Consequences of Maritime Critical Infrastructure Accidents: Environmental Impacts Modeling-Identification-Prediction-Optimization-Mitigation 0128196750, 9780128196755

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Consequences of Maritime Critical Infrastructure Accidents: Environmental Impacts Modeling-Identification-Prediction-Optimization-Mitigation
 0128196750, 9780128196755

Table of contents :
Cover
Consequences of Maritime Critical Infrastructure Accidents
Copyright
Contents
Preface
Acknowledgment
1 Introduction to the analysis of critical infrastructure accident consequences
1.1 Basic notions of process of initiating events
1.2 Basic notions of process of environment threats
1.3 Basic notions of process of environment degradation
2 Shipping as a critical infrastructure
2.1 Overview of ship accidents at Baltic Sea and world sea waters
2.2 Consequences of maritime transport of chemicals for people, environment, and infrastructure
2.2.1 Initiating events caused by world sea ship traffic accidents
2.2.2 Threats from chemical releases into marine environment
2.2.2.1 Explosion of chemical substance in the ship’s accident area (H1 threat)
2.2.2.2 Fire from chemical substance in the ship’s accident area (H2 threat)
2.2.2.3 Presence of toxic substance in the ship’s accident area (H3 threat)
2.2.2.4 Presence of corrosive substance in the ship’s accident area (H4 threat)
2.2.2.5 Presence of bioaccumulative substance in the ship’s accident area (H5 threat)
2.2.2.6 Presence of other dangerous chemical substances in the ship’s accident area (H6 threat)
2.2.3 Marine environment degradation effects coming from chemical releases
3 Modeling critical infrastructure accident consequences
3.1 Modeling process of initiating events
3.2 Modeling process of environment threats
3.3 Modeling process of environment degradation
4 Identification of critical infrastructure accident consequences
4.1 Identification of process of initiating events—theoretical background
4.1.1 Estimating unknown basic parameters of process of initiating events
4.1.2 Estimating parameters of distributions of process of initiating events conditional sojourn times at initiating events...
4.1.3 Identification of distribution functions of process of initiating events conditional sojourn times at initiating even...
4.2 Identification of process of initiating events—application to Baltic Sea and world sea waters
4.2.1 States of the process of initiating events
4.2.2 Statistical identification of process of initiating events at Baltic Sea waters
4.2.2.1 Collected statistical data of process of initiating events
4.2.2.2 Evaluating basic parameters of process of initiating events
4.2.2.3 Evaluating parameters of distributions of conditional sojourn times at initiating events states
4.2.2.4 Identification of distribution functions of conditional sojourn times at initiating event states
4.2.3 Extension of statistical identification of process of initiating events for world sea waters
4.3 Identification of process of environment threats—theoretical background
4.3.1 Estimating unknown basic parameters of process of environment threats
4.3.2 Estimating parameters of distributions of process of environment threats conditional sojourn times at environment thr...
4.3.3 Identification of distribution functions of process of environment threats conditional sojourn times at environment t...
4.4 Identification of process of environment threats—application to Baltic Sea and world sea waters
4.4.1 States of process of environment threats
4.4.2 Statistical identification of process of environment threats at Baltic Sea waters
4.4.2.1 Collected statistical data of process of environment threats
4.4.2.2 Evaluating basic parameters of process of environment threats
4.4.2.3 Identification of distribution functions of conditional sojourn times at environment threats states
4.4.3 Extension of statistical identification of process of environment threats for world sea waters
4.5 Identification of process of environment degradation—theoretical background
4.5.1 Estimating unknown basic parameters of process of environment degradation
4.5.2 Estimating parameters of distributions of process of environment degradation conditional sojourn times at environment...
4.5.3 Identification of distribution functions of process of environment degradation conditional sojourn times at environme...
4.6 Identification of process of environment degradation—application to Baltic Sea and world sea waters
4.6.1 States of process of environment degradation
4.6.2 Statistical identification of process of environment degradation at Baltic Sea waters
4.6.2.1 Collected statistical data of process of environment degradation
4.6.2.2 Evaluating basic parameters of process of environment degradation
4.6.2.3 Identification of distribution functions of conditional sojourn times at environment degradation states
4.6.3 Extension of statistical identification of process of environment degradation for world sea waters
5 Prediction of critical infrastructure accident consequences
5.1 Prediction of process of initiating events
5.1.1 Prediction of process of initiating events at Baltic Sea waters
5.1.2 Prediction of process of initiating events at world sea waters
5.2 Prediction of process of environment threats
5.2.1 Prediction of process of environment threats at Baltic Sea waters
5.2.2 Prediction of process of environment threats at world sea waters
5.3 Prediction of process of environment degradation
5.3.1 Prediction of process of environment degradation at Baltic Sea waters
5.3.2 Prediction of process of environment degradation at world sea waters
5.4 Superposition of initiating events, environment threats and environment degradation processes
5.4.1 Superposition of initiating events, environment threats and environment degradation processes at Baltic Sea waters
5.4.2 Superposition of initiating events, environment threats and environment degradation processes at world sea waters
6 Modeling critical infrastructure accident losses
6.1 Critical infrastructure accident losses
6.2 Integrated impact model on critical infrastructure accident consequences related to climate–weather change process
6.2.1 Critical infrastructure accident area climate–weather change process
6.2.2 Critical infrastructure accident losses related to climate–weather impact
6.3 Critical infrastructure accident losses—application for Baltic Sea and world sea waters
6.3.1 Losses of shipping critical infrastructure accident at Baltic Sea waters without considering climate–weather change p...
6.3.1.1 Open and restricted sea waters
6.3.1.2 Gdynia and Karlskrona ports
6.3.2 Losses of shipping critical infrastructure accident at Baltic Sea waters with considering climate–weather change proc...
6.3.2.1 Open and restricted sea waters
6.3.2.2 Gdynia and Karlskrona ports
6.3.3 Losses of shipping critical infrastructure accident without considering climate–weather change process impact—extensi...
6.3.3.1 Open and restricted sea waters
6.3.3.2 Gdynia and Karlskrona ports
6.3.4 Losses of shipping critical infrastructure accident with considering the climate–weather change process impact—extens...
6.3.4.1 Open and restricted sea waters
6.3.4.2 Gdynia and Karlskrona ports
6.3.5 Losses of shipping critical infrastructure accident at Baltic Sea and world sea waters—analysis of results
7 Optimization of critical infrastructure accident losses
7.1 Critical infrastructure accident losses minimization without considering climate–weather change process impact—theoreti...
7.2 Critical infrastructure accident losses minimization with considering climate–weather change process impact—theoretical...
7.3 Critical infrastructure accident losses minimization—application for Baltic Sea and world sea waters
7.3.1 Minimization of critical infrastructure accident losses without considering climate–weather change process impact
7.3.1.1 Open and restricted sea waters
7.3.1.2 Gdynia and Karlskrona ports
7.3.2 Minimization of critical infrastructure accident losses with considering climate–weather change process impact
7.3.2.1 Open sea waters
7.3.2.2 Restricted sea waters
7.3.2.3 Gdynia and Karlskrona ports
7.3.3 Minimization of critical infrastructure accident losses without considering climate–weather change process impact—ext...
7.3.4 Minimization of critical infrastructure accident losses with considering climate–weather change process impact—extens...
8 Mitigation of critical infrastructure accident losses
8.1 Inventory of accident losses at Baltic Sea waters
8.2 Inventory of accident losses at world sea waters
8.3 Strategy for mitigation of critical infrastructure accident losses
9 Summary
Bibliography
Appendices
Appendix 1 Realizations of conditional lifetimes at states of process of initiating events at Baltic Sea waters
Appendix 2 Realizations of conditional lifetimes at states of process of initiating events at world sea waters
Appendix 3 Realizations of conditional lifetimes at states of process of environment threats at Baltic Sea waters
Appendix 4 Realizations of conditional lifetimes at states of process of environment threats at world sea waters
Appendix 5 Realizations of conditional lifetimes at states of process of environment degradation at Baltic Sea waters
Appendix 6 Realizations of conditional lifetimes at states of process of environment degradation at world sea waters
Appendix 7 Table of chi-square distribution
Appendix 8 Procedures of modeling, identification, prediction, optimization, and mitigation of critical infrastructure acci...
Modeling
Identification
Prediction
Optimization
Mitigation
Index
Back Cover

Citation preview

Consequences of Maritime Critical Infrastructure Accidents Environmental Impacts Modeling—Identification—Prediction—Optimization—Mitigation

Consequences of Maritime Critical Infrastructure Accidents Environmental Impacts Modeling—Identification—Prediction— Optimization—Mitigation Magdalena Bogalecka Gdynia Maritime University, Gdynia, Poland

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-819675-5 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Candice Janco Acquisition Editor: Louisa Munro Editorial Project Manager: Sara Valentino Production Project Manager: Selvaraj Raviraj Cover Designer: Mark Rogers Typeset by MPS Limited, Chennai, India

Contents Preface Acknowledgment

ix xi

1. Introduction to the analysis of critical infrastructure accident consequences 1 1.1 Basic notions of process of initiating events 1.2 Basic notions of process of environment threats 1.3 Basic notions of process of environment degradation

2. Shipping as a critical infrastructure 2.1 Overview of ship accidents at Baltic Sea and world sea waters 2.2 Consequences of maritime transport of chemicals for people, environment, and infrastructure 2.2.1 Initiating events caused by world sea ship traffic accidents 2.2.2 Threats from chemical releases into marine environment 2.2.3 Marine environment degradation effects coming from chemical releases

3. Modeling critical infrastructure accident consequences 3.1 Modeling process of initiating events 3.2 Modeling process of environment threats 3.3 Modeling process of environment degradation

2 3 4

7 8

14 14 15

20

21 21 25 32

4. Identification of critical infrastructure accident consequences 39 4.1 Identification of process of initiating events—theoretical background 4.1.1 Estimating unknown basic parameters of process of initiating events

40

40

4.1.2 Estimating parameters of distributions of process of initiating events conditional sojourn times at initiating events states 4.1.3 Identification of distribution functions of process of initiating events conditional sojourn times at initiating events states 4.2 Identification of process of initiating events—application to Baltic Sea and world sea waters 4.2.1 States of the process of initiating events 4.2.2 Statistical identification of process of initiating events at Baltic Sea waters 4.2.3 Extension of statistical identification of process of initiating events for world sea waters 4.3 Identification of process of environment threats—theoretical background 4.3.1 Estimating unknown basic parameters of process of environment threats 4.3.2 Estimating parameters of distributions of process of environment threats conditional sojourn times at environment threats states 4.3.3 Identification of distribution functions of process of environment threats conditional sojourn times at environment threats states 4.4 Identification of process of environment threats—application to Baltic Sea and world sea waters 4.4.1 States of process of environment threats 4.4.2 Statistical identification of process of environment threats at Baltic Sea waters 4.4.3 Extension of statistical identification of process of environment threats for world sea waters

41

44

46 46

46

56 59

59

61

65

66 67

68

72 v

vi

Contents

4.5 Identification of process of environment degradation—theoretical background 4.5.1 Estimating unknown basic parameters of process of environment degradation 4.5.2 Estimating parameters of distributions of process of environment degradation conditional sojourn times at environment degradation states 4.5.3 Identification of distribution functions of process of environment degradation conditional sojourn times at environment degradation states 4.6 Identification of process of environment degradation—application to Baltic Sea and world sea waters 4.6.1 States of process of environment degradation 4.6.2 Statistical identification of process of environment degradation at Baltic Sea waters 4.6.3 Extension of statistical identification of process of environment degradation for world sea waters

5. Prediction of critical infrastructure accident consequences

76

5.4.2 Superposition of initiating events, environment threats and environment degradation processes at world sea waters 118

77

6. Modeling critical infrastructure accident losses 78

82

83 84

85

89

95

5.1 Prediction of process of initiating events 95 5.1.1 Prediction of process of initiating events at Baltic Sea waters 95 5.1.2 Prediction of process of initiating events at world sea waters 97 5.2 Prediction of process of environment threats 98 5.2.1 Prediction of process of environment threats at Baltic Sea waters 98 5.2.2 Prediction of process of environment threats at world sea waters 101 5.3 Prediction of process of environment degradation 107 5.3.1 Prediction of process of environment degradation at Baltic Sea waters 107 5.3.2 Prediction of process of environment degradation at world sea waters 110 5.4 Superposition of initiating events, environment threats and environment degradation processes 117 5.4.1 Superposition of initiating events, environment threats and environment degradation processes at Baltic Sea waters 117

6.1 Critical infrastructure accident losses 6.2 Integrated impact model on critical infrastructure accident consequences related to climate weather change process 6.2.1 Critical infrastructure accident area climate weather change process 6.2.2 Critical infrastructure accident losses related to climate weather impact 6.3 Critical infrastructure accident losses—application for Baltic Sea and world sea waters 6.3.1 Losses of shipping critical infrastructure accident at Baltic Sea waters without considering climate weather change process impact 6.3.2 Losses of shipping critical infrastructure accident at Baltic Sea waters with considering climate weather change process impact 6.3.3 Losses of shipping critical infrastructure accident without considering climate weather change process impact—extension based on wider data collected at world sea waters 6.3.4 Losses of shipping critical infrastructure accident with considering the climate weather change process impact—extension based on wider data collected at the world sea waters 6.3.5 Losses of shipping critical infrastructure accident at Baltic Sea and world sea waters—analysis of results

121 121

122 122 123

124

124

126

132

134

137

7. Optimization of critical infrastructure accident losses 143 7.1 Critical infrastructure accident losses minimization without considering climate weather change process impact—theoretical background

143

Contents

7.2 Critical infrastructure accident losses minimization with considering climate weather change process impact—theoretical background 145 7.3 Critical infrastructure accident losses minimization—application for Baltic Sea and world sea waters 148 7.3.1 Minimization of critical infrastructure accident losses without considering climate weather change process impact 148 7.3.2 Minimization of critical infrastructure accident losses with considering climate weather change process impact 152 7.3.3 Minimization of critical infrastructure accident losses without considering climate weather change process impact—extension based on wider data collected at world sea waters 160

vii

7.3.4 Minimization of critical infrastructure accident losses with considering climate weather change process impact—extension based on wider data collected at world sea waters 162

8. Mitigation of critical infrastructure accident losses 8.1 Inventory of accident losses at Baltic Sea waters 8.2 Inventory of accident losses at world sea waters 8.3 Strategy for mitigation of critical infrastructure accident losses

9. Summary Bibliography Appendices Index

171 171 171 171

177 179 183 213

Preface The probabilistic general model of critical infrastructure accident consequences including the superposition of three models, the process of initiating events generated by a critical infrastructure accident, the process of environment threats, and the process of environment degradation is presented in the book. The designed model is adopted to the maritime transport critical infrastructure understood as a ship network operating at the sea waters. The proposed model, methods, and tools are applied to this critical infrastructure accident with chemical release consequences modeling and identification, on the basis of the statistical data coming from reports of chemical accidents at the Baltic Sea and world sea waters, and prediction. The model also includes the cost analysis of losses associated with those consequences of chemical releases. Further, under the assumption of the stress of weather influence on the ship operation condition in the form of maritime storm and/or other hard sea conditions existence, critical infrastructure accident consequences are examined, and the results are compared with the previous ones. Finally, the critical infrastructure accident losses optimization is performed, and practical suggestions and procedures of these losses mitigation are given. Chapter 1, Introduction to the analysis of critical infrastructure accident consequences, is an introduction to the subject associated with the critical infrastructure accident consequences. The concept of the general model of critical infrastructure consequences is introduced, and basic notions used in the general model as well as in three particular models of which the general one is built are presented. In Chapter 2, Shipping as a critical infrastructure, shipping belonging to the transport sector is considered a critical infrastructure. The Baltic Sea and the worldwide sea-born shipping and its environmental footprint, especially connected with the sea accidents, are presented. Moreover, the detailed notions concerned with the initiating events generated by the accident of a shipping critical infrastructure, dangerous for the environment and implicating the environment threats, which then lead to the environment degradation, are introduced. In Chapter 3, Modeling critical infrastructure accident consequences, the general semi-Markov model of critical infrastructure accident consequences is designed. The process of initiating events, the process of environment threats, and the process of environment degradation are defined. To build models of these particular processes, the vectors of initial probabilities of these processes staying at their particular states, the matrices of probabilities of these processes transitions between their particular states, the matrices of conditional distribution functions, and the matrices of conditional density functions of these processes conditional sojourn times at their particular states are defined. The distributions, such as uniform, exponential, chimney, double trapezium, and quasitrapezium, are suggested and introduced to describe those processes conditional sojourn times at the particular states. Next, the mean values of particular processes conditional sojourn times at their states having these distributions are determined. In addition, the unconditional distribution functions of the sojourn times at the particular states of these processes, the mean values of their unconditional sojourn times, and limit values of transient probabilities at the particular states are determined. Further, the superposition of the process of initiating events, the process of environment threats, and the process of environment degradation is done to create the joint probabilistic general model of critical infrastructure accident consequences. Finally, the unconditional transient probabilities and mean values of sojourn total times of the joint general process at its particular states, for the sufficiently large time, are determined. The order of Chapters 3 8 corresponds to the subtitle of the book, that is, “Modeling—Identification—Prediction— Optimization—Mitigation.” Moreover, the order of these chapters can be seen as a scheme about how to apply, step by step, proposed methods and procedures of modeling, identification, prediction, and optimization. Chapters 4 7 are constructed similarly, according to the scheme: theoretical background and its application to the Baltic Sea and world sea waters. In Chapter 4, Identification of critical infrastructure accident consequences, the theoretical background of identification of the process of initiating events, the process of environment threats, and the process of environment degradation is presented. There are methods and procedures for estimating unknown basic and distribution parameters as well as identifying distribution functions of those processes sojourn times at particular states. Next, these methods and ix

x

Preface

procedures are applied to the identification of particular processes that are parts of the general model of critical infrastructure accident consequences at the Baltic Sea and world sea waters. In Chapter 5, Prediction of critical infrastructure accident consequences, the general model of critical infrastructure accident consequences is practically applied to its prediction at the Baltic Sea and world sea waters. Namely, unconditional mean sojourn times, limit values of transient probabilities, and approximate mean values of sojourn total times at particular states of three particular processes of the general model for the fixed sufficiently large time are determined. Finally, applying the expression for total probability, the unconditional transient probabilities and mean values of sojourn total times of the joint process linking the process of initiating events, the process of environment threats, and the process of environment degradation for the fixed time are found. In Chapter 6, Modeling critical infrastructure accident losses, the functions of the environment losses associated with the process of environment degradation without and with considering the climate weather impact are defined. There are presented procedures and formulae estimating the unknown parameters of these functions of losses. Next, the expected values of the total environment loss without and with considering the climate weather impact, generated by the accident of one of ships of the shipping critical infrastructure network operating at the Baltic Sea waters, for the fixed time interval, are determined and compared each other. In addition, the indicator of resilience to the loss associated with the critical infrastructure accident related to the climate weather change is proposed. In Chapter 7, Optimization of critical infrastructure accident losses, the methods based on the results of the general model of critical infrastructure accident consequences and the linear programming are proposed to the critical infrastructure accident losses optimization without and with considering the climate weather impact. The method of the optimization of critical infrastructure accident losses determining optimal values of limit transient probabilities at particular states of the process of environment degradation that minimize the expected value of total environment losses concerned with the critical infrastructure accident for the fixed interval of time is proposed. These tools are practically tested to the minimizing losses associated with a chemical release generated by an accident of a ship critical infrastructure network operating at the Baltic Sea waters. Chapter 8, Mitigation of critical infrastructure accident losses, contains an overview of the obtained values of losses associated with the shipping critical infrastructure accident without and with considering the climate weather change impact, and results of their optimization are presented as well. Moreover, the general procedures and the new strategy assuring lower environment losses concerned with chemical releases caused by an accident of ship critical infrastructure network operating within the Baltic Sea or world sea waters are proposed. The book is completed with a summary containing the evaluation of obtained results, the perspective of future research on the considered problems, and the suggestion of other practical application of the proposed general model of critical infrastructure accident consequences. Moreover, the list of appropriate references is enclosed in bibliography. Appendices include the tables with realizations of conditional lifetimes at particular states of the process of initiating events, the process of environment threats, and the process of environment degradation used in the identification of the unknown parameters of these processes. The table of chi-square distribution necessary for the identification of these processes and summarized procedure of application of the general model of critical infrastructure accident consequences is also given. Presented in the book the theoretical background of the methods and procedures of critical infrastructure accident consequences modeling, identification, prediction and optimization, and their practical applications to the forecasting and optimization of the environment losses caused by the chemical releases generated by an accident of a ship critical infrastructure network allows one to improve the quality in the field of maritime environment protection. The need of common considering the initiating events, environment threats, and environment degradation in the accident consequences analysis is obvious and very important in the maritime transport. Linking three processes, the process of initiating events, the process of environment threats, and the process of environment degradation is an original and novel approach to the maritime critical infrastructure accident consequences analysis not used in other scientific descriptions by now. The book is offered to academics, researchers, consultants, practitioners, and other professionals to apply the proposed theoretical background in various areas of risk assessment, environment control and protection, and accident impacts analysis and management, having clear pattern in maritime sector to follow.

Acknowledgment It is a genuine gratefulness and warmest regard to acknowledge and thank Prof. Krzysztof Kołowrocki for his inspiration, ideas, and suggestions as well as for his great support and encouragement.

xi

Chapter 1

Introduction to the analysis of critical infrastructure accident consequences Chapter Outline 1.1 Basic notions of process of initiating events 1.2 Basic notions of process of environment threats

2 3

1.3 Basic notions of process of environment degradation

4

The critical infrastructure is a complex system in its operating environment, and significant features are its inside and outside dependencies, which, in the case of its degradation, have significant destructive influence on the health, safety and security, economics, and social conditions of large human communities and territories (Council Directive 2008/ 114/EC; Moteff et al., 2003; Sergiejczyk and Dziula, 2013). The critical infrastructure accident is understood as an event that changes the critical infrastructure safety state into another safety state that is worse than the critical safety state, which is dangerous for the critical infrastructure itself, its operating environment and has disastrous influence on human health and activity (Dziula et al., 2014; Kołowrocki, 2013a,b; Kołowrocki and Soszy´nska-Budny, 2012a,b; Rinaldi et al., 2001) as well. Each critical infrastructure accident can generate the initiating events, causing dangerous situations in the critical infrastructure operating surroundings. The process of these initiating events can result in this type of environment threats and lead to the dangerous environment degradations (Fig. 1.1). Thus the probabilistic joint general model of critical infrastructure accident consequences includes the models of the process of initiating events generated either by the critical infrastructure accident or by its loss of safety critical level, the process of environment threats, and the process of environment degradation (Bogalecka, 2010; Bogalecka and Kołowrocki, 2006, 2015a,b, 2016, 2017a). By using the statistical data from free-accessible reports of chemical accidents at sea (CEDRE, 2018; GISIS, 2018), the proposed model is applied to modeling, identifying, and predicting the critical infrastructure accident consequences generated by the critical infrastructure defined as a ship operating within the Baltic Sea area (Fig. 1.2) and to the cost analysis of losses associated with these consequences as well. Furthermore, under the assumption of the stress of climateweather influence on the operation conditions in the form of maritime storm and other hard sea conditions existence, the mentioned critical infrastructure accident consequences and losses associated with them are examined, and the results are compared with the previous results, not considering the weather impact. Finally, the critical infrastructure accident losses optimization and mitigation are performed, and practical suggestions and procedures for reducing the critical infrastructure accident losses are proposed. In the experiment sea area, statistical data from Fig. 1.2 are placed in the neighborhood of the maritime ferry route (Bogalecka and Kołowrocki, 2018a). The considered maritime ferry is a passenger/ro-ro cargo ship operating in the south of the Baltic Sea, between Gdynia and Karlskrona ports on a regular everyday line. The approximate maritime ferry sea water route is 135 nautical miles long. Moreover, the extension of general model of critical infrastructure accident consequences application using the wider data collected at the world sea waters and ports is given. The obtained results of the considered cases were compared with each other.

FIGURE 1.1 Interrelations of the critical infrastructure accident consequences general model. Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00001-2 © 2020 Elsevier Inc. All rights reserved.

1

2

Consequences of Maritime Critical Infrastructure Accidents

FIGURE 1.2 Maritime ferry route between Gdynia Port and Karlskrona Port.

The modeling, identification, prediction, and optimization methods, algorithms, and procedures are adapted from Kołowrocki and Soszy´nska-Budny (2011), where they were used related to operation processes of industrial systems, such as the maritime ferry technical system and the port oil piping transportation system, and significantly developed to an accident consequences analysis. To construct the general model of critical infrastructure accident consequences and to apply it practically, the basic notions concerned with these three particular processes by which the model is composed of are listed and named in Sections 1.11.3.

1.1

Basic notions of process of initiating events

EðtÞ—the semi-Markov process of initiating events, Ei —the event initiating the dangerous situation for the critical infrastructure operating environment after either the critical infrastructure accident or its loss of safety critical level, el —the state of the process of initiating events, ω—the number of initiating event states, nl ð0Þ—the number of the process of initiating events staying at the state el at the initial moment t 5 0; nð0Þ—the number of the process of initiating events realizations at the initial moment t 5 0; nlj —the number of the process of initiating events transitions from the state el into the state e j during the experimental time, nl —the total number of the process of initiating events departures from the state el during the experimental time, pl ð0Þ—the probability of initial state of the process of initiating events at the initial moment t 5 0; plj —the probability of transition between states of the process of initiating events, θlj —the random conditional sojourn time of transition between states of the process of initiating events,

Introduction to the analysis of critical infrastructure accident consequences Chapter | 1

3

H lj ðtÞ—the conditional distribution function of sojourn time of transition between states of the process of initiating events, hlj ðtÞ—the conditional density function of sojourn time of transition between states of the process of initiating events, M lj —the mean value of conditional sojourn time of transition between states of the process of initiating events, H l ðtÞ—the unconditional distribution function of sojourn time of transition at states of the process of initiating events, θl —the unconditional sojourn time at states of the process of initiating events, M l —the mean value of unconditional sojourn time at states of the process of initiating events, pl —the limit value of transient probabilities at particular states of the process of initiating events, l M^ —the mean value of sojourn total time at particular states of the process of initiating events for the fixed time.

1.2

Basic notions of process of environment threats

D—the environment area of considered critical infrastructure operating environment region that may be degraded by environment threats, Dk —the environment subarea of considered critical infrastructure operating environment region that may be degraded by environment threats, n3 —the number of subareas Dk ; Hi —the environment threat (a consequence of an initiating event) that causes the environment degradation, f i —the parameter characterizing a particular environment threat, li —the range value of a parameter f i ; SðkÞ ðtÞ—the semi-Markov process of environment threats in the subarea Dk ; Sðk=lÞ ðtÞ—the semi-Markov conditional process of environment threats in the subarea Dk , while the process of initiating events is in the state el ; sυ —the state of the process of environment threats, sυðkÞ —the state of the process of environment threats in the subarea Dk ; sυðk=lÞ —the conditional state of the process of environment threats in the subarea Dk , while the process of initiating events is in the state el ; υ—the number of environment threat states, υk —the number of environment threat states in the subarea Dk ; niðk=lÞ ð0Þ—the number of the conditional process of environment threats staying at the state siðk=lÞ at the initial moment t 5 0; nðk=lÞ ð0Þ—the number of the conditional process of environment threats realizations at the initial moment t 5 0; nijðk=lÞ —the number of the conditional process of environment threats transitions from the state siðk=lÞ into the state sjðk=lÞ during the experimental time, niðk=lÞ —the total number of the process of environment threats departures from the state siðk=lÞ during the experimental time, piðk=lÞ ð0Þ—the probability of initial state of the conditional process of environment threats at the initial moment t 5 0; pijðk=lÞ —the probability of transition between states of the conditional process of environment threats, ηijðk=lÞ —the random conditional sojourn time of transition between states of the conditional process of environment threats, ij Hðk=lÞ ðtÞ—the conditional distribution function of sojourn time of transition between states of the conditional process of environment threats,

hijðk=lÞ ðtÞ—the conditional density function of sojourn time of transition between states of the conditional process of environment threats, ij Mðk=lÞ —the mean value of conditional sojourn time of transition between states of the conditional process of environment threats, i Hðk=lÞ ðtÞ—the unconditional distribution function of sojourn time of transition at states of the conditional process of environment threats,

4

Consequences of Maritime Critical Infrastructure Accidents

ηiðk=lÞ —the unconditional sojourn time at states of the conditional process of environment threats,

i —the mean value of unconditional sojourn time at states of the conditional process of environment threats, Mðk=lÞ i pðk=lÞ —the conditional approximate value of transient probabilities at particular states of the process of environment threats in the subarea Dk , while the process of initiating events is in the state el ; i M^ ðk=lÞ —the mean value of sojourn total time at particular states of the process of environment threats for the fixed time.

1.3

Basic notions of process of environment degradation

Rm ðkÞ —the m ν ðkÞ —the

degradation effect caused by the environment threats in the subarea Dk ;

level of degradation effect Rm ðkÞ in the subarea Dk ; RðkÞ ðtÞ—the semi-Markov process of environment degradation in the subarea Dk ; Rðk=υÞ ðtÞ—the semi-Markov conditional process of environment degradation in the subarea Dk , while the process of environment threats is in the state sυðkÞ ; r ‘ —the state of the process of environment degradation, ‘ rðkÞ —the state of the process of environment degradation in the subarea Dk ; ‘ rðk=υÞ —the conditional state of the process of environment degradation of the subarea Dk , while the process of environment threats is in the state sυðkÞ ; ‘—the number of environment degradation states, ‘k —the number of environment degradation states in the subarea Dk ; i niðk=υÞ ð0Þ—the number of the conditional process of environment degradation staying at the state rðk=υÞ at the initial moment t 5 0; nðk=υÞ ð0Þ—the number of the conditional process of environment degradation realizations at the initial moment t 5 0; i nijðk=υÞ —the number of the conditional process of environment degradation transitions from the state rðk=υÞ into the j state rðk=υÞ during the experimental time, i niðk=υÞ —the total number of the process of environment degradation departures from the state rðk=υÞ during the experimental time, qiðk=υÞ ð0Þ—the probability of the initial state of the conditional process of environment degradation at the initial moment t 5 0; qijðk=υÞ —the probability of transition between states of the conditional process of environment degradation,

ζ ijðk=υÞ —the random conditional sojourn time of transition between states of the conditional process of environment degradation, Gijðk=υÞ ðtÞ—the conditional distribution function of sojourn time of transition between states of the conditional process of environment degradation, gijðk=υÞ ðtÞ—the conditional density function of sojourn time of transition between states of the conditional process of environment degradation, ij Mðk=υÞ —the mean value of conditional sojourn time of transition between states of the conditional process of environment degradation, Giðk=υÞ ðtÞ—the unconditional distribution function of sojourn time of transition at states of the conditional process of environment degradation, ζ iðk=υÞ —the unconditional sojourn time at states of the conditional process of environment degradation, i Mðk=υÞ —the mean value of unconditional sojourn time at states of the conditional process of environment degradation, qiðk=υÞ —the conditional approximate value of transient probabilities at particular states of the process of environment degradation in the subarea Dk , while the process of environment threats is in the state sυðkÞ ;

i M^ ðk=υÞ —the mean value of sojourn total time at particular states of the process of environment degradation for the fixed time, qiðkÞ —the unconditional limit transient probabilities of the joint process linking the process of initiating events, the process of environment threats, and the process of environment degradation at particular states in the subarea Dk ;

Introduction to the analysis of critical infrastructure accident consequences Chapter | 1

5

i M^ ðkÞ —the mean value of sojourn total time of the joint process linking the process of initiating events, the process of environment threats, and the process of environment degradation at particular states in the subarea Dk ; for the fixed time, LiðkÞ ðtÞ—losses associated with the process of environment degradation in the subarea Dk ; at the environment degrai dation state rðkÞ ; LðkÞ ðtÞ—losses associated with the process of environment degradation in the subarea Dk ; LðtÞ—the total value of losses associated with the process of environment degradation in all subareas Dk of the considered critical infrastructure operating in the area D; i KðkÞ ðtÞ—the cost function expressing the losses associated with the process of environment degradation in the subi area Dk ; at the environment degradation state rðkÞ ; CðtÞ—the semi-Markov climateweather change process (affecting losses associated with the process of environment degradation), cb —the state of the climateweather change process, w—the number of climateweather change process states, qb —the conditional approximate value of limit transient probabilities at the particular states of the climateweather change process, ρðkÞ —the coefficient of the climateweather change process impact on the losses associated with the process of the environment degradation in the subarea Dk ; ½LiðkÞ ðtÞðbÞ —conditional losses associated with the process of environment degradation in the subarea Dk ; at the envii ronment degradation state rðkÞ , while the climateweather change process at the critical infrastructure accident area is at the climateweather state cb ; ½LðkÞ ðtÞðbÞ —conditional losses associated with the process of environment degradation in the subarea Dk , while the climateweather change process is at the climateweather state cb ; LðkÞ ðtÞ—unconditional losses associated with the process of environment degradation in the subarea Dk ; impacted by the climateweather change process, LðtÞ—the total value of unconditional losses associated with the process of environment degradation in all subareas Dk of the considered critical infrastructure operating in the area D; impacted by the climateweather change process, RIðkÞ —the indicator of the environment of the subareas Dk resilience to losses associated with the critical infrastructure accident related to the climateweather change, RI—the indicator of the environment of the entire area D resilience to losses associated with the critical infrastructure accident related to the climateweather change, q_iðkÞ —the optimal value of limit transient probabilities qiðkÞ at the particular states of the process of environment degradation in the subarea Dk ; L_ðkÞ ðtÞ—the minimum value of losses associated with the process of environment degradation in the subarea Dk ; _ LðtÞ—the minimum total value of losses associated with the process of environment degradation in all subareas Dk of the considered critical infrastructure operating in the area D; L_ ðkÞ ðtÞ—the minimum value of losses associated with the process of environment degradation of the subarea Dk ; impacted by the climateweather change process, _ LðtÞ—the minimum total value of losses associated with the process of environment degradation in all subareas Dk of the considered critical infrastructure operating in the area D; impacted by the climateweather change process, RI_ðkÞ —the indicator of the environment of the subareas Dk resilience to losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, _ RI—the indicator of the environment of the entire area D resilience to losses associated with the critical infrastructure accident related to the climateweather change, after the optimization.

Chapter 2

Shipping as a critical infrastructure Chapter Outline 2.1 Overview of ship accidents at Baltic Sea and world sea waters 2.2 Consequences of maritime transport of chemicals for people, environment, and infrastructure 2.2.1 Initiating events caused by world sea ship traffic accidents

8 14

2.2.2 Threats from chemical releases into marine environment 2.2.3 Marine environment degradation effects coming from chemical releases

15 20

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The European Commission described critical infrastructure as an asset or system which is essential for the maintenance of vital societal functions. The damage to a critical infrastructure, its destruction, or disruption by natural disasters, terrorism, criminal activity, or malicious behavior may have a significant negative impact for the security of the European Union and the well-being of its citizens. There are two main sectors of European Critical Infrastructure that are divided into some subsectors (Council Directive 2008/114/EC): G

G

Sector I—energy: G electricity (infrastructures and facilities for generation and transmission of electricity in respect of the supplied electricity), G oil (oil production, refining, treatment, storage, and transmission by pipelines), and G gas (gas production, refining, treatment, storage, and transmission by pipelines, LNG terminal). Sector II—transport: G road transport, G rail transport, G air transport, G inland waterways transport, and G ocean and short-sea shipping and ports.

Transportation infrastructure is defined as the physical distribution systems critical in supporting the national security and economic well-being of a nation, including the national airspace systems, airlines, aircraft, airports, roads and highways, trucking and personal vehicles, ports and waterways, and the vessels operating thereon, mass transit, both rail and bus, pipelines, including natural gas, petroleum, and other hazardous materials, freight and long haul passenger rail, and delivery services (Moteff et al., 2003; Pursiainen, 2007; US President’s Commission on Critical Infrastructure Protection, 1997). According to the previous discussions, shipping is a maritime segment of the general transportation system, the safety of which strongly depends on the individual ships that are components of the maritime transportation system. Consequently, ships are components of the critical infrastructure (Blokus-Roszkowska et al., 2016a). The technical condition of a ship, crew training, and traffic safety are influential factors for the whole safety of the transportation system. In addition, the shipping subsector is a critical infrastructure network that interacts with other sectors. For example, the shipping and the wind energy sector interact with each other. Wind farms occupy some marine areas. There is usually a safety zone of 500 m around wind farms with restrictions for shipping, and a buffer zone of 500 m around the cables, where anchoring of vessels is prohibited (Bogalecka et al., 2016). On the other hand, shipping may cause accidental pollution and in this way have influence on the proper operation of other critical infrastructures. In other words, the inside and outside dependencies of the critical infrastructures are observed and defined as a linkage or connection Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00002-4 © 2020 Elsevier Inc. All rights reserved.

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Consequences of Maritime Critical Infrastructure Accidents

between two infrastructures, through which the state of one infrastructure influences or is correlated to the state of the other (Rinaldi et al., 2001). The system inside dependencies are the ones within a system itself, that is, the relationship between components and subsystems in a system causing state changes of other components and subsystems and in a consequence, resulting in changes of the system state. The system outside dependencies are those coming from the system-operating environment (external factors), including changes of the system state caused from outside the system conditions, for example, climate changes, changes of its functionality, location, other objects, government, and human decisions (regulations, economic, and public policy). The occurrence of these dependencies is defined as the critical infrastructure network cascading effect. It means the degrading effects occurring within an infrastructure and between infrastructures in their operating environment, including situations in which one infrastructure causes degradation of another one, which again causes additional degradation in other infrastructures and in their operating environment. Taking this into consideration, we can define the critical infrastructure as a complex system in its operating environment, the significant features of which are inside-system dependencies and outside-system dependencies that in the case of its degradation have a significant and destructive influence on the health, safety and security, economic, and social conditions of large human communities and territory areas. In reference to the Baltic Sea the set of ships operating within this area’s waters at a fixed moment of time (or at the fixed time interval) is called the dynamic Baltic Shipping Critical Infrastructure Network—BSCIN (Bogalecka et al., 2016). The BSCIN is an element of the Global Baltic Network of Critical Infrastructure Networks—GBNCIN that closely cooperates and interacts with the Baltic port critical infrastructure, the Baltic ship traffic, and port operation information critical infrastructure that are described in Blokus-Roszkowska et al. (2016b) and Guze and Ledo´chowski (2016), respectively, and also are elements of the GBNCIN. The objects of these critical infrastructures are vulnerable to damage, caused by external factors, occurring threats of other critical infrastructures and their networks, finally causing environment degradation.

2.1

Overview of ship accidents at Baltic Sea and world sea waters

The maritime transport is used for the transportation of goods and passengers by sea. The shipping also allows movement of large quantities of goods over long distances if the cargo delivery time does not matter. The main traffic lanes connect Asiatic, African, European, and American economic centers, crossing the Atlantic, Pacific, and Indian oceans (Fig. 2.1). These days, there are usually more than 80,000 ships in ocean and sea areas around the world at any given moment (Fig. 2.2).

FIGURE 2.1 Worldwide sea-born shipping in 2017. Generated from: www.marinetraffic.com.

Shipping as a critical infrastructure Chapter | 2

9

FIGURE 2.2 Number of moving ships on the world sea waters (September 10, 2015, 17:38). Generated from: www.marinetraffic.com.

The unusual traffic density at some regions is observed. For example, 15% of the global transportation is focused within the Baltic Sea (Andersson et al., 2016; HELCOM, 2009; Luhtala, 2010; Ojala, 2015; Posti and Ha¨kkinen, 2012). It means that there are more than 70,000 ships entering and leaving the Baltic Sea every year, and about 2000 vessels on the Baltic Sea at any time (Fig. 2.3). The main destinations for cruise ships include St. Petersburg, Copenhagen, Tallinn, Helsinki, and Stockholm (Caban et al., 2017; Ha¨nninen and Rytko¨nen, 2006; HELCOM, 2018a; Nikula and Tynkkynen, 2007; Serry, 2014). The traffic intensity in any world region and a number of ships recorded by automatic identification system may be observed in real time through the Internet (www.marinetraffic.com). More than half of the goods are regularly transported by the sea, either in a bulk or a packaged form. The United Nations Conference on Trade and Development estimates that the operation of merchant ships contributes about 380 billion dollars in freight rates within the global economy, equivalent to about 5% of the total world trade (International Chamber of Shipping, 2018). The most popular chemical cargos are petroleum hydrocarbons, palm and vegetable oils, and methanol. A lot of chemicals transported by the sea if introduced into the marine ecosystem, may create a hazard to human health and marine life, harm living resources, damage amenities, or interfere with other marine users. These chemical cargoes, because of their properties and behavior in the marine environment, are divided into two main groups: oils and other chemicals, also called hazardous and noxious substances (HNS) (IMO, 2002). The spilled oil threatens the marine ecosystem causing damages to wildlife and their habitats and poisoning exposed organisms. These effects may appear immediately after the oil release or in the later stage of spillage through long-term exposure. Moreover, the spilled oil undergoes natural processes, dependent on its properties as well as the weather condition, such as evaporation, spreading, dispersion, emulsification, dissolution, sedimentation, oxidation, and biodegradation (Fig. 2.4). Emulsification is one of the most negative effects that means formatting the water in oil emulsions, often called “chocolate mouse.” The formatted emulsion keeps the oil on the sea water surface and significantly affects the environment, also being the barrier disturbing a gas exchange between the sea water and the air (AMSA, 2003; IMO, 2005; ITOPF, 2011; Ornitz and Champ, 2002). HNS when released into the marine environment, may rapidly evaporate, float on the water surface, dissolve in the water, sink to the bottom, or reveal more than one of these behaviors at the same time (Fig. 2.5). Moreover, HNS may also react between themselves or with the sea water and air components (HELCOM, 2002; IMO, 1999, 2002; Purnell, 2009). The huge traffic is one of the reasons behind sea accidents and consequently accidental pollution (Butt et al., 2013; EMSA, 2018a, 2018b; HELCOM, 2014; ITOPF, 2019). In the field of accidental marine pollution, oil spills are one of the biggest sea accidents, and a chemical cargo release is often accompanied by a spillage of the ship’s fuel. Approximately 5.74 million tons of oil was spilled in the marine ecosystem as a result of tanker incidents and accidents from 1970 to 2018.

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Consequences of Maritime Critical Infrastructure Accidents

FIGURE 2.3 Number of moving ships on the Baltic Sea waters (March 31, 2019, 13:26). Generated from: www.marinetraffic.com.

The m/t Globe Asimi accident was the most severe ship accident that caused the Baltic Sea pollution. On November 19, 1981, in rough weather conditions, the British tanker loaded with 20,000 t of boiler fuel was leaving the port in Klaip˙eda when it hit the pier. Most of the cargo spilled into the sea (16,000 t), causing an environmental disaster. There is a lack of exhausting, detailed, and completed documents on ship accidents at the Baltic Sea before 1998 (Bogalecka et al., 2017). Annual reports on shipping accidents in the Baltic Sea area have been compiled by the Helsinki Commission (HELCOM) since the year 2000. Unfortunately, the HELCOM desisted publishing the reports. The last report on shipping accidents in the Baltic Sea area was written in 2014 (HELCOM, 2014). The HELCOM data shows more than 100 accidents and incidents at the Baltic Sea every year (Fig. 2.6). The HELCOM reported approximately 1840 accidents from 1960 vessels at the Baltic Sea from 1989 to 2013. On average, 4.7% of them (from 2004 to 2013) caused the sea environment pollution with usually no more than 0.11 t of only oil substances. The pollution from HNS has been noted only once since 1996. The leakage of 0.5 m3 of orthoxylene happened near Go¨teborg on February 13, 1996 (Ha¨kkinen and Posti, 2014). The real threat of HNS appeared on the Polish coast in 1979, when the Dutch chemical carrier m/t Anna Broere was sailing alongshore and hit the coast near Karwia (Poland). The cargo consisted of 2400 t of para-xylene, a liquid melting at 113 C. Because of the winter time (air temperature 217 C to 222 C), if the cargo had released overboard, it would have solidified immediately. Nevertheless, if the accident had happened at an air temperature higher than 113 C, it could have been necessary to evacuate people from the Polish coast because of the spreading irritating vapor of para-xylene (Bogalecka, 2009). Beaches, ports, and other coastal areas are more threatened areas of the Baltic Sea, as a lot of sea accidents occurred rather in the coastal waters such as ports—26% and their approaches—19% than open sea areas—34% in 2013. Of the

Shipping as a critical infrastructure Chapter | 2

11

FIGURE 2.4 Main processes of spilled oil in the sea water. Based on IMO, 2005. Manual on Oil Pollution: Section IV  Combating Oil Spills, IMO Publishing, London.

FIGURE 2.5 Main behavior of HNS in the sea water (G—gases, E—evaporators, F—floaters, D— dissolvers, S—sinkers). Based on IMO, 1999. Manual on Chemical Pollution: Section I  Problem Assessment and Response Arrangements, IMO Publishing, London.

reported accidents in 2013, 15 took place in icy conditions (especially northern parts of the Baltic Sea). Unfortunately, the information on icy conditions is missing for 37% of the reported accidents. The intensive transport within the Baltic Sea region and other human activities such as trade, fishing, industry, tourism in combination with special geographical, climatological, and oceanographic characteristics of this sea area (sheltered inland, shallowed and brackish sea with many coastal types, limited the exchange of water, low concentration of oxygen, cold climate) cause the Baltic Sea to be highly sensitive to the impacts of economic activities (HELCOM, 2018b; Ryde´n et al., 2003). The oil and HNS spills in a high sensitivity area cause more damages than the same quantity of spills in a low sensitivity one. The Northern Baltic Sea is an area of high sensitivity to damages but a low level of ship traffic. When accidents occur, they have a great impact because of the ice coverage, making oil and other chemical recoveries more

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Consequences of Maritime Critical Infrastructure Accidents

FIGURE 2.6 Accidents in the Baltic Sea since 2000, generated from the HELCOM map and data service. Generated from: HELCOM, 2019.

challenging. The Danish coasts are generally vulnerable because of the narrow straits and high traffic density. The risk of damage is high around the Danish islands and the Finnish archipelago (Norden, 2013). Thus the highest sensitivity is observed on the coast, in archipelagos, and in shallow water areas of the Baltic Sea. The world’s largest accidental oil spill from a ship happened on July 19, 1979, when s/s Atlantic Empress and m/t Aegean Captain collided with each other at the Caribbean Sea 10 nautical miles off Tobago where 287,000 t of oil had spilled (Hooke, 1997). However, after the collision the ships went up in flames and most of the spilled oil burned out without causing significant environmental pollution such as after accidents of m/t Exxon Valdez (Prince William Sound, United States, 1989, 37,000 t of spilled oil), m/t Prestige (coast of Galicia, 2002, 63,000 t of spilled oil) or m/t Hebei Spirit (Yellow Sea, South Korea, 2007, 11,000 t of spilled oil) with less volume of oil spillage. These accidents are recognized as the most tragic ones, as the oil remained in the environment and coastlines were hazardously impacted by it. The mentioned spillage from m/t Globe Asimi accident—the most severe ship accident at the Baltic Sea is classified as a medium one, compared to the s/s Atlantic Empress accident—the most tragic sea accident in the world (Bogalecka, 2009). However, comparing the volume of spillage is pointless as for example, the release of 16,000 t of fuel could be more dangerous for the environment of the closed Baltic Sea than the larger oil volume spilt at the open sea waters. The International Tanker Owners Pollution Federation Limited, ITOPF, is an organization established on behalf of the world’s shipowners to promote an effective response to marine spills of oil and HNS. The ITOPF also collects statistical data of these accidental spills. The location of major oil spills in the last 50 years is presented in Fig. 2.7.

Shipping as a critical infrastructure Chapter | 2

13

FIGURE 2.7 Location of major oil spills within the world sea waters since 1967. Based on ITOPF.

140 7–700 t

120

FIGURE 2.8 Number of large and medium oil spills in the years 197018. Based on ITOPF.

Number of spills

>700 t 100

80

78.8

60 45.4 40

35.8 18.1

20

6.4

1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018

0

Year

Recently, the ITOPF statistical surveys referring to 2018 show approximately 116,000 t of total volume of oil lost to the environment. They have recorded three large spills ( . 700 t) in 2018. The m/t Sanchi accident was a major one that occurred at the East China Sea in January, causing a spillage of 113,000 t of ultralight crude oil. In comparison, the ITOPF recorded two large spills ( . 700 t) in 2017. The major one occurred at the Indian Ocean in June causing over 5000 t of oil spillage. The second one involved a tanker that sank off the coast of Greece in September spilling 700 t of oil. According to the statistical data worked out by the ITOPF, 19 of the 20 largest oil spills (more than 700 t) occurred before the year 2000. The average number of spills greater than 7 t decreased from approximately 78.8 per year in the 1970s to 6.4 per year in the period of 201018 (Fig. 2.8). It further decreased from approximately 319,500 t per year in the 1970s to 17,800 t per year in the period of 201018 (Fig. 2.9).

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Consequences of Maritime Critical Infrastructure Accidents

FIGURE 2.9 Quantities of oil spilt 7 tonnes and over in the years 19702018. Based on ITOPF.

700 Thousand tonnes

600 500 400 300

319.5

200 100

117.5

113.4 19.6

17.8

1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018

0

Year

It is interesting and surprising to observe that the downward trend in oil spills continue despite an overall increase in the oil trading. About 50% of large oil spills over the last 50 years happened at sea during sailing. The collision and grounding are the most frequent reasons of accidents with the large spills of oil, whereas small spills most frequently occur during loading or unloading of cargo (ITOPF, 2019). The amount of HNS transported by sea is much smaller than oil. Thus the possibility of an accident with HNS is relatively lower than an accident with oil (Ha¨kkinen and Posti, 2013; Mamaca et al., 2009). Accidents with oil spills from ships are more thoroughly researched than accidents with HNS. There are some scientific descriptions of HNS accident assessment (CEDRE, 2018; Ellis, 2011; EMSA, 2018a; GISIS, 2018; Ha¨kkinen and Posti, 2014; Marchand, 2002; Mamaca et al., 2009; Mullai and Larsson, 2008; Rømer et al., 1993, 1995; Wern, 2002). Nevertheless, none of them are comprehensive. Taking into account the data from these several sources, it is known that sulfuric acid, sodium hydroxide, acrylonitrile, ammonium nitrate, and styrene are HNS that are the most frequently involved in sea accidents. These descriptions show different causes of the accidents. Namely, collision is pointed in Rømer et al. (1995) as the most frequent reason of accidents. In Mamaca et al. (2009), failures in aids to navigation are pointed as the most frequent one, whereas collision is the third one. On the other hand, research worked out by CEDRE (2018) shows structural damages as the main cause of accidents with HNS.

2.2 Consequences of maritime transport of chemicals for people, environment, and infrastructure 2.2.1 Initiating events caused by world sea ship traffic accidents Some maritime authorities defined and classified sea accidents. According to the International Maritime Organization (IMO), a ship accident is classified as very serious, serious, less serious, and marine incident (IMO, 2008). A very serious accident is a casualty to ships which involves a total loss of the ship, a loss of life, or severe pollution. Severe pollution is the pollution, which as evaluated by the affected coastal state or flag administration as appropriate, produces a major deleterious effect on the environment or which would have produced such an effect without prevention. A serious accident is a casualty to ships, which does not qualify as a very serious casualty, and which involves fire, explosion, collision, grounding, contact, heavy weather damage, ice damage, hull cracking, or suspected hull defect, etc., resulting in immobilization of main engines, extensive accommodation damage, severe structural damage, such as penetration of the hull under water, rendering the ship unfit to proceed, or pollution (regardless of quantity), and/or a breakdown necessitating towage or shore assistance. A less serious accident is a casualty to ships, which does not qualify as a very serious or a serious one and also includes marine incidents only for the purpose of recording useful information. The IMO (2008) points out the following initial events caused by ship accidents: collision, stranding or grounding, contact, fire or explosion, hull failure, or failure of watertight doors, ports, etc., machinery damage, damages to ship or equipment, capsizing or listing, missing (assumed lost), accidents with life-saving appliances and others. The marine traffic across the world and ship accidents were observed and analyzed. Based on the analysis and the maritime authorities’ classification of the initial events caused by sea accidents mentioned above, n1 5 7 initiating events that generate dangerous situations for the sea environment are distinguished (Bogalecka, 2010). These initial events are marked by Ei ; i 5 1; 2; . . .; 7 and defined as follows:

Shipping as a critical infrastructure Chapter | 2

G G G G G G G

15

E1 —collision (a ship striking another ship), E2 —grounding (a ship striking the sea bottom, shore, or underwater wreck), E3 —contact (a ship striking an external object, e.g., pier or floating object), E4 —fire or explosion onboard, E5 —shipping without control (drifting of ship) or missing ship, E6 —capsizing or listing of a ship, and E7 —movement of cargo in a ship.

2.2.2 Threats from chemical releases into marine environment Chemical substances transported by ships through the sea may be unexpectedly released into the marine environment as a result of initiating events caused by sea accidents presented in Section 2.2.1. Some of them are dangerous for the ships’ operating environment. These substances may behave differently when released into the marine environment. Namely, according to the physical properties such as the physical state, water solubility, density in air or water, and vapor pressure, the substance may evaporate, float on the water surface, dissolve in the water, or sink to the bottom of the sea. Moreover, released chemicals may exhibit more than one of these behaviors at the same time. For example, ammonia simultaneously evaporates and dissolves rapidly in the sea water. In addition, some chemicals such as carbide may also react with the sea water. Thus the Standard European Behavior Classification (SEBC), based on the behavior of released chemicals, includes five main categories: G G G G G

G—gases E—evaporators D—dissolvers F—floaters S—sinkers

and some subcategories of mixed groups, for example, GD—gases/dissolvers (HELCOM, 2002; IMO, 2002). The release of dangerous substances and their consequences for the environment are analyzed, and there are distinguished n2 5 6 possible environmental threats that may cause degradation in the neighborhood region of the ship’s accident area (Andersson et al., 2016; Bigelow, 1994; Burke, 2003; Carson and Mumford, 2002; Clark, 2001; Crathorne et al., 2001; GESAMP, 2014; HELCOM, 2002; IMO, 2002; Neely, 1994; Preston and Chester, 2001). These threats are marked by Hi ; i 5 1; 2; . . .; 6, and they are as follows: G G G G G G

H1 —explosion of the chemical substance in the ship’s accident area, H2 —fire from the chemical substance in the ship’s accident area, H3 —toxic substance present in the ship’s accident area, H4 —corrosive substance present in the ship’s accident area, H5 —bioaccumulative substance present in the ship’s accident area, and H6 —other dangerous chemical substances present in the ship’s accident area.

Each of the environmental threats is characterized by one parameter. We mark the parameter of threats Hi ; i 5 1; 2; . . .; 6 by f i ;i 5 1; 2; . . .; 6, and they are as follows: G G G G G G

f 1 —explosive range of the chemical substance causing the explosion, f 2 —flashpoint of the chemical substance causing the fire, f 3 —toxicity of the chemical substance, f 4 —time of occurrence of skin necrosis caused by the corrosive substance, f 5 —ability of the chemical substance to bioaccumulate in living organisms, and f 6 —ability of the chemical substance to cause other threats.

There are some common classifications of dangerous chemicals, such as the International Maritime Dangerous Goods Code—IMDG Code (IMO, 2018) and the United Nation Joint Group of Experts on the Scientific Aspects of Marine Pollution—GESAMP (GESAMP, 2014, 2016), which can be used to point a threat and its range coming from dangerous chemicals released into the marine ecosystem as a result of dangerous for environment initiating events caused by a sea accident. The IMDG Code contains the most important guidelines for the safe transportation of hazardous materials. The IMDG Code classification of dangerous goods is based on the characterizing physical hazards associated with

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Consequences of Maritime Critical Infrastructure Accidents

dangerous substances. Packaged dangerous goods are divided into nine hazard classes: (1) explosives, (2) gases (compressed, liquefied, or dissolved under pressure), (3) flammable liquids, (4) flammable solids, (5) oxidizing substances and organic peroxides, (6) toxic and infectious substances, (7) radioactive material, (8) corrosive substances, and (9) miscellaneous substances and articles. The GESAMP classification groups chemicals according to the potential hazards they pose in relation to the marine environment. The GESAMP classification provides assessment ratings for the following types of hazards: (A) bioaccumulation and biodegradation, (B) aquatic toxicity, (C) acute mammalian toxicity, (D) irritation, corrosion, and other long-term health effects, and (E) interference with other uses of the sea (in addition, the GESAMP F-rating contains remarks).

2.2.2.1 Explosion of chemical substance in the ship’s accident area (H1 threat) Explosives are able to have a rapid breakdown and simultaneously generate both high temperature and lots of gas fumes. Explosives are grouped into six divisions of class 1 of the IMDG Code in accordance with their danger—the lower the number of division is, the larger the hazard becomes (Fig. 2.10). In addition, some flammable gases, vapor of flammable liquids, and dust of flammable solids and oxidizers are able to explode, for example, when mixed with air, as well as when surrounding temperature or pressure of vapor rise. Based on Fig. 2.10, we set that f 1 parameter (explosiveness range of the substance causing the explosion) may reach 1 l 5 6 ranges as follows: G G G G G G

f 11 5 1—the released chemical substance causes the explosion and belongs to class 1.6 of the IMDG Code, f 12 5 2—the released chemical substance causes the explosion and belongs to class 1.5 of the IMDG Code, f 13 5 3—the released chemical substance causes the explosion and belongs to class 1.4 of the IMDG Code, f 14 5 4—the released chemical substance causes the explosion and belongs to class 1.3 of the IMDG Code, f 15 5 5—the released chemical substance causes the explosion and belongs to class 1.2 of the IMDG Code, and f 16 5 6—the released chemical substance causes the explosion and belongs to class 1.1 of the IMDG Code.

2.2.2.2 Fire from chemical substance in the ship’s accident area (H2 threat) Some gases, liquids, and solids are flammable. The fire could be initiated by heating these substances, their contact with an electrical spark, or an open flame. For gases, the presence of fire depends on the performance of the combustion triangle. It means that the fire will appear if simultaneously the concentration of the flammable substance is between the lower explosion limit (LEL) and upper explosion limit (UEL), the oxygen concentration in the atmosphere is sufficient for the oxidizing reaction, and the ignition source is present. The flammability grade of liquids is defined by their flashpoint (the lowest temperature making the sufficient amount of vapor to lead to the fire)—the lower the flashpoint, the larger the hazard becomes.

FIGURE 2.10 Ranking of explosiveness.

Shipping as a critical infrastructure Chapter | 2

17

FIGURE 2.11 Ranking of the flammability.

The IMDG Code classification refers to the flammability of substances. Combustible gases, liquids, and solids belong to classes 2.1, 3, and 4, respectively. Moreover, the substances of classes 5.1 (oxidizing substances) and 5.2 (organic peroxides) may initiate a fire (Fig. 2.11). Based on Fig. 2.11, we set that f 2 parameter (flashpoint of the substance causing the fire) may reach l2 5 4 ranges as follows: G G G

G

f 21 5 1—the released chemical substance causes the fire, and its flashpoint ( C) belongs to the interval ð61; 1 NÞ; f 22 5 2—the released chemical substance causes the fire and its flashpoint ( C) belongs to the interval ð23; 61i; f 23 5 3—the released chemical substance causes the fire and its flashpoint ( C) belongs to the interval ð 2 18; 23i, and f 24 5 4—the released chemical substance causes the fire, and its flashpoint ( C) belongs to the interval ð 2N; 2 18i:

2.2.2.3 Presence of toxic substance in the ship’s accident area (H3 threat) Dissolvers and substances dispersed in the sea water can cause water contamination and impairment of organisms existing in the sea water. Substances can also cause air contamination; therefore, they are toxic by inhalation. These substances impair the central nervous system, sometimes causing death. The substances may get to the organism through the skin or the respiratory system. The toxicity is defined by the average lethal concentration (LC50 —means the concentration of toxic substances dissolved in the sea water or the concentration of the toxic substances in the air that is lethal for 50% of the experimental organisms). The higher the LC50 value is, the smaller the hazard becomes (Fig. 2.12A). On the other hand, the GESAMP classification could be used to determine the toxicity of a substance. The B1- and C3-rating of the GESAMP classification (acute aquatic and mammalian inhalation toxicity, respectively) are also based on the LC50 value. The higher the B1- and C3-rating values are, the larger the hazard becomes (Fig. 2.12B). Based on Fig. 2.12, we set that f 3 parameter (toxicity of the chemical substance) may reach l3 5 6 ranges as follows: G

G

f 31 5 1—the released chemical substance causes water contamination and its LC50 (mg/dm3) belongs to the interval ð100; 1 NÞ; or the released chemical substance caused air contamination and its LC50 (mg/dm3) belongs to the interval ð10; 1 NÞ, f 32 5 2—the released chemical substance causes water contamination, and its LC50 (mg/dm3) belongs to the interval ð10; 100i; or the released chemical substance causes air contamination, and its LC50 (mg/dm3) belongs to the interval ð2; 10i,

18

Consequences of Maritime Critical Infrastructure Accidents

FIGURE 2.12 Ranking of the (A) water contamination and (B) air contamination.

G

G

G

G

f 33 5 3—the released chemical substance causes water contamination and its LC50 (mg/dm3) belongs to the ð1; 10i; or the released chemical substance causes air contamination and its LC50 (mg/dm3) belongs to the ð0:5; 2i, f 34 5 4—the released chemical substance causes water contamination and its LC50 (mg/dm3) belongs to the ð0:1; 1i; or the released chemical substance causes air contamination and its LC50 (mg/dm3) belongs to the ð0; 0:5i, f 35 5 5—the released chemical substance causes water contamination, and its LC50 (mg/dm3) belongs to the ð0:01; 0:1i, f 36 5 6—the released chemical substance causes water contamination, and its LC50 (mg/dm3) belongs to the ð0; 0:01i:

interval interval interval interval interval interval

2.2.2.4 Presence of corrosive substance in the ship’s accident area (H4 threat) Corrosive substances impair living tissue, cause burns or irritation, and may degrade other substances and materials due to their reactivity. The corrosiveness is defined by the time of a skin necrosis caused by the corrosive substances. Moreover, the IMDG Code classification as well as the GESAMP classification is based on the time of a skin necrosis.

Shipping as a critical infrastructure Chapter | 2

19

FIGURE 2.13 Ranking of the corrosiveness.

Corrosives belong to class 8 of the IMDG Code and grouped into three packaging groups corresponding to D1-range indicator of the GESAMP classification. The lower the packaging group is, the larger the hazard becomes (Fig. 2.13). Based on Fig. 2.13, we set that f 4 parameter (time of corrosive substance causing the skin necrosis) may reach 4 l 5 3 ranges as follows: G

G

G

f 41 5 1—the released chemical substance is corrosive, and its time (min) of causing the skin necrosis belongs to the interval ð60; 1 NÞ; f 42 5 2—the released chemical substance is corrosive, and its time (min) of causing the skin necrosis belongs to the interval ð3; 60i; f 43 5 3—the released chemical substance is corrosive and its time (min) of causing the skin necrosis belongs to the interval ð0; 3i:

2.2.2.5 Presence of bioaccumulative substance in the ship’s accident area (H5 threat) Bioaccumulative substances are able to concentrate on living organisms. These organisms can be dangerous for the others in the trophic web. In addition, bioaccumulative substances wipe out the seafood flavor. The bioaccumulation range is defined by bioconcentration factor (BCF) as well as logP parameter. The logP points toward the substance’s ability to dissolve in water and nonpolar solvents. The substance with better solubility in the nonpolar solvent has more intense bioaccumulation. In addition, the A1-rating of the GESAMP classification is based on logP and BCF parameters. The higher the logP; BCF, or A1-rating of the GESAMP classification is, the larger the hazard becomes (Fig. 2.14). Based on Fig. 2.14, we set that f 5 parameter (ability to bioaccumulate in living organisms) may reach l5 5 5 ranges as follows: G

G

G

G

G

f 51 5 1—the released chemical substance logP belongs to the interval ð1; 2i or the released chemical substance BCF belongs to the interval ð1; 10i, f 52 5 2—the released chemical substance logP belongs to the interval ð2; 3i or the released chemical substance BCF belongs to the interval ð10; 100i; f 53 5 3—the released chemical substance logP belongs to the interval ð3; 4i or the released chemical substance BCF belongs to the interval ð100; 500i; f 54 5 4—the released chemical substance logP belongs to the interval ð4; 5i or the released chemical substance BCF belongs to the interval ð500; 4000i; f 55 5 5—the released chemical substance logP belongs to the interval ð5; 1 NÞ or the released chemical substance BCF belongs to the interval ð4000; 1 NÞ:

20

Consequences of Maritime Critical Infrastructure Accidents

FIGURE 2.14 Ranking of the ability of the living organisms’ bioaccumulation.

TABLE 2.1 Levels of the degradation effects. Degradation effect

Level 1

Level 2

Level 3

R —increase of temperature ( C) R 2 —decrease of oxygen concentration (% in air or mg/dm3 in other subareas) R 3 —disturbance of pH regime (unit) R 4 —esthetic nuisance

ð0; 10i ð0; 2i

ð10; 20i ð2; 5i

ð20; 1 NÞ ð5; 1 NÞ

6 ð0; 1i closure of accident area is not required

R 5 —pollution

ð0; 0:5LC50 i

6 ð1; 2i closure of accident area is required for not more than 2 days ð0:5LC50 ; LC50 i

6 ð2; 14i closure of accident area is required for 2 days or more ðLC50 ; 1 NÞ

1



LC50 , Lethal concentration, see Section 2.2.2.3.

2.2.2.6 Presence of other dangerous chemical substances in the ship’s accident area (H6 threat) Sea accidents can cause other threats for the environment such as the release and presence of carcinogenic, mutagenic, immunotoxic, and other dangerous chemicals in the ship’s accident area. We set that f 6 parameter (ability to cause other threats) may reach l6 5 1 range: G

f 61 5 1—the released chemical substance causes other threats.

2.2.3 Marine environment degradation effects coming from chemical releases We distinguished mk 5 5 possible environment degradation effects in the neighborhood region of a critical infrastructure (a ship) accident area that may be caused by threats Hi ; i 5 1; 2; . . .; 6 coming from a chemical substance released into the marine environment as a result of a sea accident, described in Section 2.2.2. These environment degradations are marked by Rm ; m 5 1; 2; . . .; 5, and they are as follows: G G G G G

R1 —the increase of temperature in the accident area, R2 —the decrease of oxygen concentration in the accident area, R3 —the disturbance of pH regime in the accident area, R4 —the esthetic nuisance (caused by smells, fume, discoloration, etc.) in the accident area, and R5 —the pollution in the accident area.

In addition, we distinguished that each degradation effect may reach ν m 5 3 levels in particular subareas Dk ; k 5 1; 2; . . .; 5 as it is shown in Table 2.1. Thus each environment degradation effect for particular subareas of the accident area is expressed in the scale from 1 (slight) up to 3 (extreme).

Chapter 3

Modeling critical infrastructure accident consequences Chapter Outline 3.1 Modeling process of initiating events 3.2 Modeling process of environment threats

21 25

3.3 Modeling process of environment degradation

32

The probabilistic models of these processes of initiating events, environment threats, and environment degradation are designed. The vectors of initial probabilities of these processes staying at their particular states, the matrices of probabilities of these processes transitions between their particular states, and the matrices of conditional distribution functions and density functions of these processes conditional sojourn times at their particular states are defined. The distributions such as uniform, exponential, chimney, double trapezium, and quasitrapezium are suggested and introduced to describe the processes conditional sojourn times at the particular states. Moreover, the mean values of particular processes’ conditional sojourn times having these distributions at their states are defined and the unconditional distribution functions of sojourn times at particular states of these processes, the mean values of their unconditional sojourn times and limit values of transient probabilities at particular states are defined. Further, the superposition of these processes of initiating events, environment threats, and environment degradation is done to create the joint probabilistic general model of critical infrastructure accident consequences. Finally, the unconditional transient probabilities and the mean values of sojourn total times of the joint general process at its particular states for the sufficiently large time are determined.

3.1

Modeling process of initiating events

To model the process of initiating events, we fix the time interval tAh0; 1 NÞ as the time of a critical infrastructure operation and distinguish n1 ; n1 AN; events initiating the dangerous situation for the critical infrastructure operating environment and mark them by E1 ; E2 ; . . .; En1 : Further, we introduce a set of vectors     (3.1) E 5 e: e 5 e1 ; e2 ; . . .; en1 ; ei Af0; 1g ; where

 ei 5

1; 0;

if the initiating event Ei occurs; if the initiating event Ei does not occur;

(3.2)

for i 5 1; 2; . . .; n1 : We call vectors (3.1) the initiating events state. We may eliminate vectors that cannot occur, and we number the remaining states of the set E from l 5 1 up to ω; ωAN; where ω is the number of different elements of the set E 5 fe1 ; e2 ; . . .; eω g; where

h i el 5 el1 ; el2 ; . . .; eln1 ; l 5 1; 2; . . .; ω; and eli Af0; 1g; i 5 1; 2; . . .; n1 :

Next, we can define the process of initiating events EðtÞ on the time interval tAh0; 1 NÞ with its discrete states from the set Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00003-6 © 2020 Elsevier Inc. All rights reserved.

21

22

Consequences of Maritime Critical Infrastructure Accidents

  E 5 e1 ; e2 ; . . .; eω : After that, we assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nskaBudny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process of initiating events EðtÞ and denote by θlj its random conditional sojourn time at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; l 6¼ j: This way, the process of initiating events can be described by G

the vector of the probabilities pl ð0Þ 5 PðEð0Þ 5 el Þ; l 5 1; 2; . . .; ω; of its initial states at the moment t 5 0 ½pl ð0Þ1 3 ω 5 ½p1 ð0Þ; p2 ð0Þ; . . .; pω ð0Þ;

G

(3.3)

the matrix of probabilities of transitions between the states el and ej ; l; j 5 1; 2; . . .; ω; l 6¼ j; 2 11 3 p p12 : : : p1ω 6 p21 p22 : : : p2ω 7  lj  7; p ω3ω 5 6 4 : : : : : : 5 pω1 pω2 : : : pωω

(3.4)

where by formal agreement ’l 5 1; 2; . . .; ω; pll 5 0; G

the matrix of conditional distribution functions H lj ðtÞ 5 Pðθlj , tÞ;

tAh0; 1 NÞ;

l; j 5 1; 2; . . .; ω;

l 6¼ j;

of conditional sojourn times θlj of the process EðtÞ at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; 2 11 3 H ðtÞ H 12 ðtÞ : : : H 1ω ðtÞ 6 H 21 ðtÞ H 22 ðtÞ : : : H 2ω ðtÞ 7  lj  7; H ðtÞ ω 3 ω 5 6 (3.5) 4 : : : : : : 5 ω1 ω2 ωω H ðtÞ H ðtÞ : : : H ðtÞ where by formal agreement ’l 5 1; 2; . . .; ω; H ll ðtÞ 5 0; which is strictly related to the matrix of corresponding conditional density functions of conditional sojourn times θlj of the process EðtÞ at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; l 6¼ j; 2 11 3 h ðtÞ h12 ðtÞ : : : h1ω ðtÞ 6 h21 ðtÞ h22 ðtÞ : : : h2ω ðtÞ 7  lj  7; h ðtÞ ω 3 ω 5 6 (3.6) 4 : : : : : : 5 hω1 ðtÞ hω2 ðtÞ : : : hωω ðtÞ where by formal agreement ’l 5 1; 2; . . .; ω; hll ðtÞ 5 0: We assume that the appropriate and typical distributions suitable to describe the process EðtÞ conditional sojourn times θlj , l; j 5 1; 2; . . .; ω; l 6¼ j at the particular states are (Kołowrocki and Soszy´nska-Budny, 2011) G

the exponential distribution with a density function (Fig. 3.1) lj

h ðtÞ 5



0; αlj exp½ 2 αlj ðt 2 xlj Þ;

where 0 # αlj , 1 N;

t , xlj t $ xlj ;

(3.7)

Modeling critical infrastructure accident consequences Chapter | 3

23

´ FIGURE 3.1 The graph of the exponential distribution’s density function. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling  Identification  Prediction  Optimization. Springer, London, Dordrecht, Heidelberg, New York. G

the chimney distribution with a density function (Fig. 3.2) 8 0; > > > > > Alj > > ; > > lj > z1 2 xlj > > > > > > < Clj lj ; h ðtÞ 5 zlj2 2 zlj1 > > > > > > > Dlj > > ; > > > ylj 2 zlj2 > > > > : 0;

t , xlj xlj # t , zlj1 zlj1 # t , zlj2

(3.8)

zlj2 # t , ylj t $ ylj ;

where 0 # xlj # zlj1 # zlj2 # ylj , 1 N; Alj $ 0; C lj $ 0; Dlj $ 0; Alj 1 Clj 1 Dlj 5 1: As the mean values of the conditional sojourn times θlj are given by (Kołowrocki and Soszy´nska-Budny, 2011) ðN ðN M lj 5 E½θlj  5 tdH lj ðtÞ 5 thlj ðtÞdt; l; j 5 1; 2; . . .; ω; l 6¼ j; (3.9) 0

0

then for the distinguished distributions (3.7) and (3.8), the mean values of the process of initiating events EðtÞ conditional sojourn times θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j; at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G

G

for the exponential distribution   1 M lj 5 E θlj 5 xlj 1 lj ; α

(3.10)

    i   1h  M lj 5 E θlj 5 Alj xlj 1 zlj1 1 Clj zlj1 1 zlj2 1 Dlj zlj2 1 ylj : 2

(3.11)

for the chimney distribution

24

Consequences of Maritime Critical Infrastructure Accidents

´ FIGURE 3.2 The graphs of the chimney distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling  Identification  Prediction  Optimization. Springer, London, Dordrecht, Heidelberg, New York.

From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times θl ; l 5 1; 2; . . .; ω of the process of initiating events EðtÞ at the states el ; l 5 1; 2; . . .; ω are determined by (Kołowrocki and Soszy´nska-Budny, 2011) H l ðtÞ 5

ω X

plj H lj ðtÞ; l 5 1; 2; . . .; ω:

(3.12)

j51

Therefore the mean values E½θl  of the process of initiating events EðtÞ unconditional sojourn times θl ; l 5 1; 2; . . .; ω at the states are given by M l 5 E½θl  5

ω X

plj M lj ; l 5 1; 2; . . .; ω;

(3.13)

j51

where M lj is defined by formula (3.9) in the case of any distribution of sojourn times θlj and by formulae (3.10), (3.11) in the cases of particular distribution of these sojourn times defined, respectively, by (3.7), (3.8). The limit values of the process of initiating events EðtÞ transient probabilities at the particular states pl ðtÞ 5 PðEðtÞ 5 el Þ; tAh0; 1 NÞ; l 5 1; 2; . . .; ω;

(3.14)

Modeling critical infrastructure accident consequences Chapter | 3

25

are given by (Kołowrocki and Soszy´nska-Budny, 2011) pl 5 lim pl ðtÞ 5 t- 1 N

πl M l

ω P

j51

πj M j

; l 5 1; 2; . . .; ω;

(3.15)

l

where M is given by (3.13), while the steady probabilities πl of the vector ½πl 1 3 ω satisfy the system of equations 8 l l lj > < ½π  5 ½π ½p  ω X (3.16) πj 5 1 > : j51

and ½plj ω 3 ω is given by (3.4). l The asymptotic distribution of the sojourn total time θ^ of the process of initiating events EðtÞ in the time interval l h0; θi; θ . 0; at the state e is normal with the expected value l l M^ 5 E½θ^ Dpl θ; l 5 1; 2; . . .; ω;

(3.17)

where pl is given by (3.15).

3.2

Modeling process of environment threats

To construct the general model of the environment threats caused by the process of initiating events generated by critical infrastructure loss of required safety critical level, we distinguish the set of n2 ; n2 AN kinds of threats as the consequences of initiating events that may cause the environment degradation and denote them by H1 ; H2 ; . . .; Hn2 : We also distinguish n3 ; n3 AN environment subareas D1 ; D2 ; . . .; Dn3 of the considered critical infrastructure operating in the environment area D 5 D1 , D2 , ? , Dn3 that may be degraded by the environment threats Hi ; i 5 1; 2; . . .; n2 : The environment threats possibility of influence on the distinguished subareas is presented in Fig. 3.3. We assume that the operating area D can be affected by some of threats Hi ; i 5 1; 2; . . .; n2 , and that a particular environment threat Hi ; i 5 1; 2; . . .; n2 ; can be characterized by the parameter f i ; i 5 1; 2; . . .; n2 : Moreover, we assume that the scale of the threat Hi ; i 5 1; 2; . . .; n2 influence on area D depends on the range of its parameter value and for i particular parameter f i ; i 5 1; 2; . . .; n2 . We distinguish li ranges f i1 ; f i2 ; . . .; f il of its values. After that, we introduce a set of vectors S 5 fs: s 5 ½f 1 ; f 2 ; . . .; f n2 g; where

8 > < 0; if a threat Hi does not appear in the area D; i f 5 f ij ; if a threat Hi appears in the area D > : and its parameter is in the range f ij ; j 5 1; 2; . . .; li ;

for i 5 1; 2; . . .; n2 :

FIGURE 3.3 Illustration of environment threats possibility of influence on the critical infrastructure operating subareas.

(3.18)

26

Consequences of Maritime Critical Infrastructure Accidents

We call vectors (3.18) the environment threat state in the area D. Simultaneously, we proceed for the particular subareas Dk ; k 5 1; 2; . . .; n3 : The vector n2 1 2 sðkÞ 5 ½fðkÞ ; fðkÞ ; . . .; fðkÞ ;

k 5 1; 2; . . .; n3 ;

(3.19)

where 8 0; > < ij i fðkÞ 5 fðkÞ ; > :

if a threat Hi does not appear in the subarea Dk ; if a threat Hi appears in the subarea Dk and

(3.20)

ij its parameter is in the range fðkÞ ; j 5 1; 2; . . .; li ;

for i 5 1; 2; . . .; n2 ; k 5 1; 2; . . .; n3 is called the environment threat state in the subarea Dk : From the abovementioned definition the maximum number of the environment threat states for the subarea Dk ; k 5 1; 2; . . .; n3 is equal to       υk 5 l1ðkÞ 1 1 ; l2ðkÞ 1 1 ; . . .; lnðkÞ2 1 1 ; k 5 1; 2; . . .; n3 : Further, we number the subarea environment threat states defined by (3.19) and (3.20) and mark them by sυðkÞ for υ 5 1; 2; . . .; υk ; and form the set

k 5 1; 2; . . .; n3 ;

n o SðkÞ 5 sυðkÞ ; υ 5 1; 2; . . .; υk ;

k 5 1; 2; . . .; n3 ;

where siðkÞ 6¼ sjðkÞ

for i 6¼ j;

i; jAf1; 2; . . .; υk g:

The set SðkÞ ; k 5 1; 2; . . .; n3 is called the set of the environment threat states in the subarea Dk ; k 5 1; 2; . . .; n3 ; while a number υk is called the number of the environment threat states of this subarea. A function SðkÞ ðtÞ; k 5 1; 2; . . .; n3 ; defined on the time interval tAh0; 1 NÞ and having values in the environment threat states set, SðkÞ ; k 5 1; 2; . . .; n3 ; is called the process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 : Next, to involve the process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 with the process of initiating events, we introduced the function Sðk=lÞ ðtÞ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω;

(3.21)

defined on the time interval tAh0; 1 NÞ; depending on the states el ; l 5 1; 2; . . .; ω of the process of initiating events EðtÞ and taking its values in the set of the environment threat states SðkÞ ; k 5 1; 2; . . .; n3 : This function is called the conditional process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 ; while the process of initiating events EðtÞ is at the state el ; l 5 1; 2; . . .; ω: We assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process Sðk=lÞ ðtÞ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω and denote by ηijðk=lÞ its random conditional sojourn times at the state siðk=lÞ , while its next transition will be done to the state sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω: The process of environment threats is defined by G

the vector of probabilities

Modeling critical infrastructure accident consequences Chapter | 3

27

  piðk=lÞ ð0Þ 5 P Sðk=lÞ ð0Þ 5 siðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; of its initial states at the moment t 5 0 h i piðk=lÞ ð0Þ G

1 3 υk

h i k 5 p1ðk=lÞ ð0Þ; p2ðk=lÞ ð0Þ; . . .; pυðk=lÞ ð0Þ ;

(3.22)

the matrix of probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; :::; n3 ; l 5 1; 2; :::; ω; of transitions between the states siðk=lÞ and sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; :::; ω; 2 3 k p11 p12 : : : p1υ ðk=lÞ ðk=lÞ ðk=lÞ 6 21 2υk 7 22 6 7 ½pijðk=lÞ υk 3 υk 5 6 pðk=lÞ pðk=lÞ : : : pðk=lÞ 7; 4 : : : : : : 5 k1 k2 k υk pυðk=lÞ pυðk=lÞ : : : pυðk=lÞ

(3.23)

where piiðk=lÞ 5 0 for i 5 1; 2; . . .; υk ; G

the matrix of conditional distribution functions

  ij ðtÞ 5 P ηijðk=lÞ , t ; tAh0; 1 NÞ; Hðk=lÞ

i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; of conditional sojourn times ηijðk=lÞ of the process Sðk=lÞ ðtÞ at the state siðk=lÞ , the state sjðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; 2 11 12 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : : 6 21 22 6 ij ½Hðk=lÞ ðtÞυk 3 υk 5 6 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : : 4 : : : : : υk 1 υk 2 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : :

while the next transition will be done to 1υk Hðk=lÞ ðtÞ

3

7 2υk Hðk=lÞ ðtÞ 7; 7 5 : υk υk Hðk=lÞ ðtÞ

(3.24)

where ii Hðk=lÞ ðtÞ 5 0

for i 5 1; 2; . . .; υk ;

or equivalently by corresponding to it the matrix of conditional density functions of the conditional sojourn times ηijðk=lÞ of the process Sðk=lÞ ðtÞ at the state siðk=lÞ , while the next transition will be done to the state sjðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; 2 3 1υk 12 h11 ðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ 6 21 7 2υk 22 6 7 ½hijðk=lÞ ðtÞυk 3 υk 5 6 hðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ 7; (3.25) 4 : : : : : : 5 υk 1 υk 2 υk υk hðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ where hiiðk=lÞ ðtÞ 5 0 for i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω: We assume that the appropriate and typical distributions suitable to describe the process Sðk=lÞ ðtÞ conditional sojourn times ηijðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the particular states are (Kołowrocki and Soszy´nska-Budny, 2011)

28

Consequences of Maritime Critical Infrastructure Accidents

´ FIGURE 3.4 The graphs of the double trapezium distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling  Identification  Prediction  Optimization. Springer, London, Dordrecht, Heidelberg, New York. G

the double trapezium distribution with a density function (Fig. 3.4) 8 > 0; > >     2 3 > > ij ij ij ij ij ij > > 2 2 q z 2 x y 2 z 2 w t 2 xijðk=lÞ > ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ > ij ij 4 5 > q 1 2 qðk=lÞ U ij ; > > < ðk=lÞ zðk=lÞ 2 xijðk=lÞ yijðk=lÞ 2 xijðk=lÞ ij     2 3 hðk=lÞ ðtÞ 5 > > 2 2 qijðk=lÞ zijðk=lÞ 2 xijðk=lÞ 2 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ yijðk=lÞ 2 t > ij ij 5 > 4 > U w 1 2 w ; > ðk=lÞ ðk=lÞ > > yijðk=lÞ 2 zijðk=lÞ yijðk=lÞ 2 xijðk=lÞ > > > > : 0; where

G

t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ (3.26) zijðk=lÞ # t , yijðk=lÞ t $ yijðk=lÞ ;

0 # xijðk=lÞ # zijðk=lÞ # yijðk=lÞ , 1 N; 0 # qijðk=lÞ , 1 N; 0 # wijðk=lÞ , 1 N;     0 # qijðk=lÞ zijðk=lÞ 2 xijðk=lÞ 1 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ # 2;

the quasitrapezium distribution with a density function (Fig. 3.5) 8 0; > > > > >  Aij 2 qijðk=lÞ  > > > qij 1 ðk=lÞ > t 2 xijðk=lÞ ; > ij ij ðk=lÞ > zðk=lÞ1 2 xðk=lÞ > > < ij ij hðk=lÞ ðtÞ 5 Aðk=lÞ ; > > >  Aijðk=lÞ 2 wijðk=lÞ  ij > > ij > > w 1 y 2 t ; > ðk=lÞ > yijðk=lÞ 2 zijðk=lÞ2 ðk=lÞ > > > > : 0;

t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ1 zijðk=lÞ1 # t , zijðk=lÞ2 zijðk=lÞ2 # t , yijðk=lÞ t $ yijðk=lÞ ;

(3.27)

Modeling critical infrastructure accident consequences Chapter | 3

29

´ FIGURE 3.5 The graphs of the quasitrapezium distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling  Identification  Prediction  Optimization. Springer, London, Dordrecht, Heidelberg, New York.

where Aijðk=lÞ 5

    2 2 qijðk=lÞ zijðk=lÞ1 2 xijðk=lÞ 2 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ2 zijðk=lÞ2 2 zijðk=lÞ1 1 yijðk=lÞ 2 xijðk=lÞ

;

0 # xijðk=lÞ # zijðk=lÞ1 # zijðk=lÞ2 # yijðk=lÞ , 1 N; 0 # qijðk=lÞ , 1 N; 0 # wijðk=lÞ , 1 N; G

the exponential distribution with a density function (the graph is similar to the one presented in Fig. 3.1 than we omit it)

30

Consequences of Maritime Critical Infrastructure Accidents

8 < 0;

hijðk=lÞ ðtÞ 5 :

h



αijðk=lÞ exp 2αijðk=lÞ t 2 xijðk=lÞ

i

t , xijðk=lÞ ;

t $ xijðk=lÞ ;

(3.28)

where 0 # αijðk=lÞ , 1 N; G

the chimney distribution with a density function (the graphs omit it) 8 > 0; > > > > > Aijðk=lÞ > > > ; > > > zijðk=lÞ1 2 xijðk=lÞ > > > > ij > < Cðk=lÞ ij ; hðk=lÞ ðtÞ 5 > zijðk=lÞ2 2 zijðk=lÞ1 > > > > > > Dijðk=lÞ > > > ; > > > yijðk=lÞ 2 zijðk=lÞ2 > > > > : 0;

are similar to the ones presented in Fig. 3.2 than we t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ1 zijðk=lÞ1 # t , zijðk=lÞ2

(3.29)

zijðk=lÞ2 # t , yijðk=lÞ t $ yijðk=lÞ ;

where 0 # xijðk=lÞ # zijðk=lÞ1 # zijðk=lÞ2 # yijðk=lÞ , 1 N; ij ij Aijðk=lÞ $ 0; Cðk=lÞ $ 0; Dijðk=lÞ $ 0; Aijðk=lÞ 1 Cðk=lÞ 1 Dijðk=lÞ 5 1:

As the mean values of the conditional sojourn times ηijðk=lÞ are given by (Kołowrocki and Soszy´nska-Budny, 2011) ðN h i ðN ij ij Mðk=lÞ 5 E ηijðk=lÞ 5 tdHðk=lÞ ðtÞ 5 thijðk=lÞ ðtÞdt; (3.30) 0 0 i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; then for the distinguished distributions (3.26)(3.29), the mean values of the process of environment threat Sðk=lÞ ðtÞ conditional sojourn times ηijðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G

for the double trapezium distribution ij Mðk=lÞ

5E

h

ηijðk=lÞ 

1

i

5

wijðk=lÞ yijðk=lÞ

2

xijðk=lÞ 1 yijðk=lÞ 1 zijðk=lÞ 3  2 2 qijðk=lÞ xijðk=lÞ 2

 3  xijðk=lÞ yijðk=lÞ xijðk=lÞ 1 yijðk=lÞ wijðk=lÞ 1 qijðk=lÞ  ij ij ij ij 4 x z 5 1 ðk=lÞ ðk=lÞ 2 yðk=lÞ zðk=lÞ 1 6 xijðk=lÞ 2 yijðk=lÞ  3  3 xijðk=lÞ qijðk=lÞ 1 yijðk=lÞ wijðk=lÞ   2 ; 3 yijðk=lÞ 2 xijðk=lÞ 2

(3.31)

Modeling critical infrastructure accident consequences Chapter | 3

G

for the quasitrapezium distribution h i qij  2  2

ðk=lÞ ij zijðk=lÞ1 2 xijðk=lÞ Mðk=lÞ 5 E ηijðk=lÞ 5 2  ij ij 2  2

Aðk=lÞ 2 qðk=lÞ 2 2 zijðk=lÞ1 2 5xijðk=lÞ zijðk=lÞ1 2 xijðk=lÞ 6  ij 2  2 wij  2  2

Aðk=lÞ ðk=lÞ ij ij ij ij 1 zðk=lÞ2 2 zðk=lÞ1 yðk=lÞ 2 zðk=lÞ2 1 2 2  2

wijðk=lÞ 2 Aijðk=lÞ  ij 2 ij ij ij 2 2 zðk=lÞ2 2 5yðk=lÞ zðk=lÞ2 2 yðk=lÞ ; 6

G

(3.32)

for the exponential distribution h i ij Mðk=lÞ 5 E ηijðk=lÞ 5 xijðk=lÞ 1

G

31

1 αijðk=lÞ

;

(3.33)

for the chimney distribution h i 1h      i ij ij Mðk=lÞ 5 E ηijðk=lÞ 5 Aijðk=lÞ xijðk=lÞ 1 zijðk=lÞ1 1 Cðk=lÞ zijðk=lÞ1 1 zijðk=lÞ2 1 Dijðk=lÞ zijðk=lÞ2 1 yijðk=lÞ : 2

(3.34)

From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times ηiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats Sðk=lÞ ðtÞ at the states siðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω are determined by (Kołowrocki and Soszy´nska-Budny, 2011) i Hðk=lÞ ðtÞ 5

υk X j51

ij pijðk=lÞ Hðk=lÞ ðtÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω:

(3.35)

Therefore the mean values E½ηiðk=lÞ  of the process of environment threats Sðk=lÞ ðtÞ unconditional sojourn times ηiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the states are given by υk  X  ij i Mðk=lÞ 5 E ηiðk=lÞ 5 pijðk=lÞ Mðk=lÞ ; j51

(3.36)

i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; ij is defined by formula (3.30) in the case of any distribution of sojourn times ηijðk=lÞ and by formulae (3.31) where Mðk=lÞ (3.34) in the cases of particular distributions of these sojourn times defined, respectively, by (3.26)(3.29). The limit values of the process of environment threats Sðk=lÞ ðtÞ transient probabilities at the particular states piðk=lÞ ðtÞ 5 P Sðk=lÞ ðtÞ 5 siðk=lÞ ; (3.37) tAh0; 1 NÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω;

are given by (Kołowrocki and Soszy´nska-Budny, 2011) piðk=lÞ 5 lim piðk=lÞ ðtÞ 5 t- 1 N

i πiðk=lÞ Mðk=lÞ ; υk X j j πðk=lÞ Mðk=lÞ

(3.38)

j51

i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; i Mðk=lÞ

where equations

is given by (3.36), while the steady probabilities πiðk=lÞ of the vector ½πiðk=lÞ 1 3 υk satisfy the system of

32

Consequences of Maritime Critical Infrastructure Accidents

8 i ij i > < ½πðk=lÞ  5 ½πðk=lÞ ½pðk=lÞ  υk X > πjðk=lÞ 5 1: :

(3.39)

j51

½pijðk=lÞ υk 3 υk

and is given by (3.23). The asymptotic distribution of the sojourn total time η^ iðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ in the time interval h0; ηi; η . 0 at the state siðk=lÞ is normal with the expected value   i M^ ðk=lÞ 5 E η^ iðk=lÞ Dpiðk=lÞ η; (3.40) where piðk=lÞ is given by (3.38). Thus according to the formula for total probability and (3.14) and (3.37), the probabilities piðkÞ ðtÞ 5 PðSðtÞ 5 siðkÞ Þ;

tAh0; 1 NÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ;

(3.41)

are defined by piðkÞ ðtÞ 5

ω ω X

X P EðtÞ 5 el UP SðkÞ ðtÞ 5 siðkÞ EðtÞ 5 el 5 pl ðtÞUpiðk=lÞ ðtÞ; l51

i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ;

l51

and according to (3.15) and (3.38), their limit forms are piðkÞ 5

3.3

ω X l51

pl Upiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 :

(3.42)

Modeling process of environment degradation

The particular states of the process of the environment threats SðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 may lead to dangerous effects degrading the environment at this subarea. Thus we assume that there are mk different dangerous degradation effects for the environment subarea Dk ; k 5 1; 2; . . .; n3 , and we mark them by k R1ðkÞ ; R2ðkÞ ; . . .; Rm ðkÞ :

This way, the set

n o k RðkÞ 5 R1ðkÞ ; R2ðkÞ ; . . .; Rm ðkÞ ; k 5 1; 2; . . .; n3 ;

is the set of degradation effects for the environment in the subarea Dk : These degradation effects may attain different levels. Namely, the degradation effect Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; may reach ν m ðkÞ levels mν m

ðkÞ m2 Rm1 ðkÞ ; RðkÞ ; . . .; RðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ;

that are called the states of this degradation effect. The set n o mν m ðkÞ m1 m2 Rm ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; ðkÞ 5 RðkÞ ; RðkÞ ; . . .; RðkÞ is called the set of states of the degradation effect Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 for the environment in the subarea Dk ; k 5 1; 2; . . .; n3 : Under the abovementioned assumptions, we can introduce the environment subarea degradation process as a vector h i k RðkÞ ðtÞ 5 R1ðkÞ ðtÞ; R2ðkÞ ðtÞ; . . .; Rm (3.43) ðkÞ ðtÞ ;

Modeling critical infrastructure accident consequences Chapter | 3

33

where Rm ðkÞ ðtÞ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; are the processes of degradation effects for the environment in the subarea Dk ; k 5 1; 2; . . .; n3 defined on the time interval tAh0; 1 NÞ and having their values in the degradation effect state sets

The vector

Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 :

(3.44)

h i mk m 1 2 5 dðkÞ ; dðkÞ ; . . .; dðkÞ rðkÞ ; k 5 1; 2; . . .; n3 ;

(3.45)

where 8 0; if a degradation effect Rm ðkÞ does not appear in the subarea Dk ; > > < m dðkÞ 5 Rmj ; if a degradation effect Rm appears in the subarea D k > ðkÞ ðkÞ > : mj m and its level is equal to RðkÞ ; j 5 1; 2; . . .; ν ðkÞ ;

(3.46)

for m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 is called the degradation state in the subarea Dk ; k 5 1; 2; . . .; n3 : From the abovementioned definition the maximum number of the environment degradation states for the subarea Dk ; k 5 1; 2; . . .; n3 is equal to       k ‘k 5 ν 1ðkÞ 1 1 ; ν 2ðkÞ 1 1 ; . . .; ν m ðkÞ 1 1 ; k 5 1; 2; . . .; n3 : Further, we number the subarea degradation states defined by (3.45) and (3.46) and mark them by ‘ rðkÞ

for ‘ 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;

and form the set of degradation states

n o ‘ ; ‘ 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; RðkÞ 5 rðkÞ

where j i rðkÞ 6¼ rðkÞ

for i 6¼ j; i; jAf1; 2; . . .; ‘k g:

The set RðkÞ ; k 5 1; 2; . . .; n3 is called the set of the environment degradation states in the subarea Dk ; k 5 1; 2; . . .; n3 ; while a number ‘k is called the number of the environment degradation states of this subarea. A function RðkÞ ðtÞ; k 5 1; 2; . . .; n3 ; defined on the time interval tAh0; 1 NÞ and having values in the environment degradation states set RðkÞ ; k 5 1; 2; . . .; n3 ; is called the process of the environment degradation in the subarea Dk ; k 5 1; 2; . . .; n3 : Next, to involve the process of environment degradation with the process of the environment threats, we define the conditional process of environment degradation Rðk=υÞ ðtÞ, while the process of the environment threats SðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 is at the state sυðkÞ ; υ 5 1; 2; . . .; υk as a vector h i k ðtÞ ; (3.47) Rðk=υÞ ðtÞ 5 R1ðk=υÞ ðtÞ; R2ðk=υÞ ðtÞ; . . .; Rm ðk=υÞ where Rm ðk=υÞ ðtÞ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; defined on the time interval tAh0; 1 NÞ and having values in the degradation effect states set Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 :

34

Consequences of Maritime Critical Infrastructure Accidents

The abovementioned definition means that the conditional process of environment degradation Rðk=υÞ ðtÞ; tAh0; 1 NÞ also takes the degradation states from the set RðkÞ of the unconditional process of environment degradation RðkÞ ðtÞ; tAh0; 1 NÞ defined by (3.43). We assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the conditional process of environment degradation in the subarea Dk Rðk=υÞ ðtÞ; tAh0; 1 NÞ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; i and denote by ζ ijðk=υÞ its random conditional sojourn times in the state rðk=υÞ , while its next transition will be done to the j state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk : The process of environment degradation is defined by G

the vector of probabilities

  i qiðk=υÞ ð0Þ 5 P Rðk=υÞ ð0Þ 5 rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;

of its initial states at the moment t 5 0 h i qiðk=υÞ ð0Þ G

1 3 ‘k

h i k 5 q1ðk=υÞ ð0Þ; q2ðk=υÞ ð0Þ; . . .; q‘ðk=υÞ ð0Þ ;

(3.48)

the matrix of probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; j i of transitions between the states rðk=υÞ and rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; 2 11 3 k qðk=υÞ q12 : : : q1‘ ðk=υÞ ðk=υÞ 6 21 h i 2‘k 7 22 6 7 qijðk=υÞ 5 6 qðk=υÞ qðk=υÞ : : : qðk=υÞ 7; 4 : ‘k 3 ‘k : : : : : 5 ‘k 1 ‘k 2 ‘k ‘k qðk=υÞ qðk=υÞ : : : qðk=υÞ

(3.49)

where qiiðk=υÞ 5 0 G

for i 5 1; 2; . . .; ‘k ;

the matrix of conditional distribution functions

  Gijðk=υÞ ðtÞ 5 P ζ ijðk=υÞ , t ; tAh0; 1 NÞ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;

i , while its next transition will of conditional sojourn times ζ ijðk=υÞ of the degradation process Rðk=υÞ ðtÞ in the state rðk=υÞ j be done to the state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; 2 3 1‘k 12 G11 ðk=υÞ ðtÞ Gðk=υÞ ðtÞ : : : Gðk=υÞ ðtÞ 6 21 7 h i 2‘k 22 6 7 Gijðk=υÞ ðtÞ 5 6 Gðk=υÞ ðtÞ Gðk=υÞ ðtÞ : : : Gðk=υÞ ðtÞ 7; (3.50) 4 5 ‘k 3 ‘k : : : : : : ‘k 1 k2 k ‘k Gðk=υÞ ðtÞ G‘ðk=υÞ ðtÞ : : : G‘ðk=υÞ ðtÞ

where Giiðk=υÞ ðtÞ 5 0

for i 5 1; 2; . . .; ‘k ;

or equivalently by corresponding to it the matrix of conditional density functions of conditional sojourn times ζ ijðk=υÞ j i of the process Rðk=υÞ ðtÞ at the state rðk=υÞ ; while its next transition will be done to the state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;

Modeling critical infrastructure accident consequences Chapter | 3

2 h

i gijðk=υÞ ðtÞ

12 g11 ðk=υÞ ðtÞ gðk=υÞ ðtÞ :

:

6 21 22 6 5 6 gðk=υÞ ðtÞ gðk=υÞ ðtÞ : 4 ‘k 3 ‘k : : : ‘k 1 ‘k 2 gðk=υÞ ðtÞ gðk=υÞ ðtÞ :

: : :

k : g1‘ ðk=υÞ ðtÞ

35

3

7 k 7 : g2‘ ðk=υÞ ðtÞ 7; 5 : : k ‘k : g‘ðk=υÞ ðtÞ

(3.51)

where giiðk=υÞ ðtÞ 5 0

G

for i 5 1; 2; . . .; ‘k :

We assume that the appropriate and typical distributions suitable to describe the process Rðk=υÞ ðtÞ conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the particular states are similar to distributions describing the process of initiating events EðtÞ that graphs are presented in Section 3.1 in Figs. 3.1 and 3.2 than we omit them and in Kołowrocki and Soszy´nska-Budny (2011). There are two types of distribution with a density function: the exponential distribution with a density function 8 < 0; t , xijðk=υÞ ij h  i gðk=υÞ ðtÞ 5 (3.52) ij ij : αij exp 2αij t 2 x ; ; t $ x ðk=υÞ ðk=υÞ ðk=υÞ ðk=υÞ where 0 # αijðk=υÞ , 1 N;

G

the chimney distribution with a density function 8 > 0; > > > > > Aijðk=υÞ > > > ; > > > zijðk=υÞ1 2 xijðk=υÞ > > > > ij > < Cðk=υÞ ij ; gðk=υÞ ðtÞ 5 > zijðk=υÞ2 2 zijðk=υÞ1 > > > > > > Dijðk=υÞ > > > ; > > > yijðk=υÞ 2 zijðk=υÞ2 > > > > : 0;

t , xijðk=υÞ xijðk=υÞ # t , zijðk=υÞ1 zijðk=υÞ1 # t , zijðk=υÞ2

(3.53)

zijðk=υÞ2 # t , yijðk=υÞ t $ yijðk=υÞ ;

where 0 # xijðk=υÞ # zijðk=υÞ1 # zijðk=υÞ2 # yijðk=υÞ , 1 N; ij ij Aijðk=υÞ $ 0; Cðk=υÞ $ 0; Dijðk=υÞ $ 0; Aijðk=υÞ 1 Cðk=υÞ 1 Dijðk=υÞ 5 1:

As the mean values of the conditional sojourn times ζ ijðk=υÞ are given by (Kołowrocki and Soszy´nska-Budny, 2011) h i Ð ÐN N ij 5 E ζ ijðk=υÞ 5 0 tdGijðk=υÞ ðtÞ 5 0 tgijðk=υÞ ðtÞdt; Mðk=υÞ (3.54) i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; then for the distinguished distributions (3.52) and (3.53), the mean values of the process of environment degradation Rðk=υÞ ðtÞ conditional sojourn times ζ ijðk=υÞ i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G

for the exponential distribution h i ij Mðk=υÞ 5 E ζ ijðk=υÞ 5 xijðk=υÞ 1

1 αijðk=υÞ

;

(3.55)

36

G

Consequences of Maritime Critical Infrastructure Accidents

for the chimney distribution h i 1h      i ij ij Mðk=υÞ 5 E ζ ijðk=υÞ 5 Aijðk=υÞ xijðk=υÞ 1 zijðk=υÞ1 1 Cðk=υÞ zijðk=υÞ1 1 zijðk=υÞ2 1 Dijðk=υÞ zijðk=υÞ2 1 yijðk=υÞ : 2

(3.56)

From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times ζ iðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation Rðk=υÞ ðtÞ at the states i rðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk are determined by (Kołowrocki and Soszy´nska-Budny, 2011) Giðk=υÞ ðtÞ 5

‘k X qijðk=υÞ Gijðk=υÞ ðtÞ;

(3.57)

j51

i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk : E½ζ iðk=υÞ 

of the process of environment degradation Rðk=υÞ ðtÞ unconditional sojourn times Therefore the mean values ζ iðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the states are given by ‘k h i X ij i 5 E ζ iðk=υÞ 5 qijðk=υÞ Mðk=υÞ ; Mðk=υÞ

(3.58)

j51

i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; ij is defined by the formula (3.54) in a case of any distribution of sojourn times ζ ijðk=υÞ and by formulae where Mðk=υÞ (3.55) and (3.56) in the cases of particular distributions of these sojourn times defined, respectively, by (3.52) and (3.53). The limit values of the process of environment degradation Rðk=υÞ ðtÞ transient probabilities at the particular states   i qiðk=υÞ ðtÞ 5 P Rðk=υÞ ðtÞ 5 rðk=υÞ ; (3.59) tAh0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;

are given by (Kołowrocki and Soszy´nska-Budny, 2011) qiðk=υÞ 5 lim qiðk=υÞ ðtÞ 5 t- 1 N

i πiðk=υÞ Mðk=υÞ ‘k X j πjðk=υÞ Mðk=υÞ

; (3.60)

j51

i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;

h i i where Mðk=υÞ is given by (3.58), while the steady probabilities πiðk=υÞ of the vector πiðk=υÞ satisfy the system of 1 3 ‘k equations i h ih i 8h ij i i > π 5 π q > ðk=υÞ ðk=υÞ < ðk=υÞ > > :

‘k X j51

πjðk=υÞ 5 1

(3.61)

  and qijðk=υÞ ‘k 3 ‘k is given by (3.49). i The asymptotic distribution of the sojourn total time ζ^ ðk=υÞ of the process of environment degradation Rðk=υÞ ðtÞ in the i time interval h0; ζi; ζ . 0; at the state rðk=υÞ is normal with the expected value h i i i M^ ðk=υÞ 5 E ζ^ ðk=υÞ Dqiðk=υÞ ζ; (3.62) where qiðk=υÞ is given by (3.60). Thus according to the formula for total probability and (3.41) and (3.59), the probabilities,   i qiðkÞ ðtÞ 5 P RðtÞ 5 rðkÞ ; tAh0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;

Modeling critical infrastructure accident consequences Chapter | 3

37

are defined by qiðkÞ ðtÞ 5

υk υk

 X    X i P SðtÞ 5 sυðkÞ UP RðkÞ ðtÞ 5 rðkÞ pυðkÞ ðtÞUqiðk=υÞ ðtÞ;

SðtÞ 5 sυðkÞ 5 υ51

i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 :

υ51

Therefore according to (3.42) and (3.60), for sufficiently large t, the limit transient probabilities of the unconditional process of the environment degradation RðkÞ ðtÞ at its particular states are given by " # υk υk X ω X X υ i l υ p Upðk=lÞ qiðk=υÞ qiðkÞ D pðkÞ Uqðk=υÞ 5 (3.63) υ51 υ51 l51 i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; where pl ; pυðk=lÞ , and qiðk=υÞ are defined, respectively, by (3.15), (3.38), and (3.60). i Therefore the sojourn total time ζ^ ðkÞ of the unconditional process of the environment degradation RðkÞ ðtÞ i k 5 1; 2; . . .; n3 ; in the time interval h0; θi; θ . 0; at the state rðkÞ has the normal distribution with the expected value h i i i M^ ðkÞ 5 E ζ^ ðkÞ DqiðkÞ θ; i 5 1; 2; . . .; ‘k ; (3.64) where qiðkÞ is given by (3.63).

