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 xv1 What Is Reliability? 11.1 Reliability as a Property of Technical Objects, 11.2 Other “Ilities”, 21.3 Hierarchical Levels of Analyzed Objects, 51.4 How Can Reliability Be Measured?, 51.5 Software Reliability, 71.5.1 Case Study: Avalanche of Software Failures, 82 Unrecoverable Objects 92.1 Unit, 92.1.1 Probability of Failure-Free Operation, 92.1.2 Mean Time to Failure, 102.2 Series Systems, 112.2.1 Probability of Failure-Free Operation, 112.2.2 Mean Time to Failure, 132.3 Parallel System, 142.3.1 Probability of Failure-Free Operation, 142.3.2 Mean Time to Failure, 182.4 Structure of Type “k-out-of-n”, 202.5 Realistic Models of Loaded Redundancy, 222.5.1 Unreliable Switching Process, 232.5.2 Non-Instant Switching, 232.5.3 Unreliable Switch, 242.5.4 Switch Serving as Interface, 252.5.5 Incomplete Monitoring of the Operating Unit, 262.5.6 Periodical Monitoring of the Operating Unit, 282.6 Reducible Structures, 282.6.1 Parallel-Series and Series-Parallel Structures, 282.6.2 General Case of Reducible Structures, 292.7 Standby Redundancy, 302.7.1 Simple Redundant Group, 302.7.2 Standby Redundancy of Type “k-out-of-n”, 332.8 Realistic Models of Unloaded Redundancy, 342.8.1 Unreliable Switching Process, 342.8.2 Non-Instant Switching, 352.8.3 Unreliable Switch, 352.8.4 Switch Serving as Interface, 372.8.5 Incomplete Monitoring of the Operating Unit, 383 Recoverable Systems: Markov Models 403.1 Unit, 403.1.1 Markov Model, 413.2 Series System, 473.2.1 Turning Off System During Recovery, 473.2.2 System in Operating State During Recovery: Unrestricted Repair, 493.2.3 System in Operating State During Recovery: Restricted Repair, 513.3 Dubbed System, 533.3.1 General Description, 533.3.2 Nonstationary Availability Coefficient, 543.3.3 Stationary Availability Coefficient, 583.3.4 Probability of Failure-Free Operation, 593.3.5 Stationary Coefficient of Interval Availability, 623.3.6 Mean Time to Failure, 633.3.7 Mean Time Between Failures, 633.3.8 Mean Recovery Time, 653.4 Parallel Systems, 653.5 Structures of Type “m-out-of-n”, 664 Recoverable Systems: Heuristic Models 724.1 Preliminary Notes, 724.2 Poisson Process, 754.3 Procedures over Poisson Processes, 784.3.1 Thinning Procedure, 784.3.2 Superposition Procedure, 804.4 Asymptotic Thinning Procedure over Stochastic Point Process, 804.5 Asymptotic Superposition of Stochastic Point Processes, 824.6 Intersection of Flows of Narrow Impulses, 844.7 Heuristic Method for Reliability Analysis of Series Recoverable Systems, 874.8 Heuristic Method for Reliability Analysis of Parallel Recoverable Systems, 874.8.1 Influence of Unreliable Switching Procedure, 884.8.2 Influence of Switch’s Unreliability, 894.8.3 Periodical Monitoring of the Operating Unit, 904.8.4 Partial Monitoring of the Operating Unit, 914.9 Brief Historical Overview and Related Sources, 935 Time Redundancy 955.1 System with Possibility of Restarting Operation, 955.2 Systems with “Admissibly Short Failures”, 985.3 Systems with Time Accumulation, 995.4 Case Study: Gas Pipeline with an Underground Storage, 1005.5 Brief Historical Overview and Related Sources, 1026 “Aging” Units and Systems of “Aging” Units 1036.1 Chebyshev Bound, 1036.2 “Aging” Unit, 1046.3 Bounds for Probability of Failure-Free Operations, 1056.4 Series System Consisting of “Aging” Units, 1086.4.1 Preliminary Lemma, 1086.5 Series System, 1106.5.1 Probability of Failure-Free Operation, 1106.5.2 Mean Time to Failure of a Series System, 1126.6 Parallel System, 1146.6.1 Probability of Failure-Free Operation, 1146.6.2 Mean Time to Failure, 1176.7 Bounds for the Coefficient of Operational Availability, 1196.8 Brief Historical Overview and Related Sources, 1217 Two-Pole Networks 1237.1 General Comments, 1237.1.1 Method of Direct Enumeration, 1257.2 Method of Boolean Function Decomposition, 1277.3 Method of Paths and Cuts, 1307.3.1 Esary–Proschan Bounds, 1307.3.2 “Improvements” of Esary–Proschan Bounds, 1337.3.3 Litvak–Ushakov Bounds, 1357.3.4 Comparison of the Two Methods, 1397.4 Brief Historical Overview and Related Sources, 1408 Performance Effectiveness 1438.1 Effectiveness Concepts, 1438.2 General Idea of Effectiveness Evaluation, 1458.2.1 Conditional Case Study: Airport Traffic Control System, 1478.3 Additive Type of System Units’ Outcomes, 1508.4 Case Study: ICBM Control System, 1518.5 Systems with Intersecting Zones of Action, 1538.6 Practical Recommendation, 1588.7 Brief Historical Overview and Related Sources, 1609 System Survivability 1629.1 Illustrative Example, 1669.2 Brief Historical Overview and Related Sources, 16710 Multistate Systems 16910.1 Preliminary Notes, 16910.2 Generating Function, 16910.3 Universal Generating Function, 17210.4 Multistate Series System, 17410.4.1 Series Connection of Piping Runs, 17410.4.2 Series Connection of Resistors, 17710.4.3 Series Connections of Capacitors, 17910.5 Multistate Parallel System, 18110.5.1 Parallel Connection of Piping Runs, 18110.5.2 Parallel Connection of Resistors, 18210.5.3 Parallel Connections of Capacitors, 18210.6 Reducible Systems, 18310.7 Conclusion, 19010.8 Brief Historical Overview and Related Sources, 190Appendix A Main Distributions Related to Reliability Theory 195A.1 Discrete Distributions, 195A.1.1 Degenerate Distribution, 195A.1.2 Bernoulli Distribution, 196A.1.3 Binomial Distribution, 197A.1.4 Poisson Distribution, 198A.1.5 Geometric Distribution, 200A.2 Continuous Distributions, 201A.2.1 Intensity Function, 201A.2.2 Continuous Uniform Distribution, 202A.2.3 Exponential Distribution, 203A.2.4 Erlang Distribution, 204A.2.5 Hyperexponential Distribution, 205A.2.6 Normal Distribution, 207A.2.7Weibull–Gnedenko Distribution, 207Appendix B Laplace Transformation 209Appendix C Markov Processes 214C.1 General Markov Process, 214C.1.1 Nonstationary Availability Coefficient, 216C.1.2 Probability of Failure-Free Operation, 218C.1.3 Stationary Availability Coefficient, 220C.1.4 Mean Time to Failure and Mean Time Between Failures, 221C.1.5 Mean Recovery Time, 222C.2 Birth–Death Process, 223Appendix D General Bibliography 227Index 231

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