# Probabilistic Reliability Models

## Books

Practical Approaches to Reliability Theory in Cutting-Edge Applications

Probabilistic Reliability Models helps readers understand and properly use statistical methods

and optimal resource allocation to solve engineering problems.

The author supplies engineers with a deeper understanding of mathematical models while also

equipping mathematically oriented readers with a fundamental knowledge of the engineeringrelated

applications at the center of model building. The book showcases the use of probability

theory and mathematical statistics to solve common, real-world reliability problems. Following

an introduction to the topic, subsequent chapters explore key systems and models including:

• Unrecoverable objects and recoverable systems

• Methods of direct enumeration

• Markov models and heuristic models

• Performance effectiveness

• Time redundancy

• System survivability

• Aging units and their related systems

• Multistate systems

Detailed case studies illustrate the relevance of the discussed methods to real-world technical

projects including software failure avalanches, gas pipelines with underground storage, and

intercontinental ballistic missile (ICBM) control systems. Numerical examples and detailed

explanations accompany each topic, and exercises throughout allow readers to test their

comprehension of the presented material.

Probabilistic Reliability Models is an excellent book for statistics, engineering, and operations

research courses on applied probability at the upper-undergraduate and graduate levels. The

book is also a valuable reference for professionals and researchers working in industry who

would like a mathematical review of reliability models and the relevant applications.

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

7.1.1 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

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