Chapter 4

Identification of critical infrastructure accident consequences Chapter Outline 4.1 Identification of process of initiating events—theoretical background 40 4.1.1 Estimating unknown basic parameters of process of initiating events 40 4.1.2 Estimating parameters of distributions of process of initiating events conditional sojourn times at initiating events states 41 4.1.3 Identification of distribution functions of process of initiating events conditional sojourn times at initiating events states 44 4.2 Identification of process of initiating events—application to Baltic Sea and world sea waters 46 4.2.1 States of the process of initiating events 46 4.2.2 Statistical identification of process of initiating events at Baltic Sea waters 46 4.2.3 Extension of statistical identification of process of initiating events for world sea waters 56 4.3 Identification of process of environment threats—theoretical background 59 4.3.1 Estimating unknown basic parameters of process of environment threats 59 4.3.2 Estimating parameters of distributions of process of environment threats conditional sojourn times at environment threats states 61 4.3.3 Identification of distribution functions of process of environment threats conditional sojourn times at environment threats states 65

4.4 Identification of process of environment threats—application to Baltic Sea and world sea waters 66 4.4.1 States of process of environment threats 67 4.4.2 Statistical identification of process of environment threats at Baltic Sea waters 68 4.4.3 Extension of statistical identification of process of environment threats for world sea waters 72 4.5 Identification of process of environment degradation— theoretical background 76 4.5.1 Estimating unknown basic parameters of process of environment degradation 77 4.5.2 Estimating parameters of distributions of process of environment degradation conditional sojourn times at environment degradation states 78 4.5.3 Identification of distribution functions of process of environment degradation conditional sojourn times at environment degradation states 82 4.6 Identification of process of environment degradation— application to Baltic Sea and world sea waters 83 4.6.1 States of process of environment degradation 84 4.6.2 Statistical identification of process of environment degradation at Baltic Sea waters 85 4.6.3 Extension of statistical identification of process of environment degradation for world sea waters 89

The general model of critical-infrastructure-accident consequences including the models of the process of initiating events generated either by the critical infrastructure accident or by its loss of safety critical level, the process of environment threats and the process of environment degradation is proposed in Chapter 3, Modeling critical infrastructure accident consequences. Now, the methods of identification of unknown parameters of the models of the process of initiating events, of environment threats, and of environment degradation are proposed. There are the methods and procedures for estimating the unknown basic parameters of these processes models and identifying the distributions of their conditional sojourn times at their changing in time states. Particularly, for each process there are given the formulae estimating probabilities of staying at particular states at the initial moment, the probabilities of transitions between states, the parameters and forms of distributions fixed for the description of conditional sojourn times at states, and mean values of conditional sojourn times at states having these distribution. The states for considering particular processes are introduced, and next, the mentioned methods and procedures are applied to the identification of the particular processes that are parts of the general model of critical infrastructure accident consequences at the Baltic Sea and world sea waters.

Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00004-8 © 2020 Elsevier Inc. All rights reserved.

39

40

Consequences of Maritime Critical Infrastructure Accidents

4.1

Identification of process of initiating events—theoretical background

We assume, as in Section 3.1, that the process of initiating events is taking ω, ωAN different initiating events states e1 ; e2 ; . . .; eω : Next, we mark by E(t), tA , 0; 1 NÞ the process of initiating events, that is a function of a continuous variable t, taking discrete values in the set fe1 ; e2 ; . . .; eω g of the initiating events states. We assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process of initiating events E(t), and we mark by θlj its random conditional sojourn times at the states el when its next state is ej, l; j 5 1; 2; . . .; ω; l 6¼ j: Under these assumption, the process of initiating events may be described by the vector [pl(0)]1 3 ω of probabilities of the process of initiating events staying at the particular initiating events states at the initial moment t 5 0, the matrix [plj(t)]ω 3 ω of probabilities of transitions between initiating events states and the matrix ½H lj ðtÞω 3 ω of distribution functions of conditional sojourn times θlj of the process E(t) at its particular states or equivalently by the matrix ½hlj ðtÞω 3 ω of density functions of conditional sojourn times θlj, l,j 5 1,2,. . .,ω, l 6¼ j of the process of initiating events at its particular states. All these parameters of the process of initiating events are unknown, and before their use to the prediction of this process, characteristics have to be estimated on the basis of statistical data coming from practice.

4.1.1 Estimating unknown basic parameters of process of initiating events In order to estimate parameters of the process of initiating events E(t), first we should fix the number of states ω of the process E(t) and define the states e1 ; e2 ; . . .; eω of the set E. Further, we should fix the vector of realizations nl ð0Þ; l 5 1,2,. . .,ω of the numbers of the process E(t) transients at the particular states el at the initial moment t 5 0 ½nl ð0Þ1 3 ω 5 ½n1 ð0Þ; n2 ð0Þ; . . .; nω ð0Þ;

(4.1)

where nð0Þ 5

ω X

nl ð0Þ:

(4.2)

l51

Next, we should fix the matrix of realizations nlj ; l,j 5 1,2,. . .,ω of numbers of the process E(t) transitions from the state el into the state ej during the experimental time 2 11 3 n n12 : : : n1ω 6 n21 n22 : : : n2ω 7 7; ½nlj ω 3 ω 5 6 (4.3) 4 : : : : : : 5 nω1 nω2 : : : nωω where nll 5 0 for l 5 1; 2; . . .; ω; and the vector of realizations of total numbers nl ; l 5 1,2,. . .,ω of the process of initiating events departures from the state el ½nl 1 3 ω 5 ½n1 ; n2 ; . . .; nω ;

(4.4)

where nl 5

ω X

nlj ;

l 5 1; 2; . . .; ω:

(4.5)

j6¼l

Having these numbers, we estimate the vector ½pl ð0Þ1 3 ω 5 ½p1 ð0Þ; p2 ð0Þ; . . .; pω ð0Þ;

(4.6)

of realizations of initial probabilities pl ð0Þ; l 5 1,2,. . .,ω of the process E(t) transients at the particular states el at the initial moment t 5 0 according to the formula

Identification of critical infrastructure accident consequences Chapter | 4

pl ð0Þ 5

nl ð0Þ ; nð0Þ

l 5 1; 2; . . .; ω;

where n(0) is the total number of the process E(t) realizations at the initial moment t 5 0 given by (4.2). Next, we evaluate the matrix 2 11 3 p p12 : : : p1ω 6 p21 p22 : : : p2ω 7 7; ½plj ω 3 ω 5 6 4 : : : : : : 5 pω1 pω2 : : : pωω

41

(4.7)

(4.8)

of realizations of transitions probabilities plj ; l,j 5 1,2,. . .,ω of the process E(t) from the state el into the state ej during the experimental time, according to the formula plj 5

nlj ; nl

l; j 5 1; 2; . . .; ω;

l 6¼ j;

(4.9)

and pll 5 0;

l 5 1; 2; . . .; ω;

l

where n , l 5 1,2,. . .,ω is the realization of the total number of the process E(t) departures from the state el during the experimental time, given by (4.5). Further, we formulate and verify the hypotheses about the form of distribution functions of the process E(t) conditional sojourn times θlj ; l,j 5 1,2,. . .,ω, l 6¼ j at the state el , while the next transition is to the state ej on the basis of their realizations θljγ ; γ 5 1; 2; . . .; nlj :

4.1.2 Estimating parameters of distributions of process of initiating events conditional sojourn times at initiating events states Prior to estimating the parameters of the distributions of the conditional sojourn times of the process of initiating events at particular states, we have to determine the following empirical characteristics of the realizations of the conditional sojourn time of the process of initiating events at particular states (Kołowrocki and Soszy´nska-Budny, 2011): G

lj

the realizations of the empirical mean values θ of the conditional sojourn times θlj ; l,j 5 1,2,. . .,ω, l 6¼ j at the state el ; while the next transition is to the state ej on the base of their realizations θljγ ; γ 5 1; 2; . . .; nlj according to the formula nlj 1X θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j; (4.10) nlj γ51 γ   the number r lj of the disjoint intervals Iz 5 aljz ; bljz ; z 5 1; 2; . . .; r lj ; that include the realizations θljγ ; γ 5 1; 2; . . .; nlj of the conditional sojourn times θlj ; l,j 5 1,2,. . .,ω, l 6¼ j at the state el ; while the next transition is to the state ej according to the formula pffiffiffiffiffi r lj D nlj ; (4.11)  lj lj  lj lj the length d of the intervals Iz 5 az ; bz ; z 5 1; 2; . . .; r according to the formula lj

θ 5

G

G

lj

dlj 5

R ; lj r 21

(4.12)

where lj

R 5 max θljγ 2 1 # γ # nlj

G

min θljγ ;

1 # γ # nlj

  the ends aljz ; bljz ; z 5 1; 2; . . .; r lj of the intervals Iz 5 aljz ; bljz ; z 5 1; 2; . . .; r lj according to the formulae

(4.13)

42

Consequences of Maritime Critical Infrastructure Accidents

alj1 5 max



(4.14)

z 5 1; 2; . . .; r lj ;

(4.15)

1 # γ # nlj

bljz 5 alj1 1 zd lj ; aljz 5 bljz21 ; in such a way that

 dlj ; 0 ; 2

min θljγ 2

z 5 2; 3; . . .; r lj ;

(4.16)

D  I1 , I2 , ? , Irlj 5 alj1 ; bljrlj

and i; z 5 1; 2; . . .; r lj ;   the numbers nljz of the realizations θljγ ; γ 5 1; 2; . . .; nlj in the intervals Iz 5 aljz ; bljz ; z 5 1; 2; . . .; r lj according to the formula Ii - Iz 5 [ for all i 6¼ z;

G

nljz 5 #fγ:θljγ AIz ; γAf1; 2; . . .; nlj gg;

z 5 1; 2; . . .; r lj ;

(4.17)

where lj

r X z51

G

whereas the symbol # means the number of elements of the set. To estimate the parameters of distributions of conditional sojourn times of the process of the initiating events at the particular states distinguished in Section 3.1, we proceed, respectively, in the following way (Kołowrocki and Soszy´nska-Budny, 2011): for the exponential distribution with the density function given by (3.7), the estimates of the unknown parameters are xlj 5 alj1 ;

G

nljz 5 nlj ;

αlj 5

1 lj

θ 2 xlj

;

(4.18)

for the chimney distribution with the density function given by (3.8), the estimates of the unknown parameters are xlj 5 alj1 ; ylj 5 xlj 1 r lj d lj ;

(4.19)

and, moreover, if _lj

n 5 maxf nljz g

(4.20)

1 # z # rlj

and i, iAf1; 2; . . .; r lj g, is the number of the interval, including the largest number of realizations, such that _lj

nlji 5 n ;

(4.21)

then for i 5 1 either zlj1 5 xlj 1 ði 2 1Þd lj ; Alj 5 0; Clj 5

nlji ; Dlj 5 nlj

zlj2 5 xlj 1 id lj ; nlji11

1 ? 1 nljrlj nlj

(4.22) ;

(4.23)

Identification of critical infrastructure accident consequences Chapter | 4

43

while nlji11 5 0 or nlji11 6¼ 0

and

nlji

nlji11

$ 3;

(4.24)

or zlj1 5 xlj 1 ði 2 1Þdlj ; zlj2 5 xlj 1 ði 1 1Þdlj ; Alj 5 0; C lj 5

(4.25)

nlji12 1 ? 1 nljrlj nlji 1 nlji11 lj ; D 5 ; nlj nlj

(4.26)

while nlji11 6¼ 0

and

nlji

nlji11

, 3;

(4.27)

for i 5 2; 3; . . .; r lj 2 1 either zlj1 5 xlj 1 ði 2 1Þd lj ; zlj2 5 xlj 1 id lj ; Alj 5

(4.28)

nlji11 1 ? 1 nljrlj nlj1 1 ? 1 nlji21 nlji lj lj ; C 5 ; D 5 ; nlj nlj nlj

(4.29)

while nlji21 5 0 or nlji21 6¼ 0

and

nlji11 5 0 or nlji11 6¼ 0

and

nlji

nlji21

$3

(4.30)

$ 3;

(4.31)

and while nlji

nlji11

or zlj1 5 xlj 1 ði 2 1Þdlj ; zlj2 5 xlj 1 ði 1 1Þdlj ; Alj 5

nlj1 1 ? 1 nlji21 nlj 1 nlj ; Clj 5 i lj i11 ; Dlj 5 lj n n

nlji12

1 ? 1 nljrlj nlj

(4.32) ;

(4.33)

while nlji21 5 0 or nlji21 6¼ 0

and

nlji

nlji21

$3

(4.34)

and while nlji11 6¼ 0

and

nlji

nlji11

, 3;

(4.35)

or zlj1 5 xlj 1 ði 2 2Þd lj ; zlj2 5 xlj 1 id lj ; Alj 5

nlj1 1 ? 1 nlji22 nlji21 1 nlji lj ; C 5 ; Dlj 5 nlj nlj

nlji11

(4.36) 1 ? 1 nljrlj nlj

;

(4.37)

44

Consequences of Maritime Critical Infrastructure Accidents

while nlji21 6¼ 0

and

nlji

nlji21

,3

(4.38)

and while nlji11 5 0 or nlji11 6¼ 0

and

nlji

nlji11

$ 3;

(4.39)

or zlj1 5 xlj 1 ði 2 2Þdlj ; zlj2 5 xlj 1 ði 1 1Þdlj ; Alj 5

nlj1 1 ? 1 nlji22 nlji21 1 nlji 1 nlji11 lj ; C 5 ; Dlj 5 nlj nlj

nlji12

(4.40) 1 ? 1 nljrlj nlj

;

(4.41)

while nlji21 6¼ 0 and

nlji

nlji21

,3

(4.42)

, 3;

(4.43)

and while nlji11 6¼ 0 and

nlji

nlji11

for i 5 r lj either zlj1 5 xlj 1 ði 2 1Þd lj ; zlj2 5 xlj 1 id lj ; Alj 5

nlj1 1 ? 1 nlji21 nlj ; Clj 5 ilj ; Dlj 5 0; lj n n

(4.44) (4.45)

while nlji21 5 0 or nlji21 6¼ 0 and

nlji

nlji21

$ 3;

(4.46)

or zlj1 5 xlj 1 ði 2 2Þd lj ; zlj2 5 xlj 1 id lj ; Alj 5

nlj1 1 ? 1 nlji22 nlji21 1 nlji lj ; C 5 ; Dlj 5 0; nlj nlj

(4.47) (4.48)

while nlji21 6¼ 0 and

nlji

nlji21

, 3:

(4.49)

4.1.3 Identification of distribution functions of process of initiating events conditional sojourn times at initiating events states To formulate and then to verify the nonparametric hypothesis concerning the form of the distribution of the process of initiating events conditional sojourn time θlj at the state el when the next transition is to the state ej on the basis of its realizations θljγ ; γ 5 1; 2; . . .; nlj ; it is due to proceed according to the following scheme (Kołowrocki and Soszy´nska-Budny, 2011): G

to construct and to plot the realization of the histogram of the process of initiating events conditional sojourn time θlj at the state el (Fig. 4.1), defined by the following formula:

Identification of critical infrastructure accident consequences Chapter | 4

45

FIGURE 4.1 The graph of the realization of the histogram of the process of initiating events conditional sojourn time θlj at the state el.

nlj

h ðtÞ 5

nljz for tAI z ; nlj

(4.50)

lj

G

n

to analyze the realization of the histogram h ðtÞ; comparing it with the graphs of the density functions hlj ðtÞ distinguished in Section 3.1 distributions, to select one of them and to formulate the null hypothesis H0 concerning the unknown form of the distribution of the conditional sojourn time θlj in the following form: H0: The process of initiating events conditional sojourn time θlj at the state el while the next transition is to the state ej has the distribution with the density function hlj ðtÞ;

G

G G

G G

to join each of the intervals Iz that has the number nljz of realizations less than 4 either with the neighbor interval Iz11 or with the neighbor interval Iz21 in a way that the numbers of realizations in all intervals are not less than 4; lj to fix a new number of intervals r ; to determine new intervals D  lj lj I z 5 aljz ; bz ; z 5 1; 2; . . .; r ; lj

to fix the numbers nljz of realizations in new intervals I z ; z 5 1; 2; . . .; r ; to calculate the hypothetical probabilities that the variable θlj takes values from the interval I z under the assumption that the hypothesis H0 is true, that is, the probabilities lj

lj

pz 5 Pðθlj AI z Þ 5 Pðaljz # θlj , bz Þ 5 H lj ðbz Þ 2 H lj ðaljz Þ;

lj

z 5 1; 2; . . .; r ;

(4.51)

lj

G

where H lj ðbz Þ and H lj ðaljz Þ are the values of the distribution function H lj ðtÞ of the random variable θlj corresponding to the density function hlj ðtÞ assumed in the null hypothesis H0; lj to calculate the realization of the χ2 Pearson’s statistics U n according to the formula lj

nlj

u 5

r X ðnlj 2nlj pz Þ2 z

z51 G G

G

nlj pz

;

(4.52)

to assume the significance level α (for instance α 5 0:05) of the test; lj to fix the number r 2 j 2 1 of degrees of freedom, substituting for j, distinguished in Section 3.1 distributions, respectively, the following values: j 5 0 for the chimney distributions and j 5 1 for the exponential distribution; to read from the table of the χ2-Pearson’s distribution (Appendix 7), the value uα for the fixed values of the signifilj cance level α and the number of degrees of freedom r 2 j 2 1 such that the following equality holds lj

PðU n . uα Þ 5 α;

(4.53)

and next to determine the acceptance domain in the form of the interval h0; uα i; and the critical domain in the form of the interval ðuα ; 1 NÞ, as presented in Fig. 4.2; and G

lj

lj

to compare the obtained value un of the realization of the statistics U n with the read from the table critical value uα of the chi-square random variable and to decide on the previously formulated null hypothesis H0 in the following way: lj lj if the value un does not belong to the critical domain, that is, when un # uα , then we do not reject the hypothesis H0; lj lj otherwise, if the value un belongs to the critical domain, that is, when un . uα , we reject the hypothesis H0.

46

Consequences of Maritime Critical Infrastructure Accidents

FIGURE 4.2 The graphical interpretation of the critical interval and the acceptance interval for the chi-square goodness-of-fit test. Based on ´ Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling  Identification  Prediction  Optimization. Springer, London, Dordrecht, Heildeberg, New York.

4.2 Identification of process of initiating events—application to Baltic Sea and world sea waters The statistical identification of the process of initiating events is performed on the basis of the data coming from freeaccessible reports of chemical accidents at sea that happened around the world in a period of 11 years (CEDRE, 2018; GISIS, 2018). The identification of the process of initiating events for the world sea waters is sufficiently accurate as the set of statistical data is most populated. Unfortunately, the less-accurate identification of the process of initiating events is performed for the Baltic Sea waters as the less sufficiently numerous set of statistical data.

4.2.1 States of the process of initiating events Taking into account the kinds of initiating events defined in Section 2.2.1 and considering (3.1) and (3.2), we distinguish ω 5 16 following the states of the process of the initiating events E(t): e1 5 ½0; 0; 0; 0; 0; 0; 0; e2 5 ½1; 0; 0; 0; 0; 0; 0; e3 5 ½0; 1; 0; 0; 0; 0; 0; e4 5 ½0; 0; 1; 0; 0; 0; 0; e5 5 ½0; 0; 0; 1; 0; 0; 0; e6 5 ½0; 0; 0; 0; 1; 0; 0; e7 5 ½0; 0; 0; 0; 0; 1; 0; e8 5 ½0; 0; 0; 0; 0; 0; 1; e9 5 ½0; 1; 0; 1; 0; 0; 0; e10 5 ½0; 0; 0; 1; 1; 0; 0; e11 5 ½0; 0; 0; 0; 1; 1; 0; e12 5 ½0; 0; 0; 1; 0; 0; 1; e13 5 ½0; 0; 0; 0; 0; 1; 1; e14 5 ½0; 0; 0; 1; 1; 1; 0; e15 5 ½0; 0; 0; 0; 1; 1; 1; e16 5 ½0; 0; 0; 1; 0; 1; 0: The states of the process of the initiating events are defined as the vector where 1 means the initiating event Ei, i 5 1; 2; . . .; 6 occurs and 0 means any initiating event Ei, i 5 1; 2; . . .; 6 does not occur. For instance, the vector G G G

e1 5 ½0; 0; 0; 0; 0; 0; 0 means no initiating event dangerous for the environment takes place, e3 5 ½0; 1; 0; 0; 0; 0; 0 means the initiating event E2 (grounding) occurred, e9 5 ½0; 1; 0; 1; 0; 0; 0 means the initiating event E2 (grounding) and initiating event E4 (fire) occurred simultaneously.

4.2.2 Statistical identification of process of initiating events at Baltic Sea waters The number ω 5 16 of the process of initiating events states is fixed, and the initiating events states el ; l 5 1; 2; . . .; 16 are defined in Section 4.2.1. Moreover, it is assumed that there are possible transitions between all states of the process of initiating events.

4.2.2.1 Collected statistical data of process of initiating events The experiment was performed within the Baltic Sea waters in the years 200414. The number of the observed ship accidents that generated the distinguished states of the process of initiating events is nð0Þ 5 104: The initial moment t 5 0 of the process of initiating event for each ship is fixed at the moment when the ship was launched and began its operation process. The unknown parameters of the process of initiating events semi-Markov model are G

the initial probabilities pl ð0Þ; l 5 1; 2; . . .; 16 of the process of initiating events staying at the particular states el at the initial moment t 5 0;

Identification of critical infrastructure accident consequences Chapter | 4

G

G

47

the probabilities plj ; l; j 5 1; 2; . . .; 16 of the process of initiating events transitions from the state el into the state ej ; and the distributions of the initiating events conditional sojourn times θlj ; l; j 5 1; 2; . . .; 16; l 6¼ j at the particular states and their mean values M lj 5 E½θlj ; l; j 5 1; 2; . . .; 16; l 6¼ j:

To identify all these parameters of the process of initiating events, the statistical data about this process is needed. The statistical data for the conditional sojourn times θlj at the states of the process of initiating events are given in Appendix 1. The collected statistical data necessary for evaluating the initial transient probabilities of process of initiating events at the particular states are G

the process of initiating events observation/experiment time Θ 5 11 years ð2004  14Þ;

G

the vector of realizations nl ð0Þ; l 5 1; 2; . . .; 16 n1 ð0Þ 5 104; n2 ð0Þ 5 0; n3 ð0Þ 5 0; n4 ð0Þ 5 0; n5 ð0Þ 5 0; n6 ð0Þ 5 0; n7 ð0Þ 5 0; n8 ð0Þ 5 0; n9 ð0Þ 5 0; n10 ð0Þ 5 0; n11 ð0Þ 5 0; n12 ð0Þ 5 0; n13 ð0Þ 5 0; n14 ð0Þ 5 0; n15 ð0Þ 5 0; n16 ð0Þ 5 0; of the number of the process of initiating events staying at the particular states el at the initial moment t 5 0 according to (4.1) ½nl ð0Þ1 3 16 5 ½104; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; and

G

the number of the process of initiating events realizations at the initial moment t 5 0 according to (4.2) nð0Þ 5

16 X

nl ð0Þ 5 104 1 0 1 ? 1 0 1 0 5 104:

l51

The collected statistical data necessary for evaluating the probabilities of transitions between initiating events states are the matrix of realizations nlj ; l; j 5 1; 2; . . .; 16 of the numbers of the process E(t) transitions from the state el into the state ej ; according to (4.3), during the experimental time. The numbers of transitions that are not equal to 0 are presented, and they are as follows (Appendix 1, C-column of the table):

G

n12 5 18; n13 5 37; n14 5 5; n15 5 10; n16 5 24; n17 5 9; n18 5 1; n21 5 15; n23 5 2; n26 5 1; n27 5 2; n31 5 65; n41 5 5; n47 5 1; n51 5 9; n5 10 5 1; n61 5 1; n62 5 2; n63 5 21; n64 5 1; n71 5 6; n73 5 5; n7 13 5 1; n81 5 1; n10 1 5 1; n13 1 5 1; G

the vector of realizations of the total numbers of the process E(t) transitions from the state el ; l 5 1; 2; . . .; 16; according to (4.4), during the experimental time ½nl 1 3 16 5 ½104; 20; 65; 6; 10; 25; 12; 1; 0; 1; 0; 0; 1; 0; 0; 0:

4.2.2.2 Evaluating basic parameters of process of initiating events On the basis of the statistical data obtained in Section 4.2.2.1, using the formulae given in Section 4.1.1, it is possible to evaluate the following unknown basic parameters of the process of initiating events:

48

G

Consequences of Maritime Critical Infrastructure Accidents

the vector ½pl ð0Þ1 3 16 5 ½1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; l

G

(4.54) l

of the initial probabilities p ð0Þ; l 5 1; 2; . . .; 16 of the process of initiating events at the particular states e at the initial moment t 5 0 according to (4.6); the matrix ½plj 16 3 16 ; l; j 5 1; 2; . . .; 16 of the probabilities of transitions of the process E(t) from the state el into the state ej ; according to (4.8), during the experimental time. The probabilities of transitions that are not equal to 0 are as follows: p12 5 0:1731; p13 5 0:3558; p14 5 0:0481; p15 5 0:0961; p16 5 0:2308; p17 5 0:0865; p18 5 0:0096; p21 5 0:7500; p23 5 0:1000; p26 5 0:0500; p27 5 0:1000; p31 5 1; p41 5 0:8333; p47 5 0:1667; p51 5 0:9000; p5 10 5 0:1000;

(4.55)

p 5 0:0400; p 5 0:0800; p 5 0:8400; p 5 0:0400; 61

62

63

64

p71 5 0:5000; p73 5 0:4167; p7 13 5 0:0833; p81 5 1; p10 1 5 1; p13 1 5 1: The values of some probabilities existing in the vector (4.54) and in the matrix (4.55) besides these standing on the main diagonal, being equal to zero do not mean that the events they are concerned with cannot appear. They evaluated on the basis of real statistical data and their values may change and become more precise if the duration of the experiment is longer.

4.2.2.3 Evaluating parameters of distributions of conditional sojourn times at initiating events states On the basis of the statistical data presented in Appendix 1, using the procedure and the formulae given in Section 4.1.2, it is possible to determine the empirical parameters of the conditional sojourn times θlj of the process of initiating events at its particular states. To illustrate the application of this procedure and these formulae, we perform it for θ13 ; θ31 ; that is, for the conditional sojourn times having sufficiently numerous set of their realizations listed in Appendix 1. The results for the conditional sojourn time θ13 are G

13

the realization θ of the mean value of the conditional sojourn time θ13 of the process of initiating events state e1 when the next transition is to the state e3 according to (4.10) 37 1 X θ13 5 8; 942; 302:7; 37 γ51 γ   13 13 the number r 13 of the disjoint intervals Iz 5 a13 that include the realizations θ13 z ; bz ; z 5 1; 2; . . .; r γ ; 13 1 γ 5 1; 2; . . .; 37 of the conditional sojourn time θ at the state e when the next transition is to the state e3 ; according to (4.11) pffiffiffiffiffi r 13 D 37D6;   13 the length d13 of the intervals Iz 5 a13 z ; bz ; z 5 1; 2; . . .; 6 defined by (4.12) that after considering (4.13) 13

θ 5

G

G

13

R 5 max θ13 γ 2 1 # γ # 37

min θ13 γ 5 21; 549; 600 2 262; 800 5 21; 286; 800;

1 # γ # 37

is 13

d 13 5

R 21; 286; 800 5 4; 257; 360; 5 13 5 r 21

Identification of critical infrastructure accident consequences Chapter | 4

G

49

 13 13  13 the ends a13 z ; bz of the intervals Iz 5 az ; bz ; z 5 1; 2; . . .; 6 defined by (4.14)(4.16) that after considering min θ13 γ 2

1 # γ # 37

d13 4; 257; 360 5 2 1; 865; 880 5 262; 800 2 2 2

are 13 13 a13 1 5 maxf 2 1; 865; 880; 0g 5 0; b1 5 a1 1 4; 257; 360 5 0 1 4; 257; 360 5 4; 257; 360; 13 a13 2 5 b1 5 4; 257; 360;

13 b13 2 5 a1 1 2U4; 257; 360 5 0 1 8; 514; 720 5 8; 514; 720;

13 a13 3 5 b2 5 8; 514; 720;

13 b13 3 5 a1 1 3U4; 257; 360 5 0 1 12; 772; 080 5 12; 772; 080;

13 a13 4 5 b3 5 12; 772; 080;

13 b13 4 5 a1 1 4U4; 257; 360 5 0 1 17; 029; 440 5 17; 029; 440;

13 a13 5 5 b4 5 17; 029; 440;

13 b13 5 5 a1 1 5U4; 257; 360 5 0 1 21; 286; 800 5 21; 286; 800;

13 a13 6 5 b5 5 21; 286; 800; G

13 the number n13 z of the realizations θγ

13 b13 6 5 a1 1 6U4; 257; 360 5 0 1 25; 544; 160 5 25; 544; 160;   13 in particular intervals Iz 5 a13 z ; bz ; z 5 1; 2; . . .; 6 defined by (4.17)

13 13 13 13 13 n13 1 5 11; n2 5 9; n3 5 7; n4 5 6; n5 5 3; n6 5 1:

G

The results for the conditional sojourn time θ31 are 31 the realization θ of the mean value of the conditional sojourn time θ31 at the state e3 when the next transition is to 1 the state e according to (4.10) 65 1 X θ31 5 1321:15; 65 γ51 γ   31 31 the number r 31 of the disjoint intervals Iz 5 a31 that include the realizations θ31 γ ; z ; bz ; z 5 1; 2; . . .; r 3 31 γ 5 1; 2; . . .; 65 of the conditional sojourn time θ at the state e when the next transition is to the state e1 according to (4.11) pffiffiffiffiffi r 31 D 65D8;   31 the length d31 of the intervals Iz 5 a31 z ; bz ; z 5 1; 2; . . .; 8 defined by (4.12) that after considering (4.13) 31

θ 5

G

G

31

R 5 max θ31 γ 2 1 # γ # 65

min θ31 γ 5 11; 520 2 10 5 11; 510;

1 # γ # 65

is 31

R 11; 510 D1644:29; 5 7 r 31 2 1   31 of the intervals Iz 5 a31 z ; bz ; z 5 1; 2; . . .; 8 defined by (4.14)(4.16) that after considering d 31 5

G

31 the ends a31 z ; bz

min θ31 γ 2

1 # γ # 65

d 31 1644:29 5 2 812:145 5 10 2 2 2

are a31 1 5 maxf 2 812:145; 0g 5 0; 31 a31 2 5 b1 5 1644:29; 31 a3 5 b31 2 5 3288:58; 31 a31 5 b 4 3 5 4932:87; 31 a5 5 b31 4 5 6577:16; 31 a31 5 b 6 5 5 8221:45; 31 a7 5 b31 6 5 9865:74; 31 a31 5 b 8 7 5 11; 510:03; G

31 the number n31 z of the realizations θγ

31 b31 1 5 a1 1 1644:29 5 0 1 1644:29 5 1644:29; 31 31 b2 5 a1 1 2U1644:29 5 0 1 3288:58 5 3288:58; 31 b31 3 5 a1 1 3U1644:29 5 0 1 4932:87 5 4932:87; 31 b4 5 a31 1 1 4U1644:29 5 0 1 6577:16 5 6577:16; 31 b31 5 a 1 1 5U1644:29 5 0 1 8221:45 5 8221:45; 5 31 b6 5 a31 1 1 6U1644:29 5 0 1 9865:74 5 9865:74; 31 b31 5 a 7 1 1 7U1644:29 5 0 1 11; 510:03 5 11; 510:03; 31 31 b8 5 a1 1 8U1644:29 5 0 1 13; 154:32 5 13; 154:32;   31 in particular intervals Iz 5 a31 z ; bz ; z 5 1; 2; . . .; 8 defined by (4.17)

50

Consequences of Maritime Critical Infrastructure Accidents

31 31 31 31 31 31 31 n31 1 5 54; n2 5 4; n3 5 2; n4 5 4; n5 5 0; n6 5 0; n7 5 0; n8 5 1:

4.2.2.4 Identification of distribution functions of conditional sojourn times at initiating event states Using the procedure given in Section 4.1.3, the statistical data from Appendix 1, and the results from Section 4.2.2.3, we may verify the hypotheses on the distributions of the process of initiating events conditional sojourn times θlj ; l; j 5 1; 2; . . .; 16; l 6¼ j at the particular states. To do this, we need a sufficient number of realizations of these variables, namely, the sets of their realizations should contain at least 3040 realizations coming from the experiment. To make the procedure familiar to the reader, we perform it for the conditional sojourn time θ13 and θ31 having the sufficiently numerous set of realizations and preliminarily analyzed in Section 4.2.2.3. 13 The realization h ðtÞ of the histogram of the process of initiating events conditional sojourn time θ13 defined by (4.50) is presented in Table 4.1 and illustrated in Fig. 4.3. 13 After analyzing and comparing the realization h ðtÞ of the histogram with the graphs of the density functions h13 ðtÞ of the previously distinguished in Section 3.1 distributions, we formulate the null hypothesis H0 in the following form: H0: The process of initiating events conditional sojourn time θ13 at the state e1 when the next transition to the state 3 e has the exponential distribution with the density function of the form  0; t , x13 13 h ðtÞ 5 (4.56) 13 13 13 α exp½ 2 α ðt 2 x Þ; t $ x13 : We estimate the unknown parameters of the density function of the hypothetical exponential distribution using (4.18), and then we obtain the following results: 1

13 x13 5 a13 1 5 0; α 5

13

θ 2 x13

1 5 0:000000112: 8; 942; 302:7 2 0

5

(4.57)

Substituting the above results into (4.56), the hypothetical density function is defined in the following form: TABLE 4.1 The realization of the histogram of the process of initiating events conditional sojourn time θ13. Histogram of the conditional sojourn time θ13   13 Iz 5 a13 z ; bz 13 nz 13

h ðtÞ 5 nz13 =n13

04,257,360 11 11/37

4,257,3608,514,720 9 9/37

8,514,72012,772,080 7 7/37

12,772,08017,029,440 6 6/37

17,029,44021,286,800 3 3/37

21,286,80025,544,150 1 1/37

FIGURE 4.3 The graph of the histogram of the process of initiating events conditional sojourn time θ13.

h31(t) 1 0.8 0.6 0.4 0.2 0 60

4 51

0–

2 77

9 02

1

6 28

1

4 54

5,

–2

0 44

0 80

,

,

29

0 7,

50

,1

1,

–2

0 08

,

72

7 2,

00

,8

7,

–1

0 72

,

14

5 8,

40

,4

2,

–1

6 ,3

7

25

4,

80

,0

8,

4,

0–

20

,7

,3

7 25

86

2

2 1,

Identification of critical infrastructure accident consequences Chapter | 4

51



0; t,0 ½ ; 0:000000112exp 2 0:000000112ðt 2 0Þ t$0  0; t,0 5 0:000000112exp½ 2 0:000000112t; t $ 0:

h13 ðtÞ 5

(4.58)

Hence, the hypothetical distribution function H 13 ðtÞ of the conditional sojourn time θ13 after taking the integral of the hypothetical density function h13 ðtÞ given by (4.58) takes the following form:  ðt 0; t,0 13 13 H ðtÞ 5 h ðtÞdt 5 (4.59) 1 2 expð2 0:000000112tÞ; t $ 0: 0 Next, we join the intervals defined in the realization of the histogram h13 ðtÞ that have the numbers n13 z of realizations less than 4 into new intervals, and we perform the following steps: G

we fix the new number of intervals 13

r 5 5; G

we determine the new intervals I 1 5 h0; 4; 257; 360Þ; I 2 5 h4; 257; 360; 8; 514; 720Þ; I 3 5 h8; 514; 720; 12; 772; 080Þ; I 4 5 h12; 772; 080; 17; 029; 440Þ; I 5 5 h17; 029; 440; 1 NÞ;

G

we fix the numbers of realizations in the new intervals 13 13 13 13 n13 1 5 11; n2 5 9; n3 5 7; n4 5 6; n5 5 4;

G

we calculate, according to (4.51), the hypothetical probabilities that the variable θ13 takes values from the new intervals

p1 5 Pðθ13 AI 1 Þ 5 Pð0 # θ13 , 4; 257; 360Þ 5 H 13 ð4; 257; 360Þ 2 H 13 ð0ÞD0:379 2 0 5 0:379; p2 5 Pðθ13 AI 2 Þ 5 Pð4; 257; 360 # θ13 , 8; 514; 720Þ 5 H 13 ð8; 514; 720Þ 2 H 13 ð4; 257; 360ÞD0:615 2 0:379 5 0:236; p3 5 Pðθ13 AI 3 Þ 5 Pð8; 514; 720 # θ13 , 12; 772; 080Þ 5 H 13 ð12; 772; 080Þ 2 H 13 ð8; 514; 720ÞD0:761 2 0:615 5 0:146; p4 5 Pðθ13 AI 4 Þ 5 Pð12; 772; 080 # θ13 , 17; 029; 440Þ 5 H 13 ð17; 029; 440Þ 2 H 13 ð12; 772; 080ÞD0:852 2 0:761 5 0:091; p5 5 Pðθ13 AI 5 Þ 5 Pð17; 029; 440 # θ13 , 1 NÞ 5 1 2 H 13 ð17; 029; 440ÞD1 2 0:852 5 0:148; G

we calculate, using (4.52), the realization of the χ2-Pearson’s statistics u13 5

5 X ðn13 2n13 pz Þ2 z

z51

n13 pz

5

ð11237U0:379Þ2 ð9237U0:236Þ2 1 37U0:379 37U0:236

ð7237U0:146Þ2 ð6237U0:091Þ2 ð4237U0:148Þ2 1 1 37U0:146 37U0:091 37U0:148 D0:652 1 0:008 1 0:473 1 2:059 1 0:398 5 3:59; 1

G G

we assume the significance level α 5 0:05; we fix the number of degrees of freedom 13

r 2 j 2 1 5 5 2 1 2 1 5 3; G

we read from the table of the χ2-Pearson’s distribution (Appendix 7) the value uα for the fixed values of the signifi13 cance level α 5 0:05 and the number of degrees of freedom r 2 j 2 1 5 3; such that, according to (4.53), the following equality holds PðU 13 . uα Þ 5 α 5 0:05 that amounts to uα 5 7:81, and we determine the acceptance domain in the form of the interval h0; 7:81i; and the critical domain in the form of the interval ð7:81; 1 NÞ as presented in Fig. 4.4;

52

Consequences of Maritime Critical Infrastructure Accidents

FIGURE 4.4 The graphical interpretation of the critical and the acceptance intervals for the chi-square goodness-of-fit test.

TABLE 4.2 The realization of the histogram of the process initiating events conditional sojourn time θ31. Histogram of the conditional sojourn time θ31   31 Iz 5 a31 z ; bz nz31 31 h ðtÞ5 nz31 =n31 31 Iz 5 a31 z ; bz 31 nz 31 h ðtÞ 5 nz31 =n31

G

01644.29 54 54/65 6577.168221.45 0 0/65

1644.293288.58 4 4/65 8221.459865.74 0 0/65

3288.584932.87 2 2/65 9865.7411,510.03 0 0/65

4932.876577.16 4 4/65 11,510.0313,154.32 1 1/65

we compare the obtained value u13 5 3:59 of the realization of the statistics U13 with the read from the table critical value uα 5 7:81 of the chi-square random variable, and since the value u13 5 3.59 does not belong to the critical domain, that is, u13 5 3:59 # uα 5 7:81;

then we do not reject the hypothesis H0 that the conditional sojourn time θ13 has the exponential distribution with the density function given by (4.58). Thus after applying formulae (3.9) and (3.10), it is possible to find the mean value of the conditional sojourn time θ13:  1 1 D8; 928; 571:43: M 13 5 E θ13 5 x13 1 13 5 0 1 α 0:000000112 31

(4.60)

The realization h ðtÞ of the histogram of the process of initiating events conditional sojourn time θ31 defined by (4.50) is presented in Table 4.2 and illustrated in Fig. 4.5. 31 After analyzing and comparing the realization h ðtÞ of the histogram with the graphs of the density functions h31 ðtÞ of the previously distinguished in Section 3.1 distributions, we formulate the null hypothesis H0 in the following form: H0: The process of initiating events conditional sojourn time θ31 at the state e3 when the next transition to the state 1 e has the chimney distribution with the density function of the form 8 0; t , x31 > > > 31 > A > > ; x31 # t , z31 > 1 31 > 31 > z 2 x > 1 > > > < C31 31 z31 (4.61) h31 ðtÞ 5 z31 2 z31 ; 1 # t , z2 > 2 1 > > > > D31 > 31 > > ; z31 > 2 #t,y 31 31 > y 2 z > 2 > > : 0; t $ y31 :

Identification of critical infrastructure accident consequences Chapter | 4

53

FIGURE 4.5 The graph of the histogram of the process of initiating events conditional sojourn time θ31.

h31(t) 1 0.8 0.6 0.4 0.2 0 9

.2

44

8

0

1

32

2 –3

9

.2

4 64

7

.8

.5

88

6 –1

8

.5

3

8 28

6

21

5 –6

7

.8

2 93

4–

1 3,

1

3–

7 5.

6

98

2

.3

54

5 1,

1

5

.4

1 22

3

.0

10

8 –9

8

6

4

.7

65

2 –8

6

.1

7 57

4

5

.4

.1

77

9 –4

1

,5

11

0 0.

Since, according to (4.20) and (4.21), we have _31

_31

31 n 5 maxf n31 z g 5 54 and n1 5 n 5 54; 1#z#8

then i 5 1 is the number of interval including the largest number of realizations. Moreover, by (4.24), we get n31 2 5 4; and

n31 54 1 . 3: 5 4 n31 2

Therefore we estimate the unknown parameters of the density function of the hypothetical chimney distribution using formulae (4.19) and (4.22)(4.24), and we obtain the following results: 31 31 31 31 x31 5 a31 1 5 0; y 5 x 1 r d 5 0 1 8U1644:29 5 13; 154:32;

z31 1

5 x 1 ði 2 1Þd 5 0 1 ð1 2 1ÞU1644:29 5 0; 31

31

31 31 z31 2 5 x 1 id 5 0 1 1U1644:29 5 1644:29;

A31 5 0; C 31 5 D31 5

n31 54 1 ; 5 65 n31

31 31 n31 4 1 2 1 4 1 0 1 0 1 0 1 1 11 2 1 n3 1 ? 1 n 8 : 5 5 31 65 65 n

(4.62) (4.63) (4.64) (4.65) (4.66)

Substituting the above results into (4.61), the hypothetical density function is completely defined in the form: 8 0; t,0 > > 8 > 54=65 > > 0; t,0 > > ; 0 # t , 1644:29 > > < 1644:29 2 0 < 0:000505245; 0 # t , 1644:29 31 (4.67) 5 h ðtÞ 5 11=65 0:000014703; 1644:29 # t , 13; 154:32 > > > > ; 1644:29 # t , 13; 154:32 > : > 0; t $ 13; 154:32: > 13; 154:32 2 1644:29 > > : 0; t $ 13; 154:32 Hence, the hypothetical distribution function H 31 ðtÞ of the conditional sojourn time θ31 after taking the integral of the hypothetical density function h31 ðtÞ given by (4.67) takes the following form 8 0; t,0 > > ðt < 0:000505245t; 0 # t , 1644:29 31 31 (4.68) H ðtÞ 5 h ðtÞdt 5 0:000014703t 1 0:806593305; 1644:29 # t , 13; 154:32 > 0 > : 1; t $ 13; 154:32:

54

Consequences of Maritime Critical Infrastructure Accidents

Next, we join the intervals defined in the realization of the histogram h31 ðtÞ that have the numbers n31 z of realizations less than 4 into new intervals, and we perform the following steps: G

we fix the new number of intervals 31

r 5 3; G

we determine the new intervals I 1 5 h0; 1644:29Þ; I 2 5 h1644:29; 4932:87Þ; I 3 5 h4932:87; 13; 154:32Þ;

G

we fix the numbers of realizations in the new intervals 31 31 n31 1 5 54; n2 5 6; n3 5 5;

G

we calculate, according to (4.51), the hypothetical probabilities that the variable θ31 takes values from the new intervals p1 5 Pðθ31 AI 1 Þ 5 Pð0 # θ31 , 1644:29Þ 5 H 31 ð1644:29Þ 2 H 31 ð0ÞD0:831 2 0 5 0:831; p2 5 Pðθ31 AI 2 Þ 5 Pð1644:29 # θ31 , 4932:87Þ 5 H 31 ð4932:87Þ 2 H 31 ð1644:29ÞD0:879 2 0:831 5 0:048; p3 5 Pðθ31 AI 3 Þ 5 Pð4932:87 # θ31 , 13; 154:32Þ 5 H 31 ð13; 154:32Þ 2 H 31 ð4932:87ÞD1 2 0:879 5 0:121;

G

we calculate, using (4.52), the realization of the χ2-Pearson’s statistics 3 X ðn31 2n31 pz Þ2

ð54265U0:831Þ2 ð6265U0:048Þ2 ð5265U0:121Þ2 1 1 n31 pz 65U0:831 65U0:048 65U0:121 z51 D0:0000042 1 2:65846 1 1:04364D3:7;

u31 5

G G

z

5

we assume the significance level α 5 0:05; we fix the number of degrees of freedom 31

r 2 j 2 1 5 3 2 0 2 1 5 2; G

we read from the table of the χ2-Pearson’s distribution (Appendix 7) the value uα for the fixed values of the signifi31 cance level α 5 0:05 and the number of degrees of freedom r 2 j 2 1 5 2; such that, according to (4.53), the following equality holds PðU 31 . uα Þ 5 α 5 0:05 that amounts to uα 5 5:99, and we determine the acceptance domain in the form of the interval h0, 5.99i, and the critical domain in the form of the interval ð5:99; 1 NÞ as presented in Fig. 4.6;

G

we compare the obtained value u31 5 3:7 of the realization of the statistics U31 with the read from the table critical value uα 5 5:99 of the chi-square random variable, since the value u31 5 3:7 does not belong to the critical domain, that is, u31 5 3:7 # uα 5 5:99

FIGURE 4.6 The graphical interpretation of the critical and the acceptance intervals for the chi-square goodness-of-fit test.

Identification of critical infrastructure accident consequences Chapter | 4

55

then we do not reject the hypothesis H0 that the conditional sojourn time θ31 has the chimney distribution with the density function given by (4.67). Thus after applying formulae (3.9) and (3.11), it is possible to find the mean value of the conditional sojourn time θ31:  1





 31 31 31 31 M 31 5 E θ31 5 A31 x31 1 z31 z1 1 z31 z2 1 y31 1 1C 2 1D 2 1 5 ½0ð0 1 0Þ 1 0:831ð0 1 1644:29Þ 1 0:169ð1644:29 1 13; 154:32Þ 2 D1933:68:

(4.69)

For the remaining cases when as a result of the experiment, coming from experts, we have less than 30 realizations of the process of initiating events, we assume that these conditional sojourn times have the empirical density functions defined by o 1 n hlj ðtÞ 5 lj # γ:θljγ AIz ; γA 1; 2; . . .; nlj ; z 5 1; 2; . . .; r lj ; (4.70) n corresponding to the empirical distribution functions defined by o 1 n H lj ðtÞ 5 lj # γ:θljγ , t; γA 1; 2; . . .; nlj ; n

(4.71)

for t $ 0; l; jAf1; 2; . . .; ωg; l 6¼ j; whereas the symbol # means the number of elements of the set. For instance, the process initiating events conditional sojourn time θ63 assumed n63 5 21 values. The order sample realizations θ63 is 10, 10, 10, 15, 20, 20, 20, 30, 30, 30, 30, 30, 30, 30, 40, 60, 60, 90, 120, 360, 5760. Thus we assume that conditional sojourn time θ63 has the empirical density function given by 8 t , 10 > > 0; > > 3=21; 10 # t , 15 > > > > 1=21; 15 # t , 20 > > > > 3=21; 20 # t , 30 > > > > < 7=21; 30 # t , 40 (4.72) h63 ðtÞ 5 1=21; 40 # t , 60 > > 2=21; 60 # t , 90 > > > > > 1=21; 90 # t , 120 > > > > 1=21; 120 # t , 360 > > > > 1=21; 360 # t , 5760 > : 1=21; t $ 5760; and the distribution function given by

8 0; > > > > 3=21; > > > > 4=21; > > > > 7=21; > > > > < 14=21; H 63 ðtÞ 5 15=21; > > 17=21; > > > > > 18=21; > > > > 19=21; > > > > 20=21; > : 1;

t , 10 t , 15 t , 20 t , 30 t , 40 t , 60 t , 90 t , 120 t , 360 t , 5760 t , 1 N:

(4.73)

In the case when as a result of the experiment, limited data coming from experts, we only have the number of realizations of the process of initiating events lifetimes at the states, and all its realizations are equal to an approximate value, we assume that these conditional sojourn times have the uniform distribution in the interval from this value minus its half to this value plus its half.

56

Consequences of Maritime Critical Infrastructure Accidents

For instance, the process of initiating events conditional sojourn time θ21 assumed n21 5 15 values equal 1, we assume that it has the uniform density function given by 8 < 0; t , 0:5 h21 ðtÞ 5 1; 0:5 # t , 1:5 (4.74) : 0; t $ 1:5; and the distribution function given by

8 < 0; t , 0:5 H 21 ðtÞ 5 t; 0:5 # t , 1:5 : 1; t $ 1:5:

(4.75)

We can proceed with the remaining conditional sojourn times at the states of the process of initiating events in the same way and approximately fix their distribution. When the distributions cannot be identified statistically, it is possible to find the approximate empirical values of the mean values M lj 5 E½θlj  of the conditional sojourn times at particular states, using (4.10), that are as follows (Appendix 1, D-column of the table): M 12 D10; 249; 200; M 14 D12; 614; 400; M 15 D13; 402; 800; M 16 D8; 694; 300; M 17 D5; 869; 200; M 18 D1; 576; 800; M 21 5 1; M 23 5 22:50; M 26 5 1; M 27 5 5:50; M 41 5 1; M 47 5 1; M 51 D163:33; M 5 10 5 10; M 61 5 120; M 62 5 80; M 63 D324:05; M 64 5 15; M 71 D225:83; M 73 D21:60; M 7 13 5 1; M 81 5 5; M 10 1 5 10; M 13 1 5 10:

(4.76)

As there are no realizations of the process of initiating events conditional sojourn times at the remaining states, it is impossible to estimate their empirical conditional mean values. These approximate mean values given by (4.60), (4.69), and (4.76) are used in Section 5.1.1 for the prediction of the characteristics of the process of initiating events at the Baltic Sea waters.

4.2.3 Extension of statistical identification of process of initiating events for world sea waters For statistical identification of the process of initiating events at the world sea waters, we used the same kind and number of this process states that are defined in Section 4.2.1. The experiment was performed within the world sea and ocean waters in years 200414. The number of the observed ship accidents that generated the distinguished states of the process of initiating events is nð0Þ 5 1630: The initial moment t 5 0 of the process of initiating events for each ship is fixed at the moment when the ship was launched and began its operation process. The unknown parameters of the process of initiating events semi-Markov model are G

G

G

the initial probabilities pl ð0Þ; l 5 1; 2; . . .; 16 of the process of initiating events staying at the particular states el at the initial moment t 5 0; the probabilities plj ; l; j 5 1; 2; . . .; 16 of the process of initiating events transitions from the state el into the state ej; and the distributions of the initiating events conditional sojourn times θlj ; l; j 5 1; 2; . . .; 16; l 6¼ j at the particular states and their mean values M lj 5 E½θlj ; l; j 5 1; 2; . . .; 16; l 6¼ j:

We identify all these parameters on the basis of statistical data coming from the process of initiating events realization (Appendix 2). Using procedure and the formulae given in Section 4.1.1 and proceedings in an analogous way as in Section 4.2.2.2, we evaluate the following unknown basic parameters of the process of initiating events: G

the vector ½pl ð0Þ1 3 16 5 ½1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0;

(4.77) l

G

of the initial probabilities pl ð0Þ; l 5 1; 2; . . .; 16 of the process of initiating events at the particular states e at the initial moment t 5 0 according to (4.6); the matrix ½plj 16 3 16 ; l; j 5 1; 2; . . .; 16 of the probabilities of transitions of the process E(t) from the state el into the state ej, according to (4.8), during the experimental time. The probabilities of transitions that are not equal to 0 are as follows:

Identification of critical infrastructure accident consequences Chapter | 4

57

p12 5 0:2460; p13 5 0:2264; p14 5 0:0754; p15 5 0:1644; p16 5 0:1436; p17 5 0:1258; p18 5 0:0184; p21 5 0:5461; p23 5 0:2880; p24 5 0:0069; p25 5 0:0092; p26 5 0:0507; p27 5 0:0922; p2 10 5 0:0046; p2 11 5 0:0023; p31 5 0:9908; p32 5 0:0026; p36 5 0:0053; p39 5 0:0013; p41 5 0:7450; p42 5 0:0201; p43 5 0:0872; p46 5 0:0604; p47 5 0:0873; p51 5 0:7692; p59 5 0:0659; p5 10 5 0:1429; p5 12 5 0:0037; p5 16 5 0:0183; p61 5 0:4488; p62 5 0:0989; p63 5 0:3110; p64 5 0:0777; p6 10 5 0:0141; p6 11 5 0:0495; p71 5 0:4115; p73 5 0:5423; p74 5 0:0038; p78 5 0:0116; p7 11 5 0:0116; p7 13 5 0:0192; p81 5 0:8750; p8 13 5 0:1250; p91 5 1;

(4.78)

p10 1 5 0:5790; p10 6 5 0:3947; p10 14 5 0:0263; p11 1 5 0:4444; p11 3 5 0:5556; p12 1 5 1; p13 1 5 0:4445; p13 3 5 0:3333; p13 15 5 0:2222; p14 1 5 1; p15 3 5 1; p16 1 5 0:4000; p16 3 5 0:4000; p16 7 5 0:2000:

G

G

G

G

The values of some probabilities existing in the vector (4.77) and in the matrix (4.78), besides these standing on the main diagonal, being equal to zero do not mean that the events they are concerned with cannot appear. They are evaluated on the basis of real statistical data, and their values may change and become more precise if the duration of the experiment is longer. On the basis of statistical data presented in Appendix 2, using the procedure and formulae given in Section 4.1.2, it is possible to determine the empirical parameters of the conditional sojourn times θlj of the process of initiating events at its particular states. Moreover, using the procedure given in Section 4.1.3 and proceedings in an analogous way as in Section 4.2.2.4, we identify the forms of the particular density function hlj ðtÞ of the process of initiating events conditional sojourn times θlj, l; j 5 1; 2; . . .; 16; l 6¼ j having sufficiently numerous set of their realization at the particular states that are as follows: the conditional sojourn time θ18 has the exponential distribution with the density function  0; t,0 18 (4.79) h ðtÞ 5 0:0000000151 exp½ 2 0:0000000151t; t$0 the conditional sojourn time θ23 has the chimney distribution with the density function 8 0; t,0 > > < 0:011264916; 0 # t , 83:8 23 h ðtÞ 5 0:000066826; 83:8 # t , 921:8 > > : 0; t $ 921:8 the conditional sojourn time θ27 has the chimney distribution with the density function 8 0; t,0 > > < 0:076271186; 0 # t , 11:8 h27 ðtÞ 5 0:001694915; 11:8 # t , 70:8 > > : 0; t $ 70:8 the conditional sojourn time θ51 has the chimney distribution with the density function 8 0; t,0 > > < 0:000482693; 0 # t , 2024:625 h51 ðtÞ 5 0:000001403; 2024:625 # t , 18; 221:625 > > : 0; t $ 18; 221:625

(4.80)

(4.81)

(4.82)

58

G

G

G

G

G

G

G

Consequences of Maritime Critical Infrastructure Accidents

the conditional sojourn time θ5 10 has the chimney distribution with the density function 8 0; t,0 > > < 0:038784745; 0 # t , 23:8 5 10 h ðtÞ 5 0:000646412; 23:8 # t , 142:8 > > : 0; t $ 142:8 the conditional sojourn time θ61 has the exponential distribution with the density function  0; t,0 h61 ðtÞ 5 0:001242761 exp½ 2 0:001242761t; t $ 0 the conditional sojourn time θ62 has the chimney distribution with the density function 8 0; t,0 > > < 0:013724267; 0 # t , 57:25 h62 ðtÞ 5 0:000935745; 57:25 # t , 286:25 > > : 0; t $ 286:25 the conditional sojourn time θ63 has the chimney distribution with density function 8 0; t,0 > > < 0:000193913; 0 # t , 5039:75 63 h ðtÞ 5 0:0000005637; 5039:75 # t , 45; 357:75 > > : 0; t $ 45; 357:75 the conditional sojourn time θ71 has the chimney distribution with the density function 8 0; t,0 > > < 0:000322916; 0 # t , 3038:9 71 h ðtÞ 5 0:000000683; 3038:9 # t , 30; 389 > > : 0; t $ 30; 389 the conditional sojourn time θ73 has the chimney distribution with the density function 8 0; t,0 > > < 0:002366234; 0 # t , 392:64 h73 ðtÞ 5 0:000016421; 392:64 # t , 4711:68 > > : 0; t $ 4711:68

(4.83)

(4.84)

(4.85)

(4.86)

(4.87)

(4.88)

the conditional sojourn time θ81 has the chimney distribution with the density function 8 0; > > < 0:001723566; h81 ðtÞ 5 0:000033146; > > : 0;

t,0 0 # t , 538:75 538:75 # t , 2693:75 t $ 2693:75:

(4.89)

For the above distributions, by application either the general formula for the mean value or particular formulae given, respectively, by (3.9) and (3.10), (3.11), the mean values M lj 5 E½θlj ; l; j 5 1; 2; . . .; 16; l 6¼ j of the process of initiating events conditional sojourn times at particular states can be determined, and they amount to M 18 D6; 622; 516:56; M 23 D69:55; M 27 D9:44; M 51 D1073:05; M 5 10 D17:61; M 61 D804:66; M 62 D58:68; M 63 D2973:45; M 71 D1808:15; M 73 D361:23; M 81 5 365:00:

(4.90)

In the case when as a result of the experiment, coming from experts, there are not any suitable distribution presented in Section 3.1 to describe the process of initiating events or we have less than 30 realizations of the process of initiating events, we assume that these conditional sojourn times have the empirical distribution expressed by density and distribution function according to (4.70) and (4.71), respectively. In the case when as a result of the experiment, limited data coming from experts, we only have the number of realizations of the process of initiating events lifetimes at the states, and all its realizations are equal to an approximate value, we assume that these conditional sojourn times have the uniform distribution in the interval from this value minus its half to this value plus its half.

Identification of critical infrastructure accident consequences Chapter | 4

59

Thus we proceed in the analogous way as in Section 4.2.2.4 for the example conditional sojourn times θ63 and θ21 the density and distribution functions of which are expressed by (4.72)(4.75), respectively. When the distributions cannot be identified statistically, it is possible to find the approximate empirical values of the mean values M lj 5 E½θlj  of the conditional sojourn times at particular states, using (4.10), that are as follows (Appendix 2, D-column of the table): M 12 D8; 745; 800:50; M 13 D11; 095; 287:80; M 14 D9; 005; 707:32; M 15 D11; 267; 059:70; M D10; 655; 753:85; M 17 D12; 392; 622:44; M 21 5 1:00; M 24 D15:33; M 25 5 5:25; M 26 D1:86; M 2 10 5 3:00; M 2 11 5 1:00; M 31 D4500:64; M 32 5 65:00; M 36 5 6:50; M 39 5 45:00; M 41 5 1:00; M 42 D13:33; M 43 D79:31; M 46 D28:56; M 47 D4:77; M 59 D1071:67; M 5 12 5 15:00; M 5 16 5 33:00; M 64 D25:91; M 6 10 5 41:25; M 6 11 D53:36; M 74 5 5:00; M 78 5 12:00; M 7 11 5 4:00; M 7 13 5 7:40; M 8 13 5 4:25; M 91 D162:63; M 10 1 D1254:77; M 10 6 D143:67; M 10 14 5 60:00; M 11 1 5 502:50; M 11 3 5 54:00; M 12 1 5 30:00; M 13 1 5 85:00; M 13 3 D103:33; M 13 15 5 15:00; M 14 1 5 120:00; M 15 3 5 140:00; M 16 1 5 75:00; M 16 3 5 90:00; M 16 7 5 60:00: 16

(4.91)

As there are no realizations of the process of initiating events conditional sojourn times at the remaining states, it is impossible to estimate their empirical conditional mean values. These approximate mean values given by (4.90) and (4.91) are used in Section 5.1.2 for the prediction of the characteristics of the process of initiating events at the world sea waters.

4.3

Identification of process of environment threats—theoretical background

We assume, as in Section 3.2, that the process of environment threats in the subarea Dk k 5 1; 2; . . .; n3 is taking υk ; υk AN different states of environment threats s1ðkÞ ; s2ðkÞ ; . . .; sυðkÞk : Next, we mark by Sðk=lÞ ðtÞ; tA , 0; 1 NÞ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω the conditional process of environment threats in the subarea Dk, k 5 1; 2; . . .; n3 ; while the process of initiating events E(t) is at the state el, l 5 1; 2; . . .; ω: The conditional process Sðk=lÞ ðtÞ is a function defined on the time interval tAh0; 1 NÞ depending on the states of the k process of initiating events E(t) and taking discrete values in the set fs1ðk=lÞ ; s2ðk=lÞ ; . . .; sυðk=lÞ g of the environment threats states. We assume a semi-Markov model (Grabski, 2015; Kołowrocki, 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process of environment threats Sðk=lÞ ðtÞ and we mark by ηijðk=lÞ its random conditional sojourn times at the states siðk=lÞ , when its next state is sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω: Under these assumption, the process of environment threats Sðk=lÞ ðtÞ for each subarea Dk, k 5 1; 2; . . .; n3 may be described by the vector ½piðk=lÞ ð0Þ1 3 υk of initial probabilities of the process of environment threats staying at particular environment threats states at the initial moment t 5 0, the matrix ½pijðk=lÞ υk 3 υk of probabilities of transitions between ij environment threats states and the matrix ½Hðk=lÞ ðtÞυk 3 υk of the distribution functions of conditional sojourn times ηijðk=lÞ of the process Sðk=lÞ ðtÞ at its particular states or equivalently by the matrix ½hijðk=lÞ ðtÞυk 3 υk of the density functions of the conditional sojourn times ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats at its states. All these parameters of the process of environment threats are unknown, and before their use to the prediction of this process, characteristics have to be estimated on the basis of the statistical data coming from practice.

4.3.1 Estimating unknown basic parameters of process of environment threats To construct the general model of environment threats caused by the process of initiating events, at the beginning, we fix the environment subareas D1 ; D2 ; . . .; Dn3 that may be degraded by the environment threats. Next, in order to estimate parameters of the process of environment threats Sðk=lÞ ðtÞ; first we should fix the number of states υk of the process k Sðk=lÞ ðtÞ and define the states s1ðk=lÞ ; s2ðk=lÞ ; . . .; sυðk=lÞ of the set Sðk=lÞ : Further, we should fix the vector of realizations i nðk=lÞ ð0Þ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the numbers of the process Sðk=lÞ ðtÞ transients at the particular states siðk=lÞ at the initial moment t 5 0 h i h i k 5 n1ðk=lÞ ð0Þ; n2ðk=lÞ ð0Þ; . . .; nυðk=lÞ ð0Þ ; (4.92) niðk=lÞ ð0Þ 1 3 υk

where

60

Consequences of Maritime Critical Infrastructure Accidents

nðk=lÞ ð0Þ 5

υk X i51

niðk=lÞ ð0Þ:

(4.93)

Next, we should fix the matrix of realizations nijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of numbers of the process Sðk=lÞ ðtÞ transitions from the state siðk=lÞ into the state sjðk=lÞ during the experimental time 2 11 3 k : : : n1υ nðk=lÞ n12 ðk=lÞ ðk=lÞ 6 21 h i 2υk 7 22 6 7 (4.94) nijðk=lÞ 5 6 nðk=lÞ nðk=lÞ : : : nðk=lÞ 7; 4 : υk 3 υk : : : : : 5 k1 k2 k υk nυðk=lÞ nυðk=lÞ : : : nυðk=lÞ where niiðk=lÞ 5 0 for i 5 1; 2; . . .; υk ; and the vector of realizations of total numbers niðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats departures from the state siðk=lÞ k ½niðk=lÞ 1 3 υk 5 ½n1ðk=lÞ ; n2ðk=lÞ ; . . .; nυðk=lÞ ;

(4.95)

where niðk=lÞ 5

υk X j6¼i

nijðk=lÞ ;

i 5 1; 2; . . .; υk ;

k 5 1; 2; . . .; n3 ;

l 5 1; 2; . . .; ω:

Having these numbers, we estimate the vector h i h i k piðk=lÞ ð0Þ 5 p1ðk=lÞ ð0Þ; p2ðk=lÞ ð0Þ; . . .; pυðk=lÞ ð0Þ ;

(4.96)

(4.97)

1 3 υk

of realizations of initial probabilities piðk=lÞ ð0Þ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process Sðk=lÞ ðtÞ transients at the particular states siðk=lÞ at the initial moment t 5 0 according to the formula piðk=lÞ ð0Þ 5

niðk=lÞ ð0Þ nðk=lÞ ð0Þ

;

i 5 1; 2; . . .; υk ;

k 5 1; 2; . . .; n3 ;

l 5 1; 2; . . .; ω;

(4.98)

where nðk=lÞ ð0Þ is the total number of the process Sðk=lÞ ðtÞ realizations at the initial moment t 5 0 given by (4.93). Next, we evaluate the matrix 2 11 3 k pðk=lÞ p12 : : : p1υ ðk=lÞ ðk=lÞ 6 21 h i 2υk 7 22 6 7 (4.99) pijðk=lÞ 5 6 pðk=lÞ pðk=lÞ : : : pðk=lÞ 7 4 : υk 3 υk : : : : : 5 k1 k2 k υk pυðk=lÞ pυðk=lÞ : : : pυðk=lÞ of realizations of transitions probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process Sðk=lÞ ðtÞ from the state siðk=lÞ into the state sjðk=lÞ during the experimental time, according to the formula pijðk=lÞ 5

nijðk=lÞ niðk=lÞ

;

i; j 5 1; 2; . . .; υk ;

i 6¼ j;

k 5 1; 2; . . .; n3 ;

l 5 1; 2; . . .; ω;

(4.100)

and piiðk=lÞ 5 0;

i 5 1; 2; . . .; υk ;

where niðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω is the realization of the total number of the process Sðk=lÞ ðtÞ departures from the state siðk=lÞ during the experimental time, given by (4.96). Further, we formulate and verify the hypotheses about the form of distribution functions of the process Sðk=lÞ ðtÞ conditional sojourn time ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the state siðk=lÞ , while the next transition is to the state sjðk=lÞ on the basis of their realizations ηijðk=lÞγ ; γ 5 1; 2; . . .; nijðk=lÞ :

Identification of critical infrastructure accident consequences Chapter | 4

61

4.3.2 Estimating parameters of distributions of process of environment threats conditional sojourn times at environment threats states Prior to estimating the parameters of the distributions of the conditional sojourn times of the process of environment threats Sðk=lÞ ðtÞ at the particular states, we have to determine the following empirical characteristics of the realizations of the conditional sojourn time of the process of environment threats at the particular environment threats states (Kołowrocki and Soszy´nska-Budny, 2011): G

the realizations of the empirical mean values ηijðk=lÞ of the conditional sojourn times ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the state siðk=lÞ , while the next transition is to the state sjðk=lÞ on the base of their realizations ηijðk=lÞγ ; γ 5 1; 2; . . .; nijðk=lÞ according to the formula nij

ηijðk=lÞ G

5

ðk=lÞ 1 X

nijðk=lÞ γ51

ηijðk=lÞγ ;

i; j 5 1; 2; . . .; υk ;

i 6¼ j;

k 5 1; 2; . . .; n3 ;

l 5 1; 2; . . .; ω;

(4.101)

D  the number r ijðk=lÞ of the disjoint intervals Iz 5 aijðk=lÞz ; bijðk=lÞz ; z 5 1; 2; . . .; r ijðk=lÞ that include the realizations ηijðk=lÞγ ; γ 5 1; 2; . . .; nijðk=lÞ of the conditional sojourn times ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the

G

state siðk=lÞ , while the next transition is to the state sjðk=lÞ according to the formula qffiffiffiffiffiffiffiffiffiffi r ijðk=lÞ D nijðk=lÞ ; D  ij the length dðk=lÞ of the intervals Iz 5 aijðk=lÞz ; bijðk=lÞz ; z 5 1; 2; . . .; r ijðk=lÞ according to the formula

(4.102)

ij

ij dðk=lÞ

5

Rðk=lÞ

r ijðk=lÞ 2 1

;

(4.103)

where ij

Rðk=lÞ 5 G

max

1 # γ # nijðk=lÞ

ηijðk=lÞγ 2

min

1 # γ # nijðk=lÞ

ηijðk=lÞγ ;

(4.104)

D  the ends aijðk=lÞz ; bijðk=lÞz ; z 5 1; 2; . . .; r ijðk=lÞ of the intervals Iz 5 aijðk=lÞz ; bijðk=lÞz ; z 5 1; 2; . . .; r ijðk=lÞ according to the formula ( ) ij dðk=lÞ ij ij ; 0 ; (4.105) aðk=lÞ1 5 max min ηðk=lÞγ 2 2 1 # γ # nijðk=lÞ ij bijðk=lÞz 5 aijðk=lÞ1 1 zdðk=lÞ ;

aijðk=lÞz 5 bijðk=lÞz21 ;

z 5 1; 2; . . .; r ijðk=lÞ ;

z 5 2; 3; . . .; r ijðk=lÞ ;

in such a way that

I1 , I2 , ? , Irij 5 ðk=lÞ

aij1 ; bijrij

(4.106) (4.107)



ðk=lÞ

and Ii - Iz 5 [ for all i 6¼ z; i; z 5 1; 2; . . .; r ijðk=lÞ ;

G

D  the numbers nijðk=lÞz of the realizations ηijðk=lÞγ ; γ 5 1; 2; . . .; nijðk=lÞ in the intervals Iz 5 aijðk=lÞz ; bijðk=lÞz ; z 5 1; 2; . . .; r ijðk=lÞ according to the formula nijðk=lÞz 5 #fγ:ηijðk=lÞγ AIz ; γAf1; 2; . . .; nijðk=lÞ gg; where

z 5 1; 2; . . .; r ijðk=lÞ ;

(4.108)

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Consequences of Maritime Critical Infrastructure Accidents

rij

ðk=lÞ X

z51

nijðk=lÞz 5 nijðk=lÞ ;

whereas the symbol # means the number of elements of the set. To estimate the parameters of distributions of conditional sojourn times of the process of environment threats at the particular states distinguished in Section 3.2, we proceed, respectively, in the following way (Kołowrocki and Soszy´nskaBudny, 2011): G for the double trapezium distribution with the density function given by (3.26), the estimates of the unknown parameters are ij xijðk=lÞ 5 aijðk=lÞ1 ; yijðk=lÞ 5 xijðk=lÞ 1 r ijðk=lÞ dðk=lÞ ; qijðk=lÞ 5

wijðk=lÞ G

5

nijðk=lÞrij

ðk=lÞ

ij nijðk=lÞ dðk=lÞ

nijðk=lÞ1

ij nijðk=lÞ dðk=lÞ

;

(4.109)

; zijðk=lÞ 5 ηijðk=lÞ ;

(4.110)

for the quasitrapezium distribution with the density function given by (3.27), the estimates of the unknown parameters are xijðk=lÞ 5 aijðk=lÞ1 ;

wijðk=lÞ

5

ij yijðk=lÞ 5 xijðk=lÞ 1 r ijðk=lÞ dðk=lÞ ;

nijðk=lÞrij

ðk=lÞ

ij nijðk=lÞ dðk=lÞ

qijðk=lÞ 5

nijðk=lÞ1

ij nijðk=lÞ dðk=lÞ

;

; zijðk=lÞ1 5 ηijðk=lÞ1 ; zijðk=lÞ2 5 ηijðk=lÞ2 ;

(4.111)

(4.112)

where ηijðk=lÞ1

G

5

nðmeÞ 1 X

nðmeÞ γ51

ηijðk=lÞγ ;

ηijðk=lÞ2

5

nijðk=lÞ 2 nðmeÞ γ5nðmeÞ 11 " ij # nðk=lÞ 1 1 ðmeÞ n 5 ; 2

ηijðk=lÞγ ;

(4.113)

(4.114)

and [x] denotes the entire part of x; for the exponential distribution with the density function given by (3.28), the estimates of the unknown parameters are xijðk=lÞ 5 aijðk=lÞ1 ; αijðk=lÞ 5

G

nij

ðk=lÞ X

1

1 ηijðk=lÞ

2 xijðk=lÞ

;

(4.115)

for the chimney distribution with the density function given by (3.29), the estimates of the unknown parameters are ij xijðk=lÞ 5 aijðk=lÞ1 ; yijðk=lÞ 5 xijðk=lÞ 1 r ijðk=lÞ dðk=lÞ

and moreover, if _ij n ðk=lÞ

5

maxij

1 # z # rðk=lÞ

n o nijðk=lÞz

(4.116)

(4.117)

and i, iAf1; 2; . . .; r ijðk=lÞ g is the number of the interval including the largest number of realizations, such that _ij

nijðk=lÞi 5 n ðk=lÞ ; then

(4.118)

Identification of critical infrastructure accident consequences Chapter | 4

63

for i 5 1 either ij ij zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 1Þdðk=lÞ ; zijðk=lÞ2 5 xijðk=lÞ 1 idðk=lÞ ; ij Aijðk=lÞ 5 0; Cðk=lÞ 5

nijðk=lÞi nijðk=lÞ

; Dijðk=lÞ 5

(4.119)

nijðk=lÞi11 1 ? 1 nijðk=lÞrij

ðk=lÞ

nijðk=lÞ

;

(4.120)

while nijðk=lÞi11 5 0 or nijðk=lÞi11 6¼ 0 and

nijðk=lÞi

nijðk=lÞi11

$ 3;

(4.121)

or ij ij zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 1Þdðk=lÞ ; zijðk=lÞ2 5 xijðk=lÞ 1 ði 1 1Þdðk=lÞ ;

nijðk=lÞi 1 nijðk=lÞi11

ij Aijðk=lÞ 5 0; Cðk=lÞ 5

Dijðk=lÞ

5

;

nijðk=lÞ

nijðk=lÞi12 1 ? 1 nijðk=lÞrij

ðk=lÞ

nijðk=lÞ

(4.122) (4.123)

;

(4.124)

while nijðk=lÞi11 6¼ 0 and

nijðk=lÞi nijðk=lÞi11

, 3;

(4.125)

for i 5 2; 3; . . .; r ijðk=lÞ 2 1 either ij ij zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 1Þdðk=lÞ ; zijðk=lÞ2 5 xijðk=lÞ 1 idðk=lÞ ;

Aijðk=lÞ

5

nijðk=lÞ1 1 ? 1 nijðk=lÞi21 nijðk=lÞ

Dijðk=lÞ

5

;

ij Cðk=lÞ

nijðk=lÞi11 1 ? 1 nijðk=lÞrij

5

ðk=lÞ

nijðk=lÞ

nijðk=lÞi nijðk=lÞ

;

;

(4.126) (4.127)

(4.128)

while nijðk=lÞi21

5 0 or

nijðk=lÞi21

6¼ 0 and

nijðk=lÞi

nijðk=lÞi21

$3

(4.129)

$ 3;

(4.130)

and while nijðk=lÞi11 5 0 or nijðk=lÞi11 6¼ 0 and

nijðk=lÞi nijðk=lÞi11

or ij ij zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 1Þdðk=lÞ ; zijðk=lÞ2 5 xijðk=lÞ 1 ði 1 1Þdðk=lÞ ;

(4.131)

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Consequences of Maritime Critical Infrastructure Accidents

Aijðk=lÞ 5

nijðk=lÞ1 1 ? 1 nijðk=lÞi21 nijðk=lÞ Dijðk=lÞ

5

ij ; Cðk=lÞ 5

nijðk=lÞi 1 nijðk=lÞi11 nijðk=lÞ

nijðk=lÞi12 1 ? 1 nijðk=lÞrij

ðk=lÞi

nijðk=lÞ

;

(4.132)

;

(4.133)

while nijðk=lÞi21 5 0 or nijðk=lÞi21 6¼ 0 and

nijðk=lÞi

nijðk=lÞi21

$ 3;

(4.134)

and while nijðk=lÞi11

nijðk=lÞi

6¼ 0 and

nijðk=lÞi11

, 3;

(4.135)

or ij ij zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 2Þdðk=lÞ ; zijðk=lÞ2 5 xijðk=lÞ 1 idðk=lÞ ;

Aijðk=lÞ

5

nijðk=lÞ1 1 ? 1 nijðk=lÞi22 nijðk=lÞ

Dijðk=lÞ

5

ij Cðk=lÞ

;

5

nijðk=lÞi21 1 nijðk=lÞi nijðk=lÞ

nijðk=lÞi11 1 ? 1 nijðk=lÞrij

ðk=lÞ

nijðk=lÞ

;

(4.136)

;

(4.137)

while nijðk=lÞi21

6¼ 0 and

nijðk=lÞi

nijðk=lÞi21

,3

(4.138)

and while nijðk=lÞi11 5 0 or nijðk=lÞi11 6¼ 0 and

nijðk=lÞi nijðk=lÞi11

$ 3;

(4.139)

or ij ij zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 2Þdðk=lÞ ; zijðk=lÞ2 5 xijðk=lÞ 1 ði 1 1Þdðk=lÞ ;

Aijðk=lÞ

5

nijðk=lÞ1 1 ? 1 nijðk=lÞi22 nijðk=lÞ

Dijðk=lÞ

5

;

ij Cðk=lÞ

5

nijðk=lÞi21 1 nijðk=lÞi 1 nijðk=lÞi11 nijðk=lÞ

nijðk=lÞi12 1 ? 1 nijðk=lÞrij

ðk=lÞ

nijðk=lÞ

;

(4.140) ;

(4.141)

(4.142)

while nijðk=lÞi21 and while

6¼ 0 and

nijðk=lÞi nijðk=lÞi21

,3

(4.143)

Identification of critical infrastructure accident consequences Chapter | 4

nijðk=lÞi

nijðk=lÞi11 6¼ 0 and

nijðk=lÞi11

, 3;

65

(4.144)

for i 5 r ijðk=lÞ either ij ij zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 1Þdðk=lÞ ; zijðk=lÞ2 5 xijðk=lÞ 1 idðk=lÞ ;

Aijðk=lÞ 5

nijðk=lÞ1 1 ? 1 nijðk=lÞi21 nijðk=lÞ

nijðk=lÞi

ij ; Cðk=lÞ 5

nijðk=lÞ

(4.145)

; Dijðk=lÞ 5 0;

(4.146)

while nijðk=lÞi21 5 0 or nijðk=lÞi21 6¼ 0 and

nijðk=lÞi

nijðk=lÞi21

$ 3;

(4.147)

or ij ij ; zijðk=lÞ2 5 xijðk=lÞ 1 idðk=lÞ ; zijðk=lÞ1 5 xijðk=lÞ 1 ði 2 2Þdðk=lÞ

Aijðk=lÞ 5

nijðk=lÞ1 1 ? 1 nijðk=lÞi22 nijðk=lÞ

ij ; Cðk=lÞ 5

nijðk=lÞi21 1 nijðk=lÞi

Dijðk=lÞ 5 0;

nijðk=lÞ

(4.148) ;

(4.149) (4.150)

while nijðk=lÞi21 6¼ 0 and

nijðk=lÞi

nijðk=lÞi21

, 3:

(4.151)

4.3.3 Identification of distribution functions of process of environment threats conditional sojourn times at environment threats states To formulate and next to verify the nonparametric hypothesis concerning the form of the distribution of the process of environment threats conditional sojourn time ηijðk=lÞ at the state siðk=lÞ when its next state is sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; on the basis of its realizations ηijðk=lÞγ ; γ 5 1; 2; . . .; nijðk=lÞ ; it is due to proceed according to the following scheme (Kołowrocki and Soszy´nska-Budny, 2011): G

to construct and to plot the realization of the histogram of the process of environment threats conditional sojourn time ηijðk=lÞ at the state siðk=lÞ (Fig. 4.7) defined by the following formula nijðk=lÞ

hðk=lÞ ðtÞ 5 G

G

G

nij

nijðk=lÞz nijðk=lÞ

for tAI z ;

(4.152)

ðk=lÞ to analyze the realization of the histogram hðk=lÞ ðtÞ, comparing it with the graphs of the density functions hijðk=lÞ ðtÞ of the distinguished in Section 3.2 distributions, to select one of them and to formulate the null hypothesis H0 concerning the unknown form of the distribution of the conditional sojourn time ηijðk=lÞ in the following form: H0: The process of environment threats conditional sojourn time ηijðk=lÞ at the state siðk=lÞ , while the next transition to the state sjðk=lÞ has the distribution with the density function hijðk=lÞ ðtÞ; to join each of the intervals Iz that has the number nijðk=lÞz of realizations less than 4 either with the neighbor interval Iz11 or with the neighbor interval Iz21 this way that the numbers of realizations in all intervals are not less than 4; ij to fix a new number of intervals r ðk=lÞ ;

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Consequences of Maritime Critical Infrastructure Accidents

FIGURE 4.7 The graph of the realization of the histogram of the process of environment threats conditional sojourn time ηijðk=lÞ at the state siðk=lÞ . G

to determine new intervals ij

I z 5 , aijðk=lÞz ; bðk=lÞz ; G G

ij

z 5 1; 2; . . .; r ðk=lÞ ; ij

to fix the numbers nijðk=lÞz of realizations in new intervals I z ; z 5 1; 2; . . .; r ðk=lÞ ; to calculate the hypothetical probabilities that the variable ηijðk=lÞ takes values from the interval I z under the assumption that the hypothesis H0 is true, that is, the probabilities ij

ij

ij ij pz 5 Pðηijðk=lÞ AI z Þ 5 Pðaijðk=lÞz # ηijðk=lÞ , bðk=lÞz Þ 5 Hðk=lÞ ðbðk=lÞz Þ 2 Hðk=lÞ ðaijðk=lÞz Þ;

ij

z 5 1; 2; . . .; r ðk=lÞ ;

(4.153)

ij

G

ij ij ij where Hðk=lÞ ðbðk=lÞz Þ and Hðk=lÞ ðaijðk=lÞz Þ are the values of the distribution function Hðk=lÞ ðtÞ of the random variable ηijðk=lÞ ij corresponding to the density function hðk=lÞ ðtÞ assumed in the null hypothesis H0 ; nij to calculate the realization of the χ2-Pearson’s statistics U ðk=lÞ according to the formula ij

u

nijðk=lÞ

5

r ðk=lÞ X ðnijðk=lÞz 2nijðk=lÞ pz Þ2 z51

G G

G

G

nijðk=lÞ pz

;

(4.154)

to assume the significance level α (for instance α 5 0:05) of the test; ij to fix the number r ðk=lÞ 2 j 2 1 of degrees of freedom, substituting for j, for the distinguished in Section 3.2 distributions, respectively, the following values: j 5 0 for the double trapezium, quasitrapezium, and chimney distributions, and j 5 1 for the exponential distribution; to read from the table of the χ2-Pearson’s distribution (Appendix 7) the value uα for the fixed values of the signifiij cance level α and the number of degrees of freedom r ðk=lÞ 2 j 2 1 such that the following equality holds  ij  n P U ðk=lÞ . uα 5 α; (4.155) and next to determine the acceptance domain in the form of the interval h0; uα i; and the critical domain in the form in Fig. 4.2; of the interval ðuα ; 1 NÞ as presented nij nij to compare the obtained value u ðk=lÞ of the realization of the statistics U ðk=lÞ with the read from the table critical value uα of the chi-square random variable and to decide on the previously formulated null hypothesis H0 in the following way: if nij nij α the value u ðk=lÞ does ijnot belong to the critical domain, that is, when u ðk=lÞ # u , we do not reject the hypothesis H0; othern nij wise, if the value u ðk=lÞ belongs to the critical domain, that is, when u ðk=lÞ . uα then we reject the hypothesis H0.

4.4 Identification of process of environment threats—application to Baltic Sea and world sea waters The statistical identification of the process of environment threats is performed on the basis of the data coming from free-accessible reports of chemical accidents at sea that happened around the world in a period of 11 years (CEDRE, 2018; GISIS, 2018). The identification of the process of environment threats for the world sea waters is sufficiently accurate as the set of statistical data is most populated. Unfortunately, the less-accurate identification of the process of environment threats is performed for the Baltic Sea waters as the less sufficiently numerous set of statistical data.

Identification of critical infrastructure accident consequences Chapter | 4

67

4.4.1 States of process of environment threats We distinguished n3 5 5 subareas Dk, k 5 1; 2; . . .; n3 that may be degraded by the environment threats Hi, i 5 1; 2; . . .; 6 as follows: G G G G G

D1—air D2—water surface D3—water column D4—sea floor D5—coast (shoreline).

