Ilia Frenkel, Center for Reliability and Risk Management, Industrial Engineering and Management Department, SCE - Shamoon College of Engineering, Israel
Ilia has forty years academic experience, teaching in Russia and Israel. Currently he is a senior lecturer and Director of the Centre for Reliability and Risk Management in the Industrial Engineering and Management Department of the SCE - Shamoon College of Engineering, Israel. Previously he worked as Department Chair and Associate Professor in the Applied Mathematics and Computers Department at Volgograd Civil Engineering Institute. He is a member of the editorial board on Maintenance and Reliability, Communications in Dependability and Quality Management, and has published scientific articles and book chapters in the fields of reliability, applied statistics and production and operation management.

Alex Karagrigoriou, Department of Mathematics and Statistics, University of Cyprus
Alex is Associate Professor of Statistics, Department of Mathematics and Statistics, University of Cyprus and Professor of Probability and Statistics, University of the Aegean. He worked at the University of Maryland, the United States Department of Agriculture and the Institute of Statistical Sciences, Taiwan, and taught thirty-two courses at the Universities of Maryland, Athens, the Aegean, and Cyprus. He has been involved in the organization of eight international conferences. He has written two textbooks on statistical analysis, teaching notes for undergraduate and graduate courses, and has published more than fifty articles on statistics and applied probability. Alex has served as reviewer for the United States National Security Council and the United Kingdom Economic and Social Research Council.

Anatoly Lisnianski, Reliability Department, The Israel Electric Corporation Ltd., Israel
Anatoly is an engineering expert in the Reliability Department of The Israel Electric Corporation Ltd., Israel, an adjunct senior lecturer in Haifa University, Israel, and Scientific Supervisor of the Centre for Reliability and Risk Management in the Industrial Engineering and Management Department of the SCE - Shamoon College of Engineering, Israel. Previous to this he was Senior Researcher in Federal Scientific & Production Center "Aurora" in St-Petersburg, Russia. He is a Senior Member of IEEE, Member of Israel Society of Quality and Israel Statistical Association, and is an author of more than one hundred publications in the field of reliability and applied probability.
He has been guest editor for International Journal of Reliability, Quality and Safety Engineering.

Andre Kleyner, Global Reliability Engineering Leader, Delphi Electronics and Safety, USA
Andre has twenty-five years of engineering, research, consulting, and managerial experience specializing in the reliability of electronic and mechanical systems. He is currently a Global Reliability Engineering Leader with Delphi Electronics & Safety and an adjunct professor at Purdue University.? He is a senior member of American Society for Quality, a Certified Reliability Engineer, Certified Quality Engineer, and a Six Sigma Black Belt.? He also holds several US and foreign patents and authored multiple publications on the topics of reliability, statistics, warranty management, and lifecycle cost analysis.? Andre Kleyner is Editor of the Wiley Series in Quality & Reliability Engineering.

List of Contributors xxv

Preface xxix

Acknowledgements xxxv

Part I DEGRADATION ANALYSIS, MULTI-STATE AND CONTINUOUS-STATE SYSTEM RELIABILITY

1 Methods of Solutions of Inhomogeneous Continuous Time Markov Chains for Degradation Process Modeling 3
Yan-Fu Li, Enrico Zio and Yan-Hui Lin

1.1 Introduction 3

1.2 Formalism of ICTMC 4

1.3 Numerical Solution Techniques 5

1.3.1 The Runge–Kutta Method 5

1.3.2 Uniformization 6

1.3.3 Monte Carlo Simulation 7

1.3.4 State-Space Enrichment 9

1.4 Examples 10

1.4.1 Example of Computing System Degradation 10

1.4.2 Example of Nuclear Component Degradation 11

1.5 Comparisons of the Methods and Guidelines of Utilization 13

1.6 Conclusion 15

References 15

2 Multistate Degradation and Condition Monitoring for Devices with Multiple Independent Failure Modes 17
Ramin Moghaddass and Ming J. Zuo

2.1 Introduction 17

2.2 Multistate Degradation and Multiple Independent Failure Modes 19

2.2.1 Notation 19

2.2.2 Assumptions 20

2.2.3 The Stochastic Process Model 21

2.3 Parameter Estimation 23

2.4 Important Reliability Measures of a Condition-Monitored Device 25

2.5 Numerical Example 27

2.6 Conclusion 28

Acknowledgements 30

References 30

3 Time Series Regression with Exponential Errors for Accelerated Testing and Degradation Tracking 32
Nozer D. Singpurwalla

