of λ M /λ B is fixed, the energy efficiency decreases when the value of λinf increases. The curves in Fig. 2.22 indicate that the energy efficiency of the water-filling power

94

2 Energy Efficiency of 5G Wireless Communications

Fig. 2.20 Energy efficiency of MIMO random cellular networks versus λ M /λ B and σ

Fig. 2.21 Energy efficiency of MIMO random cellular networks versus λ M /λ B , θ and ρmin

2.4 Energy Efficient Power Control Scheme

95

Fig. 2.22 Energy efficiency of MIMO random cellular networks versus λ M /λ B and λinf

allocation scheme is always larger than for the energy efficiency of average power allocation scheme under different values of λinf in MIMO random cellular networks. The reason is that the water-filling power allocation scheme substantially reduces the sum power, by up to 80%, in comparison to the average power allocation scheme [111]. The water-filling power allocation efficiently exploits the multiuser MIMO channels (e.g., multiuser diversity), hence power reduction is more significant. So the water-filling power allocation scheme is better than the average power allocation scheme in the energy efficiency of cellular networks when the traffic rate of MSs is same.

2.4.6 Conclusions The energy efficiency of MIMO random cellular networks with different power allocation schemes is evaluated. Considering the path loss, Nakagami-m fading and shadowing effects on wireless channels, an interference model and the achievable rate of MIMO random cellular networks is first presented. Furthermore, the energy efficiency of average and water-filling power allocation schemes is proposed, respectively. Simulation results indicate that there exists maximal network energy efficiency when considering the trade-off between intensity ratios of MSs to BSs and wireless channel conditions. When the CSI is available for both transmitters and receivers, the energy efficiency of the water-filling power allocation scheme is better than the

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2 Energy Efficiency of 5G Wireless Communications

energy efficiency of average power allocation scheme in MIMO random cellular networks. Therefore, our results evaluate the impact of different power allocation schemes on the energy efficiency of MIMO random cellular networks.

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Chapter 3

Energy Efficiency of Cellular Networks

3.1 On the Energy-Efficient Deployment for Ultra-Dense Heterogeneous Networks with NLoS and LoS Transmissions 3.1.1 Introduction Ultra-dense deployment of small cell base stations (BSs), relay nodes, and distributed antennas is considered as a de facto solution for realizing the significant performance improvements needed to accommodate the overwhelming future mobile traffic demand [1]. Traditional network expansion techniques like cell splitting are often utilized by telecom operators to achieve the expected throughput, which is less efficient and proven not to keep up with the pace of traffic proliferation in the near future. Heterogeneous networks (HetNets) then become a promising and attractive network architecture to alleviate the problem. “HetNets” is a broad term that refers to the coexistence of different networks (e.g., traditional macrocells and small cell networks like femtocells and picocells), each of them constituting a network tier. Due to differences in deployment, BSs in different tiers may have different transmit powers, radio access technologies, fading environments and spatial densities. HetNets are envisioned to change the existing network architectures and have been introduced in the LTE-Advanced standardization [2, 3]. Massive work has been done in HetNet scenario mainly related to cell association scheme [4–6], cache-enabled networks [7], physical layer security [8], etc. In [4], the pertinent user association algorithms designed for HetNets, massive MIMO networks, mmWave scenarios and energy harvesting networks have been surveyed for the future fifth generation (5G) networks. Bethanabhotla et al. [5] investigated the optimal user-cell association problem for massive MIMO HetNets and illustrated how massive MIMO could also provide nontrivial advantages at the system level. The joint downlink cell association and wireless backhaul bandwidth allocation in a © Publishing House of Electronics Industry, Beijing and Springer Nature Singapore Pte Ltd. 2019 X. Ge and W. Zhang, 5G Green Mobile Communication Networks, https://doi.org/10.1007/978-981-13-6252-1_3

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two-tier HetNet was studied in [6]. In [7], Yang et al. aimed to model and evaluate the performance of the wireless HetNet where the radio access network (RAN) caching and device-to-device (D2D) caching coexist. The physical layer security of HetNets where the locations of all BSs, mobile users (MUs) and eavesdroppers are modeled as independent homogeneous Poisson Point Processes (PPPs) in [8]. From the mobile operators’ point of view, the commercial viability of network densification depends on the underlying capital and operational expenditure [9]. While the former cost may be covered by taking up a high volume of customers, with the rapid rise in the price of energy, and given that BSs are particularly powerhungry, energy efficiency (EE) has become an increasingly crucial factor for the success of dense HetNets [10]. Recently, loads of work [11–15] has investigated the EE in the 5G network scenarios. In [11], Niu et al. investigated the problem of minimizing the energy consumption via optimizing concurrent transmission scheduling and power control for the mmWave backhauling of small cells densely deployed in HetNets. A self-organized cross-layer optimization for enhancing the EE of the D2D communications without creating harmful impact on other tiers by employing a noncooperative game in a three-tier HetNet was proposed in [12]. To jointly optimize the EE and video quality, Wu et al. [13] presented an energy-quality aware bandwidth aggregation scheme. In [14], Yang et al. investigated the energy-efficient resource allocation problem for downlink heterogeneous OFDMA networks. The mobile edge computing offloading mechanisms were studied in 5G HetNets [15]. Different from most prior work analyzing network performance where the propagation path losses between the BSs and the MUs follow the same power-law model, in this section we consider the co-existence of both non-line-of-sight (NLoS) and line-of-sight (LoS) transmissions, which frequently occur in urban areas. More specifically, for a randomly selected MU, BSs deployed according to a homogeneous PPP are divided into two categories, i.e., NLoS BSs and LoS BSs, depending on the distance between BSs and MUs. It is well known that LoS transmission may occur when the distance between a transmitter and a receiver is small, and NLoS transmission is common in office environments and central business districts. Moreover, as the trend of ultra-dense network deployment, the distance between a transmitter and a receiver decreases, the probability that a LoS path exists between them increases, thereby causing a transition from NLoS transmission to LoS transmission with a higher probability [16]. In this context, Ding et al. [16] studied the coverage and capacity performance by using a multi-slop path loss model incorporating probabilistic NLoS and LoS transmissions. The coverage and capacity performance in millimeter wave cellular networks were studied in [17–19]. In [17], a three-state statistical model for each link was assumed, in which a link can either be in an NLoS, LoS or an outage state. In [18], self-backhauled millimeter wave cellular networks were characterized assuming a cell association scheme based on the smallest path loss. However, both [17, 18] assume a noise-limited network, ignoring inter-cell interference, which may not be very practical since modern wireless networks work in the interference-limited region. In [19], the coverage probability and capacity were calculated in a millimeter wave cellular network based on the smallest path loss cell association model assuming multi-path fading modeled as Nakagami-m

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fading, respectively. However, shadowing was ignored in their models, which may not be very practical for an ultra-dense heterogeneous network.

3.1.2 System Model In this section, a K-tier HetNet is considered, which consists of macrocells, picocells, femtocells, etc. BSs of each tier are assumed to be spatially distributed on the infinite plane and locations of BSs follow the independent homogeneous PPP denoted by def k  {Xk,i } with a density (aka intensity) λk , k ∈ {1, 2, . . . , K }  K, where Xk,i denotes the location of BS in the k-th tier. MUs are deployed according to another independent HPPP denoted by u with a density λu (λu  λk ). BSs belong to the same tier transmit using the same constant power Pk and sharing the same bandwidth. Besides, within a cell assume that each MU uses orthogonal multiple access method to connect to a serving BS for downlink and uplink transmissions and therefore there is no intra-cell interference in the analysis of this work. However, adjacent BSs which are not serving the connected MU may cause inter-cell interference which is the main focus of this section. It is further assumed that each MU can possibly associate with a BS belonging to any tier, i.e., open access policy is employed. Without loss of generality and from the Slivnyak’s Theorem [20], we consider the typical MU which is usually assumed to be located at the origin, as the focus of our performance analysis.

3.1.3 Signal Propagation Model The long-distance signal attenuation in tier k is modeled by a monotone, nonincreasing and continuous path loss function lk : [0, ∞] → [0, ∞] and lk decays to zero asymptotically. The fast fading coefficient for the wireless link between a BS Xk,i ∈ k and the typical MU is denoted as h Xk,i . {h Xk,i } are assumed to be random variables which are mutually independent and identically distributed (i.i.d.) and also independent of BS locations Xk,i , thus h Xk,i can be denoted as h k for the sake of simplicity. Similarly, the shadowing is denoted by gk and particularly assume that it follows a log-normal distribution with zero mean and standard deviation σk . Note that the proposed model is general enough to account for various propagation scenarios with fasting fading, shadowing, and different path loss models. To characterize shadowing effect in urban areas which is a unique scenario in our analysis, both NLoS and LoS transmissions are incorporated. That is, if the visual path between a BS Xk,i ∈ k and the typical MU is blocked by obstacles like buildings, trees, and even MUs, it is an NLoS transmission. Otherwise it is a LoS transmission. The occurrence of NLoS and LoS transmissions depends on various

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environmental factors, including geographical structure, distance, and cluster. In this work, a one-parameter distance-based NLoS/LoS transmission probability model is applied. That is,     pkNL Xk,i  + pkL Xk,i   1,

(3.1)

    where pkNL Xk,i  and pkL Xk,i  denote  the probability of the occurrence of NLoS and LoS transmissions, respectively, Xk,i  is the distance between the BS Xk,i and the typical MU.     Regarding the mathematical  form of pkL Xk,i  (or pkNL Xk,i  ), Blaunstein   and Levin [21] formulated pkL Xk,i  as a negative exponential function, i.e.,   pkL Xk,i   e−κ Xk,i  , where κ is a parameter determined by the density and the mean length of the blockages lying in the visual path between BSs and the typical MU. Bai et al. [22] extended Blaunstein’s work by using random shape theory which shows that κ is not only determined by the mean length but also the   mean width of the blockages. Renzo [17] and Bai and Heath [19] approximated pkL Xk,i  by using piece-wise  functions and step functions, respectively. Ding et al. [16] con sidered pkL Xk,i  to be a linear function and a two-piece exponential function, respectively; both are recommended by the 3GPP. It is important to note that the introduction of NLoS and LoS transmissions is essential to model practical networks, where a MU does not necessarily have to connect to the nearest BS. Instead, for many cases, MUs are associated with farther BSs with stronger signal strength. It should be noted that the occurrence of NLoS and LoS transmissions is assumed to be independent for different BS-MU pairs. Though such assumption might not be entirely realistic (e.g., NLoS transmission caused by a large obstacle may be spatially correlated), Bai et al. [19, 22] showed that the impact of the independence assumption on the SINR analysis is negligible. For a specific tier k, note that from the viewpoint of the typical MU, each BS in the infinite plane R2 is either an NLoS BS or a LoS BS to the typical MU. Accordingly, a thinning procedure on points in the PPP k is performed to model the distributions of NLoS BSs and LoS BSs, respectively. That is, each BS in k will be kept if a BS has an NLoS transmission with the typical MU, thus forming a new point process NL L denoted by NL k . While BSs in k \k form another point process denoted by k , representing the set of BSs with LoS path to the typical MU. As a consequence of the independence assumption of NLoS and LoS transmissions mentioned in the last L paragraph, NL k and  non-homogeneous PPPs with intensity  kare two independent  NL  Xk,i  and λk pkL Xk,i  , respectively. functions λk pk Based on assumptions above, the received power of the typical MU from a BS Xk,i ∈ k is defined as follows. rec Definition 1 The received power of the typical MU from a BS Xk,i ∈ k , i.e., Pk,i is

 rec Pk,i



    NL NL NL   , w.p. p NL Xk,i  X Pk ANL k,i k h k gkl k k      , w.p. pkL Xk,i  Pk ALk h Lk gkLlkL Xk,i  ,

(3.2)

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

107

 L    NL NL NL NL  Xk,i  , Pk,i where we denote Pk,i  Pk ANL  Pk ALk h Lk gkLlkL Xk,i  , k h k gk l k L ANL k and Ak denote the respective path loss for NLoS and LoS transmissions at the reference distance (usually at 1 m). For simplicity, denote BkU  Pk AU k and let   −αkU  def U  lk Xk,i   Xk,i  , where the superscript U ∈ {NL, L}  U used distinguishes NLoS and LoS transmissions and αkU denotes the path loss exponent for NLoS or LoS transmission in the k-th tier. Recently, [23, 24] took bounded path loss model and stretched exponential path loss model into consideration, in which several interesting performance trends are found and will be investigated in our future work. Remark 1 Apart from the fixed transmit power, a density-dependent transmit power Tk η/10 is further assumed and analyzed mentioned in [25], i.e., Pk (λk )  10 −αNL , where NL Ak rk k  rk  πλ1 k is the radius of an equivalent disk-shaped coverage area in the k-th tier  with an area size of rk  π1 and Tk is the per tier SINR threshold. 3.1.3.1

Cell Association Scheme

Cell association scheme [26] plays a crucial role in network performance determining BS coverage, MU hand-off regulation and even facility deployment of small cells. Conventionally, a typical MU is connected to the BS Xk,m if and only if dBm Pk,m > P dBm j,n , j  k,

(3.3)

dBm where Pk,m denotes the average received power with dBm unit from the BS Xk,m and Eq. (3.3) is known as the MARP association scheme. In practical, PkdBm is usually averaged out in time and frequency domains to cope with fluctuations caused by channel fading. In this text, a typical MU is connected to the BS Xk,m if and only if dBm Pk,m > P dBm j,n , j  k,

(3.4)

dBm where Pk,m denotes the average received power with dBm unit from the BS Xk,m and Eq. (3.4) is known as the maximum average received power (MARP) association scheme. Aided by cell range expansion (CRE), which is realized by MUs adding a positive cell range expansion bias (CREB) to the received power from BSs in different tiers, more MUs can be offloaded to small cells. That is, if a MU is associated with the BS Xk,m if and only if dB dBm dBm Pk,m + dB k,m > P j,n + j.n , j  k,

(3.5)

dB where dB k,m and j.n is the CREB with dB unit in the k-th and j-th tier. With proper CREB chosen, the coverage of BSs in some tiers is artificially expanded, allowing

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3 Energy Efficiency of Cellular Networks

MUs more flexible to be associated with BSs which may not provide the strongest received power, thus balancing traffic load to achieve spatial efficiency. However, CRE causes severe interference to the small cell MU which impairs the QoS of small cell users and thus almost blank subframes (ABS) coordination is needed between macrocell BSs and small cell BSs. However, the analysis of CRE plus ABS is challenging because (i) the association scheme is not only determined by the received power but also the current resource allocation strategy, and (ii) ignoring ABS while using CRE can impair the coverage performance. For simplicity, CRE and ABS are not going to be considered in this section, which are left as future work.

3.1.3.2

Performance Metrics

To evaluate the network performance, the following three metrics, i.e., the coverage probability, the potential throughput (PT) and the EE, are focused on. The coverage probability is the probability that  than is greater   the received SINR Xk,i  > Tk , ({λ }, {T })  Pr ∪ SINR a given threshold, i.e., p cov k k k∈K,X ∈ k k,i k   where SINRk Xk,i  is defined as follows   U U U    Xk,i  Pk AU k h k gk l k   SINRk Xk,i  , (3.6) Ik + η Ik 

K



  U U U  Xk, j  , Pk AU k h k gk l k

(3.7)

k1 Xk, j ∈k \Xk,i

where k \Xk,i is the Palm point process [27] representing the set of interfering BSs in the k-th tier and η denotes the noise power at the MU side, which is assumed to be the additive white Gaussian noise (AWGN). The PT is defined as follows [24, 28] T ({λk }, {Tk }) 

K

cond Ak pcov,k log2 (1 + Tk )

k1



K

λk pcov,k log2 (1 + Tk ),

(3.8)

k1

where the network is fully loaded due to the assumption that λu  λk , Ak is the cond is association probability that the typical MU is connected to the k-th tier, pcov,k the conditional association coverage probability and pcov,k is the per-tier coverage probability. Compared with the area spectral efficiency (ASE), which is defined as ASE({λk }, {Tk }) 

K     E λk log2 1 + SINRk Xk,i  , k1

(3.9)

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

109

the PT implicitly assumes a fixed rate transmission from all BSs in the network, and has a unit of bps/Hz/m2 , while the ASE assumes full buffers but it allows each link to adapt its rate to the optimal value for a given SINR, thus avoiding outages at low SINR and the wasting of rate at high SINR [24]. In other words, the PT is a more realistic performance metric and the ASE upper bounds the PT. In our analysis, the PT is chosen as our performance metric. The EE is defined as the ratio between the PT and the total energy consumption of the network, i.e., T ({λk }, {Tk }) , E({λk }, {Tk })  K k−1 λk (ak Pk + bk )

(3.10)

where the coefficient ak accounts for power consumption that scales with the average radiated power, and the term bk models the static power consumed by signal processing, battery backup and cooling [29]. Other performance metrics, such as the bit-error probability and per-MU data rate, can be found using the coverage probability (SINR distribution) following the methods mentioned in [30].

3.1.4 Performance Analysis In this section, we derive expressions for the considered performance metrics and study the effect of densification on these metrics. It is started by introducing the network transformation and then presenting the analytical expressions with the maximum instantaneous received power (MIRP) and MARP association schemes in the following subsections.

3.1.5 Network Transformation Before presenting our main analytical results, firstly the network transformation is introduced, which aims to unify the analysis and to reduce the complexity as well. Using the manipulation in [31, 32], we define    −1/αkNL NL Rk,i  Xk,i  · BkNL gkNL ,

(3.11)

   −1/αkL L Rk,i  Xk,i  · BkL gkL ,

(3.12)

and

respectively. Then Eq. (3.2) can be written as

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3 Energy Efficiency of Cellular Networks

rec Pk,i

⎧  −αkNL   ⎪ ⎨ P NL  h NL R NL , w.p. pkNL Xk,i  k,i k k,i   −αkL  .  ⎪ ⎩ P L  hL RL , w.p. pkL Xk,i  k,i k k,i

(3.13)

 Equivalence Theorem in [31], it is concluded that the distances  Byadopting  the NL L Rk,i (or Rk,i ) form a scaled point process for NLoS BSs (or LoS BSs), which i i   L U , U ∈ U are mutually independent still remains a PPP denoted by NL k (or k ). k k with each other, and the intensity measures and intensities are provided in Lemma 1 as below. Lemma 1 The intensity measure and intensity of U k can be formulated as λNL k (t) 

d NL

([0, t]), dt k

(3.14)

λLk (t) 

d L

([0, t]), dt k

(3.15)

and

respectively, where ⎡ ⎢ ⎢

NL k ([0, t])  EgkNL ⎣2π λk

t ( BkNL gkNL )





1/αkNL

⎥ pkNL (z)zdz ⎥ ⎦,

(3.16)

z0

and ⎡ ⎢

Lk ([0, t])  EgkL ⎢ ⎣2π λk

t ( BkL gkL )



1/αkL

⎤ ⎥ pkL (z)zdz ⎥ ⎦.

(3.17)

z0

Proof The proof can be referred to [31, Appendix 1] and thus omitted here. Aided by the network transformation and stochastic geometry tool, the coverage probability, the PT and the EE will be derived in the following.

3.1.5.1

Coverage Probability with the MIRP Association Scheme

With the MIRP association scheme, the typical MU is associated with the BS which offers the maximum instantaneous received power. Using this cell association scheme and considering Lemma 1, the general results of coverage probability in the K-tier HetNets is given by Theorem 1.

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

111

Theorem 1 When Tk ≥ 1, the coverage probability for a typical MU with the MIRP association scheme can be derived as K 



MIRP pcov ({λk }, {Tk })



αkNL

−Tk ηr λNL k (r )e

k1r 0

×

K       NL NL LMIRP Tk r αk LMIRP Tk r αk dr I NL IL j

j

j1

+

K

∞ λLk (r )

k1

e−Tk ηr

αkL

K  

    αkL MIRP αkL LMIRP T L T dr , r r NL L k k I I j

j

j1

t0

(3.18) where ⎡

∞

⎢ (s)  exp⎣− LMIRP I NL j

y0

⎤ λNL j (y) 1+y

α NL j

/s

⎥ dy ⎦,

(3.19)

and ⎡ ⎢ (s)  exp⎣− LMIRP IL

∞

⎤ λLj (y) α Lj

j

y0

1 + y /s

⎥ dy ⎦.

(3.20)

Proof See Appendix 1. In pursuit of the analytical results of the PT and the EE, the NLoS/LoS coverage probability and per-tier coverage probability are presented in the following two corollaries. Corollary 1 When Tk ≥ 1, the coverage probabilities for a typical MU which is served by NLoS BSs and LoS BSs with the MIRP association scheme are given by MIRP pcov,NL ({λk }, {Tk })



K

MIRP pNL,k ({λk }, {Tk })

k1 K 





k1r 0

−Tk ηr λNL k (r )e

αkNL

K  

    αkNL MIRP αkNL LMIRP T L T dr , r r NL L k k I I j

j

j1

(3.21) and

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3 Energy Efficiency of Cellular Networks

MIRP pcov,L ({λk }, {Tk }) 

K

MIRP pL,k ({λk }, {Tk })

k1 K 





L

αk λLk (r )e−Tk ηr

k1 t0

K  

    L L LMIRP Tk r αk LMIRP Tk r αk dr , I NL IL j

j

j1

(3.22) respectively. Proof This corollary can be derived from Theorem 1 by rearranging the terms in Eq. (3.18) and thus the proof is omitted here. Corollary 2 When Tk ≥ 1, the per-tier coverage probability for a typical MU which is covered by the k-th tier with the MIRP association scheme is given by ∞ MIRP pcov,k ({λk }, {Tk })



−Tk ηr λNL k (r )e

αkNL

r 0

×

K       NL NL LMIRP Tk r αk LMIRP Tk r αk dr I NL IL j

j

j1

∞ +

λLk (r )e−Tk ηr

t0

αkL

K       αkL MIRP αkL LMIRP T L T dr . r r NL L k k I I j

j

j1

(3.23) Proof This corollary can be derived from Theorem 1 by rearranging the terms in Eq. (3.18) and thus the proof is omitted here.

3.1.5.2

Coverage Probability with the MARP Association Scheme

With the MARP association scheme, the typical MU is associated with the BS which offers the maximum long-term averaged received power by averaging out the effect of multi-path fading h U k . With this cell association scheme, the primary results of coverage probability are given by Theorem 2. Theorem 2 The coverage probability for a typical MU with the MARP association scheme is MARP pcov ({λk }, {Tk }) 

K k1

MARP pk,NL ({λk }, {Tk }) +

K k1

MARP pk,L ({λk }, {Tk })

3.1 On the Energy-Efficient Deployment for Ultra-Dense … K 





k1r 0

×e +



K

j1

K  

λNL k (r )

113

    αkNL MARP1 αkNL LMARP1 T L T r r NL L k k I I j

j

j1       NL α NL /α Lj α NL /α NL j

Lj 0,r k + NL 0,r k −Tk ηr αk j

K ∞

    αkL MARP2 αkL LMARP2 T L T r r NL L k k I I

λLk (r )

j

j

k1r 0 j1       K

L α L /α L α L /α NL

Lj 0,r k j + NL 0,r k j −Tk ηr αk − j

×e

dr

K  

j1

dr,

(3.24)

where ⎡

∞

⎢ (s)  exp⎢ LMARP1 I NL ⎣− j



α NL /α NL j yr k

∞

⎢ LMARP1 (s)  exp⎢ IL ⎣−

λNL j (y) 1+y

α NL j

/s

α NL /α Lj yr k

∞

⎢ (s)  exp⎢ LMARP2 I NL ⎣− j

α L /α NL yr k j

⎥ dy ⎥ ⎦,

(3.25)

⎤ ⎥ dy ⎥ ⎦,

λLj (y) α Lj

j





1 + y /s

(3.26)

⎤ λNL j (y) 1+y

α NL j

/s

⎥ dy ⎥ ⎦,

(3.27)

and ⎡ ⎢ LMARP2 (s)  exp⎢ IL ⎣−

∞

α Lj

j

yr

⎤ λLj (y)

αkL /α Lj

1 + y /s

⎥ dy ⎥ ⎦.

(3.28)

Proof See Appendix 2. Remark 2 Note that different from Theorem 1, Theorem 2 can be applied to scenarios without the assumption of a particular range of SINR threshold Tk , e.g., Tk ≥ 1. Similar to the study for Theorem 1, we provide two corollaries, i.e., the NLoS/LoS coverage probability and the per-tier coverage probability, as follows. Corollary 3 The coverage probabilities for a typical MU which is served by NLoS BSs and LoS BSs with the MARP association scheme are given by

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3 Energy Efficiency of Cellular Networks

MARP pcov,NL ({λk }, {Tk }) 

K

MARP pNL,k ({λk }, {Tk })

k1 K 





λNL k (r )

k1r 0

×e

K



j1

K  

    NL NL LMARP1 Tk r αk LMARP1 Tk r αk I NL IL j

j

j1       NL α NL /α Lj α NL /α NL j

Lj 0,r k + NL 0,r k −Tk ηr αk j

dr

(3.29)

and MARP pcov,L ({λk }, {Tk }) 

K

MARP pL,k ({λk }, {Tk })

k1 K 





k1r 0

×e



K

j1

λLk (r )

K       αkL MARP2 αkL LMARP2 T L T r r NL L k k I I j

j

j1       L α L /α L α L /α NL

Lj 0,r k j + NL 0,r k j −Tk ηr αk j

dr,

(3.30)

respectively. Proof This corollary can be derived from Theorem 2 by rearranging the terms in Eq. (3.24) and thus the proof is omitted here. Corollary 4 The per-tier coverage probability for a typical MU which is covered by the k-th tier with the MARP association scheme is given by ∞ MARP pcov,k ({λk }, {Tk })



−Tk ηr λNL k (r )e

αkNL

r 0 K       NL NL LMARP1 Tk r αk LMARP1 Tk r αk × I NL IL j

j

j1

      K

α NL /α Lj α NL /α NL j

Lj 0,r k + NL 0,r k − j

dr × e j1 ∞  K       αkL MARP2 αkL + LMARP2 T L T λLk (r ) r r NL L k k I I j

j

j1       K

L α L /α L α L /α NL −

Lj 0,r k j + NL 0,r k j −Tk ηr αk j

r 0

×e

j1

dr.

(3.31)

Proof This corollary can be derived from Theorem 2 by rearranging the terms in Eq. (3.24) and thus the proof is omitted here.

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

115

Intuitively, the coverage probability with the MIRP association scheme is higher than that with the MARP association scheme. However, it can be proved mathematically which is summarized in the following corollary. Corollary 5 In the studied K-tier HetNet, the coverage probability with the MIRP association scheme is higher than that with the MARP association scheme, where the gap is determined by the intensity and the intensity measure. Proof See Appendix 3.

3.1.5.3

The PT and the EE

As the results with the MIRP and the MARP association schemes are some kind of similar and the MARP association scheme is more practical in the real network, we take the MARP association scheme as an example to evaluate the PT and the EE in the following. The PT with the MARP association scheme can be directly obtained from the coverage probability expressions using Eq. (3.8), i.e., T ({λk }, {Tk }) 

K

MARP λk pcov,k ({λk }, {Tk }) log2 (1 + Tk ).

(3.32)

k1

While the PT with the MIRP association scheme is similar except for replacing MARP MIRP ({λk }, {Tk }) by pcov,k ({λk }, {Tk }). pcov,k The EE can be derived by using Eq. (3.10) and we will only provide expressions for it when necessary.

3.1.6 Performance Optimization and Tradeoff As mentioned, from the mobile operators’ point of view, the commercial viability of network densification depends on the underlying capital and operational expenditure [9]. While the former cost may be covered by taking up a high volume of customers, with the rapid rise in the price of energy, and given that BSs are particularly power-hungry, EE has become an increasingly crucial factor for the success of dense HetNets [10]. There are two main approaches to enhance the energy consumption of cellular networks: (1) improvement in hardware and (2) energy-efficient system design. The improvement in hardware may have achieved its bottleneck due to the limit of Moore’s law, while the energy-efficient system design has a great potential in the future 5G networks. In the following, two energy-efficient optimization problems are proposed trying to obtain insights of the system design.

116

3.1.6.1

3 Energy Efficiency of Cellular Networks

Optimizing Coverage Probability with the Maximum Total Power Consumption Constraint

To pursue a further study on coverage performance, we formulate a theoretical framework which determines the optimal BS density to maximize the coverage probability while guaranteeing that the total area power consumption is lower than a given expected value P max as follows MARP OP1 : max pcov ({λk }, {Tk }) λk

s.t. C1:

K

λk (ak Pk + bk ) ≤ P max

k1

C2: λk ≥ 0, ∀k ∈ K

(3.33)

where ak and bk are defined in Eq. (3.10). Note that OP1 assumes the MARP association scheme, while the optimization problem with the MIRP association scheme is similar to OP1 and omitted here for brevity.

3.1.6.2

Optimizing the EE Under the Minimum Coverage Probability Constraint

In this subsection, another framework are formulated which determines the optimal BS density to maximize the EE while guaranteeing QoS of the network, i.e., the min as follows coverage probability is higher than a given expected value pcov OP2 : max E({λk }, {Tk }) λk

MARP min s.t. C1: pcov ({λk }, {Tk }) ≥ pcov C2 : λk ≥ 0, ∀k ∈ K

(3.34)

We will show in the simulation results that tradeoff exists between the coverage probability and the EE.

3.1.6.3

Optimization Solution

As NLoS and LoS transmissions are incorporated into our model, the coverage probability is not a monotonically increasing function with respect to BS density λk like the cases in [10, 29, 33, 34] anymore. Besides, the coverage probability function is not convex with respect to λk , either. Therefore, the optimization problem under consideration should be tackled numerically. Exhaustive search algorithms are wellsuited for tackling the problem considering that the objective function derivative is not available analytically and its accurate evaluation is resource-intensive. Brent’s

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

117

algorithm [35] and heuristic downhill simplex method [36] can be utilized to obtain the solutions of OP1 and OP2 in exponential time. To gain an analytical insight into the effect of different operational settings on the maximum energy-efficient deployment solution, in the following, we focus on the problem of finding the optimal BS density in a 2-tier HetNet.

3.1.7 Results and Insights A 2-tier HetNet is considered in our analysis. Macrocell BSs are in Tier 1 and small cell BSs are in Tier 2. We assume that P1  46 dBm, P2  24 dBm, ANL 1  2.7, L NL L NL  32.9, A  41.1, α  4.28, α  2.42, α  3.75, AL1  30.8, ANL 2 2 1 1 2 α2L  2.09, σ1NL  8 dB, σ1L  4 dB, σ2NL  4 dB, σ2L  3 dB, η  −95 dBm [18, 19, 31, 37–41] unless stated otherwise.

3.1.7.1

Validation of the Analytical Results of Coverage Probability with Monte Carlo Simulations

MIRP If fixing λ2 , the analytical and simulation results of pcov ({λk }, {Tk }) and the analytMARP ical results of pcov ({λk }, {Tk }) configured with T  1 dB are plotted in Figs. 3.1 and 3.2 respectively. As can be observed from Fig. 3.1, the analytical results match the simulation results well, which validate the accuracy of our theoretical analysis. In Fig. 3.2, aided by the utilization of a density-dependent BS transmit power, the coverage probability improves a lot as λ1 increases. Figures 3.3 and 3.4 illustrate the coverage probability versus the ratio of λ1 and λ2 , i.e., λ1 /λ2 with the MIRP and the MARP association schemes when λ1 (or λ2 ) is fixed. It is found that in Fig. 3.3, there is always a coverage peak when λ1 /λ2 is low, max max  0.3417 (or 0.3725 with the MIRP), pcov  0.6998 (or medium and high, i.e., pcov max 0.7868 with the MIRP), pcov  0.6521 (or 0.7476 with the MIRP), which indicates that there exists an optimal λ1 when implementing the network design if λ2 is fixed. And in Fig. 3.4, the optimal λ2 exists as well. However, compared with Fig. 3.3, when the fixed value of λ1 is sparse, the coverage probability firstly increases and then reaches a peak. Finally, it decreases to a certain value. When the fixed value of λ1 becomes larger, the coverage probability saturates. Based on the above observations, dense deployment of small cell BSs and macrocell BSs will lead to a better coverage probability. However, there is no need to deploy an infinite number of BSs in a finite area. When λ1 approaches infinite if λ2 remains fixed, and vice versa, the coverage probability becomes much worse. In contrast, when λ1 goes to zero if λ2 is fixed, and vice versa, the coverage probability saturates to a certain value. To have a full picture of the coverage probability with respect to λ1 and λ2 , two 3D figures are presented in Figs. 3.5 and 3.6. In Fig. 3.5, we compare the MIRP and MARP association schemes based on the fixed transmit power. It is found that

118

3 Energy Efficiency of Cellular Networks 0.8 MIRP, MIRP,

0.7

Coverage probability

MARP, MARP,

0.6

2 2

= 7.94 BSs/km2 (Ana.) = 7.94 BSs/km2 (Sim.)

2 2

= 7.94 BSs/km2 (Ana.) = 7.94 BSs/km2 (Sim.)

0.5 0.4 0.3 0.2 0.1 10-1

100

101 1

102

103

104

[BSs/km 2]

Fig. 3.1 Coverage probability versus λ1 with the MIRP and MARP association schemes, Tk  1 dB 0.8

Coverage probability

0.7 0.6 MARP, fixed power (Ana.) MARP, density-dependent power (Ana.)

0.5 0.4

2

= 7.94 BSs/km2

0.3 0.2 0.1 10-1

100

101

102

103

104

2

1

[BSs/km ]

Fig. 3.2 Coverage probability versus λ1 with the MARP association scheme, Tk  1 dB

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

119

0.8

Coverage probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 10-4

10-2

102

100

/ 1

104

106

2

Fig. 3.3 Coverage probability versus λ1 /λ2 with the MARP association scheme (the solid line) and the MIRP association scheme (the dashed line) when λ2 is fixed, Tk  1 dB 0.8

Coverage Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 10-6

10-4

10-2

100

102

104

Fig. 3.4 Coverage probability versus λ1 /λ2 with the MARP association scheme (the solid line) and the MIRP association scheme (the dashed line) when λ1 is fixed, Tk  1 dB

120

3 Energy Efficiency of Cellular Networks

Fig. 3.5 Comparison of coverage probability with the MIRP and MARP association schemes

the coverage probability with the MIRP association scheme is always greater than that with the MARP association scheme as with former association scheme BSs can provide the maximum power all the time even though it is not practical in the real networks. In Fig. 3.6, coverage probabilities based on the fixed transmit power and density-dependent transmit power are illustrated, respectively. By utilizing a density-dependent transmit power, the coverage probability improves compared with the HetNets using a fixed transmit power. Besides, it is noted that the coverage probability using a density-dependent transmit power fluctuates with BS density as illustrated in Fig. 3.6 as well as in Fig. 3.2. It is because the imperfect power control used in Remark 1 which only depends on BS densities and an approximate equivalent coverage area, the 3D coverage probability appears more unique than that using a fixed transmit power.

3.1.7.2

The PT and the EE

In this subsection, two typical energy consumption scenarios are considered, i.e., practical power consumption and ideal power consumption, denoted by S1 and S2. Recall that the definition of the EE in Eq. (3.10) have parameters {ak } and {bk }, thus we define S1 as the HetNets which are configured with {a1  22.6, a2  5.5, b1  414.2, b2  32} [29] and S2 configured with {a1  1, a2  1, b1  0, b2  0}, respectively. Note that S2 accounts for the HetNets with perfect power amplifier and ignoring the static power consumed by signal processing, battery backup, and cooling, etc. In other words, in S2 only radiated

3.1 On the Energy-Efficient Deployment for Ultra-Dense … Density-dependent transmit power

Fixed power

0.8

0.8

0.7

0.7

Coverage Probability

Coverage Probability

121

0.6 0.5 0.4 0.3 0.2

0.6 0.5 0.4 0.3 0.2 0.1

0.1 10 4

104 104 10 2

1

[BSs/km 2 ]

104 102

102 10 0

100 2

1

[BSs/km 2 ]

[BSs/km 2 ]

102 100

100 2

[BSs/km 2 ]

Fig. 3.6 Comparison of coverage probability based on different transmit power models, i.e., the fixed transmit power and density-dependent transmit power Density-dependent transmit power

3000

2000

Throughput [bps/Hz/km 2 ]

Throughput [bps/Hz/km 2 ]

Fixed power

2500 2000 1500 1000 500 0

1

[BSs/km 2 ]

100

100

2

1500

1000

500

0

[BSs/km 2 ] 1

[BSs/km 2 ]

100

100

2

[BSs/km 2 ]

Fig. 3.7 The PT versus λ1 and λ2 based on the fixed transmit power and density-dependent transmit power

power is considered. It is observed that λ1 has a greater impact on the PT than λ2 in Fig. 3.7. However, a larger λ1 cannot always provide a better EE as illustrated in Figs. 3.8 and 3.9. Therefore, there should exist a tradeoff among coverage probability, the PT and the EE, which is revealed in the following subsection.