Considering (3.19) and (3.20), we distinguish for each environment subarea Dk, k 5 1; 2; . . .; 5 the following threat states sυðkÞ ; υ 5 1; 2; . . .; υk ; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29: G

for subarea D1—air: s1ð1Þ 5 ½0; 0; 0; 0; 0; 0; s6ð1Þ 5 ½0; 0; 1; 0; 0; 0; s11 ð1Þ 5 ½0; 0; 0; 2; 0; 0; s16 ð1Þ 5 ½0; 0; 2; 3; 0; 0; s21 ð1Þ 5 ½0; 3; 2; 0; 0; 0; s26 ð1Þ 5 ½0; 0; 2; 0; 1; 1; s31 ð1Þ 5 ½0; 0; 4; 0; 1; 1;

G

s4ð1Þ 5 ½0; 3; 0; 0; 0; 0; s9ð1Þ 5 ½0; 0; 4; 0; 0; 0; s14 ð1Þ 5 ½0; 0; 0; 2; 0; 1; s19 ð1Þ 5 ½0; 0; 3; 0; 2; 0; s24 ð1Þ 5 ½0; 0; 2; 0; 0; 1; s29 ð1Þ 5 ½0; 0; 3; 2; 3; 0; s34 ð1Þ 5 ½3; 3; 0; 0; 0; 0;

s5ð1Þ 5 ½0; 4; 0; 0; 0; 0; s10 ð1Þ 5 ½0; 0; 0; 1; 0; 0; s15 ð1Þ 5 ½0; 0; 1; 3; 0; 0; s20 ð1Þ 5 ½0; 0; 3; 1; 0; 0; s25 ð1Þ 5 ½0; 0; 3; 3; 0; 0; s30 ð1Þ 5 ½0; 0; 4; 3; 0; 0; s35 ð1Þ 5 ½3; 4; 0; 0; 0; 0;

s2ð2Þ 5 ½0; 0; 1; 0; 0; 0; s7ð2Þ 5 ½0; 0; 5; 0; 0; 0; s12 ð2Þ 5 ½0; 0; 0; 0; 1; 1; s17 ð2Þ 5 ½0; 0; 2; 3; 0; 0; s22 ð2Þ 5 ½0; 0; 3; 0; 3; 1; s27 ð2Þ 5 ½0; 0; 5; 0; 5; 1; s32 ð2Þ 5 ½0; 0; 2; 3; 0; 0;

s3ð2Þ 5 ½0; 0; 0; 2; 0; 0; s8ð2Þ 5 ½0; 1; 0; 0; 0; 0; s13 ð2Þ 5 ½0; 0; 0; 2; 0; 1; s18 ð2Þ 5 ½0; 0; 3; 0; 1; 0; s23 ð2Þ 5 ½0; 0; 3; 1; 0; 0; s28 ð2Þ 5 ½3; 3; 0; 0; 0; 0; s33 ð2Þ 5 ½0; 0; 2; 0; 3; 1;

s4ð2Þ 5 ½0; 0; 2; 0; 0; 0; s9ð2Þ 5 ½0; 2; 0; 0; 0; 0; s14 ð2Þ 5 ½0; 0; 2; 0; 1; 0; s19 ð2Þ 5 ½0; 0; 3; 0; 2; 0; s24 ð2Þ 5 ½0; 0; 3; 2; 2; 0; s29 ð2Þ 5 ½0; 0; 1; 3; 0; 0;

s5ð2Þ 5 ½0; 0; 3; 0; 0; 0; s10 ð2Þ 5 ½0; 3; 0; 0; 0; 0; s15 ð2Þ 5 ½0; 0; 2; 1; 0; 1; s20 ð2Þ 5 ½0; 0; 3; 0; 2; 1; s25 ð2Þ 5 ½0; 0; 3; 2; 3; 0; s30 ð2Þ 5 ½0; 0; 2; 0; 0; 1;

s3ð3Þ 5 ½0; 0; 2; 0; 0; 0; s8ð3Þ 5 ½0; 0; 0; 0; 1; 1; s13 ð3Þ 5 ½0; 0; 2; 0; 2; 0; s18 ð3Þ 5 ½0; 0; 3; 0; 3; 0; s23 ð3Þ 5 ½0; 0; 2; 1; 0; 1; s28 ð3Þ 5 ½0; 0; 3; 2; 3; 0;

s4ð3Þ 5 ½0; 0; 3; 0; 0; 0; s9ð3Þ 5 ½0; 0; 0; 2; 0; 1; s14 ð3Þ 5 ½0; 0; 2; 0; 0; 0; s19 ð3Þ 5 ½0; 0; 3; 1; 0; 0; s24 ð3Þ 5 ½0; 0; 2; 0; 3; 1; s29 ð3Þ 5 ½0; 0; 5; 0; 5; 1;

s5ð3Þ 5 ½0; 0; 4; 0; 0; 0; s10 ð3Þ 5 ½0; 0; 1; 3; 0; 0; s15 ð3Þ 5 ½0; 0; 3; 0; 1; 0; s20 ð3Þ 5 ½0; 0; 4; 0; 0; 1; s25 ð3Þ 5 ½0; 0; 3; 0; 2; 1;

for subarea D3—water column: s1ð3Þ 5 ½0; 0; 0; 0; 0; 0; s6ð3Þ 5 ½0; 0; 5; 0; 0; 0; s11 ð3Þ 5 ½0; 0; 2; 0; 0; 1; s16 ð3Þ 5 ½0; 0; 3; 0; 0; 1; s21 ð3Þ 5 ½0; 0; 4; 0; 2; 0; s26 ð3Þ 5 ½0; 0; 3; 0; 3; 1;

G

s3ð1Þ 5 ½0; 2; 0; 0; 0; 0; s8ð1Þ 5 ½0; 0; 3; 0; 0; 0; s13 ð1Þ 5 ½0; 0; 0; 0; 2; 1; s18 ð1Þ 5 ½0; 0; 3; 0; 0; 1; s23 ð1Þ 5 ½0; 0; 4; 0; 1; 0; s28 ð1Þ 5 ½0; 0; 3; 1; 0; 1; s33 ð1Þ 5 ½0; 0; 4; 2; 2; 0;

for subarea D2—water surface: s1ð2Þ 5 ½0; 0; 0; 0; 0; 0; s6ð2Þ 5 ½0; 0; 4; 0; 0; 0; s11 ð2Þ 5 ½0; 4; 0; 0; 0; 0; s16 ð2Þ 5 ½0; 0; 2; 2; 0; 0; s21 ð2Þ 5 ½0; 0; 3; 0; 3; 0; s26 ð2Þ 5 ½0; 0; 4; 2; 0; 0; s31 ð2Þ 5 ½0; 0; 2; 0; 1; 1;

G

s2ð1Þ 5 ½0; 1; 0; 0; 0; 0; s7ð1Þ 5 ½0; 0; 2; 0; 0; 0; s12 ð1Þ 5 ½0; 0; 0; 0; 3; 0; s17 ð1Þ 5 ½0; 2; 3; 0; 0; 0; s22 ð1Þ 5 ½0; 0; 3; 2; 0; 0; s27 ð1Þ 5 ½0; 0; 2; 0; 3; 1; s32 ð1Þ 5 ½0; 0; 4; 0; 5; 1;

s2ð3Þ 5 ½0; 0; 1; 0; 0; 0; s7ð3Þ 5 ½0; 0; 0; 2; 0; 0; s12 ð3Þ 5 ½0; 0; 2; 0; 1; 0; s17 ð3Þ 5 ½0; 0; 3; 0; 2; 0; s22 ð3Þ 5 ½0; 0; 2; 0; 1; 1; s27 ð3Þ 5 ½0; 0; 3; 2; 2; 0;

for subarea D4—sea floor: s1ð4Þ 5 ½0; 0; 0; 0; 0; 0; s6ð4Þ 5 ½0; 0; 5; 0; 0; 0; s11 ð4Þ 5 ½0; 0; 2; 0; 0; 1; s16 ð4Þ 5 ½0; 0; 3; 0; 0; 1; s21 ð4Þ 5 ½0; 0; 4; 0; 2; 0; s26 ð4Þ 5 ½0; 0; 3; 0; 3; 1;

s2ð4Þ 5 ½0; 0; 1; 0; 0; 0; s7ð4Þ 5 ½0; 0; 0; 2; 0; 0; s12 ð4Þ 5 ½0; 0; 2; 0; 1; 0; s17 ð4Þ 5 ½0; 0; 3; 0; 2; 0; s22 ð4Þ 5 ½0; 0; 2; 0; 1; 1; s27 ð4Þ 5 ½0; 0; 3; 2; 2; 0;

s3ð4Þ 5 ½0; 0; 2; 0; 0; 0; s8ð4Þ 5 ½0; 0; 0; 0; 1; 1; s13 ð4Þ 5 ½0; 0; 2; 0; 2; 0; s18 ð4Þ 5 ½0; 0; 3; 0; 3; 0; s23 ð4Þ 5 ½0; 0; 2; 1; 0; 1; s28 ð4Þ 5 ½0; 0; 3; 2; 3; 0;

s4ð4Þ 5 ½0; 0; 3; 0; 0; 0; s9ð4Þ 5 ½0; 0; 0; 2; 0; 1; s14 ð4Þ 5 ½0; 0; 2; 3; 0; 0; s19 ð4Þ 5 ½0; 0; 3; 1; 0; 0; s24 ð4Þ 5 ½0; 0; 2; 0; 3; 1; s29 ð4Þ 5 ½0; 0; 5; 0; 5; 1;

s5ð4Þ 5 ½0; 0; 4; 0; 0; 0; s10 ð4Þ 5 ½0; 0; 1; 3; 0; 0; s15 ð4Þ 5 ½0; 0; 3; 0; 1; 0; s20 ð4Þ 5 ½0; 0; 4; 0; 0; 1; s25 ð4Þ 5 ½0; 0; 3; 0; 2; 1;

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for subarea D5—coast (shoreline): s1ð5Þ 5 ½0; 0; 0; 0; 0; 0; s6ð5Þ 5 ½0; 0; 5; 0; 0; 0; s11 ð5Þ 5 ½0; 0; 2; 0; 0; 1; s16 ð5Þ 5 ½0; 0; 3; 0; 0; 1; s21 ð5Þ 5 ½0; 0; 4; 0; 2; 0; s26 ð5Þ 5 ½0; 0; 3; 0; 3; 1;

s2ð5Þ 5 ½0; 0; 1; 0; 0; 0; s7ð5Þ 5 ½0; 0; 0; 2; 0; 0; s12 ð5Þ 5 ½0; 0; 2; 0; 1; 0; s17 ð5Þ 5 ½0; 0; 3; 0; 2; 0; s22 ð5Þ 5 ½0; 0; 2; 0; 1; 1; s27 ð5Þ 5 ½0; 0; 3; 2; 2; 0;

s3ð5Þ 5 ½0; 0; 2; 0; 0; 0; s8ð5Þ 5 ½0; 0; 0; 0; 1; 1; s13 ð5Þ 5 ½0; 0; 2; 0; 2; 0; s18 ð5Þ 5 ½0; 0; 3; 0; 3; 0; s23 ð5Þ 5 ½0; 0; 2; 1; 0; 1; s28 ð5Þ 5 ½0; 0; 3; 2; 3; 0;

s4ð5Þ 5 ½0; 0; 3; 0; 0; 0; s9ð5Þ 5 ½0; 0; 0; 2; 0; 1; s14 ð5Þ 5 ½0; 0; 2; 3; 0; 0; s19 ð5Þ 5 ½0; 0; 3; 1; 0; 0; s24 ð5Þ 5 ½0; 0; 2; 0; 3; 1; s29 ð5Þ 5 ½0; 0; 5; 0; 5; 1:

s5ð5Þ 5 ½0; 0; 4; 0; 0; 0; s10 ð5Þ 5 ½0; 0; 1; 3; 0; 0; s15 ð5Þ 5 ½0; 0; 3; 0; 1; 0; s20 ð5Þ 5 ½0; 0; 4; 0; 0; 1; s25 ð5Þ 5 ½0; 0; 3; 0; 2; 1;

The states of the process of environment threats are defined as the vector where 0 means any threat does not appear, but the other digit means the threat appears and its value expresses the range f ij ; i 5 1; 2; . . .; n2 ; j 5 1; 2; . . .; li of the threat Hi, i 5 1; 2; . . .; n2 : For instance, the vector G

G

G

G

s1ð1Þ 5 ½0; 0; 0; 0; 0; 0 means the sea accident has happened without the dangerous substance spill or a chemical substance has released, but the substance is not dangerous, in other words, the environment subarea D1 of the accident area is not threatened, s10 ð1Þ 5 ½0; 0; 0; 1; 0; 0 means the released substance causes the threat H4 (substance is corrosive) in the environment subarea D1 of the accident area and the corrosiveness range is slight (f 41 5 1); in other words, the substance causes the skin necrosis at the time longer than 1 hour; s11 ð1Þ 5 ½0; 0; 0; 2; 0; 0 means the released substance causes the threat H4 (substance is corrosive) in the environment subarea D1 of the accident area and the corrosiveness range is moderate (f 42 5 2); in other words, this substance causes the skin necrosis at the time longer than 3 minutes but less than 1 hour; and s15 ð1Þ 5 ½0; 0; 1; 3; 0; 0 means the released substance causes simultaneously the threat H3 and H4 (substance is toxic and corrosive) in the environment subarea D1 of the accident area and contamination range is slight (f 31 5 1), and the corrosiveness range is high (f 43 5 3).

If two or more dangerous substances have released as a result of sea accident causing the same kind of threat but with different ranges of its parameters fi, we set the parameter with the highest range. Moreover, the process of environment threats of the each subarea is involved with the process of initiating events. Namely, the state of the process of environment threats is conditional upon the state of the process of initiating event that caused the previous one. For instance, s10 ð1=2Þ means the state of the process of environment threats of the environment subarea D1 when the process of initiating event is at the state e2.

4.4.2 Statistical identification of process of environment threats at Baltic Sea waters We divided the marine environment into subareas Dk, k 5 1; 2; . . .; 5 defined in Section 4.4.1. Next, the number of the process of environment threats states υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 for each subarea Dk, k 5 1; 2; . . .; 5 are fixed and the environment threats states siðkÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 are defined in Section 4.4.1. In addition, we involve the process of initiating events with the process of environment threats. Thus we fix environment threats states siðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 for a particular subarea Dk, k 5 1; 2; . . .; 5 according to the initiating event that causes the environment threats state. Moreover, it is assumed that there are possible transitions between all states of the process of environment threats.

4.4.2.1 Collected statistical data of process of environment threats The experiment was performed within the Baltic Sea waters in years 200414. The initial moment t 5 0 of the process of environment threats is fixed at the moment when the initiating event causing ship accident generated one of the distinguished environment threats states. The unknown parameters of the process of environment threats semi-Markov model are G

G

the initial probabilities piðk=lÞ ð0Þ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats staying at the particular states siðk=lÞ at the initial moment t 5 0; the probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats transitions from the state siðk=lÞ into the state sjðk=lÞ ; and

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69

the distributions of the environment threats conditional sojourn times ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 at the particular states and their mean values ij Mðk=lÞ 5 E½ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16:

To identify all these parameters of the process of environment threats, the statistical data about this process is needed. The statistical data for the conditional sojourn times ηijðk=lÞ at the states of the process of environment threats are given in Appendix 3. The collected statistical data necessary to evaluating the initial transient probabilities of process of environment threats at the particular states are G

the process of environment threats observation/experiment time Θ 5 11 years (200414);

G

the vectors of realizations of the number niðk=lÞ ð0Þ of the process of environment threats staying at the particular states siðk=lÞ at the initial moment t 5 0 according to (4.92) h i niðk=1Þ ð0Þ 5 ½104; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; h

i niðk=2Þ ð0Þ

h

i niðk=3Þ ð0Þ

h h h h h h h h h h h h h

i niðk=4Þ ð0Þ i niðk=5Þ ð0Þ i niðk=6Þ ð0Þ i niðk=7Þ ð0Þ i niðk=8Þ ð0Þ i niðk=9Þ ð0Þ niðk=10Þ ð0Þ niðk=11Þ ð0Þ niðk=12Þ ð0Þ niðk=13Þ ð0Þ niðk=14Þ ð0Þ niðk=15Þ ð0Þ niðk=16Þ ð0Þ

1 3 υk 1 3 υk 1 3 υk 1 3 υk 1 3 υk 1 3 υk 1 3 υk 1 3 υk 1 3 υk

5 ½20; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½65; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½6; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½10; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½25; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½12; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½1; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½0; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29;

i

1 3 υk

i

1 3 υk

i

1 3 υk

i

1 3 υk

i

1 3 υk

i

1 3 υk

i

1 3 υk

5 ½1; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½0; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½0; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½1; 0; . . .; 0for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½0; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½0; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29; 5 ½0; 0; . . .; 0 for k 5 1; υ1 5 35; and k 5 2; υ2 5 33; and k 5 3; 4; 5; υk 5 29;

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the numbers of the process of environment threats realizations at the initial moment t 5 0 according to (4.93) nðk=1Þ ð0Þ 5 104; for k 5 1; 2; . . .; 5; nðk=2Þ ð0Þ 5 20; for k 5 1; 2; . . .; 5; nðk=3Þ ð0Þ 5 65; for k 5 1; 2; . . .; 5; nðk=4Þ ð0Þ 5 6; for k 5 1; 2; . . .; 5; nðk=5Þ ð0Þ 5 10; for k 5 1; 2; . . .; 5; nðk=6Þ ð0Þ 5 25; for k 5 1; 2; . . .; 5; nðk=7Þ ð0Þ 5 12; for k 5 1; 2; . . .; 5; nðk=8Þ ð0Þ 5 1; for k 5 1; 2; . . .; 5; nðk=9Þ ð0Þ 5 0; for k 5 1; 2; . . .; 5; nðk=10Þ ð0Þ 5 1; for k 5 1; 2; . . .; 5; nðk=11Þ ð0Þ 5 0; for k 5 1; 2; . . .; 5; nðk=12Þ ð0Þ 5 0; for k 5 1; 2; . . .; 5; nðk=13Þ ð0Þ 5 1; for k 5 1; 2; . . .; 5; nðk=14Þ ð0Þ 5 0; for k 5 1; 2; . . .; 5; nðk=15Þ ð0Þ 5 0; for k 5 1; 2; . . .; 5; nðk=16Þ ð0Þ 5 0; for k 5 1; 2; . . .; 5:

The collected statistical data necessary for evaluating the probabilities of transitions between states are ij i G the matrices of realizations n ðk=lÞ of the number of the process of environment threats transitions from the state sðk=lÞ j into the state sðk=lÞ ; according to (4.94), during the experimental time. The numbers of transitions that are not equal to 0 are presented only, and they are as follows (Appendix 3, C-column of the table): 27 1 n1ð1=2Þ 5 1; n27 ð1=2Þ 5 1; 27 30 1 30 1 n1ð1=3Þ 5 1; n1ð1=3Þ 5 1; n27 ð1=3Þ 5 1; nð1=3Þ 5 1; 6 1 n1ð1=8Þ 5 1; n6ð1=8Þ 5 1; 33 1 n1ð2=2Þ 5 1; n33 ð2=2Þ 5 1; 17 33 1 33 1 n1ð2=3Þ 5 1; n1ð2=3Þ 5 1; n17 ð2=3Þ 5 1; nð2=3Þ 5 1; 24 1 n1ð3=2Þ 5 1; n24 ð3=2Þ 5 1; 14 24 1 24 1 n1ð3=3Þ 5 1; n1ð3=3Þ 5 1; n14 ð3=3Þ 5 3; nð3=3Þ 5 1; 14 1 n1ð4=3Þ 5 1; n14 ð4=3Þ 5 3; G

the vectors of realization of the total number of the process of environment threats transitions from the state siðk=lÞ ; according to (4.95), during the experimental time ½nið1=2Þ 1 3 35 5 ½1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; 0; 0; 0; ½nið1=3Þ 1 3 35 5 ½2; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 1; 0; 0; 0; 0; 0; ½nið1=8Þ 1 3 35 5 ½1; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; ½nið1=lÞ 1 3 35 5 ½0; 0; . . .; 0 for l 5 1; 4; 5; 6; 7; 9; 10; 11; 12; 13; 14; 15; 16; ½nið2=2Þ 1 3 33 5 ½1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; ½nið2=3Þ 1 3 33 5 ½2; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; ½nið2=lÞ 1 3 33 5 ½0; 0; . . .; 0 for l 5 1; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; ½nið3=2Þ 1 3 29 5 ½1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; ½nið3=3Þ 1 3 29 5 ½2; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; ½nið3=lÞ 1 3 29 5 ½0; 0; . . .; 0 for l 5 1; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; ½nið4=3Þ 1 3 29 5 ½1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; ½nið4=lÞ 1 3 29 5 ½0; 0; . . .; 0 for l 5 1; 2; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; ½nið5=lÞ 1 3 29 5 ½0; 0; . . .; 0 for l 5 1; 2; . . .; 16:

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4.4.2.2 Evaluating basic parameters of process of environment threats On the basis of the statistical data obtained in Section 4.4.2.1, using the formulae given in Section 4.3.1, it is possible to evaluate the following unknown basic parameters of the process of environment threats: G

the vectors of the initial probabilities piðk=lÞ ð0Þ of the process of environment threats at the particular states at the initial moment t 5 0 according to (4.97) as follows: ½piðk=lÞ ð0Þ1 3 υk 5 ½1; 0; . . .; 0;

G

(4.156)

for k 5 1; 2; . . .; 5; l 5 1; 2; . . .; 16; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; the matrices ½pijðk=lÞ υk xυk ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the probabilities of transitions of the process Sðk=lÞ ðtÞ from the state siðk=lÞ into the state sjðk=lÞ ; according to (4.99), during the experimental time. The probabilities of transitions that are not equal to 0 are as follows: 27 1 p1ð1=2Þ 5 1; p27 ð1=2Þ 5 1;

(4.157)

27 30 1 30 1 p1ð1=3Þ 5 0:5; p1ð1=3Þ 5 0:5; p27 ð1=3Þ 5 1; pð1=3Þ 5 1;

(4.158)

6 1 p1ð1=8Þ 5 1; p6ð1=8Þ 5 1;

(4.159)

33 1 p1ð2=2Þ 5 1; p33 ð2=2Þ 5 1;

(4.160)

17 33 1 33 1 p1ð2=3Þ 5 0:5; p1ð2=3Þ 5 0:5; p17 ð2=3Þ 5 1; pð2=3Þ 5 1;

(4.161)

24 1 p1ð3=2Þ 5 1; p24 ð3=2Þ 5 1;

(4.162)

14 24 1 24 1 p1ð3=3Þ 5 0:5; p1ð3=3Þ 5 0:5; p14 ð3=3Þ 5 1; pð3=3Þ 5 1;

(4.163)

14 1 p1ð4=3Þ 5 1; p14 ð4=3Þ 5 1:

(4.164)

The values of some probabilities existing in the vectors (4.156) and in the matrices (4.157)(4.164), besides these standing on the main diagonal, being equal to zero do not mean that the events they are concerned with cannot appear. They evaluated on the basis of real statistical data and their values may change and become more precise if the duration of the experiment is longer.

4.4.2.3 Identification of distribution functions of conditional sojourn times at environment threats states As we only have the number of realizations of the process of environment threats, and all its realizations are equal to an approximate value, we assume that these conditional sojourn times have the uniform distribution in the interval from this value minus its half to this value plus its half. Thus we proceed in the analogous way as in Section 4.2.2.4, for example, conditional sojourn time θ21 the density and distribution function of which is expressed by (4.74) and (4.75), respectively. For these distributions that cannot be identified statistically, by application, the general formula for the mean value ij given by (4.101), the mean values Mðk=lÞ 5 E½ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; l 5 1; 2; . . .; 16; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment threats conditional sojourn times at particular states can be determined, and they amount to (Appendix 3, D-column of the table): 1 27 27 1 1 27 1 30 27 1 30 1 16 61 Mð1=2Þ 5 1; Mð1=2Þ 5 300; Mð1=3Þ 5 1; Mð1=3Þ 5 1; Mð1=3Þ 5 180; Mð1=3Þ 5 240; Mð1=8Þ 5 1; Mð1=8Þ 5 240; 1 33 33 1 1 17 1 33 17 1 33 1 1 24 Mð2=2Þ 5 1; Mð2=2Þ 5 1440; Mð2=3Þ 5 1; Mð2=3Þ 5 1; Mð2=3Þ 5 10; 080; Mð2=3Þ 5 1440; Mð3=2Þ 5 1; 24 1 1 14 1 24 14 1 24 1 1 14 14 1 Mð3=2Þ 5 1440; Mð3=3Þ 5 1; Mð3=3Þ 5 1; Mð3=3Þ 5 10; 080; Mð3=3Þ 5 1440; Mð4=3Þ 5 1; Mð4=3Þ 5 10; 080:

(4.165)

As there are no realizations of the process of environment threats conditional sojourn times at remaining states, it is impossible to estimate their empirical conditional mean values.

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These approximate mean values given by (4.165) are used in Section 5.2.1 for the prediction of the characteristics of the process of environment threats to the Baltic Sea waters.

4.4.3 Extension of statistical identification of process of environment threats for world sea waters To statistically identify the process of environment threats at the world sea waters, we used the same kind and number of this process states that are defined in Section 4.4.1. The experiment was performed within the world sea and ocean waters in years 200414. The initial moment t 5 0 of the process of environment threats is fixed at the moment when the initiating event causing ship accident generated one of the distinguished environment threats states. The unknown parameters of the process of environment threats semi-Markov model are G

G

G

G

the initial probabilities piðk=lÞ ð0Þ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats staying at the particular states siðk=lÞ at the initial moment t 5 0; the probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats transitions from the state siðk=lÞ into the state sjðk=lÞ ; the distributions of the environment threat conditional sojourn times ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 at the particular states and their mean values ij Mðk=lÞ 5 E½ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16: We identify all these parameters on the basis of statistical data coming from the process of environment threats realization (Appendix 4). Using procedure and the formulae given in Section 4.3.1 and proceedings in an analogous way as in Section 4.4.2.2, we evaluate the following unknown basic parameters of the process of environment threats: the vectors of the initial probabilities piðk=lÞ ð0Þ of the process of environment threats at the particular states at the initial moment t 5 0 according to (4.97), as follows: ½piðk=lÞ ð0Þ1 3 υk 5 ½1; 0; . . .; 0;

G

(4.166)

for k 5 1; 2; . . .; 5; l 5 1; 2; . . .; 16; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; the matrices ½pijðk=lÞ υk 3 υk ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the probabilities of transitions of the process Sðk=lÞ ðtÞ from the state siðk=lÞ into the state sjðk=lÞ ; according to (4.99), during the experimental time. The probabilities of transitions that are not equal to 0 are as follows: 27 1 1 27 7 p1ð1=2Þ 5 1; p7ð1=2Þ 5 1; p27 ð1=2Þ 5 0:977; pð1=2Þ 5 0:023;

(4.167)

23 27 30 1 27 1 30 1 p1ð1=3Þ 5 0:023; p1ð1=3Þ 5 0:931; p1ð1=3Þ 5 0:046; p23 ð1=3Þ 5 1; pð1=3Þ 5 1; pð1=3Þ 5 1;

(4.168)

13 27 35 13 1 27 1 35 5 p1ð1=4Þ 5 0:055; p1ð1=4Þ 5 0:890; p1ð1=4Þ 5 0:055; p51 ð1=4Þ 5 1; pð1=4Þ 5 1; pð1=4Þ 5 1; pð1=4Þ 5 1;

(4.169)

3 4 27 34 17 5 0:1; p1ð1=5Þ 5 0:3; p1ð1=5Þ 5 0:5; p1ð1=5Þ 5 0:1; p3ð1=5Þ 5 1; p1ð1=5Þ 41 71 17 7 27 1 34 4 pð1=5Þ 5 1; pð1=5Þ 5 1; pð1=5Þ 5 1; pð1=5Þ 5 1; pð1=5Þ 5 1;

(4.170)

27 1 p1ð1=6Þ 5 1; p27 ð1=6Þ 5 1;

(4.171)

27 30 1 30 1 p1ð1=7Þ 5 0:875; p1ð1=7Þ 5 0:125; p27 ð1=7Þ 5 1; pð1=7Þ 5 1;

(4.172)

3 6 7 8 27 p1ð1=8Þ 5 0:059; p1ð1=8Þ 5 0:176; p1ð1=8Þ 5 0:059; p1ð1=8Þ 5 0:059; p1ð1=8Þ 5 0:647; 31 61 7 21 81 21 1 27 1 pð1=8Þ 5 1; pð1=8Þ 5 1; pð1=8Þ 5 1; pð1=8Þ 5 1; pð1=8Þ 5 1; pð1=8Þ 5 1; 33 1 1 33 2 5 1; p2ð2=2Þ 5 1; p33 p1ð2=2Þ ð2=2Þ 5 0:977; pð2=2Þ 5 0:023;

(4.173) (4.174)

Identification of critical infrastructure accident consequences Chapter | 4

73

17 18 33 1 18 1 33 1 p1ð2=3Þ 5 0:067; p1ð2=3Þ 5 0:022; p1ð2=3Þ 5 0:911; p17 ð2=3Þ 5 1; pð2=3Þ 5 1; pð2=3Þ 5 1;

(4.175)

33 1 5 1; p33 p1ð2=4Þ ð2=4Þ 5 1;

(4.176)

33 1 p1ð2=5Þ 5 1; p33 ð2=5Þ 5 1;

(4.177)

33 1 p1ð2=6Þ 5 1; p33 ð2=6Þ 5 1;

(4.178)

17 33 1 33 1 p1ð2=7Þ 5 0:125; p1ð2=7Þ 5 0:875; p17 ð2=7Þ 5 1; pð2=7Þ 5 1;

(4.179)

2 4 5 33 5 0:071; p1ð2=8Þ 5 0:071; p1ð2=8Þ 5 0:071; p1ð2=8Þ 5 0:787; p1ð2=8Þ 21 41 51 33 1 pð2=8Þ 5 1; pð2=8Þ 5 1; pð2=8Þ 5 1; pð2=8Þ 5 1;

(4.180)

24 1 1 24 2 5 1; p2ð3=2Þ 5 1; p24 p1ð3=2Þ ð3=2Þ 5 0:977; pð3=2Þ 5 0:023;

(4.181)

14 15 24 1 5 0:067; p1ð3=3Þ 5 0:022; p1ð3=3Þ 5 0:911; p14 p1ð3=3Þ ð3=3Þ 5 1 1 24 1 p15 ð3=3Þ 5 1; pð3=3Þ 5 1;

(4.182)

24 1 5 1; p24 p1ð3=4Þ ð3=4Þ 5 1;

(4.183)

24 1 p1ð3=5Þ 5 1; p24 ð3=5Þ 5 1;

(4.184)

24 1 p1ð3=6Þ 5 1; p24 ð3=6Þ 5 1;

(4.185)

14 24 1 24 1 p1ð3=7Þ 5 0:125; p1ð3=7Þ 5 0:875; p14 ð3=7Þ 5 1; pð3=7Þ 5 1;

(4.186)

2 3 4 24 p1ð3=8Þ 5 0:071; p1ð3=8Þ 5 0:071; p1ð3=8Þ 5 0:071; p1ð3=8Þ 5 0:787; 1 1 1 1 p2ð3=8Þ 5 1; p3ð3=8Þ 5 1; p4ð3=8Þ 5 1; p24 5 1; ð3=8Þ

(4.187)

2 24 1 1 p1ð4=2Þ 5 0:046; p1ð4=2Þ 5 0:954; p2ð4=2Þ 5 1; p24 ð4=2Þ 5 1;

(4.188)

14 15 24 1 15 1 24 1 5 0:136; p1ð4=3Þ 5 0:046; p1ð4=3Þ 5 0:818; p14 p1ð4=3Þ ð4=3Þ 5 1; pð4=3Þ 5 1; pð4=3Þ 5 1;

(4.189)

24 1 5 1; p24 p1ð4=4Þ ð4=4Þ 5 1;

(4.190)

24 1 p1ð4=5Þ 5 1; p24 ð4=5Þ 5 1;

(4.191)

24 1 p1ð4=6Þ 5 1; p24 ð4=6Þ 5 1;

(4.192)

14 24 1 24 1 p1ð4=7Þ 5 0:2; p1ð4=7Þ 5 0:8; p14 ð4=7Þ 5 1; pð4=7Þ 5 1;

(4.193)

2 4 24 1 1 1 p1ð4=8Þ 5 0:2; p1ð4=8Þ 5 0:2; p1ð4=8Þ 5 0:6; p2ð4=8Þ 5 1; p4ð4=8Þ 5 1; p24 ð4=8Þ 5 1;

(4.194)

24 1 5 1; p24 p1ð5=2Þ ð5=2Þ 5 1;

(4.195)

15 24 1 24 1 p1ð5=3Þ 5 0:062; p1ð5=3Þ 5 0:938; p15 ð5=3Þ 5 1; pð5=3Þ 5 1;

(4.196)

24 1 p1ð5=4Þ 5 1; p24 ð5=4Þ 5 1;

(4.197)

24 1 p1ð5=5Þ 5 1; p24 ð5=5Þ 5 1;

(4.198)

24 1 p1ð5=6Þ 5 1; p24 ð5=6Þ 5 1;

(4.199)

24 1 p1ð5=7Þ 5 1; p24 ð5=7Þ 5 1;

(4.200)

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Consequences of Maritime Critical Infrastructure Accidents

24 1 p1ð5=8Þ 5 1; p24 ð5=8Þ 5 1:

(4.201)

The values of some probabilities existing in the vectors (4.166) and in the matrices (4.167)(4.201), besides these standing on the main diagonal, equal to zero do not mean that the events they are concerned with cannot appear. They evaluated on the basis of real statistical data, and their values may change and become more precise if the duration of the experiment is longer. On the basis of the statistical data presented in Appendix 4, using the procedure and the formulae given in Section 4.3.2, it is possible to determine the empirical parameters of the conditional sojourn times ηijðk=lÞ of the process of environment threats at its particular states. Moreover, using the procedure given in Section 4.3.3, we identify the forms of the particular density function hijðk=lÞ ðtÞ of the process of environment threats conditional sojourn times ηijðk=lÞ i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 having sufficiently numerous set of their realizations at particular states. The application of these procedures and formulae is similar to the one presented in Section 4.2.2.4 than we omit it. We have got the following results: G

G

G

G

G

G

1 the conditional sojourn time η27 ð1=2Þ has the quasitrapezium distribution with the density function 8 0 t,0 > > > > 0 # t , 1320 < 2 0:00000010959 t 1 0:000250627; 1 h27 ðtÞ 5 0:000105985; 1320 # t , 5760 ð1=2Þ > > 0:000000307975 t 2 0:00166796; 5760 # t , 6840 > > : 0; t $ 6840 27 the conditional sojourn time η1ð1=3Þ has the chimney distribution with the density function 8 0 t,0 > > < 0:003387769; 0 # t , 287:8 27 h1ð1=3Þ ðtÞ 5 0:000017373; 287:8 # t , 1726:8 > > : 0; t $ 1726:8 1 the conditional sojourn time η27 ð1=3Þ has the double trapezium distribution with the density function 8 0; t,0 > > < 2 0:000000147 t 1 0:00035461; 0 # t , 2718 27 1 hð1=3Þ ðtÞ 5 0:0000000934 t 2 0:000299711; 2718 # t , 6768 > > : 0; t $ 6768 1 the conditional sojourn time η33 ð2=2Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:000022616; 0 # t , 43; 164 33 1 hð2=2Þ ðtÞ 5 0:00000011; 43; 164 # t , 258; 984 > > : 0; t $ 258; 984 33 the conditional sojourn time η1ð2=3Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:003389888; 0 # t , 287:8 33 h1ð2=3Þ ðtÞ 5 0:000016949; 287:8 # t , 1726:8 > > : 0; t $ 1726:8 1 the conditional sojourn time η33 ð2=3Þ has the exponential distribution with the density function  0; t,0 1 ðtÞ 5 h33 ð2=3Þ 0:0000898 exp½ 2 0:0000898t; t $ 0

(4.202)

(4.203)

(4.204)

(4.205)

(4.206)

(4.207)

Identification of critical infrastructure accident consequences Chapter | 4

G

G

G

1 the conditional sojourn time η24 ð3=2Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:000022616; 0 # t , 43; 164 24 1 hð3=2Þ ðtÞ 5 0:00000011; 43; 164 # t , 258; 984 > > : 0; t $ 258; 984 24 the conditional sojourn time η1ð3=3Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:003389888; 0 # t , 287:8 1 24 hð3=3Þ ðtÞ 5 0:000016949; 287:8 # t , 1726:8 > > : 0; t $ 1726:8

75

(4.208)

(4.209)

1 the conditional sojourn time η24 ð3=3Þ has the exponential distribution with the density function

 1 h24 ð3=3Þ ðtÞ 5

0; 0:00008066 exp½ 2 0:00008066t;

t,0 t $ 0:

(4.210)

For the above distributions, by application, either the general formula for the mean value or particular formulae ij given, respectively, by (3.30) and (3.31)(3.34), the mean values Mðk=lÞ 5 E½ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; l 5 1; 2; . . .; 16; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment threats conditional sojourn times at particular states can be determined, and they amount to 27 1 1 27 27 1 Mð1=2Þ D3685:10; Mð1=3Þ D165:485; Mð1=3Þ D3596:51; 33 1 1 33 33 1 Mð2=2Þ D24; 171:84; Mð2=3Þ D161:17; Mð2=3Þ D11; 135:86; 24 1 1 24 24 1 Mð3=2Þ D24; 171:84; Mð3=3Þ D161:17; Mð3=3Þ D12; 398:05:

(4.211)

In the case when as a result of the experiment, coming from experts, we have less than 30 realizations of the process of environment threats, we assume that these conditional sojourn times have the empirical density functions defined by n oo 1 n hijðk=lÞ ðtÞ 5 ij # γ:ηijðk=lÞγ AIz ; γA 1; 2; . . .; nijðk=lÞ ; z 5 1; 2; . . .; r ijðk=lÞ ; (4.212) nðk=lÞ corresponding to the empirical distribution functions defined by n oo 1 n ij Hðk=lÞ ðtÞ 5 ij # γ:ηijðk=lÞγ , t; γA 1; 2; . . .; nijðk=lÞ ; nðk=lÞ

(4.213)

for t $ 0; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; whereas the symbol # means the number of elements of the set. On the other hand, in the case when, as a result of the experiment, limited data come from experts, we only have the number of realizations of the process of environment threats lifetimes at the states, and all its realizations are equal to an approximate value, we assume that these conditional sojourn times have the uniform distribution in the interval from this value minus its half to this value plus its half. Thus we proceed in the analogous way as in Section 4.2.2.4 for the example conditional sojourn times θ63 and θ21 the density and distribution functions of which are expressed by (4.72)(4.75), respectively. When the distributions cannot be identified statistically, it is possible to find the approximate empirical values of ij the mean values Mðk=lÞ 5 E½ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; l 5 1; 2; . . .; 16; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the conditional sojourn times at particular states, using (4.101), that are as follows (Appendix 4, D-column of the table):

76

Consequences of Maritime Critical Infrastructure Accidents

1 27 71 27 7 1 23 1 30 23 1 30 1 Mð1=2Þ 5 1; Mð1=2Þ 5 240; Mð1=2Þ 5 240; Mð1=3Þ 5 1; Mð1=3Þ 5 1; Mð1=3Þ 5 240; Mð1=3Þ 5 240; 1 13 1 27 1 35 51 13 1 27 1 35 5 Mð1=4Þ 5 1; Mð1=4Þ 5 1; Mð1=4Þ 5 1; Mð1=4Þ 5 300; Mð1=4Þ 5 120; Mð1=4Þ 5 2546:25; Mð1=4Þ 5 120; 13 14 1 27 1 34 3 17 41 71 17 7 Mð1=5Þ 5 1; Mð1=5Þ 5 1; Mð1=5Þ 5 1; Mð1=5Þ 5 1; Mð1=5Þ 5 1; Mð1=5Þ 5 1147:50; Mð1=5Þ 5 5; Mð1=5Þ 5 300; 27 1 34 4 1 27 27 1 1 27 1 30 27 1 Mð1=5Þ 5 1488; Mð1=5Þ 5 15; Mð1=6Þ 5 1; Mð1=6Þ 5 5760; Mð1=7Þ 5 1; Mð1=7Þ 5 1; Mð1=7Þ 5 2185:8; 30 1 13 16 17 18 1 27 31 61 Mð1=7Þ 5 240; Mð1=8Þ 5 1; Mð1=8Þ 5 1; Mð1=8Þ 5 1; Mð1=8Þ 5 1; Mð1=8Þ 5 1; Mð1=8Þ 5 40; Mð1=8Þ 5 1100; 7 21 81 21 1 27 1 1 33 21 33 2 Mð1=8Þ 5 15; Mð1=8Þ 5 240; Mð1=8Þ 5 5; Mð1=8Þ 5 2648:4; Mð2=2Þ 5 1; Mð2=2Þ 5 280; 800; Mð2=2Þ 5 2880; 1 17 1 18 17 1 18 1 1 33 33 1 1 33 Mð2=3Þ 5 1; Mð2=3Þ 5 1; Mð2=3Þ 5 24; 480; Mð2=3Þ 5 43; 200; Mð2=4Þ 5 1; Mð2=4Þ 5 31; 050; Mð2=5Þ 5 1; 33 1 1 33 33 1 1 17 1 33 17 1 33 1 Mð2=5Þ 5 38; 016; Mð2=6Þ 5 1; Mð2=6Þ 5 12; 240; Mð2=7Þ 5 1; Mð2=7Þ 5 1; Mð2=7Þ 5 1440; Mð2=7Þ 5 8640; 12 14 15 1 33 21 41 51 Mð2=8Þ 5 1; Mð2=8Þ 5 1; Mð2=8Þ 5 1; Mð2=8Þ 5 1; Mð2=8Þ 5 7200; Mð2=8Þ 5 1440; Mð2=8Þ 5 1440; 33 1 1 24 21 24 2 1 14 1 15 14 1 Mð2=8Þ 5 12; 133:63; Mð3=2Þ 5 1; Mð3=2Þ 5 280; 800; Mð3=2Þ 5 2880; Mð3=3Þ 5 1; Mð3=3Þ 5 1; Mð3=3Þ 5 24; 480; 15 1 1 24 24 1 1 24 24 1 1 24 24 1 5 43; 200; Mð3=4Þ 5 1; Mð3=4Þ 5 31; 050; Mð3=5Þ 5 1; Mð3=5Þ 5 38; 016; Mð3=6Þ 5 1; Mð3=6Þ 5 12; 240; Mð3=3Þ 1 14 1 24 14 1 24 1 12 13 14 1 24 Mð3=7Þ 5 1; Mð3=7Þ 5 1; Mð3=7Þ 5 1440; Mð3=7Þ 5 8640; Mð3=8Þ 5 1; Mð3=8Þ 5 1; Mð3=8Þ 5 1; Mð3=8Þ 5 1; 21 31 41 24 1 12 1 24 Mð3=8Þ 5 7200; Mð3=8Þ 5 1440; Mð3=8Þ 5 1440; Mð3=8Þ 5 10; 497:27; Mð4=2Þ 5 1; Mð4=2Þ 5 549:33; 21 24 1 1 14 1 15 1 24 14 1 Mð4=2Þ 5 280; 800; Mð4=2Þ 5 35; 382:86; Mð4=3Þ 5 1; Mð4=3Þ 5 1; Mð4=3Þ 5 440:78; Mð4=3Þ 5 24; 480; 15 1 24 1 1 24 24 1 1 24 24 1 Mð4=3Þ 5 43; 200; Mð4=3Þ 5 32; 230:59; Mð4=4Þ 5 1; Mð4=4Þ 5 70; 765:71; Mð4=5Þ 5 1; Mð4=5Þ 5 91; 440; 1 24 24 1 1 14 1 24 14 1 24 1 12 Mð4=6Þ 5 720:5; Mð4=6Þ 5 20; 880; Mð4=7Þ 5 1; Mð4=7Þ 5 90:75; Mð4=7Þ 5 1440; Mð4=7Þ 5 18; 630; Mð4=8Þ 5 1; 14 1 24 21 41 24 1 1 24 24 1 Mð4=8Þ 5 1; Mð4=8Þ 5 1; Mð4=8Þ 5 7200; Mð4=8Þ 5 1440; Mð4=8Þ 5 34; 560; Mð5=2Þ 5 351:32; Mð5=2Þ 5 26; 441:05; 1 15 1 24 15 1 24 1 1 24 24 1 1 24 Mð5=3Þ 5 1; Mð5=3Þ 5 288:87; Mð5=3Þ 5 43; 200; Mð5=3Þ 5 22; 368; Mð5=4Þ 5 1; Mð5=4Þ 5 64; 260; Mð5=5Þ 5 1; 24 1 1 24 24 1 1 24 24 1 1 24 24 1 Mð5=5Þ 5 172; 800; Mð5=6Þ 5 1440; Mð5=6Þ 5 17; 280; Mð5=7Þ 5 1; Mð5=7Þ 5 11; 520; Mð5=8Þ 5 1; Mð5=8Þ 5 87; 480:

(4.214) As there are no realizations of the process of environment threats conditional sojourn times at the remaining states, it is impossible to estimate their empirical conditional mean values. These approximate mean values given by (4.211) and (4.214) are used in Section 5.2.2 for the prediction of the characteristics of the process of environment threats at the world sea waters.

4.5

Identification of process of environment degradation—theoretical background

We assume, as in Section 3.3, that the process of environment degradation in the subarea Dk, k 5 1; 2; . . .; n3 is taking ‘k 1 2 ‘k , ‘k AN different states of degradation effects rðkÞ ; rðkÞ ; . . .; rðkÞ : Next, we mark by Rðk=υÞ ðtÞ; tAh0; 1 NÞ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk the conditional process of the environment degradation in the subarea Dk, k 5 1; 2; . . .; n3 , while the process of environment threats SðkÞ ðtÞ in the subarea Dk is at the state sυðkÞ ; υ 5 1; 2; . . .; υk : The conditional process Rðk=υÞ ðtÞ is a function defined on the time interval tAh0; 1 NÞ depending on the states of ‘k 1 2 the process of environment threats SðkÞ ðtÞ and taking discrete values in the set frðk=υÞ ; rðk=υÞ ; . . .; rðk=υÞ g of the degradation effects states. We assume a semi-Markov model (Grabski, 2015; Kołowrocki, 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process of environment degradation Rðk=υÞ ðtÞ and j i we mark by ζ ijðk=υÞ its random conditional sojourn times at the states rðk=υÞ , when its next state is rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1,2,. . .,n3, υ 5 1; 2; . . .; υk : Under these assumption, the process of environment degradation Rðk=υÞ ðtÞ for each subarea Dk, k 5 1; 2; . . .; n3 may be described by the vector ½qiðk=υÞ ð0Þ1 3 ‘k of initial probabilities of the process of environment degradation staying at particular degradation effects states at the initial moment t 5 0, the matrix ½qijðk=υÞ ‘k 3 ‘k of probabilities of transitions between degradation effects states and the matrix ½Gijðk=υÞ ðtÞ‘k 3 ‘k of the distribution functions of conditional sojourn times ζ ijðk=υÞ of the process Rðk=υÞ ðtÞ at its particular states or equivalently by the matrix ½gijðk=υÞ ðtÞ‘k 3 ‘k of the density functions of the conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation at its states.

Identification of critical infrastructure accident consequences Chapter | 4

77

These all parameters of the process of environment degradation are unknown, and before their use to the prediction of this process, characteristics have to be estimated on the basis of statistical data coming from practice.

4.5.1 Estimating unknown basic parameters of process of environment degradation In order to estimate the parameters of the process of environment degradation Rðk=υÞ ðtÞ; first we should fix the number ‘k 1 2 of states ‘k of the process Rðk=υÞ ðtÞ and define the states rðk=υÞ ; rðk=υÞ ; . . .; rðk=υÞ of the set Rðk=υÞ : Further, we should fix the i vector of realizations nðk=υÞ ð0Þ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the numbers of the process Rðk=υÞ ðtÞ i transients at the particular states rðk=υÞ at the initial moment t 5 0 h i h i k 5 n1ðk=υÞ ð0Þ; n2ðk=υÞ ð0Þ; . . .; n‘ðk=υÞ ð0Þ ; (4.215) niðk=υÞ ð0Þ 1 3 ‘k

where n ðk=υÞ ð0Þ 5

‘k X i51

niðk=υÞ ð0Þ:

(4.216)

Next, we should fix the matrix of realizations nijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the j i numbers of process Rðk=υÞ ðtÞ transitions from the state rðk=υÞ into the state rðk=υÞ during the experimental time 2 3 k n12 : : : n1‘ n11 ðk=υÞ ðk=υÞ ðk=υÞ 6 21 h i 2‘k 7 22 6 7 (4.217) 5 6 nðk=υÞ nðk=υÞ : : : nðk=υÞ 7; nijðk=υÞ 4 : ‘k 3 ‘k : : : : : 5 ‘k 1 ‘k 2 k ‘k nðk=υÞ nðk=υÞ : : : n‘ðk=υÞ where niiðk=υÞ 5 0 for i 5 1; 2; . . .; ‘k ; and the vector of realizations of total numbers niðk=υÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of i environment degradations departures from the state rðk=υÞ h i h i k niðk=υÞ 5 n1ðk=υÞ ; n2ðk=υÞ ; . . .; n‘ðk=υÞ ; (4.218) 1 3 ‘k

where niðk=υÞ 5

‘k X j6¼i

nijðk=υÞ ;

i 5 1; 2; . . .; ‘k ;

k 5 1; 2; . . .; n3 ;

υ 5 1; 2; . . .; υk :

Having these numbers, we estimate the vector h i h i k 5 q1ðk=υÞ ð0Þ; q2ðk=υÞ ð0Þ; . . .; q‘ðk=υÞ ð0Þ ; qiðk=υÞ ð0Þ 1 3 ‘k

(4.219)

(4.220)

of realizations of the initial probabilities qiðk=υÞ ð0Þ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process Rðk=υÞ ðtÞ i transients at the particular states rðk=υÞ at the initial moment t 5 0 according to the formula qiðk=υÞ ð0Þ 5

niðk=υÞ ð0Þ nðk=υÞ ð0Þ

;

i 5 1; 2; . . .; ‘k ;

k 5 1; 2; . . .; n3 ;

υ 5 1; 2; . . .; υk ;

(4.221)

where nðk=υÞ ð0Þ is the total number of the process Rðk=υÞ ðtÞ realizations at the initial moment t 5 0, given by (4.216). Next, we evaluate the matrix

78

Consequences of Maritime Critical Infrastructure Accidents

2

q11 ðk=υÞ

6 21 6 ½qijðk=υÞ ‘k 3 ‘k 5 6 qðk=υÞ 4 : ‘k 1 qðk=υÞ

q12 ðk=υÞ

:

: :

q22 ðk=υÞ : ‘k 2 qðk=υÞ

: : :

: : : : : :

k q1‘ ðk=υÞ

3

7 k 7 q2‘ ðk=υÞ 7 : 5 k ‘k q‘ðk=υÞ

(4.222)

of realizations of the transitions probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process j i Rðk=υÞ ðtÞ from the state rðk=υÞ into the state rðk=υÞ during the experimental time according to the formula qijðk=υÞ 5

nijðk=υÞ niðk=υÞ

;

i; j 5 1; 2; . . .; ‘k ;

i 6¼ j;

k 5 1; 2; . . .; n3 ;

υ 5 1; 2; . . .; υk ;

(4.223)

and qiiðk=υÞ 5 0;

i 5 1; 2; . . .; ‘k ;

where niðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk is the realization of the total number of the process Rðk=υÞ ðtÞ i departures from the state rðk=υÞ during the experimental time, given by (4.219). Further, we formulate and verify the hypotheses about the form of distribution functions of the process Rðk=υÞ ðtÞ coni ditional sojourn time ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the state rðk=υÞ , while the next tranj ij ij sition is to the state rðk=υÞ on the basis of their realizations ζ ðk=υÞγ ; γ 5 1; 2; . . .; nðk=υÞ :

4.5.2 Estimating parameters of distributions of process of environment degradation conditional sojourn times at environment degradation states Prior to estimating the parameters of the distributions of the conditional sojourn times of the process of environment degradation Rðk=υÞ ðtÞ at the particular states, we have to determine the following empirical characteristics of the realizations of the conditional sojourn time of the process of environment degradation at the particular environment degradation states (Kołowrocki and Soszy´nska-Budny, 2011): G

ij

the realizations of the empirical mean values ζ ðk=υÞ of the conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; j i k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the state rðk=υÞ , while the next transition is to the state rðk=υÞ on the base of their ij ij realizations ζ ðk=υÞγ ; γ 5 1; 2; . . .; nðk=υÞ according to the formula nij

ij ζ ðk=υÞ G

G

5

ðk=υÞ 1 X

nijðk=υÞ

γ51

ζ ijðk=υÞγ ;

i; j 5 1; 2; . . .; ‘k ;

i 6¼ j;

k 5 1; 2; . . .; n3 ;

υ 5 1; 2; . . .; υk ;

(4.224)

D  the number r ijðk=υÞ of the disjoint intervals Iz 5 aijðk=υÞz ; bijðk=υÞz ; z 5 1; 2; . . .; r ijðk=υÞ that include the realizations ζ ijðk=υÞγ ; γ 5 1; 2; . . .; nijðk=υÞ of the conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at j i the state rðk=υÞ , while the next transition is to the state rðk=υÞ according to the formula qffiffiffiffiffiffiffiffiffiffi r ijðk=υÞ D nijðk=υÞ ; (4.225) D  ij the length dðk=υÞ of the intervals Iz 5 aijðk=υÞz ; bijðk=υÞz ; z 5 1; 2; . . .; r ijðk=υÞ according to the formula ij

ij dðk=υÞ

5

Rðk=υÞ

r ijðk=υÞ 2 1

;

(4.226)

where ij

Rðk=υÞ 5 G

max

1 # γ # nijðk=υÞ

ζ ijðk=υÞγ 2

min

1 # γ # nijðk=υÞ

ζ ijðk=υÞγ ;

(4.227)

D  the ends aijðk=υÞz ; bijðk=υÞz ; z 5 1; 2; . . .; r ijðk=υÞ of the intervals Iz 5 aijðk=υÞz ; bijðk=υÞz ; z 5 1; 2; . . .; r ijðk=υÞ according to the formula

Identification of critical infrastructure accident consequences Chapter | 4

( aijðk=υÞ1

5 max

min

1 # γ # nijðk=υÞ

ζ ijðk=υÞγ

2

ij dðk=υÞ

2

79

) ; 0 ;

(4.228)

ij bijðk=υÞz 5 aijðk=υÞ1 1 zdðk=υÞ ; z 5 1; 2; . . .; r ijðk=υÞ ;

(4.229)

aijðk=υÞz 5 bijðk=υÞz21 ; z 5 2; 3; . . .; r ijðk=υÞ ;

(4.230)

in such a way that I1 , I2 , ? , Irij

ðk=υÞ

 5 aijðk=υÞ1 ; bijðk=υÞrij ðk=υÞ

and Ii - Iz 5 [ for all i 6¼ z; i; z 5 1; 2; . . .; r ijðk=υÞ ;

G

D  the numbers nijðk=υÞz of the realizations ζ ijðk=υÞγ ; γ 5 1; 2; . . .; nijðk=υÞ in the intervals Iz 5 aijðk=υÞz ; bijðk=υÞz ; z 5 1; 2; . . .; r ijðk=υÞ according to the formula nijðk=υÞz 5 #fγ:ζ ijðk=υÞγ AIz ; γAf1; 2; . . .; nijðk=υÞ gg;

z 5 1; 2; . . .; r ijðk=υÞ ;

(4.231)

where rijðk=υÞ

X z51

nijðk=υÞz 5 nijðk=υÞ ;

whereas the symbol # means the number of elements of the set. To estimate the parameters of distributions of conditional sojourn times of the process of environment degradation at the particular states distinguished in Section 3.3, we proceed, respectively, in the following way (Kołowrocki and Soszy´nska-Budny, 2011): G for the exponential distribution with the density function given by (3.52), the estimates of the unknown parameters are xijðk=υÞ 5 aijðk=υÞ1 ; αijðk=υÞ 5 G

1 ij ζ ðk=υÞ

2 xijðk=υÞ

;

(4.232)

for the chimney distribution with the density function given by (3.53), the estimates of the unknown parameters are ij xijðk=υÞ 5 aijðk=υÞ1 ; yijðk=υÞ 5 xijðk=υÞ 1 r ijðk=υÞ dðk=υÞ ;

and moreover, if _ij n ðk=υÞ

5

n maxij

1 # z # r ðk=υÞ

nijðk=υÞz

(4.233)

o (4.234)

and iAf1; 2; . . .; r ijðk=υÞ g is the number of the interval including the largest number of realizations, that is, such that _ij

nijðk=υÞi 5 n ðk=υÞ ;

(4.235)

ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 1Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ 1 idðk=υÞ ;

(4.236)

then for i 5 1 either

Aijðk=υÞ

5 0;

ij Cðk=υÞ

5

nijðk=υÞi nijðk=υÞ

;

(4.237)

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Consequences of Maritime Critical Infrastructure Accidents

Dijðk=υÞ

nijðk=υÞi11 1 ? 1 nijðk=υÞrij

ðk=υÞ

5

nijðk=υÞ

;

(4.238)

while nijðk=υÞi11

5 0 or

nijðk=υÞi11

nijðk=υÞi

6¼ 0 and

nijðk=υÞi11

$ 3;

(4.239)

or ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 1Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ 1 ði 1 1Þdðk=υÞ ;

Aijðk=υÞ

ij Cðk=υÞ

5 0;

Dijðk=υÞ

5

5

nijðk=υÞi 1 nijðk=υÞi11 nijðk=υÞ

nijðk=υÞi12 1 ? 1 nijðk=υÞrij

ðk=υÞ

nijðk=υÞ

;

(4.240) (4.241)

;

(4.242)

while nijðk=υÞi11

nijðk=υÞi

6¼ 0 and

nijðk=υÞi11

, 3;

(4.243)

for i 5 2; 3; . . .; r ijðk=υÞ 2 1 either ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 1Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ 1 idðk=υÞ ;

Aijðk=υÞ 5

nijðk=υÞ1 1 ? 1 nijðk=υÞi21 nijðk=υÞ

Dijðk=υÞ

5

ij ; Cðk=υÞ 5

nijðk=υÞi11 1 ? 1 nijðk=υÞr ij

ðk=υÞ

nijðk=υÞ

nijðk=υÞi nijðk=υÞ

(4.244)

;

(4.245)

;

(4.246)

while nijðk=υÞi21 5 0 or nijðk=υÞi21 6¼ 0

nijðk=υÞi

and

nijðk=υÞi21

$3

(4.247)

$ 3;

(4.248)

and while nijðk=υÞi11

5 0 or

nijðk=υÞi11

6¼ 0

nijðk=υÞi

and

nijðk=υÞi21

or ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 1Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ 1 ði 1 1Þdðk=υÞ ;

Aijðk=υÞ

5

nijðk=υÞ1 1 ? 1 nijðk=υÞi21 nijðk=υÞ

;

ij Cðk=υÞ

5

nijðk=υÞi 1 nijðk=υÞi11 nijðk=υÞ

;

(4.249) (4.250)

Identification of critical infrastructure accident consequences Chapter | 4

Dijðk=υÞ

5

nijðk=υÞi12 1 ? 1 nijðk=υÞrij

ðk=υÞ

nijðk=υÞ

;

81

(4.251)

while nijðk=υÞi21

nijðk=υÞi21

5 0 or

nijðk=υÞi

6¼ 0 and

nijðk=υÞi21

$3

(4.252)

and while nijðk=υÞi11 6¼ 0

nijðk=υÞi

and

, 3;

nijðk=υÞi11

(4.253)

or ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 2Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ 1 idðk=υÞ ;

Aijðk=υÞ 5

nijðk=υÞ1 1 ? 1 nijðk=υÞi22 nijðk=υÞ

Dijðk=υÞ

5

;

ij Cðk=υÞ 5

nijðk=υÞi21 1 nijðk=υÞi nijðk=υÞ

nijðk=υÞi11 1 ? 1 nijðk=υÞrij

ðk=υÞ

nijðk=υÞ

;

(4.254) ;

(4.255)

(4.256)

while nijðk=υÞi21 6¼ 0 and

nijðk=υÞi nijðk=υÞi21

,3

(4.257)

and while nijðk=υÞi11 5 0 or nijðk=υÞi11 6¼ 0 and

nijðk=υÞi

$ 3;

nijðk=υÞi11

(4.258)

or ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 2Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ ði 1 1Þdðk=υÞ ;

Aijðk=υÞ 5 ij Cðk=υÞ

5

Dijðk=υÞ

nijðk=υÞ1 1 ? 1 nijðk=υÞi22 nijðk=υÞ

;

(4.260)

nijðk=υÞi21 1 nijðk=υÞi 1 nijðk=υÞi11

5

nijðk=υÞ

nijðk=υÞi12 1 ? 1 nijðk=υÞrij

ðk=υÞ

nijðk=υÞ

(4.259)

;

;

(4.261)

(4.262)

while nijðk=υÞi21

6¼ 0 and

nijðk=υÞi nijðk=υÞi21

,3

(4.263)

82

Consequences of Maritime Critical Infrastructure Accidents

and while nijðk=υÞi

nijðk=υÞi11 6¼ 0 and

nijðk=υÞi11

, 3;

(4.264)

for i 5 r ijðk=υÞ either ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 1Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ 1 idðk=υÞ ;

Aijðk=υÞ

5

nijðk=υÞ1 1 ? 1 nijðk=υÞi21 nijðk=υÞ

;

ij Cðk=υÞ

5

nijðk=υÞi nijðk=υÞ

(4.265)

;

(4.266)

Dijðk=υÞ 5 0;

(4.267)

while nijðk=υÞi21

5 0 or

nijðk=υÞi21

6¼ 0 and

nijðk=υÞi nijðk=υÞi21

$ 3;

(4.268)

or ij ij zijðk=υÞ1 5 xijðk=υÞ 1 ði 2 2Þdðk=υÞ ; zijðk=υÞ2 5 xijðk=υÞ 1 idðk=υÞ ;

Aijðk=υÞ 5

nijðk=υÞ1 1 ? 1 nijðk=υÞi22 nijðk=υÞ

ij ; Cðk=υÞ 5

nijðk=υÞi21 1 nijðk=υÞi

Dijðk=υÞ 5 0;

nijðk=υÞ

(4.269) ;

(4.270) (4.271)

while nijðk=υÞi21 6¼ 0 and

nijðk=υÞi

nijðk=υÞi21

, 3:

(4.272)

4.5.3 Identification of distribution functions of process of environment degradation conditional sojourn times at environment degradation states To formulate and next to verify the nonparametric hypothesis concerning the form of the distribution of the process of j i environment degradation conditional sojourn time ζ ijðk=υÞ at the state rðk=υÞ when its next state is rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; ij ij i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; on the basis of its realizations ζ ðk=υÞγ ; γ 5 1; 2; . . .; nðk=υÞ ; it is due to proceed according to the following scheme (Kołowrocki and Soszy´nska-Budny, 2011): G

to construct and to plot the realization of the histogram of the process of environment degradation conditional i sojourn time ζ ijðk=υÞ at the state rðk=υÞ (Fig. 4.8) defined by the following formula: nij

ðk=υÞ gðk=υÞ ðtÞ 5

G

nij

nijðk=υÞz nijðk=υÞ

for tAI z ;

(4.273)

ðk=υÞ to analyze the realization of the histogram gðk=υÞ ðtÞ; comparing it with the graphs of the density functions gijðk=υÞ ðtÞ distinguished in Section 3.3 distributions, to select one of them and to formulate the null hypothesis H0 concerning the unknown form of the distribution of the conditional sojourn time ζ ijðk=υÞ in the following form: i H0: The process of environment degradation conditional sojourn times ζ ijðk=υÞ at the state rðk=υÞ , while the next j ij transition to the state rðk=υÞ has the distribution with the density function gðk=υÞ ðtÞ;

Identification of critical infrastructure accident consequences Chapter | 4

83

FIGURE 4.8 The graph of the realization of the histogram of the process of environment degradation conditional sojourn time ζ ijðk=υÞ at the i state rðk=υÞ . G

G G

G G

to join each of the intervals Iz that has the number nijðk=υÞz of realizations less than 4 either with the neighbor interval Iz11 or with the neighbor interval Iz21 in a way that the numbers of realizations in all intervals are not less than 4; ij to fix a new number of intervals r ðk=υÞ ; to determine new intervals D  ij ij I z 5 aijðk=υÞz ; bðk=υÞz ; z 5 1; 2; . . .; r ðk=υÞ ; ij

to fix the numbers nijðk=υÞz of realizations in new intervals I z ; z 5 1; 2; . . .; r ðk=υÞ ; to calculate the hypothetical probabilities that the variable ζ ijðk=υÞ takes values from the interval I z under the assumption that the hypothesis H0 is true, that is, the probabilities ij

ij

qz 5 Pðζ ijðk=υÞ AI z Þ 5 Pðaijðk=υÞz # ζ ijðk=υÞ , bðk=υÞz Þ 5 Gijðk=υÞ ðbðk=υÞz Þ 2 Gijðk=υÞ ðaijðk=υÞz Þ;

ij

z 5 1; 2; . . .; r ðk=υÞ ;

(4.274)

ij

G

where Gijðk=υÞ ðbðk=υÞz Þ and Gijðk=υÞ ðaijðk=υÞz Þ are the values of the distribution function Gijðk=υÞ ðtÞ of the random variable ζ ijðk=υÞ corresponding to the density function gijðk=υÞ ðtÞ assumedij in the null hypothesis H0 ; n to calculate the realization of the χ2-Pearson’s statistics U ðk=υÞ according to the formula ij

u

nijðk=υÞ

X ðnijðk=υÞz 2nijðk=υÞ qz Þ2

r ðk=υÞ

5

z51 G G

G

G

G

nijðk=υÞ qz

;

(4.275)

to assume the significance level α (for instance α 5 0:05) of the test; ij to fix the number r ðk=υÞ 2 j 2 1 of degrees of freedom, substituting for j, distinguished in Section 3.3 distributions, respectively, the values are as follows: j 5 0 for the chimney distribution and j 5 1 for the exponential distribution; to read from the table of the χ2-Pearson’s distribution (Appendix 7), the value uα for the fixed values of the signifiij cance level α and the number of degrees of freedom r ðk=υÞ 2 j 2 1 such that the following equality holds  ij  n P U ðk=υÞ . uα 5 α; (4.276) to determine the acceptance domain in the form of the interval h0; uα i; and the critical domain in the form of the interval ðuα ; 1 NÞ as presented inij Fig. 4.2; and n nij to compare the obtained value u ðk=υÞ of the realization of the statistics U ðk=υÞ with the read from the table critical value uα of the chi-square random variable and to decide on the previously formulated ijnull hypothesis H0 in the folnij n lowing way: if the value u ðk=υÞ does not belong to the critical domain, that is, when u ðk=υÞ # uijα , we do not reject the nijðk=υÞ n hypothesis H0; otherwise if the value u belongs to the critical domain, that is, when u ðk=υÞ . uα , we reject the hypothesis H0.

4.6 Identification of process of environment degradation—application to Baltic Sea and world sea waters The statistical identification of the process of environment degradation is performed on the basis of the data coming from free-accessible reports of chemical accidents at sea that happened around the world in a period of 11 years

84

Consequences of Maritime Critical Infrastructure Accidents

(CEDRE, 2018; GISIS, 2018). The identification of the process of environment degradation for the world sea waters is sufficiently accurate as the set of statistical data is most populated. Unfortunately, the less accurate identification of the process of environment degradation is performed for the Baltic Sea waters as the less sufficiently numerous set of statistical data.

4.6.1 States of process of environment degradation Considering (3.45) and (3.46), we distinguish for each environment subarea Dk, k 5 1; 2; . . .; 5 the following degradation ‘ states rðkÞ ; ‘ 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23: G

for subarea D1—air 1 2 3 4 rð1Þ 5 ½0; 0; 0; 0; 0; rð1Þ 5 ½0; 0; 0; 0; 1; rð1Þ 5 ½0; 0; 0; 0; 2; rð1Þ 5 ½0; 0; 0; 0; 3; 5 6 7 8 rð1Þ 5 ½0; 0; 0; 1; 0; rð1Þ 5 ½0; 0; 0; 1; 1; rð1Þ 5 ½0; 0; 0; 1; 2; rð1Þ 5 ½0; 0; 0; 2; 2; 9 10 11 12 5 ½0; 0; 0; 2; 3; rð1Þ 5 ½0; 0; 0; 3; 3; rð1Þ 5 ½0; 0; 1; 0; 0; rð1Þ 5 ½0; 0; 1; 0; 2; rð1Þ 13 14 15 16 5 ½0; 0; 1; 1; 1; rð1Þ 5 ½0; 0; 2; 2; 2; rð1Þ 5 ½0; 0; 2; 2; 3; rð1Þ 5 ½0; 0; 2; 3; 3; rð1Þ 17 18 19 20 5 ½0; 1; 0; 0; 0; rð1Þ 5 ½0; 2; 0; 0; 0; rð1Þ 5 ½0; 3; 0; 0; 0; rð1Þ 5 ½1; 0; 0; 0; 0; rð1Þ 21 22 23 24 5 ½1; 1; 0; 1; 0; rð1Þ 5 ½1; 1; 0; 2; 2; rð1Þ 5 ½2; 1; 0; 0; 0; rð1Þ 5 ½2; 2; 0; 2; 0; rð1Þ 25 26 27 28 rð1Þ 5 ½2; 2; 0; 3; 3; rð1Þ 5 ½3; 2; 0; 0; 0; rð1Þ 5 ½3; 3; 0; 3; 0; rð1Þ 5 ½3; 3; 0; 3; 3; 29 30 5 ½2; 0; 0; 0; 0; rð1Þ 5 ½3; 1; 0; 0; 0; rð1Þ

G

for subarea D2—water surface: 1 2 3 4 5 ½0; 0; 0; 0; 0; rð2Þ 5 ½0; 0; 0; 0; 1; rð2Þ 5 ½0; 0; 0; 0; 2; rð2Þ 5 ½0; 0; 0; 0; 3; rð2Þ 5 6 7 8 5 ½0; 0; 0; 1; 0; rð2Þ 5 ½0; 0; 0; 1; 1; rð2Þ 5 ½0; 0; 0; 1; 2; rð2Þ 5 ½0; 0; 0; 2; 0; rð2Þ 9 10 11 12 5 ½0; 0; 0; 2; 2; rð2Þ 5 ½0; 0; 0; 2; 3; rð2Þ 5 ½0; 0; 0; 3; 3; rð2Þ 5 ½0; 0; 1; 0; 1; rð2Þ 13 14 15 16 rð2Þ 5 ½0; 0; 1; 1; 1; rð2Þ 5 ½0; 0; 1; 1; 2; rð2Þ 5 ½0; 0; 2; 0; 1; rð2Þ 5 ½0; 0; 2; 0; 2; 17 18 19 20 5 ½0; 0; 2; 1; 2; rð2Þ 5 ½0; 0; 2; 2; 3; rð2Þ 5 ½0; 0; 2; 3; 3; rð2Þ 5 ½0; 0; 3; 0; 2; rð2Þ 21 22 23 24 5 ½0; 0; 3; 0; 3; rð2Þ 5 ½0; 0; 3; 1; 2; rð2Þ 5 ½0; 0; 3; 2; 3; rð2Þ 5 ½1; 0; 0; 0; 0; rð2Þ 25 26 27 28 5 ½1; 0; 3; 0; 3; rð2Þ 5 ½1; 1; 0; 2; 2; rð2Þ 5 ½2; 0; 3; 0; 3; rð2Þ 5 ½2; 1; 0; 0; 0; rð2Þ

G

for subarea D3—water column: 1 2 3 4 rð3Þ 5 ½0; 0; 0; 0; 0; rð3Þ 5 ½0; 0; 0; 0; 1; rð3Þ 5 ½0; 0; 0; 0; 2; rð3Þ 5 ½0; 0; 0; 0; 3; 5 6 7 8 5 ½0; 0; 0; 1; 0; rð3Þ 5 ½0; 0; 0; 1; 1; rð3Þ 5 ½0; 0; 0; 1; 2; rð3Þ 5 ½0; 0; 0; 2; 0; rð3Þ 9 10 11 12 5 ½0; 0; 0; 2; 2; rð3Þ 5 ½0; 0; 0; 2; 3; rð3Þ 5 ½0; 0; 0; 3; 3; rð3Þ 5 ½0; 0; 1; 0; 1; rð3Þ 13 14 15 16 5 ½0; 0; 1; 1; 1; rð3Þ 5 ½0; 0; 1; 1; 2; rð3Þ 5 ½0; 0; 2; 0; 1; rð3Þ 5 ½0; 0; 2; 0; 2; rð3Þ 17 18 19 20 5 ½0; 0; 2; 1; 2; rð3Þ 5 ½0; 0; 2; 2; 3; rð3Þ 5 ½0; 0; 2; 3; 3; rð3Þ 5 ½0; 0; 3; 0; 2; rð3Þ 21 22 23 24 5 ½0; 0; 3; 0; 3; rð3Þ 5 ½0; 0; 3; 1; 2; rð3Þ 5 ½0; 0; 3; 2; 3; rð3Þ 5 ½1; 0; 0; 0; 0; rð3Þ 25 26 27 28 5 ½1; 0; 3; 0; 3; rð3Þ 5 ½1; 1; 0; 2; 2; rð3Þ 5 ½2; 0; 3; 0; 3; rð3Þ 5 ½2; 1; 0; 0; 0; rð3Þ

Identification of critical infrastructure accident consequences Chapter | 4

G

85

for subarea D4—sea floor: 1 2 3 4 rð4Þ 5 ½0; 0; 0; 0; 0; rð4Þ 5 ½0; 0; 0; 0; 1; rð4Þ 5 ½0; 0; 0; 0; 2; rð4Þ 5 ½0; 0; 0; 0; 3; 5 6 7 8 rð4Þ 5 ½0; 0; 0; 1; 0; rð4Þ 5 ½0; 0; 0; 1; 1; rð4Þ 5 ½0; 0; 0; 1; 2; rð4Þ 5 ½0; 0; 0; 2; 0; 9 10 11 12 5 ½0; 0; 0; 2; 2; rð4Þ 5 ½0; 0; 0; 2; 3; rð4Þ 5 ½0; 0; 0; 3; 3; rð4Þ 5 ½0; 0; 1; 0; 1; rð4Þ 13 14 15 16 rð4Þ 5 ½0; 0; 1; 1; 1; rð4Þ 5 ½0; 0; 1; 1; 2; rð4Þ 5 ½0; 0; 2; 0; 1; rð4Þ 5 ½0; 0; 2; 0; 2; 17 18 19 20 5 ½0; 0; 2; 1; 2; rð4Þ 5 ½0; 0; 2; 2; 3; rð4Þ 5 ½0; 0; 2; 3; 3; rð4Þ 5 ½0; 0; 3; 0; 2; rð4Þ 21 22 23 24 5 ½0; 0; 3; 0; 3; rð4Þ 5 ½0; 0; 3; 1; 2; rð4Þ 5 ½0; 0; 3; 2; 3; rð4Þ 5 ½0; 1; 0; 1; 1; rð4Þ 25 26 27 28 5 ½0; 1; 0; 2; 2; rð4Þ 5 ½0; 1; 0; 3; 3; rð4Þ 5 ½1; 0; 0; 0; 0; rð4Þ 5 ½1; 0; 3; 0; 3; rð4Þ 29 30 31 5 ½1; 1; 0; 2; 2; rð4Þ 5 ½2; 0; 3; 0; 3; rð4Þ 5 ½2; 1; 0; 0; 0; rð4Þ

G

for subarea D5—coast (shoreline): 1 2 3 4 rð5Þ 5 ½0; 0; 0; 0; 0; rð5Þ 5 ½0; 0; 0; 0; 1; rð5Þ 5 ½0; 0; 0; 0; 2; rð5Þ 5 ½0; 0; 0; 0; 3; 5 6 7 8 rð5Þ 5 ½0; 0; 0; 1; 0; rð5Þ 5 ½0; 0; 0; 1; 1; rð5Þ 5 ½0; 0; 0; 1; 2; rð5Þ 5 ½0; 0; 0; 2; 0; 9 10 11 12 5 ½0; 0; 0; 2; 2; rð5Þ 5 ½0; 0; 0; 2; 3; rð5Þ 5 ½0; 0; 0; 3; 3; rð5Þ 5 ½0; 0; 1; 0; 1; rð5Þ 13 14 15 16 5 ½0; 0; 1; 1; 1; rð5Þ 5 ½0; 0; 1; 1; 2; rð5Þ 5 ½0; 0; 2; 0; 1; rð5Þ 5 ½0; 0; 2; 0; 2; rð5Þ 17 18 19 20 5 ½0; 0; 2; 1; 2; rð5Þ 5 ½0; 0; 2; 2; 3; rð5Þ 5 ½0; 0; 2; 3; 3; rð5Þ 5 ½0; 0; 3; 0; 2; rð5Þ 21 22 23 5 ½0; 0; 3; 0; 3; rð5Þ 5 ½0; 0; 3; 1; 2; rð5Þ 5 ½0; 0; 3; 2; 3: rð5Þ

The states of the process of environment degradations are defined as the vector where 0 means any degradation effect does not appear, but the other digit means the level of appeared degradation effect. For instance, the vector G r 1 5 ½0; 0; 0; 0; 0 means there are no degradation effects and the environment subarea D1 of the accident area is not ð1Þ degraded, G r 17 5 ½0; 1; 0; 0; 0 means the released substance causes the degradation effect R2 (the oxygen concentration has chanð1Þ ged) in the environment subarea D1 and the oxygen concentration decreased the value up to 1 level (no more than 2%), G r 18 5 ½0; 2; 0; 0; 0 means the released substance causes the degradation effect R2 (the oxygen concentration has chanð1Þ ged) in the environment subarea D1, and the oxygen concentration decreased the value up to 2 level (more than 2% but less than 5%), G r 23 5 ½2; 1; 0; 0; 0 means the released substance causes simultaneously the degradation effect R1 and R2 (the temperð1Þ ature and the oxygen concentration have changed) in the environment subarea D1, and the temperature increased the value up to 2 level (more than 10 C but less than 20 C), and the oxygen concentration decreased the value up to 1 level (no more than 2%). Moreover, the process of environment degradation of the each subarea is involved with the process of environment threats, namely, the state of the process of environment degradation is conditional upon the state of the process of envi2 ronment threats that caused the previous one. For instance, rð5=3Þ means the state of the process of environment degradation of the environment subarea D5 when the process of environment threats is at the state s3ð5Þ :

4.6.2 Statistical identification of process of environment degradation at Baltic Sea waters The number of the process of environment degradation states ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 for each subi area Dk, k 5 1; 2; . . .; 5 are fixed and the degradation effects states rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 are defined in Section 4.6.1. In addition, we involve the process of environment threats with i the process of environment degradation. Thus we fix environment degradation effects states rðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29

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for a particular subarea Dk, k 5 1; 2; . . .; 5 according to the environment threat that cause the environment degradation effects state. Moreover, it is assumed that there are possible transitions between all states of the process of environment degradation.

4.6.2.1 Collected statistical data of process of environment degradation The experiment was performed within the Baltic Sea waters in years 200414. The initial moment t 5 0 of the process of environment degradation was fixed at the moment when the threat caused by ship accident generated one of the distinguished degradation effects states. The unknown parameters of the process of environment degradation semi-Markov model are the initial probabilities qiðk=υÞ ð0Þ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation staying at the i particular states rðk=υÞ at the initial moment t 5 0; ij G the probabilities q ðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘ 1 5 30; ‘2 5 28; ‘ 3 5 28; ‘ 4 5 31; ‘ 5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation transitions j i from the state rðk=υÞ into the state rðk=υÞ ; ij G the distributions of the environment degradation conditional sojourn times ζ ðk=υÞ ; i; j 5 1; 2; . . .; ‘ k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; ij υ5 5 29 at the particular states and their mean values Mðk=υÞ 5 E½ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29: To identify all these parameters of the process of environment degradation, the statistical data about this process is needed. The statistical data for the conditional sojourn times ζ ijðk=υÞ at the states of the processes of environment degradation are given in Appendix 5. The collected statistical data necessary to evaluating the initial transient probabilities of process of environment degradation at the particular states are: G the process of environment degradation observation/experiment time G

Θ 5 11 years (200414); G

niðk=υÞ ð0Þ

the vector of realizations of the number of the processes of environment degradation staying at the particui lar states rðk=υÞ at the initial moment t 5 0 according to (4.215) h i niðk=υÞ ð0Þ 5 ½108; 0; . . .; 0 for k 5 1; υ 5 1; ‘1 5 30; h

i niðk=υÞ ð0Þ

h

i niðk=υÞ ð0Þ

h

i niðk=υÞ ð0Þ

h

i niðk=υÞ ð0Þ

h

i niðk=υÞ ð0Þ

h

1 3 ‘k 1 3 ‘k 1 3 ‘k 1 3 ‘k 1 3 ‘k 1 3 ‘k

5 ½107; 0; . . .; 0 for k 5 2; υ 5 1; ‘2 5 28; and k 5 3; υ 5 1; ‘3 5 28; 5 ½105; 0; . . .; 0 for k 5 4; υ 5 1; ‘4 5 31; 5 ½104; 0; . . .; 0 for k 5 5; υ 5 1; ‘5 5 23; 5 ½2; 0; . . .; 0 for k 5 1; υ 5 27; ‘1 5 30; and k 5 2; υ 5 33; ‘2 5 28; and k 5 3; υ 5 24; ‘3 5 28; 5 ½1; 0; . . .; 0 for k 5 1; υ 5 6; 30; ‘1 5 30; and k 5 2; υ 5 17; ‘2 5 28; and

k 5 3; υ 5 14; ‘3 5 28; and k 5 4; υ 5 14; ‘4 5 31; i 5 ½0; 0; . . .; 0 for k 5 1; υ 5 2; 3; 4; 5; 7; 8; . . .; 26; 28; 29; . . .; 35; ‘1 5 30; and k 5 2; niðk=υÞ ð0Þ 1 3 ‘k

υ 5 1; 2; . . .16; 18; 19; . . .; 32; ‘2 5 28; and k 5 3; υ 5 1; 2; . . .; 13; 15; 16; . . .; 23; 25; 26; . . .; 29; ‘3 5 28; and k 5 4; υ 5 1; 2; . . .; 13; 15; 16; . . .; 29; ‘4 5 31; k 5 5; υ 5 2; 3; . . .29; ‘5 5 23;

Identification of critical infrastructure accident consequences Chapter | 4

G

87

the numbers of the process of environment degradation realizations at the initial moment t 5 0 according to (4.216) nð1=υÞ ð0Þ 5 108; for υ 5 1; nð1=υÞ ð0Þ 5 2; for υ 5 27; nð1=υÞ ð0Þ 5 1; for υ 5 6; 30; nð1=υÞ ð0Þ 5 0; for υ 5 2; 3; 4; 5; 7; 8. . .; 26; 28; 29; 31; 32; . . .; 35; nð2=υÞ ð0Þ 5 107; for υ 5 1; nð2=υÞ ð0Þ 5 2; for υ 5 33; nð2=υÞ ð0Þ 5 1; for υ 5 17; nð2=υÞ ð0Þ 5 0; for υ 5 2; 3; . . .; 16; 18; 19; . . .; 32; nð3=υÞ ð0Þ 5 107; for υ 5 1; nð3=υÞ ð0Þ 5 2; for υ 5 24; nð3=υÞ ð0Þ 5 1; for υ 5 14; nð3=υÞ ð0Þ 5 0; for υ 5 2; 3; . . .; 13; 15; 16; . . .23; 25; 26; . . .; 29; nð4=υÞ ð0Þ 5 105; for υ 5 1; nð4=υÞ ð0Þ 5 1; for υ 5 14; nð4=υÞ ð0Þ 5 0; for υ 5 2; 3; . . .; 13; 15; 16; . . .23; 25; 26; . . .; 29; nð5=υÞ ð0Þ 5 104; for υ 5 1; nð5=υÞ ð0Þ 5 0; for υ 5 2; 3; . . .; 29:

The collected statistical data necessary for evaluating the probabilities of transitions between states are: the matrices of realizations nijðk=υÞ of the number of the process of environment degradation transitions from j i the state rðk=υÞ into the state rðk=υÞ ; according to (4.217), during the experimental time. The numbers of transitions that are not equal to 0 are presented only and they are as follows (Appendix 5, C-column of the table):

G

2 1 n1ð1=6Þ 5 1; n2ð1=6Þ 5 1; 6 1 n1ð1=27Þ 5 2; n6ð1=27Þ 5 2; 11 1 n1ð1=30Þ 5 1; n11 ð1=30Þ 5 1; 27 1 16 12 21 16 25 21 27 25 n1ð2=17Þ 5 1; n12 ð2=17Þ 5 1; nð2=17Þ 5 1; nð2=17Þ 5 1; nð2=17Þ 5 1; nð2=17Þ 5 1; 6 1 n1ð2=33Þ 5 2; n6ð2=33Þ 5 2; 27 1 16 12 21 16 25 21 27 25 n1ð3=14Þ 5 1; n12 ð3=14Þ 5 1; nð3=14Þ 5 1; nð3=14Þ 5 1; nð3=14Þ 5 1; nð3=14Þ 5 1; 6 1 n1ð3=24Þ 5 2; n6ð3=24Þ 5 2; 30 1 16 12 21 16 28 21 30 28 n1ð4=14Þ 5 1; n12 ð4=14Þ 5 1; nð4=14Þ 5 1; nð4=14Þ 5 1; nð4=14Þ 5 1; nð4=14Þ 5 1; G

the vectors of realization of the total number of the process of environment degradation transitions from the state i rðk=υÞ ; according to (4.218), during the experimental time

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Consequences of Maritime Critical Infrastructure Accidents

h h h h h h h h h

nið1=υÞ nið1=υÞ nið2=υÞ nið2=υÞ nið3=υÞ nið3=υÞ nið4=υÞ nið4=υÞ nið5=υÞ

i 1 3 30

i

1 3 30

i

1 3 28

i

1 3 28

i

1 3 28

i

1 3 28

i

1 3 31

i

1 3 31

i

1 3 23

5 ½1; 0; . . .; 0; for υ 5 6; 27; 30; 5 ½0; 0; . . .; 0; for υ 5 1; 2; . . .; 5; 7; 8; . . .; 26; 28; 29; 31; 32; . . .; 35; 5 ½1; 0; . . .; 0; for υ 5 17; 33; 5 ½0; 0; . . .; 0; for υ 5 1; 2; . . .; 16; 18; 19; . . .; 32; 5 ½1; 0; . . .; 0; for υ 5 14; 24; 5 ½0; 0; . . .; 0; for υ 5 1; 2; . . .; 13; 15; 16; . . .; 23; 25; 26; . . .; 29; 5 ½1; 0; . . .; 0; for υ 5 14; 5 ½0; 0; . . .; 0; for υ 5 1; 2; . . .; 13; 15; 16; . . .; 29; 5 ½0; 0; . . .; 0; for υ 5 1; 2; . . .; 29:

4.6.2.2 Evaluating basic parameters of process of environment degradation On the basis of the statistical data obtained in Section 4.6.2.1, using the formulae given in Section 4.5.1, it is possible to evaluate the following unknown basic parameters of the process of environment degradation: G

the vectors of the initial probabilities qiðk=υÞ ð0Þ of the processes of environment degradation at the particular states at the initial moment t 5 0 according to (4.220) as follows: ½qiðk=υÞ ð0Þ1 3 ‘k 5 ½1; 0; . . .; 0;

i 5 1; 2; . . .; ‘k ;

k 5 1; k 5 2; k 5 3; k 5 4; k 5 5; h i qiðk=υÞ ð0Þ

i 5 1; 2; . . .; ‘k ;

(4.277)

for

1 3 ‘k

υ 5 1; 6; 27; 30; ‘1 5 30; υ 5 1; 17; 33; ‘2 5 28; υ 5 1; 14; 24; ‘3 5 28; υ 5 1; 14; ‘4 5 31; υ 5 1; ‘5 5 23; 5 ½0; 0; . . .; 0;

(4.278)

for remaining cases: k 5 1; k 5 2; k 5 3; k 5 4; k 5 5; G

υ 5 2; 3; 4; 5; 7; 8; . . .; 26; 28; 29; 31; 32; . . .; 35; ‘1 5 30; υ 5 2; 3; . . .; 16; 18; 19; . . .; 32; ‘2 5 28; υ 5 2; 3; . . .; 13; 15; 16; . . .; 23; 25; 26; . . .; 29; ‘3 5 28; υ 5 2; 3; . . .; 13; 15; 16; . . .; 29; ‘4 5 31; υ 5 1; 2; . . .; 29; ‘5 5 23;

the matrices ½qijðk=υÞ ‘k 3 ‘k ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the probabilities of transitions of the process Rðk=υÞ ðtÞ j i from the state rðk=υÞ into the state rðk=υÞ ; according to (4.222), during the experimental time. The probabilities of transitions that are not equal to 0 are as follows: 2 1 q1ð1=6Þ 5 1; q2ð1=6Þ 5 1;

(4.279)

6 1 q1ð1=27Þ 5 1; q6ð1=27Þ 5 1;

(4.280)

Identification of critical infrastructure accident consequences Chapter | 4

89

11 1 q1ð1=30Þ 5 1; q11 ð1=30Þ 5 1;

(4.281)

27 1 16 12 21 16 25 21 27 25 q1ð2=17Þ 5 1; q12 ð2=17Þ 5 1; qð2=17Þ 5 1; qð2=17Þ 5 1; qð2=17Þ 5 1; qð2=17Þ 5 1;

(4.282)

6 1 q1ð2=33Þ 5 1; q6ð2=33Þ 5 1;

(4.283)

27 1 16 12 21 16 25 21 27 25 q1ð3=14Þ 5 1; q12 ð3=14Þ 5 1; qð3=14Þ 5 1; qð3=14Þ 5 1; qð3=14Þ 5 1; qð3=14Þ 5 1;

(4.284)

6 1 q1ð3=24Þ 5 1; q6ð3=24Þ 5 1;

(4.285)

30 1 16 12 21 16 28 21 30 28 q1ð4=14Þ 5 1; q12 ð4=14Þ 5 1; qð4=14Þ 5 1; qð4=14Þ 5 1; qð4=14Þ 5 1; qð4=14Þ 5 1:

(4.286)

The values of some probabilities existing in the vectors (4.277) and (4.278) and in the matrices (4.279)(4.286), besides of these standing on the main diagonal, equal to zero do not mean that the events they are concerned with, cannot appear. They evaluated on the basis of real statistical data and their values may change and become more precise if the duration of the experiment is longer.

4.6.2.3 Identification of distribution functions of conditional sojourn times at environment degradation states As we only have the number of realizations of the process of environment degradation and all its realizations are equal to an approximate value, we assume that these conditional sojourn times have the uniform distribution in the interval from this value minus its half to this value plus its half. Thus we proceed in the analogous way as in Section 4.2.2.4 for the example conditional sojourn time θ21 which density and distribution function is expressed by (4.74) and (4.75), respectively. For these distributions that cannot be identified statistically, by application the general formula for the mean value ij given by (4.224), the mean values Mðk=υÞ 5 E½ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation conditional sojourn times at particular states can be determined and they amount to (Appendix 5, D-column of the table): 12 21 16 61 1 11 11 1 1 27 12 1 Mð1=6Þ 5 1; Mð1=6Þ 5 240; Mð1=27Þ 5 1; Mð1=27Þ 5 240; Mð1=30Þ 5 1; Mð1=30Þ 5 240; Mð2=17Þ 5 1; Mð2=17Þ 5 2880; 16 12 21 16 25 21 27 25 16 61 1 27 Mð2=17Þ 5 3780; Mð2=17Þ 5 2880; Mð2=17Þ 5 300; Mð2=17Þ 5 240; Mð2=33Þ 5 1; Mð2=33Þ 5 1440; Mð3=14Þ 5 1; 12 1 16 12 21 16 25 21 27 25 16 61 5 2880; Mð3=14Þ 5 3780; Mð3=14Þ 5 2880; Mð3=14Þ 5 300; Mð3=14Þ 5 240; Mð3=24Þ 5 1; Mð3=24Þ 5 1440; Mð3=14Þ 1 30 12 1 16 12 21 16 28 21 30 28 Mð4=14Þ 5 1; Mð4=14Þ 5 2880; Mð4=14Þ 5 3780; Mð4=14Þ 5 2880; Mð4=14Þ 5 300; Mð4=14Þ 5 240:

(4.287) As there are no realizations of the process of environment degradation conditional sojourn times at remaining states, it is impossible to estimate their empirical conditional mean values. These approximate mean values given by (4.287) are used in Section 5.3.1 for the prediction of the characteristics of the process of environment degradation at the Baltic Sea waters.