3.1 Introduction 32

3.2 Preliminaries: Statement of the Problem 33

3.2.1 Relevance to Accelerated Testing, Degradation and Risk 33

3.3 Estimation and Prediction by Least Squares 34

3.4 Estimation and Prediction by MLE 35

3.4.1 Properties of the Maximum Likelihood Estimator 35

3.5 The Bayesian Approach: The Predictive Distribution 37

3.5.1 The Predictive Distribution of YT+1 when λ > A 38

3.5.2 The Predictive Distribution of YT+1 when λ ≤ A 39

3.5.3 Alternative Prior for β 40

Acknowledgements 42

References 42

4 Inverse Lz-Transform for a Discrete-State Continuous-Time Markov Process and Its Application to Multi-State System Reliability Analysis 43
Anatoly Lisnianski and Yi Ding

4.1 Introduction 43

4.2 Inverse Lz-Transform: Definitions and Computational Procedure 44

4.2.1 Definitions 44

4.2.2 Computational Procedure 47

4.3 Application of Inverse Lz-Transform to MSS Reliability Analysis 50

4.4 Numerical Example 52

4.5 Conclusion 57

References 58

5 OntheLz-Transform Application for Availability Assessment of an Aging Multi-State Water Cooling System for Medical Equipment 59
Ilia Frenkel, Anatoly Lisnianski and Lev Khvatskin

5.1 Introduction 59

5.2 Brief Description of the Lz-Transform Method 61

5.3 Multi-state Model of the Water Cooling System for the MRI Equipment 62

5.3.1 System Description 62

5.3.2 The Chiller Sub-System 64

5.3.3 The Heat Exchanger Sub-System 66

5.3.4 The Pump Sub-System 67

5.3.5 The Electric Board Sub-System 69

5.3.6 Model of Stochastic Demand 71

5.3.7 Multi-State Model for the MRI Cooling System 73

5.4 Availability Calculation 75

5.5 Conclusion 76

Acknowledgments 76

References 77

6 Combined Clustering and Lz-Transform Technique to Reduce the Computational Complexity of a Multi-State System Reliability Evaluation 78
Yi Ding

6.1 Introduction 78

6.2 The Lz-Transform for Dynamic Reliability Evaluation for MSS 79

6.3 Clustering Composition Operator in the Lz-Transform 81

6.4 Computational Procedures 83

6.5 Numerical Example 83

6.6 Conclusion 85

References 85

7 Sliding Window Systems with Gaps 87
Gregory Levitin

7.1 Introduction 87

7.2 The Models 89

7.2.1 The k/eSWS Model 89

7.2.2 The mCSWS Model 89

7.2.3 The mGSWS Model 90

7.2.4 Interrelations among Different Models 90

7.3 Reliability Evaluation Technique 91

7.3.1 Determining u-functions for Individual Elements and their Groups 91

7.3.2 Determining u-functions for all the Groups of r Consecutive Elements 92

7.3.3 Detecting the System Failure 93

7.3.4 Updating the Counter 94

7.3.5 Recursive Determination of System Failure Probability 95

7.3.6 Computational Complexity Reduction 95

7.3.7 Algorithm for System Reliability Evaluation 95

7.4 Conclusion 96

References 96

8 Development of Reliability Measures Motivated by Fuzzy Sets for Systems with Multi- or Infinite-States 98
Zhaojun (Steven) Li and Kailash C. Kapur

