Network Reliability (eBook)

Preface 6
Contents 8
Notations and Abbreviations 10
1 Probability-Reminder 11
1.1 Algebra of Events 11
1.2 Random Variables 12
1.3 The Law of Total Probability 16
1.4 Order Statistics 16
1.5 Average, Variance 18
References 19
2 Networks and Examples 20
2.1 Nodes, Edges, Terminals 20
2.2 Series-Parallel Networks 26
References 28
3 Direct Network Reliability Calculation 29
3.1 Static Reliability and Structure Function 29
3.2 Counting Paths and Cuts 33
3.3 Using Inclusion/Exclusion Formula 35
3.4 Pivotal Formula 37
3.5 Burtin-Pittel Approximation 37
3.6 Network Resilience 39
3.7 Static and Dynamic Reliability 41
References 42
4 Elementary Methods for Reliability Evaluation 43
4.1 Exact Method for Computing Network Reliability 43
4.2 Crude Monte Carlo (CMC) 45
4.3 Relative Error of CMC 47
4.4 CMC for Network Dynamic Reliability Evaluation 49
4.5 Checking Network State 50
References 55
5 Element Importance 56
5.1 Birnbaum Importance Measure (BIM) 56
5.2 Direct Computation of BIM 57
5.3 Monte Carlo for Computing BIM 61
References 63
6 Destruction Monte Carlo 64
6.1 Individual and Comparative Network Reliability Analysis 64
6.2 Network Destruction 66
6.3 BIM Spectrum 70
6.4 CD-Spectrum and BIM Spectra Monte Carlo 72
References 76
7 Lomonosov's Turnip 77
7.1 Evolution Process 77
7.2 Lomonosov's Algorithm 80
7.3 Turnip for Networks with Unreliable Nodes 82
7.4 Convolution of Exponents 85
References 86
8 Examples of Network Analysis 87
8.1 Network Structure and Resilience Against Node Attack 87
8.2 Road/Highway Reinforcement 90
8.3 Flow in Network with Unreliable Edges 93
References 96