4.6.3 Extension of statistical identification of process of environment degradation for world sea waters To statistically identify the process of environment degradation at the world sea waters, we used the same kind and number of this process states that are defined in Section 4.6.1. The experiment was performed within the world sea and ocean waters in years 200414. The initial moment t 5 0 of the process of environment degradation was fixed at the moment when the threat caused by ship accident generated one of the distinguished degradation effects states. The unknown parameters of the process of environment degradation semi-Markov model are G

the initial probabilities qiðk=υÞ ð0Þ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation staying at the i particular states rðk=υÞ at the initial moment t 5 0;

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Consequences of Maritime Critical Infrastructure Accidents

the probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation transitions j i from the state rðk=υÞ into the state rðk=υÞ ; ij G the distributions of the environment degradation conditional sojourn times ζ ðk=υÞ ; i; j 5 1; 2; . . .; ‘ k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; ij υ5 5 29 at the particular states and their mean values Mðk=υÞ 5 E½ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29: We identify all these parameters on the basis of statistical data coming from the process of environment degradation realization (Appendix 6). Using procedure and the formulae given in Section 4.5.1 and proceedings in an analogous way as in Section 4.6.2.2, we obtain the following results: G the vectors of the initial probabilities qi ðk=υÞ ð0Þ of the process of environment degradation at the particular states at the initial moment t 5 0 according to (4.220) as follows: h i 5 ½1; 0; . . .; 0; i 5 1; 2; . . .; ‘k ; (4.288) qiðk=υÞ ð0Þ G

1 3 ‘k

for k 5 1; k 5 2; k 5 3; k 5 4; k 5 5;

υ 5 1; 3; 4; 5; 6; 7; 8; 13; 17; 21; 23; 27; 30; 34; 35; ‘1 5 30; υ 5 1; 2; 4; 5; 17; 18; 33; ‘2 5 28; υ 5 1; 2; 3; 4; 14; 15; 24; ‘3 5 28; υ 5 1; 2; 4; 14; 15; 24; ‘4 5 31; υ 5 1; 2; 4; 14; 15; 24; ‘5 5 23; ½qiðk=υÞ ð0Þ1 3 ‘k 5 ½0; 0; . . .; 0;

i 5 1; 2; . . .; ‘k ;

(4.289)

for remaining cases: k 5 1; k 5 2; k 5 3; k 5 4; k 5 5; G

υ 5 2; 9; 10; 11; 12; 14; 15; 16; 18; 19; 20; 22; 24; 25; 26; 28; 29; 31; 32; 33; ‘1 5 30; υ 5 3; 6; 7; . . .; 16; 19; 20; 21; . . .; 32; ‘2 5 28; υ 5 5; 6; . . .; 13; 16; 17; . . .; 23; 25; 26; . . .; 29; ‘3 5 28; υ 5 3; 5; 6; . . .; 13; 16; 17; . . .; 23; 25; 26; . . .; 31; ‘4 5 31; υ 5 3; 5; 6; . . .; 13; 16; 17; . . .; 23; 25; 26; . . .; 29; ‘5 5 23;

the matrices ½qijðk=υÞ ‘k x‘k ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the probabilities of transitions of the process Rðk=υÞ ðtÞ j i from the state rðk=υÞ into the state rðk=υÞ ; according to (4.222), during the experimental time. The probabilities of transitions that are not equal to 0 are as follows: 28 1 6 25 22 28 25 q1ð1=3Þ 5 1; q6ð1=3Þ 5 1; q22 ð1=3Þ 5 1; qð1=3Þ 5 1; qð1=3Þ 5 1; 27 30 1 21 1 24 21 27 24 29 20 30 29 q1ð1=4Þ 5 0:75; q1ð1=4Þ 5 0:25; q20 ð1=4Þ 5 1; qð1=4Þ 5 1; qð1=4Þ 5 1; qð1=4Þ 5 1; qð1=4Þ 5 1; qð1=4Þ 5 1;

(4.290) (4.291)

26 1 23 20 26 23 q1ð1=5Þ 5 1; q20 ð1=5Þ 5 1; qð1=5Þ 5 1; qð1=5Þ 5 1;

(4.292)

2 19 1 1 18 17 19 18 q1ð1=6Þ 5 0:667; q1ð1=6Þ 5 0:333; q2ð1=6Þ 5 1; q17 ð1=6Þ 5 1; qð1=6Þ 5 1; qð1=6Þ 5 1;

(4.293)

2 10 1 1 6 8 q1ð1=7Þ 5 0:5; q1ð1=7Þ 5 0:5; q2ð1=7Þ 5 1; q6ð1=7Þ 5 1; q8ð1=7Þ 5 1; q10 ð1=7Þ 5 1;

(4.294)

15 1 14 13 15 14 q1ð1=8Þ 5 1; q13 ð1=8Þ 5 1; qð1=8Þ 5 1; qð1=8Þ 5 1;

(4.295)

6 1 q1ð1=13Þ 5 1; q6ð1=13Þ 5 1;

(4.296)

28 1 6 25 22 28 25 q1ð1=17Þ 5 1; q6ð1=17Þ 5 1; q22 ð1=17Þ 5 1; qð1=17Þ 5 1; qð1=17Þ 5 1;

(4.297)

20 1 q1ð1=21Þ 5 1; q20 ð1=21Þ 5 1;

(4.298)

2 1 q1ð1=23Þ 5 1; q2ð1=23Þ 5 1;

(4.299)

6 1 q1ð1=27Þ 5 1; q6ð1=27Þ 5 1;

(4.300)

Identification of critical infrastructure accident consequences Chapter | 4

91

11 1 q1ð1=30Þ 5 1; q11 ð1=30Þ 5 1;

(4.301)

26 1 23 20 26 23 q1ð1=35Þ 5 1; q20 ð1=35Þ 5 1; qð1=35Þ 5 1; qð1=35Þ 5 1;

(4.302)

5 1 q1ð2=1Þ 5 1; q5ð2=1Þ 5 1;

(4.303)

2 1 q1ð2=2Þ 5 1; q2ð2=2Þ 5 1;

(4.304)

2 1 q1ð2=4Þ 5 1; q2ð2=4Þ 5 1;

(4.305)

18 1 14 13 18 14 q1ð2=5Þ 5 1; q13 ð2=5Þ 5 1; qð2=5Þ 5 1; qð2=5Þ 5 1;

(4.306)

25 27 1 16 12 21 16 25 21 27 25 q1ð2=17Þ 5 0:5; q1ð2=17Þ 5 0:5; q12 ð2=17Þ 5 1; qð2=17Þ 5 1; qð2=17Þ 5 1; qð2=17Þ 5 1; qð2=17Þ 5 1;

(4.307)

11 1 6 9 q1ð2=18Þ 5 1; q6ð2=18Þ 5 1; q9ð2=18Þ 5 1; q11 ð2=18Þ 5 1;

(4.308)

6 9 11 23 1 1 5 0:480; q1ð2=33Þ 5 0:256; q1ð2=33Þ 5 0:256; q1ð2=33Þ 5 0:008; q2ð2=33Þ 5 1; q6ð2=33Þ 5 1; q1ð2=33Þ 96 11 9 12 2 15 12 20 15 22 20 23 22 qð2=33Þ 5 1; qð2=33Þ 5 1; qð2=33Þ 5 1; qð2=33Þ 5 1; qð2=33Þ 5 1; qð2=33Þ 5 1; qð2=33Þ 5 1;

(4.309)

2 1 5 1; q2ð3=2Þ 5 1; q1ð3=2Þ

(4.310)

2 1 q1ð3=3Þ 5 1; q2ð3=3Þ 5 1;

(4.311)

18 1 14 13 18 14 q1ð3=4Þ 5 1; q13 ð3=4Þ 5 1; qð3=4Þ 5 1; qð3=4Þ 5 1;

(4.312)

25 27 1 16 12 21 16 25 21 27 25 q1ð3=14Þ 5 0:5; q1ð3=14Þ 5 0:5; q12 ð3=14Þ 5 1; qð3=14Þ 5 1; qð3=14Þ 5 1; qð3=14Þ 5 1; qð3=14Þ 5 1;

(4.313)

11 1 6 9 q1ð3=15Þ 5 1; q6ð3=15Þ 5 1; q9ð3=15Þ 5 1; q11 ð3=15Þ 5 1;

(4.314)

6 9 11 23 1 1 9 5 0:648; q1ð3=24Þ 5 0:296; q1ð3=24Þ 5 0:048; q1ð3=24Þ 5 0:008; q2ð3=24Þ 5 1; q6ð3=24Þ 5 0:961; q6ð3=24Þ 5 0:039; q1ð3=24Þ 96 9 11 11 9 12 2 15 12 20 15 22 20 22 qð3=24Þ 5 0:857; qð3=24Þ 5 0:143; qð3=24Þ 5 1; qð3=24Þ 5 1; qð3=24Þ 5 1; qð3=24Þ 5 1; qð3=24Þ 5 1; q23 ð3=24Þ 5 1;

(4.315) 2 q1ð4=2Þ

5 1;

1 q2ð4=2Þ

5 1;

(4.316)

18 1 14 13 18 14 q1ð4=4Þ 5 1; q13 ð4=4Þ 5 1; qð4=4Þ 5 1; qð4=4Þ 5 1;

(4.317)

28 30 1 16 12 21 16 28 21 30 28 q1ð4=14Þ 5 0:5; q1ð4=14Þ 5 0:5; q12 ð4=14Þ 5 1; qð4=14Þ 5 1; qð4=14Þ 5 1; qð4=14Þ 5 1; qð4=14Þ 5 1;

(4.318)

11 1 6 9 q1ð4=15Þ 5 1; q6ð4=15Þ 5 1; q9ð4=15Þ 5 1; q11 ð4=15Þ 5 1;

(4.319)

6 9 11 1 9 24 5 0:912; q1ð4=24Þ 5 0:070; q1ð4=24Þ 5 0:018; q6ð4=24Þ 5 0:600; q6ð4=24Þ 5 0:390; q6ð4=24Þ 5 0:010; q1ð4=24Þ 96 9 11 9 25 11 9 11 26 24 6 qð4=24Þ 5 0:610; qð4=24Þ 5 0:280; qð4=24Þ 5 0:110; qð4=24Þ 5 0:294; qð4=24Þ 5 0:706; qð4=24Þ 5 1; 6 25 24 26 9 26 25 q25 ð4=24Þ 5 0:857; qð4=24Þ 5 0:143; qð4=24Þ 5 0:917; qð4=24Þ 5 0:083;

(4.320)

5 1 q1ð5=1Þ 5 1; q5ð5=1Þ 5 1;

(4.321)

11 1 6 9 q1ð5=15Þ 5 1; q6ð5=15Þ 5 1; q9ð5=15Þ 5 1; q11 ð5=15Þ 5 1;

(4.322)

6 9 11 1 5 0:745; q1ð5=24Þ 5 0:177; q1ð5=24Þ 5 0:078; q6ð5=24Þ 5 0:699; q1ð5=24Þ 69 96 9 11 9 qð5=24Þ 5 0:301; qð5=24Þ 5 0:761; qð5=24Þ 5 0:239; q11 ð5=24Þ 5 1:

(4.323)

The values of some probabilities existing in the vectors (4.288)(4.289) and in the matrices (4.290)(4.323), besides these standing on the main diagonal, equal to zero do not mean that the events they are concerned with cannot appear. They evaluated on the basis of real statistical data, and their values may change and become more precise if the duration of the experiment is longer. On the basis of the statistical data presented in Appendix 6, using the procedure and the formulae given in Section 4.5.2, it is possible to determine the empirical parameters of the conditional sojourn times ζ ijðk=υÞ of the process of environment degradation at the particular environment degradation states. Moreover, using the procedure given in

92

Consequences of Maritime Critical Infrastructure Accidents

Section 4.5.3, we identify the forms of the particular density function gijðk=υÞ ðtÞ of the process of environment degradation conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 having sufficiently numerous set of their realizations at particular states. The application of these procedures and formulae is similar to the one presented in Section 4.2.2.4 than we omit it. We have got the following results: 61 G the conditional sojourn time ζ ð2=33Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:0000162911; 0 # t , 59; 898 1 (4.324) g6ð2=33Þ ðtÞ 5 0:00000008976; 59; 898 # t , 329; 439 > > : 0; t $ 329; 439 G

G

G

G

G

G

G

G

6 the conditional sojourn time ζ 9ð2=33Þ has the exponential distribution with the density function  0; t,0 6 ðtÞ 5 g9ð2=33Þ 0:000175142 exp½ 2 0:000175142t; t$0 9 the conditional sojourn time ζ 11 ð2=33Þ has the exponential distribution with the density function  0; t,0 11 9 gð2=33Þ ðtÞ 5 0:000197971 exp½ 2 0:000197971t; t $ 0 1 the conditional sojourn time ζ 6ð3=24Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:00003003; 0 # t , 31; 957 61 gð3=24Þ ðtÞ 5 0:000000126; 31; 957 # t , 351; 527 > > : 0; t $ 351; 527 6 the conditional sojourn time ζ 9ð3=24Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:000059866; 0 # t , 13; 920 96 gð3=24Þ ðtÞ 5 0:000004789; 13; 920 # t , 48; 720 > > : 0; t $ 48; 720 6 the conditional sojourn time ζ 1ð4=24Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:000984632; 0 # t , 839:83 6 g1ð4=24Þ ðtÞ 5 0:000034348; 839:83 # t , 5878:81 > > : 0; t $ 5878:81 1 the conditional sojourn time ζ 6ð4=24Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:0000242253; 0 # t , 35; 485:7 61 gð4=24Þ ðtÞ 5 0:00000056502; 35; 485:7 # t , 283; 885:6 > > : 0; t $ 283; 885:6 9 the conditional sojourn time ζ 6ð4=24Þ has the exponential distribution with the density function  0; t,0 69 gð4=24Þ ðtÞ 5 0:000571517 exp½ 2 0:000571517t; t $ 0 6 the conditional sojourn time ζ 9ð4=24Þ has the exponential distribution with the density function  0; t,0 96 gð4=24Þ ðtÞ 5 0:000094208 exp½ 2 0:000094208t; t $ 0

(4.325)

(4.326)

(4.327)

(4.328)

(4.329)

(4.330)

(4.331)

(4.332)

Identification of critical infrastructure accident consequences Chapter | 4

G

G

G

6 the conditional sojourn time ζ 1ð5=24Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:001462496; 0 # t , 575:8 16 gð5=24Þ ðtÞ 5 0:000054844; 575:8 # t , 3454:8 > > : 0; t $ 3454:8 1 the conditional sojourn time ζ 6ð5=24Þ has the chimney distribution with the density function 8 0; t,0 > > < 0:000023999; 0 # t , 38; 400 61 gð5=24Þ ðtÞ 5 0:0000003404; 38; 400 # t , 268; 800 > > : 0; t $ 268; 800

93

(4.333)

(4.334)

6 the conditional sojourn time ζ 9ð5=24Þ has the exponential distribution with the density function

 6 ðtÞ 5 g9ð5=24Þ

0; 0:000073209 exp½ 2 0:000073209t;

t,0 t $ 0:

(4.335)

For the above distributions, by application, either the general formula for the mean value or particular formulae ij given, respectively, by (3.54) and (3.55), (3.56), the mean values Mðk=υÞ 5 E½ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation conditional sojourn times at particular states can be determined, which amount to 61 96 11 9 Mð2=33Þ D33; 935:21; Mð2=33Þ D5709:65; Mð2=33Þ D5051:25; 61 96 16 Mð3=24Þ D16; 687:31; Mð3=24Þ D11; 018:72; Mð4=24Þ D928:66; 61 69 96 D37; 664:56; Mð4=24Þ D1749:73; Mð4=24Þ D10; 614:81; Mð4=24Þ 16 61 96 Mð5=24Þ D564:28; Mð5=24Þ D29; 741:18; Mð5=24Þ D13; 659:52:

(4.336)

In the case when as a result of the experiment, coming from experts, there are not any suitable distribution presented in Section 3.3 to describe the process of environment degradation or we have less than 30 realizations of the process of environment degradation, we assume that these conditional sojourn times have the empirical density functions defined by o 1 n gijðk=υÞ ðtÞ 5 ij # γ:ζ ijðk=υÞγ AIz ; γAf1; 2; . . .; nijðk=υÞ g ; z 5 1; 2; . . .; r ijðk=υÞ ; (4.337) nðk=υÞ corresponding to the empirical distribution functions defined by o 1 n Gijðk=υÞ ðtÞ 5 ij # γ:ζ ijðk=υÞγ , t; γAf1; 2; . . .; nijðk=υÞ g ; nðk=υÞ

(4.338)

for t $ 0; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; whereas the symbol # means the number of elements of the set. On the other hand, in the case when as a result of the experiment, limited data come from experts, we only have the number of realizations of the process of environment degradation lifetimes at the states and all its realizations are equal to an approximate value, we assume that these conditional sojourn times have the uniform distribution in the interval from this value minus its half to this value plus its half. Thus we proceeds in analogous way as in Section 4.2.2.4 for the example conditional sojourn times θ63 and θ21 the density and distribution functions of which are expressed by (4.72)(4.75), respectively. When the distributions cannot be identified statistically, it is possible to find the approximate empirical values of ij the mean values Mðk=υÞ 5 E½ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; k 5 1; 2; . . .; 5; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the conditional sojourn times at particular states using (4.224) that are as follows (Appendix 6, D-column of the table):

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Consequences of Maritime Critical Infrastructure Accidents

1 28 61 22 6 25 22 28 25 1 27 1 30 20 1 Mð1=3Þ 5 1; Mð1=3Þ 5 180; Mð1=3Þ 5 120; Mð1=3Þ 5 120; Mð1=3Þ 5 300; Mð1=4Þ 5 1; Mð1=4Þ 5 1; Mð1=4Þ 5 20; 21 1 24 21 27 24 29 20 30 29 1 26 20 1 23 20 Mð1=4Þ 5 680; Mð1=4Þ 5 920; Mð1=4Þ 5 2020; Mð1=4Þ 5 20; Mð1=4Þ 5 40; Mð1=5Þ 5 1; Mð1=5Þ 5 30; Mð1=5Þ 5 60; 26 23 12 1 19 21 17 1 18 17 19 18 12 Mð1=5Þ 5 360; Mð1=6Þ 5 1; Mð1=6Þ 5 1; Mð1=6Þ 5 1560; Mð1=6Þ 5 30; Mð1=6Þ 5 30; Mð1=6Þ 5 180; Mð1=7Þ 5 1; 1 10 21 61 86 10 8 1 15 13 1 14 13 5 1; Mð1=7Þ 5 15; Mð1=7Þ 5 180; Mð1=7Þ 5 120; Mð1=7Þ 5 420; Mð1=8Þ 5 1; Mð1=8Þ 5 60; Mð1=8Þ 5 60; Mð1=7Þ 15 14 16 61 1 28 61 22 6 25 22 Mð1=8Þ 5 120; Mð1=13Þ 5 1; Mð1=13Þ 5 120; Mð1=17Þ 5 1; Mð1=17Þ 5 180; Mð1=17Þ 5 120; Mð1=17Þ 5 120; 28 25 1 20 20 1 12 21 16 61 1 11 Mð1=17Þ 5 300; Mð1=21Þ 5 1; Mð1=21Þ 5 10; Mð1=23Þ 5 1; Mð1=23Þ 5 240; Mð1=27Þ 5 1; Mð1=27Þ 5 1; Mð1=30Þ 5 1; 11 1 1 26 20 1 23 20 26 23 15 51 12 5 260; Mð1=35Þ 5 1; Mð1=35Þ 5 30; Mð1=35Þ 5 60; Mð1=35Þ 5 360; Mð2=1Þ 5 1; Mð2=1Þ 5 600; Mð2=2Þ 5 1; Mð1=30Þ 21 12 21 1 18 13 1 14 13 18 14 1 25 Mð2=2Þ 5 2880; Mð2=4Þ 5 1; Mð2=4Þ 5 1440; Mð2=5Þ 5 1; Mð2=5Þ 5 780; Mð2=5Þ 5 360; Mð2=5Þ 5 300; Mð2=17Þ 5 1; 1 27 12 1 16 12 21 16 25 21 27 25 1 11 Mð2=17Þ 5 1; Mð2=17Þ 5 2640; Mð2=17Þ 5 2145; Mð2=17Þ 5 1530; Mð2=17Þ 5 255; Mð2=17Þ 5 270; Mð2=18Þ 5 1; 61 96 11 9 16 19 1 11 1 23 Mð2=18Þ 5 10; 080; Mð2=18Þ 5 11; 520; Mð2=18Þ 5 21; 600; Mð2=33Þ 5 1; Mð2=33Þ 5 1; Mð2=33Þ 5 1; Mð2=33Þ 5 1; 21 61 96 11 9 12 2 15 12 5 246; 240; Mð2=33Þ 5 33; 935:21; Mð2=33Þ D5709:65; Mð2=33Þ D5051:25; Mð2=33Þ 5 14; 400; Mð2=33Þ 5 7200; Mð2=33Þ 20 15 22 20 23 22 12 21 12 21 5 10; 080; Mð2=33Þ 5 1440; Mð2=33Þ 5 1440; Mð3=2Þ 5 1; Mð3=2Þ 5 2880; Mð3=3Þ 5 1; Mð3=3Þ 5 1440; Mð2=33Þ 1 18 13 1 14 13 18 14 1 25 1 27 12 1 16 12 Mð3=4Þ 5 1; Mð3=4Þ 5 780; Mð3=4Þ 5 360; Mð3=4Þ 5 300; Mð3=14Þ 5 1; Mð3=14Þ 5 1; Mð3=14Þ 5 2640; Mð3=14Þ 5 2145; 21 16 25 21 27 25 1 11 61 96 11 9 Mð3=14Þ 5 1530; Mð3=14Þ 5 255; Mð3=14Þ 5 270; Mð3=15Þ 5 1; Mð3=15Þ 5 10; 080; Mð3=15Þ 5 11; 520; Mð3=15Þ 5 21; 600; 16 19 1 11 1 23 21 61 69 Mð3=24Þ 5 1; Mð3=24Þ 5 1; Mð3=24Þ 5 1; Mð3=24Þ 5 1; Mð3=24Þ 5 246; 240; Mð3=24Þ D16; 687:31; Mð3=24Þ 5 2448; 96 9 11 11 9 12 2 15 12 20 15 D11; 018:72; Mð3=24Þ 5 3780; Mð3=24Þ 5 8177:14; Mð3=24Þ 5 14; 400; Mð3=24Þ 5 7200; Mð3=24Þ 5 10; 080; Mð3=24Þ 22 20 23 22 12 21 1 18 13 1 14 13 18 14 Mð3=24Þ 5 1440; Mð3=24Þ 5 1440; Mð4=2Þ 5 1; Mð4=2Þ 5 2880; Mð4=4Þ 5 1; Mð4=4Þ 5 780; Mð4=4Þ 5 360; Mð4=4Þ 5 300; 1 28 1 30 12 1 16 12 21 16 28 21 30 28 Mð4=14Þ 5 1; Mð4=14Þ 5 1; Mð4=14Þ 5 2640; Mð4=14Þ 5 2145; Mð4=14Þ 5 1530; Mð4=14Þ 5 300; Mð4=14Þ 5 270; 1 11 61 96 11 9 16 19 1 11 Mð4=15Þ 5 1; Mð4=15Þ 5 10; 080; Mð4=15Þ 5 11; 520; Mð4=15Þ 5 21; 600; Mð4=24Þ D928:66; Mð4=24Þ 5 1; Mð4=24Þ 5 1; 61 69 6 24 96 9 11 9 25 D37; 664:56; Mð4=24Þ D1749:73; Mð4=24Þ 5 4320; Mð4=24Þ D10; 614:81; Mð4=24Þ 5 2010; Mð4=24Þ 5 7080; Mð4=24Þ 11 9 11 26 24 6 25 6 25 24 26 9 5 6189:6; Mð4=24Þ 5 5160; Mð4=24Þ 5 5760; Mð4=24Þ 5 15; 120; Mð4=24Þ 5 30; 240; Mð4=24Þ 5 30; 763:64; Mð4=24Þ 26 25 15 51 1 11 61 96 11 9 5 10; 080; Mð5=1Þ 5 1; Mð5=1Þ 5 600; Mð5=15Þ 5 1; Mð5=15Þ 5 10; 080; Mð5=15Þ 5 11; 520; Mð5=15Þ 5 21; 600; Mð4=24Þ 16 19 1 11 61 69 96 Mð5=24Þ D564:28; Mð5=24Þ 5 1; Mð5=24Þ 5 1; Mð5=24Þ D29; 741:18; Mð5=24Þ 5 1418:18; Mð5=24Þ D13; 659:52; 9 11 11 9 Mð5=24Þ 5 1483:64; Mð5=24Þ 5 27; 552:

(4.339) As there are no realizations of the process of environment degradation conditional sojourn times at the remaining states, it is impossible to estimate their empirical conditional mean values. These approximate mean values given by (4.336) and (4.339) are used in Section 5.3.2 for the prediction of the characteristics of the process of environment degradation at the world sea waters.

Chapter 5

Prediction of critical infrastructure accident consequences Chapter Outline 5.1 Prediction of process of initiating events 5.1.1 Prediction of process of initiating events at Baltic Sea waters 5.1.2 Prediction of process of initiating events at world sea waters 5.2 Prediction of process of environment threats 5.2.1 Prediction of process of environment threats at Baltic Sea waters 5.2.2 Prediction of process of environment threats at world sea waters 5.3 Prediction of process of environment degradation

95 95 97 98 98 101 107

5.3.1 Prediction of process of environment degradation at Baltic Sea waters 107 5.3.2 Prediction of process of environment degradation at world sea waters 110 5.4 Superposition of initiating events, environment threats and environment degradation processes 117 5.4.1 Superposition of initiating events, environment threats and environment degradation processes at Baltic Sea waters 117 5.4.2 Superposition of initiating events, environment threats and environment degradation processes at world sea waters 118

The unknown parameters of particular processes that are the parts of the general model of critical infrastructure accident consequences are identified in Chapter 4, Identification of critical infrastructure accident consequences. Now, the general model of critical infrastructure accident consequences is adapted to the prediction of critical infrastructure accident consequences at the Baltic Sea and world sea waters through the determining the characteristics of the processes of initiating events, environment threats, and environment degradation such as unconditional mean sojourn times, limit values of transient probabilities staying at their states, and approximate mean values of sojourn total times at particular states for the fixed time. Finally, applying the expression for total probability, the unconditional transient probabilities and mean values of sojourn total times of the joint process linking the processes of initiating events, environment threats, and environment degradation at particular states, for the fixed sufficiently large time, are fined.

5.1

Prediction of process of initiating events

We assume, as in Section 4.2, that we have identified the unknown parameters of the semi-Markov model of the process of initiating events EðtÞ; tAh0; 1 NÞ: G

G

G

the initial probabilities pl ð0Þ; l 5 1; 2; . . .; ω of the process of initiating events staying at the particular states el at the moment t 5 0; the probabilities plj ; l; j 5 1; 2; . . .; ω; l 6¼ j of the process of initiating events transitions from the state el into the state ej ; and the distributions of the process of initiating events conditional sojourn times θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j at the particular states and their mean values M lj 5 E½θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j;

then we can predict its basic characteristics.

5.1.1 Prediction of process of initiating events at Baltic Sea waters The matrix of probabilities plj ; l; j 5 1; 2; . . .; 16 of the process of initiating events transitions from the state el into the state ej is estimated in (4.55). The mean values M lj of the conditional sojourn times θlj ; l; j 5 1; 2; . . .; 16 of the process Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00005-X © 2020 Elsevier Inc. All rights reserved.

95

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of initiating events at particular states are determined, and their values are given by (4.60), (4.69), and (4.76). This way, the process of initiating events for the Baltic Sea waters is approximately defined, and we may predict its main characteristics. After considering these results and applying (3.13), we conclude that the unconditional mean sojourn times of the process of initiating events at particular states are as follows: M 1 5 E½θ1  5 p12 M 12 1 p13 M 13 1 p14 M 14 1 p15 M 15 1 p16 M 16 1 p17 M 17 1 p18 M 18 5 0:1731U10; 249; 200 1 0:3558U8; 928; 571:43 1 0:0481U12; 614; 400 10:0962U13; 402; 800 1 0:2308U8; 694; 300 1 0:0865U5; 869; 200 1 0:0096U1; 576; 800D9; 376; 491:76; M 2 5 E½θ2  5 p21 M 21 1 p23 M 23 1 p26 M 26 1 p27 M 27 5 0:75U1 1 0:1U22:50 1 0:05U1:00 1 0:1U5:50 5 3:6; M 3 5 E½θ3  5 p31 M 31 5 1U1933:69 5 1933:69; M 4 5 E½θ4  5 p41 M 41 1 p47 M 47 5 0:8333U1 1 0:1667U1:00 5 1:0; 5 M 5 E½θ5  5 p51 M 51 1 p5 10 M 5 10 5 0:9U163:33 1 0:1U10:00D148:0; M 6 5 E½θ6  5 p61 M 61 1 p62 M 62 1 p63 M 63 1 p64 M 64 5 0:04U120:00 1 0:08U80:00 1 0:84U324:05 1 0:04U15:00D284:0; M 7 5 E½θ7  5 p71 M 71 1 p73 M 73 1 p7 13 M 7 13 5 0:5U225:83 1 0:4167U21:60 1 0:0833U1:00D122:0; M 8 5 E½θ8  5 p81 M 81 5 1U5:00 5 5:00; M 10 5 E½θ10  5 p10 1 M 10 1 5 1U10:00 5 10:00; M 13 5 E½θ13  5 p13 1 M 13 1 5 1U10:00 5 10:00:

(5.1)

To find the limit values of the transient probabilities pl ; l 5 1; 2; . . .; 16 of the process of initiating events EðtÞ at its particular states, first, we have to solve the system of equations (3.16) that in our case takes the following form: 8 l l lj > < ½π 1 3 16 5 ½π 1 3 16 ½p 16 3 16 16 X πj 5 1; > : j51

where ½πl 1 3 16 5 ½π1 ; π2 ; . . .; π16  and the elements of the matrix ½plj 16 3 16 are given by (4.55). The system of equations consists of the following equations: π1 5 0:7500Uπ2 1 π3 1 0:8333Uπ4 1 0:9000Uπ5 1 0:0400Uπ6 10:5000Uπ7 1 π8 1 π10 1 π13 ; π2 5 0:1731Uπ1 1 0:0800Uπ16 ; 3 π 5 0:3558Uπ1 1 0:1000Uπ2 1 0:8400Uπ6 1 0:4167Uπ7 ; π4 5 0:0481Uπ1 1 0:0400Uπ6 ; π5 5 0:0962Uπ1 ; 6 π 5 0:2308Uπ1 1 0:0500Uπ2 ; 7 π 5 0:0865Uπ1 1 0:1000Uπ2 1 0:1667Uπ4 ; π8 5 0:0096Uπ1 ; π10 5 0:1000Uπ5 ; π13 5 0:0833Uπ7 ; 1 2 3 4 π 1 π 1 π 1 π 1 π5 1 π6 1 π7 1 π8 1 π10 1 π13 5 1:

(5.2)

Solving the system of equations, we get π1 5 0:42449; π2 5 0:08164; π3 5 0:26533; π4 5 0:02450; π5 5 0:04079; π6 5 0:10205; π7 5 0:04897; π8 5 0:00407; π10 5 0:00408; π13 5 0:00408:

(5.3)

Therefore according to (3.15) and considering (5.1) we get the approximate limit values of transient probabilities at the particular states of the process of initiating events

Prediction of critical infrastructure accident consequences Chapter | 5

p1 5 0:999860710821154; p2 5 0:000000073830730; p3 5 0:000128885740640; p4 5 0:000000006154570; p5 5 0:000001516516305; p6 5 0:000007280530278; p7 5 0:000001500795773; p8 5 0:000000005112062; p10 5 0:000000010249244; p13 5 0:000000010249244:

97

(5.4)

l

Further, by (3.17) and considering (5.4), the approximate mean values of the sojourn total times θ^ of the process of initiating events EðtÞ during the fixed time θ 5 1 month 5 43,200 minutes at the particular states el expressed in minutes are 1 2 3 4 5 6 M^ 5 43; 193:98271; M^ 5 0:00319; M^ 5 5:56786; M^ 5 0:00027; M^ 5 0:06551; M^ 5 0:31452; 7 8 10 13 M^ 5 0:06483; M^ 5 0:00022; M^ 5 0:00044; M^ 5 0:00044:

(5.5)

5.1.2 Prediction of process of initiating events at world sea waters Proceedings in analogous way as in Section 5.1.1, we predict the process of initiating events within the world sea waters. The matrix of probabilities plj ; l; j 5 1; 2; . . .; 16 of the process of initiating events transitions from the state el into the state ej is estimated in (4.78). The mean values M lj of the conditional sojourn times θlj ; l; j 5 1; 2; . . .; 16 of the process of initiating events at particular states are determined, and their values are given by (4.90) and (4.91). This way, the process of initiating events for the world sea waters is approximately defined, and we may predict its main characteristics. After considering these results and applying (3.13), we conclude that the unconditional mean sojourn times of the process of initiating events at particular states are as follows: M 1 D10; 405; 787:49; M 2 D21:71; M 3 D4459:50; M 4 D10:07; M 5 D899:19; M 6 D1296:91; M 7 D940:30; M 8 D319:91; M 9 D162:63; M 10 D784:63; M 11 D253:31; M 12 5 30:00; M 13 D75:56; M 14 5 120:00; M 15 5 140:00; M 16 5 78:00:

(5.6)

To find the limit values of the transient probabilities pl ; l 5 1; 2; . . .; 16 of the process of initiating events EðtÞ at its particular states, first, we have to solve the system of equations (3.16) that in our case takes the following form:  l  lj  8  l > < π 1 3 16 5 π 1 3 16 p 16 3 16 16 X πj 5 1; > : j51

where ½πl 1 3 16 5 ½π1 ; π2 ; . . .; π16  and the elements of the matrix ½plj 16 3 16 are given by (4.78). The system of equations consists of the following equations: π1 5 0:5461Uπ2 1 0:9908Uπ3 1 0:7450Uπ4 1 0:7692Uπ5 1 0:4488Uπ6 10:4115Uπ7 1 0:8750Uπ8 1 π9 1 0:5790Uπ10 1 0:4444Uπ11 1 π12 10:4445Uπ13 1 π14 1 0:4000Uπ16 ; π2 5 0:2460Uπ1 1 0:0026Uπ3 1 0:0201Uπ4 1 0:0989Uπ6 ; 3 1 2 π 5 0:2264Uπ 1 0:2880Uπ 1 0:0872Uπ4 1 0:3110Uπ6 1 0:5423Uπ7 10:5556Uπ11 1 0:3333Uπ13 1 π15 1 0:4000Uπ16 ; π4 5 0:0754Uπ1 1 0:0069Uπ2 1 0:0777Uπ6 1 0:0038Uπ7 ; π5 5 0:1644Uπ1 1 0:0092Uπ2 ; 6 1 π 5 0:1436Uπ 1 0:0507Uπ2 1 0:0053Uπ3 1 0:0604Uπ4 1 0:3947Uπ10 ; π7 5 0:1258Uπ1 1 0:0922Uπ2 1 0:0873Uπ4 1 0:2000Uπ16 ; π8 5 0:0184Uπ1 1 0:0116Uπ7 ; π9 5 0:0013Uπ3 1 0:0659Uπ5 ; 10 π 5 0:0046Uπ2 1 0:1429Uπ5 1 0:0141Uπ6 ; π11 5 0:0023Uπ2 1 0:0495Uπ6 1 0:0116Uπ7 ; π12 5 0:0037Uπ5 ; 13 π 5 0:0192Uπ7 1 0:1250Uπ8 ; π14 5 0:0263Uπ10 ; π15 5 0:2222Uπ13 ; π16 5 0:0183Uπ5 ; 1 π 1 π2 1 ? 1 π16 5 1: (5.7)

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Solving the system of equations, we get π1 5 0:41596; π2 5 0:11083; π3 5 0:19241; π4 5 0:03807; π5 5 0:06940; π6 5 0:07319; π7 5 0:06612; π8 5 0:00842; π9 5 0:00482; π10 5 0:01146; 11 π 5 0:00465; π12 5 0:00026; π13 5 0:00232; π14 5 0:00030; π15 5 0:00052; π16 5 0:00127:

(5.8)

Therefore according to (3.15) and considering (5.6), we get the approximate limit values of transient probabilities at the particular states of the process of initiating events p1 5 0:999747226936710; p2 5 0:000000555751755; p5 5 0:000014413671674; p6 5 0:000021924276591; 9 p 5 0:000000181055360; p10 5 0:000002076888651; p13 5 0:000000040489613; p14 5 0:000000008315075;

p3 5 0:000198188063477; p7 5 0:000014360281962; p11 5 0:000000272062681; p15 5 0:000000016814930;

p4 5 0:000000088547445; p8 5 0:000000622162160; p12 5 0:000000001801600; p16 5 0:000000022880316: (5.9) l Further, by (3.17) and considering (5.9), the approximate mean values of the sojourn total times θ^ of the process of initiating events EðtÞ during the fixed time θ 5 1 month 5 43,200 minutes at the particular states el expressed in minutes are 1 2 3 4 5 6 M^ 5 43; 189:08020; M^ 5 0:02401; M^ 5 8:56172; M^ 5 0:00383; M^ 5 0:62267; M^ 5 0:94713; 7 8 9 10 11 12 M^ 5 0:62036; M^ 5 0:02688; M^ 5 0:00782; M^ 5 0:08972; M^ 5 0:01175; M^ 5 0:00008; 13 14 15 16 M^ 5 0:00175; M^ 5 0:00036; M^ 5 0:00073; M^ 5 0:00099:

5.2

(5.10)

Prediction of process of environment threats

We assume, as in Section 4.4, that we have identified the unknown parameters of the semi-Markov model of the process of environment threats Sðk=lÞ ðtÞ; tAh0; 1 NÞ for particular environment subareas Dk ; k 5 1; 2; . . .; n3 , while the process of initiating events EðtÞ is at the state el ; l 5 1; 2; . . .; ω: G

G

G

the initial probabilities piðk=lÞ ð0Þ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats staying at the particular states siðk=lÞ at the moment t 5 0; the probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats transitions from the state siðk=lÞ into the state sjðk=lÞ ; and the distributions of the process of environment threats conditional sojourn times ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; ij k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the particular states and their mean values Mðk=lÞ 5 E½ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω;

then we can predict its basic characteristics.

5.2.1 Prediction of process of environment threats at Baltic Sea waters The matrices of probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; j l 5 1; 2; . . .; 16 of the process of environment threats transitions from the state siðk=lÞ into the state sðk=lÞ are estimated in ij ij (4.157)(4.164). The mean values Mðk=lÞ of the conditional sojourn times ηðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats at particular states are determined, and their values are given by (4.165). This way, the process of environment threats for the Baltic Sea waters is approximately defined, and we may predict its main characteristics. After considering these results and applying (3.36), we conclude that the unconditional mean sojourn times of the process of environment threats at particular states are as follows: h i 1 27 1 27 Mð1=2Þ 5 E η1ð1=2Þ 5 p1ð1=2Þ Mð1=2Þ 5 1U1 5 1; h i (5.11) 27 27 1 27 1 5 E η27 Mð1=2Þ ð1=2Þ 5 pð1=2Þ Mð1=2Þ 5 1U300 5 300;

Prediction of critical infrastructure accident consequences Chapter | 5

h i 1 27 1 27 30 1 30 Mð1=3Þ 5 E η1ð1=3Þ 5 p1ð1=3Þ Mð1=3Þ 1 p1ð1=3Þ Mð1=3Þ 5 0:5U1 1 0:5U1 5 1; h i 27 27 27 1 27 1 Mð1=3Þ 5 E ηð1=3Þ 5 pð1=3Þ Mð1=3Þ 5 1U180 5 180; h i 30 30 1 30 1 5 E η30 Mð1=3Þ ð1=3Þ 5 pð1=3Þ Mð1=3Þ 5 1U240 5 240; h i 1 6 16 Mð1=8Þ 5 E η1ð1=8Þ 5 p1ð1=8Þ Mð1=8Þ 5 1U1 5 1; h i 6 1 61 5 E η6ð1=8Þ 5 p6ð1=8Þ Mð1=8Þ 5 1U240 5 240; Mð1=8Þ h i 1 33 1 33 5 E η1ð2=2Þ 5 p1ð2=2Þ Mð2=2Þ 5 1U1 5 1; Mð2=2Þ h i 33 33 1 33 1 5 E η33 Mð2=2Þ ð2=2Þ 5 pð2=2Þ Mð2=2Þ 5 1U1440 5 1440; h i 1 17 1 17 33 1 33 5 E η1ð2=3Þ 5 p1ð2=3Þ Mð2=3Þ 1 p1ð2=3Þ Mð2=3Þ 5 0:5U1 1 0:5U1 5 1; Mð2=3Þ h i 17 5 E η17 Mð2=3Þ 5 p17 1 M 17 1 5 1U10; 080 5 10; 080; hð2=3Þ i ð2=3Þ ð2=3Þ 33 33 1 33 1 5 E η33 Mð2=3Þ ð2=3Þ 5 pð2=3Þ Mð2=3Þ 5 1U1440 5 1440; h i 1 24 1 24 Mð3=2Þ 5 E η1ð3=2Þ 5 p1ð3=2Þ Mð3=2Þ 5 1U1 5 1; h i 24 24 1 24 1 5 E η24 Mð3=2Þ ð3=2Þ 5 pð3=2Þ Mð3=2Þ 5 1U1440 5 1440; h i 1 14 1 14 24 1 24 5 E η1ð3=3Þ 5 p1ð3=3Þ Mð3=3Þ 1 p1ð3=3Þ Mð3=3Þ 5 0:5U1 1 0:5U1 5 1; Mð3=3Þ h i 14 5 E η14 Mð3=3Þ 5 p14 1 M 14 1 5 1U10; 080 5 10; 080; hð3=3Þ i ð3=3Þ ð3=3Þ 24 24 1 24 1 5 E η24 Mð3=3Þ ð3=3Þ 5 pð3=3Þ Mð3=3Þ 5 1U1440 5 1440; h i 1 14 1 14 Mð4=3Þ 5 E η1ð4=3Þ 5 p1ð4=3Þ Mð4=3Þ 5 1U1 5 1; h i 14 14 1 14 1 5 E η14 Mð4=3Þ ð4=3Þ 5 pð4=3Þ Mð4=3Þ 5 1U10; 080 5 10; 080:

99

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

(5.17)

(5.18)

To find the limit values of the transient probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats Sðk=lÞ ðtÞ at its particular states, first, we have to solve the systems of equations (3.39) 8 i ij i > < ½πðk=lÞ 1 3 υk 5 ½πðk=lÞ 1 3 υk ½pðk=lÞ υk 3 υk υk X > πjðk=lÞ 5 1; : j51

where k k ½πυðk=lÞ 1 3 υk 5 ½π1ðk=lÞ ; π2ðk=lÞ ; . . .; πυðk=lÞ 

and the elements of the matrices ½pijðk=lÞ υk xυk are given by (4.157)(4.164). The systems of equations take the forms 8 1 27 > < πð1=2Þ 5 πð1=2Þ 1 π27 (5.19) ð1=2Þ 5 πð1=2Þ > : π1 1 π27 5 1 ð1=2Þ ð1=2Þ 8 1 30 πð1=3Þ 5 π27 > ð1=3Þ 1 πð1=3Þ > > < π27 5 0:5Uπ1 ð1=3Þ ð1=3Þ (5.20) π30 5 0:5Uπ1ð1=3Þ > > ð1=3Þ > : 1 30 πð1=3Þ 1 π27 ð1=3Þ 1 πð1=3Þ 5 1

100

Consequences of Maritime Critical Infrastructure Accidents

8 1 6 > < πð1=8Þ 5 πð1=8Þ π6ð1=8Þ 5 π1ð1=8Þ > : π1 1 π6 5 1 ð1=8Þ ð1=8Þ 8 1 33 > < πð2=2Þ 5 πð2=2Þ 1 π33 ð2=2Þ 5 πð2=2Þ > : π1 1 π33 5 1 ð2=2Þ ð2=2Þ

8 1 33 πð2=3Þ 5 π17 > ð2=3Þ 1 πð2=3Þ > > < π17 5 0:5Uπ1 ð2=3Þ ð2=3Þ 5 0:5Uπ1ð2=3Þ > π33 > ð2=3Þ > : 1 33 πð2=3Þ 1 π17 ð2=3Þ 1 πð2=3Þ 5 1 8 1 24 > < πð3=2Þ 5 πð3=2Þ 1 π24 ð3=2Þ 5 πð3=2Þ > : π1 1 π24 5 1 ð3=2Þ ð3=2Þ 8 1 14 πð3=3Þ 5 πð3=3Þ 1 π24 > ð3=3Þ > > < π14 5 0:5Uπ1 ð3=3Þ ð3=3Þ π24 5 0:5Uπ1ð3=3Þ > > ð3=3Þ > : 1 24 πð3=3Þ 1 π14 ð3=3Þ 1 πð3=3Þ 5 1 8 1 14 > < πð4=3Þ 5 πð4=3Þ 1 π14 ð4=3Þ 5 πð4=3Þ > : π1 1 π14 5 1: ð4=3Þ ð4=3Þ

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)

(5.26)

Solving the particular systems of equations, we get π1ð1=2Þ 5 0:5; π27 ð1=2Þ 5 0:5;

(5.27)

30 π1ð1=3Þ 5 0:5; π27 ð1=3Þ 5 0:25; πð1=3Þ 5 0:25;

(5.28)

π1ð1=8Þ 5 0:5; π6ð1=8Þ 5 0:5;

(5.29)

π1ð2=2Þ 5 0:5; π33 ð2=2Þ 5 0:5;

(5.30)

33 π1ð2=3Þ 5 0:5; π17 ð2=3Þ 5 0:25; πð2=3Þ 5 0:25;

(5.31)

π1ð3=2Þ 5 0:5; π24 ð3=2Þ 5 0:5;

(5.32)

24 π1ð3=3Þ 5 0:5; π14 ð3=3Þ 5 0:25; πð3=3Þ 5 0:25;

(5.33)

π1ð4=3Þ 5 0:5; π14 ð4=3Þ 5 0:5:

(5.34)

Therefore according to (3.38) and considering (5.11)(5.18), we get the approximate limit values of transient probabilities that are not equal to 0 at the particular states of the process of environment threats p1ð1=2Þ 5 0:003322259136213; p27 ð1=2Þ 5 0:996677740863787;

(5.35)

30 p1ð1=3Þ 5 0:004739336492891; p27 ð1=3Þ 5 0:426540284360190; pð1=3Þ 5 0:568720379146919;

(5.36)

p1ð1=8Þ 5 0:004149377593361; p6ð1=8Þ 5 0:995850622406639;

(5.37)

p1ð2=2Þ 5 0:000693962526024; p33 ð2=2Þ 5 0:999306037473976;

(5.38)

33 p1ð2=3Þ 5 0:000173580975525; p17 ð2=3Þ 5 0:874848116646416; pð2=3Þ 5 0:124978302378059;

(5.39)

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101

p1ð3=2Þ 5 0:000693962526024; p24 ð3=2Þ 5 0:999306037473976;

(5.40)

24 p1ð3=3Þ 5 0:000173580975525; p14 ð3=3Þ 5 0:874848116646416; pð3=3Þ 5 0:124978302378059;

(5.41)

p1ð4=3Þ 5 0:000099196508283; p14 ð4=3Þ 5 0:999900803491717:

(5.42)

Further, by (3.40) and considering (5.35)(5.42), the approximate mean values of the sojourn total times η^ iðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ during the fixed time η 5 1 month 5 43,200 minutes at the particular states siðk=lÞ expressed in minutes are 1 27 M^ ð1=2Þ 5 143:52; M^ ð1=2Þ 5 43; 056:48;

(5.43)

1 27 30 M^ ð1=3Þ 5 204:74; M^ ð1=3Þ 5 18; 426:54; M^ ð1=3Þ 5 24; 568:72;

(5.44)

1 6 M^ ð1=8Þ 5 179:25; M^ ð1=8Þ 5 43; 020:75;

(5.45)

1 33 M^ ð2=2Þ 5 29:98; M^ ð2=2Þ 5 43; 170:02;

(5.46)

1 17 33 M^ ð2=3Þ 5 7:50; M^ ð2=3Þ 5 37; 793:44; M^ ð2=3Þ 5 5399:06;

(5.47)

1 24 M^ ð3=2Þ 5 29:98; M^ ð3=2Þ 5 43; 170:02;

(5.48)

1 14 24 M^ ð3=3Þ 5 7:50; M^ ð3=3Þ 5 37; 793:44; M^ ð3=3Þ 5 5399:06;

(5.49)

1 14 M^ ð4=3Þ 5 4:29; M^ ð4=3Þ 5 43; 195:71:

(5.50)

5.2.2 Prediction of process of environment threats at world sea waters Proceedings in analogous way as in Section 5.2.1, we predict the process of environment threats within the world sea waters. The matrices of probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats transitions from the state siðk=lÞ into the state sjðk=lÞ are estiij mated in (4.167)(4.201). The mean values Mðk=lÞ of the conditional sojourn times ηijðk=lÞ i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats at particular states are determined, and their values are given by (4.211) and (4.214). This way, the process of environment threats for the world sea waters is approximately defined, and we may predict its main characteristics. After considering these results and applying (3.36), we conclude that the unconditional mean sojourn times of the process of environment threats at particular states are as follows: 1 7 27 Mð1=2Þ 5 1; Mð1=2Þ 5 240; Mð1=2Þ 5 3605:86;

(5.51)

1 23 27 30 Mð1=3Þ 5 154:14; Mð1=3Þ 5 240; Mð1=3Þ 5 3593:55; Mð1=3Þ 5 240;

(5.52)

1 5 13 27 35 Mð1=4Þ 5 1; Mð1=4Þ 5 300; Mð1=4Þ 5 120; Mð1=4Þ 5 2546:25; Mð1=4Þ 5 120;

(5.53)

1 3 4 7 17 27 34 Mð1=5Þ 5 1; Mð1=5Þ 5 1; Mð1=5Þ 5 1147:50; Mð1=5Þ 5 5; Mð1=5Þ 5 300; Mð1=5Þ 5 1488; Mð1=5Þ 5 15;

(5.54)

1 27 5 1; Mð1=6Þ 5 5760; Mð1=6Þ

(5.55)

1 27 30 Mð1=7Þ 5 1; Mð1=7Þ 5 2185:8; Mð1=7Þ 5 240;

(5.56)

1 3 6 7 8 21 27 5 1; Mð1=8Þ 5 40; Mð1=8Þ 5 1100; Mð1=8Þ 5 15; Mð1=8Þ 5 240; Mð1=8Þ 5 5; Mð1=8Þ 5 2648:4; Mð1=8Þ

(5.57)

1 2 33 5 1; Mð2=2Þ 5 280; 800; Mð2=2Þ 5 23; 682:13; Mð2=2Þ

(5.58)

1 17 18 33 Mð2=3Þ 5 146:91; Mð2=3Þ 5 24; 480; Mð2=3Þ 5 43; 200; Mð2=3Þ 5 11; 135:86;

(5.59)

1 33 Mð2=4Þ 5 1; Mð2=4Þ 5 31; 050;

(5.60)

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Consequences of Maritime Critical Infrastructure Accidents

1 33 Mð2=5Þ 5 1; Mð2=5Þ 5 38; 016;

(5.61)

1 33 Mð2=6Þ 5 1; Mð2=6Þ 5 12; 240;

(5.62)

1 17 33 Mð2=7Þ 5 1; Mð2=7Þ 5 1440; Mð2=7Þ 5 8640;

(5.63)

1 2 4 5 33 5 1; Mð2=8Þ 5 7200; Mð2=8Þ 5 1440; Mð2=8Þ 5 1440; Mð2=8Þ 5 12; 133:63; Mð2=8Þ

(5.64)

1 2 24 5 1; Mð3=2Þ 5 280; 800; Mð3=2Þ 5 23; 682:13; Mð3=2Þ

(5.65)

1 14 15 24 Mð3=3Þ 5 146:91; Mð3=3Þ 5 24; 480; Mð3=3Þ 5 43; 200; Mð3=3Þ 5 12; 398:05;

(5.66)

1 24 Mð3=4Þ 5 1; Mð3=4Þ 5 31; 050;

(5.67)

1 24 Mð3=5Þ 5 1; Mð3=5Þ 5 38; 016;

(5.68)

1 24 Mð3=6Þ 5 1; Mð3=6Þ 5 12; 240;

(5.69)

1 14 24 Mð3=7Þ 5 1; Mð3=7Þ 5 1440; Mð3=7Þ 5 8640;

(5.70)

1 2 3 4 24 Mð3=8Þ 5 1; Mð3=8Þ 5 7200; Mð3=8Þ 5 1440; Mð3=8Þ 5 1440; Mð3=8Þ 5 10; 497:27;

(5.71)

1 2 24 5 524:11; Mð4=2Þ 5 280; 800; Mð4=2Þ 5 35; 382:86; Mð4=2Þ

(5.72)

1 14 15 24 Mð4=3Þ 5 360:74; Mð4=3Þ 5 24; 480; Mð4=3Þ 5 43; 200; Mð4=3Þ 5 32; 230:59;

(5.73)

1 24 Mð4=4Þ 5 1; Mð4=4Þ 5 70; 765:71;

(5.74)

1 24 Mð4=5Þ 5 1; Mð4=5Þ 5 91; 440;

(5.75)

1 24 Mð4=6Þ 5 720:5; Mð4=6Þ 5 20; 880;

(5.76)

1 14 24 Mð4=7Þ 5 72:8; Mð4=7Þ 5 1440; Mð4=7Þ 5 18; 630;

(5.77)

1 2 4 24 Mð4=8Þ 5 1; Mð4=8Þ 5 7200; Mð4=8Þ 5 1440; Mð4=8Þ 5 34; 560;

(5.78)

1 24 Mð5=2Þ 5 351:32; Mð5=2Þ 5 26; 441:05;

(5.79)

1 15 24 Mð5=3Þ 5 271:02; Mð5=3Þ 5 43; 200; Mð5=3Þ 5 22; 368;

(5.80)

1 24 Mð5=4Þ 5 1; Mð5=4Þ 5 64; 260;

(5.81)

1 24 Mð5=5Þ 5 1; Mð5=5Þ 5 172; 800;

(5.82)

1 24 Mð5=6Þ 5 1440; Mð5=6Þ 5 17; 280;

(5.83)

1 24 Mð5=7Þ 5 1; Mð5=7Þ 5 11; 520;

(5.84)

1 24 Mð5=8Þ 5 1; Mð5=8Þ 5 87; 480:

(5.85)

pijðk=lÞ ;

To find the limit values of the transient probabilities i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; 5; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29; l 5 1; 2; . . .; 16 of the process of environment threats Sðk=lÞ ðtÞ at its particular states, first, we have to solve the systems of equations (3.39) 8 i ij i > < ½πðk=lÞ 1 3 υk 5 ½πðk=lÞ 1 3 υk ½pðk=lÞ υk 3 υk υk X > πjðk=lÞ 5 1; : j51

where k k ½πυðk=lÞ 1 3 υk 5 ½π1ðk=lÞ ; π2ðk=lÞ ; . . .; πυðk=lÞ 

Prediction of critical infrastructure accident consequences Chapter | 5

103

and the elements of the matrices ½pijðk=lÞ υk 3 υk are given by (4.167)(4.201). Solving the particular systems of equations, we get π1ð1=2Þ D0:4943; π7ð1=2Þ D0:0114; π27 ð1=2Þ D0:4943;

(5.86)

27 30 π1ð1=3Þ 5 0:5000; π23 ð1=3Þ 5 0:0115; πð1=3Þ 5 0:4655; πð1=3Þ D0:0230;

(5.87)

27 35 π1ð1=4Þ 5 0:4866; π5ð1=4Þ D0:0268; π13 ð1=4Þ D0:0268; πð1=4Þ D0:4330; πð1=4Þ D0:0268;

(5.88)

π1ð1=5Þ D0:4348; π3ð1=5Þ D0:0435; π4ð1=5Þ D0:1738; π7ð1=5Þ D0:0435; 27 34 π17 ð1=5Þ D0:0435; πð1=5Þ D0:2174; πð1=5Þ D0:0435;

(5.89)

π1ð1=6Þ 5 0:5; π27 ð1=6Þ 5 0:5;

(5.90)

30 π1ð1=7Þ 5 0:5000; π27 ð1=7Þ 5 0:4375; πð1=7Þ 5 0:0625;

(5.91)

π1ð1=8Þ D0:48570; π3ð1=8Þ D0:02865; π6ð1=8Þ D0:08550; π7ð1=8Þ D0:02865; 27 π8ð1=8Þ D0:02865; π21 ð1=8Þ D0:02865; πð1=8Þ D0:31420;

(5.92)

π1ð2=2Þ D0:4943; π2ð2=2Þ D0:0114; π33 ð2=2Þ D0:4943;

(5.93)

18 33 π1ð2=3Þ 5 0:5000; π17 ð2=3Þ D0:0335; πð2=3Þ D0:0110; πð2=3Þ 5 0:4555;

(5.94)

π1ð2=4Þ 5 0:5; π33 ð2=4Þ 5 0:5;

(5.95)

π1ð2=5Þ 5 0:5; π33 ð2=5Þ 5 0:5;

(5.96)

π1ð2=6Þ 5 0:5; π33 ð2=6Þ 5 0:5;

(5.97)

33 π1ð2=7Þ 5 0:5000; π17 ð2=7Þ 5 0:4375; πð2=7Þ 5 0:0625;

(5.98)

π1ð2=8Þ 5 0:5000; π2ð2=8Þ 5 0:0355; π4ð2=8Þ 5 0:0355; π5ð2=8Þ 5 0:0355; π33 ð2=8Þ 5 0:3935;

(5.99)

π1ð3=2Þ D0:4943; π2ð3=2Þ D0:0114; π24 ð3=2Þ D0:4943;

(5.100)

15 24 π1ð3=3Þ 5 0:5000; π14 ð3=3Þ D0:0335; πð3=3Þ D0:0110; πð3=3Þ D0:4555;

(5.101)

π1ð3=4Þ 5 0:5; π24 ð3=4Þ 5 0:5;

(5.102)

π1ð3=5Þ 5 0:5; π24 ð3=5Þ 5 0:5;

(5.103)

π1ð3=6Þ 5 0:5; π24 ð3=6Þ 5 0:5;

(5.104)

24 π1ð3=7Þ 5 0:5000; π14 ð3=7Þ 5 0:4375; πð3=7Þ 5 0:0625;

(5.105)

π1ð3=8Þ 5 0:5000; π2ð3=8Þ 5 0:0355; π3ð3=8Þ 5 0:0355; π4ð3=8Þ 5 0:0355; π24 ð3=8Þ 5 0:3935;

(5.106)

π1ð4=2Þ 5 0:500; π2ð4=2Þ 5 0:023; π24 ð4=2Þ 5 0:477;

(5.107)

15 24 π1ð4=3Þ 5 0:500; π14 ð4=3Þ 5 0:068; πð4=3Þ 5 0:023; πð4=3Þ 5 0:409;

(5.108)

π1ð4=4Þ 5 0:5; π24 ð4=4Þ 5 0:5;

(5.109)

π1ð4=5Þ 5 0:5; π24 ð4=5Þ 5 0:5;

(5.110)

π1ð4=6Þ 5 0:5; π24 ð4=6Þ 5 0:5;

(5.111)

24 π1ð4=7Þ 5 0:5; π14 ð4=7Þ 5 0:1; πð4=7Þ 5 0:4;

(5.112)

π1ð4=8Þ 5 0:5; π2ð4=8Þ 5 0:1; π4ð4=8Þ 5 0:1; π24 ð4=8Þ 5 0:3;

(5.113)

π1ð5=2Þ 5 0:5; π24 ð5=2Þ 5 0:5;

(5.114)

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Consequences of Maritime Critical Infrastructure Accidents

24 π1ð5=3Þ 5 0:500; π15 ð5=3Þ 5 0:031; πð5=3Þ 5 0:469;

(5.115)

π1ð5=4Þ 5 0:5; π24 ð5=4Þ 5 0:5;

(5.116)

π1ð5=5Þ 5 0:5; π24 ð5=5Þ 5 0:5;

(5.117)

π1ð5=6Þ 5 0:5; π24 ð5=6Þ 5 0:5;

(5.118)

π1ð5=7Þ 5 0:5; π24 ð5=7Þ 5 0:5;

(5.119)

π1ð5=8Þ 5 0:5; π24 ð5=8Þ 5 0:5:

(5.120)

Therefore according to (3.38) and considering (5.51)(5.85), we get the approximate limit values of transient probabilities that are not equal to 0 at the particular states of the process of environment threats p1ð1=2Þ 5 0:000276824647437; p7ð1=2Þ 5 0:001532252145231; p27 ð1=2Þ 5 0:998190923207332;

(5.121)

27 p1ð1=3Þ 5 0:043835911892547; p23 ð1=3Þ 5 0:001569834135506; pð1=3Þ 5 0:951454585700935; 30 pð1=3Þ 5 0:003139668271012;

(5.122)

p1ð1=4Þ 5 0:000435442144920; p5ð1=4Þ 5 0:007194728411754; p13 ð1=4Þ 5 0:002877891364702; 35 p27 5 0:986614046713922; p 5 0:002877891364702; ð1=4Þ ð1=4Þ

(5.123)

p1ð1=5Þ 5 0:000809193690969; p3ð1=5Þ 5 0:000080956590518; p4ð1=5Þ 5 0:371163634671754; 27 p7ð1=5Þ 5 0:000404782952589; p17 ð1=5Þ 5 0:024286977155353; pð1=5Þ 5 0:602040106081050; p34 ð1=5Þ 5 0:001214348857767;

(5.124)

p1ð1=6Þ 5 0:000173580975525; p27 ð1=6Þ 5 0:999826419024475;

(5.125)

30 p1ð1=7Þ 5 0:000514515776340; p27 ð1=7Þ 5 0:984050010933460; pð1=7Þ 5 0:015435473290200;

(5.126)

p1ð1=8Þ 5 0:000519321952217; p3ð1=8Þ 5 0:001225330362859; p6ð1=8Þ 5 0:100560489203204; p7ð1=8Þ 5 0:000459498886072; p8ð1=8Þ 5 0:007351982177153; p21 ð1=8Þ 5 0:000153166295357; 27 pð1=8Þ 5 0:889730211123138;

(5.127)

p1ð2=2Þ 5 0:000033157381296; p2ð2=2Þ 5 0:214729428310395; p33 ð2=2Þ 5 0:785237414308309;

(5.128)

18 p1ð2=3Þ 5 0:011404073946944; p17 ð2=3Þ 5 0:127319487610230; pð2=3Þ 5 0:073775998088456; p33 ð2=3Þ 5 0:787500440354370;

(5.129)

p1ð2=4Þ 5 0:000032205081962; p33 ð2=4Þ 5 0:999967794918038;

(5.130)

p1ð2=5Þ 5 0:000026304021885; p33 ð2=5Þ 5 0:999973695978115;

(5.131)

p1ð2=6Þ 5 0:000081692672167; p33 ð2=6Þ 5 0:999918307327833;

(5.132)

33 p1ð2=7Þ 5 0:000427167876975; p17 ð2=7Þ 5 0:538231524989321; pð2=7Þ 5 0:461341307133704;

(5.133)

p1ð2=8Þ 5 0:000097410376222; p2ð2=8Þ 5 0:049796184324711; p4ð2=8Þ 5 0:009959236864942; p5ð2=8Þ 5 0:009959236864942; p33 ð2=8Þ 5 0:930187931569183;

(5.134)

p1ð3=2Þ 5 0:000033157381296; p2ð3=2Þ 5 0:214729428310395; p24 ð3=2Þ 5 0:785237414308309;

(5.135)

15 p1ð3=3Þ 5 0:010469571021354; p14 ð3=3Þ 5 0:116886335895330; pð3=3Þ 5 0:067730449245758; p24 ð3=3Þ 5 0:804913643837558;

(5.136)

Prediction of critical infrastructure accident consequences Chapter | 5

105

p1ð3=4Þ 5 0:000032205081962; p24 ð3=4Þ 5 0:999967794918038;

(5.137)

p1ð3=5Þ 5 0:000026304021885; p24 ð3=5Þ 5 0:999973695978115;

(5.138)

p1ð3=6Þ 5 0:000081692672167; p24 ð3=6Þ 5 0:999918307327833;

(5.139)

24 p1ð3=7Þ 5 0:000427167876975; p14 ð3=7Þ 5 0:538231524989321; pð3=7Þ 5 0:461341307133704;

(5.140)

p1ð3=8Þ 5 0:000111382990928; p2ð3=8Þ 5 0:056938984962282; p3ð3=8Þ 5 0:011387796992456; p4ð3=8Þ 5 0:011387796992456; p24 ð3=8Þ 5 0:920174038061878;

(5.141)

p1ð4=2Þ 5 0:011104929242627; p2ð4=2Þ 5 0:273683291753946; p24 ð4=2Þ 5 0:715211779003427;

(5.142)

15 p1ð4=3Þ 5 0:011258403715360; p14 ð4=3Þ 5 0:103904136833938; pð4=3Þ 5 0:062018905203649; 24 pð4=3Þ 5 0:822818554247053;

(5.143)

p1ð4=4Þ 5 0:000014130938121; p24 ð4=4Þ 5 0:999985869061879;

(5.144)

p1ð4=5Þ 5 0:000010936013386; p24 ð4=5Þ 5 0:999989063986614;

(5.145)

p1ð4=6Þ 5 0:033355709358580; p24 ð4=6Þ 5 0:966644290641420;

(5.146)

24 p1ð4=7Þ 5 0:004769142078508; p14 ð4=7Þ 5 0:018866935695194; pð4=7Þ 5 0:976363922226298;

(5.147)

p1ð4=8Þ 5 0:000044513687959; p2ð4=8Þ 5 0:064099710661028; p4ð4=8Þ 5 0:012819942132206; p24 ð4=8Þ 5 0:923035833518807;

(5.148)

p1ð5=2Þ 5 0:013112688425847; p24 ð5=2Þ 5 0:986887311574153;

(5.149)

24 p1ð5=3Þ 5 0:011325246951561; p15 ð5=3Þ 5 0:111923627167956; pð5=3Þ 5 0:876751125880483;

(5.150)

p1ð5=4Þ 5 0:000015561538102; p24 ð5=4Þ 5 0:999984438461898;

(5.151)

p1ð5=5Þ 5 0:000005787003547; p24 ð5=5Þ 5 0:999994212996453;

(5.152)

p1ð5=6Þ 5 0:076923076923077; p24 ð5=6Þ 5 0:923076923076923;

(5.153)

p1ð5=7Þ 5 0:000086798021005; p24 ð5=7Þ 5 0:999913201978995;

(5.154)

p1ð5=8Þ 5 0:000011431053600; p24 ð5=8Þ 5 0:999988568946400:

(5.155)

Further, by (3.40) and considering (5.121)(5.155), the approximate mean values of the sojourn total times η^ iðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ during the fixed time η 5 1 month 5 43,200 minutes at the particular states siðk=lÞ expressed in minutes are 1 7 27 M^ ð1=2Þ 5 11:96; M^ ð1=2Þ 5 66:19; M^ ð1=2Þ 5 43; 121:85;

(5.156)

1 23 27 30 M^ ð1=3Þ 5 1893:71; M^ ð1=3Þ 5 67:82; M^ ð1=3Þ 5 41; 102:84; M^ ð1=3Þ 5 135:63;

(5.157)

1 5 13 27 35 M^ ð1=4Þ 5 18:81; M^ ð1=4Þ 5 310:81; M^ ð1=4Þ 5 124:32; M^ ð1=4Þ 5 42; 621:73; M^ ð1=4Þ 5 124:32;

(5.158)

1 3 4 7 M^ ð1=5Þ 5 34:96; M^ ð1=5Þ 5 3:50; M^ ð1=5Þ 5 16; 034:27; M^ ð1=5Þ 5 17:49; 17 27 34 M^ ð1=5Þ 5 1049:20; M^ ð1=5Þ 5 26; 008:13; M^ ð1=5Þ 5 52:46;

(5.159)

1 27 M^ ð1=6Þ 5 7:50; M^ ð1=6Þ 5 43; 192:50;

(5.160)

106

Consequences of Maritime Critical Infrastructure Accidents

1 27 30 M^ ð1=7Þ 5 22:23; M^ ð1=7Þ 5 42; 510:96; M^ ð1=7Þ 5 666:81; 1 3 6 7 M^ ð1=8Þ 5 22:43; M^ ð1=8Þ 5 52:93; M^ ð1=8Þ 5 4344:21; M^ ð1=8Þ 5 19:85; 8 21 27 M^ ð1=8Þ 5 317:61; M^ ð1=8Þ 5 6:62; M^ ð1=8Þ 5 38; 436:35;

(5.161) (5.162)

1 2 33 M^ ð2=2Þ 5 1:43; M^ ð2=2Þ 5 9276:31; M^ ð2=2Þ 5 33; 922:26;

(5.163)

1 17 18 33 M^ ð2=3Þ 5 492:66; M^ ð2=3Þ 5 5500:20; M^ ð2=3Þ 5 3187:12; M^ ð2=3Þ 5 34; 020:02;

(5.164)

1 33 M^ ð2=4Þ 5 1:39; M^ ð2=4Þ 5 43; 198:61;

(5.165)

1 33 M^ ð2=5Þ 5 1:14; M^ ð2=5Þ 5 43; 198:86;

(5.166)

1 33 M^ ð2=6Þ 5 3:53; M^ ð2=6Þ 5 43; 196:47;

(5.167)

1 17 33 M^ ð2=7Þ 5 18:45; M^ ð2=7Þ 5 23; 251:60; M^ ð2=7Þ 5 19; 929:94;

(5.168)

1 2 4 5 33 M^ ð2=8Þ 5 4:21; M^ ð2=8Þ 5 2151:20; M^ ð2=8Þ 5 430:24; M^ ð2=8Þ 5 430:24; M^ ð2=8Þ 5 40; 184:12;

(5.169)

1 2 24 M^ ð3=2Þ 5 1:43; M^ ð3=2Þ 5 9276:31; M^ ð3=2Þ 5 33; 922:26;

(5.170)

1 14 15 24 M^ ð3=3Þ 5 452:29; M^ ð3=3Þ 5 5049:49; M^ ð3=3Þ 5 2925:96; M^ ð3=3Þ 5 34; 772:27;

(5.171)

1 24 M^ ð3=4Þ 5 1:39; M^ ð3=4Þ 5 43; 198:61;

(5.172)

1 24 M^ ð3=5Þ 5 1:14; M^ ð3=5Þ 5 43; 198:86;

(5.173)

1 24 M^ ð3=6Þ 5 3:53; M^ ð3=6Þ 5 43; 196:47;

(5.174)

1 14 24 M^ ð3=7Þ 5 18:45; M^ ð3=7Þ 5 23; 251:60; M^ ð3=7Þ 5 19; 929:94;

(5.175)

1 2 3 4 24 M^ ð3=8Þ 5 4:81; M^ ð3=8Þ 5 2459:76; M^ ð3=8Þ 5 491:95; M^ ð3=8Þ 5 491:95; M^ ð3=8Þ 5 39; 751:52;

(5.176)

1 2 24 M^ ð4=2Þ 5 479:73; M^ ð4=2Þ 5 11; 823:12; M^ ð4=2Þ 5 30; 897:15;

(5.177)

1 14 15 24 M^ ð4=3Þ 5 486:36; M^ ð4=3Þ 5 4488:66; M^ ð4=3Þ 5 2679:22; M^ ð4=3Þ 5 35; 545:76;

(5.178)

1 24 M^ ð4=4Þ 5 0:61; M^ ð4=4Þ 5 43; 199:39;

(5.179)

1 24 M^ ð4=5Þ 5 0:47; M^ ð4=5Þ 5 43; 199:53;

(5.180)

1 24 M^ ð4=6Þ 5 1440:97; M^ ð4=6Þ 5 41; 759:03;

(5.181)

1 14 24 M^ ð4=7Þ 5 206:03; M^ ð4=7Þ 5 815:05; M^ ð4=7Þ 5 42; 178:92;

(5.182)

1 2 4 24 M^ ð4=8Þ 5 1:92; M^ ð4=8Þ 5 2769:11; M^ ð4=8Þ 5 553:82; M^ ð4=8Þ 5 39; 875:15;

(5.183)

1 24 M^ ð5=2Þ 5 566:47; M^ ð5=2Þ 5 42; 633:53;

(5.184)

1 15 24 M^ ð5=3Þ 5 489:25; M^ ð5=3Þ 5 4835:10; M^ ð5=3Þ 5 37; 875:65;

(5.185)

1 24 M^ ð5=4Þ 5 0:67; M^ ð5=4Þ 5 43; 199:33;

(5.186)

1 24 M^ ð5=5Þ 5 0:25; M^ ð5=5Þ 5 43; 199:75;

(5.187)

1 24 M^ ð5=6Þ 5 3323:08; M^ ð5=6Þ 5 39; 876:92;

(5.188)

Prediction of critical infrastructure accident consequences Chapter | 5

5.3

107

1 24 M^ ð5=7Þ 5 3:75; M^ ð5=7Þ 5 43; 196:25;

(5.189)

1 24 M^ ð5=8Þ 5 0:49; M^ ð5=8Þ 5 43; 199:51:

(5.190)

Prediction of process of environment degradation

We assume, as in Section 4.6, that we have identified the unknown parameters of the semi-Markov model of the process of environment degradation Rðk=υÞ ðtÞ; tAh0; 1 NÞ for particular environment subareas Dk ; k 5 1; 2; . . .; n3 , while the process of environment threats SðkÞ ðtÞ is at the state sυðkÞ ; υ 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 : G

G

G

the initial probabilities qiðk=υÞ ð0Þ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process environment degradai tion staying at the particular states rðk=υÞ at the moment t 5 0; the probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradaj i tion transitions from the state rðk=υÞ into the state rðk=υÞ ; the distributions of the process of environment degradation conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; ij k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the particular states and their mean values Mðk=υÞ 5 E½ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;

then we can predict its basic characteristics.

5.3.1 Prediction of process of environment degradation at Baltic Sea waters The matrices of probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation transitions from j ij i the state rðk=υÞ into the state rðk=υÞ are estimated in (4.279)(4.286). The mean values Mðk=υÞ of the conditional sojourn ij times ζ ðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation at particular states are determined, and their values are given by (4.287). This way, the process of environment degradation for the Baltic Sea waters is approximately defined, and we may predict its main characteristics. After considering these results and applying (3.58), we conclude that the unconditional mean sojourn times of the process of environment degradation at particular states are as follows: h i 1 2 12 Mð1=6Þ 5 E ζ 1ð1=6Þ 5 q1ð1=6Þ Mð1=6Þ 5 1U1 5 1; h i (5.191) 2 2 1 21 5 E ζ ð1=6Þ 5 q2ð1=6Þ Mð1=6Þ 5 1U240 5 240; Mð1=6Þ h i 1 6 16 5 E ζ 1ð1=27Þ 5 q1ð1=27Þ Mð1=27Þ 5 1U1 5 1; Mð1=27Þ h i (5.192) 6 6 61 61 Mð1=27Þ 5 E ζ ð1=27Þ 5 qð1=27Þ Mð1=27Þ 5 1U240 5 240; h i 1 11 1 11 5 E ζ 1ð1=30Þ 5 q1ð1=30Þ Mð1=30Þ 5 1U1 5 1; Mð1=30Þ h i (5.193) 11 11 11 1 11 1 Mð1=30Þ 5 E ζ ð1=30Þ 5 qð1=30Þ Mð1=30Þ 5 1U240 5 240; h i 1 27 1 27 5 E ζ 1ð2=17Þ 5 q1ð2=17Þ Mð2=17Þ 5 1U1 5 1; Mð2=17Þ h i 12 1 12 1 5 E ζ 12 Mð2=17Þ 5 q12 ð2=17Þ Mð2=17Þ 5 1U2880 5 2880; h ð2=17Þ i 16 12 16 12 5 E ζ 16 Mð2=17Þ 5 q16 ð2=17Þ Mð2=17Þ 5 1U3780 5 3780; h ð2=17Þ i (5.194) 21 21 16 21 16 5 E ζ 21 Mð2=17Þ ð2=17Þ 5 qð2=17Þ Mð2=17Þ 5 1U2880 5 2880; h i 25 21 25 21 5 E ζ 25 Mð2=17Þ 5 q25 ð2=17Þ Mð2=17Þ 5 1U300 5 300; h ð2=17Þ i 27 27 25 27 25 5 E ζ 27 Mð2=17Þ ð2=17Þ 5 qð2=17Þ Mð2=17Þ 5 1U240 5 240;

108

Consequences of Maritime Critical Infrastructure Accidents

h i 1 6 16 Mð2=33Þ 5 E ζ 1ð2=33Þ 5 q1ð2=33Þ Mð2=33Þ 5 1U1 5 1; h i 6 6 61 61 Mð2=33Þ 5 E ζ ð2=33Þ 5 qð2=33Þ Mð2=33Þ 5 1U1440 5 1440; h i 1 27 1 27 5 E ζ 1ð3=14Þ 5 q1ð3=14Þ Mð3=14Þ 5 1U1 5 1; Mð3=14Þ h i 12 12 12 1 12 1 Mð3=14Þ 5 E ζ ð3=14Þ 5 qð3=14Þ Mð3=14Þ 5 1U2880 5 2880; h i 16 12 16 12 5 E ζ 16 Mð3=14Þ 5 q16 ð3=14Þ Mð3=14Þ 5 1U3780 5 3780; h ð3=14Þ i 21 5 E ζ 21 Mð3=14Þ 5 q21 16 M 21 16 5 1U2880 5 2880; hð3=14Þ i ð3=14Þ ð3=14Þ 25 21 25 21 5 E ζ 25 Mð3=14Þ 5 q25 ð3=14Þ Mð3=14Þ 5 1U300 5 300; h ð3=14Þ i 27 27 25 27 25 5 E ζ 27 Mð3=14Þ ð3=14Þ 5 qð3=14Þ Mð3=14Þ 5 1U240 5 240; h i 1 6 16 5 E ζ 1ð3=24Þ 5 q1ð3=24Þ Mð3=24Þ 5 1U1 5 1; Mð3=24Þ h i 6 1 61 5 E ζ 6ð3=24Þ 5 q6ð3=24Þ Mð3=24Þ 5 1U1440 5 1440; Mð3=24Þ h i 1 30 1 30 5 E ζ 1ð4=14Þ 5 q1ð4=14Þ Mð4=14Þ 5 1U1 5 1; Mð4=14Þ h i 12 1 12 1 5 E ζ 12 Mð4=14Þ 5 q12 ð4=14Þ Mð4=14Þ 5 1U2880 5 2880; h ð4=14Þ i 16 12 16 12 5 E ζ 16 Mð4=14Þ 5 q16 ð4=14Þ Mð4=14Þ 5 1U3780 5 3780; h ð4=14Þ i 21 5 E ζ 21 Mð4=14Þ 5 q21 16 M 21 16 5 1U2880 5 2880; hð4=14Þ i ð4=14Þ ð4=14Þ 28 21 28 21 5 E ζ 28 Mð4=14Þ 5 q28 ð4=14Þ Mð4=14Þ 5 1U300 5 300; h ð4=14Þ i 30 30 28 30 28 5 E ζ 30 Mð4=14Þ ð4=14Þ 5 qð4=14Þ Mð4=14Þ 5 1U240 5 240:

(5.195)

(5.196)

(5.197)

(5.198)

To find the limit values of the transient probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation Rðk=υÞ ðtÞ at its particular states, first, we have to solve the systems of equations (3.61) 8 ij i i > > < ½πðk=υÞ 1 3 ‘k 5 ½πðk=υÞ 1 3 ‘k ½qðk=υÞ ‘k 3 ‘k ‘k X > πjðk=υÞ 5 1; > : j51

where k k ½π‘ðk=υÞ 1 3 ‘k 5 ½π1ðk=υÞ ; π2ðk=υÞ ; . . .; π‘ðk=υÞ 

and the elements of the matrices ½qijðk=υÞ ‘k 3 ‘k are given by (4.279)(4.286). The systems of equations take the forms 8 1 2 > < πð1=6Þ 5 πð1=6Þ π2ð1=6Þ 5 π1ð1=6Þ (5.199) > : π1 1 π2 5 1 ð1=6Þ ð1=6Þ 8 1 6 > < πð1=27Þ 5 πð1=27Þ π6ð1=27Þ 5 π1ð1=27Þ (5.200) > : π1 6 1 π 5 1 ð1=27Þ ð1=27Þ 8 1 11 > < πð1=30Þ 5 πð1=30Þ 1 π11 (5.201) ð1=30Þ 5 πð1=30Þ > : π1 11 1 π 5 1 ð1=30Þ ð1=30Þ

Prediction of critical infrastructure accident consequences Chapter | 5

8 1 πð2=17Þ 5 π12 > ð2=17Þ > > > > π12 5 π16 > ð2=17Þ ð2=17Þ > > > 16 21 > π 5 π > ð2=17Þ < ð2=17Þ 25 π21 ð2=17Þ 5 πð2=17Þ > > 25 > πð2=17Þ 5 π27 > ð2=17Þ > > > 1 > π27 5 π > ð2=17Þ ð2=17Þ > > : π1 12 16 21 25 27 ð2=17Þ 1 πð2=17Þ 1 πð2=17Þ 1 πð2=17Þ 1 πð2=17Þ 1 πð2=17Þ 5 1 8 1 6 > < πð2=33Þ 5 πð2=33Þ 6 πð2=33Þ 5 π1ð2=33Þ > : π1 6 ð2=33Þ 1 πð2=33Þ 5 1 8 1 πð3=14Þ 5 π12 > ð3=14Þ > > > > π12 5 π16 > ð3=14Þ ð3=14Þ > > > π16 5 π21 > > ð3=14Þ < ð3=14Þ 25 π21 ð3=14Þ 5 πð3=14Þ > > 25 > πð3=14Þ 5 π27 > ð3=14Þ > > > 27 1 > π 5 π > ð3=14Þ ð3=14Þ > > : π1 12 16 21 25 27 ð3=14Þ 1 πð3=14Þ 1 πð3=14Þ 1 πð3=14Þ 1 πð3=14Þ 1 πð3=14Þ 5 1 8 1 2 > < πð3=24Þ 5 πð3=24Þ 2 πð3=24Þ 5 π1ð3=24Þ > : π1 2 ð3=24Þ 1 πð3=24Þ 5 1 8 π1ð4=14Þ 5 π12 > ð4=14Þ > > > 12 16 > π 5 π > ð4=14Þ ð4=14Þ > > > 16 21 > π 5 π > ð4=14Þ ð4=14Þ < 21 πð4=14Þ 5 π28 ð4=14Þ > > 28 30 > π 5 π > ð4=14Þ ð4=14Þ > > > 30 1 > π 5 π > ð4=14Þ ð4=14Þ > > : π1 12 16 21 28 30 1 π 1 π 1 π ð4=14Þ ð4=14Þ ð4=14Þ 1 πð4=14Þ 1 πð4=14Þ 5 1: ð4=14Þ

109

(5.202)

(5.203)

(5.204)

(5.205)

(5.206)

Solving the particular systems of equations, we get π1ð1=6Þ 5 0:5; π2ð1=6Þ 5 0:5;

(5.207)

π1ð1=27Þ 5 0:5; π6ð1=27Þ 5 0:5;

(5.208)

π1ð1=30Þ 5 0:5; π11 ð1=30Þ 5 0:5;

(5.209)

16 21 25 27 π1ð2=17Þ D0:1667; π12 ð2=17Þ D0:1667; πð2=17Þ D0:1667; πð2=17Þ D0:1667; πð2=17Þ D0:1666; πð2=17Þ D0:1666;

π1ð2=33Þ 5 0:5; π6ð2=33Þ 5 0:5; 16 21 25 27 π1ð3=14Þ D0:1667; π12 ð3=14Þ D0:1667; πð3=14Þ D0:1667; πð3=14Þ D0:1667; πð3=14Þ D0:1666; πð3=14Þ D0:1666;

π1ð3=24Þ 5 0:5; π2ð3=24Þ 5 0:5; 16 21 28 30 π1ð4=14Þ D0:1667; π12 ð4=14Þ D0:1667; πð4=14Þ D0:1667; πð4=14Þ D0:1667; πð4=14Þ D0:1666; πð4=14Þ D0:1666:

(5.210) (5.211) (5.212) (5.213) (5.214)

Therefore according to (3.60) and considering (5.191)(5.198), we get the approximate limit values of transient probabilities that are not equal to 0 at the particular states of the process of environment degradation

110

Consequences of Maritime Critical Infrastructure Accidents

q1ð1=6Þ 5 0:004149377593361; q2ð1=6Þ 5 0:995850622406639;

(5.215)

q1ð1=27Þ 5 0:004149377593361; q6ð1=27Þ 5 0:995850622406639;

(5.216)

q1ð1=30Þ 5 0:004149377593361; q11 ð1=30Þ 5 0:995850622406639;

(5.217)

16 q1ð2=17Þ 5 0:000099199695891; q12 ð2=17Þ 5 0:285695124165349; qð2=17Þ 5 0:374974850467021; 25 27 q21 ð2=17Þ 5 0:285695124165349; qð2=17Þ 5 0:029742056392439; qð2=17Þ 5 0:023793645113951;

q1ð2=33Þ 5 0:000693962526024; q6ð2=33Þ 5 0:999306037473976; 16 q1ð3=14Þ 5 0:000099199695891; q12 ð3=14Þ 5 0:285695124165349; qð3=14Þ 5 0:374974850467021; 21 25 27 qð3=14Þ 5 0:285695124165349; qð3=14Þ 5 0:029742056392439; qð3=14Þ 5 0:023793645113951;

q1ð3=24Þ 5 0:000693962526024; q6ð3=24Þ 5 0:999306037473976; 16 q1ð4=14Þ 5 0:000099199695891; q12 ð4=14Þ 5 0:285695124165349; qð4=14Þ 5 0:374974850467021; 21 28 qð4=14Þ 5 0:285695124165349; qð4=14Þ 5 0:029742056392439; q30 ð4=14Þ 5 0:023793645113951:

(5.218) (5.219) (5.220) (5.221) (5.222)

i Further, by (3.62) and considering (5.215)(5.222), the approximate mean values of the sojourn total times ζ^ ðk=υÞ of the process of environment degradation Rðk=υÞ ðtÞ during the fixed time ζ 5 1 month 5 43,200 minutes at the particular i states rðk=υÞ expressed in minutes are 1 2 M^ ð1=6Þ 5 179:25; M^ ð1=6Þ 5 43; 020:75;

(5.223)

1 6 M^ ð1=27Þ 5 179:25; M^ ð1=27Þ 5 43; 020:75;

(5.224)

1 11 M^ ð1=30Þ 5 179:25; M^ ð1=30Þ 5 43; 020:75;

(5.225)

1 12 16 21 M^ ð2=17Þ 5 4:29; M^ ð2=17Þ 5 12; 342:03; M^ ð2=17Þ 5 16; 198:91; M^ ð2=17Þ 5 12; 342:03; 25 27 M^ ð2=17Þ 5 1284:86; M^ ð2=17Þ 5 1027:89;

(5.226)

1 6 M^ ð2=33Þ 5 29:98; M^ ð2=33Þ 5 43; 170:02;

(5.227)

1 12 16 21 M^ ð3=14Þ 5 4:29; M^ ð3=14Þ 5 12; 342:03; M^ ð3=14Þ 5 16; 198:91; M^ ð3=14Þ 5 12; 342:03; 25 27 M^ ð3=14Þ 5 1284:86; M^ ð3=14Þ 5 1027:89;

(5.228)

1 6 M^ ð3=24Þ 5 29:98; M^ ð3=24Þ 5 43; 170:02;

(5.229)

1 12 16 21 M^ ð4=14Þ 5 4:29; M^ ð4=14Þ 5 12; 342:03; M^ ð4=14Þ 5 16; 198:91; M^ ð4=14Þ 5 12; 342:03; 28 30 M^ ð4=14Þ 5 1284:86; M^ ð4=14Þ 5 1027:89:

(5.230)

5.3.2 Prediction of process of environment degradation at world sea waters Proceedings in analogous way as in Section 5.3.1, we predict the process of environment degradation within the world sea waters. The matrices of probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation transij ij i tions from the state rðk=υÞ into the state rðk=υÞ are estimated in (4.290)(4.323). The mean values Mðk=υÞ of the condiij tional sojourn times ζ ðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation at particular states are determined, and their values are given by (4.336) and (4.339). This way, the process of environment degradation for the world sea waters is approximately defined, and we may predict their main characteristics. After considering

Prediction of critical infrastructure accident consequences Chapter | 5

111

these results and applying (3.58), we conclude that the unconditional mean sojourn times of the process of environment degradation at particular states are as follows: 1 6 22 25 28 5 1; Mð1=3Þ 5 180; Mð1=3Þ 5 120; Mð1=3Þ 5 120; Mð1=3Þ 5 300; Mð1=3Þ 1 20 21 24 27 29 30 Mð1=4Þ 5 1; Mð1=4Þ 5 20; Mð1=4Þ 5 680; Mð1=4Þ 5 920; Mð1=4Þ 5 2020; Mð1=4Þ 5 20; Mð1=4Þ 5 40;

(5.231) (5.232)

1 20 23 26 Mð1=5Þ 5 1; Mð1=5Þ 5 30; Mð1=5Þ 5 60; Mð1=5Þ 5 360;

(5.233)

1 2 17 18 19 Mð1=6Þ 5 1; Mð1=6Þ 5 1560; Mð1=6Þ 5 30; Mð1=6Þ 5 30; Mð1=6Þ 5 180;

(5.234)

1 2 6 8 10 Mð1=7Þ 5 1; Mð1=7Þ 5 15; Mð1=7Þ 5 180; Mð1=7Þ 5 120; Mð1=7Þ 5 420;

(5.235)

1 13 14 15 Mð1=8Þ 5 1; Mð1=8Þ 5 60; Mð1=8Þ 5 6; Mð1=8Þ 5 120;

(5.236)

1 6 Mð1=13Þ 5 1; Mð1=13Þ 5 120;

(5.237)

1 6 22 25 28 Mð1=17Þ 5 1; Mð1=17Þ 5 180; Mð1=17Þ 5 120; Mð1=17Þ 5 120; Mð1=17Þ 5 300;

(5.238)

1 20 Mð1=21Þ 5 1; Mð1=21Þ 5 10;

(5.239)

1 2 Mð1=23Þ 5 1; Mð1=23Þ 5 240;

(5.240)

1 6 Mð1=27Þ 5 1; Mð1=27Þ 5 2982:34;

(5.241)

1 11 Mð1=30Þ 5 1; Mð1=30Þ 5 260;

(5.242)

1 20 23 26 Mð1=35Þ 5 1; Mð1=35Þ 5 30; Mð1=35Þ 5 60; Mð1=35Þ 5 360;

(5.243)

1 5 Mð2=1Þ 5 1; Mð2=1Þ 5 600;

(5.244)

1 2 Mð2=2Þ 5 1; Mð2=2Þ 5 2880;

(5.245)

1 2 Mð2=4Þ 5 1; Mð2=4Þ 5 1440;

(5.246)

1 13 14 18 Mð2=5Þ 5 1; Mð2=5Þ 5 780; Mð2=5Þ 5 360; Mð2=5Þ 5 300;

(5.247)

1 12 16 21 25 27 5 1; Mð2=17Þ 5 2640; Mð2=17Þ 5 2145; Mð2=17Þ 5 1530; Mð2=17Þ 5 255; Mð2=17Þ 5 270; Mð2=17Þ

(5.248)

1 6 9 11 5 1; Mð2=18Þ 5 10; 080; Mð2=18Þ 5 11; 520; Mð2=18Þ 5 21; 600; Mð2=18Þ

(5.249)

1 2 6 9 5 1; Mð2=33Þ 5 246; 240; Mð2=33Þ 5 33; 935:21; Mð2=33Þ 5 5709:65; Mð2=33Þ 11 12 15 20 Mð2=33Þ 5 5051:25; Mð2=33Þ 5 14; 400; Mð2=33Þ 5 7200; Mð2=33Þ 5 10; 080; 22 23 Mð2=33Þ 5 1440; Mð2=33Þ 5 1440;

(5.250)

1 2 5 1; Mð3=2Þ 5 2880; Mð3=2Þ

(5.251)

1 5 Mð3=3Þ 5 1; Mð3=3Þ 5 1440;

(5.252)

1 13 14 18 Mð3=4Þ 5 1; Mð3=4Þ 5 780; Mð3=4Þ 5 360; Mð3=4Þ 5 300;

(5.253)

1 12 16 21 25 27 5 1; Mð3=14Þ 5 2640; Mð3=14Þ 5 2145; Mð3=14Þ 5 1530; Mð3=14Þ 5 255; Mð3=14Þ 5 270; Mð3=14Þ 1 6 9 11 5 1; Mð3=15Þ 5 10; 080; Mð3=15Þ 5 11; 520; Mð3=15Þ 5 21; 600; Mð3=15Þ 1 2 6 9 11 5 1; Mð3=24Þ 5 246; 240; Mð3=24Þ 5 16; 131:98; Mð3=24Þ 5 9983:58; Mð3=24Þ 5 2185:8; Mð3=24Þ 12 15 20 22 23 5 14; 400; Mð3=24Þ 5 7200; Mð3=24Þ 5 10; 080; Mð3=24Þ 5 1440; Mð3=24Þ 5 1440; Mð3=24Þ

(5.254) (5.255) (5.256)

1 2 5 1; Mð4=2Þ 5 2880; Mð4=2Þ

(5.257)

1 13 14 18 Mð4=4Þ 5 1; Mð4=4Þ 5 780; Mð4=4Þ 5 360; Mð4=4Þ 5 300;

(5.258)

112

Consequences of Maritime Critical Infrastructure Accidents

1 12 16 21 28 30 Mð4=14Þ 5 1; Mð4=14Þ 5 2640; Mð4=14Þ 5 2145; Mð4=14Þ 5 1530; Mð4=14Þ 5 300; Mð4=14Þ 5 270;

(5.259)

1 6 9 11 5 1; Mð4=15Þ 5 10; 080; Mð4=15Þ 5 11; 520; Mð4=15Þ 5 21; 600; Mð4=15Þ

(5.260)

1 6 9 11 5 847:03; Mð4=24Þ 5 23; 281:14; Mð4=24Þ 5 7816:63; Mð4=24Þ 5 2300:69; Mð4=24Þ 24 25 26 Mð4=24Þ 5 5760; Mð4=24Þ 5 17; 282:16; Mð4=24Þ 5 29; 046:90;

(5.261)

1 5 5 1; Mð5=1Þ 5 600; Mð5=1Þ

(5.262)

1 6 9 11 Mð5=15Þ 5 1; Mð5=15Þ 5 10; 080; Mð5=15Þ 5 11; 520; Mð5=15Þ 5 21; 600;

(5.263)

1 6 9 11 5 420:64; Mð5=24Þ 5 21; 215:96; Mð5=24Þ 5 10; 749:49; Mð5=24Þ 5 27; 552: Mð5=24Þ

(5.264)

To find the limit values of the transient probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; 5; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; υ 5 1; 2; . . .; υk ; υ1 5 35; υ2 5 33; υ3 5 29; υ4 5 29; υ5 5 29 of the process of environment degradation Rðk=υÞ ðtÞ at its particular states, first, we have to solve the systems of equations (3.61) 8 ij i i > > < ½πðk=υÞ 1 3 ‘k 5 ½πðk=υÞ 1 3 ‘k ½qðk=υÞ ‘k 3 ‘k ‘k X > πjðk=υÞ 5 1; > : j51

where k k ½π‘ðk=υÞ 1 3 ‘k 5 ½π1ðk=υÞ ; π2ðk=υÞ ; . . .; π‘ðk=υÞ 

and the elements of the matrices ½qijðk=υÞ ‘k 3 ‘k are given by (4.290)(4.323). Solving the particular systems of equations, we get 25 28 π1ð1=3Þ 5 0:2; π6ð1=3Þ 5 0:2; π22 ð1=3Þ 5 0:2; πð1=3Þ 5 0:2; πð1=3Þ 5 0:2;

(5.265)

21 24 π1ð1=4Þ 5 0:2500; π20 ð1=4Þ 5 0:0625; πð1=4Þ 5 0:1875; πð1=4Þ 5 0:1875; 29 30 π27 ð1=4Þ 5 0:1875; πð1=4Þ 5 0:0625; πð1=4Þ 5 0:0625;

(5.266)

23 26 π1ð1=5Þ 5 0:25; π20 ð1=5Þ 5 0:25; πð1=5Þ 5 0:25; πð1=5Þ 5 0:25;

(5.267)

18 19 π1ð1=6Þ D0:3751; π2ð1=6Þ D0:2502; π17 ð1=6Þ D0:1249; πð1=6Þ D0:1249; πð1=6Þ D0:1249;

(5.268)

π1ð1=7Þ D0:3332; π2ð1=7Þ D0:1667; π6ð1=7Þ D0:1667; π8ð1=7Þ D0:1667; π10 ð1=7Þ D0:1667;

(5.269)

14 15 π1ð1=8Þ 5 0:25; π13 ð1=8Þ 5 0:25; πð1=8Þ 5 0:25; πð1=8Þ 5 0:25;

(5.270)

π1ð1=13Þ 5 0:5; π6ð1=13Þ 5 0:5;

(5.271)

25 28 π1ð1=17Þ 5 0:2; π6ð1=17Þ 5 0:2; π22 ð1=17Þ 5 0:2; πð1=17Þ 5 0:2; πð1=17Þ 5 0:2;

(5.272)

π1ð1=21Þ 5 0:5; π20 ð1=21Þ 5 0:5;

(5.273)

π1ð1=23Þ 5 0:5; π2ð1=23Þ 5 0:5;

(5.274)

π1ð1=27Þ 5 0:5; π6ð1=27Þ 5 0:5;

(5.275)

π1ð1=30Þ 5 0:5; π11 ð1=30Þ 5 0:5;

(5.276)

23 26 π1ð1=35Þ 5 0:25; π20 ð1=35Þ 5 0:25; πð1=35Þ 5 0:25; πð1=35Þ 5 0:25;

(5.277)

π1ð2=1Þ 5 0:5; π5ð2=1Þ 5 0:5;

(5.278)

π1ð2=2Þ 5 0:5; π2ð2=2Þ 5 0:5;

(5.279)

Prediction of critical infrastructure accident consequences Chapter | 5

113

π1ð2=4Þ 5 0:5; π2ð2=4Þ 5 0:5;

(5.280)

14 18 π1ð2=5Þ 5 0:25; π13 ð2=5Þ 5 0:25; πð2=5Þ 5 0:25; πð2=5Þ 5 0:25;

(5.281)

16 21 25 27 π1ð2=17Þ D0:1818; π12 ð2=17Þ D0:1818; πð2=17Þ D0:1818; πð2=17Þ D0:1818; πð2=17Þ D0:1818; πð2=17Þ D0:0910;

π1ð2=18Þ 5 0:25; π6ð2=18Þ 5 0:25; π9ð2=18Þ 5 0:25; π11 ð2=18Þ 5 0:25; π1ð2=33Þ D0:35610; π2ð2=33Þ D0:00285; π6ð2=33Þ D0:35330; π9ð2=33Þ D0:18230; π11 ð2=33Þ D0:09120; 12 15 20 22 23 πð2=33Þ D0:00285; πð2=33Þ D0:00285; πð2=33Þ D0:00285; πð2=33Þ D0:00285; πð2=33Þ D0:00285;

(5.282) (5.283) (5.284)

π1ð3=2Þ 5 0:5; π2ð3=2Þ 5 0:5;

(5.285)

π1ð3=3Þ 5 0:5; π2ð3=3Þ 5 0:5;

(5.286)

14 18 π1ð3=4Þ 5 0:25; π13 ð3=4Þ 5 0:25; πð3=4Þ 5 0:25; πð3=4Þ 5 0:25;

(5.287)

16 21 25 27 π1ð3=14Þ D0:1818; π12 ð3=14Þ D0:1818; πð3=14Þ D0:1818; πð3=14Þ D0:1818; πð3=14Þ D0:1818; πð3=14Þ D0:0910;

π1ð3=15Þ 5 0:25; π6ð3=15Þ 5 0:25; π9ð3=15Þ 5 0:25; π11 ð3=15Þ 5 0:25; 12 π1ð3=24Þ D0:3787; π2ð3=24Þ D0:0030; π6ð3=24Þ D0:3909; π9ð3=24Þ D0:1698; π11 ð3=24Þ D0:0426; πð3=24Þ D0:0030; 20 22 23 π15 ð3=24Þ D0:0030; πð3=24Þ D0:0030; πð3=24Þ D0:0030; πð3=24Þ D0:0030;

