Probabilistic reliability models / Igor Ushakov.

Includes bibliographical references and index.

Preface--xiii
Acronyms and Notations-- xv

1. What Is Reliability?-- 1
1.1 Reliability as a Property of Technical Objects, 1
1 2 Other "Ilities", 2
1.3 Hierarchical Levels of Analyzed Objects, 5
1.4 How Can Reliability Be Measured?, 5
1.5 Software Reliability, 7
1.5.1 Case Study: Avalanche of Software Failures, 8

2 Unrecoverable Objects-- 9
2.1 Unit, 9
2.1.1 Probability of Failure-Free Operation, 9
2.1.2 Mean Time to Failure, 10
2.2 Series Systems, 11
2.2.1 Probability of Failure-Free Operation, 11
2.2.2 Mean Time to Failure, 13
2.3 Parallel System, 14
2.3.1 Probability of Failure-Free Operation, 14
2.3.2 Mean Time to Failure, 18
2.4 Structure of Type "k-out-of-n", 20
2.5 Realistic Models of Loaded Redundancy, 22
2.5.1 Unreliable Switching Process, 23
2.5.2 Non-Instant Switching, 23
2.5.3 Unreliable Switch, 24
2.5.4 Switch Serving as Interface, 25
2.5.5 Incomplete Monitoring of the Operating Unit, 26
2.5.6 Periodical Monitoring of the Operating Unit, 28
2.6 Reducible Structures, 28
2.6.1 Parallel-Series and Series-Parallel Structures, 28
2.6.2 General Case of Reducible Structures, 29
2.7 Standby Redundancy, 30
2.7.1 Simple Redundant Group, 30
2.7.2 Standby Redundancy of Type "k-out-of-n", 33
2.8 Realistic Models of Unloaded Redundancy, 34
2.8.1 Unreliable Switching Process, 34
2.8.2 Non-Instant Switching, 35
2.8.3 Unreliable Switch, 35
2.8.4 Switch Serving as Interface, 37
2.8.5 Incomplete Monitoring of the Operating Unit, 38

3 Recoverable Systems: Markov Models 40
3.1 Unit, 40
3.1.1 Markov Model, 41
3.2 Series System, 47
3.2.1 Turning Off System During Recovery, 47
3.2.2 System in Operating State During Recovery: Unrestricted
Repair, 49
3.2.3 System in Operating State During Recovery: Restricted
Repair, 51
3.3 Dubbed System, 53
3.3.1 General Description, 53
3.3.2 Nonstationary Availability Coefficient, 54
3.3.3 Stationary Availability Coefficient, 58
3.3.4 Probability of Failure-Free Operation, 59
3.3.5 Stationary Coefficient of Interval Availability, 62
3.3.6 Mean Time to Failure, 63
3.3.7 Mean Time Between Failures, 63
3.3.8 Mean Recovery Time, 65
3.4 Parallel Systems, 65
3.5 Structures of Type "m-out-of-n", 66

4 Recoverable Systems: Heuristic Models 72
4.1 Preliminary Notes, 72
4.2 Poisson Process, 75
4.3 Procedures over Poisson Processes, 78
4.3.1 Thinning Procedure, 78
4.3.2 Superposition Procedure, 80
4.4 Asymptotic Thinning Procedure over Stochastic Point
Process, 80
4.5 Asymptotic Superposition of Stochastic Point Processes,
82
4.6 Intersection of Flows of Narrow Impulses, 84
4.7 Heuristic Method for Reliability Analysis of Series
Recoverable Systems, 87
4.8 Heuristic Method for Reliability Analysis of Parallel
Recoverable Systems, 87
4.8.1 Influence of Unreliable Switching Procedure, 88
4.8.2 Influence of Switch's Unreliability, 89
4.8.3 Periodical Monitoring of the Operating Unit, 90
4.8.4 Partial Monitoring of the Operating Unit, 91
4.9 Brief Historical Overview and Related Sources, 93