122

3 Energy Efficiency of Cellular Networks Density-dependent transmit power, S1

Fixed power, S1

0.015

0.02

EE [bps/Hz/W]

EE [bps/Hz/W]

0.025

0.015 0.01 0.005

0.01

0.005

0

0

10 0

100

100

100

Fig. 3.8 The EE versus λ1 and λ2 based on the fixed transmit power and density-dependent transmit power in scenario S1 Fixed power, S2

Density-dependent transmit power, S2

3 15000

EE [bps/Hz/W]

EE [bps/Hz/W]

2.5 2 1.5 1 0.5 0

10000

5000

0 10 10 10 0

10 4

4

2

10 0

10 10 0

2

10 0

Fig. 3.9 The EE versus λ1 and λ2 based on the fixed transmit power and density-dependent transmit power in scenario S2

3.1.7.3

Optimal Deployment Solutions

In this subsection, the optimal deployment solutions for OP1 and OP2 are presented. Regarding OP1, Figs. 3.10, 3.11 and 3.12 offer the optimal coverage probability, the optimal λ2 and the optimal λ1 with respect to P max , respectively. From Fig. 3.10, we conclude that the maximum coverage probability increases with the increase of P max and finally becomes invariant with P max . The reason behind this is that a larger P max provides more flexible BS deployment choice which will finally approach the optimal BS deployment without the constraint of power consumption. Besides, the maximum coverage probability of HetNets with a density-dependent transmit power is more sensitive than that with a fixed transmit power. By comparison, the maximum

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

123

Maximum Coverage Probability Pr(SINR>T)

0.7

0.65

0.6

0.55 Density-dependent transmit power, S1 Density-dependent transmit power, S2 Fixed transmit power, S1 Fixed transmit power, S2 Fixed transmit power, S3

0.5

0.45

0.4 0

1000

2000

3000

4000

5000

6000

7000

8000

9000 10000

2

Total Power Constraints [W/km ]

Fig. 3.10 OP1: The Maximum coverage probability versus P max in scenarios S1, S2 and S3 (S3: {a1  10.3, a2  5.5, b1  156, b2  32})

coverage probability in S2 is superior to that in S1 when P max is small and inferior to that in S1 when P max becomes large. The optimal λ2 in S1 grows up to a certain value with the increase of P max , after which the optimal λ2 reaches its saturation, as illustrated in Fig. 3.11. While in S2, the optimal λ2 has an opposite tendency compared with that in S1. It is because, in S1, static power consumption takes up most of the total power, especially for the macrocell BSs. Therefore, deploying more small cell BSs can save much more energy. If ignoring the static power consumption, i.e., S2 is considered, every single macrocell BS can provide a better coverage performance than every single small cell BS, thus more macrocell BSs should be deployed in this scenario as shown in Fig. 3.12. Regarding OP2, Figs. 3.13, 3.14 and 3.15 present the maximum EE, the optimal λ2 and λ1 , respectively. It is observed in Fig. 3.13 that the maximum EE is a min min . It is because a smaller pcov corresponds to decreasing function with respect to pcov a less constraint to the network deployment, as a result choosing proper BS densities becomes much more feasible for mobile operators. The tendency of the red curve, i.e., utilizing a density-dependent transmit power in S2, is greatly different from the rest. It is because the corresponding curve of the EE versus λ1 and λ2 as illustrated in Figs. 3.8 and 3.9 is different from that of the rest. It is also noted that the optimal EE min min , e.g., when 0.43 ≤ pcov ≤ 0.48 for the red curve and is not strictly related to pcov min when 0.30 ≤ pcov ≤ 0.656 for the black curve, the optimal EE remains the same. It is because, in these regimes, the optimal λ1 and λ2 can guarantee the coverage min . To be specific, when probability is greater a bit more than the threshold, i.e., pcov 2 2 4 we deploy λ1  10 BS/km and λ2  316.2 BS/km , the coverage probability is min  0.43, the deployment of BSs, i.e., λ1  104 BS/km2 and 0.48. And if we set pcov

124

3 Energy Efficiency of Cellular Networks

10 1 Density-dependent transmit power, S1 Density-dependent transmit power, S2 Fixed transmit power, S1 Fixed transmit power, S2 Fixed transmit power, S3

Optimal

2

2 [BS/km ]

10 2

10 0

10 -1 0

1000

2000

3000

4000

5000

6000

7000

8000

9000 10000

Total Power Constraints [W/km 2 ]

Fig. 3.11 OP1: The optimal λ2 versus {a1  10.3, a2  5.5, b1  156.2, b2  32})

P max

in scenarios S1, S2 and S3 (S3:

10 1

Optimal

1

2

[BS/km ]

Density-dependent transmit power, S1 Density-dependent transmit power, S2 Fixed transmit power, S1 Fixed transmit power, S2 Fixed transmit power, S3

10 0

10 -1 0

1000

2000

3000 4000

5000

6000

7000

8000

9000 10000

Total Power Constraints [W/km2 ]

Fig. 3.12 OP1: The optimal λ1 versus {a1  10.3, a2  5.5, b1  156.2, b2  32})

P max

in scenarios S1, S2 and S3 (S3:

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

125

104 Density-dependent transmit power, S1 Density-dependent transmit power, S2 Fixed transmit power, S1 Fixed transmit power, S2

Optimal EE [bps/Hz/W]

103

102

101

100

10-1

10-2 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

p min cov

min in scenarios S1 and S2 Fig. 3.13 OP2: The maximum EE versus pcov

λ2  316.2 BS/km2 , can guarantee the minimal coverage probability thus keeps min is greater than a certain value, e.g., the optimal EE the same. Moreover, when pcov 0.6762 of the red curve, there is no feasible solution to achieve the optimal EE as the QoS of the network, i.e., the coverage probability, cannot be guaranteed. The optimin in Fig. 3.14. In Fig. 3.15, mal λ2 is also a decreasing function with respect to pcov min when pcov is small, the network is not constrained by the coverage performance and min is larger, deploying more small cell BSs can achieve a better EE. While when pcov mobile operators have to deploy more macrocell BSs to guarantee the network coverage performance, which results in a worse EE. The tendency of the red curve in min is small, the optimal λ1 decreases Fig. 3.15 is rather different from others. When pcov min with the increase of pcov , then a “flip-flop phenomenon” appears, i.e., the optimal λ1 jumps to a high value to guarantee the coverage performance and then decreases to a low value to achieve high EE. Besides, comparing Figs. 3.14 and 3.15, it is found min is small, that to achieve the optimal EE, an adjustment of λ1 is needed when pcov min ≤ 0.42, while λ2 may keep the same; an adjustment of λ2 is needed i.e., 0.30 ≤ pcov min min is medium, i.e., 0.42 ≤ pcov ≤ 0.65, while λ1 may keep the same; an when pcov min min is large, i.e., pcov ≥ 0.65. adjustment of λ1 as well as λ2 is needed when pcov

3.1.8 Conclusions and Future Work We investigated network performance of downlink ultra-dense HetNets and study the maximum energy-efficient BS deployment incorporating both NLoS and LoS transmissions. Through analysis, we found that the coverage probability with the

126

3 Energy Efficiency of Cellular Networks

Density-dependent transmit power, S1 Density-dependent transmit power, S2 Fixed transmit power, S1 Fixed transmit power, S2

103

Optimal

2

[BS/km2 ]

104

102

101 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.65

0.7

pmin cov min in scenarios S1 and S2 Fig. 3.14 OP2: The optimal λ2 versus pcov

104

Density-dependent transmit power, S1 Density-dependent transmit power, S2 Fixed transmit power, S1 Fixed transmit power, S2

102

Optimal

1

2 [BS/km ]

103

101

100

10-1 0.3

0.35

0.4

0.45

0.5

0.55

0.6

pmin cov min in scenarios S1 and S2 Fig. 3.15 OP2: The optimal λ1 versus pcov

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

127

MIRP association scheme is better than that with the MARP association scheme and by utilizing a density-dependent transmit power, the coverage probability improves when densities of macrocell BSs and small cell BSs are sparse or medium compared with the HetNets using the fixed transmit power. Moreover, we formulated two optimization problems to achieve the maximum energy-efficient deployment solution with certain minimum service criteria. Simulation results show that there are tradeoffs among the coverage probability, the total power consumption and the EE. In detail, the maximum coverage probability with ideal power consumption is superior to that with practical power consumption when the total power constraint is small and inferior to that with practical power consumption when the total power constraint becomes large. Furthermore, the maximum EE is a decreasing function with respect to the coverage probability constraint. In our future work, networks with idle mode capability and multiple-antennas are also worth further studying.

3.1.9 Appendix 3.1.9.1

Appendix 1: Proof of Theorem 1

The coverage probability in a K-tier HetNet with the MIRP association scheme is defined as follows     MIRP (3.35) pcov ∪ SINRk Xk,o  > Tk . ({λk }, {Tk })  Pr k∈K,Xk,o ∈k

MIRP As we consider both NLoS and LoS transmissions, pcov ({λk }, {Tk }) can be further expressed by     MIRP   pcov ({λk }, {Tk })  E I ∪ SINRk Xk,o > Tk k∈K,Xk,o ∈k ⎫ ⎧ K ⎬ ⎨      Xk,o  > Tk  I SINRNL E k ⎭ ⎩ k1 Xk,o ∈NL k $ %& ' I ⎫ ⎧ K ⎬ ⎨      I SINRLk Xk,o  > Tk + E , (3.36) ⎭ ⎩ k1 Xk,o ∈Lk $ %& ' II

where I(·) is the indicator function, (a) follows from [42, Lemma 1] under the assumpL tion that Tk  1∀k and the independence between NL k and k , Part I and II in

128

3 Energy Efficiency of Cellular Networks

Eq. (3.36) can be comprehended as the probability that the typical MU is covered by NLoS BSs and LoS BSs, respectively. Proof For Part I in Eq. (3.36), we have ⎫ ⎧ ⎪ K ⎬ ⎨    ⎪ (a) NL Part I  I SINRNL > Tk E Rk,o k ⎪ ⎪ ⎭ k1 ⎩ NL NL Rk,o ∈k

K ∞ (b)



(

λNL k (r ) Pr

k1r 0

K ∞ (c)



) −αkNL h NL k r > Tk dr

K

K NL L j1 I j + j1 I j + η

−Tk ηr λNL k (r )e

αkNL

K       NL NL LMIRP Tk r αk LMIRP Tk r αk dr , I NL IL j

k1r 0

j

(3.37)

j1

to NL where (a) is due to the transformation from NL k k , (b) follows NL from Campbell theorem [20] and variable substitution, i.e., Rk,o → r and NL L



−α j −α j def NL def NL L L I j  i:r ∈NL  h i r j,i and I j  i:r j,i ∈NL h i r j,i are the aggregate interferj,i

j

j

*

 ence from NLoS BSs and LoS BSs in the j-th tier, respectively, where NL j   NL NL NL ∼ exp(1), LMIRP (s) and LMIRP (s) denote the j \ 0, Rk,o , (c), is due to h k I NL IL j

j

Laplace transform of I jNL and I jL evaluated at s with the MIRP association scheme, (s) as follows respectively. Using the definition of Laplace transform, we derive LMIRP I jNL   NL LMIRP (s)  E I jNL e−s I j I jNL ⎤ ⎡  −α NL ⎥ ⎢  j (a) −sh NL r j,i ⎥ NL e  ENL ⎢ E h ⎦ ⎣ j

*

i:r j,i ∈ j

NL

(b)

(

 exp



+

,

1 1 + sy −α j

NL



y0

NL ⎢  exp⎣−s 1/α j

− 1 λNL j (y)dy

  ∞ λNL ys 1/αNL j j

y0

)

1+y

α NL j

⎤ ⎥ dy ⎦,

(3.38)

where (a) follows from the independence between the fading random variables (RVs), i.e., h NL j , (b) follows from probability generating functional (PGFL) of PPP [20]. (s) is obtained as follows Similarly, LMIRP IL j

3.1 On the Energy-Efficient Deployment for Ultra-Dense …



∞

L ⎢ LMIRP (s)  exp⎣−s 1/α j IL

129

⎤ L

λLj (ys 1/α j )

j

1+y

y0

α Lj

⎥ dy ⎦.

(3.39)

Using a similar approach compared with Part I, Part II can also be easily obtained as follows K 



Part II 

L

αk λLk (r )e−Tk ηr

k1t0

K       L L LMIRP Tk r αk LMIRP Tk r αk dr , I NL IL j

j

(3.40)

j1

where LMIRP (s) and LMIRP (s) are defined in Eqs. (3.38) and (3.39), respectively. I jNL I jL Then, the proof is completed.

3.1.9.2

Appendix 2: Proof of Theorem 2

Using the law of total probability, we can calculate coverage probability MARP pcov ({λk }, {Tk }) as MARP pcov ({λk }, {Tk }) 

K

pkNL ({λk }, {Tk }) +

k1

K

pkL ({λk }, {Tk }),

(3.41)

k1

where the first part and the second part on the right side of the equation denote the conditional coverage probability that the typical MU is in the coverage of NLoS BSs and LoS BSs, respectively, by observing that the two events are disjoint. Given that the typical MU is served by an NLoS  and the maximum average received power  BS NL . Then is denote by PkNL , i.e., PkNL  max Pk.i     - NL .  . NL NL pk λk , {Tk }  Pr SINRk > Tk ∩ ∩ PkNL > P Lj , j∈K  - NL . NL NL ∩ Yk ∩ Pk > P j j∈K\k    . NL  EYkNL Pr SINRk > Tk | ∩ PkNL > P Lj , j∈K  - NL . NL NL ∩ Yk ∩ Pk > P j j∈K\k ⎫ ⎪ ⎪ ⎪  ⎬ NL L NL NL NL , (3.42) × Pr ∩ {Pk > P j }, ∩ {Pk > P j }|Yk ⎪ j∈K j∈K\k ⎪ $ %& '⎪ ⎭ I

130

3 Energy Efficiency of Cellular Networks

where YkNL is the equivalent distance between the typical MU and the BS providing the maximum average received power to the typical MU in NL k , i.e.,  −αkNL NL   −α NL , and also note that PkNL  YkNL k . We label YkNL  arg max R NL ∈NL Rk.i k.i k the formulas before and after the product sign “×” as Part II and Part I, respectively. For Part I,   . . NL |Y Pr ∩ PkNL > P Lj , ∩ PkNL > P NL j k j∈K j∈K\k       NL  Pr PkNL > P Lj |YkNL Pr PkNL > P NL j |Yk j∈K





j∈K\k

 −αNL  −αLj NL  Pr YkNL k > Y Lj |Yk ,

(3.43)

j∈K

where YkL , similar to the definition of YkNL , is the equivalent distance between the typical MU and the BS providing the maximum average received power to the typical  −αkL  −αL L and also note that PkL  YkL k . MU in Lk , i.e., YkL  arg max R L ∈L Rk,i k,i

k

For Part II, we know that SINRNL k 

K j1

NL h NL k Pk

. I jNL + Kj1 I jL +η

The conditional coverage

probability can be derived as follows ⎡ ⎤  NL −αkNL h NL Yk (a) k Part II  Pr ⎣ K > Tk |E⎦

K NL L j1 I j + j1 I j + η ⎡ ⎞ ⎤ ⎛ K K NL   α NL k ⎝  Pr ⎣h NL I jNL + I jL + η⎠|E⎦ k > Tk Yk j1 (b) −Tk ηr αkNL

e

K  

LMARP1 I NL j



j1

   NL αkNL Tk r αk LMARP1 T , r L k I

(3.44)

j

j1 def

NL where in (a) event E (∩ j∈K {PkNL > P Lj }, ∩ j∈K\k {PkNL > P NL j }) ∩ Yk , (b) folNL NL MARP1 lows from h k ∼ exp(1) and variable substitution, i.e., Yk → r , L I NL (s) and j

LMARP1 (s) denote the Laplace transform of I jNL and I jL evaluated at $s$ with the IL j

MARP association scheme, respectively. Like Appendix 1, we derive LMARP1 (s) as I jNL follows ⎡ ⎤   ⎢  ⎥ 1 NL ⎥ LMARP1 (s)  E I jNL e−s I j  ENL ⎢ NL ⎦ I jNL ⎣ −α j j NL 1 + sr j,i i:r j,i ∈ j *

3.1 On the Energy-Efficient Deployment for Ultra-Dense …

⎡ ⎢  exp⎢ ⎣−

(a)



∞

y(YkNL )

131

λNL j (y)

αkNL /α NL j

1+y

α NL j

/s

⎥ dy ⎥ ⎦,

(3.45)

  αNL /αNL  NL NL and in (a) the lower limit of integral is YkNL k j where NL j   j \ 0, Yk MARP1 which guarantees that PkNL > P NL (s) j , ∀ j ∈ K\k in event E is true. Similarly, L I jL is calculated by ⎡ ⎤ ∞  L   ⎢ ⎥ λ j (y) L (a) ⎥, LMARP1 (s)  E I jL e−s I j  exp⎢ dy (3.46) L I jL ⎣− ⎦ αj 1 + y /s α NL /α Lj y(YkNL ) k *

αNL /αL  where in (a) the lower limit of integral is YkNL k j which guarantees that {PkNL > P Lj }, ∀ j ∈ K in event E is true. Finally, note that the value of pkNL ({λk }, {Tk }) in Eq. (3.42) should be calculated by taking the expectation with respect to YkNL in terms of its PDF, which is given by   NL f YkNL (ε)  λNL k (ε) exp − k ([0, ε]) .

(3.47)

as in [31]. By substituting Eqs. (3.43), (3.44), (3.45), (3.46), and (3.47) into Eq. (3.42), we can derive the conditional probability pkNL ({λk }, {Tk }). Given that the typical MU is connected to a LoS BS, the conditional coverage probability pkL ({λk }, {Tk }) can be derived using the similar way as the above. Thus the proof is completed.

3.1.9.3

Appendix 3: Proof of Corollary 5

MIRP MARP By comparing pcov ({λk }, {Tk }) and pcov ({λk }, {Tk }) in Theorems 1 and 2, it is noticed that the difference between them lies in the Laplace transform and the term

K

αkL /α Lj

αkL /α NL j

])+ j ([0,r ])] . Thus, we prove this corollary by taking the cove− j1 [ j ([0,r erage probability for a typical MU which is served LoS BSs for an example. L

NL

MIRP MARP pcov ({λk }, {Tk }) > pcov ({λk }, {Tk })

⇔e



K

j1

      α L /α L α L /α NL K  

Lj 0,r k j + NL 0,r k j j

    L L LMARP2 Tk r αk LMARP2 Tk r αk / I NL IL j

j1 K       L L LMIRP Tk r αk LMIRP Tk r αk 0

Lσ 2 (−2π jω)

−∞



· L I (−2π jω) · xi

psuc (γ0 , )  E

⎧ ⎨ ⎩

e

−1 2π jω K γ ·y·{L(r )} 0

2

(3.64)

dω +∞

1 · 2π

2π λ B r e−π λ B r dr ·

Lσ 2 (−2π jω)

−∞

r >0

·L I (−2π jω) ·

 −1 2π jω K γ0 ·S y j (xi ){L(r )} e

2π jω

xi



−1

4π 2 jω

−1

⎫ ⎪ ⎬ dω

⎪ ⎭

2π λ B r e−π λ B r dr 2

 r >0



+∞ 6 +∞

−∞ f (y) · e

·

−1 2π jω K γ ·y·{L(r )} 0

6 +∞ dy − −∞ f (y)dy

4π 2 jω

−∞ +∞

Lσ 2 (−2π jω) · L I (−2π jω)dω

×

(3.65)

xi

−∞



4 4  4 6 +∞ 2π jω K ·y·{L(r )}−1 +∞44 6 +∞ γ 0 dy − −∞ f (y)dy 44 4 −∞ f (y) · e 4 4 4 4 4π 2 jω 4 4 r >0 −∞ 4 4 4 4 4 4 · 4Lσ 2 (−2π jω)4 · 44L I (−2π jω)44dωdr < ∞. 



psuc (γ0 , ) 

xi

(3.66)

2π λ B r e−πλ B r dr 2

r >0 +∞ 6 +∞ −∞

·

f (y) · e



2π jω Kγ ·y·{L(r )}−1 0

4π 2

−∞

dy −

6 +∞ −∞

f (y)dy



· Lσ 2 (−2π jω) · L Ix (−2π jω)dω i    +∞ L −1  − L Sy j (xi ) (0) Sy j (xi ) −2π jωK · {γ0 L(r )} −πλ B r 2  2π λ B r e 2 4π jω −∞

r >0

· Lσ 2 (−2π jω) · L Ix (−2π jω)dωdr

(3.67)

i

 psuc (γ0 , ) 

2π λ B r e

−πλ B r 2

+∞ L

Sy j (xi ) (2π js)

− L Sy j (xi ) (0)

4π 2

js −∞     · Lσ 2 2π jγ0 L(r )s/K  · L Ix 2π jγ0 L(r )s/K  dsdr r >0

i

(3.68)

3.2 Spatial Spectrum and Energy Efficiency of Random Cellular …

143

The Laplace transform of the noise is expressed as [58] Lσ 2 (s)  e−σ s . 2

(3.69)

According to the definition of Ix i in (3.60b), we follow a basic technique in [77] to obtain the outage probability condition on having interferers, which consists of two steps: (1) Consider a finite network, say on disk of radius a centered at the origin, and condition on having a constant number of nodes in this finite area, for example nodes. Assume that the nodes’ locations are independent identical distribution. (2) Let the disk radius go to infinity, while keeping the node density, i.e., the ratio of the number of nodes to the network area, constant. Step 1: Conditioning on having nodes (i.e., interferers) in the disk of radius a, the Laplace transform of aggregate interference is denoted by 4 . L Ix ,a (s)  E e−s Ixi , ,a 4# B(ori, a)  , i

(3.70)

where B(ori, a) is the set of nodes located on a disk with radius a centered at the origin ori and # B(ori, a)  means that the number of nodes in B(ori, a) is . These nodes (i.e., interferers) are uniformly distributed on the disk with radial density 7 f R (r ) 

2r i f 0  r  a a2

0

other wise

(3.71)

Therefore, the Laplace transform L Ix ,a (s) is the product of the individual Laplace i transform, which is given by (3.72). 4   4 −s I L I ,a (s)  E e xi , ,a 4# B(ori, a)  xi ⎧⎛ ⎞ ⎫ ⎪ ⎪ ⎨ a 2r ⎬    −1 ⎠ E ⎝ exp −s K · S (x ){L(r )} dr y l i 2 ⎪ ⎪ a ⎩ ⎭

(3.72)

0

Assume that the path loss law is L(r )  r b , then the individual Laplace transform is derived as follows [78] a

  2r exp −s K  · S yl (xi ){L(r )}−1 dr 2 a

0

a 

  2r exp −s K  · S yl (xi )r −b dr a2

0

2 − 2 a

∞ 1 a

  y −3 exp −s K  · S yl (xi )y b dy

144

3 Energy Efficiency of Cellular Networks 

 2 1 1  2 s K · S yl (xi ) b ·  − , s K  b S yl (xi ) b b a

(3.73)

where (·) denotes Gamma function. Step 2: When a goes to infinity, (3.73) is further derived by a

  2r exp −s K  · S yl (xi ){L(r )}−1 dr 2 a→∞ a 0  2 1 2  s K  · S yl (xi ) b ·  − , 0 b b  2 2 1  s K  · S yl (xi ) b ·  − b b lim

(3.74)

Then L Ixi , (s) is transformed into (3.75). By substituting (3.68) and (3.75) into (3.61), the outage probability conditioned on having interferers is derived by (3.76). 

L I ,a (s)  lim L Ix a→∞

xi

 

 2 1  2 s K · S yl (xi ) b ·  − b b     2 E S y (xi ) S yl (xi ) b l

(s)  E S y (xi ) l i , ,a

 1   2 2 sK b ·  − b b

 pout (γ0 , )  1 −

2π λ B r e

−πλ B r 2

+∞ L

Sy j (xi ) (2π js)

(3.75)

− L Sy j (xi ) (0)

4π 2 js

−∞

r >0



    · Lσ 2 2π jγ0 L(r )s/K  × L Ix 2π jγ0 L(r )s/K  dsdr i

 1− r >0

2π λ B r e−πλ B r

2

+∞ L

Sy j (xi ) (2π js)

−∞

4π 2

− L Sy j (xi ) (0) js

  2 1 2π jγ0 σ 2 r −b s × 2π jγ0 r −b s b · exp −  K b     2  2 · − E Syl (xi ) S yl (xi ) b dsdr b

3.2.3.3

(3.76)

Blocking Probability with Rayleigh Fading

When wireless signals are assumed to suffer Rayleigh fading, the PDF of signal is expressed as   f (x)  x/σ 2 · exp −x 2 /2σ 2 (0  x  ∞). And the PDF of the signal power is expressed as

(3.77)

3.2 Spatial Spectrum and Energy Efficiency of Random Cellular …

145

  f (y)  1/2σ 2 · exp −y/2σ 2 (0  y  ∞).

(3.78)

It is obvious that the PDF of the signal power follows an exponential distribution. Without loss of generality, let S yl (xi ) ∼ exp(1). Then the Laplace transform of S yl (xi ) is derived by 1 , s+1

L Syl (xi ) (s) 

(3.79)

and then E



S yl (xi )

 2 b 

  1+

2 . b

(3.80)

Moreover, the Laplace transform of the aggregate interference is derived by  L Ix ,a (s)  i

 1    b2 2 sK · − b b



 2 . · 1+ b

(3.81)

By substituting (3.81) into (3.76), the outage probability is derived as (3.82).  +∞ pout (γ0 , )  1 −  ×

r >0 −∞

  λ B r e−πλ B r − · exp −2π jγ0 σ 2 r −b s/K  2π js + 1 2

  b2 1 2 −b 2π γ0 r js ·  − b b



  1+

2 dsdr b

(3.82)

Assume that a call will be dropped if the call cannot be served immediately in PVT random cellular networks. As a consequence, if the total number of active MUs exceeds the number of available channels in the typical cell Cori , a call will be blocked due to a lack of sufficient channel resource. Therefore, the blocking probability is derived by pb 



π (m, n) 

mnC

1 λ χ η

i jC

m

  1 C α m! n β

n

,

(3.83a)

with 1−ε α  , β ε  NI NI ε  lim p (1 − p) N I − , pout (γ0 , ) N I →∞ 1

(3.83b) (3.83c)

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3 Energy Efficiency of Cellular Networks

 +∞ pout (γ0 , )  1 −  ×

r >0 −∞

  λ B r e−πλ B r · exp −2π jγ0 σ 2 r −b s/K  2π js + 1 2



  b2 1 2 −b 2π γ0 r js ·  − b b



 2 dsdr,  1+ b

(3.83d)

Furthermore, the mean sojourn time of a MU can be derived by Little’s theorem [79] D where N  Cori .

3.2.3.4

C m1

N , λ

(3.84)

m · π (m, n) is the mean number of served MUs in the typical cell

Performance Analysis

Assuming Rayleigh fading channels, the blocking probability and the mean sojourn time of PVT random cellular networks can be numerically computed. Following the simulation configuration in [42, 50, 74], default parameters for a PVT random cellular network are configured as follows: BS intensity is λ B  0.2 per square kilometers; the maximum number of available channels in a typical cell Cori is C  20; the arrival rate of calls is λ  1 min−1 ; the BS transmitting power is Pyi  30 dBm; the mean noise power is σ 2  0 dBm; the path loss exponent is b  4; the antenna gain is K  31.54 dB for an urban microcell environment. Figure 3.18 shows the call blocking probability with respect to the SINR threshold considering different maximum number of channel numbers C in a typical cell Cori . When the maximum number of available channels is fixed, it is observed that the blocking probability increases with the increase of SINR threshold. The reason is that the successful decoded signal is reduced for MUs when the SINR threshold is increased. When the SINR threshold is fixed, the blocking probability increases with the decrease of the maximum available channels number in the cell Cori . That is because a call is more likely to be dropped when channel resource is insufficient. Figure 3.19 illustrates the blocking probability versus the SINR threshold for different path loss exponents b in the typical cell Cori . When the SINR threshold is fixed, the blocking probability decreases with the increase of the path loss exponent. It is well known that the path loss exponent affects both desired signals and interference signals. However, these curves imply that the path loss exponent has a more significant attenuation on the aggregated interference in PVT random cellular networks. Figure 3.20 shows the mean sojourn time versus the SINR threshold for different arrival rates of calls. When the SINR threshold is fixed, the mean sojourn time decreases with the increase of arrival rate of calls. When the arrival rate of calls are

3.2 Spatial Spectrum and Energy Efficiency of Random Cellular …

147

Fig. 3.18 Blocking probability versus SINR threshold with different maximum channel numbers C

Fig. 3.19 Blocking probability versus SINR threshold with different b

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3 Energy Efficiency of Cellular Networks

Fig. 3.20 Mean sojourn time versus SINR threshold with different λ

fixed, the mean sojourn time decreases with the increase of SINR threshold. That is because the number of available channels decreases with the increase of SINR threshold. As a result, the service ability of cellular network is decreased and then the mean sojourn time of a MU “staying” in the PVT random cellular network is decreased.

3.2.4 Spatial Spectrum and Energy Efficiency of PVT Random Cellular Networks In this section, we further evaluate the spatial spectrum and energy efficiency of PVT random cellular networks.

3.2.4.1

Spatial Spectrum and Energy Efficiency

Assume that the bandwidth of a typical cell Cori is B, the throughput of the typical cell Cori is then given by  . Tthr oughput  (1 − pb )B · E log2 1 + S I N R y j (xi ) ·



m · π (m, n).

0mnC

(3.85)

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149

Without loss of generality, the number of interferers on a wireless link of PVT random cellular networks is assumed as . Therefore, the link capacity between a MU and the associated BS is defined as  . E log2 1 + S I N R y j (xi ) +∞     P log2 1 + SINRyj (xi ) > t dt  0

+∞   P SINRyj (xi ) > 2t − 1 dt  0

+∞    1 − Pout 2t − 1, dt 

(3.86)

0

A simple proof of (3.86) is given as follows. Proof Consider a random value x which has continues PDF, then its expectation is given by +∞

0

+∞ x f (x)d x + x f (x)d x,

−∞

0

x f (x)d x 

E{x}  −∞

(3.87)

Furthermore, the two terms of the right side of (3.87) are written as 0

0 x f (x)d x  −

−∞

−∞

⎛ ⎝

x

⎛ 0 ⎞  0  y 0 ⎝ dy ⎠ f (x)d x  − f (x)d xd y  − F(y)dy, −∞ −∞

x

⎞ dy ⎠ f (x)d x 

0

−∞

(3.88a) +∞+∞ 0

y

+∞ f (x)d xd y  [1 − F(y)]dy,

(3.88b)

0

where F(·) denotes the cumulative probability function (CDF) operation. As we know S I N R y j (xi )  0. Substituting S I N R y j (xi ) into (3.88b), the result of (3.86) is obtained. Based on (3.85), the spatial spectrum efficiency of the PVT random cellular network is derived as (3.89) according to [65], where λ B is the BS density of PVT random cellular networks. SS E  λ B · Tthr oughput

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3 Energy Efficiency of Cellular Networks

+∞    1 − Pout 2t − 1, dt ·  (1 − pb )Bλ B ·



m · π (m, n),

0mnC

0

(3.89) The energy efficiency of the typical cell Cori during the whole life time is derived as follows ϕ

E total , Dtotal

(3.90)

where E total denotes the BS energy consumption in the life time and Dtotal denotes the BS throughput in the life time. In the entire life time, the BS energy consumption includes the operation energy and the embodied energy [9]. Moreover, the embodied energy includes the initial embodied energy consumed in factories and the maintenance embodied energy in the life time. Therefore, the BS energy consumption in the life time is expressed as E total  E E Minit + E E Mmaint + E E Moper ,

(3.91)

where E E Minit denotes the initial embodied energy, E E Mmaint denotes the maintenance embodied energy, and E E Moper is the operation energy, which is a linear function of the total transmission power over all occupied channels and is given as follows ⎡ ⎤ m · π (m, n) + k ⎦ × tli f etime , (3.92) E E Moper  ⎣h · Pchl 0mnC

where Pchl is the transmission power over a wireless channel, tli f etime is the life time of a BS, h and k are linear coefficients of the total transmission power. The total throughput of BS in the life time is expressed as Dtotal  tli f etime · Tthr oughput .

(3.93)

Therefore, the energy efficiency of the typical cell Cori is derived as (3.94). tli f etime · Tthr oughput  .

+ E E Mmaint + h · Pchl 0mnC m · π (m, n) + k · tli f etime 

ϕ E E Minit

(3.94) Based on the Palm theory, the result of (3.94) can be extended to the whole PVT random cellular network.

3.2 Spatial Spectrum and Energy Efficiency of Random Cellular …

3.2.4.2

151

Numerical Results and Discussion

Based on the spatial spectrum and energy efficiency analysis, numerical results are illustrated in this subsection. Based on default parameters used in Sect. 3.2.3.4, some parameters used for comparing grid cellular networks (i.e., regular hexagonal cellular networks) and PVT random cellular networks are configured as follows [9, 74, 80]: the channel bandwidth is B  0.1 MHz, the embodied energy is configured as E E Minit + E E Mmaint  85 GJ, the transmission power over a wireless channel is Pchl  1 W, h  7.84, k  71.5, without loss of generality, the BS life time tli f etime is configured as 1 year, e.g., 365 days. Figure 3.21 illustrates the spatial spectrum efficiency of PVT and grid cellular networks with respect to the path loss exponent and the BS density in cellular networks, in which “PVT model” labels the PVT random cellular network and “Grid model” represents the regular hexagonal cellular network. When the BS density λ B is fixed, numerical results of PVT and grid cellular networks consistently show that the spatial spectrum efficiency increases with the increase of the path loss exponent. When the path loss exponent b is fixed, numerical results of PVT and grid cellular networks consistently illustrate that the spatial spectrum efficiency increases with the increase of the BS density. Comparing with PVT model results and grid model results in Fig. 3.21, it is shown that values corresponding to PVT random cellular networks are obviously less than values corresponding to grid cellular networks. This result is confirmed in [59] which shows that the average transmission rate of PPP random cellular networks is less than the average transmission rate of grid cellular networks. Considering that the PVT random cellular network forms a special case of the PPP random cellular networks, the spatial spectrum efficiency of PVT random cellular network is less than the spatial spectrum efficiency of grid cellular network when the transmission bandwidth is fixed in PVT and grid cellular networks. In Fig. 3.22, the effect of the call arrival rate λ on the spatial spectrum efficiency of PVT and grid cellular networks is investigated. When the path loss exponent is fixed, numerical results of PVT and grid cellular networks consistently demonstrate that the spatial spectrum efficiency decreases with the increase of the call arrival rate. When the call arrival rate increases, the blocking probability in the cell is correspondingly increased. The increase of blocking probability in the cell in turn reduces the spatial spectrum efficiency of PVT and grid cellular networks. Figure 3.23 shows the spatial spectrum efficiency with respect to the call arrival rate considering different SINR thresholds in PVT and grid cellular networks. When the call arrival rate in cellular scenarios is fixed, the spatial spectrum efficiency decreases with the increase of the SINR threshold. When the SINR threshold is fixed, there exist thresholds for different call arrival rates in cellular scenarios. Below the threshold, the spatial spectrum efficiency increases with the increase in the call arrival rate and above the threshold the spatial spectrum efficiency decreases with the increase in the call arrival rate. Numerical results of PVT and grid cellular networks consistently validate that there exists maximum spatial spectrum efficiency values considering different call arrival rates in cellular scenarios.

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3 Energy Efficiency of Cellular Networks

Fig. 3.21 Spatial spectrum efficiency with respect to the path loss exponent b considering different BS densities λ B in PVT and grid cellular networks

Fig. 3.22 Spatial spectrum efficiency with respect to the path loss exponent b considering different call arrival rates λ in PVT and grid cellular networks

3.2 Spatial Spectrum and Energy Efficiency of Random Cellular …

153

Fig. 3.23 Spatial spectrum efficiency with respect to call arrival rate λ considering different SINR thresholds γ0 in PVT and grid cellular networks

The impact of the call arrival rate on the energy efficiency of PVT and grid random cellular networks is evaluated in Fig. 3.24. When the SINR threshold is fixed, there exists thresholds for different call arrive rates in cellular scenarios. Below the threshold, the energy efficiency increases and above the threshold the energy efficiency decreases with the increase in the call arrival rate. Numerical results of PVT and grid cellular networks consistently validate that there exist maximum energy efficiency values considering different call arrive rates in cellular scenarios. Therefore, to achieve an optimal spectrum and energy efficiency of cellular networks, the call arrive rate and the SINR threshold in a cell should be considered carefully by telecommunications operators. Figure 3.25 depicts the energy efficiency with respect to the path loss exponent considering different BS densities in PVT and grid cellular networks. When the BS density of cellular networks is fixed, numerical results of PVT and grid cellular networks consistently confirm that the energy efficiency increases with the increase of the path loss exponent. With the increase of the path loss exponent, both the desired signal and the interference are exponentially attenuated over wireless channels. Since the distance between the interfering transmitters and the receiver is longer than the distance between the desired BS and the receiver, the interference will experience larger attenuation than the desired signal when the path loss exponent increases. Therefore, the outage probability decreases with the increase of the path loss exponent. This result implies that the energy and spatial spectrum efficiency increase with the increase of the path loss exponent in Figs. 3.22, 3.23 and 3.24. When the path

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3 Energy Efficiency of Cellular Networks

Fig. 3.24 Energy efficiency with respect to call arrival rate λ considering different SINR thresholds γ0 in PVT and grid cellular networks

Fig. 3.25 Energy efficiency with respect to path loss exponent b considering different BS densities λ B in PVT and grid cellular networks

loss exponent is fixed, the energy efficiency decreases with the increase of the BS density in PVT and grid cellular networks.

3.2 Spatial Spectrum and Energy Efficiency of Random Cellular …

155

3.2.5 Conclusions To evaluate the spatial spectrum and energy efficiency in network level, the Markov chain is first integrated into the PVT random cellular networks in this section. Based on the Markov chain based channel access model, spatial spectrum and energy efficiency are analyzed for PVT random cellular networks. To derive these models, a Markov chain is first presented for modeling of wireless channel access in a typical PVT cell. Moreover, taking into account the path loss and Rayleigh fading effects over wireless channels, the outage probability and the blocking probability are derived for a typical PVT cell. Furthermore, the spatial spectrum and energy efficiency are obtained for PVT random cellular networks. Numerical results have shown that the call arrival rate in a PVT cell and the BS density of PVT random cellular networks have adverse effects on the spatial spectrum efficiency of PVT random cellular networks. Moreover, the path loss exponent and the SINR threshold have great impact on the energy efficiency of PVT cellular networks. In the end, our results provide insights into the evaluation of spatial spectrum and energy efficiency of PVT random cellular networks considering different call arrival rates in cellular scenarios.