(5.288) (5.289) (5.290)

π1ð4=2Þ 5 0:5; π2ð4=2Þ 5 0:5;

(5.291)

14 18 π1ð4=4Þ 5 0:25; π13 ð4=4Þ 5 0:25; πð4=4Þ 5 0:25; πð4=4Þ 5 0:25;

(5.292)

16 21 28 30 π1ð4=14Þ D0:1818; π12 ð4=14Þ D0:1818; πð4=14Þ D0:1818; πð4=14Þ D0:1818; πð4=14Þ D0:1818; πð4=14Þ D0:0910;

(5.293)

π1ð4=15Þ 5 0:25; π6ð4=15Þ 5 0:25; π9ð4=15Þ 5 0:25; π11 ð4=15Þ 5 0:25;

(5.294)

24 π1ð4=24Þ D0:2305; π6ð4=24Þ D0:3842; π9ð4=24Þ D0:2307; π11 ð4=24Þ D0:0687; πð4=24Þ D0:0080; 26 π25 ð4=24Þ D0:0294; πð4=24Þ D0:0485;

(5.295)

π1ð5=1Þ 5 0:5; π5ð5=1Þ 5 0:5;

(5.296)

π1ð5=15Þ 5 0:25; π6ð5=15Þ 5 0:25; π9ð5=15Þ 5 0:25; π11 ð5=15Þ 5 0:25;

(5.297)

π1ð5=24Þ 5 0:25; π6ð5=24Þ 5 0:25; π9ð5=24Þ 5 0:25; π11 ð5=24Þ 5 0:25:

(5.298)

Therefore according to (3.60) and considering (5.231)(5.264) we get the approximate limit values of transient probabilities that are not equal to 0 at the particular states of the process of environment degradation q1ð1=3Þ 5 0:001386962552011; q6ð1=3Þ 5 0:249653259361998; q22 ð1=3Þ 5 0:166435506241331; 25 28 qð1=3Þ 5 0:166435506241331; qð1=3Þ 5 0:416088765603329;

(5.299)

21 q1ð1=4Þ 5 0:000365497076023; q20 ð1=4Þ 5 0:001827485380117; qð1=4Þ 5 0:186403508771930; 24 27 qð1=4Þ 5 0:252192982456140; qð1=4Þ 5 0:553728070175439; q29 ð1=4Þ 5 0:001827485380117; 30 qð1=4Þ 5 0:003654970760234;

(5.300)

23 q1ð1=5Þ 5 0:002217294900221; q20 ð1=5Þ 5 0:066518847006652; qð1=5Þ 5 0:133037694013304; q26 ð1=5Þ 5 0:798226164079823;

(5.301)

q1ð1=6Þ 5 0:000891687433483; q2ð1=6Þ 5 0:927849388263434; q17 ð1=6Þ 5 0:008907365537886; 19 q18 5 0:008907365537885; q 5 0:053444193227312; ð1=6Þ ð1=6Þ

(5.302)

114

Consequences of Maritime Critical Infrastructure Accidents

q1ð1=7Þ 5 0:002712080724285; q2ð1=7Þ 5 0:020352814679096; q6ð1=7Þ 5 0:244233776149155; q8ð1=7Þ 5 0:162822517432770; q10 ð1=7Þ 5 0:569878811014694;

(5.303)

14 q1ð1=8Þ 5 0:005347593582888; q13 ð1=8Þ 5 0:320855614973262; qð1=8Þ 5 0:032085561497326; 15 qð1=8Þ 5 0:641711229946524;

(5.304)

q1ð1=13Þ 5 0:008264462809917; q6ð1=13Þ 5 0:991735537190083;

(5.305)

q1ð1=17Þ 5 0:001386962552011; q6ð1=17Þ 5 0:249653259361998; q22 ð1=17Þ 5 0:166435506241331; 25 28 qð1=17Þ 5 0:166435506241331; qð1=17Þ 5 0:416088765603329;

(5.306)

q1ð1=21Þ 5 0:090909090909091; q20 ð1=21Þ 5 0:909090909090909;

(5.307)

q1ð1=23Þ 5 0:004149377593361; q2ð1=23Þ 5 0:995850622406639;

(5.308)

q1ð1=27Þ 5 0:000335194781688; q6ð1=27Þ 5 0:999664805218312;

(5.309)

q1ð1=30Þ 5 0:003831417624521; q11 ð1=30Þ 5 0:996168582375479;

(5.310)

23 q1ð1=35Þ 5 0:002217294900221; q20 ð1=35Þ 5 0:066518847006652; qð1=35Þ 5 0:133037694013304; q26 ð1=35Þ 5 0:798226164079823;

(5.311)

q1ð2=1Þ 5 0:001663893510815; q5ð2=1Þ 5 0:998336106489185;

(5.312)

q1ð2=2Þ 5 0:000347101700798; q2ð2=2Þ 5 0:999652898299202;

(5.313)

q1ð2=4Þ 5 0:000693962526024; q2ð2=4Þ 5 0:999306037473976;

(5.314)

14 q1ð2=5Þ 5 0:000693962526024; q13 ð2=5Þ 5 0:541290770298404; qð2=5Þ 5 0:249826509368494; q18 ð2=5Þ 5 0:208188757807078;

(5.315)

16 q1ð2=17Þ 5 0:000149116888447; q12 ð2=17Þ 5 0:393668585500819; qð2=17Þ 5 0:319855725719415; 21 25 27 qð2=17Þ 5 0:228148839324338; qð2=17Þ 5 0:038024806554056; qð2=17Þ 5 0:020152926012925;

(5.316)

q1ð2=18Þ 5 0:000023147612324; q6ð2=18Þ 5 0:233327932223791; q9ð2=18Þ 5 0:266660493970047; q11 ð2=18Þ 5 0:499988426193838;

(5.317)

q1ð2=33Þ 5 0:000024916927851; q2ð2=33Þ 5 0:049105030314844; q6ð2=33Þ 5 0:838912565588299; 12 q9ð2=33Þ 5 0:072831403073114; q11 ð2=33Þ 5 0:032234150016616; qð2=33Þ 5 0:002871639199698; 15 20 qð2=33Þ 5 0:001435819599849; qð2=33Þ 5 0:002010147439789; q22 ð2=33Þ 5 0:000287163919970; 23 qð2=33Þ 5 0:000287163919970;

(5.318)

q1ð3=2Þ 5 0:000347101700798; q2ð3=2Þ 5 0:999652898299202;

(5.319)

q1ð3=3Þ 5 0:000693962526024; q2ð3=3Þ 5 0:999306037473976;

(5.320)

14 q1ð3=4Þ 5 0:000693962526024; q13 ð3=4Þ 5 0:541290770298404; qð3=4Þ 5 0:249826509368494; q18 ð3=4Þ 5 0:208188757807078;

(5.321)

16 q1ð3=14Þ 5 0:000149116888447; q12 ð3=14Þ 5 0:393668585500819; qð3=14Þ 5 0:319855725719415; 21 25 27 qð3=14Þ 5 0:228148839324338; qð3=14Þ 5 0:038024806554056; qð3=14Þ 5 0:020152926012925;

(5.322)

q1ð3=15Þ 5 0:000023147612324; q6ð3=15Þ 5 0:233327932223791; q9ð3=15Þ 5 0:266660493970047; q11 ð3=15Þ 5 0:499988426193838;

(5.323)

Prediction of critical infrastructure accident consequences Chapter | 5

q1ð3=24Þ 5 0:000042373940330; q2ð3=24Þ 5 0:082657716399501; q6ð3=24Þ 5 0:705597268529304; 12 q9ð3=24Þ 5 0:189682617425731; q11 ð3=24Þ 5 0:010418940701696; qð3=24Þ 5 0:004833784584766; 15 20 qð3=24Þ 5 0:002416892292383; qð3=24Þ 5 0:003383649209336; q22 ð3=24Þ 5 0:000483378458477; 23 qð3=24Þ 5 0:000483378458476;

115

(5.324)

q1ð4=2Þ 5 0:000347101700798; q2ð4=2Þ 5 0:999652898299202;

(5.325)

14 q1ð4=4Þ 5 0:000693962526024; q13 ð4=4Þ 5 0:541290770298404; qð4=4Þ 5 0:249826509368494; 18 qð4=4Þ 5 0:208188757807078;

(5.326)

16 q1ð4=14Þ 5 0:000148122944977; q12 ð4=14Þ 5 0:391044574740492; qð4=14Þ 5 0:317723716976649; 21 28 30 qð4=14Þ 5 0:226628105815512; qð4=14Þ 5 0:044436883493238; qð4=14Þ 5 0:020018596029132;

(5.327)

q1ð4=15Þ 5 0:000023147612324; q6ð4=15Þ 5 0:233327932223791; q9ð4=15Þ 5 0:266660493970047; q11 ð4=15Þ 5 0:499988426193838;

(5.328)

q1ð4=24Þ 5 0:014944737158927; q6ð4=24Þ 5 0:684668207853979; q9ð4=24Þ 5 0:138033884146611; q11 ð4=24Þ 5 0:012098552156107; 24 25 qð4=24Þ 5 0:003527207664885; qð4=24Þ 5 0:038892325438421; q26 ð4=24Þ 5 0:107835085581070;

(5.329)

q1ð5=1Þ 5 0:001663893510815; q5ð5=1Þ 5 0:998336106489185;

(5.330)

q1ð5=15Þ 5 0:000023147612324; q6ð5=15Þ 5 0:233327932223791; q9ð5=15Þ 5 0:266660493970047; q11 ð5=15Þ 5 0:499988426193838;

(5.331)

q1ð5=24Þ 5 0:007017907978048; q6ð5=24Þ 5 0:353964565771115; q9ð5=24Þ 5 0:179343218978116; q11 ð5=24Þ 5 0:459674307272721:

(5.332)

i Further, by (3.62) and considering (5.299)(5.332), the approximate mean values of the sojourn total times ζ^ ðk=υÞ of the process of environment degradation Rðk=υÞ ðtÞ during the fixed time ζ 5 1 month 5 43,200 minutes at the particular i states rðk=υÞ expressed in minutes are 1 6 22 25 28 M^ ð1=3Þ 5 59:92; M^ ð1=3Þ 5 10; 785:02; M^ ð1=3Þ 5 7190:01; M^ ð1=3Þ 5 7190:01; M^ ð1=3Þ 5 17; 975:03;

(5.333)

1 20 21 24 M^ ð1=4Þ 5 15:79; M^ ð1=4Þ 5 78:95; M^ ð1=4Þ 5 8052:63; M^ ð1=4Þ 5 10; 894:74; 27 29 30 M^ ð1=4Þ 5 23; 921:10; M^ ð1=4Þ 5 78:95; M^ ð1=4Þ 5 157:89;

(5.334)

1 20 23 26 M^ ð1=5Þ 5 95:79; M^ ð1=5Þ 5 2873:61; M^ ð1=5Þ 5 5747:23; M^ ð1=5Þ 5 34; 483:37;

(5.335)

1 2 17 18 19 M^ ð1=6Þ 5 38:52; M^ ð1=6Þ 5 40; 083:09; M^ ð1=6Þ 5 384:80; M^ ð1=6Þ 5 384:80; M^ ð1=6Þ 5 2308:79;

(5.336)

1 2 6 8 10 M^ ð1=7Þ 5 117:16; M^ ð1=7Þ 5 879:24; M^ ð1=7Þ 5 10; 550:90; M^ ð1=7Þ 5 7033:93; M^ ð1=7Þ 5 24; 618:76;

(5.337)

1 13 14 15 M^ ð1=8Þ 5 231:02; M^ ð1=8Þ 5 13; 860:96; M^ ð1=8Þ 5 1386:10; M^ ð1=8Þ 5 27; 721:93;

(5.338)

1 6 M^ ð1=13Þ 5 357:03; M^ ð1=13Þ 5 42; 842:98;

(5.339)

1 6 22 25 28 M^ ð1=17Þ 5 59:92; M^ ð1=17Þ 5 10; 785:02; M^ ð1=17Þ 5 7190:01; M^ ð1=17Þ 5 7190:01; M^ ð1=17Þ 5 17; 975:03;

(5.340)

1 20 M^ ð1=21Þ 5 3927:27; M^ ð1=21Þ 5 39; 272:73;

(5.341)

1 2 M^ ð1=23Þ 5 179:25; M^ ð1=23Þ 5 43; 020:75;

(5.342)

116

Consequences of Maritime Critical Infrastructure Accidents

1 6 M^ ð1=27Þ 5 14:48; M^ ð1=27Þ 5 43; 185:52;

(5.343)

1 11 M^ ð1=30Þ 5 165:52; M^ ð1=30Þ 5 43; 034:48;

(5.344)

1 20 23 26 M^ ð1=35Þ 5 95:79; M^ ð1=35Þ 5 2873:61; M^ ð1=35Þ 5 5747:23; M^ ð1=35Þ 5 34; 483:37;

(5.345)

1 5 M^ ð2=1Þ 5 71:88; M^ ð2=1Þ 5 43; 128:12;

(5.346)

1 2 M^ ð2=2Þ 5 14:99; M^ ð2=2Þ 5 43; 185:01;

(5.347)

1 2 M^ ð2=4Þ 5 29:98; M^ ð2=4Þ 5 43; 170:02;

(5.348)

1 13 14 18 M^ ð2=5Þ 5 29:98; M^ ð2=5Þ 5 23; 383:76; M^ ð2=5Þ 5 10; 792:51; M^ ð2=5Þ 5 8993:75;

(5.349)

1 12 16 21 M^ ð2=17Þ 5 6:44; M^ ð2=17Þ 5 17; 006:48; M^ ð2=17Þ 5 13; 817:77; M^ ð2=17Þ 5 9856:03; 25 27 M^ ð2=17Þ 5 1642:67; M^ ð2=17Þ 5 870:61;

(5.350)

1 6 9 11 M^ ð2=18Þ 5 1:00; M^ ð2=18Þ 5 10; 079:77; M^ ð2=18Þ 5 11; 519:73; M^ ð2=18Þ 5 21; 599:50;

(5.351)

1 2 6 9 M^ ð2=33Þ 5 1:08; M^ ð2=33Þ 5 2121:34; M^ ð2=33Þ 5 36; 241:02; M^ ð2=33Þ 5 3146:32; 11 12 15 20 22 23 M^ ð2=33Þ 5 1392:52; M^ ð2=33Þ 5 124:05; M^ ð2=33Þ 5 62:03; M^ ð2=33Þ 5 86:84; M^ ð2=33Þ 5 12:41; M^ ð2=33Þ 5 12:41;

(5.352)

1 2 M^ ð3=2Þ 5 14:99; M^ ð3=2Þ 5 43; 185:01;

(5.353)

1 2 M^ ð3=3Þ 5 29:98; M^ ð3=3Þ 5 43; 170:02;

(5.354)

1 13 14 18 M^ ð3=4Þ 5 29:98; M^ ð3=4Þ 5 23; 383:76; M^ ð3=4Þ 5 10; 792:51; M^ ð3=4Þ 5 8993:75;

(5.355)

1 12 16 21 M^ ð3=14Þ 5 6:44; M^ ð3=14Þ 5 17; 006:48; M^ ð3=14Þ 5 13; 817:77; M^ ð3=14Þ 5 9856:03; 25 27 M^ ð3=14Þ 5 1642:67; M^ ð3=14Þ 5 870:61;

(5.356)

1 6 9 11 M^ ð3=15Þ 5 1:00; M^ ð3=15Þ 5 10; 079:77; M^ ð3=15Þ 5 11; 519:73; M^ ð3=15Þ 5 21; 599:50;

(5.357)

1 2 6 9 11 M^ ð3=24Þ 5 1:83; M^ ð3=24Þ 5 3570:81; M^ ð3=24Þ 5 30; 481:80; M^ ð3=24Þ 5 8194:29; M^ ð3=24Þ 5 450:10; 12 15 20 22 23 M^ ð3=24Þ 5 208:82; M^ ð3=24Þ 5 104:41; M^ ð3=24Þ 5 146:17; M^ ð3=24Þ 5 20:88; M^ ð3=24Þ 5 20:88;

(5.358)

1 2 M^ ð4=2Þ 5 14:99; M^ ð4=2Þ 5 43; 185:01;

(5.359)

1 13 14 18 M^ ð4=4Þ 5 29:98; M^ ð4=4Þ 5 23; 383:76; M^ ð4=4Þ 5 10; 792:51; M^ ð4=4Þ 5 8993:75;

(5.360)

1 12 16 21 M^ ð4=14Þ 5 6:40; M^ ð4=14Þ 5 16; 893:13; M^ ð4=14Þ 5 13; 725:66; M^ ð4=14Þ 5 9790:33; 28 30 M^ ð4=14Þ 5 1919:67; M^ ð4=14Þ 5 864:80;

(5.361)

1 6 9 11 M^ ð4=15Þ 5 1:00; M^ ð4=15Þ 5 10; 079:77; M^ ð4=15Þ 5 11; 519:73; M^ ð4=15Þ 5 21; 599:50;

(5.362)

1 6 9 11 M^ ð4=24Þ 5 645:61; M^ ð4=24Þ 5 29; 577:67; M^ ð4=24Þ 5 5963:06; M^ ð4=24Þ 5 522:66; 24 25 26 M^ ð4=24Þ 5 152:38; M^ ð4=24Þ 5 1680:15; M^ ð4=24Þ 5 4658:48;

(5.363)

1 5 M^ ð5=1Þ 5 71:88; M^ ð5=1Þ 5 43; 128:12;

(5.364)

1 6 9 11 M^ ð5=15Þ 5 1:00; M^ ð5=15Þ 5 10; 079:77; M^ ð5=15Þ 5 11; 519:73; M^ ð5=15Þ 5 21; 599:50;

(5.365)

Prediction of critical infrastructure accident consequences Chapter | 5

1 6 9 11 M^ ð5=24Þ 5 303:17; M^ ð5=24Þ 5 15; 291:27; M^ ð5=24Þ 5 7747:63; M^ ð5=24Þ 5 19; 857:93:

117

(5.366)

5.4 Superposition of initiating events, environment threats and environment degradation processes The processes of initiating event, environment threats, and environment degradation are predicted in Sections 5.15.3. Then, we may find the unconditional limit transient probabilities qiðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 and the mean values of sojourn total times for the fixed sufficiently large time of the joint (the superposition) processes of initiating events, environment threats, and environment degradation.

5.4.1 Superposition of initiating events, environment threats and environment degradation processes at Baltic Sea waters Considering (5.4), (5.35)(5.42), and (5.215)(5.222) and applying (3.63), the unconditional approximate limit transient probabilities qiðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 at the particular states of the process of environment degradation are given by (the limit transient probabilities that are not equal to 0 are presented only) for the subarea Dk ; k 5 1     q1ð1Þ 5 p1 Up1ð1=1Þ 1 p2 Up1ð1=2Þ 1 p3 Up1ð1=3Þ 1 p8 Up1ð1=8Þ Uq1ð1=1Þ 1 p8 Up6ð1=8Þ Uq1ð1=6Þ     3 27 1 3 30 1 1 p2 Up27 1 p Up 1 p Up Uq ð1=2Þ ð1=3Þ ð1=27Þ ð1=3Þ Uqð1=30Þ

G

5 ð0:999860710821154U1 1 0:000000073830730U0:003322259136213 1 0:000128885740640U0:004739336492891 1 0:000000005112062U0:004149377593361ÞU1 1 ð0:000000005112062U0:995850622406639ÞU0:004149377593361 1 ð0:000000073830730U0:996677740863787 1 0:000128885740640U0:426540284360190ÞU0:004149377593361 1 ð0:000128885740640U0:568720379146919ÞU0:004149377593361 5 0:999872179003445;   q2ð1Þ 5 p8 Up6ð1=8Þ Uq2ð1=6Þ 5 ð0:000000005112062U0:995850622406639ÞU0:995850622406639 5 0:000000005069726;   3 27 6 q6ð1Þ 5 p2 Up27 ð1=2Þ 1 p Upð1=3Þ Uqð1=27Þ 5 ð0:000000073830730U0:996677740863787 1 0:000128885740640U0:426540284360190ÞU0:995850622406639 5 0:000054820128704;   3 30 11 q11 ð1Þ 5 p Upð1=3Þ Uqð1=30Þ 5 ð0:000128885740640U0:568720379146919ÞU0:995850622406639 5 0:000072995798125; G

(5.367) (5.368)

(5.369)

(5.370)

for the remaining subareas Dk ; k 5 2; . . .; 5 q1ð2Þ 5 0:999871085266778; q6ð2Þ 5 0:000016170471066; q12 ð2Þ 5 0:000032213681563; 21 25 q16 5 0:000042280457051; q 5 0:000032213681563; q ð2Þ ð2Þ ð2Þ 5 0:000003353578877; 27 qð2Þ 5 0:000002682863102;

(5.371)

q1ð3Þ 5 0:999871085266778; q6ð3Þ 5 0:000016170471066; q12 ð3Þ 5 0:000032213681563; 16 21 qð3Þ 5 0:000042280457051; qð3Þ 5 0:000032213681563; q25 ð3Þ 5 0:000003353578877; q27 5 0:000002682863102; ð3Þ

(5.372)

118

Consequences of Maritime Critical Infrastructure Accidents

16 q1ð4Þ 5 0:999871139828532; q12 ð4Þ 5 0:000036818375059; qð4Þ 5 0:000048324117265; 21 28 qð4Þ 5 0:000036818375059; qð4Þ 5 0:000003832946714; q30 ð4Þ 5 0:000003066357371;

(5.373)

q1ð5Þ 5 1:

(5.374)

The results (5.367)(5.374) can play an essential and practically important role in the minimization of critical infrastructure accident consequences in Sections 6.3.1 and 6.3.2 and the losses mitigation. i Further, by (3.64) and considering (5.367)(5.374), the approximate mean values of the sojourn total time ζ^ ðkÞ of the unconditional process of environment degradation RðkÞ ðtÞ during the fixed time θ 5 1 month 5 43,200 minutes at pari ticular states rðkÞ expressed in minutes are 1 2 6 11 M^ ð1Þ 5 43; 194:47813; M^ ð1Þ 5 0:00022; M^ ð1Þ 5 2:36823; M^ ð1Þ 5 3:15342;

(5.375)

1 6 12 16 M^ ð2Þ 5 43; 194:43088; M^ ð2Þ 5 0:69856; M^ ð2Þ 5 1:39163; M^ ð2Þ 5 1:82652; 21 25 27 M^ ð2Þ 5 1:39163; M^ ð2Þ 5 0:14487; M^ ð2Þ 5 0:11590;

(5.376)

1 6 12 16 M^ ð3Þ 5 43; 194:43088; M^ ð3Þ 5 0:69856; M^ ð3Þ 5 1:39163; M^ ð3Þ 5 1:82652; 21 25 27 M^ ð3Þ 5 1:39163; M^ ð3Þ 5 0:14487; M^ ð3Þ 5 0:11590;

(5.377)

1 12 16 M^ ð4Þ 5 43; 194:43324; M^ ð4Þ 5 1:59055; M^ ð4Þ 5 2:08760; 21 28 30 M^ ð4Þ 5 1:59055; M^ ð4Þ 5 0:16558; M^ ð4Þ 5 0:13247;

(5.378)

1 M^ ð5Þ 5 43; 200:

(5.379)

5.4.2 Superposition of initiating events, environment threats and environment degradation processes at world sea waters Proceeding in an analogous way as in Section 5.4.1, we calculate the unconditional approximate limit transient probabilities qiðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 at the particular states of the process of environment degradation, for the particular subareas Dk ; k 5 1; 2; . . .; 5; at the world sea waters. Namely, considering (5.9), (5.121)(5.155), and (5.299)(5.332) and applying (3.63), we obtained the following results (the limit transient probabilities that are not equal to 0 are presented only): q1ð1Þ 5 0:999758661877160; q2ð1Þ 5 0:000000368024153; q6ð1Þ 5 0:000234503153208; q8ð1Þ 5 0:000000001135173; 11 q10 ð1Þ 5 0:000000003973106; qð1Þ 5 0:000000840669180; 14 q13 ð1Þ 5 0:000000001467634; qð1Þ 5 0:000000000146763; 15 qð1Þ 5 0:000000002935267; q17 ð1Þ 5 0:000000000557289; 18 qð1Þ 5 0:000000000557289; q19 ð1Þ 5 0:000000003343732; 21 q20 5 0:000000009922697; q ð1Þ 5 0:000000997227226; ð1Þ 22 qð1Þ 5 0:000000058584258; q23 ð1Þ 5 0:000000000118657; 25 q24 5 0:000001349189777; q ð1Þ ð1Þ 5 0:000000058263165; 27 q26 5 0:000000000711942; q ð1Þ 5 0:000002962351467; ð1Þ 28 qð1Þ 5 0:000000146460645; q29 ð1Þ 5 0:000000009776737; q30 5 0:000000019553475; ð1Þ

(5.380)

Prediction of critical infrastructure accident consequences Chapter | 5

q1ð2Þ 5 0:999749861320829; q5ð2Þ 5 0:000002264763052; q9ð2Þ 5 0:000018475286439; q12 ð2Þ 5 0:000013550968354; q14 ð2Þ 5 0:000000001547990; q16 ð2Þ 5 0:000010543199308; q20 ð2Þ 5 0:000000402306137; q22 ð2Þ 5 0:000000057472305; q25 ð2Þ 5 0:000001253387330;

119

q2ð2Þ 5 0:000009984221534; q6ð2Þ 5 0:000171309578824; q11 ð2Þ 5 0:000013761858129; q13 ð2Þ 5 0:000000003353979; q15 ð2Þ 5 0:000000287361526; q18 ð2Þ 5 0:000000001289992; q21 ð2Þ 5 0:000007520323982; q23 ð2Þ 5 0:000000057472305; q27 ð2Þ 5 0:000000664287985;

(5.381)

q1ð3Þ 5 0:999751944191735; q2ð3Þ 5 0:000016989451252; q6ð3Þ 5 0:000146779291559; q9ð3Þ 5 0:000042195540139; 12 q11 ð3Þ 5 0:000008832641765; qð3Þ 5 0:000013146320356; 14 q13 ð3Þ 5 0:000000003835076; qð3Þ 5 0:000000001770035; 15 qð3Þ 5 0:000000492036938; q16 ð3Þ 5 0:000009881825265; 20 q18 5 0:000000001475029; q ð3Þ ð3Þ 5 0:000000688851714; 21 qð3Þ 5 0:000007048574664; q22 ð3Þ 5 0:000000098407388; 25 q23 5 0:000000098407388; q ð3Þ ð3Þ 5 0:000001174762444; 27 qð3Þ 5 0:000000622617253;

(5.382)

q1ð4Þ 5 0:999756082711471; q6ð4Þ 5 0:000149222927982; q11 ð4Þ 5 0:000008731753325; q13 ð4Þ 5 0:000000004317380; q16 ð4Þ 5 0:000006628826921; q21 ð4Þ 5 0:000004728254167; q25 ð4Þ 5 0:000008313641854; 28 qð4Þ 5 0:000000927108660;

q2ð4Þ 5 0:000000191913748; q9ð4Þ 5 0:000032783820854; q12 ð4Þ 5 0:000008158556210; q14 ð4Þ 5 0:000000001992637; q18 ð4Þ 5 0:000000001660531; q24 ð4Þ 5 0:000000753977576; q26 ð4Þ 5 0:000023050878822; 30 qð4Þ 5 0:000000417657862;

(5.383)

q1ð5Þ 5 0:999751426543813; q5ð5Þ 5 0:000003933082185; q6ð5Þ 5 0:000084474766319; q9ð5Þ 5 0:000046093518504; q11 ð5Þ 5 0:000114072089179:

(5.384)

The results (5.380)(5.384) can play an essential and practically important role in the minimization of critical infrastructure accident consequences in Sections 6.3.3 and 6.3.4 and the losses mitigation. i Further, by (3.64) and considering (5.380)(5.384), the approximate mean values of the sojourn total time ζ^ ðkÞ of the unconditional process of environment degradation RðkÞ ðtÞ during the fixed time θ 5 1 month 5 43,200 minutes at pari ticular states rðkÞ expressed in minutes are 1 2 6 8 10 M^ ð1Þ 5 43; 189:57419; M^ ð1Þ 5 0:01590; M^ ð1Þ 5 10:13054; M^ ð1Þ 5 0:00005; M^ ð1Þ 5 0:00017; 11 13 14 15 17 18 M^ ð1Þ 5 0:03632; M^ ð1Þ 5 0:00006; M^ ð1Þ 5 0:00001; M^ ð1Þ 5 0:00013; M^ ð1Þ 5 0:00002; M^ ð1Þ 5 0:00002;

19 20 21 22 23 24 M^ ð1Þ 5 0:00014; M^ ð1Þ 5 0:00043; M^ ð1Þ 5 0:04308; M^ ð1Þ 5 0:00253; M^ ð1Þ 5 0:00001; M^ ð1Þ 5 0:05828; 25 26 27 28 29 30 M^ ð1Þ 5 0:00252; M^ ð1Þ 5 0:00003; M^ ð1Þ 5 0:12797; M^ ð1Þ 5 0:00633; M^ ð1Þ 5 0:00042; M^ ð1Þ 5 0:00084;

(5.385)

1 2 5 6 9 11 M^ ð2Þ 5 43; 189:19401; M^ ð2Þ 5 0:43132; M^ ð2Þ 5 0:09784; M^ ð2Þ 5 7:40057; M^ ð2Þ 5 0:79813; M^ ð2Þ 5 0:59451; 12 13 14 15 16 18 (5.386) M^ ð2Þ 5 0:58540; M^ ð2Þ 5 0:00014; M^ ð2Þ 5 0:00007; M^ ð2Þ 5 0:01241; M^ ð2Þ 5 0:45547; M^ ð2Þ 5 0:00006; 20 21 22 23 25 27 M^ ð2Þ 5 0:01738; M^ ð2Þ 5 0:32488; M^ ð2Þ 5 0:00248; M^ ð2Þ 5 0:00248; M^ ð2Þ 5 0:05415; M^ ð2Þ 5 0:02870;

120

Consequences of Maritime Critical Infrastructure Accidents

1 2 6 9 11 M^ ð3Þ 5 43; 189:28399; M^ ð3Þ 5 0:73394; M^ ð3Þ 5 6:34087; M^ ð3Þ 5 1:82285; M^ ð3Þ 5 0:38157; 12 13 14 15 16 18 M^ ð3Þ 5 0:56792; M^ ð3Þ 5 0:00017; M^ ð3Þ 5 0:00008; M^ ð3Þ 5 0:02126; M^ ð3Þ 5 0:42689; M^ ð3Þ 5 0:00006;

20 M^ ð3Þ

5 0:02976;

21 M^ ð3Þ

5 0:30450;

22 M^ ð3Þ

5 0:00425;

23 M^ ð3Þ

5 0:00425;

25 M^ ð3Þ

5 0:05075;

27 M^ ð3Þ

(5.387)

5 0:02690;

1 2 6 9 11 12 M^ ð4Þ 5 43; 189:46277; M^ ð4Þ 5 0:00829; M^ ð4Þ 5 6:44643; M^ ð4Þ 5 1:41626; M^ ð4Þ 5 0:37721; M^ ð4Þ 5 0:35245; 13 14 16 18 21 24 (5.388) M^ ð4Þ 5 0:00019; M^ ð4Þ 5 0:00009; M^ ð4Þ 5 0:28637; M^ ð4Þ 5 0:00007; M^ ð4Þ 5 0:20426; M^ ð4Þ 5 0:03257; 25 26 28 30 M^ ð4Þ 5 0:35915; M^ ð4Þ 5 0:99580; M^ ð4Þ 5 0:04005; M^ ð4Þ 5 0:01804;

1 5 6 9 11 M^ ð5Þ 5 43; 189:26163; M^ ð5Þ 5 0:16991; M^ ð5Þ 5 3:64931; M^ ð5Þ 5 1:99124; M^ ð5Þ 5 4:92791:

(5.389)

Chapter 6

Modeling critical infrastructure accident losses Chapter Outline 6.1 Critical infrastructure accident losses 121 6.2 Integrated impact model on critical infrastructure accident consequences related to climateweather change process 122 6.2.1 Critical infrastructure accident area climateweather change process 122 6.2.2 Critical infrastructure accident losses related to climateweather impact 123 6.3 Critical infrastructure accident losses—application for Baltic Sea and world sea waters 124 6.3.1 Losses of shipping critical infrastructure accident at Baltic Sea waters without considering climateweather change process impact 124

6.3.2 Losses of shipping critical infrastructure accident at Baltic Sea waters with considering climateweather change process impact 126 6.3.3 Losses of shipping critical infrastructure accident without considering climateweather change process impact—extension based on wider data collected at world sea waters 132 6.3.4 Losses of shipping critical infrastructure accident with considering the climateweather change process impact—extension based on wider data collected at the world sea waters 134 6.3.5 Losses of shipping critical infrastructure accident at Baltic Sea and world sea waters—analysis of results 137

The general model of critical infrastructure accident consequences is applied to the forecasting environment losses associated with chemical release generated by dynamic ship critical infrastructure network operating at the Baltic Sea waters in the neighborhood of the maritime ferry (roro ship) route between Gdynia and Karlskrona ports on regular everyday line (Fig. 1.2). Thus functions of the environment losses associated with the process of environment degradation without and with considering the climateweather impact are introduced. Next, the cost analysis of these environment losses, without and with considering the climateweather impact, is performed and compared with each other. Moreover, the indicator of resilience to the loss associated with the critical infrastructure accident related to the climateweather change is proposed and determined.

6.1

Critical infrastructure accident losses

We denote by LiðkÞ ðtÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;

(6.1)

the losses associated with the process of the environment degradation RðkÞ ðtÞ; tAh0; 1 NÞ; k 5 1; 2; . . .; n3 ; i in the subarea Dk ; k 5 1; 2; . . .; n3 at the environment degradation state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 in the time interval h0; ti: Thus the approximate expected value of the losses in the time interval h0; ti associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk can be defined by

LðkÞ ðtÞD

‘k X

qiðkÞ ULiðkÞ ðtÞ for k 5 1; 2; . . .; n3

(6.2)

i51

Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00006-1 © 2020 Elsevier Inc. All rights reserved.

121

122

Consequences of Maritime Critical Infrastructure Accidents

where qiðkÞ mean the limit transient probabilities of the unconditional process of the environment degradation at its particular states and are defined by (3.63), and LiðkÞ ðtÞ; tA , 0; 1 NÞ are defined by (6.1). The losses associated with particular environment degradation states are involved with negative consequences in the accident area. The types of consequences are various for different kinds of accident and accidental area. For instance, in the shipping, the closure of port and fishery area and people death can be considered the negative consequences. The losses can be expressed by the cost of the negative consequences in the case, such as the closure of port and fishery area (Etkin, 1999; Goldstein and Ritterling, 2001; Kontovas et al., 2011; Liu and Wirtz, 2006; Psaraftis, 2008; Ventikos et al., 2019). In the case of negative consequences such as people death, the losses can be expressed as the number of life losses. Here we only consider the accident consequences that can be expressed by cost (Bogalecka and Kołowrocki, 2017a, 2018c). Under the assumption, if we fix the number of various kinds of accident consequences by ξ and the cost function of this consequence lasting t by i ½KðkÞ ðtÞðjÞ ;

(6.3)

for j 5 1; 2; . . .; ξ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 than the loss for the subarea Dk ; k 5 1; 2; . . .; n3 is expressed by the total cost of all consequences lasting t in the subarea Dk ; k 5 1; 2; . . .; n3 and is given by LiðkÞ ðtÞD

ξ X

i ½KðkÞ ðtÞðjÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 :

(6.4)

j51

Hence, according to (6.2), losses associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk, k 5 1; 2; . . .; n3 are given by " # ξ ‘k X X i i ðjÞ LðkÞ ðtÞD qðkÞ ½KðkÞ ðtÞ ; k 5 1; 2; . . .; n3 : (6.5) i51

j51

Furthermore, the total expected value of the losses for the fixed time ϕ; ϕ $ 0; associated with the process of the environment degradation R(t) in all subareas of the considered critical infrastructure operating in the environment area D can be evaluated by LðϕÞD

n3 X

LðkÞ ðϕÞ;

(6.6)

k51

where LðkÞ ðϕÞ are given by (6.5) for t 5 ϕ:

6.2 Integrated impact model on critical infrastructure accident consequences related to climateweather change process 6.2.1 Critical infrastructure accident area climateweather change process Critical infrastructure accident area climateweather change process parameters are (Kołowrocki et al., 2017b) G G

the number of climateweather states w; the vector ½qb ð0Þ1xw of the initial probabilities qb ð0Þ 5 PðCð0Þ 5 cb Þ; b 5 1; 2; . . .; w;

G

of the climateweather change process C(t) staying at particular climateweather states cb at the moment t 5 0; the matrix ½qbl wxw of the probabilities of transitions qbl ; b; l 5 1; 2; . . .; w; b 6¼ l;

G

of the climateweather change process C(t) from the climateweather state cb into the climateweather state cl ; and the matrix ½Nbl wxw of the mean values Nbl 5 E½C bl ; b; l 5 1; 2; . . .; w; b 6¼ l;

Modeling critical infrastructure accident losses Chapter | 6

123

of the climateweather change process C(t) conditional sojourn times Cbl at the climateweather state cb when the next climateweather state is cl : The critical infrastructure operating area climateweather change process characteristic is (Kołowrocki et al., 2017b) the vector ½qb 1xw 5 ½q1 ; q2 ; . . .; qw 

(6.7)

of the limit values of transient probabilities qb ðtÞ 5 PðCðtÞ 5 cb Þ; tA , 0; 1 NÞ; b 5 1; 2; . . .; w; of the climateweather change process C(t) at the particular states cb.

6.2.2 Critical infrastructure accident losses related to climateweather impact We denote the losses associated with the process of the environment degradation RðkÞ ðtÞ; tAh0; 1 Ni; k 5 1; 2; . . .; n3 ; i in the subarea Dk ; k 5 1; 2; . . .; n3 ; at the environment degradation state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; in the time interval h0; ti, while the climateweather change process CðtÞ at the critical infrastructure accident area is at the climateweather state cb ; b 5 1; 2; . . .; w (Bogalecka and Kołowrocki, 2017b) by ½LiðkÞ ðtÞðbÞ ; tA , 0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; b 5 1; 2; . . .; w:

(6.8)

The losses ½LiðkÞ ðtÞðbÞ are the conditional losses while the climateweather change process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; w and defined by ½LiðkÞ ðtÞðbÞ 5 ½ρiðkÞ ðbÞ ULiðkÞ ðtÞ; tA , 0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; b 5 1; 2; . . .; w;

(6.9)

½ρiðkÞ ðbÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; b 5 1; 2; . . .; w;

(6.10)

where

are the coefficients of the climateweather change process impact on the losses associated with the process of the envii ronment degradation in the subarea Dk ; k 5 1; 2; . . .; n3 ; at the environment degradation state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; in the time interval h0; ti, while the climateweather change process CðtÞ at the critical infrastructure accident area is at the climateweather state cb ; b 5 1; 2; . . .; w: Thus by (6.5) and (6.9) the conditional approximate expected value of the losses in the time interval h0; ti associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 , while the climateweather change process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; w can be defined by ½LðkÞ ðtÞðbÞ D

‘k X qiðkÞ U½LiðkÞ ðtÞðbÞ ; k 5 1; 2; . . .; n3 ; b 5 1; 2; . . .; w;

(6.11)

i51

where qiðkÞ are given by (3.63) and ½LiðkÞ ðtÞðbÞ ; tA , 0; 1 NÞ are defined by (6.9). Further, applying the formula for total probability, the unconditional approximate expected value of the losses, impacted by the climateweather change process CðtÞ in the time interval h0; ti; associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 can be expressed by LðkÞ ðtÞD

w X

 ðbÞ qb U LðkÞ ðtÞ ; k 5 1; 2; . . .; n3 ;

(6.12)

b51

where qb are given by (6.7), and ½LðkÞ ðtÞðbÞ ; tA , 0; 1 NÞ are determined by (6.11). Hence, according to (6.11), we have LðkÞ ðtÞD

‘k w X X b51 i51

qb UqiðkÞ U½LiðkÞ ðtÞðbÞ ; k 5 1; 2; . . .; n3 :

(6.13)

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Consequences of Maritime Critical Infrastructure Accidents

Finally, the total expected value of the losses, impacted by the climateweather change process C(t) in the fixed time interval h0; ϕi; associated with the process of the environment degradation RðtÞ in all subareas of the considered critical infrastructure operating in the environment area D can be evaluated by LðϕÞD

n3 X

LðkÞ ðϕÞ;

(6.14)

k51

where LðkÞ ðϕÞ are given by (6.12) for t 5 ϕ: Thus considering (6.9), the coefficient of the climateweather change process impact on the losses associated with the process of the environment degradation in the subarea Dk, k 5 1; 2; . . .; n3 in the time interval h0; ϕi may by defined as ρðkÞ 5

LðkÞ ðϕÞ ; ϕA , 0; 1 NÞ; k 5 1; 2; . . .; n3 ; LðkÞ ðϕÞ

(6.15)

where LðkÞ ðϕÞ are the losses related to the climateweather impact determined by (6.12), and LðkÞ ðϕÞ are the losses without considering climateweather impact determined by (6.2). Similarly, the coefficient of the climateweather change process impact on the total losses associated with the process of the environment degradation in the entire considered area D in the time interval h0; ϕi may by defined as ρ5

LðϕÞ ; ϕA , 0; 1 NÞ; LðϕÞ

(6.16)

where LðϕÞ are the total losses related to the climateweather impact determined by (6.14), and LðϕÞ are the total losses without considering climateweather impact determined by (6.6). Other practically interesting characteristics of the environment degradation caused by critical infrastructure accident consequences related to the climateweather are the indicators of the environment in the subareas Dk ; k 5 1; 2; . . .; n3 resilience to the losses associated with the critical infrastructure accident related to the climateweather change that are proposed to be defined by RIðkÞ 5

1 ; k 5 1; 2; . . .; n3 ; ρðkÞ

(6.17)

where ρðkÞ are determined by (6.15). Similarly, the indicator of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change is proposed to be defined by RI 5

1 ; ρ

(6.18)

where ρ is determined by (6.16).

6.3 Critical infrastructure accident losses—application for Baltic Sea and world sea waters 6.3.1 Losses of shipping critical infrastructure accident at Baltic Sea waters without considering climateweather change process impact Based on the procedure and formulae given in Section 6.1, we estimate the losses of shipping critical infrastructure accident at the Baltic Sea open waters, restricted waters as well as Gdynia and Karlskrona ports.

6.3.1.1 Open and restricted sea waters According to (6.4) and the information coming from experts, we determine the losses LiðkÞ ðtÞ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 for the particular subareas Dk ; k 5 1; 2; . . .; 5; during the time t 5 1 hour, expressed by the approximate costs of accident consequences associated with the particular environment i degradation state rðkÞ , k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 that amount (in PLN)

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125

L1ð1Þ ð1Þ 5 0; L2ð1Þ ð1Þ 5 5000; L6ð1Þ ð1Þ 5 5000; L11 ð1Þ ð1Þ 5 7000;

(6.19)

16 L1ð2Þ ð1Þ 5 0; L6ð2Þ ð1Þ 5 10; 000; L12 ð2Þ ð1Þ 5 15; 000; Lð2Þ ð1Þ 5 20; 000; 21 25 27 Lð2Þ ð1Þ 5 25; 000; Lð2Þ ð1Þ 5 27; 000; Lð2Þ ð1Þ 5 30; 000;

(6.20)

16 L1ð3Þ ð1Þ 5 0; L6ð3Þ ð1Þ 5 15; 000; L12 ð3Þ ð1Þ 5 20; 000; Lð3Þ ð1Þ 5 25; 000; 25 27 L21 ð3Þ ð1Þ 5 30; 000; Lð3Þ ð1Þ 5 30; 000; Lð3Þ ð1Þ 5 30; 000;

(6.21)

16 21 L1ð4Þ ð1Þ 5 0; L12 ð4Þ ð1Þ 5 15; 000; Lð4Þ ð1Þ 5 25; 000; Lð4Þ ð1Þ 5 30; 000; 28 30 Lð4Þ ð1Þ 5 30; 000; Lð4Þ ð1Þ 5 30; 000;

(6.22)

L1ð5Þ ð1Þ 5 0:

(6.23)

Hence considering (5.367)(5.374), after applying (6.2), the losses associated with the process of the environment degradation RðkÞ ðtÞ of the particular subareas Dk ; k 5 1; 2; . . .; 5; during the time t 5 1 hour, amount (in PLN) Lð1Þ ð1Þ 5 0:999872179003445U0 1 0:000000005069726U5000 1 0:000054820128704U5000 1 0:000072995798125U7000D0:785;

(6.24)

Lð2Þ ð1Þ 5 0:999871085266778U0 1 0:000016170471066U10; 000 1 0:000032213681563U15; 000 1 0:000042280457051U20; 000 1 0:000032213681563U25; 000 1 0:000003353578877U27; 000 1 0:000002682863102U30; 000D2:467;

(6.25)

Lð3Þ ð1Þ 5 0:999871085266778U0 1 0:000016170471066U15; 000 1 0:000032213681563U20; 000 1 0:000042280457051U25; 000 1 0:000032213681563U30; 000 1 0:000003353578877U30; 000 1 0:000002682863102U30; 000D3:091;

(6.26)

Lð4Þ ð1Þ 5 0:999871139828532U0 1 0:000036818375059U15; 000 1 0:000048324117265U25; 000 1 0:000036818375059U30; 000 1 0:000003832946714U30; 000 1 0:000003066357371U30; 000D3:072;

(6.27)

Lð5Þ ð1Þ 5 1U0 5 0:

(6.28)

Considering the results (6.24)(6.28), after applying (6.6), the total expected value of losses associated with the process of the environment degradation R(t) in all subareas of the Baltic Sea open and restricted waters, during the time t 5 1 hour, amounts (in PLN) Lð1Þ 5

5 X

LðkÞ ð1Þ 5 0:785 1 2:467 1 3:091 1 3:072 1 0 5 9:415:

(6.29)

k51

6.3.1.2 Gdynia and Karlskrona ports We established the losses of shipping critical infrastructure accident at the Gdynia and Karlskrona ports (e.g., ports of the Baltic Sea) in the same way as in Section 6.3.1.1. According to (6.4) and the information coming from experts, we determine the losses LiðkÞ ðtÞ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 for the particular subareas Dk ; k 5 1; 2; . . .; 5; during the time t 5 1 hour, expressed by the approximate costs of accident consequences i associated with particular environment degradation state rðkÞ , k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 that amount (in PLN) L1ð1Þ ð1Þ 5 0; L2ð1Þ ð1Þ 5 10; 260; L6ð1Þ ð1Þ 5 10; 260; L11 ð1Þ ð1Þ 5 12; 260;

(6.30)

16 L1ð2Þ ð1Þ 5 0; L6ð2Þ ð1Þ 5 15; 260; L12 ð2Þ ð1Þ 5 20; 260; Lð2Þ ð1Þ 5 25; 260; 21 25 27 Lð2Þ ð1Þ 5 30; 260; Lð2Þ ð1Þ 5 32; 260; Lð2Þ ð1Þ 5 35; 260;

(6.31)

16 L1ð3Þ ð1Þ 5 0; L6ð3Þ ð1Þ 5 20; 260; L12 ð3Þ ð1Þ 5 25; 260; Lð3Þ ð1Þ 5 30; 260; 21 25 27 Lð3Þ ð1Þ 5 35; 260; Lð3Þ ð1Þ 5 35; 260; Lð3Þ ð1Þ 5 35; 260;

(6.32)

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Consequences of Maritime Critical Infrastructure Accidents

16 21 L1ð4Þ ð1Þ 5 0; L12 ð4Þ ð1Þ 5 20; 260; Lð4Þ ð1Þ 5 30; 260; Lð4Þ ð1Þ 5 35; 260; 28 30 Lð4Þ ð1Þ 5 35; 260; Lð4Þ ð1Þ 5 37; 260;

(6.33)

L1ð5Þ ð1Þ 5 0:

(6.34)

Hence considering (5.367)(5.374), after applying (6.2), the losses associated with the process of the environment degradation RðkÞ ðtÞ of the particular subareas Dk ; k 5 1; 2; . . .; 5; during the time t 5 1 hour, amount (in PLN) Lð1Þ ð1Þ 5 0:999872179003445U0 1 0:000000005069726U10; 260 1 0:000054820128704U10; 260 1 0:000072995798125U12; 260 D1:457;

(6.35)

Lð2Þ ð1Þ 5 0:999871085266778U0 1 0:000016170471066U15; 260 1 0:000032213681563U20; 260 1 0:000042280457051U25; 260 1 0:000032213681563U30; 260 1 0:000003353578877U32; 260 1 0:000002682863102U35; 260D3:145;

(6.36)

Lð3Þ ð1Þ 5 0:999871085266778U0 1 0:000016170471066U20; 260 1 0:000032213681563U25; 260 1 0:000042280457051U30; 260 1 0:000032213681563U35; 260 1 0:000003353578877U35; 260 1 0:000002682863102U35; 260D3:769;

(6.37)

Lð4Þ ð1Þ 5 0:999871139828532U0 1 0:000036818375059U20; 260 1 0:000048324117265U30; 260 1 0:000036818375059U35; 260 1 0:000003832946714U35; 260 1 0:000003066357371U35; 260 D3:750;

(6.38)

Lð5Þ ð1Þ 5 1U0 5 0:

(6.39)

Considering the results (6.35)(6.39), after applying (6.6), the total expected value of losses associated with the process of the environment degradation R(t) in all subareas of Gdynia and Karlskrona ports, during the time t 5 1 hour, amounts (in PLN) Lð1Þ 5

5 X

LðkÞ ð1Þ 5 1:457 1 3:145 1 3:769 1 3:750 1 0 5 12:121:

(6.40)

k51

6.3.2 Losses of shipping critical infrastructure accident at Baltic Sea waters with considering climateweather change process impact We consider that the climateweather change process (Kołowrocki and Soszy´nska-Budny, 2017; Kołowrocki et al., 2017a, 2017b) affects the losses associated with the process of the environment degradation (Bogalecka and Kołowrocki, 2017b). Based on the procedure and formulae given in Section 6.2.2, we estimate the losses of shipping critical infrastructure accident impacted by the climateweather change process at the Baltic Sea open waters, restricted waters as well as Gdynia and Karlskrona ports.

6.3.2.1 Open and restricted sea waters To analyze the climateweather impact on the losses, we suppose that there are w 5 6 climateweather states cb ; b 5 1; 2; . . .; 6 dependent on the wave height and the wind speed, distinguished for the ship operating area at the Baltic Sea open and restricted waters (GMU Safety Interactive Platform, 2018; Kuligowska, 2017): G G G G G G

c1—the wave height belongs to the interval h0, 2) m, and the wind speed belongs to the interval h0, 17) m/s; c2—the wave height belongs to the interval h2, 5) m, and the wind speed belongs to the interval h0, 17) m/s; c3—the wave height belongs to the interval h5, 14) m, and the wind speed belongs to the interval h0, 17) m/s; c4—the wave height belongs to the interval h0, 2) m, and the wind speed belongs to the interval h17, 33) m/s; c5—the wave height belongs to the interval h2, 5) m, and the wind speed belongs to the interval h17, 33) m/s; c6—the wave height belongs to the interval h5, 14) m, and the wind speed belongs to the interval h17, 33) m/s.

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127

According to the information coming from experts, the coefficients (6.10) ½ρiðkÞ ðbÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of the climateweather impact on losses at the climateweather change process states cb ; b 5 1; 2; . . .; 6 at the open and restricted sea waters for particular subareas Dk ; k 5 1; 2; . . .; 5 are as follows: ½ρið1Þ ðbÞ 5 1:0; for b 5 1; 2; 3; i 5 1; 2; . . .; 30; ½ρið1Þ ðbÞ 5 2:0; for b 5 4; 5; 6; i 5 1; 2; . . .; 30; ½ρið2Þ ðbÞ 5 1:0; ½ρið2Þ ðbÞ 5 2:0; ½ρið2Þ ðbÞ 5 2:5; ½ρið2Þ ðbÞ 5 1:8; ½ρið2Þ ðbÞ 5 3:0;

(6.41)

b 5 1; i 5 1; 2; . . .; 28; b 5 2; i 5 1; 2; . . .; 28; b 5 3; 5; i 5 1; 2; . . .; 28; b 5 4; i 5 1; 2; . . .; 28; b 5 6; i 5 1; 2; . . .; 28;

(6.42)

½ρið3Þ ðbÞ 5 1:0; for b 5 1; 4; i 5 1; 2; . . .; 28; ½ρið3Þ ðbÞ 5 2:0; for b 5 2; 5; i 5 1; 2; . . .; 28; ½ρið3Þ ðbÞ 5 3:0; for b 5 3; 6; i 5 1; 2; . . .; 28;

(6.43)

½ρið4Þ ðbÞ 5 1:0; for b 5 1; 2; . . .; 6; i 5 1; 2; . . .; 31;

(6.44)

½ρið5Þ ðbÞ 5 1:0; for b 5 1; 2; . . .; 6; i 5 1; 2; . . .; 23:

(6.45)

for for for for for

Further, considering (6.19)(6.23) and (6.41)(6.45) after applying (6.9), we calculate the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ of the particular subareas Dk ; k 5 1; 2; . . .; 5 of the open and restricted sea waters, at the environment degi radation state rðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; while the process CðtÞ is at the state cb ; b 5 1; 2; . . .; 6: For instance, we get the following results (in PLN) for the subarea Dk, k 5 1 (air): ½Lið1Þ ð1ÞðbÞ D1:0U0 5 0; for b 5 1; 2; . . .; 6; i 5 1; ½Lið1Þ ð1ÞðbÞ D1:0U5000 5 5000; for b 5 1; 2; 3; i 5 2; 6; ½Lið1Þ ð1ÞðbÞ D1:0U7000 5 7000; for b 5 1; 2; 3; i 5 11; ½Lið1Þ ð1ÞðbÞ D2:0U5000 5 10; 000; for b 5 4; 5; 6; i 5 2; 6; ½Lið1Þ ð1ÞðbÞ D2:0U7000 5 14; 000; for b 5 4; 5; 6; i 5 11:

(6.46)

In a similar way, we obtain the losses associated with the process of the environment degradation RðkÞ ðtÞ in the subi area Dk ; k 5 2; 3; 4; 5 at the environment degradation state rðkÞ ; k 5 2; 3; 4; 5 i 5 1; 2; . . .; ‘k ; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; impacted by the climateweather change process CðtÞ then we omit it. Next, according to (6.11), the conditional approximate expected values of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk, k 5 1 of the open and restricted sea waters, while the process CðtÞ is at the climateweather state cb, b 5 1; 2; . . .; 6 are (in PLN) ½Lð1Þ ð1ÞðbÞ D0:999872179003445U0 1 0:000000005069726U5000 1 0:000054820128704U5000 1 0:000072995798125U7000 D0:785; for b 5 1; 2; 3;

(6.47)

½Lð1Þ ð1ÞðbÞ D0:999872179003445U0 1 0:000000005069726U10; 000 1 0:000054820128704U10; 000 1 0:000072995798125U14; 000 D1:570; for b 5 4; 5; 6:

(6.48)

The value of losses associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk, k 5 1, without considering the climateweather impact, calculated in (6.24) is Lð1Þ ð1ÞD 0.785 PLN. The value of losses impacted by the climateweather change process increases up to ½Lð1Þ ð1ÞðbÞ D 1.570 PLN only when the climateweather change process is at the state cb, b 5 4; 5; 6: The reason for this is a rough wind speed at these states in the opposite to wind speed when the climateweather change process is at the state cb, b 5 1; 2; 3; causing the stagnation in the air then the

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Consequences of Maritime Critical Infrastructure Accidents

air degradation effects do not spread off. The wave height is not significant for the critical infrastructure accident losses in the subarea Dk, k 5 1 (air). In analogous way, we obtain the conditional approximate expected value of the losses (in PLN) impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, for the remaining subareas Dk ; k 5 2; 3; 4; 5 of the open and restricted sea waters, associated with the process of the environment degradation RðkÞ ðtÞ, while the process C(t) is at the climateweather state cb, b 5 1; 2; . . .; 6: ½Lð2Þ ð1ÞðbÞ D2:467 for b 5 1; ½Lð2Þ ð1ÞðbÞ D4:934 for b 5 2; ½Lð2Þ ð1ÞðbÞ D6:167 for b 5 3; 5; ½Lð2Þ ð1ÞðbÞ D4:440 for b 5 4; ½Lð2Þ ð1ÞðbÞ D7:401 for b 5 6;

(6.49)

½Lð3Þ ð1ÞðbÞ D3:091 for b 5 1; 4; ½Lð3Þ ð1ÞðbÞ D6:183 for b 5 2; 5; ½Lð3Þ ð1ÞðbÞ D9:274 for b 5 3; 6;

(6.50)

½Lð4Þ ð1ÞðbÞ D3:072 for b 5 1; 2; . . .; 6;

(6.51)

½Lð5Þ ð1ÞðbÞ 5 0 for b 5 1; 2; . . .; 6:

(6.52)

Further, taking into account the following approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the open sea waters (GMU Safety Interactive Platform, 2018): q1 5 0:834; q2 5 0:149; q3 5 0; q4 5 0; q5 5 0:015; q6 5 0:002;

(6.53)

and considering (6.47)(6.52), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk, k 5 1; 2; . . .; 5; while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the open sea waters are (in PLN) Lð1Þ ð1ÞD0:834U0:785 1 0:149U0:785 1 0U0:785 1 0U1:570 10:015U1:570 1 0:002U1:570D0:798; Lð2Þ ð1ÞD0:834U2:467 1 0:149U4:934 1 0U6:167 1 0U4:440 10:015U6:167 1 0:002U7:401D2:900; Lð3Þ ð1ÞD0:834U3:091 1 0:149U6:183 1 0U9:274 1 0U4:440 10:015U6:183 1 0:002U9:274D3:611; Lð4Þ ð1ÞD0:834U3:072 1 0:149U3:072 1 0U3:072 1 0U3:072 10:015U3:072 1 0:002U3:072D3:072; Lð5Þ ð1ÞD0:834U0 1 0:149U0 1 0U0 1 0U0 1 0:015U0 1 0:002U0 5 0:

(6.54)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðtÞ in the entire area D of the open sea waters amounts (in PLN) 5 X Lð1Þ 5 LðkÞ ð1Þ 5 0:798 1 2:900 1 3:611 1 3:072 1 0 5 10:381: (6.55) k51

Thus considering (6.24)(6.28) and (6.54) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the open sea waters, related to the climateweather change impact, are 0:785 5 0:984 5 98:4%; RIð1Þ 5 0:798 2:467 5 0:851 5 85:1%; RIð2Þ 5 2:900 3:091 (6.56) RIð3Þ 5 5 0:856 5 85:6%; 3:611 3:072 5 1 5 100%; RIð4Þ 5 3:072 RIð5Þ 2 n=a as Lð5Þ ð1Þ 5 0 and Lð5Þ ð1Þ 5 0;

Modeling critical infrastructure accident losses Chapter | 6

129

and considering (6.29) and (6.55) and according to (6.16) and (6.18), the indicator RI of the environment in the entire area D resilience to the losses associated with the critical infrastructure accident at the open sea waters, related to the climateweather change impact, is RI 5

9:415 5 0:907 5 90:7%: 10:381

(6.57)

Similarly, taking into account the following approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the restricted sea waters (GMU Safety Interactive Platform, 2018): q1 5 0:827; q2 5 0:155; q3 5 0:004; q4 5 0; q5 5 0:007; q6 5 0:007;

(6.58)

and considering (6.47)(6.52), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5, while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the restricted sea waters are (in PLN) Lð1Þ ð1ÞD0:827U0:785 1 0:155U0:785 1 0:004U785 1 0U1:570 10:007U1:570 1 0:007U1:570D0:796; Lð2Þ ð1ÞD0:827U2:467 1 0:155U4:934 1 0:004U6:167 1 0U4:440 10:007U6:167 1 0:007U7:401D2:925; Lð3Þ ð1ÞD0:827U3:091 1 0:155U6:183 1 0:004U9:274 1 0U3:091 10:007U6:183 1 0:007U9:274D3:660; Lð4Þ ð1ÞD0:827U3:072 1 0:155U3:072 1 0:004U3:072 1 0U3:072 10:007U3:072 1 0:007U3:072D3:072; Lð5Þ ð1ÞD0:827U0 1 0:155U0 1 0U004U0 1 0U0 10:007U0 1 0:007U0D0:

(6.59)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðtÞ in all subareas D of the restricted sea waters, amounts (in PLN) Lð1Þ 5

5 X

LðkÞ ð1Þ 5 0:796 1 2:925 1 3:660 1 3:072 1 0 5 10:453:

(6.60)

k51

Thus considering (6.24)(6.28) and (6.59) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the restricted sea waters, related to the climateweather change impact, are 0:785 5 0:986 5 98:6%; 0:796 2:467 5 0:843 5 84:3%; RIð2Þ 5 2:925 3:091 RIð3Þ 5 5 0:845 5 84:5%; 3:660 3:072 RIð4Þ 5 5 1 5 100%; 3:072

RIð1Þ 5

(6.61)

RIð5Þ 2 n=a as Lð5Þ ð1Þ 5 0 and Lð5Þ ð1Þ 5 0; and considering (6.29) and (6.60) and according to (6.16) and (6.18), the indicator RI of the environment in the entire area D resilience to the losses associated with the critical infrastructure accident at the restricted sea waters, related to the climateweather change impact, is RI 5

9:415 5 0:901 5 90:1%: 10:453

(6.62)

130

Consequences of Maritime Critical Infrastructure Accidents

6.3.2.2 Gdynia and Karlskrona ports To analyze the climateweather impact on the losses, we suppose that there are w 5 6 climateweather states cb ; b 5 1; 2; . . .; 6 dependent on the wind speed and the wind direction, distinguished for the ship operating area at the Baltic Sea port waters (GMU Safety Interactive Platform, 2018; Kuligowska, 2017): G

G

G

G

G

G

c1—the wind speed belongs to the interval h0, 17) m/s, and the wind direction belongs to the interval h0, 22.5) degree or h67.5, 112.5) degree or h337.5, 360) degree in the case of the Gdynia Port, and h0, 67.5) degree or h292.5, 360) degree in the case of the Karlskrona Port. c2—the wind speed belongs to the interval h17, 33) m/s, and the wind direction belongs to the interval h0, 22.5) degree or h67.5, 112.5) degree or h337.5, 360) degree in the case of the Gdynia Port, and h0, 67.5) degree or h292.5, 360) degree in the case of the Karlskrona Port. c3—the wind speed belongs to the interval h0, 17) m/s, and the wind direction belongs to the interval h22.5, 67.5) degree or h112.5, 247.5) degree in the case of the Gdynia Port and h67.5, 157.5) degree in the case of the Karlskrona Port. c4—the wind speed belongs to the interval h17, 33) m/s, and the wind direction belongs to the interval h22.5, 67.5) degree or h112.5, 247.5) degree in the case of the Gdynia Port and h67.5, 157.5) degree in the case of the Karlskrona Port. c5—the wind speed belongs to the interval h0, 17) m/s, and the wind direction belongs to the interval h247.5, 337.5) degree in the case of the Gdynia Port and h157.5, 292.5) degree in the case of the Karlskrona Port. c6—the wind speed belongs to the interval h17, 33) m/s, and the wind direction belongs to the interval h247.5, 337.5) degree in the case of the Gdynia Port and h157.5, 292.5) degree in the case of the Karlskrona Port.

According to the information coming from experts, the coefficients (6.10) ½ρiðkÞ ðbÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of the climateweather impact on losses at the climateweather change process states cb ; b 5 1; 2; . . .; 6 at the Gdynia and Karlskrona ports are as follows: ½ρið1Þ ðbÞ 5 1:0 for b 5 1; 3; 5; i 5 1; 2; . . .; 30; ½ρið1Þ ðbÞ 5 2:0 for b 5 2; 4; 6; i 5 1; 2; . . .; 30;

(6.63)

½ρið2Þ ðbÞ 5 1:0 for b 5 1; 3; 5; i 5 1; 2; . . .; 28; ½ρið2Þ ðbÞ 5 2:0 for b 5 2; 4; 6; i 5 1; 2; . . .; 28;

(6.64)

½ρið3Þ ðbÞ 5 1:0 for b 5 1; 3; 5; i 5 1; 2; . . .; 28; ½ρið3Þ ðbÞ 5 1:5 for b 5 2; 4; 6; i 5 1; 2; . . .; 28;

(6.65)

½ρið4Þ ðbÞ 5 1:0 for b 5 1; 2; . . .; 6; i 5 1; 2; . . .; 31;

(6.66)

½ρið5Þ ðbÞ 5 1:0 for b 5 1; 2; . . .; 6; i 5 1; 2; . . .; 23:

(6.67)

Next, considering (6.19)(6.23) and (6.63)(6.67) after applying (6.9), and further (6.11), we calculate the conditional approximate expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5 of the Gdynia and Karlskrona ports, while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6: The results are as follows (in PLN): ½Lð1Þ ð1ÞðbÞ D1:457 for b 5 1; 3; 5; ½Lð1Þ ð1ÞðbÞ D2:915 for b 5 2; 4; 6;

(6.68)

½Lð2Þ ð1ÞðbÞ D3:145 for b 5 1; 3; 5; ½Lð2Þ ð1ÞðbÞ D6:290 for b 5 2; 4; 6;

(6.69)

½Lð3Þ ð1ÞðbÞ D3:769 for b 5 1; 3; 5; ½Lð3Þ ð1ÞðbÞ D5:654 for b 5 2; 4; 6;

(6.70)

½Lð4Þ ð1ÞðbÞ D3:750 for b 5 1; 2; . . .; 6;

(6.71)

½Lð5Þ ð1Þ

ðbÞ

5 0 for b 5 1; 2; . . .; 6:

(6.72)

Further, taking into account the following approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the Gdynia Port (GMU Safety Interactive Platform, 2018):

Modeling critical infrastructure accident losses Chapter | 6

q1 5 0:394; q2 5 0:010; q3 5 0:473; q4 5 0:006; q5 5 0:017; q6 5 0;

131

(6.73)

and considering (6.68)(6.72), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5; while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the Gdynia Port are (in PLN) Lð1Þ ð1ÞD1:481; Lð2Þ ð1ÞD3:195; Lð3Þ ð1ÞD3:800; Lð4Þ ð1ÞD3:750; Lð5Þ ð1Þ 5 0:

(6.74)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðtÞ in the entire area D of the Gdynia Port amounts (in PLN) Lð1Þ 5 12:226:

(6.75)

Thus considering (6.35)(6.39) and (6.74) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the Gdynia Port, related to the climateweather change impact, are 1:457 5 0:984 5 98:4%; 1:481 3:145 5 0:984 5 98:4%; RIð2Þ 5 3:195 3:769 5 0:992 5 99:2%; RIð3Þ 5 3:800 3:750 5 1 5 100%; RIð4Þ 5 3:750

RIð1Þ 5

(6.76)

RIð5Þ 2 n=a as Lð5Þ ð1Þ 5 0 and Lð5Þ ð1Þ 5 0; and considering (6.40) and (6.75) and according to (6.16) and (6.18), the indicator RI of the environment in the entire area D resilience to the losses associated with the critical infrastructure accident at the Gdynia Port related to the climateweather change impact is RI 5

12:121 5 0:991 5 99:1%: 12:226

(6.77)

Similarly, taking into account the following approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the Karlskrona Port (GMU Safety Interactive Platform, 2018): q1 5 0:364; q2 5 0:005; q3 5 0:417; q4 5 0:016; q5 5 0:197; q6 5 0:001;

(6.78)

and considering (6.68)(6.72), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5, while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the Karlskrona Port are (in PLN) Lð1Þ ð1ÞD1:489; Lð2Þ ð1ÞD3:214; Lð3Þ ð1ÞD3:811; Lð4Þ ð1ÞD3:750; Lð5Þ ð1Þ 5 0:

(6.79)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðtÞ in the entire area D of the Karlskrona Port amounts (in PLN) Lð1Þ 5 12:264:

(6.80)

Thus considering (6.35)(6.39) and (6.79) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the Karlskrona Port, related to the climateweather change impact, are

132

Consequences of Maritime Critical Infrastructure Accidents

1:457 5 0:979 5 97:9%; 1:489 3:145 5 0:979 5 97:9%; RIð2Þ 5 3:214 3:769 5 0:989 5 98:9%; RIð3Þ 5 3:811 3:750 5 1 5 100%; RIð4Þ 5 3:750

RIð1Þ 5

(6.81)

RIð5Þ 2 n=a as Lð5Þ ð1Þ 5 0 and Lð5Þ ð1Þ 5 0; and considering (6.40) and (6.80) and according to (6.16) and (6.18), the indicator RI of the environment in the entire area D resilience to the losses associated with the critical infrastructure accident at the Karlskrona Port, related to the climateweather change impact, is RI 5

12:121 5 0:988 5 98:8%: 12:264

(6.82)

6.3.3 Losses of shipping critical infrastructure accident without considering climateweather change process impact—extension based on wider data collected at world sea waters Based on the procedure and formulae given in Section 6.1, proceeding in the same way as in Section 6.3.1, we estimate the losses of shipping critical infrastructure accident at the Baltic Sea open waters, restricted waters as well as Gdynia and Karlskrona ports using the wider data collected at the world sea waters.

6.3.3.1 Open and restricted sea waters According to (6.4) and the information coming from experts, we determine the losses LiðkÞ ðtÞ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 for the particular subareas Dk ; k 5 1; 2; . . .; 5 of the open and restricted sea waters during the time t 5 1 hour, expressed by the approximate costs of accident consequences associated i with particular environment degradation state rðkÞ , k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 that amount (in PLN) L1ð1Þ ð1Þ 5 0; L2ð1Þ ð1Þ 5 5000; L6ð1Þ ð1Þ 5 5000; L8ð1Þ ð1Þ 5 6500; L10 ð1Þ ð1Þ 5 7500; 11 14 15 ð1Þ 5 10; 000; L ð1Þ 5 12; 000; L ð1Þ 5 12; 500; L17 Lð1Þ ð1Þ 5 7000; L13 ð1Þ ð1Þ ð1Þ 5 2000 ð1Þ ð1Þ 19 20 21 22 L18 ð1Þ 5 2500; L ð1Þ 5 3000; L ð1Þ 5 2000; L ð1Þ 5 2000; L ð1Þ 5 8000; ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ 23 24 25 26 27 Lð1Þ ð1Þ 5 3000; Lð1Þ ð1Þ 5 4000; Lð1Þ ð1Þ 5 10; 000; Lð1Þ ð1Þ 5 3500; Lð1Þ ð1Þ 5 7000; 29 30 L28 ð1Þ ð1Þ 5 11; 000; Lð1Þ ð1Þ 5 2500; Lð1Þ ð1Þ 5 4000;

(6.83)

L1ð2Þ ð1Þ 5 0; L2ð2Þ ð1Þ 5 7000; L5ð2Þ ð1Þ 5 5000; L6ð2Þ ð1Þ 5 10; 000; L9ð2Þ ð1Þ 5 8000; 12 13 14 L11 ð2Þ ð1Þ 5 12; 000; Lð2Þ ð1Þ 5 15; 000; Lð2Þ ð1Þ 5 15; 000; Lð2Þ ð1Þ 5 17; 000; 15 16 18 Lð2Þ ð1Þ 5 18; 000; Lð2Þ ð1Þ 5 20; 000; Lð2Þ ð1Þ 5 22; 000; L20 ð2Þ ð1Þ 5 24; 000; 21 22 23 25 Lð2Þ ð1Þ 5 25; 000; Lð2Þ ð1Þ 5 25; 000; Lð2Þ ð1Þ 5 26; 000; Lð2Þ ð1Þ 5 27; 000; L27 ð2Þ ð1Þ 5 30; 000;

(6.84)

L1ð3Þ ð1Þ 5 0; L2ð3Þ ð1Þ 5 13; 000; L6ð3Þ ð1Þ 5 15; 000; L9ð3Þ ð1Þ 5 18; 000; L11 ð3Þ ð1Þ 5 20; 000; 13 14 15 ð1Þ 5 20; 000; L ð1Þ 5 20; 000; L ð1Þ 5 23; 000; L ð1Þ 5 22; 000; L12 ð3Þ ð3Þ ð3Þ ð3Þ 16 18 20 21 Lð3Þ ð1Þ 5 25; 000; Lð3Þ ð1Þ 5 30; 000; Lð3Þ ð1Þ 5 28; 000; Lð3Þ ð1Þ 5 30; 000; 23 25 27 L22 ð3Þ ð1Þ 5 28; 000; Lð3Þ ð1Þ 5 30; 000; Lð3Þ ð1Þ 5 30; 000; Lð3Þ ð1Þ 5 30; 000;

(6.85)

L1ð4Þ ð1Þ 5 0; L2ð4Þ ð1Þ 5 12; 000; L6ð4Þ ð1Þ 5 13; 000; L9ð4Þ ð1Þ 5 15; 000; L11 ð4Þ ð1Þ 5 18; 000; 13 14 16 L12 ð1Þ 5 15; 000; L ð1Þ 5 15; 000; L ð1Þ 5 17; 000; L ð1Þ 5 25; 000; ð4Þ ð4Þ ð4Þ ð4Þ 18 21 24 25 Lð4Þ ð1Þ 5 28; 000; Lð4Þ ð1Þ 5 30; 000; Lð4Þ ð1Þ 5 25; 000; Lð4Þ ð1Þ 5 28; 000; 28 30 L26 ð4Þ ð1Þ 5 30; 000; Lð4Þ ð1Þ 5 30; 000; Lð4Þ ð1Þ 5 30; 000;

(6.86)

Modeling critical infrastructure accident losses Chapter | 6

L1ð5Þ ð1Þ 5 0 ðfor open sea watersÞ

133

(6.87)

and L1ð5Þ ð1Þ 5 0; L5ð5Þ ð1Þ 5 1000; L6ð5Þ ð1Þ 5 5000; L9ð5Þ ð1Þ 5 10; 000; L11 ð5Þ ð1Þ 5 15; 000 ðfor restricted sea watersÞ:

(6.88)

Hence considering (5.380)(5.384), after applying (6.2), the losses associated with the process of the environment degradation RðkÞ ðtÞ of the particular subareas Dk ; k 5 1; 2; . . .; 5; during the time t 5 1 hour, amount (in PLN) Lð1Þ ð1ÞD1:211; Lð2Þ ð1ÞD2:781; Lð3Þ ð1ÞD4:170; Lð4Þ ð1ÞD4:005; Lð5Þ ð1Þ 5 0 ðfor open watersÞ; and Lð5Þ ð1ÞD2:598 ðfor restricted sea watersÞ:

(6.89)

Considering the previous results, after applying (6.6), the total expected value of losses associated with the process of the environment degradation RðtÞ in all subareas of the considered critical infrastructure operating environment area D, during the time t 5 1 hour, amounts (in PLN) Lð1Þ 5 12:167 ðfor open sea watersÞ

(6.90)

Lð1Þ 5 14:765 ðfor restricted sea watersÞ:

(6.91)

and

6.3.3.2 Gdynia and Karlskrona ports According to (6.4) and the information coming from experts, we determine the losses LiðkÞ ðtÞ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 for the particular subareas Dk ; k 5 1; 2; . . .; 5 of Gdynia and Karlskrona ports, during the time t 5 1 hour, expressed by the approximate costs of accident consequences associated i with particular environment degradation state rðkÞ , k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 that amount (in PLN) L1ð1Þ ð1Þ 5 0; L2ð1Þ ð1Þ 5 10; 260; L6ð1Þ ð1Þ 5 10; 260; L8ð1Þ ð1Þ 5 11; 760; 11 13 14 L10 ð1Þ ð1Þ 5 12; 760; Lð1Þ ð1Þ 5 12; 260; Lð1Þ ð1Þ 5 15; 260; Lð1Þ ð1Þ 5 17; 260; 15 17 18 19 Lð1Þ ð1Þ 5 17; 760; Lð1Þ ð1Þ 5 7260; Lð1Þ ð1Þ 5 7760; Lð1Þ ð1Þ 5 8260; L20 ð1Þ ð1Þ 5 7260; 21 22 23 24 25 Lð1Þ ð1Þ 5 7260; Lð1Þ ð1Þ 5 13; 260; Lð1Þ ð1Þ 5 8260; Lð1Þ ð1Þ 5 9260; Lð1Þ ð1Þ 5 15; 260; 27 28 29 30 L26 ð1Þ ð1Þ 5 8760; Lð1Þ ð1Þ 5 12; 260; Lð1Þ ð1Þ 5 16; 260; Lð1Þ ð1Þ 5 7760; Lð1Þ ð1Þ 5 9260;

(6.92)

L1ð2Þ ð1Þ 5 0; L2ð2Þ ð1Þ 5 12; 260; L5ð2Þ ð1Þ 5 10; 260; L6ð2Þ ð1Þ 5 15; 260; L9ð2Þ ð1Þ 5 13; 260; 12 13 14 L11 ð2Þ ð1Þ 5 17; 260; Lð2Þ ð1Þ 5 20; 260; Lð2Þ ð1Þ 5 20; 260; Lð2Þ ð1Þ 5 22; 260; 15 16 18 Lð2Þ ð1Þ 5 23; 260; Lð2Þ ð1Þ 5 25; 260; Lð2Þ ð1Þ 5 27; 260; L20 ð2Þ ð1Þ 5 29; 260; 22 23 25 ð1Þ 5 30; 260; L ð1Þ 5 30; 260; L ð1Þ 5 31; 260; L ð1Þ 5 32; 260; L27 L21 ð2Þ ð2Þ ð2Þ ð1Þ 5 35; 260; ð2Þ ð2Þ

(6.93)

L1ð3Þ ð1Þ 5 0; L2ð3Þ ð1Þ 5 18; 260; L6ð3Þ ð1Þ 5 20; 260; L9ð3Þ ð1Þ 5 23; 260; L11 ð3Þ ð1Þ 5 25; 260; 12 13 14 15 Lð3Þ ð1Þ 5 25; 260; Lð3Þ ð1Þ 5 25; 260; Lð3Þ ð1Þ 5 28; 260; Lð3Þ ð1Þ 5 27; 260; 18 20 21 L16 ð3Þ ð1Þ 5 30; 260; Lð3Þ ð1Þ 5 35; 260; Lð3Þ ð1Þ 5 33; 260; Lð3Þ ð1Þ 5 35; 260; 23 25 27 L22 ð3Þ ð1Þ 5 33; 260; Lð3Þ ð1Þ 5 35; 260; Lð3Þ ð1Þ 5 35; 260; Lð3Þ ð1Þ 5 35; 260;

(6.94)

L1ð4Þ ð1Þ 5 0; L2ð4Þ ð1Þ 5 17; 260; L6ð4Þ ð1Þ 5 18; 260; L9ð4Þ ð1Þ 5 20; 260; L11 ð4Þ ð1Þ 5 23; 260; 13 14 16 L12 ð1Þ 5 20; 260; L ð1Þ 5 20; 260; L ð1Þ 5 22; 260; L ð1Þ 5 30; 260; ð4Þ ð4Þ ð4Þ ð4Þ 18 21 24 25 Lð4Þ ð1Þ 5 33; 260; Lð4Þ ð1Þ 5 35; 260; Lð4Þ ð1Þ 5 30; 260; Lð4Þ ð1Þ 5 33; 260; 28 30 L26 ð4Þ ð1Þ 5 35; 260; Lð4Þ ð1Þ 5 35; 260; Lð4Þ ð1Þ 5 35; 260;

(6.95)

L1ð5Þ ð1Þ 5 0; L5ð5Þ ð1Þ 5 6260; L6ð5Þ ð1Þ 5 10; 260; L9ð5Þ ð1Þ 5 15; 260; L11 ð5Þ ð1Þ 5 20; 260:

(6.96)

Hence considering (5.380)(5.384), after applying (6.2), the losses associated with the process of the environment degradation RðkÞ ðtÞ of the particular subarea Dk ; k 5 1; 2; . . .; 5; during the time t 5 1 hour, amount (in PLN)

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Lð1Þ ð1ÞD2:481; Lð2Þ ð1ÞD4:097; Lð3Þ ð1ÞD5:475; Lð4Þ ð1ÞD5:288; Lð5Þ ð1Þ 5 3:906:

(6.97)

Considering the previous results, after applying (6.6), the total expected value of losses associated with the process of the environment degradation RðtÞ in all subareas of the considered critical infrastructure operating environment area D, during the time t 5 1 hour, amounts (in PLN) Lð1Þ 5 21:247:

(6.98)

6.3.4 Losses of shipping critical infrastructure accident with considering the climateweather change process impact—extension based on wider data collected at the world sea waters Based on the procedure and formulae given in Section 6.2.2, proceeding in the same way as in Section 6.3.2, we estimate the losses of shipping critical infrastructure accident impacted by the climateweather change process at the Baltic Sea open waters, restricted waters as well as Gdynia and Karlskrona ports using the wider data collected at the world sea waters.

6.3.4.1 Open and restricted sea waters To analyze the climateweather impact on the losses, we suppose that there are w 5 6 climateweather states cb ; b 5 1; 2; . . .; 6 dependent on the wave height and the wind speed, distinguished for the ship operating area at the Baltic Sea open and restricted waters (GMU Safety Interactive Platform, 2018; Kuligowska, 2017) also defined in Section 6.3.2.1. According to the information coming from experts, the coefficients (6.10) ½ρiðkÞ ðbÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of the climateweather impact on losses at the climateweather change process states cb ; b 5 1; 2; . . .; 6 at the open and restricted sea waters are given by (6.41)(6.45). Considering (6.41)(6.45) and (6.83)(6.88) after applying (6.9), and further (6.11), we calculate the conditional approximate expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5 of the open and restricted sea waters, while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6: The results are as follows (in PLN): ½Lð1Þ ð1ÞðbÞ D1:211 for b 5 1; 2; 3; ½Lð1Þ ð1ÞðbÞ D2:422 for b 5 4; 5; 6;

(6.99)

½Lð2Þ ð1ÞðbÞ D2:781 for b 5 1; ½Lð2Þ ð1ÞðbÞ D5:562 for b 5 2; ½Lð2Þ ð1ÞðbÞ D6:953 for b 5 3; 5; ½Lð2Þ ð1ÞðbÞ D5:006 for b 5 4; ½Lð2Þ ð1ÞðbÞ D8:343 for b 5 6;

(6.100)

½Lð3Þ ð1ÞðbÞ D4:170 for b 5 1; 4; ½Lð3Þ ð1ÞðbÞ D8:340 for b 5 2; 5; ½Lð3Þ ð1ÞðbÞ D12:510 for b 5 3; 6;

(6.101)

½Lð4Þ ð1ÞðbÞ D4:005 for b 5 1; 2; . . .; 6;

(6.102)

½Lð5Þ ð1ÞðbÞ 5 0 for b 5 1; 2; . . .; 6 ðfor open sea watersÞ and ½Lð5Þ ð1ÞðbÞ 5 2:598 for b 5 1; 2; . . .; 6 ðfor restricted sea watersÞ:

(6.103)

Further, taking into account the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the open sea waters (GMU Safety Interactive Platform, 2018) also given by (6.53) and considering (6.99)(6.103), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5; while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the open sea waters are (in PLN) Lð1Þ ð1ÞD1:232; Lð2Þ ð1ÞD3:269; Lð3Þ ð1ÞD4:871; Lð4Þ ð1ÞD4:005; Lð5Þ ð1ÞD0:

(6.104)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðtÞ in the entire area D of the open sea waters amounts (in PLN)

Modeling critical infrastructure accident losses Chapter | 6

Lð1Þ 5 1:232 1 3:269 1 4:871 1 4:005 1 0 5 13:377:

135

(6.105)

Thus considering (6.89) and (6.104) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the open sea waters, related to the climateweather change impact, are 1:211 5 0:983 5 98:3%; 1:232 2:781 RIð2Þ 5 5 0:851 5 85:1%; 3:269 4:170 RIð3Þ 5 5 0:856 5 85:6%; 4:871 4:005 5 1 5 100%; RIð4Þ 5 4:005 RIð1Þ 5

(6.106)

RIð5Þ 2 n=a as Lð5Þ ð1Þ 5 0 and Lð5Þ ð1Þ 5 0; and considering (6.90) and (6.105), and according to (6.16) and (6.18), the indicator RI of the environment of the entire area D resilience to the losses associated with the critical infrastructure accident at the open sea waters, related to the climateweather change impact, is RI 5

12:167 5 0:910 5 91:0%: 13:377

(6.107)

Similarly, taking into account the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the restricted sea waters (GMU Safety Interactive Platform, 2018) also given by (6.58) and considering (6.99)(6.103), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5; while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the restricted sea waters are (in PLN) Lð1Þ ð1ÞD1:228; Lð2Þ ð1ÞD3:297; Lð3Þ ð1ÞD4:937; Lð4Þ ð1ÞD4:005; Lð5Þ ð1ÞD2:598:

(6.108)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðtÞ in the entire area D of the restricted sea waters amounts (in PLN) Lð1Þ 5 1:228 1 3:297 1 4:937 1 4:005 1 2:598 5 16:065:

(6.109)

Thus considering (6.89) and (6.108) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the restricted sea waters, related to the climateweather change impact, are 1:211 5 0:986 5 98:6%; 1:228 2:781 5 0:844 5 84:4%; RIð2Þ 5 3:297 4:170 RIð3Þ 5 5 0:845 5 84:5%; 4:937 4:005 5 1 5 100%; RIð4Þ 5 4:005 2:598 5 1 5 100%; RIð5Þ 5 2:598

RIð1Þ 5

(6.110)

and considering (6.91) and (6.109), and according to (6.16) and (6.18), the indicator RI of the environment of the entire area D resilience to the losses associated with the critical infrastructure accident at the restricted sea waters, related to the climateweather change impact, is

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RI 5

14:765 5 0:919 5 91:9%: 16:065

(6.111)

6.3.4.2 Gdynia and Karlskrona ports To analyze the climateweather impact on the losses, we suppose that there are w 5 6 climateweather states cb ; b 5 1; 2; . . .; 6 dependent on the wind speed and the wind direction, distinguished for the ship operating area at the Baltic Sea port waters (GMU Safety Interactive Platform, 2018; Kuligowska, 2017) also defined in Section 6.3.2.2. According to the information coming from experts, the coefficients (6.10) ½ρiðkÞ ðbÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of the climateweather impact on losses at the climateweather change process states cb ; b 5 1; 2; . . .; 6 at the Gdynia and Karlskrona ports are given by (6.63)(6.67). Considering (6.63)(6.67) and (6.92)(6.96) after applying (6.9), and further (6.11), we calculate the conditional approximate expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5 of the Gdynia and Karlskrona ports, while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6: The results are as follows (in PLN): ½Lð1Þ ð1ÞðbÞ D2:481 for b 5 1; 3; 5; ½Lð1Þ ð1ÞðbÞ D4:961 for b 5 2; 4; 6;

(6.112)

½Lð2Þ ð1ÞðbÞ D4:097 for b 5 1; 3; 5; ½Lð2Þ ð1ÞðbÞ D8:193 for b 5 2; 4; 6;

(6.113)

½Lð3Þ ð1ÞðbÞ D5:475 for b 5 1; 3; 5; ½Lð3Þ ð1ÞðbÞ D8:212 for b 5 2; 4; 6;

(6.114)

½Lð4Þ ð1ÞðbÞ D5:288 for b 5 1; 2; . . .; 6;

(6.115)

ðbÞ

½Lð5Þ ð1Þ D3:906 for b 5 1; 2; . . .; 6:

(6.116)

Further, taking into account the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the Gdynia Port (GMU Safety Interactive Platform, 2018) also given by (6.73) and considering (6.112)(6.116), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5, while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the Gdynia Port are (in PLN) Lð1Þ ð1ÞD2:520; Lð2Þ ð1ÞD4:162; Lð3Þ ð1ÞD5:519; Lð4Þ ð1ÞD5:288; Lð5Þ ð1Þ 5 3:906:

(6.117)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðtÞ in the entire area D of the Gdynia Port amounts (in PLN) Lð1Þ 5 21:395: (6.118) Thus considering (6.97) and (6.117) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the Gdynia Port, related to the climateweather change impact, are 2:481 5 0:985 5 98:5%; 2:520 4:097 RIð2Þ 5 5 0:984 5 98:4%; 4:162 5:475 5 0:992 5 99:2%; RIð3Þ 5 5:519 5:288 5 1 5 100%; RIð4Þ 5 5:288 3:906 5 1 5 100%; RIð5Þ 5 3:906

RIð1Þ 5

(6.119)

Modeling critical infrastructure accident losses Chapter | 6

137

and considering (6.98) and (6.118), and according to (6.16) and (6.18), the indicator RI of the environment of the entire area D resilience to the losses associated with the critical infrastructure accident at the Gdynia Port, related to the climateweather change impact, is RI 5

21:247 5 0:993 5 99:3%: 21:395

(6.120)

Similarly, taking into account the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the Karlskrona Port (GMU Safety Interactive Platform, 2018) also given by (6.78) and considering (6.112)(6.116), and applying (6.12), we calculate the unconditional approximate expected value of the environmental losses impacted by the climateweather change process CðtÞ; during the time t 5 1 hour, associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; 5; while the process CðtÞ is at the climateweather state cb ; b 5 1; 2; . . .; 6 that for the Karlskrona Port are (in PLN) Lð1Þ ð1ÞD2:535; Lð2Þ ð1ÞD4:187; Lð3Þ ð1ÞD5:535; Lð4Þ ð1ÞD5:288; Lð5Þ ð1Þ 5 3:906:

(6.121)

Hence after applying (6.14), the total expected value of the losses impacted by the climateweather change process C(t), during the time t 5 1 hour, associated with the process of the environment degradation R(t) in the entire area D of the Karlskrona Port amounts (in PLN) Lð1Þ 5 21:451:

(6.122)

Thus considering (6.97) and (6.121) and according to (6.15) and (6.17), the indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident at the Karlskrona Port, related to the climateweather change impact, are 2:481 5 0:979 5 97:9%; 2:535 4:097 RIð2Þ 5 5 0:979 5 97:9%; 4:187 5:475 5 0:989 5 98:9%; RIð3Þ 5 5:535 5:288 5 1 5 100%; RIð4Þ 5 5:288 3:906 5 1 5 100%; RIð5Þ 5 3:906 RIð1Þ 5

(6.123)

and considering (6.98) and (6.122), and according to (6.16) and (6.18), the indicator RI of the environment of the entire area D resilience to the losses associated with the critical infrastructure accident at the Karlskrona Port, related to the climateweather change impact, is RI 5

21:247 5 0:991 5 99:1%: 21:451

(6.124)

6.3.5 Losses of shipping critical infrastructure accident at Baltic Sea and world sea waters—analysis of results In Sections 6.3.1 and 6.3.3 the general model of critical infrastructure accident consequences is applied to cost analysis of losses associated with consequences generated by the critical infrastructure defined as a ship operating at the Baltic Sea. Moreover, the losses of critical infrastructure accident consequences impacted by the climateweather change process are calculated. These losses are compared with losses without considering the climateweather impact. The summary of obtained results and resilience indicators for the Baltic Sea waters are presented in Tables 6.16.4. In Tables 6.16.4, there are presented values of losses associated with the process of the environment degradation for the time t 5 1 hour. We assume that the each value of these losses is the same at any given moment of the particular

138

Consequences of Maritime Critical Infrastructure Accidents

TABLE 6.1 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the open sea waters (results based on data collected within the Baltic Sea).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 0:785 Lð2Þ ð1Þ 5 2:467 Lð3Þ ð1Þ 5 3:091 Lð4Þ ð1Þ 5 3:072 Lð5Þ ð1Þ 5 0 Lð1Þ 5 9:415

Lð1Þ ð1Þ 5 0:798 Lð2Þ ð1Þ 5 2:900 Lð3Þ ð1Þ 5 3:611 Lð4Þ ð1Þ 5 3:072 Lð5Þ ð1Þ 5 0 Lð1Þ 5 10:381

RIð1Þ 5 0:984 RIð2Þ 5 0:851 RIð3Þ 5 0:856 RIð4Þ 5 1:000 n/a RI 5 0:907

TABLE 6.2 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the restricted sea waters (results based on data collected within the Baltic Sea).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 0:785 Lð2Þ ð1Þ 5 2:467 Lð3Þ ð1Þ 5 3:091 Lð4Þ ð1Þ 5 3:072 Lð5Þ ð1Þ 5 0 Lð1Þ 5 9:415

Lð1Þ ð1Þ 5 0:796 Lð2Þ ð1Þ 5 2:925 Lð3Þ ð1Þ 5 3:660 Lð4Þ ð1Þ 5 3:072 Lð5Þ ð1Þ 5 0 Lð1Þ 5 10:453

RIð1Þ 5 0:986 RIð2Þ 5 0:843 RIð3Þ 5 0:845 RIð4Þ 5 1:000 n/a RI 5 0:901

TABLE 6.3 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the Gdynia Port (results based on data collected within the Baltic Sea).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 1:457 Lð2Þ ð1Þ 5 3:145 Lð3Þ ð1Þ 5 3:769 Lð4Þ ð1Þ 5 3:750 Lð5Þ ð1Þ 5 0 Lð1Þ 5 12:121

Lð1Þ ð1Þ 5 1:481 Lð2Þ ð1Þ 5 3:195 Lð3Þ ð1Þ 5 3:800 Lð4Þ ð1Þ 5 3:750 Lð5Þ ð1Þ 5 0 Lð1Þ 5 12:226

RIð1Þ 5 0:984 RIð2Þ 5 0:984 RIð3Þ 5 0:992 RIð4Þ 5 1:000 n/a RI 5 0:991

Modeling critical infrastructure accident losses Chapter | 6

139

TABLE 6.4 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the Karlskrona Port (results based on data collected within the Baltic Sea).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 1:457 Lð2Þ ð1Þ 5 3:145 Lð3Þ ð1Þ 5 3:769 Lð4Þ ð1Þ 5 3:750 Lð5Þ ð1Þ 5 0 Lð1Þ 5 12:121

Lð1Þ ð1Þ 5 1:489 Lð2Þ ð1Þ 5 3:214 Lð3Þ ð1Þ 5 3:811 Lð4Þ ð1Þ 5 3:750 Lð5Þ ð1Þ 5 0 Lð1Þ 5 12:264

RIð1Þ 5 0:979 RIð2Þ 5 0:979 RIð3Þ 5 0:989 RIð4Þ 5 1:000 n/a RI 5 0:988

Open sea waters

Restricted waters 100 Losses (PLN)

Losses (PLN)

100

50

50

0

0

0

1

2

3

4

5

6

No impacted by C-WCP

7

8

0

9 10 Time (h)

1

2

3

4

5

6

7

9

10

Time (h)

Impacted by C-WCP

No impacted by C-WCP

Impacted by C-WCP

Karlskrona Port

Gdynia Port

100

100 Losses (PLN)

Losses (PLN)

8

50

50

0

0 0

1

2

3

4

No impacted by C-WCP

5

6

7

8

0

9 10 Time (h)

Impacted by C-WCP

1

2

3

4

5

No impacted by C-WCP

6

7

8

9 10 Time (h)

Impacted by C-WCP

FIGURE 6.1 The graph of the function of environmental total losses associated with the process of the environment degradation not impacted and impacted by climateweather change process—C-WCP (results based on data collected within the Baltic Sea).

TABLE 6.5 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the open sea waters (results based on data collected within the world sea waters).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 1:211 Lð2Þ ð1Þ 5 2:781 Lð3Þ ð1Þ 5 4:170 Lð4Þ ð1Þ 5 4:005 Lð5Þ ð1Þ 5 0 Lð1Þ 5 12:167

Lð1Þ ð1Þ 5 1:232 Lð2Þ ð1Þ 5 3:269 Lð3Þ ð1Þ 5 4:871 Lð4Þ ð1Þ 5 4:005 Lð5Þ ð1Þ 5 0 Lð1Þ 5 13:377

RIð1Þ 5 0:983 RIð2Þ 5 0:851 RIð3Þ 5 0:856 RIð4Þ 5 1:000 n/a RI 5 0:910

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Consequences of Maritime Critical Infrastructure Accidents

environment degradation state duration. Thus the every next hour of environment degradation state duration causes the losses, the value of which is a multiple of state time duration and value LðkÞ ð1Þ or LðkÞ ð1Þ, respectively. Taking this into account, the graphs of the time function of environmental total losses associated with the process of the environment degradation not impacted and impacted by climateweather change process (results based on data collected within the Baltic Sea waters) are presented in Fig. 6.1.