8.1 Introduction 98

8.2 Models for Components and Systems Using Fuzzy Sets 100

8.2.1 Binary Reliability and Multi-State Reliability Model 100

8.2.2 Definition of Fuzzy Reliability 101

8.2.3 Fuzzy Unreliability: A Different Perspective 102

8.2.4 Evolution from Binary State to Multi-State and to Fuzzy State Reliability Modeling 102

8.3 Fuzzy Reliability for Systems with Continuous or Infinite States 103

8.4 Dynamic Fuzzy Reliability 104

8.4.1 Time to Fuzzy Failure Modeled by Fuzzy Random Variable 105

8.4.2 Stochastic Performance Degradation Model 106

8.4.3 Membership Function Evaluation for the Expectation of Time to Fuzzy Failure 107

8.4.4 Performance Measures for Dynamic Fuzzy Reliability 108

8.5 System Fuzzy Reliability 110

8.6 Examples and Applications 111

8.6.1 Reliability Performance Evaluation Based on Time to Fuzzy Failure 111

8.6.2 Example for System Fuzzy Reliability Modeling 113

8.6.3 Numerical Results 115

8.7 Conclusion 117

References 118

9 Imperatives for Performability Design in the Twenty-First Century 119
Krishna B. Misra

9.1 Introduction 119

9.2 Strategies for Sustainable Development 120

9.2.1 The Internalization of Hidden Costs 120

9.2.2 Mitigation Policies 121

9.2.3 Dematerialization 121

9.2.4 Minimization of Energy Requirement 124

9.3 Reappraisal of the Performance of Products and Systems 124

9.4 Dependability and Environmental Risk are Interdependent 126

9.5 Performability: An Appropriate Measure of Performance 126

9.5.1 Performability Engineering 127

9.6 Towards Dependable and Sustainable Designs 129

9.7 Conclusion 130

References 130

Part II NETWORKS AND LARGE-SCALE SYSTEMS

10 Network Reliability Calculations Based on Structural Invariants 135
Ilya B. Gertsbakh and Yoseph Shpungin

10.1 First Invariant: D-Spectrum, Signature 135

10.2 Second Invariant: Importance Spectrum. Birnbaum Importance Measure (BIM) 139

10.3 Example: Reliability of a Road Network 141

10.4 Third Invariant: Border States 142

10.5 Monte Carlo to Approximate the Invariants 144

10.6 Conclusion 146

References 146

11 Performance and Availability Evaluation of IMS-Based Core Networks 148
Kishor S. Trivedi, Fabio Postiglione and Xiaoyan Yin

11.1 Introduction 148

11.2 IMS-Based Core Network Description 149

11.3 Analytic Models for Independent Software Recovery 151

11.3.1 Model 1: Hierarchical Model with Top-Level RBD and Lower-Level MFT 152

11.3.2 Model 2: Hierarchical Model with Top-Level RBD and Lower-Level FT 153

11.3.3 Model 3: Hierarchical Model with Top-Level RBD and Lower-Level SRN 154

11.4 Analytic Models for Recovery with Dependencies 155

11.4.1 Model 4: Hierarchical Model with Top-Level RBD, Middle-Level MFT and Lower-Level CTMC 155

11.4.2 Model 5: Alternative Approach for Model 4 based on UGF 156

11.4.3 Model 6: Hierarchical Model with Top-Level RBD and Lower-Level SRN 158

11.5 Redundancy Optimization 158

11.6 Numerical Results 159

11.6.1 Model Comparison 159

11.6.2 Influences of Performance Demand and Redundancy Configuration 162

11.7 Conclusion 165

References 165

12 Reliability and Probability of First Occurred Failure for Discrete-Time Semi-Markov Systems 167
Stylianos Georgiadis, Nikolaos Limnios and Irene Votsi

12.1 Introduction 167

12.2 Discrete-Time Semi-Markov Model 168

12.3 Reliability and Probability of First Occurred Failure 170

12.3.1 Rate of Occurrence of Failures 171

12.3.2 Steady-State Availability 171

12.3.3 Probability of First Occurred Failure 172

12.4 Nonparametric Estimation of Reliability Measures 172

12.4.1 Estimation of ROCOF 173

12.4.2 Estimation of the Steady-State Availability 174

12.4.3 Estimation of the Probability of First Occurred Failure 175

12.5 Numerical Application 176

12.6 Conclusion 178

References 179

13 Single-Source Epidemic Process in a System of Two Interconnected Networks 180
Ilya B. Gertsbakh and Yoseph Shpungin

13.1 Introduction 180

13.2 Failure Process and the Distribution of the Number of Failed Nodes 181

13.3 Network Failure Probabilities 184

13.4 Example 185

13.5 Conclusion 187

13.A Appendix D: Spectrum (Signature) 188

References 189

Part III MAINTENANCE MODELS

14 Comparisons of Periodic and Random Replacement Policies 193
Xufeng Zhao and Toshio Nakagawa

14.1 Introduction 193

14.2 Four Policies 195

14.2.1 Standard Replacement 195

14.2.2 Replacement First 195

14.2.3 Replacement Last 196

14.2.4 Replacement Over Time 196

14.3 Comparisons of Optimal Policies 197

14.3.1 Comparisons of T ∗ S and T ∗ F, T ∗ L, and T ∗ O 197

14.3.2 Comparisons of T ∗ O and T ∗ F, T ∗ L 198

14.3.3 Comparisons of T ∗ F and T ∗ L 199

14.4 Numerical Examples 1 199

14.5 Comparisons of Policies with Different Replacement Costs 201

14.5.1 Comparisons of T ∗ S, and T ∗ F, T ∗ L 201

14.5.2 Comparisons of T ∗ S and T ∗ O 201

14.6 Numerical Examples 2 202

14.7 Conclusion 203

Acknowledgements 204

References 204

15 Random Evolution of Degradation and Occurrences of Words in Random Sequences of Letters 205
Emilio De Santis and Fabio Spizzichino