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular Networks: A Mean Field Game Approach 3.3.1 Introduction To improve the spectral efficiency in the limited frequency bands, the full duplex transmission technology is emerging as one of key technologies for the fifth generation (5G) cellular networks [79, 81–84]. Considering the battery constraint of the user equipment (UE), the half-duplex transmission technology is still configured for UEs. In this case, it is unbalanced for downlink and uplink transmissions in 5G cellular networks with full duplex transmissions. On the other hand, the energy harvesting technology is proposed to periodically charge the battery of radio remote unit (RRU) in 5G cellular networks [85, 86]. Moreover, the RRU transmission power is constrained by the battery volume of RRUs. Therefore, it is a great challenge for optimizing the transmission power strategy of RRU to not only keep the coverage of the network at a stable level but also improve the energy efficiency of 5G cellular networks with full duplex transmissions. To improve the spectrum and energy efficiency, the full-duplex transmission and energy harvesting technologies are emerging for 5G wireless communication systems [87]. The full-duplex transmission technology can be used to transmit and receive wireless signals in the same frequency and then improve the spectrum efficiency of wireless communications [88]. However, the self-interference is an inevitable problem for wireless full-duplex communications [89]. Some potential self-interference cancellation technologies were discussed in [90–92]. On the other

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3 Energy Efficiency of Cellular Networks

hand, two nonconvex power allocation algorithms were developed to improve the spectrum and energy efficiency of wireless full-duplex communication systems with massive MIMO antennas [93]. The path-following algorithms were developed for jointly designing the energy harvesting time and beamforming to maximize the sum rate and energy efficiency of wireless full-duplex transmission systems [94]. By proving the result that the relation of the energy and spectrum efficiency is quasiconcave for cellular networks with full-duplex transmissions, optimal energy efficiency oriented resource allocation algorithms were developed for cellular networks with spectrum efficiency constraints [95]. The energy harvesting technologies have been proposed to collect and save energy in the air for optimizing the energy efficiency of wireless networks [96]. When small cell BSs are harvested with energy from the environmental power, a discrete single-controller discounted two-player stochastic game was proposed to address the problem of downlink power control in a two-tier microcell-small cell network under co-channel deployment [97]. By modelling the energy harvesting and consumption as a probabilistic framework for small cell networks, a bandit framework was proposed to solve the user association problem in a distributed manner [98]. To overcome the uncertainty of the environmental conditions that affects the harvesting energy, two online energy trading approaches, one decentralized and one centralized were proposed to minimize the nonrenewable energy consumption in a multi-tier cellular network [99]. Based on the energy harvesting and full-duplex transmission technologies, the cell association and BS power allocation scheme were proposed to optimize the energy efficiency of cellular networks [100]. By modelling the battery dynamics of an energy harvesting small cell BS as a discrete-time Markov chain and considering a practical power consumption model, a tractable model was proposed to optimize the energy efficiency of heterogeneous cellular networks [101]. With the massive MIMO and millimeter wave transmission technologies adopted for 5G wireless communications, small cell BSs have to be densely deployed in 5G cellular networks [1]. Since 5G cellular networks are composed of a large number of small cell BSs [102–105], the mean field game theory can be used for the performance investigation of 5G cellular networks. The mean field game theory is the study of strategic decision making in very large populations of small interacting agents [106–108]. Moreover, the mean field game theory has been used for describing wireless networks [109–111]. The interaction between the primary user and a large number of secondary users was formulated as a hierarchical mean field game problem for cognitive radio networks [109]. Taking into account the limited energy available to mobile transmitters and the effects of channel fading, the energy-efficient distributed power control was formed as a mean field game for wireless networks [100]. Furthermore, the mean field game is utilized for improving the energy efficiency of wireless networks [112, 113]. By weaving notions from the Lyapunov optimization and mean-field theory, a new approach for joint power control and user scheduling was proposed for optimizing the energy efficiency in ultra-dense small cell networks [112]. Based on a mean-field game theoretic framework with the interference meanfield approximation, a novel energy and interference aware power control policy was

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular …

157

proposed to optimize the energy efficiency of ultra-dense device-to-device (D2D) Networks [113]. However, in all the aforementioned capacity studies, only simple scenarios, such as BSs with single antenna or half duplex technologies, were analyzed by the mean field game theory and a simple power control scheme was developed for traditional cellular networks. Besides, the network energy efficiency adopting full duplex and energy harvesting technologies has not been fully investigated. Moreover, detailed investigation of the interaction among BS power control strategy in 5G cellular networks with full duplex and energy harvesting technologies is surprisingly rare in the open literature.

3.3.2 System Model In this section BSs and UEs are assumed to be randomly located in an infinite plane. Moreover, the motion of users is isotropic and relatively slow, such that during an observation period, e.g., a time slot, the relative positions of UEs and BSs are assumed to be stationary. Following the studies in [3, 114, 115], the locations of UEs and BSs Poisson point processes, which are 3 are modeled as two independent3 denoted as UE  {UEi : i  1, 2, 3, . . .} and BS  {BSk : k  1, 2, 3, . . .} with intensities λUE and λBS , respectively. The detailed network topology is illustrated in Fig. 3.26a. Every BS is equipped with a baseband unit (BBU) and a RRU. To improve the coverage efficiency of the BS, the BBU and RRU are separated, i.e., the RRU is located at the outdoor position and the BBU is located at the indoor position. The BBU and RRU are connected by optical fiber links. In this case, the BBU is usually supplied by the traditional power lines. However, the power supply of RRU is an issue considering the deployment problem of electricity lines in outdoor scenarios. Based on the solution in [85], in this section every separated RRU is assumed to be equipped with a battery which can be charged by the energy harvesting method. The basic structure of BBU and RRU is shown in Fig. 3.26b. Considering the volume limit of battery, the maximum transmission power of a RRU is configured as pT-RRUmax . The transmission power of the RRU at the kth BS is denoted by pT-RRUk , k  1, 2, 3, …. The value range of pT-RRUk is given by 0 ≤ pT-RRUk ≤ pT-RRUmax . The full duplex technology is adopted at BSs and then the transmitting and receiving signals can be operated by BSs over the same frequency bands. Moreover, the self-interference is inevitable between the transmitting and receiving antennas at BSs. The half-duplex technology is adopted for UEs. The full duplex transmission scheme is illustrated in Fig. 3.27. In this section we focus on how to optimize the energy efficiency of cellular networks with full duplex and energy harvesting technologies in the time slot t, 0 ≤ t ≤ T , where T is the battery charging period of RRUs.

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3 Energy Efficiency of Cellular Networks

Fig. 3.26 Cellular networks with energy harvesting

Fig. 3.27 Full duplex transmission scheme

3.3.3 Formulation of Energy Efficiency 3.3.3.1

SINR for Downlinks

Since the half-duplex scheme is adopted at UEs, the self-interference is ignored for UEs in this section. Considering the random cellular network topology in Fig. 3.26a, the interference from adjacent BSs and the wireless channel noise are included into the received signals at UEs. Therefore, the signal-to-interference-and-noise ratio (SINR) of downlink between the desired BS BSk and the UE UEi  1, 2, 3, . . . , at the time slot t is expressed as SINRDL ki (t) 

(t) · rRRU-UE (t)−α pT-RRUk (t) · hRRU-UE ki ki m∈N+ ,m k

, pT −R RUm (t) · h mR iRU −U E (t) · rmRiRU −U E (t)−α + N0 (3.95)

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular …

159

where pT-RRUk (t) is the RRU transmission power of the desired BS BSk , hRRU-UE (t) ki is the small scale fading between the RRU of desired BS BSk and the UE UEi , (t) is the distance between the RRU of desired BS

BSk and the UE UEi , α rRRU-UE ki is the path loss fading, N0 is the wireless channel noise, m∈N+ ,m k pT −R RUm (t) · h mR iRU −U E (t) ·rmRiRU −U E (t)−α is the aggregated interference from RRUs of interfering (t) BSs, pT-RRUm (t) is the RRU transmission power of the interfering BS BSm , hRRU-UE mi is the small scale fading between the RRU of interfering BS BSm and the UE UEi , (t) is the distance between the RRU of interfering BS BSm and the UE UEi . rRRU-UE mi 3.3.3.2

SINR of Uplinks

Since the full duplex scheme is adopted for BSs, the self-interference is considered for the SINR of uplinks in this section. Hence, the SINR of uplink between the desired UE UEi and the BS BSk at the time slot t is expressed by SINRUL ki (t) 

(t) · rRRU-UE (t)−α pT-UE (t) · hRRU-UE ki ki pT-RRUk (t) · hself (t) + IUL k (t) + N0

,

(3.96)

where hself (t) is the gain of the self-interference channels at RRUs, pT-RRUk (t) · hself (t) is the self-interference at the BS BSk , hRRU-RRU (t) is the small scale fading between mk (t) is the the RRU of interfering BS BSm and the RRU of desired BS BSk , rRRU-RRU mk

(t)  p distance between the RRU of interfering BS BSk , IUL + k m∈N ,m k T-RRUm (t) · RRU-RRU −α (t) · r (t) is the aggregated interference from adjacent BSs except hRRU-RRU mk mk for the desired BS BSk .

3.3.3.3

Weighted Energy Efficiency

When the full duplex scheme is adopted for BSs and the half duplex scheme is adopted for UEs, the energy efficiency of downlinks and uplinks needs to be estimated for the energy efficiency of BSs. The weighted energy efficiency of a cell with the BS BSk is expressed as

UBSk (t) 

i DL

EEDL ki (t) DL

NDL

+β ·

i UL

EEUL ki (t) UL

NUL

,

(3.97)

where β is the weighted factor for uplinks and downlinks, the index of i DL is denoted for the downlink UE, the index of i UL is denoted for the uplink UE. EEDL kiDL (t) is the energy efficiency of downlinks between the desired BS BSk and the UE UEiDL at the time slot t, NDL is the active number of UEs in downlinks, EEUL kiUL (t) is the energy efficiency of uplinks between the desired UE UEiUL and the BS BSk at the time slot t, NUL is the active number of UEs in uplinks.

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3 Energy Efficiency of Cellular Networks

Without loss of generality, the total power consumption of the BS BSk is divided into the RRU transmission power pT-RRUk (t) and the static power pstatic , i.e., pBSk (t)  pT-RRUk (t) + pstatic . The static power pstatic is fixed for all BSs. To simplify derivations, bandwidths of downlinks and uplinks are normalized as 1. Hence, the energy efficiency of downlinks between the desired BS BSk and the UE UEi at the time slot t is extended as EEDL ki (t) 

ln(1 + SINRDL ki (t)) pT-RRUk (t) + pstatic

.

(3.98)

Moreover, the energy efficiency of uplinks between the desired UE UEi and the BS BSk at the time slot t is extended as EEUL ki (t) 

ln(1 + SINRUL ki (t)) pT-UE (t) + pUE-static

.

(3.99)

where pUE-static is the static power at UEs and is fixed for all UEs, pT-UE (t) is the transmission power at UEs. The total power consumption of the UE includes the static power and the transmission power at UEs.

3.3.3.4

Energy Efficiency Formulation of BSs

Based on the system model in Fig. 3.26a, the RRUs are supported by batteries. In this case, the RRU transmission power of the BS BSk is ranged by pT-RRUk (t) ∈ [0, pT-RRUmax ], where pT-RRUmax is the maximum transmission power supported by the battery. Since the wireless transmission is supported by the battery energy of RRUs, the residual battery volume of RRUs is depended on the transmission power and is changed with the transmission time. The maximum volume of the battery equipped for a RRU is configured as Emax which is a constant for all batteries. The residual volume of the battery used for a RRU of BS BSk is denoted as ek (t) at the time t. Hence, the value of ek (t) is ranged as 0 ≤ ek (t) ≤ Emax . Without loss of generality, the batteries of RRUs are assumed to be charged by the energy harvesting method after a period T . The state equation of battery volume at the RRU is defined as follows: Definition 2 (State equation) The battery state of RRU at the BS BSk is denoted by the residual battery volume of RRU ek (t), the change of the residual battery volume of RRU is expressed by the following state equation: dek (t)  −pT-RRUk (t)dt, 0 ≤ t ≤ T.

(3.100)

In this section the transmission power of RRUs is assumed to be adaptively adjusted by the residual battery volume of RRUs. Therefore, the RRU transmission power of the BS BSk pT-RRUk (t) is replaced as pT-RRUk (t, ek (t)), which is simply

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular …

161

denoted as pT-RRUk (t, ek ). When the pT-RRUk (t) is replaced by pT-RRUk (t, ek ) in (3.98) and (3.99), the weighted energy efficiency of a cell associated with the BS BSk is denoted as UBSk (pT-RRUk (t, ek )). Considering the energy harvesting technology adopted for RRUs, the battery volume of RRU is recharged after a time interval T . Hence, the battery volume of RRU is limited in the time interval T . To maximize the energy efficiency of BSs in the time interval T , not only the weighted energy efficiency of cells but also the residual battery volume of RRU at BSs need to be considered for optimizing the transmission power. Based on (3.99) and (3.100), the maximum energy efficiency of cell with the BS BSk in the time interval T is formulated as ⎡ max

pT-RRUk (0→T )

E⎣

T

⎤   UBSk pT-RRUk (t, ek ) dt ⎦,

0

 0 ≤ pT-RRUk (t, ek ) ≤ pT-RRUmax , s.t. dek (t)  −pT-RRUk (t, ek )dt, 0 ≤ t ≤ T.

(3.101)

where (3.101) is the optimal function maximizing the average energy efficiency of BSs in the time interval T , pT-RRUk (0 → T ) is the RRU transmission power strategy of the BS BSk in the time interval T . Based on the stochastic optimal control theory in [116], the optimal RRU transmission power strategy p∗T-RRUk (0 → T ) of the BS BSk is given to achieve the result of (3.7). The expression of p∗T-RRUk (0 → T ) is expressed as follows: ⎡ p∗T-RRUk (0 → T )  arg

max

pT-RRUk (0→T )

E⎣

T

⎤   UBSk pT-RRUk (t, ek ) dt ⎦.

(3.102)

0

To illustrate the maximum of energy efficiency of cells with different RRU transmission power strategies from the time t to T , the value function Vk (t, ek ) of the BS BSk is defined, which is given as ⎡ Vk (t, ek ) 

max

pT-RRUk (t→T )

E⎣

T

⎤   UBSk pT-RRUk (τ, ek ) dτ ⎦, t ∈ [0, T ].

(3.103)

t

Based on the Bellman principle of optimality in [117], the optimal strategy of RRU transmission power satisfies that the RRU transmission power strategy is always the optimal strategy from any start time to the end time. Therefore, the optimal strategy of RRU transmission power is defined as follows: Definition 3 (The optimal strategy of RRU transmission power) When p∗T-RRUk (t → T ) is the optimal strategy of RRU transmission power at the BS BSk , for any time slot t ∈ [0, T ], the following equation is always satisfied

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3 Energy Efficiency of Cellular Networks

⎡ E⎣

T UBSk



⎤  p∗T-RRUk (τ, ek ) dτ ⎦  Vk (t, ek ),

(3.104)

t

where p∗T-RRUk (τ, ek ) is the RRU transmission power of the BS BSk at the time slot τ when the optimal strategy of RRU transmission power p∗T-RRUk (t → T ) is adopted for the BS BSk . The result of (3.104) implies that the (3.101) is always achieved the maximum in the time internal T when the optimal RRU transmission power strategy p∗T-RRUk (t → T ) is adopted at the BS BSk .

3.3.4 Network Energy Efficiency Optimization 3.3.4.1

Differential Game

The maximum energy efficiency in (3.101) is formulated for the single cell. However, the optimal RRU transmission power strategy is influenced between each other by the interference from adjacent RRUs in 5G cellular networks. Hence, the optimal RRU transmission power strategy is not independent from each other. Therefore, we use the differential game theory [118] to optimize the RRU transmission power strategy in 5G cellular networks. In this section ϕG is defined for the differential game among the RRU transmission power strategy in 5G cellular networks. Definition 4 (Differential game of RRU transmission power strategy ϕG ) Every BS is assumed as a player for a cellular network. The utility function of player k is expressed as: ⎡ T ⎤    UFk  E⎣ UBSk pT-RRUk (t, ek ) dt ⎦,

(3.105)

0

with the object function: max

pT-RRUk (0→T )

UFk

 0 ≤ pT-RRUk (t, ek ) ≤ pT-RRUmax , s.t. dek (t)  −pT-RRUk (t, ek )dt, 0 ≤ t ≤ T. Every BS in the cellular network, i.e., every player in the differential game system, always tries to achieve the maximum utility function of player by adjusting the transmission power of RRU. In this case, the RRU transmission power strategy of BSs is adaptively adjusted by a game method. In the end, every BS of cellular network will approach to a stable state, i.e., the RRU transmission power strategy finally selected by every BS is the optimal for cellular networks. As a consequent,

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the optimal RRU transmission power strategy of every BS in cellular network is called as the Nash equilibrium. The Nash equilibrium of differential game of RRU transmission power strategy is defined as: Definition 5 (Nash Equilibrium of the Differential game of RRU transmission power strategy ϕG ) If and only if the following equation, i.e., (3.106) is satisfied, the strategy set of all players p∗  [p∗T-RRU1 (0 → T ), . . . , p∗T-RRUk (0 → T ), . . .] is the Nash equilibrium of differential game of RRU transmission power strategy, ⎡ p∗T-RRUk (0 → T )  arg

max

pT-RRUk (0→T )

E⎣

T

⎤   UBSk pT-RRUk (t, ek ) dt ⎦,

(3.106)

0

subject to: dek (t)  −pT-RRUk (t, ek )dt, 0 ≤ t ≤ T. Assumed that p∗−k is the strategy set of players without the player k, i.e., the BS BSk . Based on the Nash equilibrium in (3.106), the BS BSk adopting the strategy p∗T-RRUk (0 → T ) can achieve the maximum utility function when other BSs adopt the strategy set p∗−k . When all BSs adopt the strategy set p∗ based on the Nash equilibrium, the value of utility function at a BS will be less than the maximum of utility function when the BS changes the current RRU transmission power strategy. Therefore, all BSs would not like to change the current RRU transmission power strategy based on the Nash equilibrium. As a consequence, the RRU transmission power of all BSs approaches to a stable state. To derive the solution of Nash equilibrium ϕG for the RRU transmission power strategy of BSs, the existence of the Nash equilibrium ϕG is firstly proved by the following expression: Proof Based on the results in [119], the Nash equilibrium is existed in the differential game if the Hamilton-Jacobi-Bellman (HJB) equations of all players can be solved. The HJB equation of BS BSk is expressed as   ∂Vk (t, ek ) ∂Vk (t, ek ) UBSk (pT-RRUk (t, ek )) − pT-RRUk (t, ek ) · + max  0. pT-RRUk (t,ek ) ∂t ∂e (3.107)   (t,ek ) To easily analyze the characteristic of (3.107), we set H ek , ∂Vk∂e    ∂Vk (t,ek ) maxpT-RRUk (t,ek ) UBSk (pT-RRUk (t, ek )) − pT-RRUk (t, ek ) · ∂e as the Hamiltonian function. Hence, (3.107) can be solved only if the Hamiltonian function can be solved. Based on the results in [120], the condition that the Hamiltonian function can be solved is that the Hamiltonian function must be a smooth function, i.e., the derivative of the Hamiltonian function on pT-RRUk (t, ek ) exists in the available domain.

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3 Energy Efficiency of Cellular Networks

Substitute (3.95), (3.96), (3.97), (3.98) and (3.99) into the Hamiltonian function, we obtain the following expression:  ∂Vk (t, ek ) H ek , ∂e ⎛ pT-RRUk (t,ek )·hRRU-UE (t)·rRRU-UE (t)−α ki ki DL DL

RRU-UE (t)−α +N ) (t)·r ⎜ 1 ln(1 + m∈N + ,m k pT-RRUm (t,em )·hRRU-UE 0 mi mi DL DL  max ⎜ pT-RRUk (t,ek )⎝ NDL p (t, e ) + p k T-RRUk static i DL

+ β + · NUL i

ln 1 +

pT-UE ·hRRU-UE (t)·rRRU-UE (t)−α ki ki UL UL

pT-RRUk (t,ek )·hself (t)+ pT-RRUm (t,em )·hRRU-RRU (t)·rRRU-RRU (t)−α +N0 mk mk m∈N + ,m k

pT-UE + pUE-static

UL

−pT-RRUk (t, ek ) ·

,

∂Vk (t, ek ) ∂e

(3.108)

In this section pT-RRUk (t, ek ) is configured as the differential variable for the Hamil(t,ek ) tonian function. The term of ∂Vk∂e is independent on the RRU transmission (t,ek ) power strategy of BS, so the derivative of ∂Vk∂e on pT-RRUk (t, ek ) is zero. NDL and NUL are considered as constants for the differential variable pT-RRUk (t, ek ).



RRU-UE (t) · rRRU-UE (t)−α and m∈N + ,m k pT-RRUm (t, em ) · m iDL m∈N + ,m k pT-RRUm (t, em ) · hm iDL hRRU-RRU (t) · rRRU-RRU (t)−α are independent on pT-RRUk (t, ek ). As a consequence, mk mk the Hamiltonian function is an elementary function when pstatic > 0. Therefore, the derivative of the Hamiltonian function on pT-RRUk (t, ek ) exists in the available domain, i.e., the Hamiltonian function is a smooth function. Furthermore, the Hamiltonian function can be solved and then (3.107) can also be solved, i.e., the Nash equilibrium can be solved by the HJB equations. In the end, the existence of the Nash equilibrium ϕG is proved for the RRU transmission power strategy of BSs.

3.3.4.2

Mean Field Game

Generally, the solution of Nash equilibrium can be derived by solving a set of partial differential equations. Moreover, the number of partial differential equations used to solve the Nash equilibrium is equal to the number of BSs in the cellular network. For a real cellular network, the number of BSs is very large. Hence, the number of partial differential equations is too large to solve the Nash equilibrium for a real cellular network. On the other hand, the distribution function of the residual battery volume of RRU at BSs is approximated as a continuous function when the number of BSs is large enough in the cellular network. In this case, the impact of the RRU transmission power strategy from the single BS on the RRU transmission power strategy of another BS can be ignored and then only the impact of the cellular network on the RRU transmission power strategy of the single BS needs to be considered.

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165

As a consequence, the differential game can be replaced by the mean field game to optimize the energy efficiency of cellular networks. (1) Mean Field of Network Energy Efficiency: To describe the probability density function (PDF) of residual battery volume of RRUs, the mean field of network energy efficiency is defined as 1 I{ek (t)e} , N→∞ N k1 N

m(t, e)  lim

(3.109)

where I{ek (t)e} is equal to 1 when the residual battery volume of RRU at the BS BSk is equal to e, otherwise I{ek (t)e} is equal to 0. N is the number of BSs in the cellular network. In other words, m(t, e) can be used to describe the distribution of the residual battery volume of RRUs in the cellular network on the time slot t. When the number of BSs is large, the mean field of network energy efficiency can be assumed as a continuous function. (2) Interference in Mean Field of Network Energy Efficiency: Considering the mean field definition in (3.109), Imean-RRUk (t) is denoted as the interference aggregated at the BS BSk and Imean-UEiDL (t) is denoted as the interference aggregated at the user UEiDL which is associated with the BS BSk . The detail expressions of Imean-RRUk (t) and Imean-UEiDL (t) are derived as ⎡ ⎤ pT-RRU (t, ek ) · hRRU-RRU (t) · rRRU-RRU (t)−α ⎦ Imean-RRUk (t)  E⎣ m m m

k

k

m k

⎡ ⎤    E pT-RRUm (t, ek ) · E[hRRU-RRU (t)] · E⎣ rRRU-RRU (t)−α ⎦ mk mk m k

⎡ Imean-UEiDL (t)  E⎣



m k



(3.110)

pT-RRUm (t, ek ) · hRRU-UE (t) · rRRU-UE (t)−α ⎦ mi mi DL

DL



⎤    RRU-UE   E pT-RRUm (t, ek ) · E hm i (t) · E⎣ rRRU-UE (t)−α ⎦ mi DL

m k

DL

(3.111) where E[pT-RRUm (t, ek )] is the expectation of RRU transmission power from interfering BSs on the time slot t. Based on the system model and properties (t) and rRRU-UE (t) are the of Poisson distribution [3], the distribution of rRRU-RRU mk m iDL RRU-RRU RRU-UE same. When rm k (t) < 1 or rm i (t) < 1, the path loss fading is conDL figured as α  1. Moreover, wireless channels of hRRU-RRU (t) and hRRU-UE (t) mk m iDL are assumed to be governed by the exponent distribution with the mean as one

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3 Energy Efficiency of Cellular Networks

[121]. Based on the Campbell’s theorem [122] and the properties of cumulative distribution function (CDF) in Poisson distribution [3], Imean-RRUk (t) and Imean-UEiDL (t) are further derived as ⎡ ⎤ rRRU-RRU (t)−α ⎦ Imean-RRUk (t)  E[pT-RRUm (t, ek )] · E⎣ mk m k



   E pT-RRUm (t, ek ) ·

rRRU-RRU (t)−α d(S) mk

S\{k}

⎡ 1    ⎣  E pT-RRUm (t, ek ) · rRRU-RRU (t)drRRU-RRU (t) mk mk 0

⎤ ∞  RRU-RRU −α RRU-RRU rm k (t) drm k (t)⎦ + 1

Imean-UEiDL

   1 1  E pT-RRUm (t, ek ) · 2π λBS · + , 2 α−2 ⎡ ⎤   (t)  E pT-RRU (t, ek ) · E⎣ rRRU-UE (t)−α ⎦ m m

   E pT-RRUm (t, ek ) ·



(3.112)

i DL

m k

rRRU-UE (t)−α d(S) mi DL

S\{k}

⎡ 1     E pT-RRUm (t, ek ) · ⎣ rRRU-UE (t)drRRU-UE (t)n mi mi DL

DL

0

⎤ ∞ −α rRRU-UE (t) drRRU-UE (t)⎦ + mi mi DL

DL

1

   E pT-RRUm (t, ek ) · 2π λBS ·



1 1 + 2 α−2

(3.113)

where S\{k} denotes the set of BSs in a cellular network without the BS BSk . pT (t, e) is configured as the average RRU transmission power with the residual battery volume e on the time slot t, i.e., pT (t, e) 

p limN→∞ N1 N k1 T-RRUk (t, ek ) · I{ek (t)e} . Hence, the expectation of the RRU transmission power in the cellular network can be derived as    pT (t, e) · m(t, e)de. (3.114) E pT-RRUm (t, ek )  e∈ε

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167

Furthermore, the average interference in the mean field of network energy efficiency is derived as   1 1 + · pT (t, e) · m(t, e)de. Imean-RRUk (t)  Imean-UEiDL (t)  2π λBS · 2 α−2 e∈ε

(3.115) Based on the results in [123], the interference fluctuation in the small scale can be smoothed by the averaging operation when a large number of BSs exists in the cellular network. Hence, Imean-RRUk (t) and Imean-UEiDL (t) are considered as a stationary value in the time slot t. (3) Energy Efficiency of Links in Mean field of Network Energy Efficiency: EEDL-mean (t) denotes the energy efficiency of downlinks in the mean field of k network energy efficiency. Considering multi-user accessing with a BS, the (t) needs to average the energy efficiency of all users calculation of EEDL-mean k (t) is derived as accessing with the BS. In detail, EEDL-mean k   

DL DL   E ln 1 + SINR (t) kiDL i DL EEkiDL (t) (t)   E EEDL , EEDL-mean k kiDL (t)  NDL pT-RRUk (t, ek ) + pstatic (3.116a) with       DL ω E ln 1 + SINRki (t)  Pr SINRDL ki (t) > e − 1 dω, DL

DL

(3.116b)

ω>0

  ω (t) > e −1 Pr SINRDL kiDL    DL ω RRU-UE  ErRRU-UE Pr SINR (t) > e − 1|r (t) (t) k k i DL i DL ki DL     ω RRU-UE Pr SINRDL (t) f rRRU-UE (t)  ki (t) > e − 1|rki ki DL

rRRU-UE (t)>0 k i DL

(





Pr

DL

DL

pT −R RUk (t, ek ) · h kRi RU −U E (t) · rkRi RU −U E (t)−α DL

DL

Imean−U Ei DL (t) + N0

rkR RU −U E (t)>0 i DL



  > eω − 1|rkRi RU −U E (t) f rRRU-UE (t) drRRU-UE (t)  ki ki DL

DL

DL

  (t)α (e − 1) · Imean-UEiDL (t) + N0 · rRRU-UE ki

DL

rRRU-UE (t)>0 k i DL

ω

>

DL

pT-RRUk (t, ek )

 Pr hRRU-UE (t) ki

(t) |rRRU-UE kiDL

)

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3 Energy Efficiency of Cellular Networks

  f rRRU-UE (t) drRRU-UE (t) kiDL kiDL   + ,  (eω − 1) · Imean−U Ei DL (t) + N0 · rkRi RU −U E (t)α DL  exp − pT −R RUk (t, ek ) rkR RU −U E (t)>0 i DL

  f rRRU-UE (t) drRRU-UE (t) ki ki DL

(3.116c)

DL

−λBS π(rRRU-UE (t))2

ki DL where f(rRRU-UE (t))  2π λBS rRRU-UE (t)e is the PDF between the UE kiDL kiDL UEiDL and the closest BS. In the end, the energy efficiency of downlinks in the mean field of network energy efficiency is expressed as

(t)  EEDL-mean k

1 , pT-RRUk (t, ek ) + pstatic

(3.117)

with 

+



1 

exp −

  , (t)α (eω − 1) · Imean-UEiDL (t) + N0 · rRRU-UE ki DL

pT-RRUk (t, ek )

ω>0 rRRU-UE (t)>0 k i DL

(t)e 2π λBS rRRU-UE ki

 2 −λBS π rRRU-UE (t) k i DL

DL

drRRU-UE (t)dω. ki DL

EEUL-mean (t) denotes the energy efficiency of uplinks in the mean field of network k (t) is similarly energy efficiency. Based on the derivation process in (3.117), EEUL-mean k derived as   

UL UL   E ln 1 + SINR (t) kiUL i UL EEkiUL (t) (t)   E EEUL , EEUL-mean k kiUL (t)  NUL pT-RRUk (t, ek ) + pstatic (3.118a) with       UL ω E ln 1 + SINRki (t)  Pr SINRUL ki (t) > e − 1 dω, UL

UL

ω>0

     ω UL ω RRU-UE (t) Pr SINRUL ki UL (t) > e − 1  ErRRU-UE (t) Pr SINRki UL (t) > e − 1|rki UL ki UL    UL  Pr SINRk (t) > eω − 1|rRRU-UE (t) f rRRU-UE (t) k k i UL

rRRU-UE (t)>0 k i UL

i UL

i UL

(3.118b)

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular … ⎛



⎜ exp⎝−

 rRRU-UE (t)>0 k

169

  ⎞ (t)α (eω − 1) · Imean-RRUk (t) + pT-RRUk (t) · hself (t) + N0 · rRRU-UE ki UL ⎟ ⎠ pT-RRUk (t, ek )

i UL

 (t) drRRU-UE (t). f rRRU-UE k k i UL

(3.119)

i UL

In the end, EEUL-mean (t) is expressed as k (t)  EEUL-mean k

2 , pT-UE (t, ek ) + pUE-static

(3.120)

with 



2 

 2 −λBS π rRRU-UE (t) RRU - UE ki UL 2π λBS rki (t)e UL

- UE ω>0 rRRU (t)>0 k

+

i UL

exp −

  , (t)α (eω − 1) · Imean-RRUk (t) + N0 + pT-RRUk (t, ek ) · hself (t) · rRRU-UE ki UL

pT-RRUk (t, ek )

(t)dω. drRRU-UE ki UL

Based on the results of (3.117) and (3.119), the weighted energy efficiency of the BS BSk , i.e., Umean BSk (pT-RRUk (t, ek ), m(t, e)) in the mean field m(t, e) of network energy efficiency is expressed as   DL-mean Umean (t) + β · EEUL-mean (t) BSk pT-RRUk (t, ek ), m(t, e)  EEk k 1 2  +β · , pT-RRUk (t, ek ) + pstatic pT-UE (t, ek ) + pUE-static

(3.121)

with 



1 

+ exp −

ω>0 rRRU-UE (t)>0 k

  , (t)α (eω − 1) · Imean-UEiDL (t) + N0 · rRRU-UE ki DL

pT-RRUk (t, ek )

i DL

 2 −λBS π rRRU-UE (t) RRU-UE ki DL (t)e drRRU-UE (t)dω, 2π λBS rki kiDL DL  2   −λBS π rRRU-UE (t) RRU-UE ki UL 2π λBS rki (t)e 2  UL ω>0 rRRU-UE (t)>0 k

+

i UL

exp −

  , (t)α (eω − 1) · Imean-RRUk (t) + N0 + pT-RRUk (t, ek ) · hself (t) · rRRU-UE ki

(t)dω. drRRU-UE ki UL

UL

pT-RRUk (t, ek )

170

3.3.4.3

3 Energy Efficiency of Cellular Networks

Mean Field Game Equations

When the differential game is replaced by the mean field game to optimize the energy efficiency of cellular networks, the differential game of RRU transmission power strategy ϕG is replaced by the mean field game of RRU transmission power strategy ϕG-mean . Definition 6 (Mean field game of RRU transmission power strategy ϕG-mean ) Every BS is assumed as a player for the cellular network. The utility function of the player k is expressed as: UFmean k

⎡ T ⎤    ⎦  E⎣ Umean BSk pT-RRUk (t, ek ), m(t, e) dt ,

(3.122)

0

with the object function: max

pT-RRUk (0→T )

UFmean k

 0 ≤ pT-RRUk (t, ek ) ≤ pT-RRUmax , s.t. dek (t)  −pT-RRUk (t, ek )dt, 0 ≤ t ≤ T. Based on the properties of mean field game [124, 125], the solution of Nash equilibrium ϕG-mean can be derived by ⎧     ∂ Vk (t,ek ) mean ⎨ ∂ Vk (t,ek ) + max U 0 p (t, e ), m(t, e) − p (t, e ) · k k T-RRUk T-RRUk BSk ∂t ∂e pT-RRUk (t,ek )   ⎩ ∂ m(t,ek ) − ∂t∂ m(t, ek )pT-RRUk (t, ek )  0 ∂t (3.123) To derive the Nash equilibrium of mean field game ϕG-mean , every BS needs to solve (3.123). The RRU transmission power strategy of every BS is the same in the mean field game model. Moreover, every point is independent and similar in the Poisson point process. Therefore, the solution of Nash equilibrium with mean field game of RRU transmission power strategy ϕG-mean is the same for every BS in the cellular network. Based on the mean field game theory, (3.123) can be simplified as ⎧   ⎨ ∂ V(t,e) + max Umean (p(t, e), m(t, e)) − p(t, e) · ∂ V(t,e)  0, BS ∂t ∂e p(t,e) (3.124) ⎩ ∂ m(t,e) − ∂ (m(t, e)p(t, e))  0, ∂t

∂t

where p(t, e) is the RRU transmission power at a BS, Umean BS (p(t, e), m(t, e)) is the weighted energy efficiency at a BS, V(t, e) is the value function of a BS. Equations in (3.124) are the HJB and Fokker-Planck-Kolmogorov (FPK) equations respectively. Based on the solutions of (3.124) the Nash equilibrium point p∗mean and the mean filed

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171

m∗ at the Nash equilibrium point can be obtained for the mean field game of RRU transmission power strategy ϕG-mean .

3.3.5 Algorithm Design of Mean Field Game Although there are no analytical solutions for (3.124) [106, 124], we can obtain the numerical solutions of (3.124) by the finite difference method and the Lax-Friedrichs method [126]. The time period T is discretized into (X + 1) time points, i.e., the time point set t  {0, 1 ∗ δt, 2 ∗ δt, 3 ∗ δt, . . . , X ∗ δt}, where δt  TX . The maximum volume of battery at the RRU is discretized into (Y + 1) energy points, i.e., the energy . Based on the point set e  {0, 1 ∗ δe, 2 ∗ δe, 3 ∗ δe, . . . , Y ∗ δe}, where δe  Emax Y time point set and the energy point set, V(t, e) and m(t, e) are expressed by V(x, y) and m(x, y), where 0 ≤ x ≤ X and 0 ≤ y ≤ Y .

3.3.5.1

Numerical Solution of FPK and HJB

Based on the Lax-Friedrichs method, (3.124b) is extended as 1 [m(x, y − 1) + m(x, x + 1)] 2  δt  p(x, x + 1)m(x, x + 1) − p(x, x − 1)m(x, x − 1) . (3.125) + 2(δe)

m(x + 1, y) 

Utilizing the discretization method,

∂V(t,e) ∂t

and

∂V(t,e) ∂e

are expressed as

V(x, y) − V(x − 1, y) ∂V(t, e)  , ∂t δt ∂V(t, e) V(x, y) − V(x, y − 1)  . ∂e δe

(3.126) (3.127)

Substitute (3.125) and (3.126) into (3.124a), we have the following result:  V(x, y) − V(x − 1, y) + max Umean BS (p(x, y), m(x, y)) p(x,y) δt V(x, y) − V(x, y − 1)  0. −p(x, y) · δe

(3.128)

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3 Energy Efficiency of Cellular Networks

3.3.5.2

Algorithm Design for Numerical Solution of Mean Field Game

Based on (3.125) and (3.128), the equilibrium of mean field game (EMFG) Algorithm is developed to obtain the Nash equilibrium p∗mean and the mean filed m∗ at the Nash equilibrium. Equilibrium of Mean Field Game Algorithm (EMFG Algorithm) Initialization: Initialize V(X, :), m(:, :) Repeat: For x  X: −1: 0 do For y  Y: −1:  0 do Solving max Umean BS (p(x, y), m(x, y)) − p(x, y) · p(x,y)

V(x,y)−V(x,y−1) δe

 , get p(x, y)

End for Update V(x − 1, :) using:  V(x, y) − V(x − 1, y) + max Umean BS (p(x, y), m(x, y)) − p(x, y) p(x,y) δt V(x, y) − V(x, y − 1) 0 · δe End for Update m(:, :) using Equation: 1 [m(x, y − 1) + m(x, y + 1)] 2  δt  p(x, y + 1)m(x, y + 1) − p(x, y − 1)m(x, y − 1) + 2(δe)

m(x + 1, y) 

Until Convergence, Obtain p∗mean  p(x, y), m∗  m(x, y) Output: p∗mean , m∗ .

3.3.6 Numerical Simulations of Mean Field Game Based on the EMFG Algorithm, the value of V(t, e) at the end of the period T needs to be configured in advance. To save of battery energy at RRUs, the value of V(t, e) at the end of the period T is configured as V(X, y ∗ δe)  0.05 ∗ exp(y ∗ δe) in this section. Moreover, the residual battery volume of RRUs is assumed to be governed by a uniform distribution at the initial stage of simulations, i.e., m(:, :)  X1+1 . The detail parameters of numerical simulations are list in Table 3.1.

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular … Table 3.1 Parameters for numerical simulations

173

Parameter

Default value

Parameter

Default value

pstatic

6W

α

3

λBS

0.0005/m2

N0

10−8 W

pT-UE

0.1 W

pUE-static

0.5 W

hself

0.0004

pT-RRUmax

1W

Emax

2J

T

1S

When the weighted factor for uplinks is configured as β  1, the RRU transmission power with respect to the time and residual battery volume of RRU is illustrated in Fig. 3.28. From Fig. 3.28a, the RRU transmission power decreases with increasing time and decreasing residual battery volume at RRU. To clearly explain the results in Fig. 3.28a, three time slots, i.e., t  0, t  0.5 and t  1 are selected and plotted in Fig. 3.28b. When the time slot is fixed in Fig. 3.28b, the RRU transmission power increases with increasing the residual battery volume of RRU. When the residual battery volume of RRU is fixed in Fig. 3.28b, the RRU transmission power decreases with increasing the time slot. Figure 3.29 shows the RRU transmission power with respect to different initial battery volumes of RRU and weighted factors for uplinks. When the weighted factor is fixed, the RRU transmission power with initial residual battery volume 2 J is larger than or equal to the RRU transmission power with initial residual battery volume 1 J. When the initial residual battery volume is fixed, the RRU transmission power decreases with increasing the weighted factor for uplinks. To analyze the distribution of residual battery volumes of RRUs in the cellular network, the mean field with respect to the residual battery volume and the time is depicted in Fig. 3.30. Without loss of generality, the weighted factor of uplinks is configured as β  1. Based on the results in Fig. 3.30a, the mean field with the

Fig. 3.28 RRU transmission power with respect to the time and residual battery volume of RRU

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3 Energy Efficiency of Cellular Networks

Fig. 3.29 RRU transmission power with respect to different initial battery volumes of RRU and weighted factors for uplinks

Fig. 3.30 Mean field with respect to the residual battery volume and the time

low residual battery volume of RRU increases with the increase of time. To clearly explain the results in Fig. 3.30a, three time slots, i.e., t  0, t  0.5 and t  1 are selected and plotted in Fig. 3.30b. When the time slot t  0 is ignored and the time slot is fixed in Fig. 3.30b, the mean field decreases with increasing the residual battery volume of RRU. When the residual battery volume is less than or equal to 1.2 J in Fig. 3.30b, the mean field with the time slot t  1 is larger than the mean field with the time slots t  0 and t  1. When the residual battery volume is larger than or equal to 1.6 J in Fig. 3.30b, the mean field with the time slot t  0 is larger than the mean field with the time slots t  0.5 and t  1. These results indicate that the number of RRUs with high residual battery volume, i.e., the residual battery volume is larger than or equal to 1.6 J, is larger than the number of RRUs with low residual battery volume, i.e., the residual battery volume is less than 1.6 J in the initial time. In the end of a period T , the number of RRUs with low residual battery volume, i.e., the residual battery volume is less than or equal to 1.2 J, is larger than the number of RRUs with high residual battery volume, i.e., the residual battery volume is larger than 1.2 J.