TABLE 6.6 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the restricted sea waters (results based on data collected within the world sea waters).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 1:211 Lð2Þ ð1Þ 5 2:781 Lð3Þ ð1Þ 5 4:170 Lð4Þ ð1Þ 5 4:005 Lð5Þ ð1Þ 5 2:598 Lð1Þ 5 14:765

Lð1Þ ð1Þ 5 1:228 Lð2Þ ð1Þ 5 3:297 Lð3Þ ð1Þ 5 4:937 Lð4Þ ð1Þ 5 4:005 Lð5Þ ð1Þ 5 2:598 Lð1Þ 5 16:065

RIð1Þ 5 0:986 RIð2Þ 5 0:844 RIð3Þ 5 0:845 RIð4Þ 5 1:000 RIð5Þ 5 1:000 RI 5 0:919

TABLE 6.7 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the Gdynia Port (results based on data collected within the world sea waters).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 2:481 Lð2Þ ð1Þ 5 4:097 Lð3Þ ð1Þ 5 5:475 Lð4Þ ð1Þ 5 5:288 Lð5Þ ð1Þ 5 3:906 Lð1Þ 5 21:247

Lð1Þ ð1Þ 5 2:520 Lð2Þ ð1Þ 5 4:162 Lð3Þ ð1Þ 5 5:519 Lð4Þ ð1Þ 5 5:288 Lð5Þ ð1Þ 5 3:906 Lð1Þ 5 21:395

RIð1Þ 5 0:985 RIð2Þ 5 0:984 RIð3Þ 5 0:992 RIð4Þ 5 1:000 RIð5Þ 5 1:000 RI 5 0:993

TABLE 6.8 Comparison of shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, without and with considering the climateweather impact and resilience indicators for the Karlskrona Port (results based on data collected within the world sea waters).

Total

Losses without climateweather change process impact (PLN)

Losses with climateweather change process impact (PLN)

Resilience indicator

Lð1Þ ð1Þ 5 2:481 Lð2Þ ð1Þ 5 4:097 Lð3Þ ð1Þ 5 5:475 Lð4Þ ð1Þ 5 5:288 Lð5Þ ð1Þ 5 3:906 Lð1Þ 5 21:247

Lð1Þ ð1Þ 5 2:535 Lð2Þ ð1Þ 5 4:187 Lð3Þ ð1Þ 5 5:535 Lð4Þ ð1Þ 5 5:288 Lð5Þ ð1Þ 5 3:906 Lð1Þ 5 21:451

RIð1Þ 5 0:979 RIð2Þ 5 0:979 RIð3Þ 5 0:989 RIð4Þ 5 1:000 RIð5Þ 5 1:000 RI 5 0:991

Modeling critical infrastructure accident losses Chapter | 6

Restricted waters

Open sea waters

100 Losses (PLN)

Losses (PLN)

100

50

50

0

0 0

1

2

3

4

5

6

No impacted by C-WCP

7

8

0

9 10 Time (h)

1

2

3

4

5

6

7

8

9

10

Time (h) No impacted by C-WCP

Impacted by C-WCP

Impacted by C-WCP Karlskrona Port

Gdynia Port 100 Losses (PLN)

100 Losses (PLN)

141

50

50

0

0 0

1

2

3

4

No impacted by C-WCP

5

6

7

8

9 10 Time (h)

Impacted by C-WCP

0

1

2

3

4

No impacted by C-WCP

5

6

7

8

9 10 Time (h)

Impacted by C-WCP

FIGURE 6.2 The graph of the function of environmental total losses associated with the process of the environment degradation not impacted and impacted by climateweather change process—C-WCP (results based on data collected within the world sea waters).

In Sections 6.3.2 and 6.3.4, similar analysis is carried out for the world sea waters, and its results are presented in Tables 6.56.8. These data allow to the comparable conclusion as for the Baltic Sea given earlier. The graphs of the time function of environmental total losses associated with the process of the environment degradation not impacted and impacted by climateweather change process (results based on data collected within the world sea waters) are presented in Fig. 6.2. The previous results point to a more significant impact of the climateweather change process within the Baltic Sea open and restricted waters than Gdynia and Karlskrona ports. The reason for this can be explained that the wave height and the wind speed are parameters considered in the state of the climateweather change process at the open and restricted sea waters, whereas the wind speed and the wind direction are parameters considered in the state of the climateweather change process at Gdynia and Karlskrona ports. It confirms that a wind direction that is considered in the states of the climateweather change process only for Gdynia and Karlskrona ports has a little significant impact on a value of losses associated with the process of the environment degradation. The results obtained in Section 6.3 are applied in Section 7.3 to the accident consequences cost optimization through the accident losses minimizing.

Chapter 7

Optimization of critical infrastructure accident losses Chapter Outline 7.1 Critical infrastructure accident losses minimization without considering climateweather change process impact— theoretical background 143 7.2 Critical infrastructure accident losses minimization with considering climateweather change process impact— theoretical background 145 7.3 Critical infrastructure accident losses minimization— application for Baltic Sea and world sea waters 148 7.3.1 Minimization of critical infrastructure accident losses without considering climateweather change process impact 148

7.3.2 Minimization of critical infrastructure accident losses with considering climateweather change process impact 152 7.3.3 Minimization of critical infrastructure accident losses without considering climateweather change process impact—extension based on wider data collected at world sea waters 160 7.3.4 Minimization of critical infrastructure accident losses with considering climateweather change process impact—extension based on wider data collected at world sea waters 162

The procedure based on the results of general model of critical infrastructure accident consequences and linear programming is applied to optimization of environment losses, associated with chemical release generated by ships’ critical infrastructure network operating at the Baltic Sea waters, either not impacted or impacted by the climateweather change process as estimated in Chapter 6, Modeling critical infrastructure accident losses. The optimal values of limit transient probabilities at particular states of the process of environment degradation that minimize the expected value of total environment losses for the fixed time interval are found.

7.1 Critical infrastructure accident losses minimization without considering climateweather change process impact—theoretical background From the linear equation (6.2), we can see that the mean value of expected critical infrastructure accident losses L(k)(t), tAh0; 1 NÞ; associated with the process of the environment degradation R(k)(t) in the subarea Dk ; k 5 1; 2; . . .; n3 ; is determined by the limit value of transient probabilities qiðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 of the process of the environi ment degradation at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; and the mean value of the critical infrastructure accii dent losses LðkÞ ðtÞ associated with the process of the environment degradation R(k)(t) in the subarea Dk ; k 5 1; 2; . . .; n3 ; i at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 . Therefore the optimization based on the linear programming (Klabjan and Adelman, 2006; Kołowrocki and Soszy´nska-Budny, 2011; Tang et al., 2007; Vercellis, 2009) of the critical infrastructure accident losses associated with the process of the environment degradation R(k)(t) in the subarea Dk, k 5 1; 2; . . .; n3 can be proposed. Namely, we may look for the corresponding optimal values q_iðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 of the limit transient probabilities qiðkÞ ; i i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 of the process of the environment degradation at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 to minimize the mean value of critical infrastructure accident losses L(k)(t) in the subarea Dk, k 5 1; 2; . . .; n3 (Bogalecka and Kołowrocki, 2018b).

Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00007-3 © 2020 Elsevier Inc. All rights reserved.

143

144

Consequences of Maritime Critical Infrastructure Accidents

Thus we may formulate the optimization problem as a linear programming model with the objective function of the following form: LðkÞ ðtÞD

‘k X

qiðkÞ ULiðkÞ ðtÞ;

tAh0; 1 NÞ;

k 5 1; 2; . . .; n3 ;

(7.1)

i51

with the following bound constraints: ‘k X

q iðkÞ # qiðkÞ # _ q iðkÞ ;

qiðkÞ 5 1;

k 5 1; 2; . . .; n3 ;

(7.2)

i51

where LiðkÞ ðtÞ, LiðkÞ ðtÞ $ 0; i 5 1; 2; . . .; ‘k are fixed mean values of the losses associated with the process of the environi ment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 ; at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; for a fixed t, tAh0; 1 NÞ and q iðkÞ ; 0 # _ q iðkÞ # 1; q iðkÞ # _ q iðkÞ ; q iðkÞ ; 0 # q iðkÞ # 1; and _ i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;

(7.3)

are lower and upper bounds of the unknown transient probabilities qiðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 , respectively. Now, we can obtain the optimal solution of the linear programming problem formulated by (7.1)(7.3), that is, we can find the optimal values q_iðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 of the limit transient probabilities qiðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 that minimize the objective functions given by (7.1). First, we arrange the mean values of the losses LiðkÞ ðtÞ, i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; associated with the process of i the environment degradation R(k)(t) in the subarea Dk, k 5 1; 2; . . .; n3 ; at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; in increasing order i‘

1 2 LiðkÞ ðtÞ # LiðkÞ ðtÞ # ? # LðkÞk ðtÞ;

k 5 1; 2; . . .; n3 ;

tAh0; 1 NÞ;

where ij Af1; 2; . . .; ‘k g for j 5 1; 2; . . .; ‘k : Then, we substitute j j j xjðkÞ 5 qðkÞ ; x jðkÞ 5 q ðkÞ ; _ x jðkÞ 5 _ q ðkÞ

i

i

i

(7.4)

for j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; and we minimize with respect to xjðkÞ ; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 the linear form (7.1) that after this transformation takes the form LðkÞ ðtÞ 5

‘k X

i

j xjðkÞ ULðkÞ ðtÞ

(7.5)

j51

for a fixed t, tAh0; 1 NÞ with the following bound constraints: x jðkÞ # xjðkÞ # _ x jðkÞ ;

‘k X

xjðkÞ 5 1;

k 5 1; 2; . . .; n3 ;

(7.6)

j51 i

i

j j where LðkÞ ðtÞ, LðkÞ ðtÞ $ 0; j 5 1; 2; . . .; ‘k are fixed mean values of the losses associated with the process of the environi ment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 ; at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; for a fixed t, tAh0; 1 NÞ; arranged in increasing order and

x jðkÞ ; 0 # x jðkÞ # 1; and _ x jðkÞ ; 0 # _ x jðkÞ # 1; x jðkÞ # _ x jðkÞ ; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;

(7.7)

are lower and upper bounds of the unknown probabilities xjðkÞ ; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; respectively. To find the optimal values of xjðkÞ ; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; we define x ðkÞ 5

‘k X j51

and

x jðkÞ ; ybðkÞ 5 1 2 x ðkÞ

(7.8)

Optimization of critical infrastructure accident losses Chapter | 7

x ðkÞ;0 5 0; _ x ðkÞ;0 5 0; and x ðkÞ;J 5

J X

x jðkÞ ; _ x ðkÞ;J 5

j51

for J 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 :

J X _j x ðkÞ j51

145

(7.9)

Next, we find the largest value JAf0; 1; . . .; ‘k g such that _ x ðkÞ;J

2 x ðkÞ;J , y^ðkÞ

(7.10)

and we fix the optimal solution that minimize (7.5) in the following way: G

If J 5 0, the optimal solution is x_1ðkÞ 5 y^ðkÞ 1 x 1ðkÞ and x_jðkÞ 5 x jðkÞ

G

for j 5 2; 3; . . .; ‘k

(7.11)

If 0 , J , ‘k ; the optimal solution is x_jðkÞ 5 _ x jðkÞ

for j 5 1; 2; . . .; J;

_ J11 x_J11 ðkÞ 5 y^ðkÞ 2 x ðkÞ;J 1 x ðkÞ;J 1 x ðkÞ

(7.12) (7.13)

and x_jðkÞ 5 x jðkÞ G

for j 5 J 1 2; J 1 3; . . .; ‘k

(7.14)

If J 5 ‘k ; the optimal solution is x_jðkÞ 5 _ x jðkÞ

for j 5 1; 2; . . .; ‘k :

(7.15)

Finally, after making the inverse to (7.4) substitution, we get the optimal limit transient probabilities i

j q_ðkÞ 5 x_jðkÞ

for j 5 1; 2; . . .; ‘k ;

(7.16)

that minimize the mean values of the losses, associated with the process of the environment degradation R(k)(t) in the subarea Dk ; k 5 1; 2; . . .; n3 ; defined by the linear form (7.1), giving its minimum value in the following form: L_ðkÞ ðtÞ 5

‘k X

q_iðkÞ LiðkÞ ðtÞ

i51

(7.17)

for k 5 1; 2; . . .; n3 and a fixed t; tAh0; 1 NÞ:

7.2 Critical infrastructure accident losses minimization with considering climateweather change process impact—theoretical background From the linear equation (6.13), we can see that the mean value of expected critical infrastructure accident losses LðkÞ ðtÞ; tAh0; 1 NÞ; associated with the process of the environment degradation R(k)(t) in the subarea Dk ; k 5 1; 2; . . .; n3 ; impacted by the climateweather change process C(t) is determined by the limit value of transient probabilities qb ; b 5 1; 2; . . .; w of the climateweather change process C(t) at the particular climateweather state cb, b 5 1; 2; . . .; w; the limit value of transient probabilities qiðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 of the process of the environi ment degradation at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; and by the mean value of the critical infrastructure b i accident losses ½LðkÞ ðtÞ associated with the process of the environment degradation R(k)(t) in the subarea Dk ; i k 5 1; 2; . . .; n3 at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; impacted by the climateweather change process C(t). Therefore the optimization based on the linear programming (Klabjan and Adelman, 2006; Kołowrocki and Soszy´nska-Budny, 2011; Tang et al., 2007; Vercellis, 2009) of the critical infrastructure accident losses associated with the process of the environment degradation R(k)(t) in the subarea Dk ; k 5 1; 2; . . .; n3 with considering the climateweather change process C(t) can be proposed. Namely, we may look for the corresponding optimal values qb q_iðkÞ ; i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 of the limit transient probabilities qb qiðkÞ ; i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; i k 5 1; 2; . . .; n3 of the process of the environment degradation at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 to minimize the mean value of critical infrastructure accident losses LðkÞ ðtÞ impacted by the climateweather change process C(t) in the subarea Dk, k 5 1; 2; . . .; n3 (Bogalecka and Kołowrocki, 2018b).

146

Consequences of Maritime Critical Infrastructure Accidents

Thus we may formulate the optimization problem as a linear programming model with the objective function of the following form: LðkÞ ðtÞD

‘k w X h iðbÞ X qb UqiðkÞ U LiðkÞ ðtÞ ; b51 i51

k 5 1; 2; . . .; n3 ;

(7.18)

tAh0; 1 NÞ;

with the following bound constraints ‘k w X X

q iðkÞ ; qb q iðkÞ # qb qiðkÞ # qb _

qb qiðkÞ 5 1;

k 5 1; 2; . . .; n3 ;

(7.19)

i51

b51

where ½LiðkÞ ðtÞðbÞ ; ½LiðkÞ ðtÞðbÞ $ 0; i 5 1; 2; . . .; ‘k are fixed mean values of the losses associated with the process of the i environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 ; at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; impacted by the climateweather change process CðtÞ for a fixed t, tAh0; 1 NÞ and q iðkÞ ; 0 # qb _ q iðkÞ # 1; qb q iðkÞ # qb _ q iðkÞ ; qb q iðkÞ ; 0 # qb q iðkÞ # 1; and qb _ i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 ;

(7.20)

are lower and upper bounds of the unknown transient probabilities qb qiðkÞ ; i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 ; respectively. Now, we can obtain the optimal solution of the linear programming problem formulated by (7.18)(7.20), that is, we can find the optimal values qb q_iðkÞ i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 of the limit transient probabilities qb qiðkÞ ; i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 that minimize the objective functions given by (7.18). First, we arrange the mean values of the losses ½LiðkÞ ðtÞðbÞ ; i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 ; associated i with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 ; at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; impacted by the climateweather change process CðtÞ in increasing order h iðbÞ h iðbÞ h i iðbÞ ‘ Uw 1 2 LiðkÞ ðtÞ # LiðkÞ ðtÞ # ? # LðkÞk ðtÞ ; k 5 1; 2; . . .; n3 ; tAh0; 1 NÞ; where ij Af1; 2; . . .; ‘k g for j 5 1; 2; . . .; ‘k Uw: Next, we substitute j j j xjðkÞ 5 qb qðkÞ ; x jðkÞ 5 qb q ðkÞ ; _ x jðkÞ 5 qb _ q ðkÞ

i

i

i

(7.21)

for j 5 1; 2; . . .; ‘k Uw and we minimize with respect to xjðkÞ ; j 5 1; 2; . . .; ‘k Uw the linear form (7.18) that after this transformation takes the form ‘k Uw X

LðkÞ ðtÞ 5

h iðbÞ ij xjðkÞ U LðkÞ ðtÞ

(7.22)

j51

for a fixed t, tAh0; 1 NÞ; with the following bound constraints: x jðkÞ # xjðkÞ # _ x jðkÞ ;

‘k Uw X

xjðkÞ 5 1;

k 5 1; 2; . . .; n3 ;

(7.23)

j51 j j where ½LðkÞ ðtÞðbÞ ; ½LðkÞ ðtÞðbÞ $ 0; j 5 1; 2; . . .; ‘k Uw are fixed mean values of the losses associated with the process of the i environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 ; at the state rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; impacted by the climateweather change process C(t) for a fixed t, tAh0; 1 NÞ; arranged in increasing order, and

i

i

x jðkÞ ; 0 # x jðkÞ # 1; and _ x jðkÞ ; 0 # _ x jðkÞ # 1; x jðkÞ # _ x jðkÞ ; j 5 1; 2; . . .; ‘k Uw; k 5 1; 2; . . .; n3 ;

(7.24)

are lower and upper bounds of the unknown probabilities xjðkÞ ; j 5 1; 2; . . .; ‘k Uw; k 5 1; 2; . . .; n3 ; respectively. To find the optimal values of xjðkÞ ; j 5 1; 2; . . .; ‘k Uw; k 5 1; 2; . . .; n3 ; we define x ðkÞ 5

‘k Uw X j51

x jðkÞ ; ybðkÞ 5 1 2 x ðkÞ

(7.25)

Optimization of critical infrastructure accident losses Chapter | 7

147

and x ðkÞ;0 5 0; and x ðkÞ;J 5 x ðkÞ;0 5 0; _

J X

x ðkÞ;J 5 x jðkÞ ; _

j51

J X _j x ðkÞ

(7.26)

j51

for J 5 1; 2; . . .; ‘k Uw: Next, we find the largest value JAf0; 1; . . .; ‘k Uwg such that _ x ðkÞ;J

2 x ðkÞ;J , y^ðkÞ

(7.27)

and we fix the optimal solution that minimize (7.22) in the following way: G

If J 5 0; the optimal solution is x_1ðkÞ 5 y^ðkÞ 1 x 1ðkÞ and x_jðkÞ 5 x jðkÞ for

G

j 5 2; 3; . . .; ‘k Uw

(7.28)

If 0 , J , ‘k Uw; the optimal solution is x_jðkÞ 5 _ x jðkÞ for

j 5 1; 2; . . .; J;

_ x_J11 ðkÞ 5 y^ðkÞ 2 x ðkÞ;J

1 x ðkÞ;J 1 x J11 ðkÞ ;

(7.29) (7.30)

and x_jðkÞ 5 x jðkÞ G

for j 5 J 1 2; J 1 3; . . .; ‘k Uw

(7.31)

If J 5 ‘k Uw; the optimal solution is x_jðkÞ 5 _ x jðkÞ

for j 5 1; 2; . . .; ‘k Uw:

(7.32)

Finally, after making the inverse to (7.21) substitution, we get the optimal limit transient probabilities i

j qb q_ðkÞ 5 x_jðkÞ

for j 5 1; 2; . . .; ‘k Uw;

(7.33)

that minimize the mean values of the losses associated with the process of the environment degradation RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 ; impacted by the climateweather change process CðtÞ; defined by the linear form (7.18), giving its minimum value in the following form L_ðkÞ ðtÞ 5

‘k w X X b51

h iðbÞ qb q_iðkÞ LiðkÞ ðtÞ

i51

(7.34)

for k 5 1; 2; . . .; n3 and a fixed t; tAh0; 1 NÞ: Thus the indicators of the environment in the subareas Dk ; k 5 1; 2; . . .; n3 resilience to the losses associated with the critical infrastructure accident related to the climateweather change after the optimization take a form RI_ðkÞ 5

L_ðkÞ ðtÞ ; L_ ðtÞ

tAh0; 1 NÞ;

k 5 1; 2; . . .; n3 ;

(7.35)

ðkÞ

where L_ðkÞ ðtÞ and L_ ðkÞ ðtÞ are determined by (7.17) and (7.34), respectively. Similarly, the indicator of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change after the optimization takes a form n3 P

RI_ 5

k51 n3 P

L_ðkÞ ðtÞ L_ ðkÞ ðtÞ

;

tAh0; 1 NÞ;

k51

where L_ðkÞ ðtÞ and L_ ðkÞ ðtÞ are determined by (7.17) and (7.34), respectively.

(7.36)

148

Consequences of Maritime Critical Infrastructure Accidents

7.3 Critical infrastructure accident losses minimization—application for Baltic Sea and world sea waters 7.3.1 Minimization of critical infrastructure accident losses without considering climateweather change process impact The losses associated with the process of the environment degradation at the Baltic Sea waters are determined in Section 6.3.1. In this section, based on the procedure and formulae given in Section 7.1, we minimize mean values of these losses of shipping critical infrastructure accident at the Baltic Sea open waters, restricted waters, as well as Gdynia and Karlskrona ports.

7.3.1.1 Open and restricted sea waters Taking into account the losses LiðkÞ ð1Þ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; during the time t 5 1 hour, associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; . . .; 5; at the i state rðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 given in (6.19)(6.23), the objective function defined by (7.1) for particular subareas Dk ; k 5 1; 2; . . .; 5 of the Baltic Sea open and restricted waters takes the forms (the parts with qiðkÞ 5 0 are omitted): Lð1Þ ð1ÞD Lð2Þ ð1ÞD

28 X

30 X qið1Þ ULið1Þ ð1Þ 5 q1ð1Þ U0 1 q2ð1Þ U5000 1 q6ð1Þ U5000 1 q11 ð1Þ U7000; i51

16 21 25 27 qið2Þ ULið2Þ ð1Þ 5 q1ð2Þ U0 1 q6ð2Þ U10; 000 1 q12 ð2Þ U15; 000 1 qð2Þ U20; 000 1qð2Þ U25; 000 1 qð2Þ U27; 000 1 qð2Þ U30; 000;

i51

Lð3Þ ð1ÞD

28 X

16 21 25 27 qið3Þ ULið3Þ ð1Þ 5 q1ð3Þ U0 1 q6ð3Þ U15; 000 1 q12 ð3Þ U20; 000 1 qð3Þ U25; 000 1qð3Þ U30; 000 1 qð3Þ U30; 000 1 qð3Þ U30; 000;

i51

Lð4Þ ð1ÞD

31 X

16 21 28 30 qið4Þ ULið4Þ ð1Þ 5 q1ð4Þ U0 1 q12 ð4Þ U15; 000 1 qð4Þ U25; 000 1 qð4Þ U30; 000 1qð4Þ U30; 000 1 qð4Þ U30; 000;

i51

Lð5Þ ð1ÞD

23 X qið5Þ ULið5Þ ð1Þ 5 q1ð5Þ U0: i51

(7.37) We do not minimize the above objective function for the subarea D5 of the open and restricted sea waters as Lð5Þ ð1Þ 5 0 PLN. The lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 coming from experts, respectively, are (q iðkÞ 5 0 and _ q iðkÞ 5 0 are omitted) q 1ð1Þ 5 0:99981; q 2ð1Þ 5 0:000000001; q 6ð1Þ 5 0:00001; q 11 ð1Þ 5 0:00001; _1 q ð1Þ 5 0:99989; _ q 2ð1Þ 5 0:00000001; _ q 6ð1Þ 5 0:0001; _ q 11 ð1Þ 5 0:0001; 1 6 12 16 21  27 q ð2Þ 5 0:99981; q ð2Þ 5 0:00001; q ð2Þ 5 0:00001; q ð2Þ 5 0:00001; q ð2Þ 5 0:00001; q 25 ð2Þ 5 0:000001; q ð2Þ 5 0:000001; _1 _6 _12 _16 _21 _25 _27 q ð2Þ 5 0:99989; q ð2Þ 5 0:0001; q ð2Þ 5 0:0001; q ð2Þ 5 0:0001; q ð2Þ 5 0:0001; q ð2Þ 5 0:00001; q ð2Þ 5 0:00001;  16  21  25  27 q 1ð3Þ 5 0:99981; q 6ð3Þ 5 0:00001; q 12 ð3Þ 5 0:00001; q ð3Þ 5 0:00001; q ð3Þ 5 0:00001; q ð3Þ 5 0:000001; q ð3Þ 5 0:000001; _1 _6 _12 _16 _21 _25 _27 q ð3Þ 5 0:99989; q ð3Þ 5 0:0001; q ð3Þ 5 0:0001; q ð3Þ 5 0:0001; q ð3Þ 5 0:0001; q ð3Þ 5 0:00001; q ð3Þ 5 0:00001;  16  21  28  30 q 1ð4Þ 5 0:99981; q 12 ð4Þ 5 0:00001; q ð4Þ 5 0:00001; q ð4Þ 5 0:00001; q ð4Þ 5 0:000001; q ð4Þ 5 0:000001; _1 _12 _16 _21 _28 _30 q ð4Þ 5 0:99989; q ð4Þ 5 0:0001; q ð4Þ 5 0:0001; q ð4Þ 5 0:0001; q ð4Þ 5 0:00001; q ð4Þ 5 0:00001: (7.38)

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Hence, according to (7.2) and (7.3), we assume the following bound constraints: 0:99981 # q1ð1Þ # 0:99989; 0:000000001 # q2ð1Þ # 0:00000001; 0:00001 # q6ð1Þ # 0:0001; 0:00001 # q11 ð1Þ # 0:0001; 30 X qið1Þ 5 1; i51

16 0:99981 # q1ð2Þ # 0:99989; 0:00001 # q6ð2Þ # 0:0001; 0:00001 # q12 ð2Þ # 0:0001; 0:00001 # qð2Þ # 0:0001; 25 27 0:00001 # q21 ð2Þ # 0:0001; 0:000001 # qð2Þ # 0:00001; 0:000001 # qð2Þ # 0:00001; 28 X qið2Þ 5 1; i51

16 0:99981 # q1ð3Þ # 0:99989; 0:00001 # q6ð3Þ # 0:0001; 0:00001 # q12 ð3Þ # 0:0001; 0:00001 # qð3Þ # 0:0001; 21 25 27 0:00001 # qð3Þ # 0:0001; 0:000001 # qð3Þ # 0:00001; 0:000001 # qð3Þ # 0:00001; 28 X qið3Þ 5 1; i51

16 0:99981 # q1ð4Þ # 0:99989; 0:00001 # q12 ð4Þ # 0:0001; 0:00001 # qð4Þ # 0:0001; 28 30 0:00001 # q21 ð4Þ # 0:0001; 0:000001 # qð4Þ # 0:00001; 0:000001 # qð4Þ # 0:00001; 31 X qið4Þ 5 1: i51

(7.39) q_iðkÞ

qiðkÞ ;

Now, before we find optimal values of the limit transient probabilities k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 that minimize the objective functions (7.37), we arrange the mean values of the losses LiðkÞ ð1Þ; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 associated with the process of the environment degradai tion RðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; 3; 4; at the state rðkÞ ; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; i i ‘4 5 31 in increasing order (LðkÞ ð1Þ are omitted if qðkÞ 5 0) L1ð1Þ ð1Þ # L2ð1Þ ð1Þ # L6ð1Þ ð1Þ # L11 ð1Þ ð1Þ; 1 6 12 16 21 27 Lð2Þ ð1Þ # Lð2Þ ð1Þ # Lð2Þ ð1Þ # Lð2Þ ð1Þ # Lð2Þ ð1Þ # L25 ð2Þ ð1Þ # Lð2Þ ð1Þ; 1 6 12 16 21 25 Lð3Þ ð1Þ # Lð3Þ ð1Þ # Lð3Þ ð1Þ # Lð3Þ ð1Þ # Lð3Þ ð1Þ # Lð3Þ ð1Þ # L27 ð3Þ ð1Þ; 1 12 16 21 28 30 Lð4Þ ð1Þ # Lð4Þ ð1Þ # Lð4Þ ð1Þ # Lð4Þ ð1Þ # Lð4Þ ð1Þ # Lð4Þ ð1Þ: Next, according to (7.4), we substitute (xiðkÞ are omitted if qiðkÞ 5 0) x1ð2Þ x1ð3Þ

x1ð1Þ 5 q1ð1Þ ; x2ð1Þ 5 q2ð1Þ ; x3ð1Þ 5 q6ð1Þ ; x4ð1Þ 5 q12 ð1Þ ; 1 2 6 3 12 4 16 5 21 7 27 5 qð2Þ ; xð2Þ 5 qð2Þ ; xð2Þ 5 qð2Þ ; xð2Þ 5 qð2Þ ; xð2Þ 5 qð2Þ ; x6ð2Þ 5 q25 ð2Þ ; xð2Þ 5 qð2Þ ; 1 2 6 3 12 4 16 5 21 6 25 7 5 qð3Þ ; xð3Þ 5 qð3Þ ; xð3Þ 5 qð3Þ ; xð3Þ 5 qð3Þ ; xð3Þ 5 qð3Þ ; xð3Þ 5 qð3Þ ; xð3Þ 5 q27 ð3Þ ; 1 1 2 12 3 16 4 21 5 28 6 30 xð4Þ 5 qð4Þ ; xð4Þ 5 qð4Þ ; xð4Þ 5 qð4Þ ; xð4Þ 5 qð4Þ ; xð4Þ 5 qð4Þ ; xð4Þ 5 qð4Þ ;

(7.40)

and x 1ð1Þ 5 q 1ð1Þ 5 0:99981; x 2ð1Þ 5 q 2ð1Þ 5 0:000000001; x 3ð1Þ 5 q 6ð1Þ 5 0:00001; x 4ð1Þ 5 q 12 ð1Þ 5 0:00001; _1 x ð1Þ 5 _ q 1ð1Þ 5 0:99989; _ x 2ð1Þ 5 _ q 2ð1Þ 5 0:00000001; _ x 3ð1Þ 5 _ q 6ð1Þ 5 0:0001; _ x 4ð1Þ 5 _ q 12 ð1Þ 5 0:0001; 1 1 2 6 3 12 4 16 x ð2Þ 5 q ð2Þ 5 0:99981; x ð2Þ 5 q ð2Þ 5 0:00001; x ð2Þ 5 q ð2Þ 5 0:00001; x ð2Þ 5 q ð2Þ 5 0:00001; _1 _1 6 7  25  27 x 5ð2Þ 5 q 21 ð2Þ 5 0:00001; x ð2Þ 5 q ð2Þ 5 0:000001; x ð2Þ 5 q ð2Þ 5 0:000001; x ð2Þ 5 q ð2Þ 5 0:99989; _2 _4 _16 _5 _21 x ð2Þ 5 _ q 6ð2Þ 5 0:0001; _ x 3ð2Þ 5 _ q 12 ð2Þ 5 0:0001; x ð2Þ 5 q ð2Þ 5 0:0001; x ð2Þ 5 q ð2Þ 5 0:0001; _6 _25 _7 _27 1 1 2 x ð2Þ 5 q ð2Þ 5 0:00001; x ð2Þ 5 q ð2Þ 5 0:00001; x ð3Þ 5 q ð3Þ 5 0:99981; x ð3Þ 5 q 6ð3Þ 5 0:00001; 4 5 6  16  21  25 x 3ð3Þ 5 q 12 ð3Þ 5 0:00001; x ð3Þ 5 q ð3Þ 5 0:00001; x ð3Þ 5 q ð3Þ 5 0:00001; x ð3Þ 5 q ð3Þ 5 0:000001; _1 _1 _2 _6 _3 7 27 q 12 x ð3Þ 5 q ð3Þ 5 0:000001; x ð3Þ 5 q ð3Þ 5 0:99989; x ð3Þ 5 q ð3Þ 5 0:0001; x ð3Þ 5 _ ð3Þ 5 0:0001; _4 _16 _5 _21 _6 _25 _7 _27 x ð3Þ 5 q ð3Þ 5 0:0001; x ð3Þ 5 q ð3Þ 5 0:0001; x ð3Þ 5 q ð3Þ 5 0:00001; x ð3Þ 5 q ð3Þ 5 0:00001; 3 4  16  21 x 1ð4Þ 5 q 1ð4Þ 5 0:99981; x 2ð4Þ 5 q 12 ð4Þ 5 0:00001; x ð4Þ 5 q ð4Þ 5 0:00001; x ð4Þ 5 q ð4Þ 5 0:00001; _1 _1 _2 5 28 6 30 q 12 x ð4Þ 5 q ð4Þ 5 0:000001; x ð4Þ 5 q ð4Þ 5 0:000001; x ð4Þ 5 q ð4Þ 5 0:99989; x ð4Þ 5 _ ð4Þ 5 0:0001; _3 _16 _4 _21 _5 _28 _6 _30 x ð4Þ 5 q ð4Þ 5 0:0001; x ð4Þ 5 q ð4Þ 5 0:0001; x ð4Þ 5 q ð4Þ 5 0:00001; x ð4Þ 5 q ð4Þ 5 0:00001;

(7.41)

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and we minimize with respect to xjðkÞ ; k 5 1; 2; 3; 4; j 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 the linear forms (7.37) that according to (7.5) and (7.6) take the form Lð1Þ ð1Þ 5 x1ð1Þ U0 1 x2ð1Þ U5000 1 x3ð1Þ U5000 1 x4ð1Þ U7000; 1 2 Lð2Þ ð1Þ 5 xð2Þ U0 1 xð2Þ U10; 000 1 x3ð2Þ U15; 000 1 x4ð2Þ U20; 000 1x5ð2Þ U25; 000 1 x6ð2Þ U27; 000 1 x7ð2Þ U30; 000; Lð3Þ ð1Þ 5 x1ð3Þ U0 1 x2ð3Þ U15; 000 1 x3ð3Þ U20; 000 1 x4ð3Þ U25; 000 1x5ð3Þ U30; 000 1 x6ð3Þ U30; 000 1 x7ð3Þ U30; 000; Lð4Þ ð1Þ 5 x1ð4Þ U0 1 x2ð4Þ U15; 000 1 x3ð4Þ U25; 000 1 x4ð4Þ U30; 000 1x5ð4Þ U30; 000 1 x6ð4Þ U30; 000;

(7.42)

with the following bound constraints 0:99981 # x1ð1Þ # 0:99989; 0:000000001 # x2ð1Þ # 0:00000001; 0:00001 # x3ð1Þ # 0:0001; 0:00001 # x4ð1Þ # 0:0001; 30 X xjð1Þ 5 1; j51

0:99981 # x1ð2Þ # 0:99989; 0:00001 # x2ð2Þ # 0:0001; 0:00001 # x3ð2Þ # 0:0001; 4 0:00001 # xð2Þ # 0:0001; 0:00001 # x5ð2Þ # 0:0001; 0:000001 # x6ð2Þ # 0:00001; 0:000001 # x7ð2Þ # 0:00001; 28 X xjð2Þ 5 1; j51

0:99981 # x1ð3Þ # 0:99989; 0:00001 # x2ð3Þ # 0:0001; 0:00001 # x3ð3Þ # 0:0001; 0:00001 # x4ð3Þ # 0:0001; 0:00001 # x5ð3Þ # 0:0001; 0:000001 # x6ð3Þ # 0:00001; 0:000001 # x7ð3Þ # 0:00001; 28 X xjð3Þ 5 1; j51

0:99981 # x1ð4Þ # 0:99989; 0:00001 # x2ð4Þ # 0:0001; 0:00001 # x3ð4Þ # 0:0001; 0:00001 # x4ð4Þ # 0:0001; 0:000001 # x5ð4Þ # 0:00001; 0:000001 # x6ð4Þ # 0:00001; 31 X xjð4Þ 5 1: j51

(7.43) According to (7.8), we calculate x ð1Þ 5

30 X

x jð1Þ D0:999830001; ybð1Þ 5 1 2 x ð1Þ 5 1 2 0:999830001 5 0:000169999;

j51

x ð2Þ 5

28 X x jð2Þ D0:999852; ybð2Þ 5 1 2 x ð2Þ 5 1 2 0:999852 5 0:000148; j51

28 X x ð3Þ 5 x jð3Þ D0:999852; ybð3Þ 5 1 2 x ð3Þ 5 1 2 0:999852 5 0:000148;

(7.44)

j51 31 X x ð4Þ 5 x jð4Þ D0:999842; ybð4Þ 5 1 2 x ð4Þ 5 1 2 0:999842 5 0:000158; j51

and according to (7.9), we find x ð1Þ0 5 0; _ x ð1Þ0 5 0; _ x ð1Þ0 2 x ð1Þ0 5 0; _ x ð1Þ1 5 0:99981; x ð1Þ1 5 0:99989; _ x ð1Þ1 2 x ð1Þ1 5 0:00008; x ð1Þ2 5 0:999810001; _ x ð1Þ2 5 0:99989001; _ x ð1Þ2 2 x ð1Þ2 5 0:000080009; x ð1Þ3 5 0:999820001; _ x ð1Þ3 5 0:99999001; _ x ð1Þ3 2 x ð1Þ3 5 0:000170009; x ð2Þ0 5 0; _ x ð2Þ0 5 0; _ x ð2Þ0 2 x ð2Þ0 5 0; x ð2Þ1 5 0:99981; _ x ð2Þ1 5 0:99989; _ x ð2Þ1 2 x ð2Þ1 5 0:00008; _ x ð2Þ2 5 0:99982; x ð2Þ2 5 0:99999; _ x ð2Þ2 2 x ð2Þ2 5 0:00017; x ð3Þ0 5 0; _ x ð3Þ0 5 0; _ x ð3Þ0 2 x ð3Þ0 5 0; x ð3Þ1 5 0:99981; _ x ð3Þ1 5 0:99989; _ x ð3Þ1 2 x ð3Þ1 5 0:00008; _ x ð3Þ2 5 0:99982; x ð3Þ2 5 0:99999; _ x ð3Þ2 2 x ð3Þ2 5 0:00017; x ð4Þ0 5 0; _ x ð4Þ0 5 0; _ x ð4Þ0 2 x ð4Þ0 5 0; x ð4Þ1 5 0:99981; _ x ð4Þ1 5 0:99989; _ x ð4Þ1 2 x ð4Þ1 5 0:00008; _ x ð4Þ2 5 0:99982; x ð4Þ2 5 0:99999; _ x ð4Þ2 2 x ð4Þ2 5 0:00017:

(7.45)

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From the above, as according to (7.44), the appropriate inequality (7.10) takes the forms _ x ð1ÞJ

2 x ð1ÞJ , 0:000169999;

(7.46)

then it follows that the largest value JAf0; 1; . . .; 30g such that this inequality holds is J 5 2; _ x ð2ÞJ

2 x ð2ÞJ , 0:000148;

(7.47)

then it follows that the largest value JAf0; 1; . . .; 28g such that this inequality holds is J 5 1; _ x ð3ÞJ

2 x ð3ÞJ , 0:000148;

(7.48)

then it follows that the largest value JAf0; 1; . . .; 28g such that this inequality holds is J 5 1; _ x ð4ÞJ

2 x ð4ÞJ , 0:000158;

(7.49)

then it follows that the largest value JAf0; 1; . . .; 31g such that this inequality holds is J 5 1: Therefore, we fix the optimal solution that minimizes linear functions (7.42) according to the appropriate rule (7.12)(7.14). Namely, we get x 1ð1Þ 5 0:99989; x_2ð1Þ 5 _ x 2ð1Þ 5 0:00000001; x_1ð1Þ 5 _ 1 x ð1Þ2 1 x 3ð1Þ D0:000169999 2 0:99989001 1 0:999810001 1 0:00001 5 0:00009999; x_4ð1Þ 5 x 4ð1Þ 5 0:00001; x 1ð2Þ 5 0:99989; x_1ð2Þ 5 _ x_2ð2Þ 5 ybð2Þ 2 _ x ð2Þ1 1 x ð2Þ1 1 x 2ð2Þ D0:000148 2 0:99989 1 0:99981 1 0:00001 5 0:000078; 3 x_ð2Þ 5 x 3ð2Þ 5 0:00001; x_4ð2Þ 5 x 4ð2Þ 5 0:00001; x_5ð2Þ 5 x 5ð2Þ 5 0:00001; x_6ð2Þ 5 x 6ð2Þ 5 0:000001; x_7ð2Þ 5 x 7ð2Þ 5 0:000001; x_1ð3Þ 5 _ x 1ð3Þ 5 0:99989; x_2ð3Þ 5 ybð3Þ 2 _ x ð3Þ1 1 x ð3Þ1 1 x 2ð3Þ D0:000148 2 0:99989 1 0:99981 1 0:00001 5 0:000078; 3 x_ð3Þ 5 x 3ð3Þ 5 0:00001; x_4ð3Þ 5 x 4ð3Þ 5 0:00001; x_5ð3Þ 5 x 5ð3Þ 5 0:00001; x_6ð3Þ 5 x 6ð3Þ 5 0:000001; x7ð3Þ 5 x 7ð3Þ 5 0:000001; x 1ð4Þ 5 0:99989; x_1ð4Þ 5 _ _ x_2ð4Þ 5 ybð4Þ 2 x ð4Þ1 1 x ð4Þ1 1 x 2ð4Þ D0:000148 2 0:99989 1 0:99981 1 0:00001 5 0:000088; 3 x_ð4Þ 5 x 3ð4Þ 5 0:00001; x_4ð4Þ 5 x 4ð4Þ 5 0:00001; x_5ð4Þ 5 x 5ð4Þ 5 0:000001; x_6ð4Þ 5 x 6ð4Þ 5 0:000001:

x_3ð1Þ 5 ybð1Þ 2 _ x ð1Þ2

(7.50)

Finally, after making the substitution inverse to (7.40), we get the optimal transient probabilities 4 q_1ð1Þ 5 x_1ð1Þ 5 0:99989; q_2ð1Þ 5 x_2ð1Þ 5 0:00000001; q_6ð1Þ 5 x_3ð1Þ 5 0:00009999; q_11 ð1Þ 5 x_ð1Þ 5 0:00001; 1 1 6 2 12 3 q_ð2Þ 5 x_ð2Þ 5 0:99989; q_ð2Þ 5 x_ð2Þ 5 0:000078; q_ð2Þ 5 x_ð2Þ 5 0:00001; 16 4 5 6 7 _25 _27 q_ð2Þ 5 x_ð2Þ 5 0:00001; q_21 ð2Þ 5 x_ð2Þ 5 0:00001; q ð2Þ 5 x_ð2Þ 5 0:000001; ð2Þ 5 x_ð2Þ 5 0:000001; q 1 1 6 2 12 3 q_ð3Þ 5 x_ð3Þ 5 0:99989; q_ð3Þ 5 x_ð3Þ 5 0:000078; q_ð3Þ 5 x_ð3Þ 5 0:00001; 16 4 5 6 7 _25 _27 q_ð3Þ 5 x_ð3Þ 5 0:00001; q_21 ð3Þ 5 x_ð3Þ 5 0:00001; q ð3Þ 5 x_ð3Þ 5 0:000001; ð3Þ 5 x_ð3Þ 5 0:000001; q 1 1 12 2 16 3 q_ð4Þ 5 x_ð4Þ 5 0:99989; q_ð4Þ 5 x_ð4Þ 5 0:000088; q_ð4Þ 5 x_ð4Þ 5 0:00001; 4 5 6 _28 _30 q_21 ð4Þ 5 x_ð4Þ 5 0:00001; q ð4Þ 5 x_ð4Þ 5 0:000001; q ð4Þ 5 x_ð4Þ 5 0:000001

(7.51)

that minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; 3; 4 and expressed by the linear forms (7.37) giving, according to (7.17) and considering (7.51), their optimal value (in PLN): L_ð1Þ ð1ÞD0:570; L_ð2Þ ð1ÞD1:437; L_ð3Þ ð1ÞD1:980; L_ð4Þ ð1ÞD1:930:

(7.52)

From the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of the open and restricted sea waters: _ 5 Lð1Þ

4 X k51

L_ðkÞ ð1ÞD5:917:

(7.53)

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7.3.1.2 Gdynia and Karlskrona ports Taking into account the losses LiðkÞ ð1Þ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; during the time t 5 1 hour, associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; . . .; 5; at the i state rðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 given in (6.30)(6.34), the objective function defined by (7.1) for particular subareas Dk ; k 5 1; 2; . . .; 5 of Gdynia and Karlskrona ports takes the forms (the parts with qiðkÞ 5 0 are omitted): Lð1Þ ð1ÞD

30 X qið1Þ ULið1Þ ð1Þ 5 q1ð1Þ U0 1 q2ð1Þ U10; 260 1 q6ð1Þ U10; 260 1 q11 ð1Þ U12; 260; i51

Lð2Þ ð1ÞD

28 X i51

Lð3Þ ð1ÞD

28 X i51

Lð4Þ ð1ÞD

31 X i51

16 qið2Þ ULið2Þ ð1Þ 5 q1ð2Þ U0 1 q6ð2Þ U15; 260 1 q12 ð2Þ U20; 260 1 qð2Þ U25; 260 25 27 1 q21 ð2Þ U30; 260 1 qð2Þ U32; 260 1 qð2Þ U35; 260; 16 qið3Þ ULið3Þ ð1Þ 5 q1ð3Þ U0 1 q6ð3Þ U20; 260 1 q12 ð3Þ U25; 260 1 qð3Þ U30; 260

(7.54)

25 27 1 q21 ð3Þ U35; 260 1 qð3Þ U35; 260 1 qð3Þ U35; 260; 16 21 qið4Þ ULið4Þ ð1Þ 5 q1ð4Þ U0 1 q12 ð4Þ U20; 260 1 qð4Þ U30; 260 1 qð4Þ U35; 260 30 1 q28 ð4Þ U35; 260 1 qð4Þ U35; 260; 23 X Lð5Þ ð1ÞD qið5Þ ULið5Þ ð1Þ 5 q1ð5Þ U0: i51

We do not minimize the above objective function for the subarea D5 of Gdynia and Karlskrona ports as Lð5Þ ð1Þ 5 0 PLN. Considering the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 coming from experts and given by (7.38), and proceedings in the same way as in Section 7.3.1.1, we determined the optimal values q_iðkÞ ; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 of limit transient probabilities qiðkÞ ; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 that are also the same as those given by (7.51). Thus, we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; 3; 4 and expressed by the linear forms (7.54) giving, according to (7.17) and considering (7.51), their optimal value (in PLN): L_ð1Þ ð1ÞD1:149; L_ð2Þ ð1ÞD2:016; L_ð3Þ ð1ÞD2:559; L_ð4Þ ð1ÞD2:509:

(7.55)

From the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of Gdynia and Karlskrona ports 4 X _ 5 Lð1Þ L_ðkÞ ð1ÞD8:233: (7.56) k51

7.3.2 Minimization of critical infrastructure accident losses with considering climateweather change process impact The losses associated with the process of the environment degradation impacted by the climateweather change process at the Baltic Sea waters are determined in Section 6.3.2. In this section, based on the procedure and formulae given in Section 7.2, we minimize mean values of these losses of shipping critical infrastructure accident at the Baltic Sea open waters, restricted waters as well as Gdynia and Karlskrona ports.

7.3.2.1 Open sea waters In order to ensure the transparency of this section, we carry out step by step the procedure of minimizing the mean values of losses associated with the process of the environment degradation impacted by the climateweather change process, for example, subarea Dk, k 5 1 (air) of the Baltic Sea open waters.

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Taking into account the losses ½Lið1Þ ð1ÞðbÞ ; b 5 1; 2; . . .; 6; i 5 1; 2; . . .; 30; during the time t 5 1 hour, associated i with the process of the environment degradation in the subarea D1 at the state rð1Þ ; i 5 1; 2; . . .; 30; impacted by the climateweather change process CðtÞ; given in (6.46), the objective function defined by (7.18) for subarea D1 of the Baltic Sea open waters takes the form (the parts with qið1Þ 5 0 are omitted) Lð1Þ ð1ÞD

6 X 30 X

h iðbÞ qb Uqið1Þ U Lið1Þ ð1Þ 5 q1 q1ð1Þ U0 1 q1 q2ð1Þ U5000 1 q1 q6ð1Þ U5000 1 q1 q11 ð1Þ U7000

b51 i51

1 2 1 q2 q1ð1Þ U0 1 q2 q2ð1Þ U5000 1 q2 q6ð1Þ U5000 1 q2 q11 ð1Þ U7000 1 q3 qð1Þ U0 1 q3 qð1Þ U5000 6 11 1 2 6 1 q3 qð1Þ U5000 1 q3 qð1Þ U7000 1 q4 qð1Þ U0 1 q4 qð1Þ U10; 000 1 q4 qð1Þ U10; 000 1 q4 q11 ð1Þ U14; 000 1 2 6 11 1 1 q5 qð1Þ U0 1 q5 qð1Þ U10; 000 1 q5 qð1Þ U10; 000 1 q5 qð1Þ U14; 000 1 q6 qð1Þ U0 1 q6 q2ð1Þ U10; 000 1 q6 q6ð1Þ U10; 000 1 q6 q11 ð1Þ U14; 000:

(7.57)

Considering the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the open sea waters given by (6.53) as well as based on the lower q ið1Þ and upper _ q ið1Þ bounds of the unknown transient probabilities qið1Þ , i 5 1; 2; . . .; 30 coming from experts and given by (7.38), we calculate lower qb q ið1Þ and upper qb _ q ið1Þ i 5 1; 2; . . .; 30; b 5 1; 2; . . .; 6 bounds of the unknown _i i i transient probabilities (qb q ð1Þ and qb q ð1Þ are omitted if q ð1Þ 5 0 and _ q ið1Þ 5 0): q1 q 1ð1Þ 5 0:83384154; q2 q 1ð1Þ 5 0:14897169; q3 q 1ð1Þ 5 0; q3 q 2ð1Þ 5 0; q5 q 1ð1Þ 5 0:01499715; q6 q 1ð1Þ 5 0:00199962;

q1 q 2ð1Þ 5 0:000000000834; q1 q 6ð1Þ 5 0:00000834; q1 q 11 ð1Þ 5 0:00000834; q2 q 2ð1Þ 5 0:000000000149; q2 q 6ð1Þ 5 0:00000149; q2 q 11 ð1Þ 5 0:00000149; 6 11 1 2 6 q3 q ð1Þ 5 0; q3 q ð1Þ 5 0; q4 q ð1Þ 5 0; q4 q ð1Þ 5 0; q4 q ð1Þ 5 0; q4 q 11 ð1Þ 5 0; q5 q 2ð1Þ 5 0:000000000015; q5 q 6ð1Þ 5 0:00000015; q5 q 11 5 0:00000015; ð1Þ q6 q 2ð1Þ 5 0:000000000002; q6 q 6ð1Þ 5 0:00000002; q6 q 11 ð1Þ 5 0:00000002

and q1 _ q 1ð1Þ 5 0:83390826; q1 _ q 2ð1Þ 5 0:00000000834; q1 _ q 6ð1Þ 5 0:0000834; q 1ð1Þ 5 0:14898361; q2 _ q 2ð1Þ 5 0:00000000149; q2 _ q 6ð1Þ 5 0:0000149; q2 _ _1 _2 _6 _11 _1 q3 q ð1Þ 5 0; q3 q ð1Þ 5 0; q3 q ð1Þ 5 0; q3 q ð1Þ 5 0; q4 q ð1Þ 5 0; q4 _ q 2ð1Þ 5 0; q5 _ q 1ð1Þ 5 0:01499835; q5 _ q 2ð1Þ 5 0:00000000015; q5 _ q 6ð1Þ 5 0:0000015; _1 _2 _6 q6 q ð1Þ 5 0:00199978; q6 q ð1Þ 5 0:00000000002; q6 q ð1Þ 5 0:0000002;

q1 _ q 11 ð1Þ 5 0:0000834; _11 q2 q ð1Þ 5 0:0000149; q4 _ q 6ð1Þ 5 0; q4 _ q 11 ð1Þ 5 0; _11 q5 q ð1Þ 5 0:0000015; q6 _ q 11 ð1Þ 5 0:0000002:

Therefore according to (7.19) and (7.20), we assume the following bound constraints: 0:83384154 # q1 q1ð1Þ # 0:83390826; 0:000000000834 # q1 q2ð1Þ # 0:00000000834; 0:00000834 # q1 q6ð1Þ # 0:0000834; 1 2 0:00000834 # q1 q11 ð1Þ # 0:0000834; 0:14897169 # q2 qð1Þ # 0:14898361; 0:000000000149 # q2 qð1Þ # 0:00000000149; 6 11 1 0:00000149 # q2 qð1Þ # 0:0000149; 0:00000149 # q2 qð1Þ # 0:0000149; 0:01499715 # q5 qð1Þ # 0:01499835; 0:000000000015 # q5 q2ð1Þ # 0:00000000015; 0:00000015 # q5 q6ð1Þ # 0:0000015; 0:00000015 # q5 q11 ð1Þ # 0:0000015; 1 2 0:00199962 # q6 qð1Þ # 0:00199978; 0:000000000002 # q6 qð1Þ # 0:00000000002; 0:00000002 # q6 q6ð1Þ # 0:0000002; 0:0000000 # q6 q11 ð1Þ # 20:0000002; 6 30 XX qb qið1Þ 5 1: b51 i51

(7.58) Now, before we find optimal values qb q_ið1Þ of the limit transient probabilities qb qið1Þ ; i 5 1; 2; . . .; 30; b 5 1; 2; . . .; 6 that minimize the objective function (7.57), we arrange the mean values of the losses ½Lið1Þ ð1ÞðbÞ ; i 5 1; 2; . . .; 30; i ; b 5 1; 2; . . .; 6; associated with the process of the environment degradation in the subarea D1 at the state rð1Þ i 5 1; 2; . . .; 30; in increasing order (½Lið1Þ ð1ÞðbÞ are omitted if qið1Þ 5 0)

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h

ið1Þ h ið2Þ h ið3Þ h ið4Þ h ið5Þ h ið6Þ L1ð1Þ ð1Þ # L1ð1Þ ð1Þ # L1ð1Þ ð1Þ # L1ð1Þ ð1Þ # L1ð1Þ ð1Þ # L1ð1Þ ð1Þ h ið1Þ h ið1Þ h ið2Þ h ið2Þ h ið3Þ h ið3Þ # L2ð1Þ ð1Þ # L6ð1Þ ð1Þ # L2ð1Þ ð1Þ # L6ð1Þ ð1Þ # L2ð1Þ ð1Þ # L6ð1Þ ð1Þ h ið1Þ h ið2Þ h ið3Þ h ið4Þ h ið4Þ h ið5Þ # L11 # L11 # L11 # L2ð1Þ ð1Þ # L6ð1Þ ð1Þ # L2ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ h ið5Þ h ið6Þ h ið6Þ h ið4Þ h ið5Þ h ið6Þ 11 11 # L6ð1Þ ð1Þ # L2ð1Þ ð1Þ # L6ð1Þ ð1Þ # L11 ð1Þ # L ð1Þ # L ð1Þ : ð1Þ ð1Þ ð1Þ

Next, according to (7.21), we substitute (xið1Þ are omitted, if qið1Þ 5 0) x1ð1Þ 5 q1 q1ð1Þ ; x2ð1Þ 5 q2 q1ð1Þ ; x3ð1Þ 5 q3 q1ð1Þ ; x4ð1Þ 5 q4 q1ð1Þ ; x5ð1Þ 5 q5 q1ð1Þ ; x6ð1Þ 5 q6 q1ð1Þ ; x7ð1Þ 5 q1 q2ð1Þ ; x8ð1Þ 5 q1 q6ð1Þ ; 6 11 2 12 6 13 11 14 11 15 11 16 2 x9ð1Þ 5 q2 q2ð1Þ ; x10 ð1Þ 5 q2 qð1Þ ; xð1Þ 5 q3 qð1Þ ; xð1Þ 5 q3 qð1Þ ; xð1Þ 5 q1 qð1Þ ; xð1Þ 5 q2 qð1Þ ; xð1Þ 5 q3 qð1Þ ; xð1Þ 5 q4 qð1Þ ; (7.59) 17 6 18 2 19 6 20 2 21 6 22 11 23 11 24 xð1Þ 5 q4 qð1Þ ; xð1Þ 5 q5 qð1Þ ; xð1Þ 5 q5 qð1Þ ; xð1Þ 5 q6 qð1Þ ; xð1Þ 5 q6 qð1Þ ; xð1Þ 5 q4 qð1Þ ; xð1Þ 5 q5 qð1Þ ; xð1Þ 5 q6 q11 ð1Þ and x 5ð1Þ

x 1ð1Þ 5 q1 q 1ð1Þ 5 0:83384154; x 2ð1Þ 5 q2 q 1ð1Þ 5 0:14897169; x 3ð1Þ 5 q3 q 1ð1Þ 5 0; x 4ð1Þ 5 q4 q 1ð1Þ 5 0; 5 0:01499715; x 6ð1Þ 5 q6 q 1ð1Þ 5 0:00199962; x 7ð1Þ 5 q1 q 2ð1Þ 5 0:000000000834; x 8ð1Þ 5 q1 q 6ð1Þ 5 0:00000834;  6ð1Þ 5 0:00000149; x 11  2ð1Þ 5 0; x 9ð1Þ 5 q2 q 2ð1Þ 5 0:000000000149; x 10 ð1Þ 5 q2 q ð1Þ 5 q3 q 12 6 13 11 14 11 x ð1Þ 5 q3 q ð1Þ 5 0; x ð1Þ 5 q1 q ð1Þ 5 0:00000834; x ð1Þ 5 q2 q ð1Þ 5 0:00000149; 15 16  2ð1Þ 5 0; x 17  6ð1Þ 5 0; x 18  2ð1Þ 5 0:000000000015; x ð1Þ 5 q3 q 11 ð1Þ 5 0; x ð1Þ 5 q4 q ð1Þ 5 q4 q ð1Þ 5 q5 q 19 6 20 2 21 6  11 x ð1Þ 5 q5 q ð1Þ 5 0:00000015; x ð1Þ 5 q6 q ð1Þ 5 0:000000000002; x ð1Þ 5 q6 q ð1Þ 5 0:00000002; x 22 ð1Þ 5 q4 q ð1Þ 5 0; 11 24 11    x 23 5 q 5 0:00000015; x 5 q 5 0:00000002; q q 5 ð1Þ 6 ð1Þ ð1Þ ð1Þ _1 x ð1Þ 5 q1 _ q 1ð1Þ 5 0:83390826; _ x 2ð1Þ 5 q2 _ q 1ð1Þ 5 0:14898361; _ x 3ð1Þ 5 q3 _ q 1ð1Þ 5 0; _1 _1 _1 _4 _5 _6 x ð1Þ 5 q4 q ð1Þ 5 0; x ð1Þ 5 q5 q ð1Þ 5 0:01499835; x ð1Þ 5 q6 q ð1Þ 5 0:00199978; _7 x ð1Þ 5 q1 _ q 2ð1Þ 5 0:00000000834; _ x 8ð1Þ 5 q1 _ q 6ð1Þ 5 0:0000834; _2 _6 _2 _9 _10 x ð1Þ 5 q2 q ð1Þ 5 0:00000000149; x ð1Þ 5 q2 q ð1Þ 5 0:0000149; _ x 11 ð1Þ 5 q3 q ð1Þ 5 0; _11 _11 _12 _14 x ð1Þ 5 q3 _ q 6ð1Þ 5 0; _ x 13 ð1Þ 5 q1 q ð1Þ 5 0:0000834; x ð1Þ 5 q2 q ð1Þ 5 0:0000149; _11 _2 _6 _15 _16 _17 _18 x ð1Þ 5 q3 q ð1Þ 5 0; x ð1Þ 5 q4 q ð1Þ 5 0; x ð1Þ 5 q4 q ð1Þ 5 0; x ð1Þ 5 q5 _ q 2ð1Þ 5 0:00000000015; _2 _19 x ð1Þ 5 q5 _ q 6ð1Þ 5 0:0000015; _ x 20 ð1Þ 5 q6 q ð1Þ 5 0:00000000002; _6 _11 _11 _11 _21 _22 _23 x ð1Þ 5 q6 q ð1Þ 5 0:0000002; x ð1Þ 5 q4 q ð1Þ 5 0; x ð1Þ 5 q5 q ð1Þ 5 0:0000015; _ x 24 ð1Þ 5 q6 q ð1Þ 5 0:0000002;

5 q5 q 1ð1Þ

(7.60) and we minimize with respect to the form

xjð1Þ ;

j 5 1; 2; . . .; 180 the linear form (7.57) that according to (7.22) and (7.23) takes

Lð1Þ ð1Þ 5 x1ð1Þ U0 1 x2ð1Þ U0 1 x3ð1Þ U0 1 x4ð1Þ U0 1 x5ð1Þ U0 1 x6ð1Þ U0 1 x7ð1Þ U5000 1 x8ð1Þ U5000 11 12 13 14 1 x9ð1Þ U5000 1 x10 ð1Þ U5000 1 xð1Þ U5000 1 xð1Þ U5000 1 xð1Þ U7000 1 xð1Þ U7000 15 16 17 18 19 1 xð1Þ U7000 1 xð1Þ U10; 000 1 xð1Þ U10; 000 1 xð1Þ U10; 000 1 xð1Þ U10; 000 21 22 23 24 1 x20 ð1Þ U10; 000 1 xð1Þ U10; 000 1 xð1Þ U14; 000 1 xð1Þ U14; 000 1 xð1Þ U14; 000

(7.61)

with the following bound constraints: 0:83384154 # x1ð1Þ # 0:83390826; 0:14897169 # x2ð1Þ # 0:14898361; 0:01499715 # x5ð1Þ # 0:01499835; 0:00199962 # x6ð1Þ # 0:00199978; 0:000000000834 # x7ð1Þ # 0:00000000834; 0:00000834 # x8ð1Þ # 0:0000834; 0:000000000149 # x9ð1Þ # 0:00000000149; 0:00000149 # x10 ð1Þ # 0:0000149; 14 0:00000834 # x13 # 0:0000834; 0:00000149 # x # 0:0000149; ð1Þ ð1Þ 19 0:000000000015 # x18 # 0:00000000015; 0:00000015 # x ð1Þ ð1Þ # 0:0000015; 20 0:00000000000 # xð1Þ # 20:00000000002; 0:00000002 # x21 ð1Þ # 0:0000002; 24 0:00000015 # x23 # 0:0000015; 0:00000002 # x # 0:0000002; ð1Þ ð1Þ 180 X j xð1Þ 5 1: j51

(7.62)

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According to (7.25), we calculate x ð1Þ 5

180 X

x jð1Þ D0:999830001;

j51

(7.63)

ybð1Þ 5 1 2 x ð1Þ 5 1  0:999830001 5 0:000169999 and according to (7.26), we find x ð1Þ0 5 0; _ x ð1Þ0 5 0; _ x ð1Þ0 2 x ð1Þ0 5 0; _ x ð1Þ1 5 0:83384154; x ð1Þ1 5 0:83390826; _ x ð1Þ1 2 x ð1Þ1 5 0:00006672; x ð1Þ2 5 0:98281323; _ x ð1Þ2 5 0:98289187; _ x ð1Þ2 2 x ð1Þ2 5 0:00007864; x ð1Þ3 5 0:98281323; _ x ð1Þ3 5 0; 98289187; _ x ð1Þ3 2 x ð1Þ3 5 0:00007864; x ð1Þ4 5 0:98281323; _ x ð1Þ4 5 0; 98289187; _ x ð1Þ4 2 x ð1Þ4 5 0:00007864; x ð1Þ5 5 0:99781038; _ x ð1Þ5 5 0; 99789022; _ x ð1Þ5 2 x ð1Þ5 5 0:00007984; x ð1Þ6 5 0:99981; _ x ð1Þ6 5 0; 99989; _ x ð1Þ6 2 x ð1Þ6 5 0:00008; x ð1Þ7 5 0:999810000834; _ x ð1Þ7 5 0:99989000834; _ x ð1Þ7 2 x ð1Þ7 5 0:000080007506; _ x ð1Þ8 5 0:999818340834; x ð1Þ8 5 0:99997340834; _ x ð1Þ8 2 x ð1Þ8 5 0:000155067506; x ð1Þ9 5 0:999818340983; _ x ð1Þ9 5 0:99997340983; _ x ð1Þ9 2 x ð1Þ9 5 0:000155068847; x ð1Þ10 5 0:999819830983; _ x ð1Þ10 5 0:99998830983; _ x ð1Þ10 2 x ð1Þ10 5 0:000168478847; x ð1Þ11 5 0:999819830983; _ x ð1Þ11 5 0:99998830983; _ x ð1Þ11 2 x ð1Þ11 5 0:000168478847; x ð1Þ12 5 0:999819830983; _ x ð1Þ12 5 0:99998830983; _ x ð1Þ12 2 x ð1Þ12 5 0:000168478847; x ð1Þ13 5 0:999828170983; _ x ð1Þ13 5 1:00007170983; _ x ð1Þ13 2 x ð1Þ13 5 0:000243538847:

(7.64)

From the above, as according to (7.43), the appropriate inequality (7.27) takes the form _ x ð1ÞJ

2 x ð1ÞJ , 0:000169999;

(7.65)

then it follows that the largest value JAf0; 1; . . .; 180g such that this inequality holds is J 5 12. Therefore we fix the optimal solution that minimizes linear function (7.61) according to the appropriate rule (7.29)(7.31). Namely, we get x 1ð1Þ 5 0:83390826; x_2ð1Þ 5 _ x 2ð1Þ 5 0:14898361; x_3ð1Þ 5 _ x 3ð1Þ 5 0; x_4ð1Þ 5 _ x 4ð1Þ 5 0; x_1ð1Þ 5 _ _5 _6 _7 5 6 7 x_ð1Þ 5 x ð1Þ 5 0:01499835; x_ð1Þ 5 x ð1Þ 5 0:00199978; x_ð1Þ 5 x ð1Þ 5 0:00000000834; _8 _10 _11 8 11 x 9ð1Þ 5 0:00000000149; x_10 x_ð1Þ 5 x ð1Þ 5 0:0000834; x_9ð1Þ 5 _ ð1Þ 5 x ð1Þ 5 0:0000149; x_ð1Þ 5 x ð1Þ 5 0; _12 _ 13 12 13 x_ð1Þ 5 x ð1Þ 5 0; x_ð1Þ 5 ybð1Þ 2 x ð1Þ12 1 x ð1Þ12 1 x ð1Þ D0:000169999  0:99998830983 1 0; 999819830983 1 0:00000834 5 0:000009860153; 14 15 16 17 15 16 17  x_14 5 x ð1Þ ð1Þ 5 0:00000149; x_ð1Þ 5 x ð1Þ 5 0; x_ð1Þ 5 x ð1Þ 5 0; x_ð1Þ 5 x ð1Þ 5 0; 18 19 19 x_18 ð1Þ 5 x ð1Þ 5 0:000000000015; x_ð1Þ 5 x ð1Þ 5 0:00000015; 20 21 22 20 21 x_ð1Þ 5 x ð1Þ 5 0:000000000002; x_ð1Þ 5 x ð1Þ 5 0:00000002; x_22 ð1Þ 5 x ð1Þ 5 0; 23 24 24 x_23 ð1Þ 5 x ð1Þ 5 0:00000015; x_ð1Þ 5 x ð1Þ 5 0:00000002:

(7.66)

Finally, after making the substitution inverse to (7.59), we get the optimal transient probabilities 13 q1 q_1ð1Þ 5 x_1ð1Þ 5 0:83390826; q1 q_2ð1Þ 5 x_7ð1Þ 5 0:00000000834; q1 q_6ð1Þ 5 x_8ð1Þ 5 0:0000834; q1 q_11 ð1Þ 5 x_ð1Þ 5 0:000009860153; 14 _11 q2 q_1ð1Þ 5 x_2ð1Þ 5 0:14898361; q2 q_2ð1Þ 5 x_9ð1Þ 5 0:00000000149; q2 q_6ð1Þ 5 x_10 ð1Þ 5 x_ð1Þ 5 0:00000149; ð1Þ 5 0:0000149; q2 q 1 3 2 11 6 12 11 15 1 4 q3 q_ð1Þ 5 x_ð1Þ 5 0; q3 q_ð1Þ 5 x_ð1Þ 5 0; q3 q_ð1Þ 5 x_ð1Þ 5 0; q3 q_ð1Þ 5 x_ð1Þ 5 0; q4 q_ð1Þ 5 x_ð1Þ 5 0; q4 q_2ð1Þ 5 x_16 ð1Þ 5 0; 6 17 11 22 1 5 2 18 q4 q_ð1Þ 5 x_ð1Þ 5 0; q4 q_ð1Þ 5 x_ð1Þ 5 0; q5 q_ð1Þ 5 x_ð1Þ 5 0:01499835; q5 q_ð1Þ 5 x_ð1Þ 5 0:000000000015; 23 _11 _1ð1Þ 5 x_6ð1Þ 5 0:00199978; q6 q_2ð1Þ 5 x_20 q5 q_6ð1Þ 5 x_19 ð1Þ 5 x_ð1Þ 5 0:00000015; q6 q ð1Þ 5 0:000000000002; ð1Þ 5 0:00000015; q5 q 6 21 11 24 q6 q_ð1Þ 5 x_ð1Þ 5 0:00000002; q6 q_ð1Þ 5 x_ð1Þ 5 0:00000002;

(7.67) that minimize mean values of the losses associated with the process of the environment degradation in the subarea D1 impacted by the climateweather change process and expressed by the linear form (7.57) giving, according to (7.34) and considering (7.67), its optimal value (in PLN) L_ ð1Þ ð1ÞD0:575:

(7.68)

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Thus considering (7.52) and (7.68) and according to (7.35), the indicator RI_ð1Þ of the environment in the subarea D1 resilience to the losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amounts to RI_ð1Þ 5

0:570 5 0:991 5 99:1%: 0:575

(7.69)

For remaining subareas Dk ; k 5 2; 3; 4; 5; we proceed in an analogous way. The objective function defined by (7.18) for these subareas takes the forms (the parts with qiðkÞ 5 0 are omitted): Lð2Þ ð1ÞD

6 X 28 h iðbÞ X qb Uqið2Þ U Lið2Þ ð1Þ 5 q1 q1ð2Þ U0 1 q1 q6ð2Þ U10; 000 1 q1 q12 ð2Þ U15; 000 b51

i51

21 25 27 1 1 q1 q16 ð2Þ U20; 000 1 q1 qð2Þ U25; 000 1 q1 qð2Þ U27; 000 1 q1 qð2Þ U30; 000 1 q2 qð2Þ U0 6 12 16 21 25 1 q2 qð2Þ U20; 000 1 q2 qð2Þ U30; 000 1 q2 qð2Þ U40; 000 1 q2 qð2Þ U50; 000 1 q2 qð2Þ U54; 000 1 6 12 16 1 q2 q27 ð2Þ U60; 000 1 q3 qð2Þ U0 1 q3 qð2Þ U25; 000 1 q3 qð2Þ U37; 500 1 q3 qð2Þ U50; 000 21 25 27 1 1 q3 qð2Þ U62; 500 1 q3 qð2Þ U67; 500 1 q3 qð2Þ U75; 000 1 q4 qð2Þ U0 1 q4 q6ð2Þ U18; 000 16 21 25 27 1 q4 q12 ð2Þ U27; 000 1 q4 qð2Þ U36; 000 1 q4 qð2Þ U45; 000 1 q4 qð2Þ U48; 600 1 q4 qð2Þ U54; 000 1 6 12 16 21 1 q5 qð2Þ U0 1 q5 qð2Þ U25; 000 1 q5 qð2Þ U37; 500 1 q5 qð2Þ U50; 000 1 q5 qð2Þ U62; 500 27 1 6 12 1 q5 q25 ð2Þ U67; 500 1 q5 qð2Þ U75; 000 1 q6 qð2Þ U0 1 q6 qð2Þ U30; 000 1 q6 qð2Þ U45; 000 16 21 25 27 1 q6 qð2Þ U60; 000 1 q6 qð2Þ U75; 000 1 q6 qð2Þ U81; 000 1 q6 qð2Þ U90; 000;

Lð3Þ ð1ÞD

6 X 28 h iðbÞ X qb Uqið3Þ U Lið3Þ ð1Þ 5 q1 q1ð3Þ U0 1 q1 q6ð3Þ U15; 000 1 q1 q12 ð3Þ U20; 000 b51

i51

21 25 27 1 1 q1 q16 ð3Þ U25; 000 1 q1 qð3Þ U30; 000 1 q1 qð3Þ U30; 000 1 q1 qð3Þ U30; 000 1 q2 qð3Þ U0 16 21 25 1 q2 q6ð3Þ U30; 000q2 q12 ð3Þ U40; 000 1 q2 qð3Þ U50; 000 1 q2 qð3Þ U60; 000 1 q2 qð3Þ U60; 000 27 1 6 12 16 1 q2 qð3Þ U60; 000 1 q3 qð3Þ U0 1 q3 qð3Þ U45; 000 1 q3 qð3Þ U60; 000 1 q3 qð3Þ U75; 000 25 27 1 6 1 q3 q21 ð3Þ U90; 000 1 q3 qð3Þ U90; 000 1 q3 qð3Þ U90; 000 1 q4 qð3Þ U0 1 q4 qð3Þ U15; 000 12 16 21 25 1 q4 qð3Þ U20; 000 1 q4 qð3Þ U25; 000 1 q4 qð3Þ U30; 000 1 q4 qð3Þ U30; 000 1 q4 q27 ð3Þ U30; 000 1 6 12 16 21 1 q5 qð3Þ U0 1 q5 qð3Þ U30; 000 1 q5 qð3Þ U40; 000 1 q5 qð3Þ U50; 000 1 q5 qð3Þ U60; 000 27 1 6 12 1 q5 q25 ð3Þ U60; 000 1 q5 qð3Þ U60; 000 1 q6 qð3Þ U0 1 q6 qð3Þ U45; 000 1 q6 qð3Þ U60; 000 21 25 27 1 q6 q16 ð3Þ U75; 000 1 q6 qð3Þ U90; 000 1 q6 qð3Þ U90; 000 1 q6 qð3Þ U90; 000;

Lð4Þ ð1ÞD

(7.70)

(7.71)

6 X 31 h iðbÞ X 16 qb Uqið4Þ U Lið4Þ ð1Þ 5 q1 q1ð4Þ U0 1 q1 q12 ð4Þ U15; 000 1 q1 qð4Þ U25; 000 b51

i51

28 30 1 12 1 q1 q21 ð4Þ U30; 000 1 q1 qð4Þ U30; 000 1 q1 qð4Þ U30; 000 1 q2 qð4Þ U0 1 q2 qð4Þ U15; 000 16 21 28 30 1 q2 qð4Þ U25; 000 1 q2 qð4Þ U30; 000 1 q2 qð4Þ U30; 000 1 q2 qð4Þ U30; 000 1 q3 q1ð4Þ U0 16 21 28 30 1 q3 q12 ð4Þ U15; 000 1 q3 qð4Þ U25; 000 1 q3 qð4Þ U30; 000 1 q3 qð4Þ U30; 000 1 q3 qð4Þ U30; 000 1 12 16 21 28 1 q4 qð4Þ U0 1 q4 qð4Þ U15; 000 1 q4 qð4Þ U25; 000 1 q4 qð4Þ U30; 000 1 q4 qð4Þ U30; 000 1 12 16 21 1 q4 q30 ð4Þ U30; 000 1 q5 qð4Þ U0 1 q5 qð4Þ U15; 000 1 q5 qð4Þ U25; 000 1 q5 qð4Þ U30; 000 28 30 1 12 1 q5 qð4Þ U30; 000 1 q5 qð4Þ U30; 000 1 q6 qð4Þ U0 1 q6 qð4Þ U15; 000 1 q6 q16 ð4Þ U25; 000 21 28 30 1 q6 qð4Þ U30; 000 1 q6 qð4Þ U30; 000 1 q6 qð4Þ U30; 000;

Lð5Þ ð1ÞD

6 X 23 X b51

i51

h iðbÞ qb Uqið5Þ U Lið5Þ ð1Þ 5 q1 q1ð5Þ U0 1 q2 q1ð5Þ U0 1 q3 q1ð5Þ U0

(7.72)

(7.73)

1 q4 q1ð5Þ U0 1 q5 q1ð5Þ U0 1 q6 q1ð5Þ U0:

We do not minimize the objective function (7.73) for the subarea D5 of the open sea waters as Lð5Þ ð1Þ 5 0 PLN. Considering the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the open sea waters given by (6.53) as well as based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘2 5 28; ‘3 5 28; ‘4 5 31 coming from experts and given by (7.38), and proceedings in the same way as earlier, we determined the optimal values qb q_iðkÞ ; b 5 1; 2; . . .; 6; k 5 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘2 5 28; ‘3 5 28; ‘4 5 31 of limit transient

Optimization of critical infrastructure accident losses Chapter | 7

157

probabilities qb qiðkÞ ; b 5 1; 2; . . .; 6; k 5 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘2 5 28; ‘3 5 28; ‘4 5 31: Thus we minimize the mean values of losses associated with the process of the environment degradation in the subarea Dk, k 5 2; 3; 4 impacted by the climateweather change process CðtÞ and expressed by the linear forms (7.70)(7.72) giving, according to (7.34), their optimal value (in PLN) L_ ð2Þ ð1ÞD1:570; L_ ð3Þ ð1ÞD2:141; L_ ð4Þ ð1ÞD1:930:

(7.74)

Finally, considering (7.68) and (7.74), we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of the open sea waters, impacted by the climateweather change process CðtÞ _ 5 Lð1Þ

4 X

L_ ðkÞ ð1ÞD6:216:

(7.75)

k51

Thus considering (7.52) and (7.74) and according to (7.35), the indicators RI_ðkÞ of the environment in the subarea Dk ; k 5 2; 3; 4 resilience to the losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amount to RI_ð2Þ 5 0:915; RI_ð3Þ 5 0:925; RI_ð4Þ 5 1;

(7.76)

and considering (7.53) and (7.75) and according to (7.36), the indicator RI_ of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amounts to RI_ 5 0:952:

(7.77)

7.3.2.2 Restricted sea waters We optimize the mean value of losses impacted by the climateweather change process C(t) of shipping critical infrastructure accident for the Baltic Sea restricted waters in the same abovementioned way, as presented in Section 7.3.2.1. The objective function defined by (7.18) for particular subareas Dk, k 5 1; 2; . . .; 5 of the Baltic Sea restricted waters takes the same forms as those of the Baltic Sea open waters given by (7.57) and (7.70)(7.73). We also do not minimize the objective function (7.73) for the subarea D5 of the restricted sea waters as Lð5Þ ð1Þ 5 0 PLN. Considering the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the restricted sea waters given by (6.58) as well as based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 coming from experts and given by (7.38), and proceedings in the same way as in Section 7.3.2.1, we determined the optimal values qb q_iðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 of limit transient probabilities qb qiðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31: Thus we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; 3; 4 impacted by the climateweather change process CðtÞ and expressed by the linear forms (7.57) and (7.70)(7.72) giving, according to (7.34), their optimal value (in PLN) L_ ð1Þ ð1ÞD0:570; L_ ð2Þ ð1ÞD1:577; L_ ð3Þ ð1ÞD2:157; L_ ð4Þ ð1ÞD1:930:

(7.78)

Finally, considering (7.78), we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of the restricted sea waters, impacted by the climateweather change process CðtÞ _ 5 Lð1Þ

4 X

L_ ðkÞ ð1ÞD6:234:

(7.79)

k51

Thus considering (7.52) and (7.78) and according to (7.35), the indicators RI_ðkÞ of the environment in the subarea Dk ; k 5 1; 2; 3; 4 resilience to the losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amount to RI_ð1Þ 5 1; RI_ð2Þ 5 0:911; RI_ð3Þ 5 0:918; RI_ð4Þ 5 1;

(7.80)

158

Consequences of Maritime Critical Infrastructure Accidents

and considering (7.53) and (7.79) and according to (7.36), the indicator RI_ of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amounts to RI_ 5 0:949:

(7.81)

7.3.2.3 Gdynia and Karlskrona ports We optimize the mean value of losses impacted by the climateweather change process CðtÞ of shipping critical infrastructure accident for Gdynia and Karlskrona ports in the same way, presented in Section 7.3.2.1. The objective function defined by (7.18) for particular subareas Dk ; k 5 1; 2; . . .; 5 of Gdynia and Karlskrona ports takes the forms (the parts with qiðkÞ 5 0 are omitted): Lð1Þ ð1ÞD

6 X 30 X

h iðbÞ qb Uqið1Þ U Lið1Þ ð1Þ 5 q1 q1ð1Þ U0 1 q1 q2ð1Þ U10; 260 1 q1 q6ð1Þ U10; 260 1 q1 q11 ð1Þ U12; 260

b51 i51

1 2 1 q2 q1ð1Þ U0 1 q2 q2ð1Þ U20; 520 1 q2 q6ð1Þ U20; 520 1 q2 q11 ð1Þ U24; 520 1 q3 qð1Þ U0 1 q3 qð1Þ U10; 260 6 11 1 2 6 1 q3 qð1Þ U10; 260 1 q3 qð1Þ U12; 260 1 q4 qð1Þ U0 1 q4 qð1Þ U20; 520 1 q4 qð1Þ U20; 520 1 q4 q11 ð1Þ U24; 520 1 2 6 11 1 2 1 q5 qð1Þ U0 1 q5 qð1Þ U10; 260 1 q5 qð1Þ U10; 260 1 q5 qð1Þ U12; 260 1 q6 qð1Þ U0 1 q6 qð1Þ U20; 520 1 q6 q6ð1Þ U20; 520 1 q6 q11 ð1Þ U24; 520;

Lð2Þ ð1ÞD

6 X 28 h iðbÞ X qb Uqið2Þ U Lið2Þ ð1Þ 5 q1 q1ð2Þ U0 1 q1 q6ð2Þ U15; 260 1 q1 q12 ð2Þ U20; 260 b51

i51

21 25 27 1 1 q1 q16 ð2Þ U25; 260 1 q1 qð2Þ U30; 260 1 q1 qð2Þ U32; 260 1 q1 qð2Þ U35; 260 1 q2 qð2Þ U0 16 21 25 1 q2 q6ð2Þ U30; 520 1 q2 q12 ð2Þ U40; 520 1 q2 qð2Þ U40; 520 1 q2 qð2Þ U60; 520 1 q2 qð2Þ U64; 520 27 1 6 12 16 1 q2 qð2Þ U70; 520 1 q3 qð2Þ U0 1 q3 qð2Þ U15; 260 1 q3 qð2Þ U20; 260 1 q3 qð2Þ U25; 260 25 27 1 6 1 q3 q21 ð2Þ U30; 260 1 q3 qð2Þ U32; 260 1 q3 qð2Þ U35; 260 1 q4 qð2Þ U0 1 q4 qð2Þ U30; 520 12 16 21 25 1 q4 qð2Þ U40; 520 1 q4 qð2Þ U40; 520 1 q4 qð2Þ U60; 520 1 q4 qð2Þ U64; 520 1 q4 q27 ð2Þ U70; 520 1 6 12 16 21 1 q5 qð2Þ U0 1 q5 qð2Þ U15; 260 1 q5 qð2Þ U20; 260 1 q5 qð2Þ U25; 260 1 q5 qð2Þ U30; 260 27 1 6 12 1 q5 q25 ð2Þ U32; 260 1 q5 qð2Þ U35; 260 1 q6 qð2Þ U0 1 q6 qð2Þ U30; 520 1 q6 qð2Þ U40; 520 21 25 27 1 q6 q16 ð2Þ U40; 520 1 q6 qð2Þ U60; 520 1 q6 qð2Þ U64; 520 1 q6 qð2Þ U70; 520;

Lð3Þ ð1ÞD

(7.83)

6 X 28 h iðbÞ X qb Uqið3Þ U Lið3Þ ð1Þ 5 q1 q1ð3Þ U0 1 q1 q6ð3Þ U20; 260 1 q1 q12 ð3Þ U25; 260 b51

i51

21 25 27 1 1 q1 q16 ð3Þ U30; 260 1 q1 qð3Þ U35; 260 1 q1 qð3Þ U35; 260 1 q1 qð3Þ U35; 260 1 q2 qð3Þ U0 16 21 25 1 q2 q6ð3Þ U30; 390 1 q2 q12 ð3Þ U37; 890 1 q2 qð3Þ U45; 390 1 q2 qð3Þ U52; 890 1 q2 qð3Þ U52; 890 27 1 6 12 16 1 q2 qð3Þ U52; 890 1 q3 qð3Þ U0 1 q3 qð3Þ U20; 260 1 q3 qð3Þ U25; 260 1 q3 qð3Þ U30; 260 25 27 1 6 1 q3 q21 ð3Þ U35; 260 1 q3 qð3Þ U35; 260 1 q3 qð3Þ U35; 260 1 q4 qð3Þ U0 1 q4 qð3Þ U30; 390 12 16 21 25 1 q4 qð3Þ U37; 890 1 q4 qð3Þ U45; 390 1 q4 qð3Þ U52; 890 1 q4 qð3Þ U52; 890 1 q4 q27 ð3Þ U52; 890 16 21 1 q5 q1ð3Þ U0 1 q5 q6ð3Þ U20; 260 1 q5 q12 U25; 260 1 q q U30; 260 1 q q U35; 260 5 ð3Þ 5 ð3Þ ð3Þ 25 27 1 6 12 1 q5 qð3Þ U35; 260 1 q5 qð3Þ U35; 260 1 q6 qð3Þ U0 1 q6 qð3Þ U30; 390 1 q6 qð3Þ U37; 890 21 25 27 1 q6 q16 ð3Þ U45; 390 1 q6 qð3Þ U52; 890 1 q6 qð3Þ U52; 890 1 q6 qð3Þ U52; 890;

Lð4Þ ð1ÞD

(7.82)

(7.84)

6 X 31 h iðbÞ X 16 qb Uqið4Þ U Lið4Þ ð1Þ 5 q1 q1ð4Þ U0 1 q1 q12 ð4Þ U20; 260 1 q1 qð4Þ U30; 260 b51

i51

28 30 1 12 1 q1 q21 ð4Þ U35; 260 1 q1 qð4Þ U35; 260 1 q1 qð4Þ U35; 260 1 q2 qð4Þ U0 1 q2 qð4Þ U20; 260 16 21 28 30 1 q2 qð4Þ U30; 260 1 q2 qð4Þ U35; 260 1 q2 qð4Þ U35; 260 1 q2 qð4Þ U35; 260 1 q3 q1ð4Þ U0 16 21 28 30 1 q3 q12 ð4Þ U20; 260 1 q3 qð4Þ U30; 260 1 q3 qð4Þ U35; 260 1 q3 qð4Þ U35; 260 1 q3 qð4Þ U35; 260 1 12 16 21 28 1 q4 qð4Þ U0 1 q4 qð4Þ U20; 260 1 q4 qð4Þ U30; 260 1 q4 qð4Þ U35; 260 1 q4 qð4Þ U35; 260 1 12 16 21 1 q4 q30 ð4Þ U35; 260 1 q5 qð4Þ U0 1 q5 qð4Þ U20; 260 1 q5 qð4Þ U30; 260 1 q5 qð4Þ U35; 260 28 30 1 12 1 q5 qð4Þ U35; 260 1 q5 qð4Þ U35; 260 1 q6 qð4Þ U0 1 q6 qð4Þ U20; 260 1 q6 q16 ð4Þ U30; 260 28 30 1 q6 q21 U35; 260 1 q q U35; 260 1 q q U35; 260; 6 ð4Þ 6 ð4Þ ð4Þ

(7.85)

Optimization of critical infrastructure accident losses Chapter | 7

Lð5Þ ð1ÞD

6 X 23 X b51

i51

h iðbÞ qb Uqið5Þ U Lið5Þ ð1Þ 5 q1 q1ð5Þ U0 1 q2 q1ð5Þ U0 1 q3 q1ð5Þ U0

159

(7.86)

1 q4 q1ð5Þ U0 1 q5 q1ð5Þ U0 1 q6 q1ð5Þ U0:

We do not minimize the objective function (7.86) for the subarea D5 of Gdynia and Karlskrona ports as Lð5Þ ð1Þ 5 0 PLN. Considering the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the Gdynia Port given by (6.73) as well as based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 coming from experts and given by (7.38), and proceedings in the same way as in Section 7.3.2.1, we determined the optimal values qb q_iðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 of limit transient probabilities qb qiðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31: Thus we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk, k 5 1,2,3,4 impacted by the climateweather change process C(t) and expressed by the linear forms (7.82)(7.85) giving, according to (7.34), their optimal value (in PLN) L_ ð1Þ ð1ÞD1:155; L_ ð2Þ ð1ÞD2:031; L_ ð3Þ ð1ÞD2:568; L_ ð4Þ ð1ÞD2:509:

(7.87)

Finally, considering (7.87), we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of the Gdynia Port impacted by the climateweather change process C(t) _ 5 Lð1Þ

4 X

L_ ðkÞ ð1ÞD8:263:

(7.88)

k51

Thus considering (7.55) and (7.87) and according to (7.35), the indicators RI_ðkÞ of the environment in the subarea Dk ; k 5 1; 2; 3; 4 resilience to the losses associated with the critical infrastructure accident related to the climate weather change, after the optimization, amount to RI_ð1Þ 5 0:995; RI_ð2Þ 5 0:993; RI_ð3Þ 5 0:997; RI_ð4Þ 5 1; (7.89) _ and considering (7.56) and (7.88) and according to (7.36), the indicator RI of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amounts to RI_ 5 0:996:

(7.90)

Similarly, considering the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process CðtÞ at the climateweather states cb ; b 5 1; 2; . . .; 6 for the Karlskrona Port given by (6.78) as well as based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; coming from experts and given by (7.38), and proceedings in the same way as in Section 7.3.2.1, we determined the optimal values qb q_iðkÞ ; b 5 1; 2; . . .; 6 k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31 of limit transient probabilities qb qiðkÞ ; b 5 1; 2; . . .; 6 k 5 1; 2; 3; 4; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31: Thus we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; 3; 4 impacted by the climateweather change process C(t) and expressed by the linear forms (7.82)(7.85) giving, according to (7.34), their optimal value (in PLN) L_ ð1Þ ð1ÞD1:157; L_ ð2Þ ð1ÞD2:037; L_ ð3Þ ð1ÞD2:572; L_ ð4Þ ð1ÞD2:509:

(7.91)

Finally, considering (7.91), we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of the Karlskrona Port impacted by the climateweather change process C(t) _ 5 Lð1Þ

4 X

L_ ðkÞ ð1ÞD8:275:

(7.92)

k51

Thus considering (7.55) and (7.91) and according to (7.35), the indicators RI_ðkÞ of the environment in the subarea Dk ; k 5 1; 2; 3; 4 resilience to the losses associated with the critical infrastructure accident related to the climate weather change, after the optimization, amount to RI_ð1Þ 5 0:993; RI_ð2Þ 5 0:990; RI_ð3Þ 5 0:995; RI_ð4Þ 5 1;

(7.93)

160

Consequences of Maritime Critical Infrastructure Accidents

and considering (7.56) and (7.92) and according to (7.36), the indicator RI_ of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amounts to RI_ 5 0:995:

(7.94)

7.3.3 Minimization of critical infrastructure accident losses without considering climateweather change process impact—extension based on wider data collected at world sea waters Based on the procedure and formulae given in Section 7.1, proceeding in the same way as in Section 7.3.1, we optimize the losses of shipping critical infrastructure accident at the Baltic Sea open waters, restricted waters as well as Gdynia and Karlskrona ports using the wider data collected at the world sea waters. Taking into account the losses LiðkÞ ð1Þ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; during the time t 5 1 hour, associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; . . .; 5; at i the state rðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 given in (6.83)(6.88), the objective function defined by (7.1) for particular subareas Dk ; k 5 1; 2; . . .; 5 of the Baltic Sea open and restricted waters takes the forms (the parts with qiðkÞ 5 0 are omitted): Lð1Þ ð1ÞD

30 X

qið1Þ ULið1Þ ð1Þ 5 q1ð1Þ U0 1 q2ð1Þ U5000 1 q6ð1Þ U5000 1 q8ð1Þ U6500

i51

11 13 14 15 1 q10 ð1Þ U7500 1 qð1Þ U7000 1 qð1Þ U10; 000 1 qð1Þ U12; 000 1 qð1Þ U12; 500 17 18 19 20 21 1 qð1Þ U2000 1 qð1Þ U2500 1 qð1Þ U3000 1 qð1Þ U2000 1 qð1Þ U2000 23 24 25 26 1 q22 ð1Þ U8000 1 qð1Þ U3000 1 qð1Þ U4000 1 qð1Þ U10; 000 1 qð1Þ U3500 27 28 29 30 1 qð1Þ U7000 1 qð1Þ U11; 000 1 qð1Þ U2500 1 qð1Þ U4000;

Lð2Þ ð1ÞD

(7.95)

28 X qið2Þ ULið2Þ ð1Þ 5 q1ð2Þ U0 1 q2ð2Þ U7000 1 q5ð2Þ U5000 1 q6ð2Þ U10; 000 i51

12 13 14 15 1 q9ð2Þ U8000 1 q11 ð2Þ U12; 000 1 qð2Þ U15; 000 1 qð2Þ U15; 000 1 qð2Þ U17; 000 1 qð2Þ U18; 000 16 18 20 21 22 1 qð2Þ U20; 000 1 qð2Þ U22; 000 1 qð2Þ U24; 000 1 qð2Þ U25; 000 1 qð2Þ U25; 000 25 27 1 q23 ð2Þ U26; 000 1 qð2Þ U27; 000 1 qð2Þ U30; 000;

Lð3Þ ð1ÞD

28 X

(7.96)

qið3Þ ULið3Þ ð1Þ 5 q1ð3Þ U0 1 q2ð3Þ U13; 000 1 q6ð3Þ U15; 000

i51

12 13 14 1 q9ð3Þ U18; 000 1 q11 ð3Þ U20; 000 1 qð3Þ U20; 000 1 qð3Þ U20; 000 1 qð3Þ U23; 000 16 18 20 21 1 q15 ð3Þ U22; 000 1 qð3Þ U25; 000 1 qð3Þ U30; 000 1 qð3Þ U28; 000 1 qð3Þ U30; 000 22 23 25 27 1 qð3Þ U28; 000 1 qð3Þ U30; 000 1 qð3Þ U30; 000 1 qð3Þ U30; 000;

Lð4Þ ð1ÞD

31 X

(7.97)

qið4Þ ULið4Þ ð1Þ 5 q1ð4Þ U0 1 q2ð4Þ U12; 000 1 q6ð4Þ U13; 000

i51

12 13 14 1 q9ð4Þ U15; 000 1 q11 ð4Þ U18; 000 1 qð4Þ U15; 000 1 qð4Þ U15; 000 1 qð4Þ U17; 000 18 21 24 25 1 q16 ð4Þ U25; 000 1 qð4Þ U28; 000 1 qð4Þ U30; 000 1 qð4Þ U25; 000 1 qð4Þ U28; 000 26 28 30 1 qð4Þ U30; 000 1 qð4Þ U30; 000 1 qð4Þ U30; 000;

Lð5Þ ð1ÞD

23 X qið5Þ ULið5Þ ð1Þ 5 q1ð5Þ U0 1 q5ð5Þ U0 1 q6ð5Þ U0 1 q9ð5Þ U0 1 q11 ð5Þ U0 ðfor open sea watersÞ;

(7.98)

(7.99)

i51

Lð5Þ ð1ÞD

23 X qið5Þ ULið5Þ ð1Þ 5 q1ð5Þ U0 1 q5ð5Þ U1000 1 q6ð5Þ U5000 1 q9ð5Þ U10; 000 i51

1 q11 ð5Þ U15; 000

ðfor restricted sea watersÞ:

We do not minimize the objective function (7.99) for the subarea D5 of the open waters as Lð5Þ ð1Þ 5 0 PLN.

