Probabilistic Reliability Models

IGOR USHAKOV, PhD, is Senior Consultant at Advanced Logistics Developments in Tel Aviv, Israel. He has published extensively in his areas of research interest, which include operations research, applied statistics, and probabilistic modeling. Dr. Ushakov is the author of Handbook of Reliability Engineering as well as the coauthor of Probabilistic Reliability Engineering and Statistical Reliability Engineering, all published by Wiley.

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