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular …

175

Fig. 3.31 Mean field with respect to the time and the weighted factor for uplinks

When the initial residual battery volume of RRUs are configured as 0, 1 and 2 J, the mean field with respect to the time and the weighted factor for uplinks is shown in Fig. 3.31. When the time is fixed, the mean field with the initial residual battery volume 0 J is larger than or equal to the mean field with the initial residual battery volume 1 and 2 J. When the residual battery volume is fixed as 0 J, the mean field increases with the increase of time. When the residual battery volume is fixed as 2 J, the mean field quickly falls into zero with increasing the time. When the residual battery volume is fixed as 1 J and the weighted factor β is fixed as 0.5, the mean field first increases with increasing the time and then the mean field decreases with increasing the time after the time is larger than 0.7 s. The reason of this change is that the number of RRUs with the residual battery volume 1 J is added by the RRUs with the initial residual battery volume 2 J in the start time, i.e., the time is less than 0.7 s. When the time is larger than or equal to 0.7 s, the number of RRUs with the residual battery volume 1 J is reduced by the energy consuming for RRU transmission power. Moreover, few RRUs with the initial residual battery volume 2 J is added into the number of RRUs with the residual battery volume 1 J. When the residual battery volume is fixed as 1 J and the weighted factor β is fixed as 0.5 and 1, the mean field increases with increasing the time. To evaluate the performance of EMFG Algorithm, the network energy efficiency is defined as (t) 



m(t, e) · Umean BS (p(t, e), m(t, e)).

(3.129)

e

Without loss of generality, the initial RRU transmission power is configured as 1 W in cellular networks. The network energy efficiencies adopted the fixed RRU transmission power strategy and the RRU transmission power strategy of mean field game, i.e., the EMFG algorithm, are compared in Fig. 3.32. Considering different weighted factors, the network energy efficiency under the RRU transmission power strategy of mean field game is always larger than that under the fixed RRU

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3 Energy Efficiency of Cellular Networks

Fig. 3.32 Weighted network energy efficiency with respect to the time compared with the fixed RRU transmission power strategy and the RRU transmission power of mean field game strategy

transmission power strategy. Moreover, the network energy efficiency under the RRU transmission power strategy of mean field game increases with the increase of the time in 5G cellular networks. To the contrary, the network energy efficiency adopted the fixed RRU transmission power strategy decreases with the increase of the time in 5G cellular networks. Hence, the RRU transmission power strategy of mean field game, i.e., the proposed EMFG algorithm can improve the network energy efficiency of 5G cellular networks. The coverage probability is another key metric of the RRU transmission power strategy for 5G cellular networks. The coverage probability of RRU is denoted as E[Pr(SINRDL ki (t) > δ)], where δ is the threshold of receive signal power at UEs. The average network coverage probability is expressed as 

1

N RRU

NRRU

k1

  E Pr(SINRDL ki (t) > δ) ,

(3.130)

where NRRU is the number of RRUs in cellular networks. The average network coverage probability with respect to the time considering different thresholds of receive signal power at UEs are compared for cellular networks with the fixed RRU

3.3 Energy Efficiency Optimization of 5G Full Duplex Cellular …

177

transmission power strategy and the RRU transmission power strategy of mean field game in Fig. 3.33. For different thresholds of receive signal power at UEs, e.g., δ is − 10, 0 and 10 dB, the average network coverage probability with the RRU transmission power strategy of mean field game always keeps a stable level with increasing the time and the average network coverage probability with the fixed RRU transmission power strategy decreases with increasing the time. In the initial time, the average network coverage probability with the fixed RRU transmission power strategy is larger than the average network coverage probability with the RRU transmission power strategy of mean field game in 5G cellular networks. After certain time, e.g., 0.1, 0 and 0.3 s corresponding to different thresholds of receive signal power at UEs, i.e., −10, 0 and 10 dB, the average network coverage probability with the fixed RRU transmission power strategy is less than that with the RRU transmission power strategy of mean field game in 5G cellular networks. Therefore, the RRU transmission power strategy of mean field game, i.e., the proposed EMFG algorithm, is helpful for keeping the stable of average network coverage probability in a long time scale.

Fig. 3.33 Average network coverage probability with respect to the time compared with the fixed RRU transmission power strategy and the RRU transmission power of mean field game strategy

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3.3.7 Conclusion In this section we investigate the network energy efficiency of 5G full duplex cellular networks by the mean field game theory. Based on the solution of mean field game equations, a new RRU transmission power strategy, i.e., the EMFG algorithm is developed to optimize the network energy efficiency of 5G full duplex cellular networks. Simulation results show that the network energy efficiency with the proposed EMFG algorithm is larger than the network energy efficiency with the fixed RRU transmission power algorithm in 5G full duplex cellular networks. Moreover, the cellular network adopted the proposed EMFG algorithm is helpful for keeping the average network coverage probability at a stable level. For the future work, we plan to investigate the network energy efficiency with quality of service constraints for 5G full duplex cellular networks based on the mean field game theory.

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121. Semasingh, P., and E. Hossain. 2016. Downlink power control in self-organizing dense small cells underlaying macrocells: A mean field game. IEEE Transactions on Mobile Computing 15 (2): 350–363. 122. Haenggi, M., J.G. Andrews, F. Baccelli, et al. 2009. Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE Journal on Selected Areas in Communications 27 (7): 1029–1046. 123. Al-Zahrani, A.Y., F.R. Yu, and M. Huang. 2013. A mean-field game approach for distributed interference and resource management in heterogeneous cellular networks. In 2013 IEEE Global Communication Conference (GLOBECOM), Atlanta, GA, 4964–4969. 124. Gueant, O., J.M. Lasry, P.-L. Lions. 2011. Mean field games and applications. In ParisPrinceton Lectures on Mathematical Finance, 205–266. New York, NY, USA: Springer. 125. Schulte, J.M. 2010. Adjoint methods for Hamilton-Jacobi-Bellman equations. Diploma Thesis, University of Munster, Germany. 126. Ahmed, R. 2004. Numerical schemes applied to the burgers and Buckley-Leverett equations. Ph.D. dissertation, Department of Mathematics, University of Reading, England.

Chapter 4

Energy Efficiency of 5G Multimedia Communications

4.1 Introduction As the rapid development of the information and communication technology (ICT), the energy consumption problem of ICT industry, which causes about 2% of worldwide CO2 emissions yearly and burdens the electrical bill of network operators [1], has drawn universal attention. Motivated by the demand for improving the energy efficiency in mobile multimedia communication systems, various resource allocation optimization schemes aiming at enhancing the energy efficiency have become one of the mainstreams in mobile multimedia communication systems, including transmission power allocation [2, 3], bandwidth allocation [4–6], sub-channel allocation [7], and etc. Multi-input multi-output (MIMO) technologies can create independent parallel channels to transmit data streams, which improves spectrum efficiency and system capacity without increasing the bandwidth requirement [8]. Orthogonal-frequencydivision-multiplexing (OFDM) technologies eliminate the multipath effect by transforming frequency selective channels into flat channels. As a combination of MIMO and OFDM technologies, the MIMO-OFDM technologies are widely used in mobile multimedia communication systems. However, how to improve energy efficiency with quality of service (QoS) constraint is an indispensable problem in MIMOOFDM mobile multimedia communication systems. On the other hand, with the popularization of various smart devices such as smart phones, tablets, Google Glass, Apple Watch, etc., augmented reality (AR) and virtual reality (VR) have the potential to be the next mainstream general computing platform [1]. Based on high-definition (HD) video services/applications, VR is able to mimic the real world by creating a virtual world or by applying various on-spot sensory equipments, such that immersive user experiences can be enabled. In comparison, by augmenting more data information to the real world, AR is able to enhance the perception of reality in a manner of reality-plus-data. Since real-time interactions and flows of massive information are involved in AR/VR applications, it will bring new challenges to the architecture designs of future networks [2] for accommodating © Publishing House of Electronics Industry, Beijing and Springer Nature Singapore Pte Ltd. 2019 X. Ge and W. Zhang, 5G Green Mobile Communication Networks, https://doi.org/10.1007/978-981-13-6252-1_4

185

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the online AR/VR applications. To be specific, for future fifth generation (5G) wireless networks, the applications of AR/VR require innovations in the cloud-based network architectures [3], with an objective of significantly improving the network throughput, transmission latency, wireless capacity, etc.

4.2 Energy Efficiency Optimization for MIMO-OFDM Mobile Multimedia Communication Systems with QoS Constraints 4.2.1 Related Work The energy efficiency has become one of the hot studies in MIMO wireless communication systems in the last decade [9–14]. An energy efficiency model for PoissonVoronoi tessellation (PVT) cellular networks considering spatial distributions of traffic load and power consumption was proposed [9]. The energy-bandwidth efficiency tradeoff in MIMO multi-hop wireless networks was studied and the effects of different numbers of antennas on the energy-bandwidth efficiency tradeoff were investigated in [10]. An accurate closed-form approximation of the tradeoff between energy efficiency and spectrum efficiency over the MIMO Rayleigh fading channel was derived by considering different types of power consumption model [11]. A relay cooperation scheme was proposed to investigate the spectral and energy efficiencies tradeoff in multi-cell MIMO cellular networks [12]. The energy efficiencyspectral efficiency tradeoff of the uplink of a multi-user cellular V-MIMO system with decode-and-forward type protocols was studied in [13]. The tradeoff between spectral and energy efficiency was investigated in the relay-aided multi-cell MIMO cellular network by comparing both the signal forwarding and interference forwarding relaying paradigms [14]. In our earlier work, we explored the tradeoff between the operating power and the embodied power contained in the manufacturing process of infrastructure equipments from a life-cycle perspective [1]. In this section, we further investigate the energy efficiency optimization for MIMO-OFDM mobile multimedia communication systems. Based on the Wishart matrix theory [15–18], numerous channel models have been proposed in the literature for MIMO communication systems [19–26]. A closed-form joint probability density function (PDF) of eigenvalues of Wishart matrix was derived for evaluating the performance of MIMO communication systems [19]. Moreover, a closed-form expression for the marginal PDF of the ordered eigenvalues of complex noncentral Wishart matrices was derived to analyze the performance of singular value decomposition (SVD) in MIMO communication systems with Rician fading channels [20]. Based on the distribution of eigenvalues of Wishart matrix, the performance of high spectrum efficiency MIMO communication systems with multiple phase shift keying (M-PSK) modulated signals in a flat Rayleigh-fading environment was investigated in terms of symbol error probabilities [21]. Furthermore, the

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cumulative density functions (CDF) of the largest and the smallest eigenvalue of a central correlated Wishart matrix were investigated to evaluate the error probability of a MIMO maximal ratio combing (MRC) communication system with perfect channel state information at both transmitter and receiver [22]. Based on PDF and CDF of the maximum eigenvalue of double-correlated complex Wishart matrices, the exact expressions for the PDF of the output signal-to-noise ratio (SNR) were derived for MIMO-MRC communication systems with Rayleigh fading channels [23]. The closed-form expressions for the outage probability of MIMO-MRC communication systems with Rician-fading channels were derived under the condition of the largest eigenvalue distribution of central complex Wishart matrices in the noncentral case [24]. Furthermore, the closed-form expressions for the outage probability of MIMO-MRC communication systems with and without co-channel interference were derived by using CDFs of Wishart matrix [25]. Meanwhile, the PDF of the smallest eigenvalue of Wishart matrix was applied to select antennas to improve the capacity of MIMO communication systems [26]. However, most existing studies mainly worked on the joint PDF of eigenvalues of Wishart matrix to measure the channel performance for MIMO communication systems. In our study, subchannels’ gains derived from the marginal probability distribution of Wishart matrix is investigated to implement energy efficiency optimization in MIMO-OFDM mobile multimedia communication systems. In conventional mobile multimedia communication systems, many studies have been carried out [27–33]. In terms of the corresponding QoS demand of different throughput levels in MIMO communication systems, an effective antenna assignment scheme and an access control scheme were proposed in [27]. A downlink QoS evaluation scheme was proposed from the viewpoint of mobile users in orthogonal frequency-division multiple-access (OFDMA) wireless cellular networks [28]. To guarantee the QoS in wireless networks, a statistical QoS constraint model was built to analyze the queue characteristics of data transmissions [29]. The energy efficiency in fading channels under QoS constraints was analyzed in [30], where the effective capacity was considered as a measure of the maximum throughput under certain statistical QoS constraints. Based on the effective capacity of the block fading channel model, a QoS driven power and rate adaptation scheme over wireless links was proposed for mobile wireless networks [31]. Furthermore, by integrating information theory with the effective capacity, some QoS-driven power and rate adaptation schemes was proposed for diversity and multiplexing systems [32]. Simulation results showed that multichannel communication systems can achieve both high throughput and stringent QoS at the same time. Aiming at optimizing the energy consumption, the key tradeoffs between energy efficiency and link-level QoS metrics were analyzed in different wireless communication scenarios [33]. However, there has been few research work addressing the problem of optimizing the energy efficiency under different QoS constraints in MIMO-OFDM mobile multimedia communication systems. Thus, motivated by aforementioned problems, this section is devoted to the energy efficiency optimization with statistical QoS constraints in MIMO-OFDM mobile multimedia communication systems with statistical QoS constraints which uses a

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statistical exponent to measure the queue characteristics of data transmission in wireless systems. All subchannels in MIMO-OFDM communication systems are first grouped by their channel gains. On this basis, a novel subchannel grouping scheme is developed to allocate the corresponding transmission power to each of the subchannels in different groups, which simplifies the multi-channel optimization problem to a multi-target single channel optimization problem.

4.2.2 System Model The MIMO-OFDM mobile multimedia communication system is illustrated in Fig. 4.1. It has a Mr × Mt antenna matrix, N subcarriers and S OFDM symbols, where Mt is the number of transmit antennas and Mr is the number of receive antennas. We denote B as the system bandwidth and T f as the frame duration. The OFDM signals are assumed to be transmitted within a frame duration. Then the received signal of MIMO-OFDM communication system can be expressed as follows:

Fig. 4.1 MIMO-OFDM system model

4.2 Energy Efficiency Optimization for MIMO-OFDM …

yk [i]  Hk xk [i] + n,

189

(4.1)

where yk [i] and xk [i] are the received signal vector and transmitted signal vector at the kth (k  1, 2, . . . , N ) subcarrier of the ith (i  1, 2, . . . , S) OFDM symbol, respectively. Hk is the frequency-domain channel matrix at the kth subcarrier and n is the additive noise vector. Let C denote the complex space, then we have yk ∈ C Mr , xk ∈ C Mt , Hk ∈ C Mr ×Mt , and n ∈ C Mr . Without loss of generality, we assume E{nn H }  I Mr ×Mr , where E{·} denotes the expectation operator. Discrete-time channels are assumed to experience a block-fading, in which the frame duration is shorter than the channel coherence time. Based on this assumption, the channel gain is invariant within a frame duration T f , but varies independently from one frame to another. In each frame duration, the channel at each subcarrier is divided into M (M  min(Mt , Mr )) parallel SISO channels by the SVD method. As a consequence, a total number of M × N parallel space-frequency subchannels can be generated in each OFDM symbol. Transmitters are assumed to obtain the channel state information (CSI) from receivers without delay via feedback channels. Furthermore, an average transmission power constraint P¯ is configured for each subchannel in the MIMO-OFDM communication system. With this average transmission power constraint, transmitters are able to perform power control adaptively according to the feedback CSI and system QoS constraints, so that the energy efficiency in the MIMO-OFDM mobile multimedia communication system can be optimized.

4.2.3 Energy Efficiency Modeling of MIMO-OFDM Mobile Multimedia Communication Systems Applying the SVD method to the channel matrix Hk at each subcarrier, where Hk ∈ C Mr ×Mt (k  1, 2, . . . , N ), we have  Hk  Uk k VkH ,

(4.2)

where Uk ∈ C Mr ×Mr and Vk ∈ C Mt ×Mt are unitary matrices. When Mr ≥ Mt , ˜ k  [k , 0 Mr ,Mt −Mr ]; otherwise when Mr < Mt , we have we have block matrix  T ˜ k  [k , 0 Mt ,Mr −Mt ] , where k  diag(λ1,k , . . . , λ M,k ) and λm,k ≥ 0, ∀m  M denotes the subchannel gain set at the kth sub1, . . . , M, k  1, . . . , N . {λm,k }m1 carrier. In this way, the MIMO channel at each subcarrier is decomposed into M parallel SISO subchannels by SVD method. Therefore, M × N parallel space-frequency subchannels are obtained at N orthogonal subcarriers for each OFDM symbol. In traditional energy efficiency optimization researches, Shannon capacity is usually used as the index which measures the system output. However, in any practical wireless communication systems, the system capacity is obviously less than Shannon capacity, especially in the scenario with strict QoS constraint. The effective capacity of each subchannel is taken as the practical data rate with certain QoS constraint.

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The total effective capacity of M × N subchannels is configured as the system output and the total transmission power allocated to M × N subchannels is configured as the system input. As a consequence, the energy efficiency of MIMO-OFDM mobile multimedia communication systems is defined as follows Ctotal (θ )  η E{Ptotal }

M m1

N

Ce (θ )m,k , E{Ptotal } k1

(4.3)

where Ce (θ )m,k (m  1, 2, . . . , M, k  1, 2, . . . , N ) is the effective capacity of the mth subchannel over the kth subcarrier, and E{Ptotal } is the expectation of the total transmission power allocated to all M × N subchannels. θ is the QoS statistical exponent, which indicates the exponential decay rate of QoS violation probabilities [31]. A smaller θ corresponds to a slower decay rate, which implies that the multimedia communication system provides a looser QoS guarantee; while a larger θ leads to a faster decay rate, which means that a higher QoS requirement should be supported. Practical MIMO-OFDM mobile multimedia communication systems involve multiple services, such as speech and video services, which are sensitive to the delay parameter. Different services in MIMO-OFDM mobile multimedia communication systems have different QoS constraints. In view of this, the effective capacity of each subchannel depends on the corresponding QoS constraint. A statistical QoS constraint is adopted to evaluate the effective capacity of each subchannel which is calculated as the system practical output in MIMO-OFDM mobile multimedia communication systems. Assuming the fading process over wireless channels is independent among frames and keeps invariant within a frame duration, the effective capacity Ce (θ ) for a subchannel with QoS statistical exponent θ in MIMO-OFDM mobile multimedia communication systems is expressed as follows [31]    1 Ce (θ )  − log E e−θ R , θ

(4.4)

R  T f B log2 (1 + μ(θ, λ)λ),

(4.5)

where R denotes the instantaneous bit rate within a frame duration, λ denotes the subchannel gain, and μ(θ, λ) denotes the transmission power allocated to a subchannel. After SVD of channel matrices at N orthogonal subcarriers, M × N parallel subchannels are obtained. The channel gain over each of these M × N parallel subchannels follows a marginal probability distribution (MPDF). Assuming pm,k (λ) as the MPDF of channel gain over the mth (m  1, 2, . . . , M) subchannel at the kth (k  1, 2, . . . , N ) orthogonal subcarrier, then the corresponding effective capacity Ce (θ )m,k over the mth subchannel at the kth orthogonal subcarrier is derived as (4.6). ⎛∞ ⎞

1 (4.6) Ce (θ )m,k  − log⎝ e−θ T f B log2 (1+μm,k (θ,λ)λ) pm,k (λ)dλ⎠, θ 0

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191

where μm,k (θ, λ) is the transmission power allocated to the mth subchannel at the kth orthogonal subcarrier. Considering the practical power consumption limitation at transmitters, an average transmission power constraint P¯ over each subchannel is derived as (4.7). P¯ 

∞ μm,k (θ, λ) pm,k (λ)dλ (∀m  1, 2, . . . , M, k  1, 2, . . . , N ).

(4.7)

0

With the average transmission power constraint, the expectation of transmission power E{Ptotal } is given by E{Ptotal }  P¯ × M × N .

(4.8)

Substituting expressions (4.7) and (4.8) into (4.3), we derive the energy efficiency model as (4.9). M η

m1

N k1

− θ1 log

 ∞ 0

 e−θ T f B log2 (1+μm,k (θ,λ)λ) pm,k (λ)dλ

P¯ × M × N

.

(4.9)

From (4.9), the energy efficiency of MIMO-OFDM mobile multimedia communication systems depends on the MPDF pm,k (λ) (m  1, 2, . . . , M, k  1, 2, . . . , N ) over M × N subchannels. Since there is a relationship between the MPDF pm,k (λ) and statistical characteristics of the subchannel, the marginal distribution characteristics of each subchannel gain is investigated to optimize the energy efficiency in MIMO-OFDM mobile multimedia communication systems.

4.2.4 Energy Efficiency Optimization of Mobile Multimedia Communication Systems In MIMO wireless communication systems, statistical characteristics of channel gain depend on the eigenvalues’ distribution of Hermitian channel matrix HH H , where H is the channel matrix [34–36]. When the elements of H are complex valued with real and imaginary parts each governed by a normal distribution N(0, 1/2) with mean value 0 and variance value 1/2, the Hermitian channel matrix W  HH H is called a central Wishart channel matrix [15–17, 19]. In this case, E{H}  0 and wireless channels have the Rayleigh fading characteristic. If E{H}  0, W  HH H is a noncentral Wishart channel matrix and wireless channels have the Rician fading characteristic [20]. Based on SVD results of wireless channel matrix, subchannels at each orthogonal subcarrier are sorted in a descending order of channel gains. Starting from the joint PDF of eigenvalues of Wishart channel matrix, the channel gain MPDF of

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subchannels ordered at the mth position in the descending order of channel gains is derived. Furthermore, all subchannels at N subcarriers are grouped according to their MPDFs. In terms of subchannel grouping results, a closed-form solution is derived to optimize the energy efficiency of MIMO-OFDM mobile multimedia communication systems in this section.

4.2.5 Optimization Solution of Energy Efficiency To maximize the energy efficiency of MIMO-OFDM mobile multimedia communication systems with statistical QoS constraints, the optimization problem can be formulated as (4.10).  M N  ∞ −θ T B log (1+μ (θ,λ)λ)  1 f m,k 2 pm,k (λ)dλ m1 k1 − θ log 0 e ηopt  max P¯ × M × N     ∞ −θ T f B log2 (1+μm,k (θ,λ)λ) M N 1 (4.10) max − log e p (λ)dλ  m,k m1 k1 0 θ  , P¯ × M × N s.t. :

∞ ¯ ∀m  1, 2, . . . , M, k  1, 2, . . . , N . μm,k (θ, λ) pm,k (λ)dλ ≤ P, (4.11) 0

where ηopt is the optimized energy efficiency. From the problem formulation in (4.10) and (4.11), it is remarkable that the energy efficiency of MIMO-OFDM mobile multimedia communication systems depends on transmission power allocation results μm,k (θ, λ) over M × N subchannels. In this case, the optimization problem in (4.10) and (4.11) is a multi-channel optimization problem, which is intractable to obtain a closed-form solution in mathematics. In most studies on MIMO wireless communication systems, the energy efficiency optimization problem is solved by a single channel optimization model [32]. How to change the multi-channel energy efficiency optimization problem into the single channel energy efficiency optimization problem and derive a closed-form solution are great challenges. Without loss of generality, the optimized transmission power allocation of single subchannel μopt (θ, λ) is expressed as follows [32]  μopt (θ, λ) 

1 1  β+1

β λ β+1



0

β  θ T f B,

1 λ

λ≥ λ 10−3 , the transmission power allocation threshold n start to decrease with the increase of subchannel gains in subchannel groups. Figure 4.3 illustrates the transmission power allocation threshold n with respect ¯ For to each grouped subchannels considering different average power constraints P. each grouped subchannels, the transmission power allocation threshold n decreases ¯ When P¯ ≤ 0.13, the transmiswith the increase of the average power constraint P. sion power allocation threshold n increases with the increase of subchannel gains

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Fig. 4.2 Transmission power allocation threshold n with respect to each grouped subchannels considering different QoS statistical exponents θ

Fig. 4.3 Transmission power allocation threshold n with respect to each grouped subchannels considering different average power constraints P¯

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199

Fig. 4.4 Effective capacity Ctotal (θ) with respect to the QoS statistical exponent θ considering different scenarios

in subchannel groups. When P¯ > 0.13, the transmission power allocation threshold n start to decrease with the increase of subchannel gains in subchannel groups. To evaluate the energy efficiency and the effective capacity of MIMO-OFDM mobile multimedia communication systems, three typical scenarios with different antenna numbers are configured in Figs. 4.4 and 4.5: (1) Mt  2, Mr  2; (2) Mt  3, Mr  2; (3) Mt  4, Mr  4. Figure 4.4 shows the impact of QoS statistical exponents θ on the effective capacity of MIMO-OFDM mobile multimedia communication systems in three different scenarios. From curves in Fig. 4.4, the effective capacity decreases with the increase of the QoS statistical exponent θ . The reason of this result is that the larger values of θ correspond to the higher QoS requirements, which result in a smaller number of subchannels being selected to satisfy the higher QoS requirements. When the QoS statistical exponent θ is fixed, the effective capacity increases with the number of antennas in MIMO-OFDM mobile multimedia communication systems. This result indicates the channel spatial multiplexing can improve the effective capacity of MIMO-OFDM mobile multimedia communication systems. Figure 4.5 illustrates the impact of QoS statistical exponents θ on the energy efficiency of MIMO-OFDM mobile multimedia communication systems in three different scenarios. From curves in Fig. 4.5, the energy efficiency decreases with the increase of the QoS statistical exponent θ . The reason of this result is that the larger values of θ correspond to the higher QoS requirements, which result in a smaller number of subchannels are selected to satisfy the higher QoS requirements. This result conduces to the effective capacity is decreased. If the total transmission

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4 Energy Efficiency of 5G Multimedia Communications

Fig. 4.5 Energy efficiency η with respect to the QoS statistical exponent θ considering different scenarios

power is constant, the decreased effective capacity will lead to the decrease of the energy efficiency in communication systems. When the QoS statistical exponent θ is fixed, the energy efficiency increases with the number of antennas in MIMO-OFDM mobile multimedia communication systems. This result indicates that the channel spatial multiplexing can improve the energy efficiency of MIMO-OFDM mobile multimedia communication systems. When the QoS statistical exponent is fixed as θ  10−3 , the impact of the average power constraint on the energy efficiency and the effective capacity of MIMO-OFDM mobile multimedia communication systems is investigated in Fig. 4.6. From Fig. 4.6, the energy efficiency decreases with the increase of the average power constraint and the affective capacity increases with the increase of the average power constraint. This result implies there is an optimization tradeoff between the energy efficiency and effective capacity in MIMO-OFDM mobile multimedia communication systems: as the transmission power increases which leads to larger effective capacity, the energy consumption of the system also rises; therefor, the larger power input results in the decline of energy efficiency. To analyze performance of the EEOPA algorithm, the traditional average power allocation (APA) algorithm [38], i.e., every subchannel with the equal transmission power algorithm is compared with the EEOPA algorithm by Figs. 4.7, 4.8, 4.9 and 4.10. Three typical scenarios with different antenna numbers are configured in Fig. 4.7, 4.8, 4.9 and 4.10: (1) Mt  2, Mr  2; (2) Mt  3, Mr  2; (3) Mt  4, Mr  4. In Fig. 4.7, the effect of the QoS statistical exponent θ on the energy efficiency of EEOPA and APA algorithms is investigated with constant average power constraint P¯  0.1 W. Considering changes of the QoS statistical exponent, the energy efficiency of EEOPA algorithm is always higher than the energy efficiency

4.2 Energy Efficiency Optimization for MIMO-OFDM …

201

Fig. 4.6 Impact of the average power constraint on the energy efficiency η and the effective capacity Ctotal (θ)

Fig. 4.7 Energy efficiency η of EEOPA and APA algorithms as variation of QoS statistical exponent θ under different scenarios

of APA algorithm in three scenarios. In Fig. 4.8, the impact of the average power constraint on the energy efficiency of EEOPA and APA algorithms is evaluated with the fixed QoS statistical exponent θ  10−3 . Considering changes of the average power constraint, the energy efficiency of EEOPA algorithm is always higher than the energy efficiency of APA algorithm in three scenarios.

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Fig. 4.8 Energy efficiency η of EEOPA and APA algorithms as variation of average power constraint P¯ under different scenarios

Fig. 4.9 Effective capacity Ctotal (θ) of EEOPA and APA algorithms as variation of QoS statistical exponent θ under different scenarios

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Fig. 4.10 Effective capacity Ctotal (θ) of EEOPA and APA algorithm as variation of average power constraint P¯ under different scenarios

In Fig. 4.9, the effect of the QoS statistical exponent θ on the effective capacity of EEOPA and APA algorithms is compared with constant average power constraint P¯  0.1 W. Considering changes of the QoS statistical exponent, the effective capacity of EEOPA algorithm is always higher than the effective capacity of APA algorithm in three scenarios. In Fig. 4.10, the impact of the average power constraint on the effective capacity of EEOPA and APA algorithms is evaluated with the fixed QoS statistical exponent θ  10−3 . Considering changes of the average power constraint, the effective capacity of EEOPA algorithm is always higher than the effective capacity of APA algorithm in three scenarios. Based on above comparison results, our proposed EEOPA algorithm can improve the energy efficiency and effective capacity of MIMO-OFDM mobile multimedia communication systems.

4.2.7 Conclusions In this section, an energy efficiency model is proposed for MIMO-OFDM mobile multimedia communication systems with statistical QoS constraints. An energy efficiency optimization scheme is presented based on the subchannel grouping method, in which the complex multi-channel joint optimization problem is simplified into a multi-target single channel optimization problem. A closed-form solution of the energy efficiency optimization is derived for MIMO-OFDM mobile multimedia communication systems. Moreover, a novel algorithm, i.e., EEOPA, is designed to

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improve the energy efficiency of MIMO-OFDM mobile multimedia communication systems. Compared with the traditional APA algorithm, simulation results demonstrate that our proposed algorithm has advantages on improving the energy efficiency and effective capacity of MIMO-OFDM mobile multimedia communication systems with QoS constraints.

4.3 Multi-path Cooperative Communications Networks for Augmented and Virtual Reality Transmission 4.3.1 Related Work With the popularization of various smart devices such as smart phones, tablets, Google Glass, Apple Watch, etc., augmented reality (AR) and virtual reality (VR) have the potential to be the next mainstream general computing platform [39]. Based on high-definition (HD) video services/applications, VR is able to mimic the real world by creating a virtual world or by applying various pieces on-spot sensory equipment, such that immersive user experiences can be enabled. In comparison, by augmenting more data information to the real world, AR is able to enhance the perception of reality in a manner of reality-plus-data. Since real-time interactions and flows of massive information are involved in AR/VR applications, it will bring new challenges to the architecture designs of future networks [40] for accommodating the online AR/VR applications. To be specific, for future fifth generation (5G) wireless networks, the applications of AR/VR require innovations in the cloud-based network architectures [41], with an objective of significantly improving the network throughput, transmission latency, wireless capacity, etc. To support the enormous traffic demands involved in AR/VR applications, there have been a great number of research activities in 5G network studies [42–45]. N. Bhushan et al. [42] pointed out that the 5G wireless networks could meet the 1000x traffic demands over the next decade, with reasonable projections on additional spectrum availability, densification of small-cell deployments, and growth in backhaul infrastructures. Meanwhile, the massive multi-input multi-output (MIMO) technique offers huge advantages in terms of energy efficiency, spectral efficiency, robustness, and reliability [43], which allows for the use of low-cost hardware at both base stations and user terminals. Moreover, the large available bandwidth at millimeter wave (mmWave) frequencies makes mmWave transmission techniques attractive for the future 5G wireless networks [44, 45]. On the other hand, the gigantic data traffics on the next generation of Internet have also been investigated [46–48]. Walravens et al. [46] studied the high-rate, small packet traffic in an Ethernet controller and used a technique called receive descriptor recycling (RDR) to reduce the small-packet loss by 40%. Laoutaris et al. [47] proposed using this already-paid-for off-peak capacity to perform global delay-tolerant bulk data transfers on the Internet.

4.3 Multi-path Cooperative Communications Networks …

205

M. Villari et al. studied the osmotic computing by moving cloud resources closer to the end users, which is regarded as a new paradigm for edge/cloud integration [48]. To reduce the network latency for high-speed reliable services like AR/VR and surveillance, V. usage of different chromatic dispersion compensation methods were discussed to reduce the transmission delay in fiber optical networks [49]. The lowdelay rate control algorithms have been proposed to address the delay problem in [50, 51]. For instance, for the AR applications considered in [51], low-delay communications of the encoded video over a Bluetooth wireless personal area network were investigated, by using a combination of the dynamic packetisation of video slices together with the centralized and predictive rate control. Mobile AR and VR have been used and studied recently [52–55]. A. D. Hartl et al. studied the verification of holograms by using mobile AR, where a re-parametrized user interface is proposed in [52]. The privacy preserving of cloth try-on was studied in [53], by using mobile AR. In [56], S. Choi et al. proposed a prediction-based delay compensation system for head mounted display, which compensated delay up to 53 ms with 1.083° of a minimum average error. In [57], T. Langlotz et al. presented an initial approach for mitigating the effect of color blending in optical see-through head-mounted displays, by introducing a real-time radiometric compensation. Different from conventional video applications, AR/VR applications impose a strict requirement on the uplink/downlink transmission delays. Take the 360° video (an application of VR) as an example, the jitter and visual field delays, i.e., motionto-photons Latency (MTP) should be limited within 20 ms. Otherwise, the users will feel dizzy [39]. Therefore, the transmission delay is one of main challenges for future 5G wireless networks. Moreover, few studies have investigated the issue of system energy consumption for the heavy traffic and high-rate transmissions required by AR/VR applications. Motivated by these gaps, it is important to propose a 5G wireless network solution for reducing the transmission delays and system energy consumption in AR/VR applications.

4.3.2 System Model 4.3.2.1

Network Model

We consider a two-tier heterogeneous cellular network, where multiple small cell base stations (SBSs) and edge data centers (EDCs) are deployed within the coverage of a macro cell base station (MBS). The MBS and EDCs are connected to the core networks through fiber to the cell (FTTC). A SDN architecture is adopted to support the separating of data and control information in Fig. 4.11. As the network controller, the SDN controller divides the control information into MBSs and data information into SBSs and EDCs. MBSs are mainly responsible for the delivery of controlling information and routing decisions. The EDCs are data centers deployed at the edge of core networks, where massive AR/VR data are stored. All EDCs and SBSs equipped with mmWave transmission techniques comprise the 5G small cell network.

206

4 Energy Efficiency of 5G Multimedia Communications

Millimeter wave link

FTTC link

Control uplink (controller connects with each User

Control downlink (controller connects with each edge data center Macro cell BS(controller) Edge data center

Access link

Small cell BS

Remote data center User

SDN Controller

(a) Deployment scenario Data information

Control information

AR/VR data Edge data center AR/VR data

SDN controller SBS transmission MBS AR/VR data User

(b) Logical architecture Fig. 4.11 A SDN-based network architecture for 5G small cell networks

4.3 Multi-path Cooperative Communications Networks …

207

Utilizing the mmWave multi-hop transmission technique, AR/VR data streams can be delivered between EDCs and SBSs. It is assumed that the MBSs, denoted by  M , follow a Poisson Point Process (PPP) with density λ M , the SBSs, denoted by  S , follow a PPP with density λ S , the user terminals, denoted by U , follow a PPP with density λU , and the EDCs, denoted by  E , follow a PPP with density λ E , in the 2-D plane. The distributions of MBSs, SBSs, EDCs and user terminals are assume to be independent each other. To establish a clean formulation, only the coverage of a single MBS is illustrated, as shown in Fig. 4.11. For ease of exposition, the SBSs located within the coverage of the MBS is denoted as a set N , whose number is expressed as |N |. The set of EDCs located within the coverage of the MBS is denoted as M, whose number is expressed as |M|. The set of user terminals within the coverage of the MBS is denoted as U, whose number is expressed as |U|. For a user terminal with AR/VR traffic demands, a request is first sent to an associated MBS. Upon receiving a user’s request, the associated MBS searches EDCs that are located close to the requesting user. If however, there is an EDC located outside the macro cell, then the associated MBS sends the request to the MBS to which this EDC belongs, through MBS controller. Upon receiving the routing information transmitted from the MBS, the close-by EDCs transmit AR/VR data to the SBS that is located closest to the requesting user. Finally, this SBS delivers the AR/VR data to the requesting user.

4.3.2.2

Storage Strategies of AR/VR Data

Consider a library  K that consists of |K| AR/VR video contents. We define q   q1 , q2 , . . . , q|K| as the corresponding popularity distributions of these |K| AR/VR video contents that are arranged in a popularity descending order, where qk (k ∈ K and 1 ≤ k ≤ |K|) denotes the popularity of the k-th popular AR/VR video content, i.e., the probability that an arbitrary user request is for AR/VR video content k. Without loss of generality, the distribution of qk is assumed to be governed by a Zipf distribution [58], k −β qk  |K| , −β j1 j

(4.33)

where β is apositive value that characterizes the skewness of the popularity distri|K| bution, and k1 qk  1. In general, a subset of the |K| AR/VR video contents are stored in EDCs, whereas the remaining video contents can be fetched from the remote data centers (RDC). For ease of exposition, we let x  (xmk ∈ {0, 1} : m ∈ M, k ∈ K) denote the AR/VR video content placement strategy in EDCs, where xmk  1 means that AR/VR video content k (k ∈ K) is stored at EDC m (m ∈ M), whereas xmk  0 means that AR/VR video content k is not stored at EDC m. Since the storage capacity at each EDC is

208

4 Energy Efficiency of 5G Multimedia Communications

limited that cannot store all |K| video contents, generally there exist three storage strategies in existing studies. (1) First in first out (FIFO): The contents stored at the EDC form a queue in a chronological order until the storage capacity is reached. In this way, the content at the head of the queue has to be removed for accommodating the newly incoming content at the tail of the queue. (2) Least recently used (LRU): The contents are stored at the EDC according to how frequently they are requested by users recently. In this way, if the storage capacity of EDC is reached, then the least recently used content will be replaced by the newly incoming content. (3) Popularity-priority: All contents are stored at the EDC in a popularity descending order until the storage capacity is reached. In this way, the more popular content can be stored by comparing the popularity of the newly incoming content with that of the least popular content stored at EDC. In our considered network model, the popularity-priority strategy is adopted to store the most popular AR/VR video content data at each EDC. Although the popularity distribution of AR/VR video contents may vary with time, to maintain cache consistency, we consider in this section a relatively static popularity distribution within a given interval. This assumption is valid in examples including popular news containing short videos that are updated every 2–3 h, new movies that are posted every week, new music videos that are posted about every month. For ease of analysis, it is assumed that each EDC is of the same storage capacity, in which A ( A ≤ |K|) AR/VR video contents of the same size can be stored. Since the popularity-priority strategy is adopted where each EDC stores the most popular contents independently, it ends up with that the same A most popular video contents are stored at each EDC, i.e., A 

xmk  A, ∀m ∈ M

(4.34)

k1

To facilitate reading, the notations and symbols with the default values used in this section and simulations are listed in Table 4.1.

