Recent Advances in Multi-state Systems Reliability (eBook)

Anatoly Lisnianski is an engineering expert at the Reliability Department of the Israel Electric Corporation Ltd., an adjunct senior lecturer at Haifa University, and scientific supervisor of the Centre for Reliability and Risk Management at Shamoon College of Engineering, Israel. He received his MSc degree (1975) in Electrical Engineering from the University of Information Technology, Precision Mechanics and Optics, St. Petersburg, and his PhD degree (1984) in Reliability from the Federal Scientific & Production Centre “Aurora” in St. Petersburg, where he served as a senior researcher until 1989. Since 1991 he has been working at the Israel Electric Corporation, Haifa, Israel, where he has specialized in reliability and applied probability. He is a Senior Member of the IEEE and Member of the Israel Society of Quality and Israel Statistical Association. He is the author or co-author of more than 150 research papers in the field of reliability and applied probability and co-author of two books. Alex Karagrigoriou is a Professor of Probability and Statistics and Director of Graduate Studies in Statistics and Actuarial-Financial Mathematics at the Department of Mathematics of the University of the Aegean, Greece. He studied at the University of Maryland, USA (MA, 1988, PhD, 1992), worked at the United States Department of Agriculture (USDA) and the Institute of Statistical Sciences, Taiwan, and taught at the Universities of Maryland, Athens, Aegean, Cyprus and Hellenic Open University. His research activities cover areas such as statistical modeling, model selection criteria, biostatistics, information theory, time series analysis, stochastic modeling, economic demography, finance and reliability theory. He has published over 100 research articles and has extensive experience in the design and execution of research projects involving the statistical analysis of medical, biomedical, technological, socioeconomic and economic data. Ilia Frenkel is the Chair of the Center for Reliability and Risk Management and a Senior Lecturer at the Industrial Engineering and Management Department, Shamoon College of Engineering, Israel. He received his MSc in Applied Mathematics from Voronezh State University, Russia, and his PhD in Operational Research and Computer Science from the Institute of Economy, Ukrainian Academy of Science, former USSR. He has more than 40 years of academic and teaching experience at universities and institutions in Russia and Israel. He has specialized in applied statistics and reliability with applications to preventive maintenance. He is an Editor and a member of the editorial board for numerous scientific and professional journals. He has published one book and more than 50 scientific articles and book chapters in the fields of reliability, applied statistics, and production and operation management.

Preface 6
Part I Modern Mathematical Methods for Multi-state System Reliability Analysis 7
Part II Applications and Case Studies 8
Contents 11
Contributors 13
Modern Mathematical Methods for Multi-state System Reliability Analysis 16
Reliability of a Network with Heterogeneous Components 17
1 Introduction 17
2 The Principal Model: Two Types of Components 19
2.1 Network Description 19
2.2 One Type of Components 19
2.3 Two Types of Components 21
2.4 Counting (k,r)-failure Vectors 22
2.5 Counting the Number C(k,r) of (k,r)-failure Sets 24
2.6 Simulation Algorithm for Estimating F(k,r) 25
3 Reliability of a Transportation Network 26
3.1 Description of the Network. Reliable Nodes, Unreliable Edges 26
3.2 Unreliable Nodes 28
4 "03F13F4 g(k,r)"03F13F4 Matrix and ``Shock Process'' Trajectories 29
5 More Than Two Types of Components---Concluding Remarks 31
References 31
2 Reliability Analysis of Complex Multi-state System with Common Cause Failure Based on DS Evidence Theory and Bayesian Network 33
Abstract 33
1 Introduction 34
2 Multi-state Bayesian Network Under Evidence Theory 35
2.1 The Node Definition of Multi-state Bayesian Network Under Evidence Theory 35
2.2 The Multi-state Bayesian Network Reasoning Under Evidence Theory 36