(7.100)

Optimization of critical infrastructure accident losses Chapter | 7

161

Next, based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 coming from experts and proceedings in the same way as in Section 7.3.1, we determine the optimal values q_iðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of limit transient probabilities qiðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23: Thus we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk ; k 5 1; 2; . . .; 5 and expressed by the linear forms (7.95)(7.100) giving, according to (7.17), their optimal value (in PLN): G

for open sea waters L_ð1Þ ð1ÞD1:007; L_ð2Þ ð1ÞD1:592; L_ð3Þ ð1ÞD2:859; L_ð4Þ ð1ÞD2:811;

(7.101)

and from the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of open sea waters _ 5 Lð1Þ

4 X

L_ðkÞ ð1ÞD8:269

(7.102)

k51 G

for restricted sea waters L_ð1Þ ð1ÞD1:007; L_ð2Þ ð1ÞD1:592; L_ð3Þ ð1ÞD2:859; L_ð4Þ ð1ÞD2:811; L_ð5Þ ð1ÞD1:030;

(7.103)

and from the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of restricted sea waters _ 5 Lð1Þ

5 X

L_ðkÞ ð1ÞD9:299:

(7.104)

k51

Similarly, taking into account the losses LiðkÞ ð1Þ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23; during the time t 5 1 hour, associated with the process of the environment degradation in the subarea Dk, i k 5 1; 2; . . .; 5; at the state rðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 given by (6.92)(6.96) the objective function defined by (7.1) for particular subareas Dk, k 5 1; 2; . . .; 5 of Gdynia and Karlskrona ports takes the forms (the parts with qiðkÞ 5 0 are omitted) Lð1Þ ð1ÞD

30 X

qið1Þ ULið1Þ ð1Þ 5 q1ð1Þ U0 1 q2ð1Þ U10; 260 1 q6ð1Þ U10; 260 1 q8ð1Þ U11; 760

i51

11 13 14 15 1 q10 ð1Þ U12; 760 1 qð1Þ U12; 260 1 qð1Þ U15; 260 1 qð1Þ U17; 260 1 qð1Þ U17; 760 18 19 20 21 1 q17 ð1Þ U7260 1 qð1Þ U7760 1 qð1Þ U8260 1 qð1Þ U7260 1 qð1Þ U7260 22 23 24 25 1 qð1Þ U13; 260 1 qð1Þ U8260 1 qð1Þ U9260 1 qð1Þ U15; 260 1 q26 ð1Þ U8760 27 28 29 30 1 qð1Þ U12; 260 1 qð1Þ U16; 260 1 qð1Þ U7760 1 qð1Þ U9260;

Lð2Þ ð1ÞD

28 X

(7.105)

qið2Þ ULið2Þ ð1Þ 5 q1ð2Þ U0 1 q2ð2Þ U12; 260 1 q5ð2Þ U10; 260 1 q6ð2Þ U15; 260

i51

12 13 14 1 q9ð2Þ U13; 260 1 q11 ð2Þ U17; 260 1 qð2Þ U20; 260 1 qð2Þ U20; 260 1 qð2Þ U22; 260 15 16 18 20 1 qð2Þ U23; 260 1 qð2Þ U25; 260 1 qð2Þ U27; 260 1 qð2Þ U29; 260 1 q21 ð2Þ U30; 260 23 25 27 U30; 260 1 q U31; 260 1 q U32; 260 1 q U35; 260; 1 q22 ð2Þ ð2Þ ð2Þ ð2Þ

Lð3Þ ð1ÞD

28 X

(7.106)

qið3Þ ULið3Þ ð1Þ 5 q1ð3Þ U0 1 q2ð3Þ U18; 260 1 q6ð3Þ U20; 260

i51

12 13 14 1 q9ð3Þ U23; 260 1 q11 ð3Þ U25; 260 1 qð3Þ U25; 260 1 qð3Þ U25; 260 1 qð3Þ U28; 260 16 18 20 21 1 q15 ð3Þ U27; 260 1 qð3Þ U30; 260 1 qð3Þ U35; 260 1 qð3Þ U33; 260 1 qð3Þ U35; 260 22 23 25 27 1 qð3Þ U33; 260 1 qð3Þ U35; 260 1 qð3Þ U35; 260 1 qð3Þ U35; 260;

Lð4Þ ð1ÞD

31 X

(7.107)

qið4Þ ULið4Þ ð1Þ 5 q1ð4Þ U0 1 q2ð4Þ U17; 260 1 q6ð4Þ U18; 260

i51

12 13 14 1 q9ð4Þ U20; 260 1 q11 ð4Þ U23; 260 1 qð4Þ U20; 260 1 qð4Þ U20; 260 1 qð4Þ U22; 260 18 21 24 25 1 q16 ð4Þ U30; 260 1 qð4Þ U33; 260 1 qð4Þ U35; 260 1 qð4Þ U30; 260 1 qð4Þ U33; 260 26 28 30 1 qð4Þ U35; 260 1 qð4Þ U35; 260 1 qð4Þ U35; 260;

(7.108)

162

Consequences of Maritime Critical Infrastructure Accidents

Lð5Þ ð1ÞD

23 X qið5Þ ULið5Þ ð1Þ 5 q1ð5Þ U0 1 q5ð5Þ U6260 1 q6ð5Þ U10; 260 1 q9ð5Þ U15; 260 1 q11 ð5Þ U20; 260:

(7.109)

i51

Next, based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilities qiðkÞ , k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 coming from experts and proceedings in the same way as in Section 7.3.1, we determine the optimal values q_iðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of limit transient probabilities qiðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23: Thus we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk, k 5 1; 2; . . .; 5 and expressed by the linear forms (7.105)(7.109) giving, according to (7.17), their optimal value (in PLN): L_ð1Þ ð1ÞD2:111; L_ð2Þ ð1ÞD2:696; L_ð3Þ ð1ÞD3:963; L_ð4Þ ð1ÞD3:916; L_ð5Þ ð1ÞD2:134;

(7.110)

and from the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of considering ports _ 5 Lð1Þ

5 X

L_ðkÞ ð1ÞD14:820:

(7.111)

k51

7.3.4 Minimization of critical infrastructure accident losses with considering climateweather change process impact—extension based on wider data collected at world sea waters Based on the procedure and formulae given in Section 7.2, proceeding in the same way as in Section 7.3.2, we optimize the losses of shipping critical infrastructure accident impacted by the climateweather change process at the Baltic Sea open waters, restricted waters as well as Gdynia and Karlskrona ports using the wider data collected at the world sea waters. The objective function defined by (7.18) for particular subareas Dk, k 5 1; 2; . . .; 5 of the open and restricted sea waters takes the forms (the parts with qiðkÞ 5 0 are omitted): Lð1Þ ð1ÞD

6 X 30 h iðbÞ X qb Uqið1Þ U Lið1Þ ð1Þ 5 q1 q1ð1Þ U0 1 q1 q2ð1Þ U5000 1 q1 q6ð1Þ U5000 1 q1 q8ð1Þ U6500 1 q1 q10 ð1Þ U7500 b51 i51

13 14 15 17 18 19 1 q1 q11 ð1Þ U7000 1 q1 qð1Þ U10; 000 1 q1 qð1Þ U12; 000 1 q1 qð1Þ U12; 500 1 q1 qð1Þ U2000 1 q1 qð1Þ U2500 1 q1 qð1Þ U3000 21 22 23 24 25 26 1 q1 q20 ð1Þ U2000 1 q1 qð1Þ U2000 1 q1 qð1Þ U8000 1 q1 qð1Þ U3000 1 q1 qð1Þ U4000 1 q1 qð1Þ U10; 000 1 q1 qð1Þ U3500 27 28 29 30 1 2 6 1 q1 qð1Þ U7000 1 q1 qð1Þ U11; 000 1 q1 qð1Þ U2500 1 q1 qð1Þ U4000 1 q2 qð1Þ U0 1 q2 qð1Þ U5000 1 q2 qð1Þ U5000 11 13 14 15 17 1 q2 q8ð1Þ U6500 1 q2 q10 ð1Þ U7500 1 q2 qð1Þ U7000 1 q2 qð1Þ U10; 000 1 q2 qð1Þ U12; 000 1 q2 qð1Þ U12; 500 1 q2 qð1Þ U2000 18 19 20 21 22 23 24 1 q2 qð1Þ U2500 1 q2 qð1Þ U3000 1 q2 qð1Þ U2000 1 q2 qð1Þ U2000 1 q2 qð1Þ U8000 1 q2 qð1Þ U3000 1 q2 qð1Þ U4000 26 27 28 29 30 1 1 q2 q25 ð1Þ U10; 000 1 q2 qð1Þ U3500 1 q2 qð1Þ U7000 1 q2 qð1Þ U11; 000 1 q2 qð1Þ U2500 1 q2 qð1Þ U4000 1 q3 qð1Þ U0 2 6 8 10 11 13 14 1 q3 qð1Þ U5000 1 q3 qð1Þ U5000 1 q3 qð1Þ U6500 1 q3 qð1Þ U7500 1 q3 qð1Þ U7000 1 q3 qð1Þ U10; 000 1 q3 qð1Þ U12; 000 17 18 19 20 21 22 1 q3 q15 ð1Þ U12; 500 1 q3 qð1Þ U2000 1 q3 qð1Þ U2500 1 q3 qð1Þ U3000 1 q3 qð1Þ U2000 1 q3 qð1Þ U2000 1 q3 qð1Þ U8000 24 25 26 27 28 29 1 q3 q23 ð1Þ U3000 1 q3 qð1Þ U4000 1 q3 qð1Þ U10; 000 1 q3 qð1Þ U3500 1 q3 qð1Þ U7000 1 q3 qð1Þ U11; 000 1 q3 qð1Þ U2500 30 1 2 6 8 10 11 1 q3 qð1Þ U4000 1 q4 qð1Þ U0 1 q4 qð1Þ U10; 000 1 q4 qð1Þ U10; 000 1 q4 qð1Þ U13; 000 1 q4 qð1Þ U15; 000 1 q4 qð1Þ U14; 000 14 15 17 18 19 20 1 q4 q13 ð1Þ U20; 000 1 q4 qð1Þ U24; 000 1 q4 qð1Þ U25; 000 1 q4 qð1Þ U4000 1 q4 qð1Þ U5000 1 q4 qð1Þ U6000 1 q4 qð1Þ U4000 21 22 23 24 25 26 27 1 q4 qð1Þ U4000 1 q4 qð1Þ U16; 000 1 q4 qð1Þ U6000 1 q4 qð1Þ U8000 1 q4 qð1Þ U20; 000 1 q4 qð1Þ U7000 1 q4 qð1Þ U14; 000 29 30 1 2 6 8 1 q4 q28 ð1Þ U22; 000 1 q4 qð1Þ U5000 1 q4 qð1Þ U8000 1 q5 qð1Þ U0 1 q5 qð1Þ U10; 000 1 q5 qð1Þ U10; 000 1 q5 qð1Þ U13; 000 10 11 13 14 15 17 1 q5 qð1Þ U15; 000 1 q5 qð1Þ U14; 000 1 q5 qð1Þ U20; 000 1 q5 qð1Þ U24; 000 1 q5 qð1Þ U25; 000 1 q5 qð1Þ U4000 1 q5 q18 ð1Þ U5000 19 20 21 22 23 24 25 1 q5 qð1Þ U6000 1 q5 qð1Þ U4000 1 q5 qð1Þ U4000 1 q5 qð1Þ U16; 000 1 q5 qð1Þ U6000 1 q5 qð1Þ U8000 1 q5 qð1Þ U20; 000 27 28 29 30 1 2 1 q5 q26 ð1Þ U7000 1 q5 qð1Þ U14; 000 1 q5 qð1Þ U22; 000 1 q5 qð1Þ U5000 1 q5 qð1Þ U8000 1 q6 qð1Þ U0 1 q6 qð1Þ U10; 000 11 13 14 1 q6 q6ð1Þ U10; 000 1 q6 q8ð1Þ U13; 000 1 q6 q10 ð1Þ U15; 000 1 q6 qð1Þ U14; 000 1 q6 qð1Þ U20; 000 1 q6 qð1Þ U24; 000 15 17 18 19 20 21 1 q6 qð1Þ U25; 000 1 q6 qð1Þ U4000 1 q6 qð1Þ U5000 1 q6 qð1Þ U6000 1 q6 qð1Þ U4000 1 q6 qð1Þ U4000 1 q6 q22 ð1Þ U16; 000 23 24 25 26 27 1 q6 qð1Þ U6000 1 q6 qð1Þ U8000 1 q6 qð1Þ U20; 000 1 q6 qð1Þ U7000 1 q6 qð1Þ U14; 000 29 30 1 q6 q28 ð1Þ U22; 000 1 q6 qð1Þ U5000 1 q6 qð1Þ U8000;

(7.112)

Optimization of critical infrastructure accident losses Chapter | 7

Lð2Þ ð1ÞD

163

6 X 28 h iðbÞ X qb Uqið2Þ U Lið2Þ ð1Þ 5 q1 q1ð2Þ U0 1 q1 q2ð2Þ U7000 1 q1 q5ð2Þ U5000 1 q1 q6ð2Þ U10; 000 b51

i51

12 13 14 1 q1 q9ð2Þ U8000 1 q1 q11 ð2Þ U12; 000 1 q1 qð2Þ U15; 000 1 q1 qð2Þ U15; 000 1 q1 qð2Þ U17; 000 16 18 20 21 1 q1 q15 ð2Þ U18; 000 1 q1 qð2Þ U20; 000 1 q1 qð2Þ U22; 000 1 q1 qð2Þ U24; 000 1 q1 qð2Þ U25; 000 22 23 25 27 1 q1 qð2Þ U25; 000 1 q1 qð2Þ U26; 000 1 q1 qð2Þ U27; 000 1 q1 qð2Þ U30; 000 1 q2 q1ð2Þ U0 1 q2 q2ð2Þ U14; 000 1 q2 q5ð2Þ U10; 000 1 q2 q6ð2Þ U20; 000 1 q2 q9ð2Þ U16; 000 1 q2 q11 ð2Þ U24; 000 13 14 15 16 U30; 000 1 q q U30; 000 1 q q U34; 000 1 q q U36; 000 1 q q 1 q2 q12 2 ð2Þ 2 ð2Þ 2 ð2Þ 2 ð2Þ U40; 000 ð2Þ 20 21 22 23 25 1 q2 q18 ð2Þ U44; 000 1 q2 qð2Þ U48; 000 1 q2 qð2Þ U50; 000 1 q2 qð2Þ U50; 000 1 q2 qð2Þ U52; 000 1 q2 qð2Þ U54; 000 27 1 2 5 6 9 1 q2 qð2Þ U60; 000 1 q3 qð2Þ U0 1 q3 qð2Þ U17; 500 1 q3 qð2Þ U12; 500 1 q3 qð2Þ U25; 000 1 q3 qð2Þ U20; 000 12 13 14 15 16 1 q3 q11 ð2Þ U30; 000 1 q3 qð2Þ U37; 500 1 q3 qð2Þ U37; 500 1 q3 qð2Þ U42; 500 1 q3 qð2Þ U45; 000 1 q3 qð2Þ U50; 000 18 20 21 22 23 1 q3 qð2Þ U55; 000 1 q3 qð2Þ U60; 000 1 q3 qð2Þ U62; 500 1 q3 qð2Þ U62; 500 1 q3 qð2Þ U65; 000 1 q3 q25 ð2Þ U67; 500 27 1 2 5 6 9 1 q3 qð2Þ U75; 000 1 q4 qð2Þ U0 1 q4 qð2Þ U12; 600 1 q4 qð2Þ U9000 1 q4 qð2Þ U18; 000 1 q4 qð2Þ U14; 400 12 13 14 15 1 q4 q11 ð2Þ U21; 600 1 q4 qð2Þ U27; 000 1 q4 qð2Þ U27; 000 1 q4 qð2Þ U30; 600 1 q4 qð2Þ U32; 400 18 20 21 1 q4 q16 ð2Þ U36; 000 1 q4 qð2Þ U39; 600 1 q4 qð2Þ U43; 200 1 q4 qð2Þ U45; 000 22 23 25 27 1 q4 qð2Þ U45; 000 1 q4 qð2Þ U46; 800 1 q4 qð2Þ U48; 600 1 q4 qð2Þ U54; 000 1 q5 q1ð2Þ U0 1 q5 q2ð2Þ U17; 500 1 q5 q5ð2Þ U12; 500 1 q5 q6ð2Þ U25; 000 1 q5 q9ð2Þ U20; 000 1 q5 q11 ð2Þ U30; 000 12 13 14 15 1 q5 qð2Þ U37; 500 1 q5 qð2Þ U37; 500 1 q5 qð2Þ U42; 500 1 q5 qð2Þ U45; 000 1 q5 q16 ð2Þ U50; 000 20 21 22 23 1 q5 q18 U55; 000 1 q q U60; 000 1 q q U62; 500 1 q q U62; 500 1 q q 5 ð2Þ 5 ð2Þ 5 ð2Þ 5 ð2Þ U65; 000 ð2Þ 27 1 2 5 6 1 q5 q25 ð2Þ U67; 500 1 q5 qð2Þ U75; 000 1 q6 qð2Þ U0 1 q6 qð2Þ U21; 000 1 q6 qð2Þ U15; 000 1 q6 qð2Þ U30; 000 9 11 12 13 14 1 q6 qð2Þ U24; 000 1 q6 qð2Þ U36; 000 1 q6 qð2Þ U45; 000 1 q6 qð2Þ U45; 000 1 q6 qð2Þ U51; 000 16 18 20 21 1 q6 q15 ð2Þ U54; 000 1 q6 qð2Þ U60; 000 1 q6 qð2Þ U66; 000 1 q6 qð2Þ U72; 000 1 q6 qð2Þ U75; 000 22 23 25 27 1 q6 qð2Þ U75; 000 1 q6 qð2Þ U78; 000 1 q6 qð2Þ U81; 000 1 q6 qð2Þ U90; 000;

Lð3Þ ð1ÞD

(7.113)

6 X 28 h iðbÞ X qb Uqið3Þ U Lið3Þ ð1Þ 5 q1 q1ð3Þ U0 1 q1 q2ð3Þ U13; 000 1 q1 q6ð3Þ U15; 000 b51

i51

12 13 14 1q1 q9ð3Þ U18; 000 1 q1 q11 ð3Þ U20; 000 1 q1 qð3Þ U20; 000 1 q1 qð3Þ U20; 000 1 q1 qð3Þ U23; 000 15 16 18 20 1q1 qð3Þ U22; 000 1 q1 qð3Þ U25; 000 1 q1 qð3Þ U30; 000 1 q1 qð3Þ U28; 000 1 q1 q21 ð3Þ U30; 000 22 23 25 27 1q1 qð3Þ U28; 000 1 q1 qð3Þ U30; 000 1 q1 qð3Þ U30; 000 1 q1 qð3Þ U30; 000 1 q2 q1ð3Þ U0 1q2 q2ð3Þ U26; 000 1 q2 q6ð3Þ U30; 000 1 q2 q9ð3Þ U36; 000 1 q2 q11 ð3Þ U40; 000 13 14 15 U40; 000 1 q q U40; 000 1 q q U46; 000 1 q q 1q2 q12 2 ð3Þ 2 ð3Þ 2 ð3Þ U44; 000 ð3Þ 16 18 20 1q2 qð3Þ U50; 000 1 q2 qð3Þ U60; 000 1 q2 qð3Þ U56; 000 1 q2 q21 ð3Þ U60; 000 23 25 27 U56; 000 1 q q U60; 000 1 q q U60; 000 1 q q U60; 000 1 q3 q1ð3Þ U0 1q2 q22 2 ð3Þ 2 ð3Þ 2 ð3Þ ð3Þ 1q3 q2ð3Þ U39; 000 1 q3 q6ð3Þ U45; 000 1 q3 q9ð3Þ U54; 000 1 q3 q11 ð3Þ U60; 000 13 14 15 U60; 000 1 q q U60; 000 1 q q U69; 000 1 q q 1q3 q12 3 ð3Þ 3 ð3Þ 3 ð3Þ U66; 000 ð3Þ 18 20 21 1q3 q16 ð3Þ U75; 000 1 q3 qð3Þ U90; 000 1 q3 qð3Þ U84; 000 1 q3 qð3Þ U90; 000 22 23 25 27 1q3 qð3Þ U84; 000 1 q3 qð3Þ U90; 000 1 q3 qð3Þ U90; 000 1 q3 qð3Þ U90; 000 1 q4 q1ð3Þ U0 1q4 q2ð3Þ U13; 000 1 q4 q6ð3Þ U15; 000 1 q4 q9ð3Þ U18; 000 1 q4 q11 ð3Þ U20; 000 13 14 15 U20; 000 1 q q U20; 000 1 q q U23; 000 1 q q 1q4 q12 4 4 4 ð3Þ ð3Þ ð3Þ ð3Þ U22; 000 16 18 20 1q4 qð3Þ U25; 000 1 q4 qð3Þ U30; 000 1 q4 qð3Þ U28; 000 1 q4 q21 ð3Þ U30; 000 23 25 27 1q4 q22 U28; 000 1 q q U30; 000 1 q q U30; 000 1 q q U30; 000 1 q5 q1ð3Þ U0 4 4 4 ð3Þ ð3Þ ð3Þ ð3Þ 1q5 q2ð3Þ U26; 000 1 q5 q6ð3Þ U30; 000 1 q5 q9ð3Þ U36; 000 1 q5 q11 ð3Þ U40; 000 12 13 14 1q5 qð3Þ U40; 000 1 q5 qð3Þ U40; 000 1 q5 qð3Þ U46; 000 1 q5 q15 ð3Þ U44; 000 18 20 21 1q5 q16 U50; 000 1 q q U60; 000 1 q q U56; 000 1 q q 5 ð3Þ 5 ð3Þ 5 ð3Þ U60; 000 ð3Þ 23 25 27 1 1q5 q22 ð3Þ U56; 000 1 q5 qð3Þ U60; 000 1 q5 qð3Þ U60; 000 1 q5 qð3Þ U60; 000 1 q6 qð3Þ U0 2 6 9 11 1q6 qð3Þ U39; 000 1 q6 qð3Þ U45; 000 1 q6 qð3Þ U54; 000 1 q6 qð3Þ U60; 000 13 14 15 1q6 q12 ð3Þ U60; 000 1 q6 qð3Þ U60; 000 1 q6 qð3Þ U69; 000 1 q6 qð3Þ U66; 000 16 18 20 1q6 qð3Þ U75; 000 1 q6 qð3Þ U90; 000 1 q6 qð3Þ U84; 000 1 q6 q21 ð3Þ U90; 000 22 23 25 27 1q6 qð3Þ U84; 000 1 q6 qð3Þ U90; 000 1 q6 qð3Þ U90; 000 1 q6 qð3Þ U90; 000;

(7.114)

164

Consequences of Maritime Critical Infrastructure Accidents

Lð4Þ ð1ÞD

6 X 31 h iðbÞ X qb Uqið4Þ U Lið4Þ ð1Þ 5 q1 q1ð4Þ U0 1 q1 q2ð4Þ U12; 000 1 q1 q6ð4Þ U13; 000 b51

i51

12 13 14 1q1 q9ð4Þ U15; 000 1 q1 q11 ð4Þ U18; 000 1 q1 qð4Þ U15; 000 1 q1 qð4Þ U15; 000 1 q1 qð4Þ U17; 000 18 21 24 25 1q1 q16 ð4Þ U25; 000 1 q1 qð4Þ U28; 000 1 q1 qð4Þ U30; 000 1 q1 qð4Þ U25; 000 1 q1 qð4Þ U28; 000 26 28 30 1 2 1q1 qð4Þ U30; 000 1 q1 qð4Þ U30; 000 1 q1 qð4Þ U30; 000 1 q2 qð4Þ U0 1 q2 qð4Þ U12; 000 12 13 1q2 q6ð4Þ U13; 000 1 q2 q9ð4Þ U15; 000 1 q2 q11 ð4Þ U18; 000 1 q2 qð4Þ U15; 000 1 q2 qð4Þ U15; 000 14 16 18 21 1q2 qð4Þ U17; 000 1 q2 qð4Þ U25; 000 1 q2 qð4Þ U28; 000 1 q2 qð4Þ U30; 000 1 q2 q24 ð4Þ U25; 000 25 26 28 30 1q2 qð4Þ U28; 000 1 q2 qð4Þ U30; 000 1 q2 qð4Þ U30; 000 1 q2 qð4Þ U30; 000 1 q3 q1ð4Þ U0 12 1q3 q2ð4Þ U12; 000 1 q3 q6ð4Þ U13; 000 1 q3 q9ð4Þ U15; 000 1 q3 q11 ð4Þ U18; 000 1 q3 qð4Þ U15; 000 14 16 18 21 1q3 q13 ð4Þ U15; 000 1 q3 qð4Þ U17; 000 1 q3 qð4Þ U25; 000 1 q3 qð4Þ U28; 000 1 q3 qð4Þ U30; 000 24 25 26 28 1q3 qð4Þ U25; 000 1 q3 qð4Þ U28; 000 1 q3 qð4Þ U30; 000 1 q3 qð4Þ U30; 000 1 q3 q30 ð4Þ U30; 000 1q4 q1ð4Þ U0 1 q4 q2ð4Þ U12; 000 1 q4 q6ð4Þ U13; 000 1 q4 q9ð4Þ U15; 000 1 q4 q11 U18; 000 ð4Þ 12 13 14 16 18 1q4 qð4Þ U15; 000 1 q4 qð4Þ U15; 000 1 q4 qð4Þ U17; 000 1 q4 qð4Þ U25; 000 1 q4 qð4Þ U28; 000 24 25 26 28 1q4 q21 ð4Þ U30; 000 1 q4 qð4Þ U25; 000 1 q4 qð4Þ U28; 000 1 q4 qð4Þ U30; 000 1 q4 qð4Þ U30; 000 1 2 6 9 1q4 q30 ð4Þ U30; 000 1 q5 qð4Þ U0 1 q5 qð4Þ U12; 000 1 q5 qð4Þ U13; 000 1 q5 qð4Þ U15; 000 11 12 13 14 1q5 qð4Þ U18; 000 1 q5 qð4Þ U15; 000 1 q5 qð4Þ U15; 000 1 q5 qð4Þ U17; 000 1 q5 q16 ð4Þ U25; 000 18 21 24 25 1q5 qð4Þ U28; 000 1 q5 qð4Þ U30; 000 1 q5 qð4Þ U25; 000 1 q5 qð4Þ U28; 000 1 q5 q26 ð4Þ U30; 000 30 1 2 6 1q5 q28 U30; 000 1 q q U30; 000 1 q q U0 1 q q U12; 000 1 q q U13; 000 5 ð4Þ 6 ð4Þ 6 ð4Þ 6 ð4Þ ð4Þ 9 11 12 13 14 1q6 qð4Þ U15; 000 1 q6 qð4Þ U18; 000 1 q6 qð4Þ U15; 000 1 q6 qð4Þ U15; 000 1 q6 qð4Þ U17; 000 18 21 24 25 1q6 q16 ð4Þ U25; 000 1 q6 qð4Þ U28; 000 1 q6 qð4Þ U30; 000 1 q6 qð4Þ U25; 000 1 q6 qð4Þ U28; 000 28 30 1q6 q26 ð4Þ U30; 000 1 q6 qð4Þ U30; 000 1 q6 qð4Þ U30; 000;

Lð5Þ ð1ÞD

6 X 23 X b51

h iðbÞ qb Uqið5Þ U Lið5Þ ð1Þ 5 q1 q1ð5Þ U0 1 q1 q5ð5Þ U0 1 q1 q6ð5Þ U0 1 q1 q9ð5Þ U0

i51

1 5 6 9 11 1 q1 q11 ð5Þ U0 1 q2 qð5Þ U0 1 q2 qð5Þ U0 1 q2 qð5Þ U0 1 q2 qð5Þ U0 1 q2 qð5Þ U0 1 5 6 9 11 1 q3 qð5Þ U0 1 q3 qð5Þ U0 1 q3 qð5Þ U0 1 q3 qð5Þ U0 1 q3 qð5Þ U0 1 q4 q1ð5Þ U0 1 5 1 q4 q5ð5Þ U0 1 q4 q6ð5Þ U0 1 q4 q9ð5Þ U0 1 q4 q11 ð5Þ U0 1 q5 qð5Þ U0 1 q5 qð5Þ U0 6 9 11 1 5 1 q5 qð5Þ U0 1 q5 qð5Þ U0 1 q5 qð5Þ U0 1 q6 qð5Þ U0 1 q6 qð5Þ U0 1 q6 q6ð5Þ U0 1 q6 q9ð5Þ U0 1 q6 q11 ð5Þ U0 ðfor open sea watersÞ;

Lð5Þ ð1ÞD

6 X 23 X b51

(7.115)

(7.116)

h iðbÞ qb Uqið5Þ U Lið5Þ ð1Þ 5 q1 q1ð5Þ U0 1 q1 q5ð5Þ U1000 1 q1 q6ð5Þ U5000

i51

1 5 6 1 q1 q9ð5Þ U10; 000 1 q1 q11 ð5Þ U15; 000 1 q2 qð5Þ U0 1 q2 qð5Þ U1000 1 q2 qð5Þ U5000 9 11 1 5 1 q2 qð5Þ U10; 000 1 q2 qð5Þ U15; 000 1 q3 qð5Þ U0 1 q3 qð5Þ U1000 1 q3 q6ð5Þ U5000 1 5 6 1 q3 q9ð5Þ U10; 000 1 q3 q11 ð5Þ U15; 000 1 q4 qð5Þ U0 1 q4 qð5Þ U1000 1 q4 qð5Þ U5000 9 11 1 5 1 q4 qð5Þ U10; 000 1 q4 qð5Þ U15; 000 1 q5 qð5Þ U0 1 q5 qð5Þ U1000 1 q5 q6ð5Þ U5000 1 5 6 1 q5 q9ð5Þ U10; 000 1 q5 q11 ð5Þ U15; 000 1 q6 qð5Þ U0 1 q6 qð5Þ U1000 1 q6 qð5Þ U5000 9 11 1 q6 qð5Þ U10; 000 1 q6 qð5Þ U15; 000 ðfor restricted sea watersÞ:

(7.117)

We do not minimize the objective function (7.116) for the subarea D5 of the open sea waters as Lð5Þ ð1Þ 5 0 PLN.

Optimization of critical infrastructure accident losses Chapter | 7

165

Next, considering the approximate limit values of transient probabilities qb ; b 5 1; 2; . . .; 6 of the climateweather change process C(t) at the climateweather states cb, b 5 1; 2; . . .; 6 for the open and restricted sea waters, respectively, given by (6.53) and (6.58) as well as based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilii ties qðkÞ , k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 coming from experts and proceedings in the same way as in Section 7.3.2.1, we determined the optimal values qb q_iðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of limit transient probabilities qb qiðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23: Thus we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk, k 5 1; 2; . . .; 5 impacted by the climateweather change process C(t) and expressed by the linear forms (7.112)(7.117) giving, according to (7.34) their optimal value (in PLN): G

for open sea waters L_ ð1Þ ð1ÞD1:009; L_ ð2Þ ð1ÞD1:668; L_ ð3Þ ð1ÞD2:984; L_ ð4Þ ð1ÞD2:811;

(7.118)

and from the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of open sea waters impacted by the climateweather change process C(t) _ 5 Lð1Þ

4 X

L_ ðkÞ ð1ÞD8:472;

(7.119)

k51 G

for restricted sea waters _ _ _ _ L_ ð1Þ ð1ÞD1:009; Lð2Þ ð1ÞD1:672; Lð3Þ ð1ÞD2:993; Lð4Þ ð1ÞD2:811; Lð5Þ ð1ÞD1:030;

(7.120)

and from the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of restricted sea waters impacted by the climateweather change process C(t) _ 5 Lð1Þ

5 X

L_ ðkÞ ð1ÞD9:515:

(7.121)

k51

Thus considering (7.101), (7.118) and (7.103), (7.120), respectively, and according to (7.35), the indicators RI_ðkÞ of the environment in the subarea Dk, k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amount to the following: G

G

for open sea waters RI_ð1Þ 5 0:998; RI_ð2Þ 5 0:954; RI_ð3Þ 5 0:958; RI_ð4Þ 5 1

(7.122)

RI_ð1Þ 5 0:998; RI_ð2Þ 5 0:952; RI_ð3Þ 5 0:955; RI_ð4Þ 5 1; RI_ð5Þ 5 1

(7.123)

for restricted sea waters

166

Consequences of Maritime Critical Infrastructure Accidents

and considering (7.102), (7.119) and (7.104), (7.121), respectively, and according to (7.36), the indicator RI_ of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amounts to the following: RI_ 5 0:976 ðfor open sea watersÞ;

(7.124)

RI_ 5 0:977 ðfor restricted sea watersÞ:

(7.125)

and

Similarly, the objective function defined by (7.18) for particular subareas Dk, k 5 1; 2; . . .; 5 of Gdynia and Karlskrona ports takes the forms (the parts with qiðkÞ 5 0 are omitted): Lð1Þ ð1ÞD

6 X 30 h iðbÞ X qb Uqið1Þ U Lið1Þ ð1Þ 5 q1 q1ð1Þ U0 1 q1 q2ð1Þ U10; 260 1 q1 q6ð1Þ U10; 260 b51 i51

11 13 14 1 q1 q8ð1Þ U11; 760 1 q1 q10 ð1Þ U12; 760 1 q1 qð1Þ U12; 260 1 q1 qð1Þ U15; 260 1 q1 qð1Þ U17; 260 15 17 18 19 20 1 q1 qð1Þ U17; 760 1 q1 qð1Þ U7260 1 q1 qð1Þ U7760 1 q1 qð1Þ U8260 1 q1 qð1Þ U7260 22 23 24 25 1 q1 q21 ð1Þ U7260 1 q1 qð1Þ U13; 260 1 q1 qð1Þ U8260 1 q1 qð1Þ U9260 1 q1 qð1Þ U15; 260 26 27 28 29 1 q1 qð1Þ U8760 1 q1 qð1Þ U12; 260 1 q1 qð1Þ U16; 260 1 q1 qð1Þ U7760 1 q1 q30 ð1Þ U9260 1 2 6 8 10 1 q2 qð1Þ U0 1 q2 qð1Þ U20; 520 1 q2 qð1Þ U20; 520 1 q2 qð1Þ U23; 520 1 q2 qð1Þ U25; 520 13 14 15 1 q2 q11 ð1Þ U24; 520 1 q2 qð1Þ U30; 520 1 q2 qð1Þ U34; 520 1 q2 qð1Þ U35; 520 18 19 20 21 1 q2 q17 ð1Þ U14; 520 1 q2 qð1Þ U15; 520 1 q2 qð1Þ U16; 520 1 q2 qð1Þ U14; 520 1 q2 qð1Þ U14; 520 22 23 24 25 1 q2 qð1Þ U26; 520 1 q2 qð1Þ U16; 520 1 q2 qð1Þ U18; 520 1 q2 qð1Þ U30; 520 1 q2 q26 ð1Þ U17; 520 28 29 30 1 1 q2 q27 U24; 520 1 q q U32; 520 1 q q U15; 520 1 q q U18; 520 1 q 2 ð1Þ 2 ð1Þ 2 ð1Þ 3 qð1Þ U0 ð1Þ 11 1 q3 q2ð1Þ U10; 260 1 q3 q6ð1Þ U10; 260 1 q3 q8ð1Þ U11; 760 1 q3 q10 ð1Þ U12; 760 1 q3 qð1Þ U12; 260 13 14 15 17 1 q3 qð1Þ U15; 260 1 q3 qð1Þ U17; 260 1 q3 qð1Þ U17; 760 1 q3 qð1Þ U7260 1 q3 q18 ð1Þ U7760 19 20 21 22 23 1 q3 qð1Þ U8260 1 q3 qð1Þ U7260 1 q3 qð1Þ U7260 1 q3 qð1Þ U13; 260 1 q3 qð1Þ U8260 25 26 27 28 1 q3 q24 ð1Þ U9260 1 q3 qð1Þ U15; 260 1 q3 qð1Þ U8760 1 q3 qð1Þ U12; 260 1 q3 qð1Þ U16; 260 30 1 2 6 1 q3 q29 ð1Þ U7760 1 q3 qð1Þ U9260 1 q4 qð1Þ U0 1 q4 qð1Þ U20; 520 1 q4 qð1Þ U20; 520 8 10 11 13 1 q4 qð1Þ U23; 520 1 q4 qð1Þ U25; 520 1 q4 qð1Þ U24; 520 1 q4 qð1Þ U30; 520 14 17 18 19 1 q4 qð1Þ U34; 520 1 q4 q15 ð1Þ U35; 520 1 q4 qð1Þ U14; 520 1 q4 qð1Þ U15; 520 1 q4 qð1Þ U16; 520 20 21 22 23 1 q4 qð1Þ U14; 520 1 q4 qð1Þ U14; 520 1 q4 qð1Þ U26; 520 1 q4 qð1Þ U16; 520 1 q4 q24 ð1Þ U18; 520 26 27 28 29 U30; 520 1 q q U17; 520 1 q q U24; 520 1 q q U32; 520 1 q q 1 q4 q25 4 ð1Þ 4 ð1Þ 4 ð1Þ 4 ð1Þ U15; 520 ð1Þ 30 1 2 6 8 1 q4 qð1Þ U18; 520 1 q5 qð1Þ U0 1 q5 qð1Þ U10; 260 1 q5 qð1Þ U10; 260 1 q5 qð1Þ U11; 760 11 13 14 15 1 q5 q10 ð1Þ U12; 760 1 q5 qð1Þ U12; 260 1 q5 qð1Þ U15; 260 1 q5 qð1Þ U17; 260 1 q5 qð1Þ U17; 760 18 19 20 21 1 q5 q17 ð1Þ U7260 1 q5 qð1Þ U7760 1 q5 qð1Þ U8260 1 q5 qð1Þ U7260 1 q5 qð1Þ U7260 22 23 24 25 1 q5 qð1Þ U13; 260 1 q5 qð1Þ U8260 1 q5 qð1Þ U9260 1 q5 qð1Þ U15; 260 1 q5 q26 ð1Þ U8760 27 28 29 30 1 q5 qð1Þ U12; 260 1 q5 qð1Þ U16; 260 1 q5 qð1Þ U7760 1 q5 qð1Þ U9260 1 q6 q1ð1Þ U0 11 1 q6 q2ð1Þ U20; 520 1 q6 q6ð1Þ U20; 520 1 q6 q8ð1Þ U23; 520 1 q6 q10 ð1Þ U25; 520 1 q6 qð1Þ U24; 520 14 15 17 18 1 q6 q13 ð1Þ U30; 520 1 q6 qð1Þ U34; 520 1 q6 qð1Þ U35; 520 1 q6 qð1Þ U14; 520 1 q6 qð1Þ U15; 520 19 20 21 22 1 q6 qð1Þ U16; 520 1 q6 qð1Þ U14; 520 1 q6 qð1Þ U14; 520 1 q6 qð1Þ U26; 520 1 q6 q23 ð1Þ U16; 520 24 25 26 27 1 q6 qð1Þ U18; 520 1 q6 qð1Þ U30; 520 1 q6 qð1Þ U17; 520 1 q6 qð1Þ U24; 520 1 q6 q28 ð1Þ U32; 520 30 1 q6 q29 U15; 520 1 q q U18; 520; 6 ð1Þ ð1Þ

(7.126)

Optimization of critical infrastructure accident losses Chapter | 7

Lð2Þ ð1ÞD

167

6 X 28 h iðbÞ X qb Uqið2Þ U Lið2Þ ð1Þ 5 q1 q1ð2Þ U0 1 q1 q2ð2Þ U12; 260 1 q1 q5ð2Þ U10; 260 b51

i51

12 13 1 q1 q6ð2Þ U15; 260 1 q1 q9ð2Þ U13; 260 1 q1 q11 ð2Þ U17; 260 1 q1 qð2Þ U20; 260 1 q1 qð2Þ U20; 260 15 16 18 20 1 q1 q14 ð2Þ U22; 260 1 q1 qð2Þ U23; 260 1 q1 qð2Þ U25; 260 1 q1 qð2Þ U27; 260 1 q1 qð2Þ U29; 260 21 22 23 25 1 q1 qð2Þ U30; 260 1 q1 qð2Þ U30; 260 1 q1 qð2Þ U31; 260 1 q1 qð2Þ U32; 260 1 q1 q27 ð2Þ U35; 260 1 2 5 6 9 1 q2 qð2Þ U0 1 q2 qð2Þ U24; 520 1 q2 qð2Þ U20; 520 1 q2 qð2Þ U30; 520 1 q2 qð2Þ U26; 520 13 14 15 16 U34; 520 1 q2 q12 1 q2 q11 ð2Þ ð2Þ U40; 520 1 q2 qð2Þ U40; 520 1 q2 qð2Þ U44; 520 1 q2 qð2Þ U46; 520 1 q2 qð2Þ U50; 520 20 21 22 23 25 1 q2 q18 ð2Þ U54; 520 1 q2 qð2Þ U58; 520 1 q2 qð2Þ U60; 520 1 q2 qð2Þ U60; 520 1 q2 qð2Þ U62; 520 1 q2 qð2Þ U64; 520 27 1 2 5 6 9 1 q2 qð2Þ U70; 520 1 q3 qð2Þ U0 1 q3 qð2Þ U12; 260 1 q3 qð2Þ U10; 260 1 q3 qð2Þ U15; 260 1 q3 qð2Þ U13; 260 12 13 14 15 16 1 q3 q11 ð2Þ U17; 260 1 q3 qð2Þ U20; 260 1 q3 qð2Þ U20; 260 1 q3 qð2Þ U22; 260 1 q3 qð2Þ U23; 260 1 q3 qð2Þ U25; 260 18 20 21 22 23 1 q3 qð2Þ U27; 260 1 q3 qð2Þ U29; 260 1 q3 qð2Þ U30; 260 1 q3 qð2Þ U30; 260 1 q3 qð2Þ U31; 260 1 q3 q25 ð2Þ U32; 260 27 1 2 5 6 9 1 q3 qð2Þ U35; 260 1 q4 qð2Þ U0 1 q4 qð2Þ U24; 520 1 q4 qð2Þ U20; 520 1 q4 qð2Þ U30; 520 1 q4 qð2Þ U26; 520 12 13 14 15 16 1 q4 q11 ð2Þ U34; 520 1 q4 qð2Þ U40; 520 1 q4 qð2Þ U40; 520 1 q4 qð2Þ U44; 520 1 q4 qð2Þ U46; 520 1 q4 qð2Þ U50; 520 20 21 22 23 25 1 q4 q18 ð2Þ U54; 520 1 q4 qð2Þ U58; 520 1 q4 qð2Þ U60; 520 1 q4 qð2Þ U60; 520 1 q4 qð2Þ U62; 520 1 q4 qð2Þ U64; 520 27 1 2 5 6 9 1 q4 qð2Þ U70; 520 1 q5 qð2Þ U0 1 q5 qð2Þ U12; 260 1 q5 qð2Þ U10; 260 1 q5 qð2Þ U15; 260 1 q5 qð2Þ U13; 260 12 13 14 15 16 1 q5 q11 ð2Þ U17; 260 1 q5 qð2Þ U20; 260 1 q5 qð2Þ U20; 260 1 q5 qð2Þ U22; 260 1 q5 qð2Þ U23; 260 1 q5 qð2Þ U25; 260 18 20 21 22 23 1 q5 qð2Þ U27; 260 1 q5 qð2Þ U29; 260 1 q5 qð2Þ U30; 260 1 q5 qð2Þ U30; 260 1 q5 qð2Þ U31; 260 27 1 2 5 6 1 q5 q25 ð2Þ U32; 260 1 q5 qð2Þ U35; 260 1 q6 qð2Þ U0 1 q6 qð2Þ U24; 520 1 q6 qð2Þ U20; 520 1 q6 qð2Þ U30; 520 12 13 14 15 1 q6 q9ð2Þ U26; 520 1 q6 q11 ð2Þ U34; 520 1 q6 qð2Þ U40; 520 1 q6 qð2Þ U40; 520 1 q6 qð2Þ U44; 520 1 q6 qð2Þ U46; 520 16 18 20 21 1 q6 qð2Þ U50; 520 1 q6 qð2Þ U54; 520 1 q6 qð2Þ U58; 520 1 q6 qð2Þ U60; 520 23 25 27 1 q6 q22 ð2Þ U60; 520 1 q6 qð2Þ U62; 520 1 q6 qð2Þ U64; 520 1 q6 qð2Þ U70; 520;

Lð3Þ ð1ÞD

(7.127)

6 X 28 h iðbÞ X qb Uqið3Þ U Lið3Þ ð1Þ 5 q1 q1ð3Þ U0 1 q1 q2ð3Þ U18; 260 1 q1 q6ð3Þ U20; 260 b51

i51

12 13 14 1 q1 q9ð3Þ U23; 260 1 q1 q11 ð3Þ U25; 260 1 q1 qð3Þ U25; 260 1 q1 qð3Þ U25; 260 1 q1 qð3Þ U28; 260 15 16 18 20 1 q1 qð3Þ U27; 260 1 q1 qð3Þ U30; 260 1 q1 qð3Þ U35; 260 1 q1 qð3Þ U33; 260 1 q1 q21 ð3Þ U35; 260 22 23 25 27 1 q1 qð3Þ U33; 260 1 q1 qð3Þ U35; 260 1 q1 qð3Þ U35; 260 1 q1 qð3Þ U35; 260 1 q2 q1ð3Þ U0 1 q2 q2ð3Þ U27; 390 1 q2 q6ð3Þ U30; 390 1 q2 q9ð3Þ U34; 890 1 q2 q11 ð3Þ U37; 890 12 13 14 1 q2 qð3Þ U37; 890 1 q2 qð3Þ U37; 890 1 q2 qð3Þ U42; 390 1 q2 q15 ð3Þ U40; 890 18 20 21 1 q2 q16 U45; 390 1 q q U52; 890 1 q q U49; 890 1 q q 2 2 2 ð3Þ U52; 890 ð3Þ ð3Þ ð3Þ 23 25 27 1 q2 q22 U49; 890 1 q q U52; 890 1 q q U52; 890 1 q q U52; 890 1 q3 q1ð3Þ U0 2 ð3Þ 2 ð3Þ 2 ð3Þ ð3Þ 2 6 9 11 1 q3 qð3Þ U18; 260 1 q3 qð3Þ U20; 260 1 q3 qð3Þ U23; 260 1 q3 qð3Þ U25; 260 13 14 15 1 q3 q12 ð3Þ U25; 260 1 q3 qð3Þ U25; 260 1 q3 qð3Þ U28; 260 1 q3 qð3Þ U27; 260 18 20 21 1 q3 q16 ð3Þ U30; 260 1 q3 qð3Þ U35; 260 1 q3 qð3Þ U33; 260 1 q3 qð3Þ U35; 260 22 23 25 27 1 q3 qð3Þ U33; 260 1 q3 qð3Þ U35; 260 1 q3 qð3Þ U35; 260 1 q3 qð3Þ U35; 260 1 q4 q1ð3Þ U0 1 q4 q2ð3Þ U27; 390 1 q4 q6ð3Þ U30; 390 1 q4 q9ð3Þ U34; 890 1 q4 q11 ð3Þ U37; 890 13 14 15 U37; 890 1 q q U37; 890 1 q q U42; 390 1 q q 1 q4 q12 4 ð3Þ 4 ð3Þ 4 ð3Þ U40; 890 ð3Þ 18 20 21 1 q4 q16 ð3Þ U45; 390 1 q4 qð3Þ U52; 890 1 q4 qð3Þ U49; 890 1 q4 qð3Þ U52; 890 22 23 25 27 1 q4 qð3Þ U49; 890 1 q4 qð3Þ U52; 890 1 q4 qð3Þ U52; 890 1 q4 qð3Þ U52; 890 1 q5 q1ð3Þ U0 1 q5 q2ð3Þ U18; 260 1 q5 q6ð3Þ U20; 260 1 q5 q9ð3Þ U23; 260 1 q5 q11 ð3Þ U25; 260 13 14 15 U25; 260 1 q q U25; 260 1 q q U28; 260 1 q q 1 q5 q12 5 5 5 ð3Þ ð3Þ ð3Þ ð3Þ U27; 260 18 20 21 1 q5 q16 U30; 260 1 q q U35; 260 1 q q U33; 260 1 q q 5 ð3Þ 5 ð3Þ 5 ð3Þ U35; 260 ð3Þ 22 23 25 27 1 q5 qð3Þ U33; 260 1 q5 qð3Þ U35; 260 1 q5 qð3Þ U35; 260 1 q5 qð3Þ U35; 260 1 q6 q1ð3Þ U0 1 q6 q2ð3Þ U27; 390 1 q6 q6ð3Þ U30; 390 1 q6 q9ð3Þ U34; 890 1 q6 q11 ð3Þ U37; 890 12 13 14 1 q6 qð3Þ U37; 890 1 q6 qð3Þ U37; 890 1 q6 qð3Þ U42; 390 1 q6 q15 ð3Þ U40; 890 18 20 21 1 q6 q16 U45; 390 1 q q U52; 890 1 q q U49; 890 1 q q 6 ð3Þ 6 ð3Þ 6 ð3Þ U52; 890 ð3Þ 23 25 27 1 q6 q22 ð3Þ U49; 890 1 q6 qð3Þ U52; 890 1 q6 qð3Þ U52; 890 1 q6 qð3Þ U52; 890;

(7.128)

168

Consequences of Maritime Critical Infrastructure Accidents

Lð4Þ ð1ÞD

6 X 31 h iðbÞ X qb Uqið4Þ U Lið4Þ ð1Þ 5 q1 q1ð4Þ U0 1 q1 q2ð4Þ U17; 260 1 q1 q6ð4Þ U18; 260 b51

i51

12 13 14 1 q1 q9ð4Þ U20; 260 1 q1 q11 ð4Þ U23; 260 1 q1 qð4Þ U20; 260 1 q1 qð4Þ U20; 260 1 q1 qð4Þ U22; 260 18 21 24 25 1 q1 q16 ð4Þ U30; 260 1 q1 qð4Þ U33; 260 1 q1 qð4Þ U35; 260 1 q1 qð4Þ U30; 260 1 q1 qð4Þ U33; 260 26 28 30 1 2 1 q1 qð4Þ U35; 260 1 q1 qð4Þ U35; 260 1 q1 qð4Þ U35; 260 1 q2 qð4Þ U0 1 q2 qð4Þ U17; 260 12 13 1 q2 q6ð4Þ U18; 260 1 q2 q9ð4Þ U20; 260 1 q2 q11 ð4Þ U23; 260 1 q2 qð4Þ U20; 260 1 q2 qð4Þ U20; 260 14 16 18 21 1 q2 qð4Þ U22; 260 1 q2 qð4Þ U30; 260 1 q2 qð4Þ U33; 260 1 q2 qð4Þ U35; 260 1 q2 q24 ð4Þ U30; 260 25 26 28 30 1 q2 qð4Þ U33; 260 1 q2 qð4Þ U35; 260 1 q2 qð4Þ U35; 260 1 q2 qð4Þ U35; 260 1 q3 q1ð4Þ U0 12 1 q3 q2ð4Þ U17; 260 1 q3 q6ð4Þ U18; 260 1 q3 q9ð4Þ U20; 260 1 q3 q11 ð4Þ U23; 260 1 q3 qð4Þ U20; 260 14 16 18 21 1 q3 q13 ð4Þ U20; 260 1 q3 qð4Þ U22; 260 1 q3 qð4Þ U30; 260 1 q3 qð4Þ U33; 260 1 q3 qð4Þ U35; 260 24 25 26 28 1 q3 qð4Þ U30; 260 1 q3 qð4Þ U33; 260 1 q3 qð4Þ U35; 260 1 q3 qð4Þ U35; 260 1 q3 q30 ð4Þ U35; 260 1 q4 q1ð4Þ U0 1 q4 q2ð4Þ U17; 260 1 q4 q6ð4Þ U18; 260 1 q4 q9ð4Þ U20; 260 1 q4 q11 U23; 260 ð4Þ 12 13 14 16 18 1 q4 qð4Þ U20; 260 1 q4 qð4Þ U20; 260 1 q4 qð4Þ U22; 260 1 q4 qð4Þ U30; 260 1 q4 qð4Þ U33; 260 24 25 26 28 1 q4 q21 ð4Þ U35; 260 1 q4 qð4Þ U30; 260 1 q4 qð4Þ U33; 260 1 q4 qð4Þ U35; 260 1 q4 qð4Þ U35; 260 1 2 6 9 1 q4 q30 ð4Þ U35; 260 1 q5 qð4Þ U0 1 q5 qð4Þ U17; 260 1 q5 qð4Þ U18; 260 1 q5 qð4Þ U20; 260 11 12 13 14 1 q5 qð4Þ U23; 260 1 q5 qð4Þ U20; 260 1 q5 qð4Þ U20; 260 1 q5 qð4Þ U22; 260 1 q5 q16 ð4Þ U30; 260 18 21 24 25 1 q5 qð4Þ U33; 260 1 q5 qð4Þ U35; 260 1 q5 qð4Þ U30; 260 1 q5 qð4Þ U33; 260 1 q5 q26 ð4Þ U35; 260 30 1 2 6 1 q5 q28 U35; 260 1 q q U35; 260 1 q q U0 1 q q U17; 260 1 q q U18; 260 5 ð4Þ 6 ð4Þ 6 ð4Þ 6 ð4Þ ð4Þ 9 11 12 13 14 1 q6 qð4Þ U20; 260 1 q6 qð4Þ U23; 260 1 q6 qð4Þ U20; 260 1 q6 qð4Þ U20; 260 1 q6 qð4Þ U22; 260 18 21 24 25 1 q6 q16 ð4Þ U30; 260 1 q6 qð4Þ U33; 260 1 q6 qð4Þ U35; 260 1 q6 qð4Þ U30; 260 1 q6 qð4Þ U33; 260 28 30 1 q6 q26 ð4Þ U35; 260 1 q6 qð4Þ U35; 260 1 q6 qð4Þ U35; 260;

Lð5Þ ð1ÞD

6 X 23 X b51

(7.129)

h iðbÞ qb Uqið5Þ U Lið5Þ ð1Þ 5 q1 q1ð5Þ U0 1 q1 q5ð5Þ U6260 1 q1 q6ð5Þ U10; 260

i51

1 5 6 1 q1 q9ð5Þ U15; 260 1 q1 q11 ð5Þ U20; 260 1 q2 qð5Þ U0 1 q1 qð5Þ U6260 1 q1 qð5Þ U10; 260 9 11 1 5 1 q1 qð5Þ U15; 260 1 q1 qð5Þ U20; 260 1 q3 qð5Þ U0 1 q3 qð5Þ U6260 1 q3 q6ð5Þ U10; 260 1 5 6 1 q3 q9ð5Þ U15; 260 1 q3 q11 ð5Þ U20; 260 1 q4 qð5Þ U0 1 q4 qð5Þ U6260 1 q4 qð5Þ U10; 260 9 11 1 5 1 q4 qð5Þ U15; 260 1 q4 qð5Þ U20; 260 1 q5 qð5Þ U0 1 q5 qð5Þ U6260 1 q5 q6ð5Þ U10; 260 1 5 6 1 q5 q9ð5Þ U15; 260 1 q5 q11 ð5Þ U20; 260 1 q6 qð5Þ U0 1 q6 qð5Þ U6260 1 q6 qð5Þ U10; 260 9 11 1 q6 qð5Þ U15; 260 1 q6 qð5Þ U20; 260:

(7.130)

Next, considering the approximate limit values of transient probabilities qb, b 5 1; 2; . . .; 6 of the climateweather change process C(t) at the climateweather states cb, b 5 1; 2; . . .; 6 for Gdynia and Karlskrona ports, respectively, given by (6.73) and (6.78) as well as based on the lower q iðkÞ and upper _ q iðkÞ bounds of the unknown transient probabilii ties qðkÞ , k 5 1,2, . . .,5, i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 coming from experts and proceedings in the same way as in Section 7.3.2.1, we determined the optimal values qb q_iðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23 of limit transient probabilities qb qiðkÞ ; b 5 1; 2; . . .; 6; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; ‘5 5 23: Thus we minimize mean values of the losses associated with the process of the environment degradation in the subarea Dk, k 5 1; 2; . . .; 5 impacted by the climateweather change process C(t) and expressed by the linear forms (7.126)(7.130) giving, according to (7.34), their optimal value (in PLN): G

for the Gdynia Port _ _ _ _ L_ ð1Þ ð1ÞD2:116; Lð2Þ ð1ÞD2:706; Lð3Þ ð1ÞD3:973; Lð4Þ ð1ÞD3:916; Lð5Þ ð1ÞD2:134;

(7.131)

and from the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of the Gdynia Port impacted by the climateweather change process C(t)

Optimization of critical infrastructure accident losses Chapter | 7

_ 5 Lð1Þ

5 X

L_ ðkÞ ð1ÞD14:845

169

(7.132)

k51 G

for the Karlskrona Port _ _ _ _ L_ ð1Þ ð1ÞD2:118; Lð2Þ ð1ÞD2:709; Lð3Þ ð1ÞD3:977; Lð4Þ ð1ÞD3:916; Lð5Þ ð1ÞD2:134;

(7.133)

and from the above, we get the optimal value (in PLN) of total losses associated with the process of the environment degradation in the entire area D of the Karlskrona Port impacted by the climateweather change process C(t) _ 5 Lð1Þ

5 X

L_ ðkÞ ð1ÞD14:854:

(7.134)

k51

Thus considering (7.110) and (7.131), (7.133), respectively, and according to (7.35), the indicators RI_ðkÞ of the environment in the subarea Dk, k 5 1; 2; . . .; 5 resilience to the losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amount to the following: G

G

for the Gdynia Port RI_ð1Þ 5 0:998; RI_ð2Þ 5 0:996; RI_ð3Þ 5 0:998; RI_ð4Þ 5 1; RI_ð5Þ 5 1

(7.135)

RI_ð1Þ 5 0:996; RI_ð2Þ 5 0:995; RI_ð3Þ 5 0:997; RI_ð4Þ 5 1; RI_ð5Þ 5 1;

(7.136)

for the Karlskrona Port

and considering (7.111), and (7.132), (7.134), respectively, and according to (7.36), the indicator RI_ of the environment in the entire area D resilience to the total losses associated with the critical infrastructure accident related to the climateweather change, after the optimization, amounts to the following: RI_ 5 0:998 ðfor the Gdynia PortÞ;

(7.137)

RI_ 5 0:998 ðfor the Karlskrona PortÞ:

(7.138)

and

Chapter 8

Mitigation of critical infrastructure accident losses Chapter Outline 8.1 Inventory of accident losses at Baltic Sea waters 8.2 Inventory of accident losses at world sea waters

171 171

8.3 Strategy for mitigation of critical infrastructure accident losses

171

The knowledge of optimal transient probabilities q_iðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; and ‘5 5 23 at the particular environment degradation states of the process of the environment degradation may be the basis to mitigate the environment consequences in both cases without and with considering the climate weather change process impact. This knowledge allows one to compare the values of losses associated with the shipping critical infrastructure accidents without and with considering the climateweather change impact and resilience indicators before optimization determined in Section 6.3 to those after its optimization determined in Section 7.3. The proposed strategy is to intuitively organize the process of the environment degradation in the way that makes the limit transient probabilities qiðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; and ‘5 5 23 at the particular states before the optimization, given by (5.367)(5.374) for data collected at the Baltic Sea waters and by (5.380)(5.384) for wider data collected at the world sea waters, approximately convergent to their optimal values q_iðkÞ ; k 5 1; 2; . . .; 5; i 5 1; 2; . . .; ‘k ; ‘1 5 30; ‘2 5 28; ‘3 5 28; ‘4 5 31; and ‘5 5 23 established in Section 7.3.

8.1

Inventory of accident losses at Baltic Sea waters

To analyze the results of optimization performed in Sections 7.3.1 and 7.3.2, the inventories of losses associated with the shipping critical infrastructure accident without and with considering the climateweather change impact, based on the data collected at the Baltic Sea waters, before and after optimization are presented in Tables 8.18.4. Moreover, the values of resilience indicators for these losses impacted by the climateweather change before and after optimization are also given in these tables.

8.2

Inventory of accident losses at world sea waters

The results of optimization performed in Sections 7.3.3 and 7.3.4 on the basis of wider data collected at the world sea waters allow for more accurate analysis. The inventories of losses associated with the shipping critical infrastructure accident without and with considering the climateweather change impact, based on the data collected at the world sea waters, before and after optimization are presented in Tables 8.58.8. Moreover, the values of resilience indicators for these losses impacted by the climateweather change before and after optimization are also given in these tables.

8.3

Strategy for mitigation of critical infrastructure accident losses

The comparison of values of losses associated with the shipping critical infrastructure accident without and with considering the climateweather change impact and values of resilience indicators before and after losses optimization confirms and justifies the reasonableness of the critical infrastructure accident losses optimization. It can be the basis of some suggestions on new strategy assuring lower environment losses concerned with chemical releases generated by an accident of ships operating within the shipping critical infrastructure network. Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00008-5 © 2020 Elsevier Inc. All rights reserved.

171

172

Consequences of Maritime Critical Infrastructure Accidents

TABLE 8.1 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the open sea waters before and after optimization (results based on the data collected within the Baltic Sea). Before optimization LðkÞ ð1Þ 0.785 (k 5 1) 2.467 (k 5 2) 3.091 (k 5 3) 3.072 (k 5 4) 0 (k 5 5)

Lð1Þ 9.415

LðkÞ ð1Þ 0.798 (k 5 1) 2.900 (k 5 2) 3.611 (k 5 3) 3.072 (k 5 4) 0 (k 5 5)

Lð1Þ 10.381

RIðkÞ 0.984 (k 5 1) 0.851 (k 5 2) 0.856 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

RI 0.907

After optimization L_ ðkÞ ð1Þ

_ Lð1Þ

0.570 (k 5 1) 1.437 (k 5 2) 1.980 (k 5 3) 1.930 (k 5 4) 0 (k 5 5)

5.917

L_ ðkÞ ð1Þ 0.575 (k 5 1) 1.570 (k 5 2) 2.141 (k 5 3) 1.930 (k 5 4) 0 (k 5 5)

_ Lð1Þ 6.216

R _I ðkÞ

R _I

0.991 (k 5 1) 0.915 (k 5 2) 0.925 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

0.952

TABLE 8.2 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the restricted sea waters before and after optimization (results based on the data collected within the Baltic Sea). Before optimization LðkÞ ð1Þ 0.785 (k 5 1) 2.467 (k 5 2) 3.091 (k 5 3) 3.072 (k 5 4) 0 (k 5 5)

Lð1Þ 9.415

LðkÞ ð1Þ 0.796 (k 5 1) 2.925 (k 5 2) 3.660 (k 5 3) 3.072 (k 5 4) 0 (k 5 5)

L_ ðkÞ ð1Þ

_ Lð1Þ

0.570 (k 5 1) 1.437 (k 5 2) 1.980 (k 5 3) 1.930 (k 5 4) 0 (k 5 5)

5.917

L_ ðkÞ ð1Þ 0.570 (k 5 1) 1.577 (k 5 2) 2.157 (k 5 3) 1.930 (k 5 4) 0 (k 5 5)

Lð1Þ 10.453

RIðkÞ 0.986 (k 5 1) 0.843 (k 5 2) 0.845 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

RI 0.901

_ Lð1Þ 6.234

R _I ðkÞ

R _I

1.000 (k 5 1) 0.911 (k 5 2) 0.918 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

0.949

After optimization

TABLE 8.3 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the Gdynia Port before and after optimization (results based on the data collected within the Baltic Sea). Before optimization LðkÞ ð1Þ 1.457 (k 5 1) 3.145 (k 5 2) 3.769 (k 5 3) 3.750 (k 5 4) 0 (k 5 5)

Lð1Þ 12.121

LðkÞ ð1Þ 1.481 (k 5 1) 3.195 (k 5 2) 3.800 (k 5 3) 3.750 (k 5 4) 0 (k 5 5)

Lð1Þ 12.226

RIðkÞ 0.984 (k 5 1) 0.984 (k 5 2) 0.992 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

RI 0.991

After optimization L_ ðkÞ ð1Þ

_ Lð1Þ

1.149 (k 5 1) 2.016 (k 5 2) 2.559 (k 5 3) 2.509 (k 5 4) 0 (k 5 5)

8.233

L_ ðkÞ ð1Þ 1.155 (k 5 1) 2.031 (k 5 2) 2.568 (k 5 3) 2.509 (k 5 4) 0 (k 5 5)

_ Lð1Þ 8.263

R _I ðkÞ

R _I

0.995 (k 5 1) 0.993 (k 5 2) 0.997 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

0.996

Mitigation of critical infrastructure accident losses Chapter | 8

173

TABLE 8.4 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the Karlskrona Port before and after optimization (results based on the data collected within the Baltic Sea). Before optimization LðkÞ ð1Þ 1.457 (k 5 1) 3.145 (k 5 2) 3.769 (k 5 3) 3.750 (k 5 4) 0 (k 5 5)

Lð1Þ 12.121

LðkÞ ð1Þ 1.489 (k 5 1) 3.214 (k 5 2) 3.811 (k 5 3) 3.750 (k 5 4) 0 (k 5 5)

L_ ðkÞ ð1Þ

_ Lð1Þ

1.149 (k 5 1) 2.016 (k 5 2) 2.559 (k 5 3) 2.509 (k 5 4) 0 (k 5 5)

8.233

L_ ðkÞ ð1Þ 1.157 (k 5 1) 2.037 (k 5 2) 2.572 (k 5 3) 2.509 (k 5 4) 0 (k 5 5)

Lð1Þ 12.264

RIðkÞ 0.979 (k 5 1) 0.979 (k 5 2) 0.989 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

RI 0.988

_ Lð1Þ 8.275

R _I ðkÞ

R _I

0.993 (k 5 1) 0.990 (k 5 2) 0.995 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

0.995

After optimization

TABLE 8.5 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the open sea waters before and after optimization (results based on the data collected within the world sea waters). Before optimization LðkÞ ð1Þ 1.211 (k 5 1) 2.781 (k 5 2) 4.170 (k 5 3) 4.005 (k 5 4) 0 (k 5 5)

Lð1Þ 12.167

LðkÞ ð1Þ 1.232 (k 5 1) 3.269 (k 5 2) 4.871 (k 5 3) 4.005 (k 5 4) 0 (k 5 5)

Lð1Þ 13.377

RIðkÞ 0.983 (k 5 1) 0.851 (k 5 2) 0.856 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

RI 0.910

After optimization L_ ðkÞ ð1Þ

_ Lð1Þ

1.007 (k 5 1) 1.592 (k 5 2) 2.859 (k 5 3) 2.811 (k 5 4) 0 (k 5 5)

8.269

L_ ðkÞ ð1Þ 1.009 (k 5 1) 1.668 (k 5 2) 2.984 (k 5 3) 2.811 (k 5 4) 0 (k 5 5)

_ Lð1Þ 8.472

R _I ðkÞ

R _I

0.998 (k 5 1) 0.954 (k 5 2) 0.958 (k 5 3) 1.000 (k 5 4) n/a (k 5 5)

0.976

TABLE 8.6 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the restricted sea waters before and after optimization (results based on the data collected within the world sea waters). Before optimization LðkÞ ð1Þ 1.211 (k 5 1) 2.781 (k 5 2) 4.170 (k 5 3) 4.005 (k 5 4) 2.598 (k 5 5)

Lð1Þ 14.765

LðkÞ ð1Þ 1.228 (k 5 1) 3.297 (k 5 2) 4.937 (k 5 3) 4.005 (k 5 4) 2.598 (k 5 5)

Lð1Þ 16.065

RIðkÞ 0.986 (k 5 1) 0.844 (k 5 2) 0.845 (k 5 3) 1.000 (k 5 4) 1.000 (k 5 5)

RI 0.919

After optimization L_ ðkÞ ð1Þ

_ Lð1Þ

1.007 (k 5 1) 1.592 (k 5 2) 2.859 (k 5 3) 2.811 (k 5 4) 1.030 (k 5 5)

9.299

L_ ðkÞ ð1Þ 1.009 (k 5 1) 1.672 (k 5 2) 2.993 (k 5 3) 2.811 (k 5 4) 1.030 (k 5 5)

_ Lð1Þ 9.515

R _I ðkÞ

R _I

0.998 (k 5 1) 0.952 (k 5 2) 0.955 (k 5 3) 1.000 (k 5 4) 1.000 (k 5 5)

0.977

174

Consequences of Maritime Critical Infrastructure Accidents

TABLE 8.7 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the Gdynia Port before and after optimization (results based on the data collected within the world sea waters). Before optimization LðkÞ ð1Þ 2.481 (k 5 1) 4.097 (k 5 2) 5.475 (k 5 3) 5.288 (k 5 4) 3.906 (k 5 5)

Lð1Þ 21.247

LðkÞ ð1Þ 2.520 (k 5 1) 4.162 (k 5 2) 5.519 (k 5 3) 5.288 (k 5 4) 3.906 (k 5 5)

L_ ðkÞ ð1Þ

_ Lð1Þ

2.111 (k 5 1) 2.696 (k 5 2) 3.963 (k 5 3) 3.916 (k 5 4) 2.134 (k 5 5)

14.820

L_ ðkÞ ð1Þ 2.116 (k 5 1) 2.706 (k 5 2) 3.973 (k 5 3) 3.916 (k 5 4) 2.134 (k 5 5)

Lð1Þ 21.395

RIðkÞ 0.985 (k 5 1) 0.984 (k 5 2) 0.992 (k 5 3) 1.000 (k 5 4) 1.000 (k 5 5)

RI 0.993

_ Lð1Þ 14.845

R _I ðkÞ

R _I

0.998 (k 5 1) 0.996 (k 5 2) 0.998 (k 5 3) 1.000 (k 5 4) 1.000 (k 5 5)

0.998

After optimization

TABLE 8.8 Shipping critical infrastructure accident losses (in PLN), during the time t 5 1 h, and resilience indicators for the Karlskrona Port before and after optimization (results based on the data collected within the world sea waters). Before optimization LðkÞ ð1Þ 2.481 (k 5 1) 4.097 (k 5 2) 5.475 (k 5 3) 5.288 (k 5 4) 3.906 (k 5 5)

Lð1Þ 21.247

LðkÞ ð1Þ 2.535 (k 5 1) 4.187 (k 5 2) 5.535 (k 5 3) 5.288 (k 5 4) 3.906 (k 5 5)

L_ ðkÞ ð1Þ

_ Lð1Þ

2.111 (k 5 1) 2.696 (k 5 2) 3.963 (k 5 3) 3.916 (k 5 4) 2.134 (k 5 5)

14.820

L_ ðkÞ ð1Þ 2.118 (k 5 1) 2.709 (k 5 2) 3.977 (k 5 3) 3.916 (k 5 4) 2.134 (k 5 5)

Lð1Þ 21.451

RIðkÞ 0.979 (k 5 1) 0.979 (k 5 2) 0.989 (k 5 3) 1.000 (k 5 4) 1.000 (k 5 5)

RI 0.991

_ Lð1Þ 14.854

R _I ðkÞ

R _I

0.996 (k 5 1) 0.995 (k 5 2) 0.997 (k 5 3) 1.000 (k 5 4) 1.000 (k 5 5)

0.998

After optimization

From the performed analysis of the losses concerned with consequences of chemical releases at sea before and after optimization, it can be suggested to modify the process of accident initiating events and the process of environment threats in the way causing the replacing approximately their conditional mean sojourn times at their particular states before the optimization by their optimal values after the optimization. Instead of this practically difficult modification, it seems to be easier to change the process of accident initiating events and the process of environment threats characteristics. This change depends on replacing approximately the unconditional mean sojourn times of these processes at particular states before the optimization by their optimal values after the optimization. The easiest way of the modification of these processes is leading to a replacing approximately the total sojourn times of the process of accident initiating events and the process of environment threats at their particular states during the fixed time before the optimization by their optimal values after the optimization. The suggestions on the way of minimizing the environment losses coming directly from the practice are the basis for creating the general procedures and new strategies assuring the critical infrastructures accident consequences

Mitigation of critical infrastructure accident losses Chapter | 8

175

decreasing the environment losses. In practice, it includes the following proactive and reactive strategies (HELCOM, 2002; IMO, 2002; Kristiansen, 2005; Mamaca et al., 2009): G

G

G G

G

Preventive measures to eliminate or reduce accidents at sea (establish and revision of national laws and regulations, IMO conventions and resolutions, inspection, certification and auditing, maintenance of the ship and equipment, and reduction traffic congestion) Investigation of accidents and learning from experience (identify the causes and potential measures that will reduce the threats and degradation effects of similar accidents in the future) Identification of hazard and possible events that may cause threats and result in severity of degradation effects Emergency preparedness (preparation and revision of emergency action plan, high-quality equipment for combating released substances, rescuers training) Reducing the time of emergency-response process, quickly undertaking a proper decision and action, selecting the best response method (recommendation for decision-making, decision support, cooperation with external parties and exchange of information, tools for forecasting, and equipment for monitoring the spread or drift of released substances)

Chapter 9

Summary Models, methods, procedures, and tools presented in this book are supposed to be very useful in modeling, identifying, predicting, optimizing, and mitigating the losses associated with the consequences of critical infrastructure accidents. The constructed general model of critical infrastructure accident consequences is applied to the maritime critical infrastructure accident consequences caused by ships operating at sea waters and chemical releases. Initially, the general model is applied to the examination of the critical infrastructure accident consequences caused by the ships operating at the Baltic Sea. However, the lack of sufficient statistical data of ship accidents at the Baltic Sea (data coming only from 104 accidents) showed that the obtained results are only straight possible examples of the proposed methods and application procedures. More accurate results are obtained for examining the critical infrastructure accident consequences caused by the ships operating at world sea waters as they are obtained on the basis of wider data coming from 1630 accidents. Next, using these results based on wider statistical data and assuming that similar accidents can happen within the Baltic Sea area, the general model is applied to this area. The designed general model of critical accident consequences is also applied to cost analysis of losses associated with consequences of chemical releases at the Baltic Sea area. The analysis is performed without and with considering the climate weather change impact on losses. Unfortunately, the cost analysis of accidents performed in the book is not very accurate as the necessary data concerned with the cost analysis of consequences associated with the particular environment degradation and the climate weather impact are not sufficiently precise. Nevertheless, the reader can learn how to work out such analyses. Moreover, in this book, there are given procedures to optimize the accident consequences cost through the minimization of accident losses and their applications. The results of the performed optimization can be an interesting guide for emergency rescuers and services as well as other government, administrative, and technical services that bear costs to remove and/or reduce the critical infrastructure accident consequences. Only the influence of the wind speed, the wave height in the sea area, and the wind direction in ports are taken into account and examined in investigating the climate weather change impact on losses associated with the environment degradation caused by a ship accident and a chemical release, and coming from its threats. The reason for this is accessing and having only such weather parameters. Nevertheless, it seems to be possible to take into account the other weather parameters (e.g., the temperature of air, the temperature of sea water, and others) that are crucial for the range of the environment degradations caused by chemical releases and consequently significant for losses associated with them. This problem can be transferred and applied in these researches and investigations. The proposed general model of critical infrastructure accident consequences is a universal tool that can have a wide range of applications in various industrial sectors. Although the general model has been designed for the maritime critical infrastructure, it can also be applied to the identification, prediction, optimization, and mitigation of losses associated with chemical releases generated by other critical infrastructures, industrial installations, and systems. To aid this transfer and application, a guide, including detailed procedures of modeling, identification, prediction, optimization, and mitigation, has been proposed in Appendix 8. After applying the procedures from this guide and analyzing the obtained results, a new strategy assuring lower consequences of the considered critical infrastructure accident can be created through the modifying the processes of initiating events, environment threats and environment degradation.

Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00009-7 © 2020 Elsevier Inc. All rights reserved.

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Appendices

Appendix 1 Realizations of conditional lifetimes at states of process of initiating events at Baltic Sea waters lj

Transitions el -ej

Transition time θlj (min)

Number of transitions nlj

Mean value θ (min)

A

B

C

D

8935,200, 10,512,000, 4730,400, 14,191,200, 4204,800, 15,768,000, 11,563,200, 10,512,000, 4204,800, 13,665,600, 3153,600, 4204,800, 11,563,200, 18,396,000, 11,037,600, 15,768,000, 14,191,200, 7884,000 3153,600, 8935,200, 8409,600, 2102,400, 8409,600, 1576,800, 12,614,400, 4730,400, 14,716,800, 1051,200, 6307,200, 3679,200, 8409,600, 15,242,400, 2102,400, 6832,800, 262,800, 11,563,200, 9986,400, 12,614,400, 3679,200, 13,140,000, 15,768,000, 525,600, 525,600, 13,665,600, 19,972,800, 12,088,800, 16,819,200, 18,921,600, 6832,800, 9986,400, 7884,000, 5256,000, 20,498,400, 1051,200, 21,549,600 10,512,000, 18,396,000, 5781,600, 4730,400, 23,652,000 1051,200, 16,293,600, 15,768,000, 14,716,800, 14,716,800, 38,368,800, 16,819,200, 11,563,200, 2102,400, 2628,000 14,716,800, 2628,000, 3679,200, 14,716,800, 1051,200, 10,512,000, 10,512,000, 2102,400, 6832,800, 1051,200, 11,563,200, 6832,800, 9460,800, 19,447,200, 12,614,400, 22,600,800, 4730,400, 3679,200, 1051,200, 14,191,200, 1576,800, 14,716,800, 1576,800, 16,819,200 1576,800, 18,921,600, 10,512,000, 5256,000, 2628,000, 262,800, 525,600, 12,614,400, 525,600 1576,800 All 1 30, 15 1 10, 1

18

10,249,200

37

8942,302.70

5

12,614,400

10

13,402,800

24

8694,300

9

5869,200

1 15 2 1 2

1576,800 1 22.50 1 5.50

e -e 1

2

e 1 -e 3

e 1 -e 4 e 1 -e 5

e 1 -e 6

e 1 -e 7 e 1 -e 8 e 2 -e 1 e 2 -e 3 e 2 -e 6 e 2 -e 7

183

184

Appendices

e 3 -e 2

e 4 -e 1 e 4 -e 7 e 5 -e 1 e 5 -e 10 e 6 -e 1 e 6 -e 2 e 6 -e 3 e 6 -e 4 e 7 -e 1 e 7 -e 3 e 7 -e 13 e 8 -e 1 e 10 -e 1 e 13 -e 1

60, 2180, 5040, 360, 820, 30, 240, 240, 120, 120, 30, 11,520, 255,760, 120, 1440, 60, 30, 1440, 5760, 720, 720, 720, 30, 600, 720, 20, 180, 10, 1440, 4320, 180, 300, 120, 4320, 660, 900, 1440, 1440, 1440, 780, 1200, 2160, 720, 2160, 600, 5760, 2880, 1440, 540, 720, 1200, 20, 360, 660, 300, 120, 1440, 1440, 60, 120, 1440, 1200, 1440, 1440 All 1 1 60, 240, 20, 120, 120, 150, 60, 20, 680 10 120 40, 120 30, 40, 30, 30, 30, 30, 20, 10, 10, 30, 60, 15, 360, 30, 60, 120, 20, 90, 20, 10, 5760 15 60, 30, 15, 30, 1200, 20 6, 20, 10, 12, 60 1 5 10 10

65

1321.15

5 1 9 1 1 2 21

1 1 163.33 10 120 80 324.05

1 6 5 1 1 1 1

15 225.83 21.60 1 5 10 10

Appendix 2 Realizations of conditional lifetimes at states of process of initiating events at world sea waters lj

Transitions el -ej

Transition time θlj (min)

Number of transitions nlj

Mean value θ (min)

A

B

C

D

8935,200, 2102,400, 7884,000, 11,037,600, 2102,400, 1576,800, 12,614,400, 9460,800, 8409,600, 10,512,000, 3679,200, 9460,800, 1576,800, 7884,000, 10,512,000, 262,800, 14,191,200, 5256,000, 4730,400, 4730,400, 3679,200, 4730,400, 2102,400, 13,140,000, 11,563,200, 9460,800, 1576,800, 10,512,000, 13,665,600, 6832,800, 1051,200, 12,088,800, 7358,400, 9460,800, 9986,400, 525,600, 11,037,600, 4204,800, 14,191,200, 13,140,000, 12,614,400, 4730,400, 2102,400, 7884,000, 4730,400, 1051,200, 1576,800, 9986,400, 4204,800, 3679,200, 5256,000, 10,512,000, 5256,000, 8409,600, 8409,600, 16,819,200, 1576,800, 1051,200, 3679,200, 1576,800, 12,614,400, 4204,800, 9460,800, 4204,800, 15,768,000, 14,716,800, 10,512,000, 13,665,600, 15,242,400, 1576,800, 17,870,400, 4730,400, 12,088,800, 9460,800, 18,396,000, 3153,600, 11,563,200, 15,768,000, 12,614,400, 12,614,400, 16,293,600, 4204,800, 15,242,400, 1314,000,014,191,200, 11,563,200, 12,088,800, 8935,200, 12,088,800, 7358,400, 12,088,800, 1576,800, 13,140,000, 13,665,600, 3679,200, 7358,400, 1576,800, 11,037,600, 6307,200, 9460,800, 6307,200, 1051,200, 6832,800, 7358,400, 10,512,000,

401

8745,800.5

e -e 1

2

Appendices

13,140,000, 8935,200, 7884,000, 4204,800, 13,665,600, 18,921,600, 4204,800, 6307,200, 10,512,000, 3679,200, 12,088,800, 14,191,200, 7358,400, 12,088,800, 4204,800, 262,800, 14,716,800, 17,870,400, 7358,400, 525,600, 18,921,600, 4204,800, 6307,200, 3679,200, 262,800, 12,614,400, 14,716,800, 6832,800, 3679,200, 13,665,600, 6307,200, 1576,800, 6307,200, 2102,400, 7358,400, 4730,400, 4730,400, 1051,200, 15,768,000, 11,037,600, 5781,600, 7884,000, 7358,400, 14,191,200, 525,600, 1051,200, 13,140,000, 525,600, 3153,600, 1576,800, 2102,400, 15,242,400, 2102,400, 15,242,400, 6832,800, 262,800, 15,242,400, 16,293,600, 5781,600, 4204,800, 12,614,400, 16,293,600, 7358,400, 8409,600, 15,768,000, 5256,000, 1051,200, 13,665,600, 3679,200, 2102,400, 9986,400, 4730,400, 10,512,000, 16,819,200, 1576,800, 18,396,000, 7358,400, 11,563,200, 1051,200, 1576,800, 4730,400, 1051,200, 13,140,000, 1051,200, 17,344,800, 1051,200, 14,191,200, 18,921,600, 6832,800, 12,614,400, 6832,800, 10,512,000, 15,768,000, 6307,200, 21,024,000, 11,563,200, 10,512,000, 1576,800, 12,088,800, 7884,000, 10,512,000, 10,512,000, 16,819,200, 16,293,600, 11,037,600, 13,140,000, 2628,000, 11,563,200, 7884,000, 16,819,200, 26,280,000, 15,768,000, 6832,800, 18,396,000, 3153,600, 11,037,600, 11,037,600, 12,088,800, 11,563,200, 2102,400, 10,512,000, 13,140,000, 5256,000, 6307,200, 1051,200, 3153,600, 7884,000, 17,344,800, 1576,800, 525,600, 262,800, 8409,600, 16,293,600, 18,921,600, 8935,200, 13,665,600, 14,191,200, 11,563,200, 13,665,600, 16,293,600, 16,293,600, 4730,400, 3153,600, 15,768,000, 7358,400, 10,512,000, 1576,800, 17,344,800, 16,293,600, 11,037,600, 12,614,400, 2628,000, 7358,400, 1051,200, 13,665,600, 6832,800, 4204,800, 1051,200, 9986,400, 5256,000, 2628,000, 21,549,600, 18,921,600, 1051,200, 14,191,200, 525,600, 12,088,800, 2102,400, 9986,400, 5781,600, 2102,400, 21,549,600, 2102,400, 11,037,600, 15,768,000, 12,614,400, 262,800, 11,037,600, 6832,800, 1051,200, 11,037,600, 8409,600, 15,242,400, 262,800,007,358,400, 13,665,600, 6832,800, 7884,000, 11,037,600, 7358,400, 8409,600, 13,140,000, 6832,800, 21,549,600, 525,600, 2628,000, 525,600, 4730,400, 6832,800, 16,293,600, 10,512,000, 22,075,200, 7884,000, 12,614,400, 12,088,800, 16,819,200,

185

186

Appendices

e 1 -e 3

10,512,000, 1051,200, 9986,400, 525,600, 3153,600, 1576,800, 17,870,400, 1051,200, 5781,600, 1576,800, 22,075,200, 11,037,600, 2102,400, 5256,000, 15,768,000, 11,563,200, 14,191,200, 9460,800, 8409,600, 262,800, 3153,600, 6832,800, 14,191,200, 6832,800, 1576,800, 9460,800, 8409,600, 2102,400, 2102,400, 1576,800, 2628,000, 3679,200, 24,177,600, 8935,200, 1051,200, 26,805,600, 3679,200, 7358,400, 19,972,800, 8935,200, 14,716,800, 4204,800, 2628,000, 17,344,800, 8409,600, 21,549,600, 6307,200, 7884,000, 36,266,400, 6307,200, 4204,800, 13,140,000, 13,140,000, 525,600, 525,600, 6307,200, 11,563,200, 1576,800, 19,972,800, 15,768,000, 15,768,000, 26,280,000, 22,600,800, 2628,000, 7884,000, 22,600,800, 3679,200, 2628,000, 7358,400, 10,512,000, 3153,600, 525,600, 8935,200, 4730,400, 2628,000, 9460,800, 9986,400, 1576,800, 8935,200, 3153,600, 8935,200, 1576,800, 12,614,400, 3153,600, 13,665,600, 6832,800, 7884,000, 8409,600, 3153,600, 1051,200 7884,000, 3153,600, 8935,200, 8935,200, 8409,600, 2102,400, 5256,000, 8409,600, 2628,000, 1576,800, 11,037,600, 3679,200, 1051,200, 17,870,400, 3679,200, 8935,200, 13,140,000, 4204,800, 525,600, 15,242,400, 4204,800, 3679,200, 13,140,000, 5781,600, 12,614,400, 7884,000, 13,140,000, 4730,400, 15,768,000, 9986,400, 14,716,800, 1051,200, 14,716,800, 6832,800, 16,293,600, 1051,200, 15,768,000, 11,037,600, 13,140,000, 7884,000, 7884,000, 6307,200, 11,563,200, 18,921,600, 3679,200, 8409,600, 15,242,400, 15,768,000, 2628,000, 10,512,000, 2628,000, 1051,200, 2102,400, 3679,200, 10,512,000, 4730,400, 12,088,800, 12,614,400, 8935,200, 7884,000, 15,242,400, 8935,200, 15,242,400, 14,716,800, 8409,600, 6832,800, 3679,200, 1576,800, 3153,600, 14,191,200, 20,498,400, 10,512,000, 6307,200, 2102,400, 262,800, 3679,200, 11,563,200, 5256,000, 6307,200, 9460,800, 7358,400, 12,088,800, 262,800, 9986,400, 18,396,000, 12,614,400, 3679,200, 14,716,800, 3153,600, 1576,800, 19,972,800, 1576,800, 13,140,000, 20,498,400, 13,140,000, 2102,400, 4730,400, 3153,600, 13,665,600, 17,344,800, 15,768,000, 4204,800, 9986,400, 11,037,600, 525,600, 6832,800, 1576,800, 21,549,600, 22,075,200, 9986,400, 525,600, 6832,800, 9986,400, 22,075,200, 13,665,600, 262,800, 16,819,200, 5781,600,

369

11,095,287.8

Appendices

14,191,200, 3679,200, 4204,800, 15,242,400, 6307,200, 5256,000, 2628,000, 12,614,400, 15,768,000, 4204,800, 9986,400, 19,972,800, 1051,200, 12,614,400, 20,498,400, 14,716,800, 3153,600, 525,600, 19,972,800, 11,037,600, 12,088,800, 16,819,200, 1576,800, 13,665,600, 1051,200, 525,600, 13,665,600, 8409,600, 13,140,000, 19,447,200, 9986,400, 5781,600, 15,242,400, 6307,200, 3679,200, 2102,400, 7358,400, 18,921,600, 13,140,000, 13,665,600, 1051,200, 5256,000, 525,600, 6832,800, 15,242,400, 15,242,400, 20,498,400, 23,126,400, 26,280,000, 6832,800, 17,344,800, 31,536,000, 7884,000, 14,716,800, 16,819,200, 13,665,600, 7358,400, 21,549,600, 525,600, 22,075,200, 21,024,000, 19,972,800, 23,126,400, 19,972,800, 6832,800, 15,768,000, 10,512,000, 23,652,000, 21,024,000, 18,396,000, 16,819,200, 13,140,000, 13,665,600, 525,600, 14,191,200, 9986,400, 8935,200, 2628,000, 24,177,600, 18,396,000, 15,768,000, 3153,600, 19,447,200, 26,280,000, 11,037,600, 21,024,000, 5256,000, 5256,000, 3153,600, 262,800, 7884,000, 9986,400, 9986,400, 5256,000, 10,512,000, 9460,800, 18,396,000, 18,396,000, 15,768,000, 18,396,000, 26,280,000, 5256,000, 7884,000, 17,870,400, 11,563,200, 1051,200, 16,293,600, 525,600, 9460,800, 24,177,600, 23,126,400, 1576,800, 12,088,800, 15,768,000, 23,126,400, 4730,400, 17,344,800, 21,024,000, 16,819,200, 16,293,600, 9460,800, 14,716,800, 11,563,200, 10,512,000, 14,191,200, 8409,600, 8409,600, 13,665,600, 18,921,600, 8409,600, 3153,600, 2628,000, 19,972,800, 19,972,800, 23,652,000, 2102,400, 8409,600, 7884,000, 16,293,600, 21,549,600, 12,088,800, 24,177,600, 1051,200, 17,344,800, 2628,000, 3679,200, 15,242,400, 16,293,600, 17,344,800, 11,037,600, 525,600, 525,600, 13,665,600, 21,549,600, 23,652,000, 7358,400, 11,037,600, 27,331,200, 11,037,600, 16,819,200, 15,242,400, 2628,000, 13,665,600, 24,177,600, 14,191,200, 6307,200, 9986,400, 3153,600, 21,549,600, 20,498,400, 3153,600, 12,614,400, 5256,000, 1051,200, 20,498,400, 23,652,000, 5256,000, 2102,400, 11,563,200, 14,191,200, 12,088,800, 3153,600, 19,447,200, 16,819,200, 22,075,200, 22,075,200, 14,716,800, 20,498,400, 12,614,400, 22,075,200, 14,191,200, 7358,400, 21,024,000, 19,447,200, 16,819,200, 12,088,800,

187

188

Appendices

e 1 -e 4

e 1 -e 5

8935,200, 262,800, 14,191,200, 14,716,800, 14,716,800, 7358,400, 3153,600, 6307,200, 19,447,200, 10,512,000, 14,191,200, 10,512,000, 11,563,200, 9986,400, 16,819,200, 1576,800, 11,037,600, 10,512,000, 18,921,600, 3679,200, 7358,400, 25,228,800, 525,600, 7358,400, 22,600,800, 13,665,600, 1051,200, 14,716,800, 6307,200, 1576,800, 1051,200, 16,819,200, 21,549,600, 6832,800, 10,512,000, 17,344,800, 8935,200, 19,972,800, 3153,600, 16,819,200, 2102,400, 2628,000, 1051,200, 9986,400, 17,870,400, 20,498,400, 7358,400, 23,126,400, 19,972,800, 7358,400, 17,344,800, 7884,000, 7884,000, 2102,400, 13,665,600 11,563,200, 11,563,200, 6307,200, 8935,200, 13,140,000, 5256,000, 12,088,800, 4730,400, 262,800, 10,512,000, 7884,000, 5256,000, 18,396,000, 13,140,000, 7358,400, 12,614,400, 13,665,600, 10,512,000, 11,037,600, 6307,200, 11,563,200, 5781,600, 10,512,000, 11,037,600, 13,665,600, 262,800, 12,088,800, 13,665,600, 15,242,400, 13,140,000, 8935,200, 5781,600, 13,665,600, 525,600, 18,396,000, 8935,200, 19,447,200, 20,498,400, 4204,800, 12,088,800, 13,140,000, 5781,600, 262,800, 2102,400, 262,800, 2102,400, 2628,000, 262,800, 1051,200, 14,716,800, 4204,800, 2628,000, 15,242,400, 6307,200, 7358,400, 10,512,000, 5256,000, 15,768,000, 5781,600, 4204,800, 32,587,200, 6832,800, 16,819,200, 9986,400, 10,512,000, 4730,400, 18,921,600, 4730,400, 10,512,000, 23,652,000, 525,600, 2628,000, 12,614,400, 5781,600, 525,600, 8935,200, 17,870,400, 6832,800, 11,563,200, 22,075,200, 26,280,000, 2628,000, 4730,400, 16,293,600, 2102,400, 1576,800, 7358,400, 16,293,600, 13,665,600, 262,800, 2102,400, 1051,200, 19,447,200, 18,921,600, 12,614,400, 18,396,000, 525,600, 1051,200, 17,870,400, 15,242,400, 2628,000, 8935,200, 3153,600, 9986,400, 8409,600, 525,600, 19,447,200, 262,800, 525,600, 2102,400, 4204,800, 2628,000, 8935,200, 8935,200, 19,447,200, 2628,000, 5256,000, 2628,000, 10,512,000, 7884,000, 11,037,600, 7884,000, 3153,600 15,242,400, 14,716,800, 15,242,400, 9460,800, 12,614,400, 18,921,600, 4730,400, 1051,200, 14,191,200, 7358,400, 22,600,800, 14,716,800, 9986,400, 6307,200, 12,088,800, 11,563,200, 16,293,600, 16,293,600,

123

9005,707.32

268

11,267,059.7

Appendices

8409,600, 15,768,000, 13,140,000, 14,716,800, 5781,600, 9986,400, 13,665,600, 14,716,800, 2102,400, 3153,600, 525,600, 14,191,200, 10,512,000, 11,037,600, 1576,800, 13,140,000, 16,293,600, 13,665,600, 11,037,600, 2102,400, 22,075,200, 2102,400, 4204,800, 3153,600, 12,088,800, 14,191,200, 5781,600, 15,242,400, 4730,400, 6307,200, 6307,200, 19,972,800, 9460,800, 5256,000, 1576,800, 10,512,000, 17,344,800, 6307,200, 3153,600, 525,600, 34,164,000, 1051,200, 11,037,600, 12,088,800, 23,652,000, 2628,000, 7358,400, 11,563,200, 14,191,200, 14,191,200, 1051,200, 15,768,000, 10,512,000, 13,140,000, 17,344,800, 13,140,000, 5781,600, 13,140,000, 11,563,200, 7884,000, 14,191,200, 4204,800, 5781,600, 9460,800, 12,088,800, 14,191,200, 16,819,200, 4204,800, 12,088,800, 12,614,400, 13,140,000, 10,512,000, 22,600,800, 3153,600, 14,716,800, 4730,400, 14,716,800, 5781,600, 38,368,800, 4204,800, 16,819,200, 12,088,800, 15,242,400, 22,075,200, 14,716,800, 3679,200, 13,665,600, 6832,800, 5256,000, 7358,400, 2628,000, 16,819,200, 19,972,800, 25,754,400, 12,088,800, 23,126,400, 26,280,000, 14,191,200, 4204,800, 15,768,000, 13,140,000, 15,768,000, 14,191,200, 15,768,000, 8409,600, 15,768,000, 17,344,800, 21,024,000, 15,242,400, 15,768,000, 8409,600, 18,921,600, 10,512,000, 32,061,600, 10,512,000, 23,652,000, 26,805,600, 5256,000, 14,191,200, 6832,800, 9986,400, 9986,400, 4204,800, 11,563,200, 4204,800, 16,819,200, 1051,200, 13,665,600, 20,498,400, 5256,000, 14,191,200, 9460,800, 3679,200, 4730,400, 13,140,000, 18,921,600, 3153,600, 10,512,000, 7358,400, 23,126,400, 13,140,000, 1576,800, 14,191,200, 11,037,600, 18,396,000, 11,037,600, 17,870,400, 5781,600, 1524,240, 10,512,000, 9986,400, 5781,600, 3153,600, 15,768,000, 7884,000, 26,280,000, 6832,800, 4730,400, 21,024,000, 21,549,600, 13,140,000, 16,293,600, 16,293,600, 10,512,000, 11,563,200, 630,720, 9986,400, 2628,000, 11,037,600, 16,293,600, 19,447,200, 262,800, 21,024,000, 7358,400, 14,716,800, 7358,400, 10,512,000, 13,665,600, 5256,000, 16,819,200, 21,024,000, 22,075,200, 1419,120, 1576,800, 14,191,200, 4204,800, 2102,400, 10,512,000, 1576,800, 22,600,800, 18,396,000, 2628,000, 5256,000,