4.3.3 Network Latency Model For the AR/VR service provisioning upon receiving an arbitrary user’s request, the network latency model is expressed as   r eq bh as D  DU L + D deli DL + D DL + D DL · Pin−E DC   r eq bh as f iber · (1 − Pin−E DC ) + DU L + D deli DL + D DL + D DL + D r eq

bh as f iber  DU L + D deli · (1 − Pin−E DC ), DL + D DL + D DL + D

(4.35)

4.3 Multi-path Cooperative Communications Networks …

209

Table 4.1 Notations and symbols with default values Symbols

Definition/explanation

Default values

λ M , λ S , λU

Density of MBSs, SBSs, users, respectively

5, 50, 200 km−2

PS , PE , PU

Transmission power at each SBS, EDC, user, respectively

30, 30, 23 dBm

n tM , nrM

Number of transmission and receive antennas at the MBS, respectively

4, 4

n tS

Number of transmission antennas at the SBS

8

nrE

Number of receive antennas at the EDC

2

U nU t , nr

Number of transmission and receive antennas at the user terminal, respectively

2, 2

α1 , α2

Path loss fading over microwave and millimeter wave channels, respectively

3.5, 2

θ1

Received power threshold at the MBS

−40 dBm

θ2

SINR threshold at the EDC

0 dB

θ3

Threshold of received wireless signal of millimeter wave transmission at the SBS

0 dB

θ4

Received power threshold at the user

−70 dBm

TU L

The time for transmitting a user request from a user to MBS

50 µs

deli TDL

The downlink transmission time of a packet from MBS to EDC

50 µs

as TDL

The downlink transmission time of a data packet from SBS to the user

50 µs

τmmW

Time slot of millimeter wave channels

5 µs

N0

Noise power spectral density

−174 dBm Hz−1

WmmW

Bandwidth of millimeter wave wireless links

200 MHz

μ

Service rate at the MBS

1.05 × 104 s−1

χ

Factor of user arrival rate

5 × 107 m2 s−1

σ

Standard variance of shadow fading over wireless channels

5 dB

rmmW

Maximum transmission distance of SBSs and EDCs

100 m

L

Data packet size

1024 Bytes

Z

Buffer size of SBS

1 MB

L f iber

Distance between the macro cell network and the RDC

1000 km

v f iber

Transmission rate in optic fibers

2 × 108 m s−1

r eq,1

(continued)

210

4 Energy Efficiency of 5G Multimedia Communications

Table 4.1 (continued) Symbols

Definition/explanation

Default values

β

Skewness parameter of popularity distributions

0.8

A

Number of video contents stored in the EDC

500

Rmax

Maximum distance between the EDC and the destination SBS

500 m

aM , bM

The fixed coefficient of MBS

21.45, 354 W

aS , bS

The fixed coefficient of SBS

7.84, 71 W

aE , bE

The fixed coefficient of EDC

7.84, 71 W

TLiMf etime

Lifetime of MBS

10 years

S TLi f etime

Lifetime of SBS

5 years

TLiE f etime

Lifetime of EDC

5 years

E storage

Energy consumption of one video content stored at the EDC

8 × 106 J

with Pin−E DC 

A 

qk ,

(4.36)

k1

where Pin−E DC denotes the probability that the requesting user can find the desired AR/VR data in the nearby EDCs. If the required AR/VR data can be found in EDCs, then the network latency is a sum of the delays within the macro cell network. Otherwise, the transmission delay over the optical networks should also be taken into account. r eq

(1) DU L : The delay incurred when a user terminal sends a request to the MBS r eq,1 through uplink, which consists of the uplink transmission delay DU L , the r eq,2 queuing delay and the processing delay at the MBS DU L , i.e., r eq

r eq,1

r eq,2

DU L  DU L + DU L .

(4.37)

When a user u ∈ U sends a request to the MBS, the corresponding received signal power at the MBS is given as [59] r eq,1

γU L

 PU ro−α1 |Huo |2 ,

(4.38)

where the same transmission power PU is adopted at each user u ∈ U, ro denotes the distance between the user u and MBS M B So , α1 denotes the path-

4.3 Multi-path Cooperative Communications Networks …

211

loss exponent, Huo ∈ Cnr ×n t denotes the channel gains from the user u to the MBS M B So , where we have h uo,x1 ,y1 ∼ CN (0, 1) for each entry h uo,x1 ,y1 (x1  1, 2, . . . , n rM , y1  1, 2, . . . , n Ut ), n rM and n Ut denote the number of receive antennas at the MBS and the number of transmission antennas at the user, respectively. Then the probability of successfully delivering the user request to the MBS is expressed as M

r eq,1

ρU L

U

≥ θ1 )  Pr(PU ro−α1 |Huo |2 ≥ θ1 ) n M nU 2 PU ro−α1 ( x1r 1 y1t 1 h uo,x1 ,y1 ) (a)  Pr( ≥ θ1 ) n Ut θ1 n Ut roα1 PU ro−α1 g ≥ θ1 )  Pr(g ≥ )  Pr( U PU nt ⎛U M ⎞ n t nr −1  1 ! θ1 n U r α1 "t − θ1 nUt roα1 t o  Ero ⎝ e PU ⎠ t! P U t0 ⎛U M ! " ∞ ! U α1 "t nr −1

nt  θ nU r α1 − 1 Pt +λ M πr 2 θ 2π λ r n r M 1 t U ·e  ⎝ dr , t! PU t0 r eq,1

 Pr(γU L

(4.39)

0

where θ1 denotes the received signal power threshold for successful reception at the MBS. Provided that MBSs are randomly located in a 2-D plane with density λ M , approximation (a) is obtained based on [60], where we have   n M nU 2 g ∼ Gamma n Ut n rM , 1 by letting g  x1r 1 y1t 1 h uo,x1 ,y1 . Otherwise if r eq,1 γU L < θ1 , retransmissions are performed. Then we have the average transmission delay for successfully delivering a user request to the MBS as n tM nrE

xn tM nrE 

 t π λ M r 2 x0 Gnt−1 M E yn tM n E , n r t

(4.40)

r

t1 r eq,1

where TU L denotes the time for transmitting a user request from a user to the associated MBS, and r 1eq,1 corresponds to the average number of retransmissions ρU L [61]. On the other hand, an M/M/1 queuing model is considered for calculating the r eq,2 queuing delay DU L at the MBS upon successful receiving the user’s request. Then we have r eq,2

DU L



1 , μ − λa

(4.41)

212

4 Energy Efficiency of 5G Multimedia Communications

where λa denotes the average arrival rate of users’ requests at the MBS, which is proportional to the user density in the macro cell, i.e., λa  χ λU where χ is the factor of user arrival rate. μ denotes the service rate at the MBS. (2) D deli DL : The delay incurred when the MBS sends the user request/controlling information to the EDCs that own the requested AR/VR data through downlink. When the MBS sends routing information to the EDC E DCmu , E DCmu ∈ M that owns the AR/VR data requested by user u ∈ U, the corresponding received signal-to-interference-and-noise ratio (SINR) at EDC is given by deli  γ DL

PM ro−α1 |Hou |2 , −α1 |Hlu |2 + σz2 M B Sl ∈ M ,M B Sl  M B So PM rl

(4.42)

where PM denotes the transmission power at MBS, ro denotes the distance between MBS M B So and EDC E DCmu , rl denotes the distance between MBS E M M B Sl and EDC E DCmu , σz2 denotes the noise power, Hou ∈ Cnr ×n t denotes the channel gains between MBS M B So and EDC E DCmu where we have h ou,x2 ,y2 ∼ CN (0, 1) for each entry h ou,x2 ,y2 (x2  1, 2, . . . , n rM , y2  1, 2, . . . , n Ut ), Hlu ∈ E M Cnr ×n t denotes the channel gains between MBS M B Sl and EDC E DCmu , where we have h lu,x2 ,y2 ∼ CN (0, 1) for each entry h lu,x2 ,y2 (x2  1, 2, . . . , n rM , y2  1, 2, . . . , n Ut ), where n rE and n tM denote the number of receive antennas at the EDC and the number of transmission antennas at the MBS, respectively. Based on our previous work [60] ,the probability of successfully delivering the routing information from the MBS to EDC E DCmu can be readily expressed as deli deli ρ DL  Pr(γ DL ≥ θ2 )

PM ro−α1 |Hou |2 ≥ θ2 ) −α1 |Hlu |2 + σz2 M B Sl ∈ M ,M B Sl  M B So PM rl ⎞ ⎛ ⎛ ⎛ ⎞



∞ ⎟ ⎜ ⎜ 2 1 ⎜ ⎜ ⎟ ⎟ −πλ M r 2 ⎜ 2 α1  2π λ M r e ⎜exp⎜−π λ M r θ2 ⎝1 − # M E ⎠dv ⎟ $ n t nr α1 ⎠ ⎝ ⎝ 1 + v− 2 0 − α2

 Pr( 

  + sum xn tM nrE −1 dr,

θ2

1

(4.43)

for successful reception at the EDC, where θ2 denotes the SINR threshold  M E xn tM nrE −1 ∈ C(n t nr −1)×1 , and sum xn tM nrE −1 denotes the summation of all elements in xn tM nrE −1 . For ease of exposition, we let T  xn tM nrE  x1 , x2 , . . . , xn tM nrE , T  yn tM nrE  y1 , y2 , . . . , yn tM nrE

(4.44)

4.3 Multi-path Cooperative Communications Networks …

213

%

Gn tM nrE

&T   M E M E M E 1 + n n n n n n t r t r k2 , . . . , C2nt M nr E −1 · kn tM nrE , (4.45)  n tM n rE k1 , t r 2 ⎡ ⎤ 0 1 M E ⎢ ⎥ n n r k1 0 2 t ⎢ ⎥ 1 M E ⎢ 1 n M n E 1 + n M n E k2 ⎥ n n r k1 0 t r ⎢3 t r ⎥ 3 t ⎢ ⎥, .. ⎢ ⎥ . 0 ⎢ ⎥ M nE M nE ⎣ n M n E C ntM nrE ⎦ n n r r t t M E M E n n C n n C t r 2n M n E −2 t r 2n M n E −3 t r nM nE t r t r t r M M k k · · · k 0 1 n t nrE −1 n t nrE −2 nM nE nM nE nM nE t

t

r

t

r

r

(4.46) Then xn tM nrE can be expressed by using (4.45) and (4.46) as n tM nrE

 t π λ M r 2 x0 Gt−1 y M E, M E n t nr

xn tM nrE 

n t nr

t1

(4.47)

and n tM nrE −1

  t π λ M r 2 x0 Gt−1 yn tM nrE −1 , M E

xn tM nrE −1 

n t nr −1

t1

(4.48)

where x0  e−πλ M k0 r , ⎛ 2



2 α1

k0  θ2

− α2 1

θ2 2 α



kq  θ2 1 − α2 1

θ2

(4.49) ⎞

1 ⎜ ⎟ ⎝1 − # $n tM nrE ⎠dv, α1 1 + v− 2

1 # $q # $n tM nrE dv, q ≥ 1, α1 α1 − 1+v 2 1+v 2

(4.50)

(4.51)

respectively. Then we have D deli DL 

deli TDL , deli ρ DL

(4.52)

deli where TDL denotes the downlink transmission time of a packet from the MBS to the EDC.

214

4 Energy Efficiency of 5G Multimedia Communications

(3) D bh DL : The backhaul delay, i.e., the delay incurred when the EDC delivers the requested AR/VR data to the destination SBS which associates with the requesting user. In order to reduce the transmission delay of AR/VR data in 5G small cell networks, a MCR scheme is proposed in Sect. 4.3.4.1. Where the B closest EDCs simultaneously transmit the requested AR/VR data to the destination SBS by multi-path cooperative routes. (4) D as DL : The delay incurred when the destination SBS transmits the requested AR/VR data to the requesting user. Considering the directivity of mmWave transmission technique, the interference is negligible in 5G small cell networks [62]. Then the corresponding received signal power at user u is given by as  PS ru−α2 |Hnu |2 , γ DL

(4.53)

where the same transmission power PS is adopted  at each SBS, ru denotes the distance between the destination SBS S B Snu S B Snu ∈ N and the user u, U S α2 denotes the path-loss exponent, Hnu ∈ Cnr ×n t denotes the channel gains between the SBS S B Snu and the user u, where we have h nu,x3 ,y3 ∼ CN (0, 1) for each entry h nu,x3 ,y3 (x3  1, 2, . . . , n rU , y3  1, 2, . . . , n tS ), and n rU and n tS denote the number of receive antennas at the user and the number of transmission antennas at the SBS, respectively. The probability of successfully delivering the requested AR/VR data to the requesting user is expressed as as as ρ DL  Pr(γ DL ≥ θ4 )

 Pr(PS ru−α2 |Hnu |2 ≥ θ4 ) ⎛ U " ! ! nr −1

∞ n tS θ n S r α2 S α2 "t − 4 Pt +λ S πr 2 θ 2π λ r n r S 4 t S  ⎝ ·e dr, t! PS t0

(4.54)

0

where θ4 denotes the received signal power threshold for successful reception. Then we have D as DL 

as TDL as , ρ DL

(4.55)

as denotes the downlink transmission time of a data packet from the where TDL associated SBS to the user. (5) D f iber : The fiber delay incurred in the uplink/downlink transmissions when the MBS has to fetch AR/VR data from the RDC upon a search failure in local EDCs. Considering that the transmission rate in optic fibers is v f iber and the distance between the given macro cell and the RDC is L f iber , then the fiber delay is expressed as

4.3 Multi-path Cooperative Communications Networks …

D f iber 

2L f iber v f iber

215

.

(4.56)

4.3.4 AR/VR Multi-path Cooperative Transmissions For AR/VR applications, not only the large network throughput is required for a lot of data transmission, but also the low system delay is needed to support the user interactions. In traditional networks the data is transmitted from a source to a destination by a fixed path. In this case, the maximum network throughput is restricted by the minimum transmission rate of a relay node in the fixed path. It implies that the total network throughput is constrained by the most congestion node with the minimum transmission rate in the fixed path. Moreover, the system delay is also limited by the minimum transmission rate in the bottleneck of the fixed path. To solve these issues, a multi-path cooperative routing scheme is proposed to meet the requirements of the large network throughput and low system delay from AR/VR applications.

4.3.4.1

Multi-path Cooperative Routing (MCR) Scheme

Considering the un-stability of wireless channels, it is very difficult to transmit a large of AR/VR data with the low system delay constraint by a fixed path in 5G small cell networks. On the other hand, the AR/VR data is repeatedly stored in multiple EDCs according to the content popularity. Therefore, the same AR/VR data can be cooperatively transmitted to a user from adjacent EDCs. The basic multi-path cooperative routing scheme is described as follows: (1) EDCs selection: According to the system model, multiple DECs are located in a MBS cell. When the requested AR/VR data is stored at EDCs, B EDCs simultaneously transmit the same AR/VR data to a destination SBS where is the closest to the user. In the end, the destination SBS transmits the AR/VR data to the user by the millimeter wave wireless communication link. (2) Multi-path transmission strategy: All B EDCs are incrementally ordered by the average distance ri , 1 ≤ i ≤ B between the EDC E DCi and the destination SBS. The E DC p , 1 ≤ p ≤ B, is the EDC where is far away the destination SBS with the average distance r p . The size of requested AR/VR data S is divided into B parts with the proportion to transmit the data of

1 rp B 1 i1 ri

1 rp

B

1 i1 ri

. Moreover, the EDC E DC p only need

· S. In this case, the closer EDC need to transmit

the larger AR/VR data and the far way EDC transmit a little AR/VR data. (3) Relay SBSs selection: Based on the system model, SBSs are densely deployed in the coverage of every MBS. When the maximum transmission distance of SBS with millimeter wave transmission techniques is configured as rmmW , it is

216

4 Energy Efficiency of 5G Multimedia Communications

assumed that there exist more than two SBSs in the distance rmmW . In this case, the requested AR/VR data is transmitted to the destination SBS by wireless relayed SBSs. To minimize the relay delay in SBSs, the relay route algorithm with the minimum hop number is adopted for the transmission path between the requested EDC and the destination SBS. Relay SBSs are selected by the relay route algorithm with the minimum hop number, e.g., the shortest path based geographical routing algorithm [63].

4.3.4.2

Delay Theorem of Multi-path Cooperative Routing Scheme

MCR Delay Theorem When EDCs are deployed in a 5G small cell network, the AR/VR data stored at B EDCs are simultaneously transmitted to a destination SBS by the MCR scheme. The system delay of MCR scheme is expressed by # $ -Z. 2τmmW 1 + 1.28 λλES L # # $$ , · (4.57) D bh DL   B 1 ) rmmW 1 + er f f (r√mmW i1 ri 2σ ri 

(2i − 1) · (2i − 3) · · · 1 , √ 2i λe · (i)

f (rmmW ) (dB)  PS (dB) − θ3 (dB) − N0 WmmW (dB) − 70 − 20 log10 (rmmW ),

(4.58)

(4.59)

where Z is the buffer size of SBS, L is the data packet size, · is the rounding up operation of a number, ri is the average distance between the destination SBS and the EDC E DCi . λ S and λ E are the densities of SBSs and EDCs in a coverage of MBS. In a wireless route between the EDC and the destination SBS, the wireless transmission is time slotted and one packet is transmitted in each time slot τmmW . rmmW is the maximum transmission distance of SBSs and EDCs, PS (dB) is the transmission power of a SBS, θ3 (dB) is the threshold of the signal receiver, N0 is the noise power spectral density, WmmW is the bandwidth of millimeter wave wireless links, σ is the standard variance of shadow fading over wireless channels in dB. Proof According to the MCR scheme, the destination SBS simultaneously receives the data from adjacent EDCs by multi-path wireless routes. R p is the distance between the destination SBS and the EDC E DC p . Without loss of generality, the PDF of the distance R p is assumed to be governed by [60] −λ E πr 2

f R p (r )  e

p  2 λ E πr 2 , r ( p)

(4.60)

where (·) is Gamma function. When p ∈ Z+ , Z+ is the set of positive integers, ( p)  ( p − 1)!. The average value of R p is derived by

4.3 Multi-path Cooperative Communications Networks …







rp  E Rp 

r f R p (r )dr 0

∞ 

−λ E πr 2

re 0

2(λ E π ) p  ( p)

p  2 λ E πr 2 dr r ( p)



e−λ E πr r 2 p dr 2

0

2(λ E π ) p−1 (2 p − 1)  ( p) · 2 



2(λ E π )

217



e−λ E πr r 2( p−1) dr 2

0

(2 p − 1)(2 p − 3) ( p) · 22

p−2

(2 p − 1)(2 p − 3) · · · 1 ( p) · 2 p−1





e−λ E πr r 2( p−2) dr 2

0

e−λ E πr dr 2

0

(2 p − 1)(2 p − 3) · · · 1  , √ ( p) · 2 p · λ E

(b)

(4.61)

∞ 2 where (b) is the Gaussian operation, i.e., 0 e−λ E πr dr  2√1λ . E In this section, the relay route algorithm with the minimum hop number is adopted for the transmission path between the requested EDC and the destination SBS. Therefore, the relay distance is configured as rmmW . When the average transmission distance is r p , the average hop number of r p is 0 / rp , (4.62) np  rmmW where · is the rounding up operation of a number. Considering the distributions of SBSs and EDCs, the number of SBSs in the coverage of a EDC is calculated by 1 + 1.28 λλES . Moreover, the probability that a 1 SBS selected by the EDC for relaying the AR/VR data is p1  λ S . When 1+1.28 λ

E

millimeter wave links are assumed to be line of sight (LoS) links, link   the wireless fading is expressed by L (dB)  70 + 20 log10 (rmmW ) + ζ , ζ ∼ N 0, σ 2 , where ζ is the shadow fading coefficient and σ is the standard variance of shadow fading over wireless channels in dB. The wireless transmission is successful over millimeter wave links only if PS (dB) − L (dB) − N0 WmmW (dB) ≥ θ3 (dB) is satisfied. Therefore, successful probability of wireless transmission over millimeter wave links is [64]   p2  P ζ ≤ PS (dB) − θ3 (dB) − N0 WmmW (dB) − 70 − 20 log10 (rmmW )

218

4 Energy Efficiency of 5G Multimedia Communications



"" ! ! f (rmmW ) 1 , 1 + er f √ 2 2σ

(4.63)

where f (rmmW ) (dB)  PS (dB) − θ3 (dB) − N0 WmmW (dB) − 70 − 20 log10 (rmmW ), PS (dB) is the transmission power of a SBS, θ3 (dB) is the threshold of received wireless signal. When one packet is transmitted the distance r p by the multi-hop relay method in this section, the transmission delay is expressed by τmmW Dp  n p · p1 p2 0 ! " / rp λS 2 $ # τmmW 1 + 1.28  rmmW λ E 1 + er f f (r√mmW ) 2σ ! " r p τmmW 2 λS $. # ≈ 1 + 1.28 rmmW λ E 1 + er f f (r√mmW ) 2σ

(4.64)

To decrease the effect of delay jittering caused by MCR scheme, a data buffer is configured for SBSs. When the SBS buffer size and- the . packet size are configured as Z and L, the maximum tolerable buffer delay is ZL for a multi-hop relay route between the EDC and the destination SBS when the AR/VR application is run by users. When the transmission control protocol (TCP) is used for AR/VR data, the EDC can transmit the next packet only if the current packet is successfully accepted at the destination SBS. Hence, the total delay of one packet in the multi-hop relay - . route is ZL · D p . Based on the MCR scheme, the EDC E DC p only need to transmit the data of

1 rp

B

1 i1 ri

· S. The system delay between the EDC E DC p and the destination

SBS is expressed by 1 rp

D wp   B

1 i1 ri

-Z.   BL

1 i1 ri

/ 0 Z · Dp L ! " λS 2 τmmW $, # 1 + 1.28 · rmmW λ E 1 + er f f (r√mmW ) ·

(4.65)



√ . Considering B EDCs are utilized for simultaneously transwhere ri  (2i−1)(2i−3)···1 (i)·2i · λ E mission in the MCR scheme, the system delay of MCR scheme is derived by

D bh DL 

max

p1,2,...B

D wp -Z.



max

p1,2,...B

L

B

1 i1 ri

# $ 2τmmW 1 + 1.28 λλES # # $$ · ) rmmW 1 + er f f (r√mmW 2σ

4.3 Multi-path Cooperative Communications Networks …

(c)

-Z.

  BL

1 i1 ri

# $ 2τmmW 1 + 1.28 λλES # # $$ , · ) rmmW 1 + er f f (r√mmW 2σ

219

(4.66)

where (c) is the condition that the system delay D wp does not depend on the p-th route based on the result of (4.66). The MCR delay theorem is proved. Lemma 1 When EDCs are deployed in a 5G small cell network, the AR/VR data stored at B EDCs are simultaneously transmitted to a destination SBS by the MCR scheme. The lower and upper bound of system delay in the MCR scheme is derived by # $ -Z. -Z. λS 1 + 1.28 τ mmW 2.28 L τmmW L λM # # $$ , # # $$ < D bh 3 DL < √ f (r ) f (r ) 2 2 mmW λ M rmmW 1 + er f √mmW π Rmax λ S rmmW 1 + er f √2σ 2σ (4.67) where Rmax is the maximum distance between the EDC and the destination SBS. Proof Based on the configuration of system model, densities of MBCs, SBSs and EDCs satisfy the following constraint: λ M < λ E < λ S . The upper bound of system delay in the MCR scheme is derived by # $ -Z. 2τmmW 1 + 1.28 λλES L # # $$ · D bh DL   B 1 ) rmmW 1 + er f f (r√mmW i1 ri 2σ # $ -Z. 2τmmW 1 + 1.28 λλES (d) # # $$ ≤ 1L 1 · ) rmmW 1 + er f f (r√mmW i1 ri 2σ # $ -Z. λS τmmW 1 + 1.28 λ E L # # $$ √ ) λ E rmmW 1 + er f f (r√mmW 2σ # $ -Z. λS 1 + 1.28 τ mmW (e) L λM # # $$ , 1, (e) is the condition with λ E > λ M . The average distance ri between the destination SBS and the EDC E DCi is derived by ri 

(2i − 1) · (2i − 3) · · · 1 √ 2i λ E · (i)

220

4 Energy Efficiency of 5G Multimedia Communications

(2i − 1) · (2i − 3) · · · 1 √ 2i λ E · (i − 1)! (2i − 1) · (2i − 3) · · · 1  √ 2 λ E · (2i − 2) · (2i − 4) · · · 2 1 2i − 1 2i − 3 3  √ · · ··· 2 2 λ E 2i − 2 2i − 4 1 1 > √ · 1 · 1· · · 1  √ , 2 λE 2 λE



(4.69)

i−1

Based on the result of (4.69), the lower bound of system delay in the MCR scheme is derived by # $ -Z. 2τmmW 1 + 1.28 λλES L # # $$ · D bh DL   B 1 ) rmmW 1 + er f f (r√mmW i1 ri 2σ # $ -Z. 2τmmW 1 + 1.28 λλES # # $$ > B L √ · ) rmmW 1 + er f f (r√mmW i1 2 λ E 2σ # $ -Z. λS τmmW 1 + 1.28 λ E # # $$  √L · B λ E rmmW 1 + er f f (r√mmW ) 2σ # $ -Z. τmmW 1 + 1.28 λλES (f) L # # $$ ≥ √ · 2 λ E π Rmax λ E rmmW 1 + er f f (r√mmW ) 2σ -Z. 2.28 L τmmW (g) > (4.70) # # $$ , 3 ) 2 π Rmax λ S2 rmmW 1 + er f f (r√mmW 2σ 2 , (g) is the condition with λ E < λ S . where ( f ) is the condition with B ≤ λ E π Rmax Therefore, the Lemma 1 is proved.

4.3.5 Service Effective Energy Optimization 4.3.5.1

Service Effective Energy

For the AR/VR applications, wireless transmissions are premised on the basis of QoS. The QoS is defined by QoS  1{D ≤ Dmax },

(4.71)

4.3 Multi-path Cooperative Communications Networks …

221

where 1{. . .} is an indicator function, which equals to 1 when the condition inside the bracket is satisfied and 0 otherwise; Dmax is the maximum delay threshold for AR/VR applications. On the other hand, the massive wireless traffic of AR/VR applications is transmitted in 5G small cell networks. Hence, the energy efficiency is another important metric for MCR scheme. Considering the requirement of QoS in AR/VR applications, the service effective energy (SEE) is defined by E S E E  E sys · QoS,

(4.72)

where E sys is the system energy of MCR scheme. Based on the system model in Fig. 4.11, the system energy of MCR scheme includes the energy consumed at MBSs, SBSs and EDCs. Without loss of generality, the energy consumption of MBSs and SBSs is classified into the embodied energy, i.e., the energy is contained in the manufacturing process of infrastructure equipment from a life-cycle perspective, and the operation energy, i.e., the energy is consumed for wireless traffic transmissions [65]. The energy consumption of EDCs is classified into the embodied energy, the operation energy and the storage energy, i.e., the energy consumed for video storage at EDCs. As a consequence, the system energy of MCR scheme is extended as E sys  λ M E M B S + λ S E S B S + λ E (E E DC + A · E storage )      λ M POMP · TLiMf etime + E EMM + λ S POS P · TLiS f etime + E ES M   + λ E POE P · TLiE f etime + E EE M + λ E A · E storage      λ M (a M PM + b M ) · TLiMf etime + E EMM + λ S (a S PS + b S ) · TLiS f etime + E ES M   + λ E (a E PE + b E ) · TLiE f etime + E EE M + λ E A · E storage , (4.73) where E M B S , E S B S and E E DC are the energy consumption at MBS, SBS and EDC, respectively; E storage is the energy consumption of one video content stored at the EDC; POMP , POS P and POE P are the operation power of MBS, SBS and EDC, respectively; TLiMf etime , TLiS f etime and TLiE f etime are the lifetime of MBS, SBS and EDC, respectively; E EMM , E ES M and E EE M are the embodied energy of MBS, SBS and EDC, respectively; a M and b M are the fixed coefficients of the operation power at MBSs; a S and b S are the fixed coefficient of operation power at SBSs; a E and b E are the fixed coefficients of operation power at EDCs.

4.3.5.2

Algorithm Design

Assumed that AR/VR video contents are stored in local EDCs. To save the energy consumption of MCR scheme, the optimal SEE problem is formulated by

222

4 Energy Efficiency of 5G Multimedia Communications

min E S E E  E sys · QoS

λ E ,A

s.t.

A 

xmk  A, ∀m ∈ M,

k1

(4.74)

ri ≤ Rmax PM > PS > PU where the minimum SEE is solved by the optimal density of EDCs λ E and the optimal number of video content A at EDCs. Considering the popularity of video content, the A xmk  A, ∀m ∈ M. To avoid total number of video content A is expressed by k1 the infinite distance between the EDC and the destination SBS, the maximum distance Rmax between the EDC and the destination SBS is constrained for the optimal SEE. Considering functions of MBSs, SBSs and users, the wireless transmission power of MBSs, SBSs and users are constrained by PM > PS > PU . To solve the optimal SEE problem in (4.74), two-step solution is proposed in this section. In the Step 1, the requirement of system delay is solved for the AR/VR MCR scheme. In the Step 2, the SEE is optimized for the AR/VR MCR scheme. Step 1: Based on (4.74), the requirement of system delay is formulated by max QoS λ E ,A

s.t.

A 

xmk  A, ∀m ∈ M.

k1

(4.75)

ri ≤ Rmax PM > PS > PU r eq

as Based on (4.35), the delays DU L , D deli DL and D DL are independent on the density of EDCs λ E and the number of video content A. Therefore, in this optimal algorithm the maximum delay threshold of AR/VR applications is replaced by the variable r eq as   Dmax − DU L − D deli Dmax DL − D DL . The condition inside of QoS indication function is expressed by f iber  D bh · (1 − Pin−E DC ) ≤ Dmax DL + D

(4.76)

Based on (4.56) to (4.59), the backhaul delay D bh DL decreases with the increase of the EDC density λ E and the fiber link delay D f l  D f iber · (1 − Pin−E DC ) decreases with the increase of the number of video content A. Considering two conditions λ E > 0 and 1 ≤ A ≤ |K|, A ∈, the critical value λ EA is obtained by selecting the value of A from the set {1, 2, . . . , |K|}. When a value of A  1, . . . , |K| is substituted into (4.76), a corresponding value of λ E ≥ λ EA is obtained. Therefore, λ EA

4.3 Multi-path Cooperative Communications Networks …

223

is the critical value for the available value value pair   of A. Moreover, the available of λ E and A is denoted by C(A, λ E )  ( A, λ EA )|A  1, . . . , |K| . Step 2: Based on the result of Step 1, the minimum system energy of MCR scheme is formulated by min E sys λ E ,S

s.t.(A, λ E ) ∈ C(A, λ E ),

(4.77)

where the available value pairs A and λ EA are substituted into (4.73), the optimal SEE is solved by obtaining the minimum system energy of MCR scheme. The detail algorithm is illustrated in SEE Optimization (SEEO) algorithm.

4.3.6 Simulation Results and Performance Analysis Based on the proposed system delay of the MCR scheme, the effect of various system parameters on the system delay of the MCR scheme will be analyzed and compared by numerical simulations in this section. In what follows, the default values of system model are illustrated in Table 4.1. Moreover, the performance of SEEO algorithm is simulated and analyzed in this section. Figure 4.12 shows the fiber link delay with respect to the number of video content considering different skewness parameters of popularity distributions. When the skewness parameter of popularity distribution is fixed, the fiber link delay decreases with the increase of the number of video content at EDCs. When the number of video content is fixed, the fiber link delay decreases with the increase of the skewness parameter of popularity distribution. Figure 4.13 depicts the backhaul delay with respect to the density of EDCs considering different number of cooperative EDCs. When the number of cooperative EDCs is fixed, the backhaul delay decreases with the increase of the density of EDCs. When the density of EDCs is fixed, the backhaul delay decreases with the increase of the number of cooperative EDCs.

224

4 Energy Efficiency of 5G Multimedia Communications

fl

Fiber link delay D (ms)

11 10

β =0.6

9

β =0.8 β =1

8 7 6 5 4 3 2 1 0

100

0

200

300

500

400

Number of video contents A Fig. 4.12 Fiber link delay with respect to the number of video content considering different skewness parameters of popularity distributions 160

B=1 B=3 B=5

bh

Backhaul delay DDL (ms)

140 120 100 80 60 40 20 0

5

10

15

20

25

30

35

40

45

50

λ (km-2) E

Fig. 4.13 Backhaul delay with respect to the density of EDCs considering different number of cooperative EDCs

4.3 Multi-path Cooperative Communications Networks …

225

60

rmmW=100m

bh

Backhaul delay DDL (ms)

50

rmmW=150m rmmW=200m

40

30

20

10

0

5

10

15

20

25

30

35

40

45

50

-2

λ (km ) E

Fig. 4.14 Backhaul delay with respect to the density of EDCs considering different the maximum transmission distances of SBSs and EDCs

Figure 4.14 illustrates the backhaul delay with respect to the density of EDCs considering different the maximum transmission distances of SBSs and EDCs. When the density of EDCs is fixed, the backhaul delay decreases with the increase of the maximum transmission distances of SBSs and EDCs. Figure 4.15 compares the backhaul delay with respect to the density of EDCs considering the MCR scheme and the single path route scheme. Figure 4.15a shows that the backhaul delay of MCR scheme is less than the backhaul delay of single path route scheme. Figure 4.15b describes the gains of backhaul delay between the MCR scheme and the single path route scheme. Result in Fig. 4.15b indicates that gains of backhaul delay decreases with the increase of the density of EDCs. In Fig. 4.16, the impact of buffer size of SBSs on the backhaul delay with different densities of EDCs is investigated. When the density of EDCs is fixed, the backhaul delay increases with the increase of the buffer size of SBSs. Figure 4.17 presents the backhaul delay with respect to the density of SBSs considering different maximum distances between the EDC and the destination SBS. When the maximum distance between the EDC and the destination SBS is fixed, the backhaul delay increases with the density of SBSs. When the density of SBSs is fixed, the backhaul delay decreases with the increase of the maximum distance between the EDC and the destination SBS.

226

4 Energy Efficiency of 5G Multimedia Communications

(a) 160 MCR scheme Single path route scheme

bh

Backhaul delay DDL (ms)

140 120 100 80 60 40 20 0

5

10

15

20

25

30

35

40

45

50

-2

λ (km ) E

(b) 100

bh

Gains of DDL (ms)

80

60

40

20

0

5

10 15 20 25 30 35 40 45 50 -2 λ (km ) E

Fig. 4.15 Backhaul delay with respect to the density of EDCs considering the MCR scheme and the single path route scheme

Figure 4.18 shows the system energy with respect to the density of EDCs considering different numbers of video content stored at EDCs. When the number of video content is fixed, the system energy increases with the density of EDCs. When the density of EDCs is fixed, the system energy increases with the increase of the number of video content stored at EDCs.

4.3 Multi-path Cooperative Communications Networks …

227

20 -2

18

λ =30km

16

λ =40km

14

λ =50km

E

bh

Backhaul delay DDL (ms)

-2

E

-2

E

12 10 8 6 4 2 0

1

2

3

5

4

6

7

8

9

10

Buffer size of SBS Z (MB) Fig. 4.16 Impact of buffer size of SBSs on the backhaul delay with different densities of EDCs 20 -2

18

λ =30km

16

λ =40km

14

λ =50km

E

bh

Backhaul delay DDL (ms)

-2

E

-2

E

12 10 8 6 4 2 0

1

2

3

4

5

6

7

8

9

10

Buffer size of SBS Z (MB) Fig. 4.17 Backhaul delay with respect to the density of SBSs considering different maximum distances between the EDC and the destination SBS

228

4 Energy Efficiency of 5G Multimedia Communications 6

4.2

x 10

4

-2

System energy Esys (J ⋅ m )

4.1

3.9 3.8 3.7 3.6 3.5 3.4

A=100 A=300 A=500

3.3 3.2 3.1

5

10

15

20

25

30

35

40

45

50

-2

λ (km ) E

Fig. 4.18 System energy with respect to the density of EDCs considering different numbers of video content stored at EDCs

Figure 4.19 explains the SEE with respect to the density of EDCs considering different numbers of video content stored at EDCs. Based on the SEEO algorithm, the optimal solution of A and λ EA are solved by A  144 and λ EA  9.873 km−2 . Figure 4.19a is a three-dimension figure describing the relationship among the SEE, the density of EDCs and the number of video content stored at EDCs. Figure 4.19b is a two-dimension figure describing the relationship among the SEE, the density of EDCs and the number of video content stored at EDCs. Based on the results in Fig. 4.19, the minimum SEE is achieved when the number of video content and the density of EDCs are configured as 144 and 9.873 km−2 , respectively. Figure 4.20 compares the SEE with respect to the number of video content considering the MCR scheme and the single path route scheme. Based on the curves in Fig. 4.20, the SEE of MCR scheme is always less than the SEE of single path route scheme in 5G small cell networks. When the number of video content stored at EDCs is configured as 144, the SEE of MCR scheme achieves the minimum, i.e., 3.3226 × 106 J m−2 . When the number of video content stored at EDCs is configured as 264, the SEE of single path route scheme achieves the minimum, i.e., 3.6432 × 106 J m−2 . Compared with the SEE minimum of single path route scheme, the SEE minimum of MCR scheme is reduced 11.5%.