3 Reliability Modeling of System with Multiple CCFGs 38
3.1 A Modified ? Factor Model for CCFGs 38
3.2 Model Limitation and Solution 41
3.3 The Bayesian Network Node with CCFGs 42
4 Reliability Analysis of Feeding Control System for CNC HDHLs with Multiple CCFGs 43
4.1 Fault Tree Modeling of Feeding Control System 43
4.2 The BN Modeling and CCFGs Fusion 44
4.3 Reliability Analysis of Feeding Control System by Using DSET Based BN 46
5 Conclusions 50
Acknowledgements 51
References 51
3 A D-MMAP to Model a Complex Multi-state System with Loss of Units 53
Abstract 53
1 Introduction 53
2 The System and the Model 55
2.1 The State Space 57
2.2 Analyzing Events 58
3 The Markovian Arrival Process with Marked Arrivals 61
4 The Transient Distribution 63
5 Measures 64
5.1 Availability 64
5.2 Reliability 64
5.3 Mean Sojourn Times 64
5.4 Mean Number of Events 66
6 A Numerical Example 66
7 Conclusions 69
Acknowledgements 69
Appendix 69
References 72
Modeling and Inference for Multi-state Systems 73
1 Introduction 73
2 A Special Case of Semi-Markov Multi-state Systems 74
3 Parametric Specification of the System 77
4 Maximum Likelihood Estimation 79
5 Concluding Remarks 82
References 83
Optimizing Availability and Performance of a Two-Unit Redundant Multi-state Deteriorating System 85
1 Introduction 85
2 Description of a Two-Unit System with Maintenance 87
2.1 Two-Unit System with Minimal and Major Maintenance 88
2.2 Two-Unit System with Minimal or Major Maintenance Only 90
3 Model Assumption and Sojourn Time Distributions 91
4 Semi-Markov Modelling 92
5 Asymptotic Dependability and Performance Measures 94
5.1 Asymptotic Availability 94
5.2 Expected Downtime Cost 94
5.3 Expected Cost Due to Maintenance and Unavailability 95
6 Optimization Problems 97
6.1 Maximization of the Asymptotic Availability 97
6.2 Minimization of the Overall Cost 97
6.3 Multi-objective Optimization Using Weighted Sum Approach 98
7 Numerical example 98
8 Conclusions and Future Work 109
References 118
Phase-Type Models and Their Extension to Competing Risks 120
1 Introduction 120
2 Phase-Type Distributions 121
2.1 Model Specification 122
3 Classical Competing Risks 123
3.1 Distributional Properties of Competing Risks 123
3.2 The Identifiability Problem of Competing Risks 124
4 Phase-Type Models for Competing Risks 124
4.1 Model Specification for Phase-Type Based Competing Risks 124
5 Statistical Inference in Coxian Phase-Type Models 126
5.1 Coxian Survival Models 126
5.2 Model 1: Coxian Competing Risks Model with K=2 Transient States and m=2 Absorbing States 126
5.3 Parametric Identifiability of Model 1 128
5.4 Identifiability of Coxian Phase-Type Models 129
5.5 Case Study: Pneumonia on Admission to Intensive Care Unit ( 130
6 Statistical Inference for General Phase-Type Distributions 131
References 132
7 A Study on Repairable Series Systems with Markov Repairable Units 134
Abstract 134
1 Introduction 135
2 Model Assumptions 137
3 General Repairable Series System 141
4 General Repairable Series System with Neglected Failures 143
5 Phased-Mission Repairable Series System 146
6 Phased-Mission Repairable Series System with Neglected Failures 154
7 Numerical Examples 158
8 Conclusions 169
Acknowledgements 169
References 169
8 Dynamic Performance of Series Parallel Multi-state Systems with Standby Subsystems or Repairable Binary Elements 171
Abstract 171
1 Introduction 173
2 System Model and Performance Metrics 174
2.1 Generic Model of Series Parallel MSSs 175
2.2 MSS Dynamic Performance Metrics 175
3 Obtaining Performance DSCTP for Individual Components 177
3.1 Performance DSCTP for Warm Standby Components 177
3.2 Performance DSCTP for Repairable Binary Elements 178
4 Examples of Component Performance Evaluation 180
4.1 Warm Standby Components 180
4.2 Repairable Binary Element 181
5 Obtaining Performance DSCTP for Entire MSS5 183
5.1 UGF (U-Function) Technique 183
5.2 Generalized RBD Method for Multi-state Series-Parallel System 184
6 Examples of System Performance Evaluation 185
6.1 Systems with Warm Standby Components 185
6.2 Systems with Repairable Binary Elements 186
7 Summary 188
References 189