15.1 Introduction 205

15.2 Waiting Times to Words’ Occurrences 206

15.2.1 The Markov Chain Approach 207

15.2.2 Leading Numbers and Occurrences Times 208

15.3 Some Reliability-Maintenance Models 209

15.3.1 Model 1 (Simple Machine Replacement) 209

15.3.2 Model 2 (Random Reduction of Age) 210

15.3.3 Model 3 (Random Number of Effective Repairs in a Parallel System) 211

15.3.4 Degradation and Words 212

15.4 Waiting Times to Occurrences of Words and Stochastic Comparisons for Degradation 213

15.5 Conclusions 216

Acknowledgements 217

References 217

16 Occupancy Times for Markov and Semi-Markov Models in Systems Reliability 218
Alan G. Hawkes, Lirong Cui and Shijia Du

16.1 Introduction 218

16.2 Markov Models for Systems Reliability 220

16.3 Semi-Markov Models 222

16.3.1 Joint Distributions of Operational and Failed Times 223

16.3.2 Distribution of Cumulative Times 224

16.4 Time Interval Omission 225

16.5 Numerical Examples 226

16.6 Conclusion 229

Acknowledgements 229

References 229

17 A Practice of Imperfect Maintenance Model Selection for Diesel Engines 231
Yu Liu, Hong-Zhong Huang, Shun-Peng Zhu and Yan-Feng Li

17.1 Introduction 231

17.2 Review of Imperfect Maintenance Model Selection Method 233

17.2.1 Estimation of the Parameters 234

17.2.2 The Proposed GOF Test 234

17.2.3 Bayesian Model Selection 235

17.3 Application to Preventive Maintenance Scheduling of Diesel Engines 236

17.3.1 Initial Failure Intensity Estimation 237

17.3.2 Imperfect Maintenance Model Selection 237

17.3.3 Implementation in Preventive Maintenance Decision-Making 240

17.4 Conclusion 244

Acknowledgment 245

References 245

18 Reliability of Warm Standby Systems with Imperfect Fault Coverage 246
Rui Peng, Ola Tannous, Liudong Xing and Min Xie

18.1 Introduction 246

18.2 Literature Review 247

18.3 The BDD-Based Approach 250

18.3.1 The BDD Construction 251

18.3.2 System Unreliability Evaluation 251

18.3.3 Illustrative Examples 252

18.4 Conclusion 253

Acknowledgments 254

References 254

Part IV STATISTICAL INFERENCE IN RELIABILITY

19 On the Validity of the Weibull-Gnedenko Model 259
Vilijandas Bagdonavi¡cius, Mikhail Nikulin and Ruta Levuliene

19.1 Introduction 259

19.2 Integrated Likelihood Ratio Test 261

19.3 Tests based on the Difference of Non-Parametric and Parametric Estimators of the Cumulative Distribution Function 264

19.4 Tests based on Spacings 266

19.5 Chi-Squared Tests 267

19.6 Correlation Test 269

19.7 Power Comparison 269

19.8 Conclusion 272

References 272

20 Statistical Inference for Heavy-Tailed Distributions in Reliability Systems 273
Ilia Vonta and Alex Karagrigoriou

20.1 Introduction 273

20.2 Heavy-Tailed Distributions 274

20.3 Examples of Heavy-Tailed Distributions 277

20.4 Divergence Measures 280

20.5 Hypothesis Testing 284

20.6 Simulations 286

20.7 Conclusion 287

References 287

21 Robust Inference based on Divergences in Reliability Systems 290
Abhik Ghosh, Avijit Maji and Ayanendranath Basu

21.1 Introduction 290

21.2 The Power Divergence (PD) Family 291

21.2.1 Minimum Disparity Estimation 293

21.2.2 The Robustness of the Minimum Disparity Estimators (MDEs) 294

21.2.3 Asymptotic Properties 295

21.3 Density Power Divergence (DPD) and Parametric Inference 296

21.3.1 Connections between the PD and the DPD 298

21.3.2 Influence Function of the Minimum DPD estimator 299

21.3.3 Asymptotic Properties of the Minimum DPD estimator 300

21.4 A Generalized Form: The S-Divergence 301

21.4.1 The Divergence and the Estimating Equation 301

21.4.2 Influence Function of the Minimum S-Divergence estimator 303

21.4.3 Minimum S-Divergence Estimators: Asymptotic Properties 303

21.5 Applications 304

21.5.1 Reliability: The Generalized Pareto Distribution 304

21.5.2 Survival Analysis 305

21.5.3 Model Selection: Divergence Information Criterion 306

21.6 Conclusion 306

References 306

22 COM-Poisson Cure Rate Models and Associated Likelihood-based Inference with Exponential and Weibull Lifetimes 308
N. Balakrishnan and Suvra Pal