4.3 Multi-path Cooperative Communications Networks …

229

-2

Service Effective Energy ESEE (J ⋅ m )

(a) 6

x 10 5 4 3 2 1

0 500 400 300

40 30 35

200

45 50

25 100 15 20 10 0 λ (km-2) 5 Number of video contents A E

-2

Service Effective Energy ESEE ( J ⋅ m )

(b)

6

4

x 10

3.5 3 6

3.355

2.5

x 10

3.35 3.345

2

3.34 3.335

1.5

3.33 3.325

1

3.32 3.315 9.6

9.8

10

10.2

10.4

10.6

10.8

11

0.5 0 5

10

15

20

25

30

35

40

45

50

-2 λE (km )

Fig. 4.19 SEE with respect to the density of EDCs considering different numbers of video content stored at EDCs

4.3.7 Conclusion AR/VR application is emerging for future networks and the requirement of lower latency is a great challenge for AR/VR data transmitting by wireless networks. In this section a solution with the SDN architecture is proposed for supporting AR/VR applications in 5G small cell networks. To solve the requirements of lower delay and big data size in AR/VR applications, a MCR scheme is proposed for AR/VR wireless transmissions in 5G small cell networks. Moreover, a delay theorem of MCR scheme is proposed. Furthermore, the lower and upper bound of the delay in the MCR scheme

230

4 Energy Efficiency of 5G Multimedia Communications 6

-2

Service Effective Energy ESEE ( J ⋅ m )

3.7

x 10

3.65 3.6 3.55

MCR scheme Single path route scheme

3.5 3.45 3.4 3.35 3.3 64

104 144 184 224 264 304 344 384 424 464

Number of video contents A Fig. 4.20 SEE with respect to the number of video content considering the MCR scheme and the single path route scheme

is derived. Simulation results indicate that the delay and the SEE of MCR scheme is better than the delay and SEE of single path route scheme in future 5G small cell networks.

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Chapter 5

Wireless Resource Management for Green Communications

The time-frequency spatial distribution of multi-network radiant energy e(t, f, s) depends on the topology of the network, the location of the base station, the transmit power of the antenna, and the physical environment of the wireless communication. The service requirements of the user R(t, f, s) depend on the user distribution and density, the type and price of the service, and the capabilities of the communication terminal. And the coverage of the network’s services. The current mismatch between multi-network energy distribution and user service requirements has led to problems such as low efficiency of multi-network energy utilization. This chapter first analyzes the basic model of multi-network multi-user service traffic, and then combines the base station distribution to model the interference distribution in heterogeneous networks. Finally, based on Bregman Inexact Excessive Gap, a green multi-network resource allocation method is proposed.

5.1 Introduction In the world, there are many literatures and products for research and modeling of Internet traffic, but there is relatively scare research on broadband wireless multimedia service flows in mobile communication networks. As for mobile operators considered the distribution and characteristics of wireless traffic on their networks as business secrets and are unwilling to publish and conduct open academic research and discussion. Almost all international standardization organizations can only make subjectively guessed combinations of service type and traffic estimation for wireless service flows in future mobile communication systems based on some typical data service models on the Internet when formulating technical specifications for nextgeneration mobile communication systems.

© Publishing House of Electronics Industry, Beijing and Springer Nature Singapore Pte Ltd. 2019 X. Ge and W. Zhang, 5G Green Mobile Communication Networks, https://doi.org/10.1007/978-981-13-6252-1_5

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For instance, in the IEEE 802.20 “Mobile Broadband Wireless Access” international standard, the artificially defined future broadband wireless service flow model is: file transmission and download (FTP) services take up 30%, Internet access and online game services take up 30%, video Streaming media business takes up 30%, and voice over IP (VoIP) services take up 10%. These subjectively developed broadband wireless service flow model, lacking sufficient support from measured data and theoretical research, is often far from reality, and cannot effectively support the development of key technologies and algorithms for next-generation broadband wireless mobile communication systems, and the successful launch of the corresponding standardization and industrialization work. Recent years, with the widespread commercialization of broadband mobile communication systems such as 3G and LTE on a global scale, the research on 5G key technologies, and the rich and growing popularity of data and entertainment powerful mobile communication terminals, mobile communication services rapidly shifting from “with voice services as the core” to “data communication as the core” business and business model. In order to comprehensively and thoroughly understand the impact of this trend on the distribution and characteristics of actual broadband wireless service flows at both the technical and service levels, it is necessary to carefully, deeply, systematically and theoretical research, modeling, verification and demonstration of measured broadband service flow data in current real networks. At the same time, radiant energy and interference determine the network’s ability to support user needs, so it is especially necessary to study inter-cell interference in cellular networks. In the past, people often assumed that they obey the lognormal distribution (corresponding to the logarithmic domain in dBm, and the interference obeys the Gaussian distribution). Many studies in the past have been based on this hypothesis [1, 2], but they have not been theoretically justified. In this regard, Ref. [3] shows that the rationality of this hypothesis is affected by environmental parameters: in the environment with small shadow variance, the distribution of propagation loss between inter-cell interference links is quite different from the lognormal distribution. In a cellular network, a conventional common system model is generally a Grid model composed of a circular cell or a hexagonal cell. Researchers have conducted research related to statistical modeling based on such models. It should be noted that in a cellular network, the interference source of the downlink is a base station, and the interfered nodes are randomly distributed users. Since the transmission power of the base station is relatively stable, the downlink interference experienced by the user is less affected by the scheduling. The interference source of the uplink is the user, and the same resource block is extremely likely to be assigned to different users in different scheduling periods. Since the difference in transmission power of users at different locations is large, the link propagation loss to the interfered base station is also different, so the uplink interference is affected by the scheduling strategy and exhibits large fluctuations. Therefore, the uplink and downlink statistical modeling of the cellular network needs to be performed separately. Downlink: In Ref. [4], the author uses theoretical analysis combined with numerical calculation to study the statistical distribution of the propagation gain of the

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downlink interference link in the macro cell. With the statistical histogram, the function of the known expression is performed to parameter fitting to describe the statistical distribution function of the propagation loss of the interfering link. However, the shortcoming of this paper is that it replaces the contribution of this part to the propagation gain with the expectation of path loss, which means that the randomness introduced by the user position distribution is overlooked. References [5, 6] deduced the conditional probability of the sum of downlink interference and noise on each subcarrier of OFDMA system under Rayleigh fading channel and Rice fading channel (assuming shadow fading and network load are known). It is found that the interference distribution results obtained by theoretical modeling are significantly different from the Gaussian distribution. In [7], the distribution function of downlink SINR in the home base station network using the hypothesis of cellular structure is studied by analysis and calculation, but it needs to be calculated in combination with specific scenarios. The specific closed expression is not given in the paper. In [8], the time series value of SINR on each subcarrier is obtained by simulation. The possible Probability Density Function (PDF) form is listed according to experience, and then the form that best matches the distribution in the simulation result is selected. This method still requires a lot of simulation work and does not ensure the accuracy of the analysis results. A method for calculating the first-order statistic and second-order statistic of downlink SINR is given in Ref. [9]. The autocorrelation function of SINR is derived and used to investigate the volatility of interference. Uplink: In Ref. [10], with the scheme of uplink power control and cell selection/handover in CDMA systems, by analyzing the first and second moments of interference and the assumption that the uplink interference distribution obeys the lognormal distribution. Moment-matching is performed to derive parameters that determine the distribution of the sum of the disturbances. Reference [11] used a large number of simulation results to obtain statistical histograms of uplink inter-cell interference to study the distribution of interference, and did not theoretically give specific expressions. Reference [12] analyzed the random distribution of uplink SINR in OFDM systems, but the results of inter-cell interference distribution are mainly calculated by numerical calculations, and the contribution of theoretical analysis is very limited. In [13], by discretizing a cell into a plurality of ring-shaped regions, assuming that the terminal transmit powers in these ring-shaped regions are the same, the equation of the moment-generating function derived from the uplink interference of a single cell is obtained. Reference [14] examines the probability distribution model and statistics of uplink inter-cell interference received by target base stations in different frequency bands in an OFDMA network using Soft Frequency Reuse (SFR) scheme. The propagation model assumption is relatively simple. Only the path loss model associated with the transmission distance is considered. In this regard, Ref. [15] further considers the shadow and Rayleigh fading in the propagation model, models the uplink interference probability distribution function and statistics of different frequency bands, and studies the influence of system parameters. In addition to the mesh model, there is another classic Wyner model, which is also commonly used in system modeling of cellular networks. For example, Ref. [16] studied the distribution model of uplink inter-cell interference based on this

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model, and analyzed the maximum uplink rate achievable by each cell. The Wyner model is characterized by simplicity and ease of analysis. However, studies have shown that the model only accurately models the uplink interference when there are a large number of interfering users at the same time (such as CDMA system), and the accuracy is poor in other cases [17]. Considering that statistical modeling based on grid model is difficult to come up with a concise and easy to handle closed expression, in recent years, some researchers represented by the research team of Professor Jeffrey G. Andrews of the University of Texas at Austin, based on the network node PPP model, assume that a series of studies have been conducted on statistical models and applications in cellular networks. This includes studying the distribution and coverage of uplink SINR in a single-layer cellular network [18, 19], and the downlink SINR distribution of different frequency reuse schemes [20], and then the corresponding research ideas are extended to analyze the SINR distribution and coverage of the downlink of the multi-layer network in heterogeneous networks [21–23], the downlink outage probability of each layer of users [24], the downlink transmission rate distribution and the layer selection cell bias [25] required for the maximum rate, and the resource allocation and load balancing strategy between the layers of the network [26]. Through the research and analysis of the same type of reference, Ref. [27] summarizes the application of stochastic geometry based on PPP and other point processes in statistical modeling of cellular and cognitive networks, and points out that random geometry is he only mathematical tool that provides accurate modeling analysis. It is worth noting that from the perspective of the rationality of the system model, the mesh model, the linear Wyner model and the random point process model cannot accurately describe the topology model of the multi-layer heterogeneous network. Therefore, it is necessary to set up a network for heterogeneous networks. Regarding the characteristics of heterogeneous networks, research the appropriate system model, then on the basis of this, making statistical modeling of the inter-cell interference of the uplink and downlink. In addition, in order to meet the increasing user demand of modern communication networks, a large number of heterogeneous network multi-service resource allocation algorithms have emerged. Most of them can be summarized as a minimum separable objective function with additive nonlinear constraints. The existing solution based on the sub-gradient algorithm can only achieve the convergence speed, which is incapable of facing the big data generated by mobile users in modern heterogeneous networks. To further develop a more efficient multi-service resource allocation algorithm, we consider a regularized Lagrangian function and combine it with smooth acceleration techniques. Specifically, we extend the linear equation constraints imposed by predecessors to more challenging nonlinear constraints. In order to solve this problem, we propose a Bregman-based projection algorithm, and through rigorous mathematical proof, we can get a fast convergence speed. In addition, we apply the BIEG algorithm to the multi-service resource allocation problem, which we call BIEG-RA, and introduce error control techniques and projection. The simulation results show that the fast convergence speed of the algorithm is very satisfactory when dealing with big data in heterogeneous networks.

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Therefore, this chapter firstly models the user traffic, and then studies the radiated energy (or interference), and finally gives a green resource allocation algorithm based on Bregman’s inexact over-distance.

5.2 User Traffic Model In this section, preliminary modeling of user service traffic is carried out, in cooperation with operators to initially obtain basic data services (WeChat, Easychat), voice service types and main indicators (as shown in Table 5.1) for analysis and research.

Table 5.1 Business data collection table Business type

The main parameters

Voice

Call demand distribution (call access request/minute), call duration, Irish (Erlang) dynamic distribution, congestion rate, call drop rate

Short message (SMS) and multimedia message (MMS)

Statistical distribution of number (number of pieces)/minute, length (number of bytes), loss rate

File transfer and download (FTP)

Download demand distribution (number of download requests/minute), statistical distribution of file length (bytes), statistical distribution of download length (seconds/file), transmission interruption probability

Internet access, instant messaging (QQ, MSN, Web Email) and online gaming

Internet demand distribution (number of Internet requests/minutes), statistical distribution of Internet/communication/game time (seconds/connections), service outage probability

Audio and video streaming

Streaming media transmission demand distribution (number of transmission requests/minute), statistical distribution of streaming media transmission time (seconds/file), transmission interruption probability

Voice over IP (VoIP, mainly analyzing data in WLAN)

Call demand distribution (call access request/minute), statistical distribution of call data volume and duration, congestion rate, dropped call rate

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5.2.1 Data Service According to the statistics provided by the mobile Internet application and the business using the quantitative analysis platform system, the detailed statistics of the typical data services (WeChat and Easychat) are used. (1) Comparison of users By sampling the users of the Android mobile phone platform, the total sample size is 113,150,735. Among them, WeChat pre-installed user 4,058,748, the number of users was 75,375,778, and the number of active users in the last 30 days was 8,655,456. The number of pre-installed users of Easychat is 1,419,821; the number of users is 3,032,161, and the number of active users in the past 30 days is 592,823 (Fig. 5.1). (2) Daily average number of starts The average number of daily starts of WeChat and Easychat was counted. The average number of daily starts of WeChat was 13.96, ranking first in the whole network in all kinds of applications; those of Easychat was 2.39 (Fig. 5.2). (3) Comparison of daily average duration The daily average usage time of WeChat and Easychat is counted. The average duration of WeChat is 135.76 s, ranking first in the whole network; those of Easychat is 46.89 s (Fig. 5.3).

Fig. 5.1 Comparative analysis of WeChat and Easychat users

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Fig. 5.2 Comparative analysis of WeChat and Easychat users start daily

Fig. 5.3 Comparative analysis of daily average usage time of WeChat and Easychat users

(4) Comparison of daily average traffic The daily average traffic of WeChat and Easychat is counted. The daily traffic per captial used by WeChat is 3.14 MB, and those of Easychat is 1.68 MB (Fig. 5.4). (5) Daily average traffic comparison The statistics on the WeChat user terminal in Shanghai are shown in Fig. 5.5. The statistics of Shanghai Telecom can only be obtained from some mobile phones preinstalled with statistical software. It cannot cover all users and all cellphone models. There are certain limitations to the research, but if the number of users is large enough, it is conceivable that the data is also sufficiently representative.

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Fig. 5.4 Comparative analysis of average daily traffic of WeChat and Easychat users

Fig. 5.5 Comparison of the number of users of WeChat and Easychat users

5.2.2 Voice Service Statistics and analysis of the voice indicators of all the days of the base station on July 25, 2012 are as follows. (1) Analysis of the situation of subway stations The collection data includes three subway stations adjacent to Shanghai Metro Line 2, namely WeiNing Road (WNRd) Station, BeiXinJing (BXJ) Station and SongHong Road (SHRd) Station. The drawing statistics of the business data collected by the three subway stations are as follows (Figs. 5.6, 5.7).

5.2 User Traffic Model

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Fig. 5.6 Traffic carried services (without switching)

Fig. 5.7 Traffic carried services (with switching)

From the traffic volume, there is a strong consistency in the traffic trend of the three subway stations throughout the day. The three have two distinct peaks throughout the day. At 6 o’clock in the morning, the traffic volume began to increase. The first peak appeared near 9:00 am. After that, the traffic volume dropped slightly, and a small trough appeared at noon. After noon, the traffic volume gradually increased, and the second peak appeared. At around 17:00 pm; traffic began to decline gradually, but remained at a high level until 19:00. This phenomenon is closely related to people’s daily travel habits. At about 6 o’clock in the morning, the subway started to run, and a few people started to travel. At around 9 am, it was the morning working hours in the morning, and the morning peaks for the working people. After the early peaks, the passenger traffic began to decline, and the passenger traffic during the lunch break showed at a low level. In the afternoon, the passenger traffic began to rise gradually.

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Fig. 5.8 Total number of call attempts

At around 17:00 pm, it was the off-duty time, and the evening peak for the next-time crowd. The statistics of traffic statistics and the regular work schedule of people are consistent. It can also be seen from the figure that the traffic volume of the Songhong Road subway station is basically higher than that of Weining Road Station and Beixinjing Station. This is because the number of office buildings around the Songhong Road subway station is higher than that of Beixinjing. Weining Road, at the same time, is the end of part route of Metro Line 2, and some passengers will wait for the subway to Hongqiao Railway Station and Hongqiao Airport, so this phenomenon will occur (Fig. 5.8). The total number of call attempts also showed two peaks in the morning and afternoon. The morning peak is around 9 am and the evening peak is around 17:00. The number of call attempts at Songhong Road Station was significantly higher than that of the other two subway stations (Fig. 5.9). The number of text messages and the number of call attempts are quite similar. (2) Analysis of business district situation The traffic data of the non-switching service carried by several base stations such as the telecommunications world, the international hotel and the New HuangPu (NHP) in the Shanghai People’s Square shopping district are selected, and the drawing statistics are as follows. As can be seen from Fig. 5.10, the three-sector traffic of the base stations located in the telecommunications world also exhibits strong consistency throughout the day. There was a peak at 11 am and a full-day peak at 14 pm. Although there are peaks in the morning and afternoon, it is very different from the subway station. The peak traffic volume of the subway station is closely related to the work schedule of people’s nine-to-five work. Therefore, the morning peak of traffic volume appears

5.2 User Traffic Model

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Fig. 5.9 Number of successful SMS messages sent

Fig. 5.10 Traffic carried by telecommunications world (TW) services (without switching)

at around 9:00 am, which coincides with the morning peak of the work trip; the traffic peak appears in the evening at around 17:00 pm, it coincides with the peak of the evening trip. The first peak of the traffic volume of the three sectors of the telecom world appears at around 11:00 am, the second peak appears at 14 o’clock in the afternoon, and the three sectors remain high until 18 o’clock. Traffic volume. Obviously, this is also related to people’s travel habits. At 10 o’clock in the morning, the number of shoppers in the mall began to increase, and many people at noon will have a lunch break or avoid the hot travel at noon. In general, the passenger flow in the afternoon mall will increase significantly and will continue until the evening.

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The rule of change in the whole traffic volume is consistent with people’s traveling rule. It can be seen from Fig. 5.11 that the traffic volume of the three sectors of the International Hotel is relatively balanced. The peak of the morning appeared around 10 o’clock, and the peak of the afternoon continued from 15 to 20 o’clock and reached an extreme value at 18:00. There is a difference in the distribution of international hotel traffic throughout the day and the distribution of traffic in the subway station

Fig. 5.11 Traffic carried by international hotel (IH) services (without switching) (Erl)

Fig. 5.12 Traffic carried by the New HuangPu (NHP) service (without switching) (Erl)

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and the distribution of traffic in the telecommunications world. This is also closely related to people’s eating habits. According to common sense, the beginning of the afternoon until 20 o’clock in the evening is a concentrated time for people to go out for dinner, and the start time of going out for dinner generally starts from 17:00 to 18:00, so at 18 o’clock, it will reach a peak of traffic. At 13:00 noon, most people started lunch and a break, so there will be a temporary low traffic. It can be seen from Fig. 5.12 that the traffic of the three sectors of the New HuangPu is extremely unbalanced, the traffic volume of one sector is normal, and the traffic of the other two sectors is very small, close to zero. The reason may be that there is a problem with the sector opening angle or direction, which requires optimization of the three sectors of the base station.

5.3 Downlink Average Rate and SINR Distribution in Cellular Networks By modeling the radiant energy, the distribution of the interference can be derived, and finally the distribution of the SINR is used to find the distribution of the capacity. The downlink interference modeling scenario considered in this subsection is as follows: There are six co-frequency cellular base stations transmitting signals around the considered base station and users. For a certain user, the base will mainly receive downlink signals transmitted by seven base stations, thereby A certain amount of interference occurs (as shown in Fig. 5.13). It is assumed that the coverage radius of each base station is D, with the target base station location considered as the origin, and the target user location of the service is (r, θ ). The locations of the six other base stations around are (ri , θi ), where i  1, . . . , 6. The base station antenna gain is G b , and the transmit power is Pt . The distance between the base station BSi and the target user UE can be indicated  by di  r 2 + ri2 − 2rri cos(θ − θi ), the downlink inter-cell interference caused by the BSi can be expressed as Yi  Pt G b Adi−n exp(ξ X i )  10(μi +X i )/10 ,

(5.1)

  where ξ  ln 10/10, μi  10 log Pt G b Adi−n , Yi ∼ L N (ξ μi , ξ σi ) can be seen as a lognormal distribution, and thus the Yi probability distribution function (PDF) is:   (ln yi − ξ μi )2 . f Yi (yi )  √ exp − 2(ξ σi )2 2π (ξ σi )yi 1

(5.2)

Considering the Log-normal Shadow effect and referring to the description of the √ √ correlation of the interference link in [1], X i  σi 1 − ζ Wi + σi ζ X 0 . Among them, the correlation coefficient ζ , the independent Gaussian variable Wi , W0 with

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Fig. 5.13 Downlink interference modeling scenario

a mean of 0 and a variance of 1. After considering the Log-normal Shadow effect, the correlation between link i and link j is expressed as follows:   Cov X i , X j ρi j  σi σ j  √ √ √ √ σi σ j Cov 1 − ζ Wi + ζ X 0 , 1 − ζ W j + ζ X 0  σi σ j    (1 − ζ )Cov Wi , W j + ζ  ζ i  j  (5.3) 1i j the shadow fading correlation matrix can be expressed as C    Therefore, ρi j σi σ j L×L . The sum of downlink inter-cell interference is Y 

L  i1

Yi 

L  i1

10(μi +X i ) /10 .

(5.4)

5.3 Downlink Average Rate and SINR Distribution in Cellular Networks

 Logarithmically, Z  ln Y  ln

L

249

 10

(μi +X i )/10

.

i1

According to the method in Ref. [28], the normal inverse Gaussian process can be used to study the sum of downlink interference. The probability distribution function of the normal inverse Gaussian process (NIG) is as follows:



K 1 α δ 2 + (x − μ)2 αδ exp δ α 2 − β 2 − βμ exp(βx) . (5.5) f (x)  π δ 2 + (x − μ)2 The mean, variance, slope and peak parameters of the NIG process are as follows α  3ρ 1/2 (ρ − 1)−1 V −1/2 |S|−1 , β  3(ρ − 1)−1 V −1/2 S −1 , μ  M − 3ρ −1 V 1/2 S −1 , δ  3ρ −1 (ρ − 1)1/2 V 1/2 |S|−1 .

(5.6)

With reference to [29], the approximate first four cumulants of Z can be expressed as: ˆ (1) cˆ(1) Z m Z ,

2

ˆ (2) ˆ (1) , cˆ(2) Z m Z − m Z

3

cˆ(3) ˆ (3) ˆ (1) ˆ (2) ˆ (1) , Z m Z − 3m Z m Z +2 m Z 2

2 4



cˆ(4) ˆ (1) ˆ (4) ˆ (3) ˆ (1) ˆ (2)  1 + 12mˆ (2) − 6 mˆ (1) . Z m Z − 4m Z m Z −3 m Z Z m Z Z

(5.7)

The approximate mean, variance, skewness and kurtosis of Z can be expressed as: M  cˆ(1) Z ,

V  cˆ(2) Z ,

cˆ(3) S  Z 3/2 , cˆ(2) Z

cˆ(4) K  Z 2 . cˆ(2) Z

(5.8)

After using the NIG approximation, we can know the probability distribution expression of the sum of downlink inter-cell interference as

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5 Wireless Resource Management for Green Communications

    2 2 + ln y − μ ˆ    α ˆ δ ˆ K

1 αˆ δˆ  exp δˆ αˆ 2 − βˆ 2 − βˆ μˆ exp βˆ ln y f Y ( y|r, θ )  .  2 πy δˆ2 + ln y − μˆ (5.9) Through simulation, it is found that the NIG method is used to approximate the sum of downlink inter-cell interference, which is more accurate than the traditional Log-normal approximation (as shown in Fig. 5.14). Different from previous studies [30–32], Log-normal and superimposed Lognormal have strong adaptability to the methods in this chapter. By obtaining the downlink interference model, considering that the base station transmission power is P and the target user to the target base station distance r , the SINR at various places can be actually calculated, so that the capacity distribution of the network everywhere can be inferred.

Fig. 5.14 Comparison of NIG approximation and Log-normal approximation of downlink intercell interference

5.4 Bregman-Based Inexact Excessive Gap Method …

251

5.4 Bregman-Based Inexact Excessive Gap Method for Multiservice Resource Allocation Recent years, different wireless access technologies (RATs) have been advanced to meet the needs of different users’ applications and services. In future wireless heterogeneous networks, advanced mobile terminals (MTs) can use multiple wireless access interfaces to connect base stations of different networks at the same time. Because the users’ applications and requirements are different to different networks, it is very important to design effective multi service algorithms to make resource allocation between different networks [33, 34]. Specifically, the service bandwidth of each mobile terminal is provided by different RAT networks, and can be described by additive inequalities. Therefore, decomposable accelerated architecture is the key to solve the resource allocation problem in heterogeneous networks. The former fast architecture mainly considers the separable linear equality constraints. However, the multi-service resource optimization problem of heterogeneous networks is a more general complex nonlinear inequality constraint. Specifically, to solve such problems, we need to face the following challenges: (1) how to analyze the dual error (Duality gap) and the feasible domain (Feasibility gap) under the constraint of nonlinear inequalities; (2) how to give the gap constant in the dual function of the regular Lagrangian square (Lipschitz con) under more general conditions (Lipschitz constant); (3) how to reduce the complexity of iteration and guarantee the high speed of convergence. In this section, we will overcome the challenges posed above and propose a new multi service resource allocation algorithm with high speed convergence. The main contributions summed up as follows • With the regularized Lagrangian function, we derive the duality gap and the feasibility gap with the optimal dual set, which is bounded under the Slater’s regularity condition. • We study the Lipschitz constant of the dual regularized Lagrangian function, and give a complete proof of the Lipschitz continuity under a more realistic assumption that both objective and constraint functions are once differentiable. While previous work assumes objective and constraint functions are twice differentiable [35] in Theorem 2.1. • We adopt the Bregman projection technique [15] to decompose the dual update procedure into two simple steps, and therefore greatly reduces the computational complexity in algorithm implementation. In addition, for the first time, our proposed algorithm combines accuracy control mechanism [36] with the Bregman projection iteration scheme and, under rigorous proofs, it can asymptotically achieve the optimal primal-dual pair with a fast convergence rate of O(1/k). • We apply the analytical results and the proposed algorithm to solve the multiservice resource allocation problem across heterogeneous wireless networks, and verify its faster convergence rate by comparing to classic (Table 5.2).

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5 Wireless Resource Management for Green Communications

Table 5.2 Comparison of algorithm BIEG, IDFG and Tran-Dinh’s Algorithms

BIEG

IDFG [37]

Tran-Dinh’s algorithm [38]

Objective function

Convex

Strongly convex

Convex

Objective function’s differentiability

Once

Twice

Nonsmooth

Constrains

Nonlinear inequality

Nonlinear inequality

Linear equality

Constrains’ differentiability

Once

Twice

Smooth

Technique

Excessive-gap scheme

Nesterov’s first order accelerated scheme

Excessive-gap scheme

Smoothing parameter

Two

None

Two

5.4.1 Problem Description Based on previous work in related areas [39, 40], the multiservice resource allocation problem studied can be modeled as a general convex optimization problem with nonlinear coupling inequality constraints, i.e. min s.t.

N

f i (xi )

i1 N

gi j (xi ) ≤ 0

(5.10)

i1

xi ∈ X i , i  1 . . . , N ; j  1, . . . , m, n, where f i (·) : R ni → R is the ith continuously differentiable convex objective function, and X i ⊂ R ni is the associating nonempty closed compact subset. Let X : X 1 × · · · × X i ⊂ R n be a convex set of x which is generated by the Cartesian product T  of X i , where n  n 1 +· · ·+n N . The notation x  x1T , . . . , x NT represents a column of vector R n . Assume that f i (·) and X i are only known by i, and gi j (·) : R ni → R are continuously differentiable convex constraints. For notational simplicity, we denote gi (xi )  [gi1 (xi ), gi2 (xi ), .. . , gim (xi )].   N Denote Y :  x ∈ R n  i1 gi j (xi ) ≤ 0, j  1, . . . , m , and assume that the set of feasible points are nonempty, X ∩ Y  0. Because X is compact and Y is closed, it can be derived that X ∩ Y is compact. In this way the set of primal optimal points X ∗ is non-empty. Furthermore, the corresponding dual function is defined as φ(u)     N N R+m is the associating min x∈X i1 f i (x i ) + u, i1 gi (x i ) , where u ∈ Lagrangian multiplier. Table 5.3 summarizes the key notations defined. In addition, the following assumptions are hold.

5.4 Bregman-Based Inexact Excessive Gap Method … Table 5.3 Symbol definition

253

Symbol

Definition

φ(u)

Dual function

∇ F(·)

Gradient of F(·)

[·]+

Projection operator onto the non-negative orthant

·

2-norm in the Euclidean space

·

Inner product operator   Upper bound of ∇gi j (·)

Mi j (g)

x

Slater’s vector

di (·)

Proximity function

Di

Upper bound of di (·)

σi

Strong convex parameter of di (·)

εi

Accuracy with subproblem

Vu (·)

Bregman Projection

PU (·)

Projection onto set U

U∗

Optimal compact set of the dual variable u

   Assumption 1 The gradient of ∇g   i j (xi ) is bounded on X i , i.e., there exists a   scalar Mi j (g) satisfying ∇gi j (xi ) ≤ Mi j (g) in xi ∈ X i for all i  1, . . . , N ; j  1, . . . , m. Assumption 2 Slater’s conditional is satisfied, i.e., there exists a vector

x ∈ X such that

N 

gi j x i < 0 for all i  1, . . . , N and j  1, . . . , m.

(5.11)

i1

NAs shown in (5.10), the objective function is denoted as a summation function i1 f i (x i ), and the constraints are given by a series of inequalities composed of some additive functions gi j (xi ). Both can be formulated as some concrete functional forms according to different specific goals. For examples, the optimization problem can be formulated as (1) to minimize a predefined utility of spectrum allocation under the total bandwidth constraints and the Variable Bit Rate (VBR) service demand range [39], (2) to minimize the spectral footprint (i.e., bandwidth-power product) when subject to channel power and Quality of Service (QoS) guarantees [41], (3) to maximize the network downlink capacity while meeting the rate requirement of all secondary users under power limitation and interference constraints [42], (4) or to find the minimum total power consumption that satisfy a given Signal to Interference plus Noise Ratio (SINR) threshold in multiple antenna jointly processing networks

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5 Wireless Resource Management for Green Communications

[43]. Therefore, our research focus in this section is to develop and analyze a fast algorithm for the general convex optimization problem. Our analytical approaches and results can be directly applied to tackle different challenging design objectives and implementation issues related to multi-service resource allocation across heterogeneous wireless networks. It is worth noting that the optimization problem may be non-convex in some specific scenarios. In such a case, these non-convex problems can be relaxed to convex programming by using convex approximation techniques [44].

5.4.2 Regularized Lagrangian Dual Function and δ-Excessive Gap Smoothing Technique Lagrangian dual decomposition is a powerful tool to solve coupling constraints. However, the dual form is usually nonsmooth and smoothing accelerated techniques cannot be directly applied [45]. To solve this problem, regularized Lagrangian dual function has been widely studied to obtain a smooth approach for the dual form. As a result, there exists some discrepancies (or gaps) between the original Lagrangian function and the regularized Lagrangian function. In this section, we will develop an effective approach to reduce such discrepancies/gaps and, further, analyze the duality gap and the feasibility gap of the problem defined in (5.10).

5.4.2.1

Regularized Lagrangian Function

According to [23], the regularized Lagrangian function of (5.10) is given by L(x, u; μ1 , μ2 ) 

N  i1

 f i (xi ) + u,

N  i1

 gi (xi ) + μ1

N 

di (xi ) − 12μ2 u 2 ,

i1

(5.12) where μ1 > 0 and μ2 > 0 are smoothing parameters. di (xi ) is called a proximity function [46] in closed convex set X i ⊂ R ni . It is strong convex with the convex parameter σi > 0. xic is defined as the proxcenter, i.e., xic  xi ∈ X i . The proximity function di (xi ) makes the primal function strong convex, which will cause the corresponding dual problem differentiable. According to the definition of proximity function, the following inequalities are satisfied 

0 < di∗  min di (xi ) ≤ Di  max di (xi ) < +∞, xi ∈X i  xi ∈X√i  xi − x c  ≤ 2Di σi . i

(5.13)

5.4 Bregman-Based Inexact Excessive Gap Method …

255

  The regularized optimal primal-dual pair is defined as z μ∗ 1 ,μ2  xμ∗ 1 ,μ2 , u ∗μ1 ,μ2 ∈ X × R+m . Hence, the solution z μ∗ 1 ,μ2 converges to the original optimal primal-dual pair z ∗ , as μ1 , μ2 and approach zero. Now we consider a super set U regarding the dual optimal set, i.e., every u ∗ is in U . Under the Slater’s regularity condition, the set U is bounded and can be derived as

 ⎧ ⎫  N   ⎨ − ϕ(0m ) ⎬  i1 f i x i + μ1 di x i



U  u ∈ R+m  u ≤ , (5.14) N ⎩ ⎭  min1≤ j≤m − i1 gi j x i

where x is some Slater’s vector and μ1 > 0. The proof is a direct extension of the Eq. 3.6 in [47], where a primal-dual algorithm based on regularized Lagrangian function is proposed without accelerated schemes. Let Mu denote the upper bound of u , which is given in the inequality in (5.13). Based on (5.11), the primal and dual functions need to be analyzed respectively. Primal Functions: f (x; μ2 ) 

N 

 f i (xi ) + max

u,

u∈U

i1

N  i1



 1 2 gi (xi ) − μ2 u , 2

(5.15)

where the optimal solution of the second term in (5.15) is defined as  ∗

u (x; μ2 )  arg max u∈U

u,

N  i1



 1 2 gi (xi ) − μ2 u , 2

(5.16)

 !  N and u ∗ (x; μ2 ) can be computed explicitly as u ∗ (x; μ2 )  PU i1 gi (x i ) μ2 , where PU (·) denotes the projection operator onto the convex set U . Dual functions: ⎧ ⎨ ⎩

ϕ(u; μ1 ) 

N

ϕi (u; μ1 )

i1

(5.17)

ϕi (u; μ1 )  min xi ∈X i { f i (xi ) + u, gi (xi ) + μ1 di (xi )},

where the optimal solution of xi under fixed μ1 is denoted as xi∗ (u; μ1 )  arg min { f i (xi ) + u, gi (xi ) + μ1 di (xi )}. xi ∈X i

(5.18)

The Lipschitz continuity of the dual function’s gradient plays an important role in designing the fast iteration scheme. To elucidate that (5.17) has a Lipschitz continuous gradient with a Lipschitz constant, the following Lemma is presented.

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5 Wireless Resource Management for Green Communications

Lemma 1 Let any μ1 > 0, then the well-defined ϕi (·; μ1 ) is concave and continuously differentiable on R+m . Moreover, the gradient of ϕi (·; μ1 ) which is known as   ∇u ϕi (u; μ1 )  gi xi∗ (u; μ1 ) ,

(5.19)

is Lipschitz continuous with a Lipschitz constant " # m  # m M (g) ij $ Mi2j (g). L id (μ1 )  μ σ 1 i j1 j1

(5.20)

Consequently, the well defined ϕ(u; μ1) is concave  and differentiable. FurtherN gi xi∗ (u; μ1 ) is Lipschitz continuous with a more, its gradient ∇u ϕ(u; μ1 )  i1 N Lipschitz constant L d (μ1 )  i1 L id (μ1 ). The proof is given in Appendix 1. Remark 1 Different from the Theorem 2.1 in [14] where both objective (strongly convex) and constraint functions are assumed twice differentiable, Lemma 1 is based on a more realistic assumption that both objective and constraint functions are only once differentiable and convex.

5.4.2.2

Inexact Excessive Gap Technique

The inexact excessive gap condition can be utilized to find the optimal primal-dual pair. According to the definition [46], a point (x, ¯ u) ¯ ∈ X × U satisfies the inexact excessive gap (δ-excessive gap) condition w.r.t μ1 > 0, μ2 > 0 and a given accuracy δ > 0 if ϕ(u; ¯ μ1 ) + δ ≥ f (x; ¯ μ2 ).

(5.21)

The duality gap is given by (x, ¯ u) ¯  f (x) ¯ − ϕ(u), ¯ and zero duality gap can be achieved when the optimal Lagrangian primal-dual pair (x ∗ , u ∗ ) is found. The following Lemma gives an upper bound of the duality gap and the feasibility gap of the problem (5.10). Lemma 2 Suppose (x, ¯ u) ¯ ∈ X × U satisfies the δ-excessive gap in (5.13). Then, for any u ∗ ∈ U ∗ , the following inequalities are satisfied % &+   N  N    ∗       ¯ − ϕ(u) ¯ ≤ μ1 gi (x¯ i )  ≤ f (x) Di + δ, (5.22) − u   i1  i1 and

5.4 Bregman-Based Inexact Excessive Gap Method …

   

257

 ⎧ N    1 ⎪ 2 ⎪ μ M + μ D + δ / Mu − u ∗  , ⎪ 2 u 1 i ⎪ i1 2 ⎪ ⎪ ⎪ % ⎪ &+  ⎪    ⎪ ⎪  N  ⎪ ⎪  gi (x¯i )  ⎪ ⎪   ≥ μ2 Mu ⎪ ⎪  i1  ⎪  !+  ⎨ N " ⎡ ⎤ # gi (x¯i )  N ≤⎪ #  i1   μ 2δ ⎪ ⎪ μ2 ⎣u ∗  + $ u ∗ 2 + 2 1 ⎦, Di + ⎪ ⎪ ⎪ μ2 i1 μ2 ⎪ ⎪ ⎪ ⎪ % ⎪ &+  ⎪    N ⎪ ⎪   ⎪ ⎪  ⎪ gi (x¯i )  ⎪  ≤ μ2 Mu ,  ⎩   i1

(5.23)

where [·]+  max{0m , ·} is the non-negative projection operator, and U ∗ is the dual optimal compact set. The proof is given in Appendix 2. Remark 2 Different from the Lemma 3.4 in [46] which considers only linear equality constraints in its analysis, the optimal solution xi∗ of (5.10) can be strictly feasible,   N i.e., i1 gi xi∗  0. Therefore, it is reasonable to derive a result that the feasibility    !+   N    N violation  i1 gi ( x¯ i )  rather than  i1 gi ( x¯ i ) converges to zero. Furthermore, we analyze the feasibility gap under the Slater’s condition, so Lemma 2 is more general. An accuracy control mechanism has been proposed in [46] to solve the primal subproblem (5.17) inexactly under linear equality constraints. We in this section extend the mechanism to solve the general problem (5.10) constrained by nonlinear inequalities. Let u ∈ R m . Then, the function ϕ(u; ·) defined by (5.9) is nondecreasing, concave and differentiable in R++ . Moreover, the following inequality holds ϕ(u; μ1 ) ≤ ϕ(u; μ˜ 1 ) + (μ1 − μ˜ 1 )

N    di xi∗ (u; μ˜ 1 ) ,

(5.24)

i1

where xi∗ (u; μ˜ 1 ) is defined by (5.17). Since it is only conceptual to solve (5.17) accurately via classical optimization method (e.g., Newton method will stop when it converges to a predefined small error region [48]), an approximation solution is reasonable. Thus, the point x˜ i∗ (u; μ1 ) is defined as an approximation solution of xi∗ (u; μ1 ) if x˜ i∗ (u; μ1 ) ∈ X i , i.e., x˜ i∗ (u; μ1 ) ≈ arg min { f i (xi ) + u, gi (xi ) + μ1 di (xi )}. xi ∈X i

(5.25)

And the following condition is satisfied   ∗    h i x˜ (u; μ1 ); u, μ1 − h i x ∗ (u; μ1 ); u, μ1  ≤ μ1 σi ε2 /2, i

i

i

(5.26)

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5 Wireless Resource Management for Green Communications

where h i (xi ; u, μ1 )  f i (xi ) + u, gi (xi ) + μ1 di (xi ), i  1, . . . , N , and εi ≥ 0 is the given accuracy for solving the subproblem. Due to the strong convexity of h i (·; u, μ1 ) w.r.t xi , we have       μ1 σi  x˜ ∗ (u; μ1 ) − x ∗ (u; μ1 )2 ≤ h i x˜ ∗ (u; μ1 ); u, μ1 − h i x ∗ (u; μ1 ); u, μ1  i i i i 2 μ1 σi 2 ≤ (5.27) ε . 2 i   Therefore, x˜ i∗ (u; μ1 ) − xi∗ (u; μ1 ) ≤ εi can be obtained for i  1, . . . , N . ˜ Furthermore,  ∗ an approximation gradient of (5.18) can be estimated as ∇ u ϕi (u; u 1 )  gi x˜ i (u; μ1 ) . For further analysis, some quantities are introduced  ε[σ ] 

N

i1

σi εi2

 21

 21  N Dσ  2 Di σi " i1, -2 # # N m $ M Mi j (g) i1 j1  N   c   x C d  M 2 Dσ + M  g i i   i1 . N m m Ld  Mi j (g)σi Mi2j (g). i1 j1

(5.28)

j1

With them, we propose and analyze a parallel fast algorithm called “Bregmanbased Inexact Excessive Gap” (BIEG) algorithm in the next section.