189

190

Appendices

e 1 -e 6

13,140,000, 10,512,000, 13,665,600, 7358,400, 7358,400, 27,856,800, 5781,600, 25,600, 2628,000, 19,972,800, 30,484,800, 2102,400, 1051,200, 5781,600, 1576,800, 262,800, 16,293,600, 1576,800, 11,563,200, 12,614,400, 9460,800, 16,819,200, 9986,400, 7358,400, 1471,680, 3679,200, 5781,600, 525,600, 19,972,800, 8935,200, 14,716,800, 2102,400, 19,447,200, 8935,200, 12,614,400, 9460,800, 19,972,800, 14,716,800, 1576,800, 8409,600, 16,293,600, 16,819,200, 2628,000, 9986,400, 2628,000, 17,344,800, 4204,800, 3153,600, 16,819,200, 10,512,000, 2102,400, 12,614,400, 5256,000, 2628,000, 5256,000, 7884,000, 7884,000 8409,600, 8935,200, 11,563,200, 14,716,800, 13,140,000, 16,293,600, 3679,200, 10,512,000, 19,447,200, 2628,000, 2102,400, 3679,200, 11,037,600, 11,037,600, 13,665,600, 2102,400, 10,512,000, 13,140,000, 8935,200, 8409,600, 15,768,000, 14,716,800, 14,716,800, 12,088,800, 1051,200, 1576,800, 11,037,600, 12,614,400, 5781,600, 5781,600, 15,768,000, 7884,000, 13,140,000, 5781,600, 2628,000, 7884,000, 6832,800, 262,800, 8409,600, 4204,800, 8409,600, 13,140,000, 2102,400, 11,563,200, 7884,000, 13,140,000, 18,921,600, 8409,600, 2628,000, 10,512,000, 19,972,800, 11,563,200, 6832,800, 7884,000, 10,512,000, 9460,800, 9460,800, 2102,400, 4730,400, 11,563,200, 6307,200, 832,800, 5781,600, 17,870,400, 9460,800, 14,191,200, 16,293,600, 1051,200, 5256,000, 1051,200, 2102,400, 1051,200, 6307,200, 6307,200, 17,870,400, 11,563,200, 12,088,800, 17,870,400, 6832,800, 9460,800, 262,800, 15,242,400, 12,614,400, 5256,000, 12,614,400, 10,512,000, 15,768,000, 15,768,000, 15,768,000, 19,447,200, 3153,600, 25,228,800, 17,870,400, 22,075,200, 15,768,000, 12,614,400, 4730,400, 1051,200, 12,614,400, 22,600,800, 13,665,600, 11,563,200, 4204,800, 19,972,800, 2628,000, 7358,400, 525,600, 4730,400, 7884,000, 14,716,800, 3679,200, 3679,200, 1051,200, 6832,800, 6832,800, 13,140,000, 14,191,200, 12,614,400, 1576,800, 15,242,400, 8409,600, 15,768,000, 262,800, 6832,800, 11,037,600, 15,242,400, 5256,000, 12,088,800, 14,716,800, 1576,800, 8409,600, 15,242,400, 21,549,600, 10,512,000, 10,512,000, 13,665,600, 12,088,800, 5781,600, 2628,000, 3679,200, 8409,600, 12,088,800, 18,396,000, 15,242,400, 10,512,000,

234

10,655,753.85

Appendices

e 1 -e 7

21,024,000, 2102,400, 16,819,200, 10,512,000, 15,768,000, 15,768,000, 7884,000, 10,512,000, 5256,000, 21,024,000, 10,512,000, 15,768,000, 21,024,000, 15,768,000, 10,512,000, 10,512,000, 21,024,000, 6307,200, 20,498,400, 5781,600, 15,768,000, 6832,800, 3153,600, 16,819,200, 3153,600, 18,921,600, 262,800, 1576,800, 3153,600, 19,972,800, 7358,400, 13,665,600, 17,870,400, 23,652,000, 15,242,400, 1576,800, 21,549,600, 18,921,600, 3679,200, 31,536,000, 11,037,600, 2102,400, 14,191,200, 48,880,800, 8409,600, 21,549,600, 525,600, 1576,800, 21,549,600, 7884,000, 15,242,400, 6832,800, 5781,600, 1576,800, 14,191,200, 3153,600, 21,024,000, 14,716,800, 5256,000, 1051,200, 10,512,000, 22,075,200, 8409,600, 14,191,200, 23,652,000, 11,037,600, 3153,600, 8935,200, 9986,400, 6832,800, 3679,200, 7884,000, 15,768,000, 22,075,200, 24,177,600, 4204,800, 15,768,000, 2102,400, 9460,800, 9460,800, 19,972,800, 14,716,800, 4204,800, 17,870,400, 3153,600, 17,870,400, 11,563,200, 3153,600, 16,819,200 262,800, 1576,800, 5256,000, 525,600, 4204,800, 15,242,400, 12,088,800, 8409,600, 7884,000, 15,242,400, 4204,800, 14,191,200, 8409,600, 3679,200, 15,768,000, 525,600, 7884,000, 11,037,600, 7358,400, 13,665,600, 11,037,600, 12,614,400, 9460,800, 9460,800, 17,870,400, 9460,800, 12,614,400, 25,228,800, 1576,800, 8409,600, 16,819,200, 8409,600, 13,665,600, 14,716,800, 6307,200, 4204,800, 7884,000, 11,563,200, 4204,800, 14,716,800, 26,280,000, 13,665,600, 14,716,800, 18,921,600, 19,972,800, 21,024,000, 14,716,800, 1576,800, 10,512,000, 3153,600, 15,242,400, 4730,400, 16,819,200, 5256,000, 13,665,600, 9986,400, 17,344,800, 17,344,800, 15,768,000, 5256,000, 21,024,000, 23,126,400, 5781,600, 7358,400, 9460,800, 17,870,400, 3153,600, 15,242,400, 21,024,000, 15,242,400, 3679,200, 14,716,800, 15,768,000, 6307,200, 14,191,200, 21,024,000, 16,819,200, 16,819,200, 14,716,800, 19,972,800, 15,768,000, 12,614,400, 39,945,600, 5256,000, 17,870,400, 5256,000, 19,447,200, 18,921,600, 19,972,800, 18,396,000, 14,716,800, 21,024,000, 3679,200, 5256,000, 9986,400, 9986,400, 14,191,200, 8409,600, 17,870,400, 525,600, 9460,800, 5256,000, 2628,000, 1576,800, 2102,400, 11,037,600, 11,563,200, 15,768,000, 262,800,

205

12,392,622.44

191

192

Appendices

e 1 -e 8

e 2 -e 1 e 2 -e 3

e 2 -e 4 e 2 -e 5 e 2 -e 6 e 2 -e 7 e 2 -e 10 e 2 -e 11

6307,200, 7358,400, 3679,200, 21,549,600, 16,819,200, 21,549,600, 5781,600, 12,088,800, 8409,600, 21,549,600, 14,716,800, 8409,600, 11,563,200, 6307,200, 15,768,000, 5781,600, 16,293,600, 11,037,600, 13,665,600, 21,549,600, 5781,600, 13,665,600, 19,447,200, 14,191,200, 21,549,600, 14,191,200, 3153,600, 23,126,400, 11,563,200, 16,293,600, 13,140,000, 3153,600, 21,024,000, 16,819,200, 15,242,400, 22,075,200, 19,972,800, 6307,200, 15,242,400, 2102,400, 14,191,200, 525,600, 22,075,200, 18,396,000, 19,447,200, 19,972,800, 16,819,200, 20,498,400, 16,819,200, 14,191,200, 22,075,200, 12,088,800, 14,716,800, 26,280,000, 12,088,800, 6832,800, 19,447,200, 19,447,200, 22,075,200, 22,075,200, 3679,200, 20,498,400, 2628,000, 8935,200, 16,293,600, 262,800, 8409,600, 2628,000, 21,024,000, 6832,800, 7358,400, 22,600,800, 22,075,200, 1576,800, 18,921,600, 18,921,600, 15,242,400, 19,972,800, 2628,000, 13,665,600, 525,600, 15,768,000, 22,600,800, 525,600, 23,126,400, 10,512,000, 23,126,400, 7884,000, 2102,400, 18,921,600, 10,512,000, 12,614,400, 7884,000, 525,600, 7884,000, 7884,000 11,037,600, 4204,800, 15,242,400, 3679,200, 6832,800, 11,037,600, 3679,200, 262,800, 525,600, 16,293,600, 14,191,200, 10,512,000, 9460,800, 525,600, 525,600, 4204,800, 4204,800, 21,024,000, 2628,000, 1576,800, 16,819,200, 4730,400, 1576,800, 525,600, 1051,200, 7884,000, 7884,000, 7884,000, 1576,800, 7358,400 All 1 40, 30, 10, 10, 15, 60, 60, 240, 15, 10, 10, 45, 15, 10, 20, 15, 20, 60, 10, 15, 20, 240, 240, 10, 10, 10, 10, 10, 300, 10, 10, 30, 10, 10, 10, 10, 20, 10, 10, 10, 30, 30, 10, 60, 30, 10, 30, 10, 15, 20, 35, 30, 30, 10, 15, 30, 20, 30, 15, 20, 600, 30, 1860, 20, 20, 30, 30, 60, 15, 20, 10, 20, 15, 10, 30, 15, 30, 30, 10, 10, 30, 15, 20, 10, 20, 30, 20, 30, 10, 30, 30, 15, 2, 20, 50, 840, 20, 60, 120, 10, 15, 30, 60, 15, 10, 30, 60, 30, 10, 5, 10, 15, 30, 60, 10, 15, 30, 10, 20, 15, 10, 30, 40, 60 1, 40, 5 5, 10, 5, 1 10, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 30, 60, 10, 1, 10, 10, 10, 15, 1, 1, 10, 1, 1, 1, 5, 1, 1, 1, 10, 1, 5, 10, 1, 1, 1, 1, 1, 1, 1, 60, 1, 1, 1, 1, 1, 10, 1, 1 1, 5 1

30

198,939,600

237 125

1 42.12

3 4 22

15.33 5.25 1.86

40

7.25

2 1

3 1

Appendices

e 3 -e 1

1440, 60, 120, 28,800, 2180, 5040, 360, 120, 820, 300, 30, 120, 120, 240, 240, 1440, 10,080, 15, 120, 14,400, 2880, 9705, 225, 240, 21,480, 120, 1440, 1905, 180, 30, 10, 150, 120, 1440, 2880, 10, 120, 1440, 10, 30, 30, 20, 20, 30,240, 10,080, 1440, 11,520, 192, 4320, 2880, 10, 25, 55, 20, 60, 20, 1440, 720, 1440, 2880, 720, 1440, 10, 8640, 1440, 360, 5760, 1440, 12, 120, 20, 1440, 1440, 120, 1200, 7200, 1440, 60, 2880, 120, 4320, 1200, 20,160, 7200, 8640, 1440, 30, 1440, 1440, 5760, 100, 1440, 14,400, 10, 10, 10, 1320, 720, 1440, 1440, 2880, 16,020, 2880, 20,160, 2880, 150, 20, 60, 3420, 390, 240, 5760, 120, 2880, 1440, 20,160, 2880, 720, 120, 1440, 1440, 5430, 7261, 720, 1440, 1440, 1440, 1440, 4320, 270, 180, 15, 1440, 1440, 30, 702, 2880, 2880, 600, 1440, 1440, 1440, 1800, 1440, 720, 2160, 180, 1440, 220, 20, 60, 180, 600, 4320, 1440, 10, 135, 1440, 2880, 4320, 14,400, 2880, 720, 420, 7200, 480, 4320, 1410, 1440, 30, 4320, 180, 1440, 1440, 600, 1200, 2880, 14,400, 14,400, 720, 30, 4320, 330, 5, 60, 1440, 4320, 5760, 1440, 120, 300, 150, 4320, 200, 30, 1440, 1440, 660, 5420, 120, 4320, 1440, 1440, 1440, 4320, 2880, 1440, 5760, 720, 300, 2880, 660, 120, 900, 14,400, 2880, 1440, 1440, 600, 720, 2160, 400, 180, 1440, 60, 720, 1440, 500, 1440, 840, 1440, 15, 600, 1200, 4320, 1440, 2880, 360, 18,720, 2160, 1440, 600, 780, 4320, 7200, 360, 1440, 1200, 2160, 2880, 120, 1440, 5760, 120, 30, 300, 360, 1440, 720, 1440, 720, 240, 120, 600, 1440, 2160, 600, 180, 2160, 1440, 900, 180, 1200, 720, 5760, 2880, 420, 1440, 30, 2160, 420, 5760, 1440, 540, 720, 300, 240, 420, 540, 90, 720, 1440, 1200, 1200, 1440, 1440, 2880, 720, 4320, 4320, 1440, 820, 14,400, 4320, 240, 20, 2880, 720, 720, 720, 2880, 2160, 360, 1440, 2160, 2160, 1440, 2880, 5760, 2160, 2880, 259,200, 2880, 900, 120, 28,800, 1440, 4320, 4320, 300, 5760, 1440, 780, 2160, 300, 480, 60, 2160, 450, 720, 480, 600, 47,520, 1200, 840, 5760, 2880, 4320, 14,400, 2160, 600, 4320, 60, 1440, 2600, 720, 420, 480, 150, 2160, 45, 2880, 7200, 2160, 600, 2880, 60, 1440, 1440, 660, 720, 1200, 1440, 840, 2880, 1440, 14,400, 720, 10,080, 1440, 540, 2880, 120, 4320, 800, 25,920, 5760, 1440, 1440, 4320, 2880, 2880, 1440, 660, 600, 1440, 1440, 600, 540, 1440, 600, 840, 300, 34,560, 480, 1440, 40,320, 240, 180, 570, 14,400, 10,080, 1440, 2160, 2880, 5760, 1440, 25,920, 720, 540, 720, 1440, 1440, 40,320, 120, 2880, 240, 1440, 660, 86,400, 720, 420, 5760, 1440, 2880, 4320, 1440, 4320, 180, 180, 360, 30, 1440, 1440, 720, 7200, 480, 7200, 1440, 480, 1440, 720, 820, 780, 180, 4320, 115,200,

753

4500.64

193

194

Appendices

e 3 -e 2 e 3 -e 6 e 3 -e 9 e 4 -e 1 e 4 -e 2 e 4 -e 3 e 4 -e 6 e 4 -e 7 e 5 -e 1

1440, 660, 2880, 2880, 120, 4320, 7200, 540, 2880, 4320, 5760, 420, 720, 5760, 5760, 720, 7200, 1440, 300, 1440, 2160, 720, 780, 2880, 1440, 2160, 660, 720, 5760, 360, 2880, 72,000, 3600, 720, 660, 2880, 720, 2160, 780, 7200, 4320, 1440, 2880, 1440, 1440, 7200, 17,280, 2880, 3500, 302,400, 2880, 690, 2160, 2160, 1440, 600, 2880, 780, 660, 4320, 1200, 2160, 420, 300, 2880, 1440, 3060, 800, 300, 1440, 5760, 1440, 780, 5760, 660, 720, 60, 170, 2880, 720, 360, 60, 2160, 1440, 158,400, 720, 120, 1440, 660, 2880, 2880, 5760, 2160, 2880, 1440, 1440, 720, 5760, 15, 1440, 420, 120, 360, 2880, 2880, 2880, 1440, 102,240, 1440, 7200, 6600, 60, 28,800, 5760, 420, 360, 4800, 480, 420, 480, 2880, 2880, 7200, 15, 4320, 3600, 240, 720, 4320, 60, 7200, 120, 150, 2880, 720, 5760, 4320, 7200, 120, 2880, 6600, 6480, 4320, 120, 300, 2880, 4320, 780, 840, 25,920, 600, 1440, 840, 120, 2880, 5760, 840, 2160, 600, 89,280, 8640, 2880, 5760, 600, 7200, 2880, 540, 2880, 1440, 2160, 4320, 360, 1440, 2880, 60, 2160, 120, 7200, 840, 4400, 1440, 540, 360, 2880, 1200, 60, 8640, 5760, 600, 7200, 2160, 300, 18,720, 720, 600, 720, 2160, 4320, 2880, 120, 660, 420, 7200, 8640, 5760, 10,080, 1440, 780, 680, 660, 1440, 2880, 720, 4320, 540, 10,080, 2880, 420, 10,080, 5760, 150, 4320, 420, 300, 1440, 25,920, 120, 90, 720, 840, 1440, 900, 240, 660, 1440, 840, 60, 60, 480, 420, 2880, 660, 660, 840, 120, 1200, 1320, 840, 2880, 880, 600, 440, 1440, 5760, 4320, 1320, 1440, 5760, 4320, 720, 2880, 1440, 470, 7200, 720, 1440, 1440, 890, 2880, 1200, 2160, 4320, 1440, 2880, 600, 2880, 1440, 1140, 780, 1040, 840, 60, 4320, 880, 4320, 480, 600, 1440, 43,200, 5760, 8640, 1440, 28,800, 43,200, 1440, 4320 10, 120 10, 10, 1, 5 45 All 1 5, 5, 30 1, 20, 600, 240, 10, 10, 30, 10, 5, 20, 40, 5, 40 240, 1, 1, 10, 1, 1, 1, 1, 1 1, 1, 10, 10, 15, 1, 5, 1, 5, 1, 1, 10, 1 30, 20, 70, 20, 60, 180, 10, 320, 60, 180, 60, 240, 30, 20, 20, 180, 300, 30, 60, 45, 180, 60, 150, 120, 120, 60, 3, 30, 45, 60, 120, 7200, 60, 3600, 1860, 70, 60, 180, 240, 20, 240, 180, 15, 30, 30, 300, 315, 360, 150, 135, 60, 330, 60, 30, 240, 135, 120, 20, 60, 140, 15, 10, 30, 60, 360, 60, 180, 360, 90, 120, 15, 120, 280, 30, 30, 25, 45, 90, 60, 80, 240, 90, 150, 45, 60, 300, 360, 60, 420, 50, 90, 90, 60, 20, 360, 240, 80, 250, 2880, 60, 60, 45, 660, 150,

2 4 1 111 3 13

65 6.50 45 1 13.33 79.31

9 13 210

28.56 4.77 380.18

Appendices

e 5 -e 9 e 5 -e 10 e 5 -e 12 e 5 -e 16 e 6 -e 1

e 6 -e 2 e 6 -e 3

e 6 -e 4 e 6 -e 10 e 6 -e 11 e 7 -e 1

300, 1440, 60, 40, 90, 50, 30, 30, 210, 1440, 30, 1440, 20, 300, 100, 1440, 120, 10, 60, 30, 120, 300, 15, 300, 60, 60, 360, 420, 30, 150, 510, 16,200, 240, 210, 50, 60, 60, 240, 90, 310, 120, 180, 120, 60, 300, 360, 120, 80, 120, 150, 120, 660, 60, 150, 360, 150, 180, 200, 120, 60, 1200, 60, 300, 360, 660, 60, 300, 180, 420, 60, 240, 420, 60, 60, 180, 30, 300, 360, 40, 150, 120, 120, 180, 60, 20, 80, 240, 30, 150, 80, 150, 660, 680, 30, 180, 60, 60, 150, 90, 180, 240, 300, 80, 8640, 240, 2880 5, 90, 20, 60, 120, 60, 30, 20, 942, 420, 40, 5760, 60, 25, 60, 5760, 5760, 60 1, 1, 1, 1, 20, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 120, 60, 1, 1, 1, 1, 2, 30, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1 15 15, 60, 20, 10, 60 1440, 60, 120, 585, 15, 60, 240, 60, 1440, 300, 60, 60, 1440, 60, 360, 960, 5760, 60, 1440, 1020, 120, 7200, 1440, 360, 60, 90, 180, 150, 180, 120, 1440, 90, 2160, 480, 60, 5, 240, 120, 1440, 300, 2880, 20, 560, 300, 1440, 1440, 600, 5760, 1440, 150, 600, 600, 10,080, 600, 120, 30, 620, 360, 360, 60, 60, 30, 300, 1440, 1440, 20, 20, 30, 7, 4320, 360, 70, 60, 650, 1440, 600, 60, 30, 780, 2200, 345, 2880, 600, 100, 2880, 1440, 80, 320, 2880, 90, 190, 1440, 6480, 360, 20, 1440, 150, 10, 30, 210, 120, 90, 50, 180, 120, 60, 120, 20, 720, 130, 1440, 1440, 180, 20, 80, 660, 150, 90, 120, 20, 20, 300, 15, 60, 110, 130, 90 10, 230, 30, 1, 3, 10, 40, 120, 60, 5, 60, 5, 120, 1, 10, 5, 75, 15, 10, 5, 20, 15, 30, 10, 15, 10, 10, 20 60, 30, 20, 18, 15, 40, 20, 30, 40,320, 20, 15, 10, 30, 30, 80, 60, 20, 30, 10, 10, 20, 10, 10, 10, 10, 120, 600, 15, 15, 70, 10, 10, 30, 60, 2, 15, 60, 360, 30, 1440, 20, 15, 60, 60, 120, 20, 90, 30, 60, 20, 20, 20, 20, 90, 40, 30, 10, 60, 10, 10, 60, 240, 30, 10, 10, 120, 720, 10, 30, 20, 10, 60, 20, 30, 20, 20, 90, 30, 60, 10, 20, 60, 120, 10, 30, 60, 20, 5760 10, 240, 15, 30, 15, 10, 20, 60, 20, 30, 5, 5, 30, 5, 10, 5, 10, 20, 5, 5, 15, 5 100, 45, 10, 10 1, 10, 180, 60, 10, 120, 20, 1, 10, 5, 5, 300, 15, 10 20, 30, 120, 480, 20, 10, 40, 1440, 60, 30, 30, 60, 60, 20, 1440, 100, 90, 180, 20, 15, 2160, 20, 1440, 150, 120, 2880, 420, 120, 180, 150, 120, 180, 1440, 60, 60, 10, 30, 20, 40, 45, 30, 15, 15, 30, 2880, 17,280, 90, 60, 20, 40, 50, 40, 360, 90, 1440, 30, 1440, 300, 40, 20, 360, 240, 300, 420, 45, 120, 180, 120, 90, 530, 100, 60, 30, 120, 240, 240, 60, 60, 100, 30, 135, 60, 25, 25, 90, 120, 60, 600, 60, 20, 20, 780, 60, 160, 120, 120, 90, 45, 30, 45, 1200, 20, 120, 240, 180, 20, 27,360

18

1071.67

39

10.59

1 5 127

15 33 804.66

28

35.36

88

593.75

22

25.91

4 14

41.25 53.36

107

686.26

195

196

Appendices

e 7 -e 3

6, 10, 60, 10, 4140, 120, 60, 15, 160, 4320, 60, 10, 4320, 10, 30, 1, 1530, 1440, 20, 30, 30, 10, 30, 220, 10, 10, 90, 480, 15, 30, 25, 1580, 600, 20, 20, 10, 10, 30, 60, 20, 60, 1020, 30, 30, 20, 30, 180, 10, 60, 10, 15, 30, 30, 5, 10, 10, 20, 90, 50, 20, 15, 15, 15, 20, 25, 25, 60, 10, 40, 380, 20, 15, 15, 10, 20, 20, 15, 10, 10, 10, 10, 10, 60, 1, 10, 20, 60, 60, 100, 10, 60, 10, 20, 4320, 270, 30, 15, 15, 180, 10, 30, 240, 30, 20, 15, 10, 5, 60, 10, 1, 10, 1, 30, 15, 15, 60, 60, 15, 20, 10, 10, 5, 1215, 10, 15, 20, 60, 20, 10, 120, 60, 60, 20, 120, 20, 20, 15, 30, 60, 30 5 1, 5, 30 1, 1, 10 15, 10, 1, 1, 10 120, 20, 300, 120, 25, 30, 150, 30, 60, 30, 180, 15, 5, 5, 10, 10, 20, 5, 730, 40, 2160, 5, 15, 45, 30, 20, 5, 20 1, 1, 5, 10 20, 20, 15, 10, 20, 20, 20, 30, 20, 40, 45, 2700, 20, 15, 15, 25, 15, 25, 15 19,840, 10, 60, 60, 240, 600, 150, 720, 60, 45, 60, 480, 180, 60, 60, 60, 3600, 180, 420, 60, 60, 600 920, 60, 15, 150, 80, 120, 420, 120, 20, 20, 60, 15, 15, 20, 120 60 60, 60, 2880, 120, 60, 120, 540, 180 30, 30, 20, 300, 30, 60, 20, 10, 20, 20 30 20, 10, 10, 300 240, 60, 10 10, 20 120 60, 220 120, 30 60, 120 60

e 7 -e 4 e 7 -e 8 e 7 -e 11 e 7 -e 13 e 8 -e 1 e 8 -e 13 e 9 -e 1 e 10 -e 1 e 10 -e 6 e 10 -e 14 e 11 -e 1 e 11 -e 3 e 12 -e 1 e 13 -e 1 e 13 -e 3 e 13 -e 15 e 14 -e 1 e 15 -e 3 e 16 -e 1 e 16 -e 3 e 16 -e 7

141

204.20

1 3 3 5 28

5 12 4 7.40 150.18

4 19

4.25 162.63

22

1254.77

15

143.67

1 8 10 1 4 3 2 1 2 2 2 1

60 502.5 54 30 85 103.33 15 120 140 75 90 60

Appendix 3 Realizations of conditional lifetimes at states of process of environment threats at Baltic Sea waters j

ij

ij

ij

Transitions siðk=lÞ -sðk=lÞ

Transition time ηðk=lÞ (min)

Number of transitions nðk=lÞ

Mean value ηðk=lÞ (min)

A

B

C

D

1 27 sð1=2Þ -sð1=2Þ 27 1 -sð1=2Þ sð1=2Þ 1 27 -sð1=3Þ sð1=3Þ 1 30 sð1=3Þ -sð1=3Þ 27 1 sð1=3Þ -sð1=3Þ 30 1 sð1=3Þ -sð1=3Þ 1 6 sð1=8Þ -sð1=8Þ 6 1 -sð1=8Þ sð1=8Þ 1 33 -sð2=2Þ sð2=2Þ 33 1 sð2=2Þ -sð2=2Þ 1 17 sð2=3Þ -sð2=3Þ 1 33 sð2=3Þ -sð2=3Þ 17 1 sð2=3Þ -sð2=3Þ

1

1

1

300

1

300

1

1

1

1

1

1

180

1

180

240

1

240

1

1

1

240

1

240

1

1

1

1440

1

1440

1

1

1

1

1

1

10,080

1

10,080

Appendices

33 1 sð2=3Þ -sð2=3Þ

1440

1

1 24 -sð3=2Þ sð3=2Þ

1

1

1

24 1 -sð3=2Þ sð3=2Þ

1440

1

1440

1 14 -sð3=3Þ sð3=3Þ

1

1

1

1 24 -sð3=3Þ sð3=3Þ

1

1

1

14 1 -sð3=3Þ sð3=3Þ 24 1 sð3=3Þ -sð3=3Þ 1 14 sð4=3Þ -sð4=3Þ 14 1 sð4=3Þ -sð4=3Þ

10,080

1

10,080

1440

1

1440

1

1

1

10,080

1

10,080

197

1440

Appendix 4 Realizations of conditional lifetimes at states of process of environment threats at world sea waters j

ij

ij

ij

Transitions siðk=lÞ -sðk=lÞ

Transition time ηðk=lÞ (min)

Number of transitions nðk=lÞ

Mean value ηðk=lÞ (min)

A

B

C

D

1 27 sð1=2Þ -sð1=2Þ 7 1 sð1=2Þ -sð1=2Þ 27 1 -sð1=2Þ sð1=2Þ

All 1

43

1

240

1

240

5760, 5760, 5760, 1440, 5760, 120, 180, 5760, 4320, 300, 420, 240, 5760, 60, 5760, 300, 180, 5760, 4320, 1440, 2880, 1440, 5760, 5760, 60, 4320, 2880, 5760, 300, 1440, 480, 5760, 5760, 5760, 5760, 5760, 600, 5760, 5760, 5760, 5760, 5760 240

42

3540

1

240

1 23 -sð1=3Þ sð1=3Þ

1

1

1

1 27 -sð1=3Þ sð1=3Þ

1440, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 All 1

40

36.975

2

1

240

1

240

5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 4320, 2880, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 600, 600, 480, 420, 360, 300, 300, 300, 300, 300, 240, 240, 180, 180, 120, 120 240, 240

40

2718

2

240

1 13 -sð1=4Þ sð1=4Þ

1

1

1

1 27 -sð1=4Þ sð1=4Þ

All 1

16

1

1 35 -sð1=4Þ sð1=4Þ

1

1

1

5 1 -sð1=4Þ sð1=4Þ

300

1

300

13 1 -sð1=4Þ sð1=4Þ

120

1

120

27 1 -sð1=4Þ sð1=4Þ

5760, 5760, 5760, 5760, 5760, 2880, 1440, 1440, 1440, 1440, 1440, 600, 420, 420, 180, 240 120

16

2546.25

1

120

1 3 -sð1=5Þ sð1=5Þ

1

1

1

1 4 -sð1=5Þ sð1=5Þ 1 27 sð1=5Þ -sð1=5Þ 1 34 sð1=5Þ -sð1=5Þ 3 17 sð1=5Þ -sð1=5Þ 4 1 sð1=5Þ -sð1=5Þ 7 1 -sð1=5Þ sð1=5Þ 17 7 -sð1=5Þ sð1=5Þ 27 1 sð1=5Þ -sð1=5Þ

All 1

3

1

All 1

5

1

1

1

1

1

1

1

30, 2880, 1440, 240

4

1147.50

27 7 sð1=2Þ -sð1=2Þ

1 30 -sð1=3Þ sð1=3Þ 23 1 -sð1=3Þ sð1=3Þ 27 1 sð1=3Þ -sð1=3Þ

30 1 -sð1=3Þ sð1=3Þ

35 5 -sð1=4Þ sð1=4Þ

5

1

5

300

1

300

5760, 600, 600, 240, 240

5

1488

198

Appendices

34 4 sð1=5Þ -sð1=5Þ

15

1

15

1 27 -sð1=6Þ sð1=6Þ

All 1

2

1

27 1 -sð1=6Þ sð1=6Þ

5760, 5760

2

5760

1 27 -sð1=7Þ sð1=7Þ

All 1

7

1

1 30 -sð1=7Þ sð1=7Þ

1

1

1

27 1 -sð1=7Þ sð1=7Þ

5760, 4320, 1440, 1440, 1440, 600, 300 240

7

2185.8

1

240

30 1 -sð1=7Þ sð1=7Þ 1 3 -sð1=8Þ sð1=8Þ 1 6 sð1=8Þ -sð1=8Þ 1 7 sð1=8Þ -sð1=8Þ 1 8 sð1=8Þ -sð1=8Þ 1 27 -sð1=8Þ sð1=8Þ 3 1 -sð1=8Þ sð1=8Þ 6 1 sð1=8Þ -sð1=8Þ 7 21 sð1=8Þ -sð1=8Þ 8 1 sð1=8Þ -sð1=8Þ 21 1 sð1=8Þ -sð1=8Þ 27 1 -sð1=8Þ sð1=8Þ

1

1

1

All 1

3

1

1

1

1

1

1

1

All 1

11

1

40

1

40

180, 240, 2880

3

1100

15

1

15

240

1

240

5

1

5

10,080, 5760, 5760, 5760, 720, 300, 240, 240, 120, 120, 30 All 1

11

2648.4

43

1

2 1 -sð2=2Þ sð2=2Þ

280,800

1

280,800

33 1 -sð2=2Þ sð2=2Þ

10,080, 10,080, 18,720, 20,160, 14,400, 1440, 1440, 20,160, 4320, 4320, 4320, 1440, 14,400, 1440, 14,400, 2880, 1440, 14,400, 11,520, 4320, 14,400, 7200, 20,160, 5760, 180, 8640, 5760, 14,400, 1440, 2880, 1440, 14,400, 20,160, 5760, 5760, 20,160, 2880, 7200, 216,000, 21,600, 5760, 34,560 2880

42

14,575.71

1

2880

1 17 -sð2=3Þ sð2=3Þ

All 1

3

1

1 18 -sð2=3Þ sð2=3Þ

1

1

1

1 33 -sð2=3Þ sð2=3Þ

1440, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 7200, 7200, 10,080

41

36.10

3

24,480

18 1 -sð2=3Þ sð2=3Þ

43,200

1

43,200

33 1 -sð2=3Þ sð2=3Þ

43,200, 43,200, 43,200, 43,200, 28,800, 28,800, 23,040, 21,600, 20,160, 20,160, 20,160, 14,400, 14,400, 14,400, 14,400, 10,080, 5760, 4320, 4320, 2880, 2880, 2880, 2880, 2880, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440 All 1

41

11,133.6

16

1

16

31,050

1 33 -sð2=2Þ sð2=2Þ

33 2 sð2=2Þ -sð2=2Þ

17 1 -sð2=3Þ sð2=3Þ

1 33 sð2=4Þ -sð2=4Þ 33 1 -sð2=4Þ sð2=4Þ

345,600, 43,200, 28,800, 20,160, 11,520, 7200, 5760, 4320, 4320, 4320, 2880, 2880, 2880, 1440, 1440, 10,080 All 1

5

1

33 1 -sð2=5Þ sð2=5Þ

172,800, 4320, 1440, 1440, 10,080

5

38,016

1 33 -sð2=6Þ sð2=6Þ

All 1

2

1

33 1 -sð2=6Þ sð2=6Þ

14,400, 10,080

2

12,240

1 17 -sð2=7Þ sð2=7Þ

1

1

1

1 33 -sð2=5Þ sð2=5Þ

Appendices

1 33 sð2=7Þ -sð2=7Þ

All 1

7

1

17 1 -sð2=7Þ sð2=7Þ

1440

1

1440

33 1 -sð2=7Þ sð2=7Þ

28,800, 14,400, 10,080, 2880, 1440, 1440, 1440 1

7

8640

1

1

1 4 -sð2=8Þ sð2=8Þ

1

1

1

1 5 -sð2=8Þ sð2=8Þ

1

1

1

1 33 -sð2=8Þ sð2=8Þ 2 1 sð2=8Þ -sð2=8Þ 4 1 sð2=8Þ -sð2=8Þ 5 1 sð2=8Þ -sð2=8Þ 33 1 sð2=8Þ -sð2=8Þ

All 1

11

1

7200

1

7200

1440

1

1440

1440

1

1440

46,800, 43,200, 17,280, 14,400, 2880, 2880, 2880, 1440, 1440, 240, 30 All 1

11

12,133.63

43

1

2 1 -sð3=2Þ sð3=2Þ

280,800

1

280,800

24 1 -sð3=2Þ sð3=2Þ

10,080, 10,080, 18,720, 20,160, 14,400, 1440, 1440, 20,160, 4320, 4320, 4320, 1440, 14,400, 1440, 14,400, 2880, 1440, 14,400, 11,520, 4320, 14,400, 7200, 20,160, 5760, 180, 8640, 5760, 14,400, 1440, 2880, 1440, 14,400, 20,160, 5760, 5760, 20,160, 2880, 7200, 216,000, 21,600, 5760, 34,560 2880

42

14,575.71

1

2880

1 14 -sð3=3Þ sð3=3Þ

All 1

3

1

1 15 -sð3=3Þ sð3=3Þ

1

1

1

1 24 -sð3=3Þ sð3=3Þ

1440, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 7200, 7200, 10,080

41

36.10

3

24,480

15 1 -sð3=3Þ sð3=3Þ

43,200

1

43,200

24 1 -sð3=3Þ sð3=3Þ

72,000, 43,200, 43,200, 43,200, 43,200, 28,800, 28,800, 23,040, 21,600, 20,160, 20,160, 14,400, 14,400, 14,400, 14,400, 10,080, 5760, 4320, 4320, 2880, 2880, 2880, 2880, 2880, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440, 1440 All 1

41

12,398.05

16

1

1 2 -sð2=8Þ sð2=8Þ

1 24 -sð3=2Þ sð3=2Þ

24 2 sð3=2Þ -sð3=2Þ

14 1 -sð3=3Þ sð3=3Þ

1 24 sð3=4Þ -sð3=4Þ 24 1 -sð3=4Þ sð3=4Þ

1 24 -sð3=5Þ sð3=5Þ 24 1 -sð3=5Þ sð3=5Þ 1 24 sð3=6Þ -sð3=6Þ 24 1 sð3=6Þ -sð3=6Þ 1 14 -sð3=7Þ sð3=7Þ 1 24 -sð3=7Þ sð3=7Þ 14 1 sð3=7Þ -sð3=7Þ 24 1 sð3=7Þ -sð3=7Þ 1 2 -sð3=8Þ sð3=8Þ 1 3 -sð3=8Þ sð3=8Þ 1 4 sð3=8Þ -sð3=8Þ 1 24 sð3=8Þ -sð3=8Þ 2 1 -sð3=8Þ sð3=8Þ

345,600, 43,200, 28,800, 20,160, 11,520, 7200, 5760, 4320, 4320, 4320, 2880, 2880, 2880, 1440, 1440, 10,080 All 1

16

31,050

5

1

172,800, 4320, 1440, 1440, 10,080

5

38,016

All 1

2

1

14,400, 10,080

2

12,240

1

1

1

All 1

7

1

1440

1

1440

28,800, 14,400, 10,080, 2880, 1440, 1440, 1440 1

7

8640

1

1

1

1

1

1

1

1

All 1

11

1

7200

1

7200

199

200

Appendices

3 1 sð3=8Þ -sð3=8Þ

1440

1

1440

4 1 -sð3=8Þ sð3=8Þ

1440

1

1440

24 1 -sð3=8Þ sð3=8Þ

28,800, 43,200, 17,280, 14,400, 2880, 2880, 2880, 1440, 1440, 240, 30 1

11

10,497.27

1

1

1440, 5040, 1440, 2160, 1440, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 280,800

21

549.33

1

280,800

1 2 -sð4=2Þ sð4=2Þ 1 24 -sð4=2Þ sð4=2Þ 2 1 -sð4=2Þ sð4=2Þ 24 1 -sð4=2Þ sð4=2Þ

28,800, 10,080, 17,280, 81,360, 14,400, 43,200, 4320, 15,840, 22,320, 1440, 43,200, 43,200, 86,400, 20,160, 5760, 5760, 20,160, 7200, 216,000, 21,600, 34,560 All 1

21

35,382.86

3

1

1 15 -sð4=3Þ sð4=3Þ

1

1

1

1 24 -sð4=3Þ sð4=3Þ

1440, 2880, 2880, 720, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 7200, 7200, 10,080

18

440.78

3

24,480

15 1 -sð4=3Þ sð4=3Þ

43,200

1

43,200

24 1 -sð4=3Þ sð4=3Þ

128,160, 23,040, 40,320, 42,480, 4320, 4320, 10,080, 2880, 72,000, 21,600, 14,400, 14,400, 20,160, 43,200, 14,400, 28,800, 43,200, 43,200 All 1

18

32,230.59

7

1

7

70,765.71

1 14 -sð4=3Þ sð4=3Þ

14 1 -sð4=3Þ sð4=3Þ

1 24 -sð4=4Þ sð4=4Þ 24 1 -sð4=4Þ sð4=4Þ 1 24 -sð4=5Þ sð4=5Þ 24 1 -sð4=5Þ sð4=5Þ 1 24 sð4=6Þ -sð4=6Þ 24 1 sð4=6Þ -sð4=6Þ 1 14 sð4=7Þ -sð4=7Þ 1 24 -sð4=7Þ sð4=7Þ 14 1 -sð4=7Þ sð4=7Þ 24 1 sð4=7Þ -sð4=7Þ 1 2 sð4=8Þ -sð4=8Þ 1 4 sð4=8Þ -sð4=8Þ 1 24 sð4=8Þ -sð4=8Þ 2 1 sð4=8Þ -sð4=8Þ 4 1 -sð4=8Þ sð4=8Þ 24 1 -sð4=8Þ sð4=8Þ 1 24 sð5=2Þ -sð5=2Þ 24 1 -sð5=2Þ sð5=2Þ

1 15 -sð5=3Þ sð5=3Þ 1 24 -sð5=3Þ sð5=3Þ 15 1 -sð5=3Þ sð5=3Þ 24 1 -sð5=3Þ sð5=3Þ

1 24 -sð5=4Þ sð5=4Þ 24 1 -sð5=4Þ sð5=4Þ 1 24 -sð5=5Þ sð5=5Þ 24 1 -sð5=5Þ sð5=5Þ

345,600, 43,200, 43,200, 28,800, 20,160, 4320, 10,080 All 1

2

1

172,800, 10,080

2

91,440

1440, 1

2

720.5

12,960, 28,800

2

20,880

1

1

1

360, 1, 1, 1

4

90.75

1440

1

1440

28,800, 28,440, 14,400, 2880

4

18,630

1

1

1

1

1

1

All 1

3

1

7200

1

7200

1440

1

1440

43,200, 17,280, 43,200

3

34,560

2160, 2880, 1440, 180, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 28,800, 10,080, 18,720, 14,400, 4320, 4320, 4320, 1440, 1440, 41,040, 43,200, 20,160, 5760, 5760, 20,160, 7200, 215,280, 21,600, 34,380 1

19

351.32

19

26,441.05

1

1

1440, 2880, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 43,200

15

288.87

1

43,200

15

22,368

23,040, 5760, 4320, 4320, 1440, 8640, 72,000, 21,600, 14,400, 14,400, 21,600, 43,200, 14,400, 43,200, 43,200 All 1

8

1

43,200, 20,160, 4320, 4320, 10,080, 43,200, 43,200, 345,600 1

8

64,260

1

1

172,800

1

172,800

Appendices

1 24 sð5=6Þ -sð5=6Þ

1440

1

1440

24 1 -sð5=6Þ sð5=6Þ

17,280

1

17,280

1 24 -sð5=7Þ sð5=7Þ

All 1

2

1

24 1 -sð5=7Þ sð5=7Þ

20,160, 2880

2

11,520

1 24 -sð5=8Þ sð5=8Þ

All 1

4

1

24 1 -sð5=8Þ sð5=8Þ

262,080, 43,200, 43,200, 1440

4

87,480

201

Appendix 5 Realizations of conditional lifetimes at states of process of environment degradation at Baltic Sea waters j

ij

ij

ij

i Transitions rðk=υÞ -rðk=υÞ

Transition time ζ ðk=υÞ (min)

Number of transitions nðk=υÞ

Mean value ζ ðk=υÞ (min)

A

B

C

D

1 2 rð1=6Þ -rð1=6Þ

1

1

1

2 1 -rð1=6Þ rð1=6Þ 1 6 rð1=27Þ -rð1=27Þ 6 1 rð1=27Þ -rð1=27Þ 1 11 rð1=30Þ -rð1=30Þ 11 1 -rð1=30Þ rð1=30Þ 1 27 -rð2=17Þ rð2=17Þ 12 1 rð2=17Þ -rð2=17Þ 16 12 rð2=17Þ -rð2=17Þ 21 16 rð2=17Þ -rð2=17Þ 25 21 rð2=17Þ -rð2=17Þ 27 25 -rð2=17Þ rð2=17Þ 1 6 -rð2=33Þ rð2=33Þ 6 1 rð2=33Þ -rð2=33Þ 1 27 rð3=14Þ -rð3=14Þ 12 1 rð3=14Þ -rð3=14Þ 16 12 rð3=14Þ -rð3=14Þ 21 16 rð3=14Þ -rð3=14Þ 25 21 -rð3=14Þ rð3=14Þ 27 25 rð3=14Þ -rð3=14Þ 1 6 rð3=24Þ -rð3=24Þ 6 1 rð3=24Þ -rð3=24Þ 1 30 rð4=14Þ -rð4=14Þ 12 1 rð4=14Þ -rð4=14Þ 16 12 -rð4=14Þ rð4=14Þ 21 16 -rð4=14Þ rð4=14Þ 28 21 rð4=14Þ -rð4=14Þ 30 28 rð4=14Þ -rð4=14Þ

240

1

240

1, 1

2

1

180, 300

2

240

1

1

1

240

1

240

1

1

1

2880

1

2880

3780

1

3780

2880

1

2880

300

1

300

240

1

240

1, 1

2

1

1440, 1440

2

1440

1

1

1

2880

1

2880

3780

1

3780

2880

1

2880

300

1

300

240

1

240

All 1

2

1

1440, 1440

2

1440

1

1

1

2880

1

2880

3780

1

3780

2880

1

2880

300

1

300

240

1

240

Appendix 6 Realizations of conditional lifetimes at states of process of environment degradation at world sea waters j

ij

ij

ij

i Transitions rðk=υÞ -rðk=υÞ

Transition time ζ ðk=υÞ (min)

Number of transitions nðk=υÞ

Mean value ζ ðk=υÞ (min)

A

B

C

D

1 28 rð1=3Þ -rð1=3Þ

1

1

1

6 1 -rð1=3Þ rð1=3Þ

180

1

180

22 6 -rð1=3Þ rð1=3Þ

120

1

120

25 22 -rð1=3Þ rð1=3Þ 28 25 rð1=3Þ -rð1=3Þ

120

1

120

300

1

300

202

Appendices

1 27 rð1=4Þ -rð1=4Þ

All 1

3

1

1 30 -rð1=4Þ rð1=4Þ

1

1

1

20 1 -rð1=4Þ rð1=4Þ

20

1

20

21 1 -rð1=4Þ rð1=4Þ

720, 720, 600

3

680

24 21 -rð1=4Þ rð1=4Þ

720, 1440, 600

3

920

27 24 -rð1=4Þ rð1=4Þ 29 20 rð1=4Þ -rð1=4Þ 30 29 rð1=4Þ -rð1=4Þ 1 26 rð1=5Þ -rð1=5Þ 20 1 rð1=5Þ -rð1=5Þ 23 20 -rð1=5Þ rð1=5Þ 26 23 -rð1=5Þ rð1=5Þ 1 2 rð1=6Þ -rð1=6Þ 1 19 rð1=6Þ -rð1=6Þ 2 1 rð1=6Þ -rð1=6Þ 17 1 rð1=6Þ -rð1=6Þ 18 17 rð1=6Þ -rð1=6Þ 19 18 -rð1=6Þ rð1=6Þ 1 2 rð1=7Þ -rð1=7Þ 1 10 rð1=7Þ -rð1=7Þ 2 1 rð1=7Þ -rð1=7Þ 6 1 rð1=7Þ -rð1=7Þ 8 6 rð1=7Þ -rð1=7Þ 10 8 -rð1=7Þ rð1=7Þ 1 15 -rð1=8Þ rð1=8Þ 13 1 rð1=8Þ -rð1=8Þ 14 13 rð1=8Þ -rð1=8Þ 15 14 rð1=8Þ -rð1=8Þ 1 6 rð1=13Þ -rð1=13Þ 6 1 -rð1=13Þ rð1=13Þ 1 28 -rð1=17Þ rð1=17Þ 6 1 rð1=17Þ -rð1=17Þ 22 6 rð1=17Þ -rð1=17Þ 25 22 rð1=17Þ -rð1=17Þ 28 25 rð1=17Þ -rð1=17Þ 1 20 -rð1=21Þ rð1=21Þ 20 1 -rð1=21Þ rð1=21Þ 1 2 rð1=23Þ -rð1=23Þ 2 1 rð1=23Þ -rð1=23Þ 1 6 rð1=27Þ -rð1=27Þ 6 1 rð1=27Þ -rð1=27Þ

3600, 2160, 300

3

2020

20

1

20

40

1

40

1

1

1

30

1

30

60

1

60

360

1

360

All 1

2

1

1

1

1

240, 2880

2

1560

30

1

30

30

1

30

180

1

180

1

1

1

1

1

1

15

1

15

180

1

180

120

1

120

420

1

420

1

1

1

60

1

60

60

1

60

120

1

120

1

1

1

120

1

120

1

1

1

180

1

180

120

1

120

120

1

120

300

1

300

1

1

1

10

1

10

1

1

1

240

1

240

All 1

124

1

5760, 5760, 5760, 120, 5760, 240, 720, 5760, 5760, 240, 5760, 1440, 5760, 5760, 1440, 5760, 180, 5760, 300, 5760, 120, 30, 120, 180, 5760, 5760, 5760, 1440, 4320, 300, 240, 4320, 1440, 420, 240, 5760, 60, 240, 300, 5760, 2880, 5760, 480, 300, 1440, 300, 180, 5760, 5760, 5760, 120, 180, 240, 300, 1440, 1440, 180, 120, 2880, 1440, 2880, 4320, 1440, 5760, 600, 5760, 60, 4320,

124

2982.34

Appendices

600, 2880, 300, 1440, 300, 1440, 5760, 1440, 1440, 600, 420, 1440, 1440, 300, 1440, 1440, 5760, 420, 1440, 10,080, 5760, 240, 1440, 480, 420, 5760, 5760, 600, 240, 240, 300, 360, 5760, 5760, 5760, 5760, 5760, 600, 4320, 600, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 8640, 5760, 5760, 5760, 5760, 5760, 5760, 5760, 5760 All 1

3

1

11 1 -rð1=30Þ rð1=30Þ

240, 300, 240

3

260

1 26 -rð1=35Þ rð1=35Þ

1

1

1

20 1 -rð1=35Þ rð1=35Þ 23 20 rð1=35Þ -rð1=35Þ 26 23 rð1=35Þ -rð1=35Þ 1 5 rð2=1Þ -rð2=1Þ 5 1 -rð2=1Þ rð2=1Þ 1 2 rð2=2Þ -rð2=2Þ 2 1 rð2=2Þ -rð2=2Þ 1 2 rð2=4Þ -rð2=4Þ 2 1 rð2=4Þ -rð2=4Þ 1 18 rð2=5Þ -rð2=5Þ 13 1 -rð2=5Þ rð2=5Þ 14 13 -rð2=5Þ rð2=5Þ 18 14 rð2=5Þ -rð2=5Þ 1 25 rð2=17Þ -rð2=17Þ 1 27 rð2=17Þ -rð2=17Þ 12 1 rð2=17Þ -rð2=17Þ 16 12 -rð2=17Þ rð2=17Þ 21 16 -rð2=17Þ rð2=17Þ 25 21 rð2=17Þ -rð2=17Þ 27 25 rð2=17Þ -rð2=17Þ 1 11 rð2=18Þ -rð2=18Þ 6 1 rð2=18Þ -rð2=18Þ 9 6 rð2=18Þ -rð2=18Þ 11 9 -rð2=18Þ rð2=18Þ 1 6 rð2=33Þ -rð2=33Þ 1 9 rð2=33Þ -rð2=33Þ 1 11 rð2=33Þ -rð2=33Þ 1 23 rð2=33Þ -rð2=33Þ 2 1 rð2=33Þ -rð2=33Þ 6 1 -rð2=33Þ rð2=33Þ

30

1

30

60

1

60

360

1

360

1 11 rð1=30Þ -rð1=30Þ

1

1

1

600

1

600

1

1

1

2880

1

2880

1

1

1

1440

1

1440

1

1

1

780

1

780

360

1

360

300

1

300

All 1

2

1

All 1

2

1

3780, 2280, 240, 4260

4

2640

3780, 2880, 480, 1440

4

2145

2880, 1440, 360, 1440

4

1530

300, 300, 360, 60

4

255

240, 300

2

270

1

1

1

10,080

1

10,080

11,520

1

11,520

21,600

1

21,600

All 1

60

1

All 1

32

1

All 1

32

1

1

1

1

246,240

1

246,240

7200, 720, 7200, 1380, 7920, 1440, 2880, 25,920, 7200, 8640, 14,400, 13,500, 12,960, 2880, 24,480, 1440, 23,040, 1440, 14,400, 1440, 30, 5760, 1440, 4320, 7200, 5760, 2880, 3600, 4320, 5760, 9360, 4320, 4320, 1440, 11,520, 1440, 1440, 1440, 5760, 2880, 8640,

124

11,319.44

203

204

Appendices

9 6 rð2=33Þ -rð2=33Þ

11 9 rð2=33Þ -rð2=33Þ

12 2 rð2=33Þ -rð2=33Þ 15 12 -rð2=33Þ rð2=33Þ 20 15 rð2=33Þ -rð2=33Þ 22 20 rð2=33Þ -rð2=33Þ 23 22 rð2=33Þ -rð2=33Þ 1 2 -rð3=2Þ rð3=2Þ 2 1 -rð3=2Þ rð3=2Þ 1 2 rð3=3Þ -rð3=3Þ 2 1 rð3=3Þ -rð3=3Þ 1 18 rð3=4Þ -rð3=4Þ 13 1 rð3=4Þ -rð3=4Þ 14 13 rð3=4Þ -rð3=4Þ 18 14 -rð3=4Þ rð3=4Þ 1 25 rð3=14Þ -rð3=14Þ 1 27 rð3=14Þ -rð3=14Þ 12 1 rð3=14Þ -rð3=14Þ

1440, 240, 1440, 2880, 1440, 12,240, 15,840, 11,520, 2880, 1440, 2880, 1440, 2880, 2880, 720, 1440, 10,080, 2880, 10,080, 7200, 5760, 10,080, 4320, 5160, 180, 5760, 2880, 2880, 1440, 2880, 1440, 1440, 2880, 5760, 1440, 4320, 1440, 2880, 2880, 4320, 1440, 2880, 1440, 10,080, 1440, 4320, 14,400, 5760, 5760, 4320, 1440, 1440, 7200, 2880, 1440, 1440, 1440, 1440, 1440, 5760, 8640, 10,080, 720, 720, 2880, 10,080, 2880, 4320, 20,160, 28,800, 28,800, 2880, 7200, 184,320, 299,520, 17,280, 14,400, 28,800, 720, 23,040, 33,120, 18,720, 139,680 4320, 2160, 1440, 60, 1440, 10,080, 5760, 7200, 2880, 900, 1440, 4320, 2160, 5760, 7200, 7200, 9360, 720, 4320, 720, 2880, 2130, 5760, 2160, 5760, 1440, 1440, 1440, 4320, 2880, 1440, 7200, 600, 2880, 2880, 14,400, 4320, 11,520, 8640, 4320, 7200, 4320, 8640, 5760, 5760, 720, 720, 7200, 1440, 5760, 7200, 10,080, 5760, 10,080, 14,400, 20,160, 8640, 5760, 10,080, 3600, 4320, 6768, 14,400, 28,800 2880, 2880, 1440, 360, 7200, 5760, 7200, 2880, 5040, 8640, 5040, 2160, 7200, 2880, 5760, 2880, 4320, 4320, 2880, 1440, 2880, 7200, 5760, 17,280, 10,080, 2880, 2880, 4320, 7200, 3600, 10,080, 4320 14,400 7200

64

5709.66

32

5051.25

1

14,400

1

7200

10,080

1

10,080

1440

1

1440

1440

1

1440

1

1

1

2880

1

2880

1

1

1

1440

1

1440

1

1

1

780

1

780

360

1

360

300

1

300

All 1

2

1

All 1

2

1

3780, 2280, 240, 4260

4

2640

Appendices

16 12 rð3=14Þ -rð3=14Þ

3780, 2880, 480, 1440

4

2145

21 16 -rð3=14Þ rð3=14Þ

2880, 1440, 360, 1440

4

1530

25 21 -rð3=14Þ rð3=14Þ

300, 300, 360, 60

4

255

27 25 -rð3=14Þ rð3=14Þ

240, 300

2

270

1 11 -rð3=15Þ rð3=15Þ

1

1

1

6 1 -rð3=15Þ rð3=15Þ 9 6 rð3=15Þ -rð3=15Þ 11 9 rð3=15Þ -rð3=15Þ 1 6 rð3=24Þ -rð3=24Þ 1 9 rð3=24Þ -rð3=24Þ 1 11 -rð3=24Þ rð3=24Þ 1 23 rð3=24Þ -rð3=24Þ 2 1 rð3=24Þ -rð3=24Þ 6 1 rð3=24Þ -rð3=24Þ

10,080

1

10,080

11,520

1

11,520

21,600

1

21,600

All 1

81

1

All 1

37

1

All 1

6

1

1

1

1

246,240

1

246,240

10,080, 2880, 8640, 1440, 10,080, 1440, 2880, 33,840, 11,520, 8640, 14,400, 14,400, 14,400, 2880, 28,800, 1440, 20,160, 1440, 14,400, 1440, 30, 5760, 1440, 10,080, 7200, 5760, 2880, 4320, 4320, 5760, 10,080, 4320, 4320, 1440, 11,520, 1440, 1440, 1440, 2160, 2880, 8640, 1440, 240, 1440, 2880, 1440, 14,400, 12,960, 11,520, 2880, 1440, 2880, 1440, 2880, 4320, 720, 1440, 11,520, 2880, 10,080, 7200, 5760, 10,080, 4320, 5760, 180, 8640, 2880, 5760, 1440, 2880, 1440, 2880, 2880, 57,600, 1440, 14,400, 1440, 2880, 2880, 4320, 1440, 2880, 1440, 14,400, 1440, 4320, 14,400, 8640, 5760, 4320, 1440, 1440, 8640, 7200, 1440, 1440, 1440, 1440, 1440, 5760, 11,520, 10,080, 1440, 1440, 2880, 10,080, 2880, 5760, 17,280, 28,800, 28,800, 2880, 7200, 175,680, 319,600, 11,520, 8640, 10,080, 5760, 17,280, 18,720, 14,400, 97,920 2160, 2880, 1440, 4320, 1440 4320, 2880, 1440, 7200, 5760, 7200, 5760, 8640, 5760, 7200, 7200, 9360, 4320, 2880, 4320, 5760, 14,400, 1440, 4320, 2880, 1440, 7200, 14,400, 7200, 5760, 4320, 5760, 8640, 8640, 10,080, 4320, 4320, 10,080, 1440, 5760, 7200, 10,080, 5760, 10,080, 20,160, 21,600, 11,520, 8640, 14,400, 7200, 17,280, 14,400, 43,200

124

11,610.08

5

2448

48

8415

6 9 rð3=24Þ -rð3=24Þ 9 6 -rð3=24Þ rð3=24Þ

205

206

Appendices

9 11 rð3=24Þ -rð3=24Þ

10,080, 1440, 720, 4320, 1440, 3600, 7200, 1440 7200, 5040, 2880, 5760, 7200, 5760, 10,080, 4320, 3600, 14,400, 8640, 3600, 7200, 28,800 14,400

8

3780

14

8177.14

15 12 -rð3=24Þ rð3=24Þ

7200

1

14,400

1

20 15 -rð3=24Þ rð3=24Þ

7200

10,080

1

10,080

22 20 -rð3=24Þ rð3=24Þ

1440

1

1440

23 22 -rð3=24Þ rð3=24Þ

1440

1

1440

1 2 -rð4=2Þ rð4=2Þ 2 1 rð4=2Þ -rð4=2Þ 1 18 rð4=4Þ -rð4=4Þ 13 1 rð4=4Þ -rð4=4Þ 14 13 rð4=4Þ -rð4=4Þ 18 14 -rð4=4Þ rð4=4Þ 1 28 -rð4=14Þ rð4=14Þ 1 30 rð4=14Þ -rð4=14Þ 12 1 rð4=14Þ -rð4=14Þ 16 12 rð4=14Þ -rð4=14Þ 21 16 rð4=14Þ -rð4=14Þ 28 21 -rð4=14Þ rð4=14Þ 30 28 -rð4=14Þ rð4=14Þ 1 11 rð4=15Þ -rð4=15Þ 6 1 rð4=15Þ -rð4=15Þ 9 6 rð4=15Þ -rð4=15Þ 11 9 rð4=15Þ -rð4=15Þ 1 6 rð4=24Þ -rð4=24Þ

1

1

1

2880

1

2880

11 9 -rð3=24Þ rð3=24Þ

12 2 -rð3=24Þ rð3=24Þ

1

1

1

780

1

780

360

1

360

300

1

300

1, 1

2

1

1, 1

2

1

3780, 2280, 4260, 240

4

2640

3780, 2880, 1440, 480

4

2145

2880, 1440, 1440, 360

4

1530

300, 300, 60, 360

4

300

240, 300

2

270

1

1

1

10,080

1

10,080

11,520

1

11,520

21,600

1

21,600

1440, 1440, 5040, 2880, 360, 720, 1440, 1440, 2160, 1440, 1440, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 All 1

52

381.56

4

1

1 11 -rð4=24Þ rð4=24Þ

1

1

1

6 1 -rð4=24Þ rð4=24Þ

28,800, 43,200, 28,800, 4320, 86,400, 17,280, 8640, 51,840, 7200, 2880, 40,320, 16,200, 8220, 10,080, 10,080, 8640, 4320, 4320, 4320, 5760, 4320, 8640, 15,840, 10,800, 14,400, 1440, 4320, 18,720, 2880, 18,720, 720, 34,560, 5760, 2880, 10,080, 4320, 51,840, 2880, 10,080, 1440, 1440, 12,960, 2880, 10,080, 14,400, 24,480, 2880, 7200, 74,880, 249,120, 7200, 1440, 4320, 10,080, 2880, 10,080, 50,400 2880, 2160, 720, 420, 720, 7200, 1440, 1440, 720, 1440, 1440, 2880, 1440, 720, 720, 4320, 4320, 4320, 1440, 1440, 720, 720, 2880, 720, 720, 1440, 720, 1440, 2880,

57

19,350.53

37

1749.73

1 9 -rð4=24Þ rð4=24Þ

6 9 -rð4=24Þ rð4=24Þ

Appendices

6 24 -rð4=24Þ rð4=24Þ 9 6 -rð4=24Þ rð4=24Þ

9 11 rð4=24Þ -rð4=24Þ

9 25 -rð4=24Þ rð4=24Þ 11 9 -rð4=24Þ rð4=24Þ 11 26 -rð4=24Þ rð4=24Þ

24 6 -rð4=24Þ rð4=24Þ 25 6 -rð4=24Þ rð4=24Þ 25 24 -rð4=24Þ rð4=24Þ 26 9 -rð4=24Þ rð4=24Þ

2880, 1440, 720, 1440, 720, 240, 1440, 1440 4320 5760, 7200, 7200, 11,520, 3600, 11,520, 11,520, 10,080, 8640, 7920, 4320, 10,080, 21,600, 5760, 2880, 7200, 5760, 8640, 3600, 3600, 4320, 4320, 8640, 17,280, 4320, 8640, 43,200, 28,800, 15,840, 5760, 7200, 8640, 2880, 10,080, 43,200 1440, 1440, 2880, 1440, 720, 720, 720, 4320, 2880, 1440, 3600, 2880, 2160, 1200, 2880, 1440 4320, 7200, 8640, 6480, 8640, 7200 7200, 6468, 5760, 2880, 8640 4320, 5760, 5760, 4320, 8640, 2880, 4320, 5760, 5760, 4320, 5760, 4320 4320, 7200 20,160, 28,800, 8640, 21,600, 8640, 2880 30,240

1

4320

35

10,614.86

16

2010

6

7080

5

6189.6

12

5160

2

5760

6

15,120

1

30,240

14,400, 14,400, 10,080, 10,080, 86,400, 57,600, 21,600, 7200, 31,680, 12,960, 72,000 10,080

11

30,763.64

1

10,080

1 5 -rð5=1Þ rð5=1Þ

1

1

1

5 1 -rð5=1Þ rð5=1Þ

600

1

600

1 11 -rð5=15Þ rð5=15Þ

1

1

1

6 1 -rð5=15Þ rð5=15Þ 9 6 rð5=15Þ -rð5=15Þ 11 9 rð5=15Þ -rð5=15Þ 1 6 rð5=24Þ -rð5=24Þ

10,080

1

10,080

11,520

1

11,520

21,600

1

21,600

2880, 1440, 1440, 2160, 1440, 720, 180, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1, 1, 1, 1, 1, 1, 1, 1, 1

38

270.82

9

1

All 1

4

1

7200, 10,080, 7200, 1440, 11,520, 15,840, 11,520, 2880, 10,080, 5760, 4320, 4320, 4320, 11,520, 4320, 4320, 1440, 11,520, 1440, 1440, 14,400, 2880, 4320, 20,160, 2880, 18,720, 7200, 7200, 4320, 2880, 10,080, 5760, 2880, 12,960, 1440, 1440, 11,520, 2880, 10,080, 8640, 20,160, 2880, 231,840, 72,000, 226,080, 7200, 8640, 12,960, 12,960, 8640, 41,760 1440, 720, 720, 720, 1440, 2880, 4320, 2160, 1440, 1440, 720, 1440, 720, 1440,

51

18,240

22

1418.18

26 25 -rð4=24Þ rð4=24Þ

1 9 -rð5=24Þ rð5=24Þ 1 11 -rð5=24Þ rð5=24Þ 6 1 rð5=24Þ -rð5=24Þ

6 9 rð5=24Þ -rð5=24Þ

207

208

Appendices

720, 1440, 720, 2880, 1440, 240, 1440, 720 2880, 10,080, 2880, 7200, 2880, 2880, 4320, 8640, 3600, 15,840, 20,160, 4320, 64,800, 11,520, 7200, 6480, 15,840, 5760, 4320, 4320, 7200, 4320, 10,080, 7200, 21,600, 4320, 28,800, 43,200, 43,200, 7200, 10,080, 11,520, 14,400, 8640, 50,400 720, 720, 720, 1440, 2880, 720, 2880, 1440, 1200, 2880, 720 30,240, 10,080, 5760, 11,520, 10,080, 18,720, 5760, 97,920, 72,000, 5760, 21,600, 7200, 14,400, 23,040, 79,200

9 6 -rð5=24Þ rð5=24Þ

9 11 rð5=24Þ -rð5=24Þ

11 9 -rð5=24Þ rð5=24Þ

35

13,659.43

11

1483.64

15

27,552

Appendix 7 Table of chi-square distribution

α n

0.01

0.02

0.05

0.10

0.20

0.30

0.70

0.80

0.90

0.95

0.98

0.99

0.06 0.45 1.00 1.65 2.34 3.07 3.82 4.59 5.38 6.18 6.99 7.81 8.63 9.47 10.31 11.15 12.00 12.86 13.72 14.59

0.02 0.21 0.58 1.06 1.61 2.20 2.83 3.49 4.17 4.86 5.58 6.30 7.04 7.79 8.55 9.31 10.08 10.86 11.65 12.44

0.00 0.10 0.35 0.71 1.14 1.63 2.17 2.73 3.32 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.12 10.85

0.00 0.04 0.18 0.43 0.75 1.13 1.56 2.03 2.53 3.06 3.61 4.18 4.76 5.37 5.98 6.61 7.25 7.91 8.57 9.24

0.00 0.02 0.11 0.30 0.55 0.87 1.24 1.65 2.09 2.56 3.08 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26

α

The values of u 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

6.63 9.21 11.34 13.28 15.09 16.81 18.47 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.80 36.19 37.57

5.41 7.82 9.35 11.67 13.39 15.03 16.62 18.17 19.68 21.16 22.62 24.05 25.47 26.87 28.26 29.63 30.99 32.35 33.69 35.02

3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.67 21.03 22.36 23.68 25.00 26.30 27.59 28.87 30.15 31.41

2.71 4.60 6.25 7.78 9.24 10.68 12.02 13.36 14.68 15.99 17.27 18.55 19.81 21.06 22.31 23.54 24.77 25.99 27.20 28.41

1.64 3.22 4.64 5.99 7.29 8.56 9.80 11.03 12.24 13.44 14.63 15.81 16.98 18.15 19.31 20.46 21.61 22.76 23.90 25.04

1.07 2.41 3.66 4.88 6.06 7.23 8.38 9.52 10.66 11.78 12.90 14.01 15.12 16.22 17.32 18.42 19.51 20.60 21.69 22.77

0.15 0.71 1.42 2.19 3.00 3.83 4.67 5.53 6.39 7.27 8.15 9.03 9.93 10.82 11.72 12.62 13.53 14.44 15.35 16.27

Appendices

209

Appendix 8 Procedures of modeling, identification, prediction, optimization, and mitigation of critical infrastructure accident consequences Modeling 1. Modeling the process of initiating events a. to fix initiating events Ei ; i 5 1; 2; . . .; n1 ; b. to fix or define general parameters of the process of initiating events i. the number of states of the process of initiating events ω, ii. the states of the process of initiating events e1 ; e2 ; . . .; eω . 2. Modeling the process of environment threats a. to fix subareas Dk ; k 5 1; 2; . . .; n3 that may be degraded by the environment threats (if necessary); b. to fix environment threats Hi ; i 5 1; 2; . . .; n2 ; c. to fix the parameters f i ; i 5 1; 2; . . .; n2 and their range li ;i 5 1; 2; . . .; n2 to characterize the particular environment threats Hi ; i 5 1; 2; . . .; n2 ; d. to fix or define general parameters of the process of environment threats i. the number of the environment threats states υk of the subarea Dk ; k 5 1; 2; . . .; n3 , ii. the states of the process of environment threats s1ðkÞ ; s2ðkÞ ; . . .; sυðkÞk of the subarea Dk ; k 5 1; 2; . . .; n3 . 3. Modeling the process of environment degradation a. to fix dangerous degradation effects Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; b. to fix the range of degradation effect vm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; c. to fix or define general parameters of the process of environment degradation i. the number of the environment degradation states ‘k of the subarea Dk ; k 5 1; 2; . . .; n3 , ‘k 1 2 ii. the states of the process of environment degradation rðkÞ ; rðkÞ ; . . .; rðkÞ of the subarea Dk ; k 5 1; 2; . . .; n3 .

Identification 1. Identification of the process of initiating events a. to fix the duration time of the experiment Θ; b. to collect the set of realizations of the conditional sojourn times θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j of the process of initiating events at the state el when the next transition is to the state ej ; c. to fix and collect statistical data necessary to evaluate the initial transient probabilities pl ð0Þ of the process of initiating events at the particular states at the initial moment t 5 0 i. the vector of realizations nl ð0Þ of the numbers of the process of initiating events staying at the states el ; l 5 1; 2; . . .; ω at the initial moment t 5 0; according to (4.1), ii. the number of the process of initiating events realizations nð0Þ at the moment t 5 0 according to (4.2); d. to fix and collect statistical data necessary to evaluate the probabilities plj ; l; j 5 1; 2; . . .; ω; l 6¼ j of the process of initiating events transitions between its states i. the matrix of realizations nlj ; l; j 5 1; 2; . . .; ω; l 6¼ j of the numbers of the process of initiating events transitions from the state el into the state ej ; according to (4.3), ii. the vector of realizations of the total numbers nl ; l 5 1; 2; . . .; ω of the process of initiating events transitions from the state el ; l 5 1; 2; . . .; ω; according to (4.4); e. on the basis of the statistical data obtained in (c) and (d), respectively, to estimate the unknown basic parameters of the process of initiating events i. the vector of the initial probabilities pl ð0Þ; l 5 1; 2; . . .; ω of the process of initiating events at the particular states el at the moment t 5 0; according to (4.6), ii. the matrix of realizations of the probabilities plj ; l; j 5 1; 2; . . .; ω; l 6¼ j of transitions of the process of initiating events from the state el into the state ej ; according to (4.8); f. to estimate parameters of distributions of the process of initiating events conditional sojourn times at particular states, according to (4.10)(4.17); g. to identify the distribution functions H lj ðtÞ; l; j 5 1; 2; . . .; ω; l 6¼ j of the process of initiating events conditional sojourn times θlj at the states, according to the scheme given in Section 4.1.3;

210

Appendices

h. on the basis of the statistical data obtained in (g) to define the matrix of conditional distribution functions H lj ðtÞ; l; j 5 1; 2; . . .; ω; l 6¼ j of the process of initiating events conditional sojourn times θlj at the state el , while its next transition will be done to the state ej ; according to (3.5); i. to estimate mean values M lj 5 E½θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j of the process of initiating events conditional sojourn times θlj at the particular states, according to (3.9)(3.11). 2. Identification of the process of environment threats a. to fix the duration time of the experiment Θ; b. to collect the set of realizations of the conditional sojourn times ηijðk=lÞ i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats at the state siðk=lÞ when the next transition is to the state sjðk=lÞ ; c. to fix and collect statistical data necessary to evaluate the initial transient probabilities piðk=lÞ ð0Þ of the process of environment threats at the particular states at the initial moment t 5 0 i. the vector of realizations niðk=lÞ ð0Þ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the numbers of the process of environment threats staying at the particular states siðk=lÞ at the initial moment t 5 0; according to (4.92), ii. the number of the process of environment threats realizations nðk=lÞ ð0Þ at the moment t 5 0; according to (4.93); d. to fix and collect statistical data necessary to evaluate the probabilities pijðk=lÞ of the process of environment threats transitions between its states i. the matrix of realizations nijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the numbers of the process of environment threats transitions from the state siðk=lÞ into the state sjðk=lÞ ; according to (4.94), ii. the vector of realizations of the total numbers niðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats transitions from the state siðk=lÞ ; according to (4.95); e. on the basis of the statistical data obtained in (c) and (d), respectively, to estimate the unknown basic parameters of the process of environment threats i. the vector of the initial probabilities piðk=lÞ ð0Þ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats at the particular states siðk=lÞ at the moment t 5 0; according to (4.97), ii. the matrix of realizations of the probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats transitions from the state siðk=lÞ into the state sjðk=lÞ ; according to (4.99); f. to estimate parameters of distributions of the process of environment threats conditional sojourn times at the particular states, according to (4.101)(4.108); ij g. to identify the distribution functions Hðk=lÞ ðtÞ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats conditional sojourn times ηijðk=lÞ ; at the particular states, according to scheme given in Section 4.3.3; h. on the basis of the statistical data obtained in (g) to define the matrix of conditional distribution functions ij Hðk=lÞ ðtÞ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats conditional sojourn times ηijðk=lÞ ; at the state siðk=lÞ , while its next transition will be done to the state sjðk=lÞ ; according to (3.24); ij i. to estimate mean values Mðk=lÞ 5 E½ηijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats conditional sojourn times ηijðk=lÞ at the particular states, according to (3.30)(3.34). 3. Identification of the process of environment degradation a. to fix the duration time of the experiment Θ; b. to collect the set of realizations of the conditional sojourn times ζ ijðk=υÞ i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; i υ 5 1; 2; . . .; υk of the process of environment degradation at the state rðk=υÞ when the next transition is to the state j rðk=υÞ ; c. to fix and collect statistical data necessary to evaluate the initial transient probabilities qiðk=υÞ ð0Þ of the process of environment degradation at the particular states at the initial moment t 5 0 i. the vector of realizations niðk=υÞ ð0Þ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the numbers of the proi cess of environment degradation staying at the particular states rðk=υÞ at the initial moment t 5 0; according to (4.215), ii. the number of the process of environment degradation realizations nðk=υÞ ð0Þ at the moment t 5 0; according to (4.216); d. to fix and collect statistical data necessary to evaluate the probabilities qijðk=υÞ of the process of environment degradation transitions between its states i. the matrix of realizations nijðk=υÞ i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the numbers of the j i process of environment degradation transitions from the state rðk=υÞ into the state rðk=υÞ ; according to (4.217),

Appendices

e.

f. g. h.

i.

211

ii. the vector of realizations of the total numbers niðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the i process of environment degradation transitions from the state rðk=υÞ ; according to (4.218); on the basis of the statistical data obtained in (c) and (d), respectively, to estimate the unknown basic parameters of the process of environment degradation i. the vector of the initial probabilities qiðk=υÞ ð0Þ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of i environment degradation at the particular states rðk=υÞ at the moment t 5 0; according to (4.220), ij ii. the matrix of realizations of the probabilities qðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of j i the process of environment degradation transitions from the state rðk=υÞ into the state rðk=υÞ ; according to (4.222); to estimate parameters of distributions of process of environment degradation conditional sojourn times at the particular states, according to (4.224)(4.231); to identify the distribution functions Gijðk=υÞ ðtÞ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation conditional sojourn times ζ ijðk=υÞ at the particular states, according to scheme given in Section 4.5.3; on the basis of the statistical data obtained in (g) to define the matrix of conditional distribution functions Gijðk=υÞ ðtÞ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation condij i tional sojourn times ζ ijðk=υÞ at the state rðk=υÞ , while its next transition will be done to the state rðk=υÞ ; according to (3.50); ij to estimate mean values Mðk=υÞ 5 E½ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation conditional sojourn times ζ ijðk=υÞ at the particular states, according to (3.54)(3.56).

Prediction 1. Prediction of the process of initiating events a. to estimate mean values of unconditional sojourn times E½θl ; l 5 1; 2; . . .; ω of the process of initiating events at particular states, according to (3.13); b. to solve the system of equations (3.16); c. to estimate approximate limit values of transient probabilities at the particular states of the process of initiating events, according to (3.15); l d. to estimate the approximate mean values of the sojourn total times θ^ of the process of initiating events EðtÞ in the time interval, for example, θ 5 1 month at the particular states el ; according to (3.17). 2. Prediction of the process of environment threats a. to estimate mean values of unconditional sojourn times E½ηiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats at particular states, according to (3.36); b. to solve the system of equations (3.39); c. to estimate approximate limit values of transient probabilities at the particular states of the process of environment threats, according to (3.38); d. to estimate the approximate mean values of the sojourn total times η^ iðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ in the time interval, for example, η 5 1 month at the particular states siðk=lÞ ; according to (3.40). 3. Prediction of the process of environment degradation a. to estimate mean values of unconditional sojourn times E½ζ iðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation at particular states, according to (3.58); b. to solve the system of equations (3.61); c. to estimate approximate limit values of transient probabilities at the particular states of the process of environment degradation, according to (3.60); i d. to estimate the approximate mean values of the sojourn total times ζ^ ðk=υÞ of the process of environment degradai tion Rðk=υÞ ðtÞ in the time interval, for example, ζ 5 1 month at the particular states rðk=υÞ ; according to (3.62). 4. Superposition of initiating events, environment threats, and environment degradation processes a. to estimate unconditional approximate limit transient probabilities, qiðkÞ , at the particular states of the process of environment degradation RðkÞ ðtÞ; according to (3.63); i b. to estimate the approximate mean values of the sojourn total time ζ^ ðkÞ of the unconditional process of environi ment degradation RðkÞ ðtÞ in the time interval, for example, θ 5 1 month at particular states rðkÞ , according to (3.64). 5. Prediction of accident losses

212

Appendices

i a. to fix losses LiðkÞ ðtÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 at the environment degradation state rðkÞ , according to (6.1); b. to estimate approximate expected value of the losses LðkÞ ðtÞ; k 5 1; 2; . . .; n3 associated with the process of the environment degradation RðkÞ ðtÞ of the subarea Dk ; according to (6.2); c. to estimate the total expected value of the losses LðϕÞ associated with the process of the environment degradation RðtÞ in all subareas for the fixed time ϕ; ϕ . 0; according to (6.6). 6. Prediction of accident losses impacted by the climateweather change process a. to fix coefficients ½ρiðkÞ ðbÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; b 5 1; 2; . . .; w of the climateweather change process i impact on the losses LiðkÞ ðtÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 at the environment degradation state rðkÞ , while the climateweather change process CðtÞ at the critical infrastructure accident area is at the climateweather state cb ; b. to estimate limit values of transient probabilities qb of the climateweather change process CðtÞ at the particular states cb ; c. to estimate losses ½LiðkÞ ðtÞðbÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; b 5 1; 2; . . .; w at the environment degradation state i rðkÞ , while the climateweather change process CðtÞ is at the climateweather state cb ; according to (6.9); d. to estimate conditional approximate expected value of the losses ½LðkÞ ðtÞðbÞ ; k 5 1; 2; . . .; n3 ; b 5 1; 2; . . .; w associated with the process of the environment degradation RðkÞ ðtÞ of the subarea Dk , while the climateweather change process CðtÞ is at the climateweather state cb ; according to (6.11); e. to estimate unconditional approximate expected value of the losses LðkÞ ðtÞ; k 5 1; 2; . . .; n3 associated with the process of the environment degradation RðkÞ ðtÞ of the subarea Dk impacted by the climateweather change process CðtÞ; according to (6.13); f. to estimate the total expected value of the losses LðϕÞ associated with the process of the environment degradation RðtÞ in all subareas impacted by the climateweather change process CðtÞ, for the fixed time ϕ; ϕ . 0; according to (6.14); g. to estimate indicators RIðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; n3 resilience to the losses associated with the critical infrastructure accident related to the climateweather change, according to (6.15) and (6.17); h. to estimate indicator RI of the environment in the entire area D resilience to the losses associated with the critical infrastructure accident related to the climateweather change, according to (6.16) and (6.18).

Optimization 1. Optimization of accident losses was not impacted by the climateweather change process a. to formulate the optimization problem as a linear programming model with the objective function of the form given by (7.1); b. to fix the optimal value q_iðkÞ i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 of the limit transient probabilities qiðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 that minimize the objective functions given by (7.1), using procedure given in Section 7.1; c. to estimate the optimal value of losses L_ðkÞ ðtÞ; k 5 1; 2; . . .; n3 ; according to (7.17). 2. Optimization of accident losses impacted by the climateweather change process a. to formulate the optimization problem as a linear programming model with the objective function of the form given by (7.18); b. to fix the optimal value qb q_iðkÞ i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 of the limit transient probabilities qb qiðkÞ ; i 5 1; 2; . . .; ‘k ; b 5 1; 2; . . .; w; k 5 1; 2; . . .; n3 that minimize the objective functions given by (7.18), using procedure given in Section 7.2; c. to estimate the optimal value of losses L_ ðkÞ ðtÞ; k 5 1; 2; . . .; n3 impacted by the climateweather change process, according to (7.34); d. to estimate the optimal indicator RI_ðkÞ of the environment in the subareas Dk ; k 5 1; 2; . . .; n3 resilience to the losses associated with the critical infrastructure accident related to the climateweather change, according to (7.35); e. to estimate the optimal indicator RI_ of the environment in the entire area D resilience to the losses associated with the critical infrastructure accident related to the climateweather change, according to (7.36).

Mitigation 1. To suggest the new strategy assuring lower environment losses.

Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A Accidental pollution, 7 9

B Baltic Sea waters, 1, 2f, 8 environment degradation, process of, prediction of, 107 110 environment degradation, process of, statistical identification of, 85 89 basic parameters, evaluation of, 88 89 collected statistical data, 86 88 distribution functions of conditional sojourn times, identification of, 89 environment threats, process of, prediction of, 98 101 environment threats, process of, statistical identification of, 68 72 basic parameters, evaluation of, 71 collected statistical data, 68 70 distribution functions of conditional sojourn times, identification of, 71 72 initiating events, process of, prediction of, 95 97 initiating events, process of, statistical identification of, 46 56 basic parameters, evaluation of, 47 48 collected statistical data, 46 47 distribution functions of conditional sojourn times, identification of, 50 56, 52f, 52t, 53f, 54f distributions of conditional sojourn times, evaluating parameters of, 48 50, 50f inventory of accident losses at, 171, 172t, 173t losses with considering climate weather change process impact, 126 132 analysis of results, 138t, 139t, 140t, 141 142 at Gdynia and Karlskrona ports, 130 132 at open and restricted sea waters, 126 130 losses without considering climate weather change process impact, 124 126 at Gdynia and Karlskrona ports, 125 126 at open and restricted sea waters, 124 125 realizations of conditional lifetimes at states of environment degradation, process of, 201

environment threats, process of, 196 197 initiating events, process of, 183 184 ship accidents at, 8 14, 10f, 12f superposition of processes at, 117 118 Baltic Shipping Critical Infrastructure Network (BSCIN), 8 Bioaccumulative substance presence in ship’s accident area (H5 threat), 19 ranking of, 19 20, 20f BSCIN. See Baltic Shipping Critical Infrastructure Network (BSCIN)

C Chemical releases into marine environment, threats from, 15 20 bioaccumulative substance presence in ship’s accident area (H5 threat), 19, 20f chemical substance explosion in ship’s accident area (H1 threat), 16, 16f corrosive substance presence in ship’s accident area (H4 threat), 18 19, 19f environment degradation, 20 fire from chemical substance in ship’s accident area (H2 threat), 16 17, 17f other dangerous chemical substances in ship’s accident area (H6 threat), 20 toxic substance presence in ship’s accident area (H3 threat), 17 18, 18f Chemical substance explosion in ship’s accident area (H1 threat), 16 ranking of, 16, 16f Climate weather change process at world sea waters losses minimization with considering, 162 169 losses minimization without considering, 160 162 integrated impact model on critical infrastructure accident consequences related to, 122 124 critical infrastructure accident area, 122 123 losses, related to, 123 124 losses minimization with considering, 145 147, 152 160 at Gdynia and Karlskrona Ports, 158 160 at open sea waters, 152 157 at restricted sea waters, 157 158

losses minimization without considering, 143 145, 148 152 at Gdynia and Karlskrona Ports, 152 at open and restricted sea waters, 148 151 Corrosive substance presence in ship’s accident area (H4 threat), 18 19 ranking of, 19, 19f Cost analysis, 121, 141 142 Critical infrastructure definition of, 7 shipping as, 7 20 Critical infrastructure accident consequences, analysis of, 1, 1f, 177 See also individual entries environment degradation, process of, 4 5 environment threats, process of, 3 4 initiating events, process of, 2 3

D Dangerous goods, 15 16 Density function of conditional sojourn times, identification of environment degradation, process of, 76, 79, 82 83, 90 93 environment threats, process of, 59, 62, 65 66, 74 75 initiating events, process of, 40, 42, 45 46, 50 58 Distribution function of conditional sojourn times, identification of environment degradation, process of, 82 83 environment threats, process of, 65 66 initiating events, process of, 44 45, 45f, 46f

E Emulsification, 9 Environment degradation, process of, 4 5 Environment degradation at Baltic Sea waters, realizations of conditional lifetimes at states of process of, 201 at world sea waters, realizations of conditional lifetimes at states of process of, 201 208 caused by chemical releases, 20 effects, levels of, 20t identification of, process of, 76 83

213

214

Index

Environment degradation (Continued) distributions of conditional sojourn times, estimating parameters of, 78 82, 83f distributions of conditional sojourn times, identification of, 82 83 statistical identification, at Baltic Sea waters, 85 89 statistical identification, at world sea waters, 89 94 unknown basic parameters, estimation of, 77 78 losses of critical infrastructure accidents associated with, 121 129, 131 149, 151 153, 155 157, 159 169 modeling process of, 32 37 prediction of, process of, 107 117 at Baltic Sea waters, 107 110 at world sea waters, 110 117 states of process of, 84 85 Environment threats at Baltic Sea waters, realizations of conditional lifetimes at states of process of, 196 197 at world sea waters, realizations of conditional lifetimes at states of process of, 197 201 modeling process of, 25 32, 25f, 28f, 29f of chemical releases into marine environment bioaccumulative substance presence in ship’s accident area (H5 threat), 19, 20f chemical substance explosion in ship’s accident area (H1 threat), 16, 16f corrosive substance presence in ship’s accident area (H4 threat), 18 19, 19f environment degradation, 20 fire from chemical substance in ship’s accident area (H2 threat), 16 17, 17f other dangerous chemical substances in ship’s accident area (H6 threat), 20 toxic substance presence in ship’s accident area (H3 threat), 17 18, 18f prediction of, process of, 98 107 at Baltic Sea waters, 98 101 at world sea waters, 101 107 process of, 3 4 states of process of, 67 68 theoretical background of identification of, process of, 59 66 distributions of conditional sojourn times, estimating parameters of, 61 65, 66f distributions of conditional sojourn times, identification of, 65 66 statistical identification, at Baltic Sea waters, 68 72 statistical identification, at world sea waters, 72 76 unknown basic parameters, estimation of, 59 60 European Commission, 7 European Critical Infrastructure, sectors of, 7

F Fire from chemical substance in ship’s accident area (H2 threat), 16 17 ranking of, 17, 17f

G

GBNCIN. See Global Baltic Network of Critical Infrastructure Networks (GBNCIN) Gdynia port, losses of shipping critical infrastructure accidents at, 125 126, 133 134 climate weather change process impact, 130 132, 136 141, 138t, 140t GESAMP. See United Nation Joint Group of Experts on the Scientific Aspects of Marine Pollution (GESAMP) Global Baltic Network of Critical Infrastructure Networks (GBNCIN), 8

H Hazardous and noxious substances (HNS), 9 12, 11f HELCOM. See Helsinki Commission (HELCOM) Helsinki Commission (HELCOM), 10 HNS. See Hazardous and noxious substances (HNS)

I Identification of critical infrastructure accident consequences, 39, 209 211 environment degradation, process of, 76 83 distributions of conditional sojourn times, estimating parameters of, 78 82, 83f distributions of conditional sojourn times, identification of, 82 83 states, 84 85 statistical identification, at Baltic Sea waters, 85 89 statistical identification, at world sea waters, 89 94 unknown basic parameters, estimation of, 77 78 environment threats, process of, 59 66 distributions of conditional sojourn times, estimating parameters of, 61 65, 66f distributions of conditional sojourn times, identification of, 65 66 states of, 67 68 statistical identification, at Baltic Sea waters, 68 72 statistical identification, at world sea waters, 72 76 unknown basic parameters, estimation of, 59 60 initiating events, process of, 40 45 distributions of conditional sojourn times, estimating parameters of, 41 44 distributions of conditional sojourn times, identification of, 44 45, 45f, 46f

states of, 46 statistical identification, at Baltic Sea waters, 46 56 statistical identification, at world sea waters, 46 47 unknown basic parameters, estimation of, 40 41, 50t IMDG Code. See International Maritime Dangerous Goods Code (IMDG Code) IMO. See International Maritime Organization (IMO) Initiating events at Baltic Sea waters, realizations of conditional lifetimes at states of process of, 183 184 at world sea waters, realizations of conditional lifetimes at states of process of, 184 196 caused by world sea ship traffic accidents, 14 15 identification of, process of, 40 45 distributions of conditional sojourn times, estimating parameters of, 41 44 distributions of conditional sojourn times, identification of, 44 45, 45f, 46f statistical identification, at Baltic Sea waters, 46 56 statistical identification, at world sea waters, 46 47 unknown basic parameters, estimation of, 40 41, 50t modeling process of, 21 25, 23f, 24f prediction of, process of, 95 98 at Baltic Sea waters, 95 97 at world sea waters, 97 98 process of, 2 3 states of process of, 46 International Maritime Dangerous Goods Code (IMDG Code), 15 19 International Maritime Organization (IMO), 14 International Tanker Owners Pollution Federation Limited (ITOPF), 12 13, 13f, 14f ITOPF. See International Tanker Owners Pollution Federation Limited (ITOPF)

K Karlskrona port, losses of shipping critical infrastructure accidents at, 125 126, 133 134 climate weather change process impact, 130 132, 136 141, 139t

L Losses minimization with considering climate weather change process impact, 145 147, 152 160 at Gdynia and Karlskrona Ports, 158 160 at open sea waters, 152 157 at restricted sea waters, 157 158 at world sea waters, 162 169

Index

without considering climate weather change process impact, 143 145, 148 152 at Gdynia and Karlskrona Ports, 152 at open and restricted sea waters, 148 151 at world sea waters, 160 162 Losses of critical infrastructure accidents, 121 122 at Baltic Sea waters with considering climate weather change process impact, 126 132 analysis of results, 138t, 139t, 140t, 141 142 Gdynia and Karlskrona ports, 130 132 open and restricted sea waters, 126 130 at Baltic Sea waters without considering climate weather change process impact, 124 126 Gdynia and Karlskrona ports, 125 126 open and restricted sea waters, 124 125 at world sea waters with considering climate weather change process impact, 134 141 analysis of results, 138t, 139t, 140t, 141 142 Gdynia and Karlskrona ports, 136 141 open and restricted sea waters, 134 136 at world sea waters without considering climate weather change process impact, 132 134 Gdynia and Karlskrona ports, 133 134 open and restricted sea waters, 132 133 climate weather change process, integrated impact model on critical infrastructure accident consequences related to, 122 124 critical infrastructure accident area, 122 123 losses, 123 124 mitigation of, 171 modeling, 121 optimization of, 143

M Maritime transport, consequences of, 14 20 environment degradation caused by chemical releases, 20 initiating events caused by world sea ship traffic accidents, 14 15 threats from chemical releases into marine environment, 15 20 bioaccumulative substance presence in ship’s accident area (H5 threat), 19, 20f chemical substance explosion in ship’s accident area (H1 threat), 16, 16f corrosive substance presence in ship’s accident area (H4 threat), 18 19, 19f fire from chemical substance in ship’s accident area (H2 threat), 16 17, 17f other dangerous chemical substances in ship’s accident area (H6 threat), 20 toxic substance presence in ship’s accident area (H3 threat), 17 18, 18f

Mitigation of critical infrastructure accident consequences, 212 losses, 171 inventory of accident losses at Baltic Sea waters, 171, 172t, 173t inventory of accident losses at world sea waters, 171, 173t, 174t strategy for, 171 175 Modeling critical infrastructure accident consequences, 21 37, 209 environment degradation, process of, 32 37 environment threats, process of, 25 32, 25f, 28f, 29f initiating events, process of, 21 25, 23f, 24f

O Optimization of critical infrastructure accident losses, 143, 212 losses minimization with considering climate weather change process impact, 145 147, 152 160 at Gdynia and Karlskrona Ports, 158 160 at open sea waters, 152 157 at restricted sea waters, 157 158 at world sea waters, 162 169 losses minimization without considering climate weather change process impact, 143 145, 148 152 at Gdynia and Karlskrona Ports, 152 at open and restricted sea waters, 148 151 at world sea waters, 160 162

P Prediction of critical infrastructure accident consequences, 95, 211 212 environment degradation, process of, 107 117 at Baltic Sea waters, 107 110 at world sea waters, 110 117 environment threats, process of, 98 107 at Baltic Sea waters, 98 101 at world sea waters, 101 107 initiating events, process of, 95 98 at Baltic Sea waters, 95 97 at world sea waters, 97 98 superposition of processes, 117 120 at Baltic Sea waters, 117 118 at world sea waters, 118 120

R Resilience indicator, 138t, 139t, 140t, 141 142

S Sea accident, 8 14 initiating events caused by, 14 15 SEBC. See Standard European Behavior Classification (SEBC) Semi-Markov model, 22, 26, 34 35, 40, 46 47, 56, 59, 68 69, 72 74, 76, 86 93, 95, 98, 107

215

Serious accident, 14 Severe pollution, 14 Ship accidents area bioaccumulative substance presence in (H5 threat), 19, 20f chemical substance explosion in (H1 threat), 16, 16f corrosive substance presence in (H4 threat), 18 19, 19f fire from chemical substance in (H2 threat), 16 17, 17f other dangerous chemical substances in ship’s accident area (H6 threat), 20 toxic substance presence in (H3 threat), 17 18, 18f at Baltic Sea and world sea waters, 8 14, 8f, 9f, 10f, 11f, 12f, 13f, 14f Shipping, 1, 2f as critical infrastructure, 7 definition of, 7 8 Spilled oil in sea water, processes of, 9, 11 12, 11f, 13f, 14f Standard European Behavior Classification (SEBC), 15 Superposition of processes prediction, 117 120 at Baltic Sea waters, 117 118 at world sea waters, 118 120

T Toxic substance presence in ship’s accident area (H3 threat), 17 18 ranking of, 17 18, 18f Transportation infrastructure, definition of, 7

U United Nation Joint Group of Experts on the Scientific Aspects of Marine Pollution (GESAMP), 15 19 United Nations Conference on Trade and Development, 9

W World sea waters losses with considering climate weather change process impact, 134 141 analysis of results, 138t, 139t, 140t, 141 142 at Gdynia and Karlskrona ports, 136 141 at open and restricted sea waters, 134 136 losses without considering climate weather change process impact, 132 134 at Gdynia and Karlskrona ports, 133 134 at open and restricted sea waters, 132 133 environment degradation, process of, at prediction of, 110 117 statistical identification of, 89 94 environment threats, process of, at prediction of, 101 107

216

Index

World sea waters (Continued) statistical identification of, 72 76 initiating events, process of, at prediction of, 97 98 statistical identification of, 46 47 inventory of accident losses at, 171, 173t, 174t

losses minimization with considering climate weather change process impact at, 162 169 losses minimization without considering climate weather change process impact at, 160 162 realizations of conditional lifetimes at states of

environment degradation, process of, 201 208 environment threats, process of, 197 201 initiating events, process of, 184 196 ship accidents at, 8 14, 8f, 9f, 10f, 11f, 12f, 13f, 14f superposition of processes at, 118 120