22.1 Introduction 308

22.2 Role of Cure Rate Models in Reliability 310

22.3 The COM-Poisson Cure Rate Model 310

22.4 Data and the Likelihood 311

22.5 EM Algorithm 312

22.6 Standard Errors and Asymptotic Confidence Intervals 314

22.7 Exponential Lifetime Distribution 314

22.7.1 Simulation Study: Model Fitting 315

22.7.2 Simulation Study: Model Discrimination 319

22.8 Weibull Lifetime Distribution 322

22.8.1 Simulation Study: Model Fitting 322

22.8.2 Simulation Study: Model Discrimination 328

22.9 Analysis of Cutaneous Melanoma Data 334

22.9.1 Exponential Lifetimes with Log-Linear Link Function 334

22.9.2 Weibull Lifetimes with Logistic Link Function 335

22.10 Conclusion 337

22.A1 Appendix A1: E-Step and M-Step Formulas for Exponential Lifetimes 337

22.A2 Appendix A2: E-Step and M-Step Formulas for Weibull Lifetimes 341

22.B1 Appendix B1: Observed Information Matrix for Exponential Lifetimes 344

22.B2 Appendix B2: Observed Information Matrix for Weibull Lifetimes 346

References 347

23 Exponential Expansions for Perturbed Discrete Time Renewal Equations 349
Dmitrii Silvestrov and Mikael Petersson

23.1 Introduction 349

23.2 Asymptotic Results 350

23.3 Proofs 353

23.4 Discrete Time Regenerative Processes 358

23.5 Queuing and Risk Applications 359

References 361

24 On Generalized Extreme Shock Models under Renewal Shock Processes 363
Ji Hwan Cha and Maxim Finkelstein

24.1 Introduction 363

24.2 Generalized Extreme Shock Models 364

24.2.1 ‘Classical’ Extreme Shock Model for Renewal Process of Shocks 364

24.2.2 History-Dependent Extreme Shock Model 365

24.3 Specific Models 367

24.3.1 Stress-Strength Model 367

24.3.2 Model A in Cha and Finkelstein (2011) 369

24.3.3 State-Dependent Shock Model 370

24.4 Conclusion 373

Acknowledgements 373

References 373

Part V SYSTEMABILITY, PHYSICS-OF-FAILURE AND RELIABILITY DEMONSTRATION

25 Systemability Theory and its Applications 377
Hoang Pham

25.1 Introduction 377

25.2 Systemability Measures 378

25.3 Systemability Analysis of k-out-of-n Systems 379

25.3.1 Variance of Systemability Calculations 380

25.4 Systemability Function Approximation 380

25.5 Systemability with Loglog Distribution 383

25.5.1 Loglog Distribution 383

25.6 Sensitivity Analysis 384

25.7 Applications: Red Light Camera Systems 385

25.8 Conclusion 387

References 387

26 Physics-of-Failure based Reliability Engineering 389
Pedro O. Quintero and Michael Pecht

26.1 Introduction 389

26.2 Physics-of-Failure-based Reliability Assessment 393

26.2.1 Information Requirements 393

26.2.2 Failure Modes, Mechanisms, and Effects Analysis (FMMEA) 396

26.2.3 Stress Analysis 397

26.2.4 Reliability Assessment 397

26.3 Uses of Physics-of-Failure 398

26.3.1 Design-for-Reliability (DfR) 398

26.3.2 Stress Testing Conditions 398

26.3.3 Qualification 399

26.3.4 Screening Conditions 399

26.3.5 Prognostics and Health Management (PHM) 399

26.4 Conclusion 400

References 400

27 Accelerated Testing: Effect of Variance in Field Environmental Conditions on the Demonstrated Reliability 403
Andre Kleyner

27.1 Introduction 403

27.2 Accelerated Testing and Field Stress Variation 404

27.3 Case Study: Reliability Demonstration Using Temperature Cycling Test 405

27.4 Conclusion 408

References 408

Index 409

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