5.4.3 Inexact Algorithm with Bregman Projection 5.4.3.1

Bregman Projection

Let d(·) denote a nonnegative and differentiable strong convex function and let u c denote the prox-center of set U , i.e., ∇d(u c ), u − u c  ≥ 0 for any u ∈ U . So, Bregman distance function is defined as [48, 49] B(z, u)  d(u) − d(z) − ∇d(z), u − z,

(5.29)

where u and z are all from U . Furthermore, the strong convex parameter σ of d(·) indicates

5.4 Bregman-Based Inexact Excessive Gap Method …

B(z, u) ≥

259

1 σ u − z 2 . 2

(5.30)

The Bregman projection of some g ∈ R m onto the set U is defined as Vu (z, g)  arg min{g, u − z + B(z, u)}.

(5.31)

u∈U

Specifically, the proximity function in this section is given by d(u)  21 u 2 with strongly convex parameter σu  1, and the explicit expression of Bregman projection can be simplified as Vu (z, g)  PU (z − g) with the prox-center u c  0m .

5.4.3.2

Starting Point and Iterative Structure

For the main iteration scheme in our proposed fast distributed algorithm, the initial starting point should satisfy the δ0 -excessive gap condition. Lemma 3 For arbitrary μ1 > 0 and ⎧ 0 ∗ m ⎨ x¯ i  x˜ i (0  ; μ1 ), N   0 0 m ⎩ u¯  Vu 0 ; −γ gi x¯ i ,

(5.32)

i1

  where γ  1/L d (μ1 ). Then, the point x¯ 0 , u¯ 0 ∈ X × R m generated by (5.32) will satisfy the δ0 -excessive gap condition w.r.t μ1 and μ2 if μ2 ≥ L d (μ1 ),

(5.33)

/ 0 2 where δ0  μ1 Cd L d ε[1] + 12ε[σ ] . The proof is given in Appendix 3. Remark 3 Since 0m ∈ U is the minimizer of d(u)  21 u 2 , the prox-center uc  0m is adopted to find the initial point in Lemma 3 according to (5.26) and (5.32). m Starting from the initial point x¯0 , u¯ 0 ∈  X × R , Lemma 4 presents our main + + iteration scheme that a new point x¯ i , u¯ satisfies the δ+ -excessive gap condition. Lemma 4 Suppose that (x, ¯ u) ¯ satisfies the δ-excessive gap condition, and a new primal-dual pair for each i is generated as

Ad (x¯ i , u; ¯ μ1 , μ2 , τ ) :

⎧ uˆ  (1 − τ )u¯ + τ u ∗ (x; ¯ μ2 )  ⎪ ⎪ ⎪ ∗ ⎪ ˆ μ1 ⎨ x¯ i+  (1− τ ) x¯ i +τ x˜ i u; ∗

N

⎪ u˜  Vu u (x; ¯ μ2 ), −τ (1 − τ )μ2 ⎪ ⎪ ⎪ i1 ⎩ + u¯  (1 − τ )u¯ + τ u˜

1 / ∗ 0 . ˆ μ1 gi x˜ i u; (5.34)

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5 Wireless Resource Management for Green Communications

For any τ ∈ (0, 1) satisfying L d (μ1 ) (1 − τ ) ≥ , τ2 μ2

(5.35)

the update rules for μ1 and μ2 are given by  μ+1  (1 − ατ )μ1 , μ+2  (1 − τ )μ2

(5.36)

where α is α

N N  /  0  ˆ μ1 / di x˜ i∗ u; Di . i1

(5.37)

i1

 + + Then, the new primal-dual  +pair+  x¯ i , u¯ satisfies the δ+ -excessive gap condition (5.22) with the parameters μ1 , μ2 . Moreover, δ+ is defined as δ+  (1 − τ )δ + η(τ, μ1 , μ2 , u, ¯ ε),

(5.38)

where   Cd μ1 μ1 τ 2 ε . ¯ ε)  Cd + τ (1 − τ ) + M u ¯ ε[1] + η(τ, μ1 , μ2 , u, d L μ2 2 [σ ] 

(5.39)

The proof is given in Appendix 4. Remark 4 Based on Lemmas 3 and 4, we for the first time combine accuracy control mechanism with the Bregman method. In our proposed fast algorithm, the parameters μ1 and μ2 are updated simultaneously, rather than alternatively in previous solutions [47]. The inequality conditions (5.34) and (5.36) can be guaranteed, if a proper update rule for sequence τk ∈ (0, 1) is set up, i.e., 1 (1 − τk ) ≥ k k Ld . 2 τk μ1 μ2

(5.40)

The explicit recursion satisfying (5.40) has been derived as τk+1 

τk



!

τk2 (1−αk τk )2 +4(1−αk τk )−τk (1−αk τk ) 2

.

(5.41)

2 3 2 3 According to the Lemma 4.2 in [47], the sequences μk1 k≥0 and μk2 k≥0 generated by (5.37) satisfy

5.4 Bregman-Based Inexact Excessive Gap Method …

⎧ ⎨

μ01 γ ≤ μk+1 ≤ (τ k+1) ∗ α∗ 1 (τ0 k+1)2/(1+α ) 0 ⎩ 0 ≤ μk+1 ≤ μ02 (1−τ0 ) 2 τ0 k+1

261

,

(5.42)

N di∗ / i1 Di , and γ is a fixed positive constant. The initial value √ 5 − 1 /2. can be derived as τ0  To ensure that the parameter δ+ defined in (5.39) is nonincreasing, an effective accuracy control mechanism is essential. When the accuracy level εik of ¯ μ1 , μ2 , τ ) is chosen such that 0 ≤ εik ≤ ε k  τk δk Q k for i  1, . . . , N , Ad (x¯ i , u; where where α ∗ 

Qk 

N

i1



  N  k μk1 τk  Cd μk1   u ¯ + N d Cd + τk (1 − τk ) + M σi , L 2 i1 μk2 

(5.43)

then the sequence {δk }k≥0 generated by (5.32) is nonincreasing.  from δk+1  δk + (ηk − τk δk ), where the function ηk   The proof derives η τk , μk1 , μk2 , u¯ k , ε¯ k is defined in (5.30). If ηk ≤ τk δk for all k ≥ 0, δk is nonincreasing with an initial value δ0 . If the initial accuracy level is chosen as ε¯ initial  δ0 /C0 in (5.26), then the Lemma 3 holds, and C0 is defined as % C0 

μ01

& N 1 Cd √ N+ σi . Ld 2 i1

(5.44)

According to (5.33) in Lemma 3, with the ε¯ initial , the initial x¯ i0 can be obtained by solving x¯ i0 ≈ arg min{ f i (xi ) + μ1 di (xi )} up to this given accuracy. After that, the xi ∈X i

0 initial Lagrangian multiplier ! u¯ is derived from the following projection procedure   0 μ N gi x¯ i0 . u¯ 0  PU L d1 i1

5.4.3.3

Algorithm Description and Convergence Analysis

  Lemma 5 The primal-dual sequence x¯ ik , u¯ k is generated by the proposed algorithm after k iterations. If the accuracy level ε˜ is chosen such that 0 ≤ δ0 ≤ c0 τ0 k + 1 for an arbitrary positive constant c0 , then the duality gap satisfies N    ∗   N  k+1   k+1 !+   k+1  μ0 i1 Di + c0     − ϕ u¯ ≤ gi x¯ i , − u  ∗  ≤ f x¯ i1 (τ0 k + 1)α and the feasibility gap satisfies

(5.45)

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5 Wireless Resource Management for Green Communications

 03 ⎧ 1 2/ 0 ∗ 0/ C f / 2μ (τ0 k + 1)α Mu − u ∗  , ⎪ ⎪  ⎪ &+  ⎪ % ⎪ % &+   N  k+1   ⎪  N  ⎪ ⎨   ≥ μk+1 Mu    k+1   gi x¯i 2     ≤ x ¯ g   i i   ⎪ i1  ⎪  i1  ⎪ +    N  ⎪  / ∗0  ⎪  α k+1 2 0 ⎪ C / μ (τ0 k + 1) ,  ⎪ x ¯ g  ≤ μk+1 2 Mu. ⎩ f   i1 i i (5.46)

BIEG Algorithm Initialization: √ (1) Provide an accuracy level δ0 ≥ 0. Set τ0  ( 5 − 1)/2, μ01  μ0 and μ02  L d /μ0 according to (23). Each i shares the Slater information gi j (x) with the others to decide the dual bound Mu according to (4). initial  δ0 /C0 . Each i computes the initial (2) Compute  0 C 0 by (34), and set ε¯ ¯ point x¯i , u0 ⎧ 0   ⎨ x¯i  x˜1∗ 0m ; u 01  N 0 0 m ⎩ u¯  Vu 0 ; −γ gi (x˜i ) i1

up to the given accuracy ε¯ initial according to (22). Iteration: for k  0, 1, . . . , each i repeats the following steps (1) (2) (3) (4) (5)

Set ε¯ k  τk δk/Q k , and update δk+1  (1 − τk )δk + Q k ε¯ k . Compute Ad x¯ik , u¯ k ; μk1 , μk2 , τk up to the accuracy ε¯ k . N N Compute αk  i1 di (x˜i∗ (uˆ k ; μk1 ))/ i1 Di . k+1 k+1  (1 − αk τk )μk+1  (1 − τk )μk2 . Update μk+1 1 and μ2 according to μ1 2 Update the step size parameter τk according to (31).

where constants C 1f and C 2f are denoted as C 1f

 L (1 − d

τ0 )Mu2

N  0 2  +2 μ Di + 2μ0 c0 i1

" # N #    2 d ∗ 0 C f  2L (1 − τ0 )u  + 2c0 L d μ0 (1 − τ0 ) + μ $2L d (1 − τ0 ) Di . (5.47) i1

Lemma 5 provides the analysis of convergence rate of our proposed fast algorithm. The proof follows Lemma 2 and the inequalities (5.48) directly, and is omitted here.

5.4 Bregman-Based Inexact Excessive Gap Method …

263

Lemma 5 tells us that if a suitable δ0 is chosen, then an asymptotic convergence rate O(1/k) can be achieved. Furthermore, if the ratio α ∗ approach 1− , i.e., α ∗ → 1− , the convergence rate of our BIEG algorithm can be accelerated.

5.4.3.4

Numerical Examples

Our theoretical analyses and results can be verified by numerical experiments on a separable convex quadratic problem, i.e., min x∈X

s.t

N  1 i1 N 

2

xiT Q i xi + qiT xi

ai j xi 2 ≤ b j

i1

xi ∈ X i ; i  1, . . . , N ; j  1, . . . , m,

(5.48)

where Q i ∈ R ni ×ni is a symmetric positive semidefinite matrix, qi ∈ R ni , b  [b1 , b2 , . . . , bm ]T ∈ R+m and ai j ∈ (0, 1) for i  1, . . . N , j  1, . . . , m. Each i associates its own utility function f i (xi )  21 xiT Q i xi + qiT xi with local constraint set X i . Problem Generation: The scale of (5.48) is set as N  10, and the number of nonlinear coupling constraints is set as m  5. The dimension n i is randomly generated from [5, 100]. The compact set X i ⊂ R ni ×2 is a random matrix whose elements are generated from [−5, 5], and its first column and second column represent the lower bound and upper bound respectively. The vector qi can be derived from qi  −Q i xi0 ,  2 N ai j xi0  where xi0 is a given feasible vector in X i . Meanwhile, b j  0.9 i1 for j  1, . . . , m. In the simulation, the quadratic proximity function is formed  2 as di (xi )  12σi xi − xic  +ri with σi ∈ (0, 1), and we can set different α ∗ by μ0  1 and adjusting ri according to (5.9). The algorithm is started2with the initial 3 −10 −4 at each δ0  10 , and each i updates the accuracy level to max ε¯ k , 10  iteration. As shown in Fig. 5.15, in view of optimality distance  f (x¯ k ) − f ∗ , our BIEG algorithm is compared with Fukushima algorithm, which is a variant of ADMM algorithm without a convergence rate proof [26]. The accuracy level of the Fukushima algorithm is fixed at ε¯ k  10−10 , and the ratio of the BIEG algorithm is set as α ∗  3/4. As seen, the BIEG algorithm can achieve the best performance of Fukushima algorithm, which is however sensitive to the penalty parameter r . For example, when r is small or large, the convergence rate of Fukushima algorithm is much slower, comparing to the BIEG algorithm. In Fig. 5.16, we also compare BIEG with Fukushima algorithm in  algorithm N m 2 + 2 view of the feasibility violation metric j1 [( i1 ai j x i − b j ) ] / b . In this aspect, Fukushima algorithm is better than our BIEG algorithm, i.e., the peak values

264

5 Wireless Resource Management for Green Communications

Fig. 5.15 Optimality distance comparison between BIEG algorithm and Fukushima algorithm with different penalty parameter r 500 iterations

Fig. 5.16 Feasibility violation comparison between BIEG algorithm and Fukushima algorithm with different penalty parameter r in first 500 iterations

5.4 Bregman-Based Inexact Excessive Gap Method …

265

(r  100, 1000, 10,000) are lower than the BIEG’s peak value and the violations are “pressed” back faster. On the other hand, the feasibility violations of Fukushima algorithms (r  1, 10) stay zero in the first 500 iterations, and the reason is that the unsuitable r leads the Fukushima algorithm to be trapped in the feasible region in a long time and the convergence rate is decreased. However, as shown in Fig. 5.18, which depicts the feasibility violation, a big a ∗ will lead to a relevant high perturbation peak of feasibility violation. Therefore,

Fig. 5.17 BIEG optimality distance under different a ∗ . a 150 iterations. b 3000 iterations

266

5 Wireless Resource Management for Green Communications

Fig. 5.18 BIEG feasibility violation under different a ∗

there is a tradeoff between convergence rate and accuracy. It is worth noting that the perturbation is insignificant (from 0.02 to 0.005), and will not affect the tendency of convergence in Fig. 5.17.

5.4.4 Multi-service Resource Allocation Across Heterogeneous Networks In this section, we apply the BIEG algorithm to the multiservice resource allocation problem in heterogeneous networks under VBR protocol [39], i.e., min

Sn M N ns

{bnms } n1 s1 m1

  −ln 1 + η¯ 1 bnms + (1 − pnms ) η¯ 2 bnms

s.t. bnms ∈ BSn , ∀s ∈ Sn , ∀n  1, . . . , N Nm Sn bnms ≤ Bmmax , ∀m ∈ M n1 s1 Nm Sn

n1 s1

(5.49)

bnms ≥ Bmmin , ∀m ∈ M,

where bnms is the bandwidth allocated from network n to MT m through Base Station (BS)/Access Point (AP)s. It is assumed that each MT subscribes to its own home network, but can also access other available networks with the multihoming capability. For N different access networks, Sn denotes the BSs/APs set associating with

5.4 Bregman-Based Inexact Excessive Gap Method …

267

network n, i.e., Sn  {1, 2, . . . , Sn }. Mns is the number of MTs which is served by s. pnms ∈ [0, 1] represents a priority corresponding to bnms , and is set by available network n, i.e., if MT m subscribes to home network n, pnms  1; otherwise pnms  β, where β ∈[0, 1).  η¯ 1 and η¯ 2 are used for scalability of bnms . For each  BS/AP s ∈ Sn , BSn  bnms  mMns bnms ≤ Cn , s ∈ Sn is the individual convex set which restricts the bandwidth limitation, where Cn is the total bandwidth of network n. Nm is the total number of networks that can provide services to MT m. Bmmax and Bmmin represent the maximum and minimum required bandwidth limitation for MT m respectively. As presented in (5.46), It is noteworthy that each MT’s maximum and minimum bandwidth constraints comprise different serving networks, the regularized Lagrangian function can be adopted by utilizing the decomposable proximity function. Due to the decomposable characteristic of regularized Lagrangian function, each MT can update its associating Lagrangian multiplier independently, and the regularized Lagrangian function of (5.46) is given as L({bnms }, u) 

Sn  Mns N  

−ln(1 + η¯ 1 bnms ) + (1 − pnms )η¯ 2 bnms

i1 s1 m1

+

M 

u 1m

m1

+

M 

,N S m  n  ,

u 2m

Bmmin



Sn Nm  

bnms

i1 s1

Sn  Mns N   ( μ1 2(bnms )2 + 0.75Dn ) i1



bnms −

i1 s1

m1

+

Bmmax

M 

s1 m1

μ2 2[t(u 1m )2 + (u 2m )2 ,

(5.50)

m1

where u  [u1T , u2T ]T , and u1 , u2 are vectors of Lagrangian multipliers corresponding to the bandwidth inequality constraints in (5.49). We have analyzed the proposed algorithm under the super set U in (5.8) for the theory integrity. However, when the upper bound Mu → ∞, the super set U becomes the nonnegative orthant, i.e., PU (·)  [·]+ , and the first condition in (5.47) will not appear. This is important for offloading the computation to MTs in a distributed way. According to the proposed algorithm, the following quantities regarding (5.50) are defined Ld 

N 

2Mn 2Mn ,

n1

% Dn 

Sn  s1

& Cn2 /2,

268

5 Wireless Resource Management for Green Communications

% D[σ 1]  2

N 

&1/2 Dn

,

n1

" # N # M  2$ (Mn )2 , n1

" # M #  2  2 ! 2 Bmmax + Bmmin , Cd  M D[σ 1] + M $

(5.51)

m1

Sn Mns . where Mn denotes the number of MTs served by network n, i.e., Mn  s1 In the next subsection, we will introduce an implementation example of BIEGResource Allocation (BIEG-RA) with some core network entities, e.g., Operations and Maintenance (OAM) entity.

5.4.4.1

Implementation

Initialization Procedure: OAMs of different networks can initialize the parameters in (5.51) through sharing relevant information with each other. Each OAM set τ0 

√ √ 5 − 1 2, δ0 , μ01  L d , μ02  1L d , and calculate C0 as   Cd √ N C0  μ01 d N + . (5.52) L 2 Then, the initial accuracy ε¯ initial  δ0 C0 is obtained and sent to their own BS/APs. After receiving the accuracy control information, each BS/AP by solving the following convex problem up to ε¯ initial Mns     μ0 −ln 1 + η¯ 1 bnms + (1 − pnms ) η¯ 2 bnms + 1 (bnms )2 , min {bnms } 2 m

s.t.

Mns 

bnms ≤ Cn .

(5.53)

m

Next, b¯ nms will be broadcasted to its connected MTs. With the bandwidth param3 2 eters from different serving network, each MT can initialize its own u 01m , u 02m as following 0

% u 01m



,N S -&+ m  n μ01  0 max −Bm , b¯ L d n s nms

5.4 Bregman-Based Inexact Excessive Gap Method …

% u 02m 

269

, -&+ Nm  Sn  μ01 0 . b¯ nms Bmmin − Ld n s

(5.54)

The initialization procedure is completed. Execution Procedure: For k  0, 1, . . . , repeat procedures. 3 2 the following By sharing relevant MTs’ information u¯ k  u¯ k1m , u¯ k2m m1,...M among OAMs, each OAM can calculate Q k by Qk 



 N

   k N μk1 τk Cd μk1   u ¯ + . C + τ + M − τ (1 ) d k k k Ld 2 μ2

(5.55)

Each OAM sets the accuracy ε¯ k  τk δk /Q k , and sends it to affiliated BS/APs. At the same time, each OAM updates δk+1  (1 − τk )δk + Q k ε¯ k . With the kth iteration information, each MT executes the following operation ⎧ %, N S - &+ m  n  ⎪ ⎪ k ∗ max ⎪ ¯ ⎪ u  bnms −Bm μk2 ⎪ ⎨ 1m n s %, - &+ , Sn Nm  ⎪ ⎪  ⎪ k ∗ ⎪ ⎪ Bmmin − b¯ nms μk2 ⎩ u 2m  n

(5.56)

s

and compute 

uˆ 1m  (1 − τk ) u¯ k1m +τk u ∗1m ,

(5.57)

uˆ 2m  (1 − τk ) u¯ k2m +τk u ∗2m .

Then, uˆ 1m , uˆ 2m will be reported to its serving BS/APs. It is worth noting that the above operations for OAMs and MTs can be executed synchronously. After receiving ε¯ k and uˆ 1m , uˆ 2mfrom corresponding OAM and MTs, each BS/AP ∗ by solving the following program up obtains the intermediate variables b˜ nms m∈Mns

to ε¯ k , Mns  2

min

{bnms }

s.t.

Mns 

   3  −ln 1 + η¯ 1 bnms + (1 − pnms ) η¯ 2 bnms + uˆ 1m − uˆ 2m bnms + μk1 2(bnms )2

m

bnms ≤ Cn .

(5.58)

m

 ∗  Then, b˜ nms

m∈Mns

will be sent to corresponding OAM and MTs. At the same ∗

time, each BS/AP updates the next value by b¯ nms  (1 − τk ) b¯ nms +τk b˜ nms . k+1

k

270

5 Wireless Resource Management for Green Communications

∗ After receiving b˜ nms from BS/APs of different serving network, u¯ k+1 1m can be updated by MTs through the following steps,

⎧ % ,N S -&+ m  n  ⎪ ⎪ k ⎪ ⎪ u˜  u ∗1m + τk (1 − τk )μk2 b¯ nms −Bmmax ⎪ ⎨ 1m n s u¯ k+1 % , -&+ 2m Sn Nm  ⎪ ⎪  ⎪ k ⎪ u˜  u ∗ + τ (1 − τ )μk B min − ⎪ , b¯ nms k k ⎩ 2m 2m 2 m n

(5.59)

s

and 

u¯ k+1 ¯ k1m +τk u˜ 1m 1m  (1 − τk ) u

(5.60)

u¯ k+1 ¯ k2m +τk u˜ 2m . 2m  (1 − τk ) u

So, the update process at MTs only involves some simple arithmetic operations, instead of complicated optimization computations. At the core network side, αk can ∗ k+1 be obtained by sharing {b˜ nms } among OAMs. Then, OAMs update μk+1 1 , μ2 , τk+1 according to (5.37) and (5.42).

5.4.4.2

Numerical Results

In this subsection, we study a scenario that a geographical region is covered by three available networks (n  1, 2, 3), i.e., 3G cellular network, 4G cellular network, and Wireless Local Area Network (WLAN) network. 50 multi-homing MTs are randomly deployed in the geographical region. Some specific simulation parameters are listed in Table 5.4.

Table 5.4 Parameter configuration

Parameter Value

Parameter Value

Parameter Value

C1

20

M12

20

M31

9

C2

2

M13

11

M32

2

C3

11

M21

3

M33

3

B min

0.256

M22

3

M34

6

B max

0.512

M23

4

M35

2

P1m

0.6

M24

12

M36

2

P2m

0.5

M25

3

M37

0

P3m

0.8

M26

0

M38

2

M11

19

M27

3

M39

1

5.4 Bregman-Based Inexact Excessive Gap Method …

271

Fig. 5.19 The iteration process of bandwidth allocation adopting. a BIEG-RA algorithm; and b subgradient algorithm

272

5 Wireless Resource Management for Green Communications

Fig. 5.20 Comparison for BIEG-RA algorithm and subgradient algorithm on a optimality distance; b feasibility violation

5.4 Bregman-Based Inexact Excessive Gap Method …

273

In Fig. 5.19a, the convergence procedure of MTs’ bandwidth allocation with the BIEG-RA algorithm is illustrated, while the iteration process of bandwidth allocation with the subgradient method [39] are shown in Fig. 5.19b. As seen, our BIEG-RA algorithm shows an extra fast convergence rate when compared with the traditional subgradient algorithm. Note that some curves are superposed with each other in Fig. 5.19, this is because some MTs are in the same condition in the simulations, i.e., they access the same home network and assisting networks, and their utility functions are the same. In Fig. 5.20, we compare the BIEG with the subgradient method [39] in view of optimality distance and feasibility violation. From the two figures, we can see that BIEG-RA outperforms the existing subgradient method on both aspects. Therefore, we believe our proposed BIEG-RA algorithm is more suitable in solving large complicated optimization problems for modern wireless networks.

5.5 Conclusion This chapter addresses the problem of mismatch between multi-network energy distribution and user service requirements, resulting in low efficiency of multi-network energy utilization. First, analyze the basic model of multi-network and multi-user service traffic. Then combine the distribution of base stations, the interference distribution in heterogeneous networks is modeled, and finally a green multi-network resource allocation method based on Bregman Inexact Excessive Gap is proposed.

Appendix 1 Let x1 and x2 represent the optimal solution under μ1 and μ2 , i.e., x1  x ∗ (μ1 ; μ1 ), x2  x ∗ (μ2 ; μ1 ). Thus the following inequality is obtained

∇u φ (u 1 ; μ1 ) − ∇u φ(u 2 ; μ1 )  N    {∇ φ (u ; μ )−∇ φ (u ; μ )}  u i 1 1 u i 2 1    i1  N      {gi j (x1i ) − gi j (x2i )} i1 N m    gi j (x1i ) − gi j (x2i ) ≤

(5.61)

i1 j1

Based on the assumption in Sect. 5.4.1, the gradient norm is bounded by a scalar Mi j (g). Thus, according to the mean value theorem, we have gi (x1i ) − gi (x2i ) ≤ Mi j (g) x1i − x2i . By combining with (5.61), the following inequality can be obtained

274

5 Wireless Resource Management for Green Communications

∇u φ (u 1 ; μ1 ) − ∇u φ(u 2 ; μ1 ) N m . ≤ Mi j (g) x1i − x2i

(5.62)

i1 j1

For the first order optimality conditions of (5.10), we obtain ⎧  m ⎪ ⎪ ⎪ u 1 j ∇gi j (x1i ) + μ1 ∇di (x1i ), x2i − x1i ≥ 0 ⎪ ∇ f i (x1i ) + ⎨ j1   . m ⎪ ⎪ ⎪ u 2 j ∇gi j (x2i ) + μ1 ∇di (x2i ), x1i − x2i ≥ 0 ⎪ ⎩ ∇ f i (x2i ) +

(5.63)

j1

By adding these inequalities of (5.63), underlying inequalities can be obtained   m m u 2 j ∇gi j (x2i ) − μ1 j ∇gi j (x1i ), x1i − x2i j1

j1

≥ ∇ f i (x1i ) − ∇ f i (x2i ), x1i − x2i  +μ1 ∇di (x1i ) − ∇di (x2i ), x1i − x2i  ≥ μ1 σi x1i − x2i 2

,

(5.64)

where we use the convexity of f i (·) and strong convexity of di (·) in the second inequality. Next, the following inequality is considered   m u 2 j ∇gi j (x2i ) + μ1 ∇di (x2i ), x1i − x2i j

− 

m

[gi j (x1i ) − gi j (x2i )](u 2 j − u 1 j )

j1 m

5. 4 u 1 j [gi j (x1i ) − gi j (x2i )] + ∇gi j (x1i ), x2i − x1i

(5.65)

j1

+

5 4 u 2 j [gi j (x2i ) − gi j (x1i )] + ∇gi j (x2i ), x1i − x2i

m

j1

≤0 The inequality of (5.65) is derived from the convexity of gi j (·), and it indicates a relationship as follows   m m u 2 j ∇gi j (x2i ) + u 1 j ∇gi j (x1i ), x1i − x2i j1 j1 . (5.66) m / 0 ≤ gij (x1i ) − gi j (x2i ) (u 2 j − u 1 j ) j1

The right hand of (5.66) suggests

Appendix 1

275 m /

0 gij (x1i ) − gi j (x2i ) (u 2 j − u 1 j ) j1. m ≤ [gi j (x1 j ) − gi j (x2i )]2 u 2 − u 1 , j1 . m ≤ Mi2j (g) x1i − x2i

u 2 − u 1 ,

(5.67)

j1

where the first inequality adopts the Hölder’s inequality. By applying (5.66), (5.67) to (5.64), we have " # m 1 # $

x1i − x2i ≤ (5.68) M 2 (g) u 2 − u 1 . μ1 σi j1 i j And after substituting (5.68) into (5.61), we obtain

∇u φ (u 1 ; μ1 ) −.∇u φ(u 2 ; μ1 ) N m m . Mi j (g) Mi2j (g) u 2 − u 1 ≤ μ1 σi i1 j1

(5.69)

j1

Then, the Lipschitz constant L id (μ1 ) can be obtained from (5.69).

Appendix 2 First, we recall the projection inequality [27], i.e., v − P (v), y − P (v) ≤ 0, ∀y ∈ . Then, we consider (u; μ1 ) in light of (5.21) φ(u; μ1 )  ≤

N

N

min { f i (xi ) + u, gi (xi ) + μ1 di (xi )}

i1 xi ∈X i

min { f i (xi ) + u, gi (xi )}+μ1

i1 xi ∈X i

 φ(u) + μ1

N

N

Di

(5.70)

i1

Di,

i1

where the inequality follows the definition of Di . After that, we consider the function

276

5 Wireless Resource Management for Green Communications

f (x; μ2 )    ≥

6

7 f (x) + max{ u, gi (x i ) − 21 μ2 u 2 } i1 N    1 2  max {cos θ u  g (x ) f (x) + i i  − 2 μ2 u }  θ∈[0,π ], u ≤Mu i1 N    1 2  f (x) + max { u  gi (x i )  − 2 μ2 u }

u ≤Mu i1   +     N f (x) + max { u  gi (x i )  − 21 μ2 u 2 },

u ≤Mu   i1 N

(5.71)

where we use the geometric definition of dot product in the second equality, and the inequality is based on the non-expansive property of [·]+ . The last term {·}1 in (5.71) is nonnegative  ⎧  +  +      N N ⎪     ⎪ 1 2 ⎪ − ≥ μ2 Mu M g (x ) μ M , g (x )    ⎪ ⎨ u  i1 i i  2 2 u  i1 i i     {·}1  (5.72) +  +   2   ⎪ ⎪  N    N ⎪ ⎪ gi (x i )  /2μ2 ,  gi (x i )  ≤ μ2 Mu .  ⎩     i1

i1

Through combining (5.70) and (5.71) together, we obtain the right hand of (5.72). Then, we consider φ(u) ≤φ(u ∗ ) 6 71 N N ∗  min f i (xi ) + u , gi (xi ) x∈X i1 i1 6 7 N gi (x i ) ≤ f (x) + u ∗ , ,  i1 N +   N N +  N +  gi (x i ) gi (x i ) − gi (x i ) , gi (x i ) + ≤ f (x) + u ∗ , i1 i1 i1 i1  N     ≤ f (x) + u ∗  g (x ) i i   i1

(5.73) where the third inequality adopts the projection inequality with + and the fourth   [·] , !+ inequality!+ follows the fact that the term N N N negative. Thus the left hand of i1 gi (x i )− i1 gi (x i ) , i1 gi (x i ) (5.22) is proved. Furthermore, the following inequality can be obtained by taking (5.70), (5.71), (5.72) into consideration

Appendix 2

277

% &+   N  N    ∗    ≤ σ + μ −u  g (x ) Di − {·}1 . i i 1    i1  i1

(5.74)

The inequality (5.23) follows from (5.73) and the specific expression of {·}1 after a few simple calculations.

Appendix 3 Because the Bregman projection is taken into consideration, the proof is different from the corresponding contents in [12] that a special mapping is considered. For brevity, we denote xi∗  xi∗ (0m ; μ1 ) and x i0  x i∗ (0m ; μ1 ). After that, the following equalities are defined 6 7 N N f i (x i0 ) + 0m , gi (x i0 ) + μ1 di (x i0 ), i1 i1 i1 6 7 N N N ∗ ∗ m m f i (xi ) + 0 , gi (xi ) + μ1 di (xi∗ ). φ(0 ; μ1 ) 

˜ m ; μ1 )  φ(0

N

i1

i1

(5.75)

i1

By considering the inexactness inequality in (5.24), the following equalities can be obtained ˜ m ; μ1 ) − φ(0m ; μ1 ) ≤ φ(0

N μ1  2 σi εi . 2 i1

(5.76)

Now we consider the following inequalities φ(u 0 ; μ1 )  2 4 5 ≥ φ(0m ; μ1 ) + 6∇φ(0m ; μ1 ), u70 − 0m − 21 L d (μ1 )u 0 − 0m  N N  2 ˜ m ; μ1 ) + ≥ φ(0 gi (xi∗ ), u 0 − 21 L d (μ1 )u 0  − μ22 σi εi2 i1 i1 7   6N  2 ˜ m ; μ1 ) +  φ(0 gi (x i0 ), u 0 − 21 L d (μ1 )u 0  i1 1 7  6 N N N μ2 0 0 ∗ 2 + gi (xi ) − gi (x i ), u − 2 σi εi , i1

i1

i1

(5.77)

2

where we use the concavity of ϕ(·; μ1 ) in the first inequality, and the second inequality follows (5.74). First, the term [·]1 of (5.75) is considered

278

5 Wireless Resource Management for Green Communications

6 7  N   0 0 1 1  0 2 m ˜ gi (x i ), u + 2 u [·]1  φ(0 ; μ1 ) − γ −γ i1 6 7  N 0 0 0 1 m m ˜ ≥ φ(0 ; μ1 ) − γ −γ gi (x i ), u + Bu (0 ; u )  i1 6 N 7  0 0 m ˜ m ; μ1 ) − 1 min − γ g (x ), u + B (0 ; u )  φ(0 i i u γ u∈U i1  7  6N 0 2 1 m m ˜ gi (x i ), u − 0 − 2γ u ≥ max φ(0 ; μ1 ) − u∈U N  6 i1 7 N N  max f i (x i0 ) + gi (x i0 ), u − 2γ1 u 2 + μ1 di (x i0 ) u∈U

i1

≥ f (x ; μ2 ) + μ1 0

N i1

i1

(5.78)

i1

di (x i0 ),

where the first equality uses the fact that γ  1/L d (μ1 ), the first inequality follows (5.28), the second equality is based on the definition of Bregman projection (5.29), the second inequality follows (5.27) and u c  0m , and the third inequality uses μ2 ≥ L d (μ1 ) in (5.31). Furthermore, [·]2 is taken into account [·]2 ≥ −

N m    gi j (x ∗ ) − gi j (x 0 )u 0  − i

i1 j1 N m 

 ≥ −u 0 

i1 j1

  Mi j (g)xi∗ − x 0  −

N m   ≥ −u 0  Mi j (g)εi −  0  i1 j1 μ 2 ≥ −u  Mε[1] − 21 ε[σ ] ,

μ1 2

N i1

μ1 2

N i1

μ1 2

N i1

σi εi2

σi εi2

(5.79)

σi εi2

where the second inequality  uses the boundness of the gradient, the third inequality follows the fact xi∗ − x 0  ≤ εi , and the last inequality   adopts Hölder’s inequality. Now we consider the following inequalities about u 0     N  0   u    PU μd1 gi (x 0 )  i   L  i1 N μ  1 ≤ gi (x i0 )  Ld  i1  N  ,  N N     0 c   c  + ≤ μL d1  g (x ) − g (x ) g (x ) i i i i   i i   i1  N i1   i1   μ1 c   ≤ L d M Dσ +  gi (x i ) ,

(5.80)

i1

where the first equality uses the definition of Bregman projection (5.21), the second inequality follows the non-expansive property of the norm operator, and the

Appendix 3

279

third inequality adopts the boundness of the gradient and Hölder’s inequality. By combining (5.76), (5.77) and (5.78), we complete the proof φ(u 0 ; μ1 )

N ≥ f (u 0 ; μ2 ) + μ1 di (x i0 ) i1 N     c  g (x ) − μL d1 M 2 Dσ + M  i i  ε[1] − 

≥ f (x 0 ; μ2 ) − σ0.

i1

μ1 2 ε 2 [σ ]

(5.81)

Appendix 4 For the clarity of the exposition, the following equalities are denoted, u 2  u ∗ (x; μ2 ), x˜i  x˜i∗ (u ; μ1 ), x i  xi∗ (x ; μ1 ). According to the definition of f (·; μ1 ) and μ+2  (1−τ )μ2 , we have 8

8

8

f (x¯ + ; μ+2 ) 1 6 N 7 N  max f i (x¯ + ) + u, gi (x¯ + ) − 21 μ+2 u 2 u∈U i1 i N 6 N 7  max f i [(1 − τ )x¯i + τ x˜i ] + u, f i [(1 − τ )x¯i + τ x˜i ] − u∈U i1 i    % N & N   μ2 2

u ≤ max (1 − τ ) f i (x¯i ) + u, gi (x¯i ) − u∈U 2 i1 i  1 % N  & N   +τ f i (x˜i ) + u, gi (x˜i ) , i1

i

1 (1−τ )μ2

u 2 2

2

(5.82) where the first inequality follows the convexities of f i (·) and gi j (·). The first term [·]1 of (5.82) can be denoted as  [·]1  − μ2

    N N  1 2 B(u 2 , u) + u 2 + u 2 , u − u 2  + f i (x i )+ u, gi (x i ) . 2 i1 i1 (5.83)

The first order optimality condition is  N  i

 gi (x i ) − μ2 u 2 , u − u 2 ≤ 0, ∀u ∈ U.