5 Time Redundancy 95
5.1 System with Possibility of Restarting Operation, 95
5.2 Systems with "Admissibly Short Failures", 98
5.3 Systems with Time Accumulation, 99
5.4 Case Study: Gas Pipeline with an Underground Storage,
100
5.5 Brief Historical Overview and Related Sources, 102

6 "Aging" Units and Systems of "Aging" Units 103
6.1 Chebyshev Bound, 103
6.2 "Aging" Unit, 104
6.3 Bounds for Probability of Failure-Free Operations, 105
6.4 Series System Consisting of "Aging" Units, 108
6.4.1 Preliminary Lemma, 108
6.5 Series System, 110
6.5.1 Probability of Failure-Free Operation, 110
6.5.2 Mean Time to Failure of a Series System, 112
6.6 Parallel System, 114
6.6.1 Probability of Failure-Free Operation, 114
6.6.2 Mean Time to Failure, 117
6.7 Bounds for the Coefficient of Operational Availability, 119
6.8 Brief Historical Overview and Related Sources, 121

7 Two-Pole Networks 123
7.1 General Comments, 123
7.1.1 Method of Direct Enumeration, 125
7.2 Method of Boolean Function Decomposition, 127
7.3 Method of Paths and Cuts, 130
7.3.1 Esary--Proschan Bounds, 130
7.3.2 "Improvements" of Esary--Proschan Bounds, 133
7.3.3 Litvak--Ushakov Bounds, 135
7.3.4 Comparison of the Two Methods, 139
7.4 Brief Historical Overview and Related Sources, 140

8 Performance Effectiveness 143
8.1 Effectiveness Concepts, 143
8.2 General Idea of Effectiveness Evaluation, 145
8.2.1 Conditional Case Study: Airport Traffic Control System,
147
8.3 Additive Type of System Units' Outcomes, 150
8.4 Case Study: ICBM Control System, 151
8.5 Systems with Intersecting Zones of Action, 153
8.6 Practical Recommendation, 158
8.7 Brief Historical Overview and Related Sources, 160

9 System Survivability 162
9.1 Illustrative Example, 166
9.2 Brief Historical Overview and Related Sources, 167

10 Multistate Systems 169
10.1 Preliminary Notes, 169
10.2 Generating Function, 169
10.3 Universal Generating Function, 172
10.4 Multistate Series System, 174
10.4.1 Series Connection of Piping Runs, 174
10.4.2 Series Connection of Resistors, 177
10.4.3 Series Connections of Capacitors, 179
10.5 Multistate Parallel System, 181
10.5.1 Parallel Connection of Piping Runs, 181
10.5.2 Parallel Connection of Resistors, 182
10.5.3 Parallel Connections of Capacitors, 182
10.6 Reducible Systems, 183
10.7 Conclusion, 190
10.8 Brief Historical Overview and Related Sources, 190

Appendix A Main Distributions Related to Reliability Theory
195
A.1 Discrete Distributions, 195
A.1.1 Degenerate Distribution, 195
A.1.2 Bernoulli Distribution, 196
A.1.3 Binomial Distribution, 197
A.1.4 Poisson Distribution, 198
A.1.5 Geometric Distribution, 200
A.2 Continuous Distributions, 201
A.2.1 Intensity Function, 201
A.2.2 Continuous Uniform Distribution, 202
A.2.3 Exponential Distribution, 203
A.2.4 Erlang Distribution, 204
A.2.5 Hyperexponential Distribution, 205
A.2.6 Normal Distribution, 207
A.2.7Weibull--Gnedenko Distribution, 207

Appendix B Laplace Transformation 209

Appendix C Markov Processes 214
C.1 General Markov Process, 214
C.1.1 Nonstationary Availability Coefficient, 216
C.1.2 Probability of Failure-Free Operation, 218
C.1.3 Stationary Availability Coefficient, 220
C.1.4 Mean Time to Failure and Mean Time Between Failures,
221
C.1.5 Mean Recovery Time, 222
C.2 Birth--Death Process, 223

Appendix D General Bibliography-- 227
Index-- 231

"Combined with many important theoretical and practical concepts, this book addresses the various applications of probabilistic reliability modeling"--