(5.84)

280

5 Wireless Resource Management for Green Communications

Therefore, the term [·]1 can be further estimated as [·]1 ≤ −μ2 B(u 2 , u) +

N i1

6 7 N f i (x¯i ) + u 2 , gi (x¯i ) − 21 μ2 u 2 2 i1

 −μ2 B(u 2 , u) + f (x; ¯ u2) ¯ μ1 ) + σ ≤ −μ2 B(u 2 , u) + φ(u; 4 5 ˆ μ1 ) + ∇φ(u; ˆ μ1 ), u¯ − uˆ + σ ≤ −μ2 B(u 2 , u) + φ(u; N   −μ2 B(u 2 , u) + φ(u; ˆ μ1 ) + gi (x˜i ), u¯ − uˆ i1  N  N   + gi (xˆi ) − gi (x˜i ), u¯ − uˆ + σ, i1

(5.85)

i1

where we use the definition of δ-excessive gap condition in the second inequality, the third inequality is based on the concavity of φ(·μ1 ), and the second equality follows ˜ the definition of ∇φ(u; μ1 ). Meanwhile, the estimation about the term [·]2 of (5.82) may lead to [·]2 

N 

 f i (x˜i ) + u, ˆ

i1

− μ1

N 

 gi (x˜i ) + μ1

i1 N 

di (x˜i ) ≤

i1

N 



N 

di (x˜i ) + u − u, ˆ

i1

f i (xˆi ) + u, ˆ

i1



N 

 gi (x˜i )

i1

 gi (xˆi ) + μ1

i1

N 

N 

di (xˆi )

i1

  N N N   μ2  2 + σi εi + u − u, ˆ gi (x˜i ) − μ1 di (x˜i ) 2 i1 i1 i1   N N N   μ2  2  φ(u; ˆ μ1 ) + u − u, ˆ gi (x˜i ) − μ1 di (x˜i ) + σi εi . 2 i1 i1 i1

(5.86)

Through substituting [·]1 and [·]2 into (5.82), we can demonstrate f (x¯ + ; μ+2 ) % ≤ max −μ2 (1 − τ )B(u 2 , u) + φ(u; ˆ μ1 ) + u∈U

% + (1 − τ )σ + (1 − τ )



 N  i1

N  i1

gi (xˆi ) −

N  i1

& gi (x˜i ), (1 − τ )(u¯ − u) ˆ + τ (u¯ − u) ˆ 

gi (x˜i ), u¯ − uˆ − μ1 τ

3 N  i1

N μ1 τ  di (x˜i ) + σi εi2 2 i1

(5.87)

&

Appendix 4

281

Furthermore, in order to explore (5.87), the term [·]3 is analyzed as follows  7 6N ˆ μ1 ) + τ gi (x˜i ), u − u 2 [·]3  max −μ2 (1 − τ )B(u 2 , u)+φ(u; u∈U i1  6N 7  τ  φ(u; ˆ μ1 ) − μ2 (1 − τ ) min − μ2 (1−τ ) gi (x˜i ), u − u 2 + B(u 2 , u) u∈U i1 7 6N ∼ gi (x˜i ), u − u 2 − μ2 (1 − τ )B(u 2 , u) φ(u; ˆ μ1 ) + τ 7 6i1 N )

u 2 − u gi (x˜i ), u˜ − u 2 − μ2 (1−τ ˜ 2 ≤ φ(u; ˆ μ1 ) + τ 2 i1 6N 7  2 ≤ φ(u; ˆ μ1 ) + gi (x˜i ), u + − uˆ − 21 L d (μ1 )u + − uˆ  6i1 7 N  2  φ(u; ˆ μ1 ) + gi (x˜i ), u + − uˆ − 21 L d (μ1 )u + − uˆ  i1 6N 7 N + + gi (x˜i ) − gi (xˆi ), u − uˆ i1 i1 6N 7 N gi (x˜i ) − gi (xˆi ), u + − uˆ ≤ φ(u + ; μ1 ) + i1

(5.88)

i1

N

≤ φ(u + ; μ+1 ) + (μ1 − μ+1 ) di (xi∗ (u + ; μ+1 )) i1 7 6N N + gi (x˜i ) − gi (xˆi ), u − uˆ , + i1

i1

where the second inequality follows the fact u + − uτ ˆ (u˜ − u 2 ), the third inequality derives from the concavity of φ(·; μ1 ), and the fourth inequality uses the concavity of φ(u; ·). In order to analyze the term [·]3 + [·]4 , we need to take underlying inequalities into consideration respectively (5.89)–(5.92). 6

N

gi (x˜i ) −

7

N

gi (xˆi ), u − uˆ + (1 − τ ) 7 N N gi (x˜i ) − gi (xˆi ), u˜ − uˆ τ i1 6

i1

+

i1

6

N

i1

gi (xˆi ) −

N

7 gi (xˆi ), u − uˆ

i1

i1

N m     Mi j (g)x˜i − xˆi  ≤ τ u˜ − uˆ 

(5.89)

i1 j1

N m   ≤ τ u˜ − uˆ  Mi j (g)εi i1 j1   ≤ τ u˜ − uˆ  Mε[1] ,

where the first equality follows line 1 and line 4 in the recursion, the first inequality of the gradient, and inequality follows   ∗is based on∗ the boundness   the second  x˜ (u;   is bounded by ≤ ε u ˜ − u ˆ ˆ μ ) − x ( u; ˆ μ ) . Actually, the term τ 1 1 i i i

282

5 Wireless Resource Management for Green Communications

  τ u˜− uˆ   N   τ  τ  Pu u 2 + μ2 (1−τ ) gi (x˜i ) − (1 − τ )u¯ − τ u 2   i1   N  τ  ≤ τ gi (x˜i ) − (1 − τ )u¯ − (1 − τ )u 2   μ2 (1−τ )    Ni1   N     τ2 1    ≤ μ2 (1−τ )  gi (x˜i ) + τ (1 − τ ) μ2 gi (x¯i ) − u¯   i1  i1  N N   1  [gi (x˜i ) − gi (x c )] + ≤ L d (μ gi (xic ) i   1) i1 i1  N  N    )  c  c + [g ( x ¯ ) − g (x )] g (x ) + τ (1−τ i i i i  i i  μ2 i1

(5.90)

i1

+τ (1 − τ ) u N  

  τ (1−τ ) 1 c   Dσ M +  gi (xi ) ≤ L d (μ1 ) + μ2 i1

+τ (1 − τ ) u , ¯

where the first equality is based on the definition of Bregman projection, the first and the second inequality follow the projection inequality and the nonexpansive property of norm, the third inequality uses, and the fourth inequality adopts the boundness of the gradient and Hölder’s inequality. We combine (5.89) and (5.90) to obtain 6

N

7

N

gi (x˜i )− gi (xˆi ), u − uˆ 7 6 N i1 N . gi (xˆi )− gi (x˜i ), u − uˆ +(1 − τ ) i1 i1 0 / ≤ μL d1 Cd + τ (1 − τ ) CL dd + M u ε[1] +

i1

(5.91)

With the definition of α, the following inequality is available N N   di (xi∗ (u + ; μ+1 )) − μ1 τ di (x˜i ) μ1 − μ+1 i1  i1  N N Di − di (x˜i ) ≤ μ1 τ α

 0.

i1

(5.92)

i1

Therefore, the conclusion of Lemma 4 can be elucidated by combining (5.87)–(5.92) together f (x + ; μ+2 ) ≤/ φ(u + ; μ+1 ) + (1 −τ )σ 0  2 + μL d1 Cd + τ (1 − τ ) CL dd + M u ε[1] + μ21 τ ε[σ ] .  φ(u + ; μ+1 ) + (1 − τ )σ + η(τ, μ1 , μ2 , u, ε)  φ(u + ; μ+1 ) + σ+ .

(5.93)

References

283

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Chapter 6

Energy Efficiency and Collaborative Optimization Theory of 5G Heterogeneous Wireless Multi Networks

6.1 Introduction In recent years, with the rapid development of the wireless communication industry, wireless communication networks are becoming more effective in the direction of network diversification, high bandwidth, high frequency band, ubiquity, synergy, overlap and application integration and more flexible to meet the needs of people’s different communication services, a variety of mature wireless access technologies are being rapidly deployed in various scenarios. At present, most hotspot areas are overlapped by multiple types of wireless network base stations. For example, macro cell base stations of LTE-A, LTE, 3G, and 2G systems, and low-power base stations such as WiFi, micro cells, and pico cells. These wireless access networks have varying capabilities in terms of network capacity, terminal power consumption, coverage, transmission rate, and mobility support. These networks have huge differences and complementarities in support and service for different services. With the deepening of wireless communication technology research and the diversification of wireless communication requirements, people gradually realize that no network can meet all the requirements of service coverage, service type, transmission rate and service cost. Compared with a single network using a specific wireless access technology, a wireless heterogeneous network that combines multiple access technologies can meet the different needs of wireless communication users [1, 2], so heterogeneous wireless network fusion will be the trend of the times. The Wireless World Research Forum (WWRF) proposes the future of wireless communication, that is, the coexistence of multiple standards, supporting multi-mode terminal mobility and gradually evolving into a heterogeneous interconnected converged network. ITUR also describes the future of wireless communication systems in M.1645. Future systems are seamless integration systems in which various cellular systems, shortrange wireless access systems, and other access systems coexist and form a common deployment (as shown in Fig. 6.1).

© Publishing House of Electronics Industry, Beijing and Springer Nature Singapore Pte Ltd. 2019 X. Ge and W. Zhang, 5G Green Mobile Communication Networks, https://doi.org/10.1007/978-981-13-6252-1_6

287

288

6 Energy Efficiency and Collaborative Optimization Theory …

Fig. 6.1 Typical wireless heterogeneous network scene graph

With the development of the mobile Internet, various mobile intelligent terminals carry more and more new applications and new services, and the scale of wireless data transmission needs is increasing rapidly, and service providers are under increasing operational pressure. In response to the ever-increasing data transmission needs of users, operators must maintain many base stations and continue to build new base stations to increase network capacity. Take China Mobile Communications Corporation as an example. In 2009, the number of GSM network base stations has exceeded 400,000. The annual energy consumption and the occupied computer room area are a huge number. In 2007, China Mobile’s operating power consumption was 8.09 billion kWh. In 2008, the total electricity consumption increased to 9.33 billion kWh. In 2009, the total electricity consumption reached 11 billion kWh [3], which is equivalent to the electricity consumption of residents in Beijing for one year [4]. Due to the mature commercialization of next-generation network technologies, the power consumption of operators will increase rapidly in the next few years. It is estimated that by the end of 2014, China will build up to 1 million 4G base stations [5]. For operators, energy-related expenditures account for nearly half of their operating costs [6]. Figure 6.2 shows a comparison of data traffic growth and operator revenue growth curves in the voice and data ages. It can be seen that in the era of mobile broadband, the gap between data growth and operating income growth is growing. This leads to the fact that on the one hand, in order to cope with the data transmission demand

6.1 Introduction

289

Fig. 6.2 Comparison of data business growth and operator revenue growth

with exponential rate increase, a large number of base stations deployed by operators will rapidly increase operators’ costs and expenses; on the other hand, there is no new growth breakthrough for operators’ business income. It is in a state of steady growth. These two reasons will eventually lead to a slowdown in the operator’s net profit. In order to solve the gap between the growth rate of operating expenses and business income, improving the energy efficiency of the base station side and reducing the energy consumption of operators are an urgent problem to be solved. The LTE-A standard states that deploying a wireless heterogeneous network is an effective means to alleviate the contradiction between data growth and energy consumption. In recent years, many researchers have analyzed how to use wireless heterogeneous networks to improve the energy efficiency of network operators. In theory and practice, wireless heterogeneous networks have proven to be a lower cost, high performance solution [7]. Massive mobile data for the mobile Internet era brings severe challenges to heterogeneous multi-networks. How to design service-driven wireless heterogeneous network collaborative architecture under heterogeneous multi-network conditions, according to user subjective and objective requirements, service distribution characteristics, and network resource status, the differentiation of service capabilities and the time-varyingness to make intelligent collaborative strategies have important implications for energy and spectrum efficiency. This chapter first introduces the new wireless collaborative architecture, secondly analyzes the typical multi-network collaboration theory and strategy, then introduces the implementation technology of wireless resource virtualization and software-defined wireless network two heterogeneous multi-network collaboration, and finally gives a summary of this chapter.

290

6 Energy Efficiency and Collaborative Optimization Theory …

6.1.1 Heterogeneous Wireless Multi-network Energy Efficiency Collaborative Optimization Architecture For the collaborative architecture of heterogeneous wireless multi-network, there is a heated debate in the 3GPP standardization process, in which China Telecom proposes two architectures [8], namely (1) distributed architecture: exchange necessary through existing or enhanced interfaces Information, through multi-network collaboration through discrete collaborative algorithms; (2) centralized architecture: through the collaborative center to information global multi-network collaboration as representatives, in the academic and industrial circles have different views. This chapter will discuss the energy-efficient priority heterogeneous multinetwork collaborative architecture in combination with existing wireless communication standards such as 2G/3G/4G/WLAN.

6.1.2 Distributed Collaborative Architecture Under the distributed collaborative architecture, the management of radio resources does not depend on a specific management entity, and the corresponding functions are distributed to the peer-to-peer radio resource management entities. Distributed management can assign the target of the system to each distributed radio resource management entity, which shares the management and calculation functions, which can reduce the computational complexity of each node and increase the reliability of the system.

6.1.3 Centralized Collaborative Architecture Under the centralized collaborative architecture, each radio access technology is managed by a centralized radio resource management control entity, and the centralized control entity can obtain traffic, load, and blocking status of all radio resource management in the managed area. It can manage these networks in a unified manner and optimize performance. However, its shortcomings are obvious. When the overmuch network managed by the concentrator, it will be difficult to manage and inefficient.

6.1.4 Hybrid Collaborative Architecture The advantages and disadvantages of centralized distributed collaboration and centralized collaboration. In this section, for the coexistence scenarios of multiple

6.1 Introduction

291

Fig. 6.3 Distributed and centralized multi-network collaborative architecture

wireless networks such as 2G/3G/4G/WLAN in the future, heterogeneous multinetwork collaborative architecture can make important improvements in three aspects (The red oval in the Fig. 6.3). (1) Baseband Virtualization Resource Pool: Baseband resource virtualization technology is used to uniformly manage and use 2G, 3G and 4G baseband hardware resources, software and wireless resources to realize coordinated signal processing, dynamic interference control and resource efficiency optimization of wireless heterogeneous networks. Reduce the number of base station rooms and energy consumption. (2) Core network convergence intelligent gateway: Integrate gateways on the core network side of 2G, 3G and LTE into a converged intelligent gateway, adopt intelligent collaborative strategy server according to user’s subjective and objective requirements, dynamic service distribution, real-time resource status and service capability of wireless heterogeneous network, etc. Resource adaptation and routing coordination strategy generated by factors, realize unified distribution of IP addresses across networks and dynamic route management, support multi-network multi-connection communication and localized transmission of massive mobile data, reduce IP address redistribution, mobility management overhead and the burden of the core network. (3) Home-integrated intelligent gateway: Integrate indoor wireless access gateways such as Femtocell and WiFi into a home-integrated intelligent gateway, and support access to the Internet through the operator’s core network and through the LAN/Metropolitan area network to achieve user authentication and authentication. The billing channel is separated from the mass data service channel, thereby significantly shortening the transmission path, improving the user experience, and reducing the burden on the core network. In the architecture of Fig. 6.4, distributed network coordination can be implemented at the home gateway level, or centralized control can be implemented on the core network convergence gateway, thereby extending the application scope and

292

6 Energy Efficiency and Collaborative Optimization Theory …

Intelligent collaborativ e policy server

Internet EPC

Converged Intelligent gateway of core network

Virtualization resource pool of baseband

MAN/LAN

GGSN

GGSN

P-GW

SGSN

SGSN

S-GW MME

BSC

RNC

2G BTS

3G Node B

SGSN

HeNB-GW Modem

LTE/ LTE-A

GGSN

APGW

Small Cell

eNodeB

HeNB

WiFi AP

Home converged intelligent gateway

Fig. 6.4 Wireless heterogeneous network collaborative architecture

effectiveness of heterogeneous collaboration, and ultimately improving the quality and efficiency of heterogeneous multi-network collaboration.

6.2 Power Reduction for Mobile Devices by Deploying Low-Power Base Stations 6.2.1 Current Energy Efficiency Metrics In order to understand which technical methods will reduce energy consumption and reduce the amount of consumption, a metric is needed to regulate, and the concept of energy efficiency will only exist if it can be measured. Energy efficiency metrics provide quantitative visual metrics for energy efficiency. In general, metrics are applied to three purposes: (1) Comparing the energy consumption of different components or systems of the same class; (2) Setting a long-term goal for energy efficiency research and development; (3) Reflect energy efficiency under certain system/network configurations and adapt to more energy efficiency configuration methods.

6.2 Power Reduction for Mobile Devices by Deploying …

293

It can be seen that the energy efficiency index provides quantitative support for the energy saving degree of the network, and provides reference for the whole process of reducing network energy consumption, which plays an important role. At present, many researchers have analyzed the energy efficiency of wireless heterogeneous networks and proposed various metrics. Among them, the most common energy efficiency indicator is the amount of bit information (bit/J) that can be transmitted per unit of energy. This metric is simple and intuitive, but it does not reflect the network performance and user experience and is not comprehensive enough. In addition, for wireless heterogeneous networks with multiple coverages of multiple network standards, the wireless network with the bit/J metrics will cause various networks to be isolated from each other, resulting in waste of resources and low energy efficiency. In response to this problem, Richter proposed the concept of area power consumption, which is defined as the amount of electricity consumed by the cell divided by the area covered by the cell, that is, the total energy consumption of various base stations per unit area [9]. When it is necessary to consider both information transmission and overall energy consumption, Wang and Shen introduced the area energy efficiency (Area Energy Efficiency) indicator, which is in bit/Joule/km2 , defined as the energy per unit area consumed. The amount of information transmitted [10]. The above two indicators incorporate the cell coverage area into the energy efficiency analysis. Yong uses the spectral efficiency in the region divided by the average energy as an energy efficiency indicator. This indicator combines spectral efficiency and energy efficiency to obtain a more comprehensive optimization goal [10]. When examining the performance of the energy-saving scheme, the energy efficiency can be converted into a relative index, that is, when the energy efficiency is defined as the other performance indicators of the system [11]. In addition, energy efficiency indicators are also proposed from the business level of the entire network, including the Energy Consumption Rating (ECR), which is the ratio of system energy consumption to capacity. The average number of users supported per watt (Performance Indicator, PI), that is, the ratio of the number of users in the cell to the total energy consumption of the base station of the cell [12]. Based on these previously proposed indicators, researchers have done a lot of research on how to improve the energy efficiency of networks through wireless heterogeneous network planning. Through simulation, Wang concluded that the energy efficiency and throughput can be improved by introducing the picocell into the macrocell [10]. Soh et al. found that the use of microcells can generally improve the energy efficiency of heterogeneous networks, but as the density of microcells increases to a certain threshold, the increase in energy efficiency tends to be saturated [11]. Quek et al. demonstrated the density ratio of picocells to macrocells that reach the maximum heterogeneous network efficiency [13]. Richter found that in a saturated load network scenario, microcells are less effective at increasing network energy efficiency [9]. Josip obtains the relationship between the configuration parameters (number of base stations, transmission power, inter-cell distance) in the heterogeneous network and the energy consumption per unit area of the heterogeneous network and the energy required per unit area of the unit area by the simulation method [14].

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6 Energy Efficiency and Collaborative Optimization Theory …

There are still some factors not considered in the previous work. In terms of energy indicators, the previous energy efficiency indicators have considered the energy efficiency of the network from different aspects, but these indicators ignore the impact of energy efficiency on the terminal side on the overall network energy efficiency. In fact, the energy efficiency at the terminal side occupies a very important position in the entire network, which includes both the user experience and the environmental impact. At present, a mainstream Android or IOS operating system smartphone usually has a single use time of less than two days. If the user continues to browse the web or run a large program, the battery power of the mobile phone will be exhausted within a few hours, and the very limited use time seriously affects the user experience. Therefore, the influence of the operator network on the power consumption speed of the terminal will gradually become an important factor for the popularity of this operator. In addition, due to the huge amount of smart terminal usage, the total energy consumption of smart terminals is not negligible. In 2012, there were six billion smartphone users worldwide, including more than one billion smartphone users and five billion non-smartphone users [15]. Through estimation, the annual power consumption of mobile phones worldwide has reached 5 billion kWh, so the terminal energy efficiency has a non-negligible impact on the environment. In summary, the energy efficiency at the terminal side should be an important consideration for the overall network energy efficiency. An energy efficient network should not only minimize the energy consumption on the base station side, but should also strive to improve the energy efficiency at the terminal side. This is a problem that was not considered in previous energy efficiency indicators. In network planning research, due to the one-sidedness of energy efficiency indicators, researchers have neglected the impact of different network base station layout on terminal energy efficiency. In fact, the energy efficiency of the terminal can usually be improved when a new low-power base station is deployed. This is due to the following two reasons: First, when a new low-power base station is deployed, the network capacity of the wireless heterogeneous network will be improved, so the terminal will complete the required data in a shorter time in the downlink transmission. The download reduces the working time of the terminal in the high power state. With the deployment of low-power base stations, the distance between some terminals and their associated base stations will be shortened. In the uplink transmission, the transmission power of the terminals will be reduced, thereby reducing the power consumption of the uplink transmission.

6.2.2 A Novel Energy Efficiency Metric Jointly Considered by Networks and Terminals The deployment of a new base station can increase the energy efficiency of the terminal on the one hand, and on the other hand, even if it is a low-power base

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station, it increases the energy consumption on the network side. Therefore, from the perspective of energy consumption, the energy consumption reduction of the terminal and the energy consumption reduction of the base station side are contradictory. When optimizing the overall network energy efficiency, it is necessary to comprehensively consider the two factors to obtain the best balance. Based on this, we propose a comprehensive consideration of the energy efficiency indicators of the terminal side and the network side. In the new base station planning, this indicator takes into account the increase in energy consumption on the network side when the new base station is deployed, and the change in the energy consumption on the terminal side after the new base station is deployed. The energy efficiency index is specifically defined as the difference between the energy saving on the terminal side and the energy consumption on the base station side after the weighting, after the new base station is deployed, compared with the network before the base station is deployed. The weight is the ratio of the energy consumption of the operating energy to the energy consumption of the user terminal from the perspective of the operator. Since the power of the user terminal is limited and greatly affects the user experience, we believe that the energy of the user terminal is more important than the energy of the network side, that is, the operator intends to spend more of its own energy consumption to improve. The user experience of the users they serve should be less than one. Specifically, this energy efficiency indicator can be expressed as:   Psave − β · Pmi

(6.1)

It can be seen that the larger the value , the more obvious the benefits of erecting this new base station. Therefore, this energy efficiency index provides a quantitative evaluation criterion for operators in the network deployment from the perspective of energy efficiency. In fact, the energy efficiency indicators we proposed are consistent with the original energy efficiency indicators. Generally speaking, if the energy efficiency index value  is higher after the system is deployed with the new base station, the system is compared with the network before the base station is deployed after the new base station is deployed according to the traditional energy efficiency index. Energy efficiency is also improved. Taking the widely used energy efficiency index as an example [11], this energy efficiency indicator is defined as Ef f 

Regional spectral efficiency . Regional average network consumption

(6.2)

First, high spectral efficiency reduces the download time of the terminal, thereby reducing the power consumption of the terminal, and thus increases as the spectral efficiency of the area increases. In addition, the deployment of new base stations will increase, which will inevitably increase the regional average network consumption. In summary, it can be known that the new energy efficiency index is positively correlated with Eq. (6.2). Compared with formula (6.2), the new energy efficiency

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index is also affected by other factors in the network, such as the energy consumption of the terminal in different cells and under different working states.

6.2.3 Network Energy Efficiency Analysis in the Case of Macro and Micro Zone Coexistence This section will analyze the relationship between the energy change at the MDs side and the low-power base station layout parameters after the low-power base station is deployed in the macro cell, and then seek the low-power cell radius for the purpose of obtaining the maximum terminal energy saving and the highest energy efficiency index. As shown in Fig. 6.5, similar to [16], we consider two types of low-power stations, that is, a HetNet with one macrocell and one low-power cell. The coverage area of the macrocell is modelled by a disk with a radius of R. The coverage area of the low-power cell is modelled by a disk with a radius of D. Multicarrier resource allocation strategy [17], that is, the entire spectrum resources are divided into two non-overlapping sets, one for macrocell and the other for low-power cell, is adopted.

Fig. 6.5 Illustration of a HetNet with one macrocell and one low-power cell

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297

Assume in a certain time slot, active MDs (mobile devices) need to download a data package with size S (bytes). We further assume that all active MDs are allocated with equal bandwidth resources at different center frequencies. Therefore, there are no interferences among MDs. If an MD is located in the coverage area of the lowpower BS, the MD will connect to the low-power BS instead of the macrocell BS, otherwise, the MD will connect to the macrocell BS. In this section, we consider two types of low-power stations, that is, a microcell BS and a picocell BS. Similar to [18], we differentiate them mainly from their different user distributions, assuming that active users under the coverage radius of the macrocell/microcell HetNet are uniformly distributed, while active users under the coverage of the macrocell/picocell HetNet are the sum of two-dimensional (2D) Gaussian distribution and uniformly distribution. The rationality lies in that a microcell is usually deployed to enhance network coverage, whereas a picocell is deployed to enhance hot spot capacity. Under the macrocell/microcell HetNet scenario, the users in macrocell are uniformly distributed with density l. Denote the Euclidean distance between the macrocell BS and the microcell BS as d. Under the macrocell/picocell HetNet scenario, users are classified into two categories. Some are the hot spot users and the others are background users. The hot spot users are 2D Gaussian distributed with standard deviation σ , whereas background users are uniformly distributed with density λ. The ratio of hot spot users out of all users is  , and that of background users is 1 −  . A standard large-scale power loss propagation model (channel model) is applied in this section, with a path loss exponent σ and path loss constant L 0 (with typical value (4π/ν)2 , where ν is the wavelength) at reference distance r0  1 m [19]. Noise power is assumed to be additive Gaussian variables with deviation γ 2 . Thus, the downlink signal-to-noise ratio (SNR) at an MD that is x metres away from a BS is SN R 

Pt x L 0 x −σ , γ2

(6.3)

where Pt x is the transmission power at the BS. The transmission rate of the MD is calculated according to the Shannon’s law. Denote the bandwidth assigned to MD as B, and then the transmission rate from the BS to the MD is Rate  BW · log2 (1 + S N R).

(6.4)

For power consumption models for MDs, different types of MDs have different power consumption patterns [15]. A general power consumption model for MDs is adopted as P ph  pPi + (1 − p)Pw ,

(6.5)

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6 Energy Efficiency and Collaborative Optimization Theory …

where Pi and Pw are the power consumption when the MD is in idle state and active state, respectively. p is the probability that the MD is in idle state. For the energy consumption of the base station, studies have shown that the energy consumption of the base station is almost independent of its load [20]. So the power consumption we set for the three base stations is [21]: Pma  a1 Ptma + Pm ma Pmi  a2 Ptmi + Pm mi .

(6.6)

where a1 and a2 are determined by the performance of the base station RF power amplifier, Ptma and Ptmi are the transmit power of each base station. Pm ma and Pm mi are the power consumption of other components in the macro base station and the micro base station, which are independent of the radio frequency power consumption, including power consumption such as signal processing and cooling. Under the macro/microcell HetNet scenario, the entire active MDs are uniformly distributed with density l. In this section, the power reduction of MDs with the deployment of a microcell is analysed, in comparison with the time when no microcell is deployed. Then an optimal value of the microcell radius is obtained to obtain the maximum power reduction for MDs.

6.2.3.1

Power Consumption for MDs Without Microcell

Under a single macrocell scenario, the number of microcell MDs is Ma  λ · π R 2 , and thus the bandwidth per MD is Bmauser  BMmaa , where Bma is the total bandwidth of the macrocell. According to the channel model, if minimum SNR required at the edge of the macrocell is SNRR, then the macrocell BS transmission power is σ Ptma  S N R R · γ 2 L −1 0 R .

(6.7)

The received SNR at the place that is r metres away from the BS is S N R(r ) 

 σ R Ptma L 0 r −σ  S N R . R γ2 r

(6.8)

Owing to the MD uniform distribution, the MD number is linearly correlated with the corresponding area size. It can be obtained that the probability that the distance from an MD to the BS is less than r meters is Fma (r ) 

r2 r < R. R2

(6.9)

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Then the probability density function (PDF) that an MD is located in a place that is r meters away from the macro BS is 2

pma (r ) 

d Rr 2 2r  2 r < R. dr R

(6.10)

As the MD needs to download the data with S bit, the transmission duration of an MD r meters away from the macrocell BS is Rama (r )  Bmauser · log2 (1 + S N Rr (r ))  Bu · ηr ,

(6.11)

where ηr  log2 (1 + S N R(r )) is the spectrum efficiency for an MD r meters away from the macrocell BS. With (6.10) and (6.11), the average transmission duration of all MDs is R Tma 

S Rama (r )

0

R pma (r )dr  0

S

2r dr . Bmauser ηr R 2

(6.12)

Thus, the total power consumption of MDs can be obtained Pmauser  Ma [Pw Tma + Pi (1 − Tma )].

6.2.3.2

(6.13)

Power Consumption for MDs with a Microcell

For the macro/microcell HetNet scenario, some MD will offload  to the macrocell. Therefore, the microcell MD number is Ma  λπ R 2 − D 2 and the microcell MD number is Mi  λπ D 2 . Correspondingly, the bandwidths per macrocell and Bma   M and Bmiuser  BMmii , where Bmi is the total bandwidth microcell MD are Bmauser  a of the microcell. If minimum required SNR at the edge of the microcell is denoted as SNRD, then the microcell BS transmission power is σ Ptmi  S N R D · γ 2 L −1 0 D .

(6.14)

Receiving SNR at a place that is r  meters away from the microcell BS is   Ptmi L 0 r −σ  S N RD · SN R r  γ2



D r



.

(6.15)

MDs in the coverage area of the microcell will connect to the microcell BS rather than the macrocell BS, and thus when an MD is located in Area 1 and Area 3 as

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shown in Fig. 6.5, similar to (6.7), the PDF that the MD is r meters away from the BS is respectively. 2r , (d − D)2 2r P{r |MD is in Area 3}  2 . R − (d + D)2 P{r |MD is in Area 1} 

(6.16) (6.17)  arccos (d 2 +r 2 −D 2 ) 2dr

( ) In Area 2, MDs that are r meters away from the macrocell BS, π proportion of the MDs will change to connect to the microcell BS. Therefore, the PDF that a macrocell MD is located at r metres away from the macrocell BS is  2 2 2 π − arccos (d +r2dr−D ) 2r × . (6.18) P{r |MD is in Area 2}  π (d + D)2 − (d − D)2 Therefore, from the entire MDs point of view, in the macrocell, the PDF of active MD as a function of r is given by ⎧ ⎨ (R 22r , r < d − D or r > d + D −D 2 )   pma (r )  π−arccos (d 2 +r2dr2 −D2 ) (6.19) 2r ⎩ , d − D < r < d + D. 2 2 π (R −D ) The PDF that a microcell MD is located at r  meters away from the microcell BS is   2r  pmi r   2 0 < r  < D. D

(6.20)

As the transmission rate of both macrocell and microcell MDs obey the Shannon’s law (6.4), the average transmission duration of macrocell and microcell MDs can be derived as  Tma (r )

R  0

  Tmi r  

D

S  Rmauser (r )

S Rmiuser (r  )

 · pma (r )dr

· pmi (r  )dr  .

(6.21)

0

The total power consumption of macrocell and microcell MDs can be expressed as     Ma [Pw · Tma + Pi · (1 − Tma )] + Mi [Pw · Tmi + Pi · (1 − Tmi )]. Puser

(6.22)

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6.2.3.3

301

Network Energy Efficiency Analysis After Micro Base Station Deployment

The power saved (Psave ) at the MDs sides after the deployment of a microcell can be obtained as  Psave  Puser − Puser



 λπ R 2 (Pw − Pi )⎣ ⎡ ×⎣

R

Bma λπ R 2

0

d+D

S

Bma 2 2 d−D λπ(R −D )

d−D 

+

S Bma η λπ(R 2 −D 2 ) r

0

R

S

⎤ 2r ⎦ dr − λπ (R 2 − D 2 )(Pw − Pi ) · ηr R 2

  π − arccos (d 2 + r 2 − D 2 )/2dr 2r dr 2 2π (R − D 2 ) · ηr

2r dr (R 2 − D 2 )

S

2r dr −λπ D 2 (Pw − Pi ) Bma 2 2) (R − D η r 2 2 λπ(R −D )

+ d+D

d+D  λ π (R − D )(Pw − Pi ) 2

2

2

2

⎡ + λ2 π 2 D 2 (Pw − Pi )⎣

d−D

R 0

S Bma ηr



D 0

2r   dr Bmi η  D2 λπ D 2 r S



  arccos (d 2 + r 2 − D 2 )/2dr 2r dr Bma ηr π S

D 2r dr − 0

⎤ S 2r  dr  ⎦. Bmi ηr 

(6.23)

where ηr  represents the spectrum efficiency when a microcell MD is located  at a place r  metres away from the microcell BS, namely, ηr   log2 1 + S N R(r  ) . To make (6.23) more tractable, we propose an approximation to use transmission rate of a representative place (analogous to the well-known “typical node (place)” in stochastic geometry) to represent the average transmission rate of the whole microcell or macrocell MDs. Denote dequal as the distance from this place to the associated BS and denote d . Through simulations (shown in Fig. 6.6), it can be found that with different α  equal R cell sizes, α barely changes. Therefore, a fixed value can be used to approximate the location of this typical user. Another approximation that when served by the macrocell, the average transmission rate of all microcell MDs equals the transmission rate of MD which is (d + 0.1D) metres away from the macrocell BS is proposed.    log2 1 + S N R R · (α)−σ , and η Dα  Thus, denote η Rα   log2 1 + S N R D · (α)−σ , (6.23) can be approximated as

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Fig. 6.6 Curves of different macro cell coverage radius values Equivalent point (α) over macrocell radius (R)

S S − λ2 π 2 D 4 (Pw − Pi ) Bma η Rα Bmi η Dα   S  + λ2 π 2 D 2 R 2 − D 2 (Pw − Pi )  −σ  Bma log2 1 + S N R R d+0.1D R ⎡ ⎤ D2 R2 S S D4 S D 2 (R 2 − D 2 )   − +  λ2 π 2 (Pw − Pi )⎣  −σ ⎦. Bma η Rα Bmi η Dα Bma log 1 + S N R d+0.1D

Psave ≈ λ2 π 2 D 2 R 2 (Pw − Pi )

2

R

R

(6.24) Denote K  λ2 π 2 S(Pw − Pi ), TMa 

1 , Td η Rα



1   −σ , TMi log2 1+S N R R ( d+0.1D ) R



1 , η Dα

then the power reduction at the MDs side is  Psave  K

 D2 R2 D4 D 2 (R 2 − D 2 ) Tma − Tmi + Td . Bma Bmi Bma

(6.25)

From (6.24), some interesting findings are as follows: (1) Power reduction for MDs Psave is linearly correlated with the square of the active MD density, download data size, working power and idle power of the MDs. (2) The benefit will be more significant for the development of a microcell when the capacity of the original macrocell (Bma ) is low. (3) More benefit will be achieved for the development of a microcell when the capacity of the microcell (Bmi ) is high.

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(4) Psave increases with the increase of distance between the microcell and the macrocell (implied by Td ). This phenomenon can be explained as the spectrum efficiency of the user is relatively low in the area far away from the macro base station. After the micro base station is deployed, the spectrum efficiency of the area can be more effectively improved, thereby saving the terminal more energy. Through (6.24), the optimal value of D to achieve the maximum power reduction  0). Ignoring the effect of the 0.1D in Td , for the MDs can be obtained from ( d dPsave D the optimum value of D is R D√ 2



Tma + Td Bma T + Td Bmi mi

.

(6.26)

Some interesting findings can be easily seen from (6.26): (1) The optimum D increases linearly with the increase of the radius of macrocell R. (2) The optimum D is positively correlated with the bandwidth ratio of microcell and macrocell (Bmi/Bma). That is to say, a more capable microcell BS fits a bigger coverage size. (3) The further the microcell is located from the macrocell, the larger the optimum D is. Available from Eq. (6.2), the energy efficiency indicator  can be expressed as follows:   Psave  − β Pmi      D2 R2 − D2 D2 R2 D4 + β Pm mi . K Tma − Tmi + Td − D σ βaS N R D γ 2 L −1 0 Bma Bmi Bma

(6.27) Denote ρ  aS NK RLD0 γ , the optimal radius of the microcell targeting the highest energy efficiency indicator can be obtained by ddD  0:    2 2 2 2 2 4   d K DBmaR Tma − BDmi Tmi + D (RBma−D ) Td d Kβρ D σ + βa Pm mi −  0. (6.28) dD dD 2

If σ  2, the optimum value of D is:  D

R 2 Bmi (Tma + Td ) − Bma Bmi βρ . 2(Bma Tmi + Bmi Td )

(6.29)

In addition to following the same rules as the optimal microcell radius for achieving maximum terminal energy savings, it can be seen that the importance of energy

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consumption β at the base station side is negatively correlated to the optimal microcell radius.

6.2.3.4

Simulation Results

The relationship of power reduction at MDs with different microcell sizes D and different distances between microcell and microcell d are shown in Fig. 6.7. In this figure, small circles stand for simulation results and solid lines are analytical results according to (6.25). It can be seen that the two curves are very close to each other. The larger d is the more power reduction can be achieved under the same configurations. This finding coincides with the analytical results. Under this simulation parameter (Table 6.1), the highest energy savings of the MDs side is 35 W, while the energy consumption of all MDs side in the entire network is 860 W. For the given configurations, power reduction for microcell MDs is close to 18%, compared with the scenario with no microcell deployments. Meanwhile, the deployment of microcell also brings about 4% of power reduction for the entire active MDs. If more data are downloaded by MDs, the power reduction gain will be more significant. The optimum D for the maximum power reduction of the MDs side when d is 2500 m is shown in Fig. 6.8. The line curves are the approximation results from (6.23) and (6.25). It can be seen that the two optimum points are both around 1500 m, and

Fig. 6.7 Relationship between terminal energy saving and micro base station position and coverage radius

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Table 6.1 Parameters configurations Macro cell parameters Carrier frequency

2.0 GHz

Bandwidth

20 MHz

Transmission power

40 W

Path loss model COST

231-Walfish–Ikegami model

Radius size

5 km

Least SNR requirement

10 dB

Low-power cell parameters Carrier frequency

2.0 GHz

Bandwidth

2 MHz

Path loss model

231-Walfish–Ikegami model

Noise

−150 dBm/Hz

User (microcell) parameters Active user density

10−5 user/m2

Working power

1.2 W

Idle power

0.6 W

Data size

100 kbit

the analytical result and the simulation result are very close to each other, with the approximation error