Optimal Imperfect Maintenance in a Multi-state System 191
1 Introduction 191
2 Modeling the System and Maintenance Policy 193
3 Cost Optimization Problem 193
4 Discrete Lifetime Distributions 196
4.1 The Discrete Modified Weibull Distribution 196
4.2 The Discrete Reduced Modified Weibull Distribution 197
5 Example for Cost Optimal Maintenance 199
6 Conclusion 204
References 204
10 Reliability Evaluation of Non-repairable Multi-state Systems Considering Survival-Death Markov Processes 206
Abstract 206
1 Introduction 207
2 Multi-state Models and Markov Processes for Non-repairable Components 208
2.1 Model I for Non-repairable Components 210
2.2 Model II for Non-repairable Components 212
3 Dynamic Reliability Evaluation for Non-repairable Multi-state Systems 213
4 System Studies 215
4.1 Example 1 215
4.2 Example 2 217
4.3 Example 3 219
5 Conclusions 222
Acknowledgements 222
References 222
11 Reliability Assessment of Systems with Dependent Degradation Processes Based on Piecewise-Deterministic Markov Process 224
Abstract 224
1 Introduction 225
2 Dynamic Reliability Models for Systems with Degradation Dependence 226
2.1 Degradation Models 226
2.1.1 PBMs 227
2.1.2 MSMs 227
2.2 Degradation Model of the System Considering Dependence 228
3 System Reliability Estimation Method 230
4 Case Study 232
5 Conclusion 235
References 235
12 Trade-Off Between Redundancy, Protection, and Imperfect False Targets in Defending Parallel Systems 237
Abstract 237
1 Introduction 238
2 Literature Review 239
3 Defense of Parallel Systems with Redundancy, Protection, and Imperfect False Targets 241
4 Conclusions and Future Research 246
Acknowledgements 247
References 247
13 Optimal Testing Resources Allocation for Improving Reliability Assessment of Non-repairable Multi-state Systems 250
Abstract 250
1 Introduction 250
2 Review of Bayesian Reliability Assessment for MSS 252
2.1 Bayesian Parameter Inference for Multi-state Components 254
2.2 Bayesian Reliability Assessment for Multi-state Systems 257
3 Optimal Testing Resources Allocation Strategy 258
3.1 Evaluating Performance of Candidate Allocation Schemes 260
3.2 Kriging Model 262
3.3 Optimization Model and Algorithm 264
4 Illustrative Examples 265
4.1 Example 1 266
4.2 Example 2 269
5 Conclusion 271
Acknowledgements 272
References 272
14 Topological Analysis of Multi-state Systems Based on Direct Partial Logic Derivatives 274
Abstract 274
1 Introduction 274
2 Structure Function 275
2.1 Modular Decomposition 276
2.2 Series and Parallel Systems 277
3 Topological Analysis of Multi-state Systems 278
3.1 Structural Importance Measures 280
3.2 Logical Differential Calculus 281
3.2.1 Chain Rule 282
4 Hand Calculation Example 284
5 Conclusion 288
Acknowledgements 289
References 289
Applications and Case Studies 291
15 Short-Term Reliability Analysis of Power Plants with Several Combined Cycle Units 292
Abstract 292
1 Introduction 293
2 Lz-Transform Method and Its Application to Reliability Analysis of Power Plant Consisting of Number CCGT Generating Units 295
2.1 A Multi-state Markov Model for a Combined Cycle Generating Unit and Lz-Transform for Its Output Generating Capacity Process 295
2.2 Reliability Analysis for Power System Consisting of Number Combine Cycle Generating Units 299
3 Numerical Example 301
4 Summary 305
References 305
16 Reliability Analysis of a Modified IEEE 6BUS RBTS Multi-state System 307
Abstract 307
1 Introduction 307
2 System Analysis—Original System 313
3 Modified System—Application of the Model 314
4 Conclusions 323
References 323
17 Lz-Transform Approach for Fault Tolerance Assessment of Various Traction Drives Topologies of Hybrid-Electric Helicopter 326
Abstract 326
1 Introduction 326
2 Comparative Analysis of Two Traction Drive Topologies of Hybrid-Electric Helicopter 328
2.1 Common Description 328
2.2 Operational Scenarios for Various Traction Drive Topologies 329
2.2.1 Serial Topology 329
2.2.2 Combined Topology 330
3 Brief Description of the Lz-Transform Method 330
4 Multi-state Modeling of the Multi Power Source Traction Drive 331
4.1 Systems’ Description 331
4.2 Elements’ Description 333
4.2.1 Elements with 2 States 333
4.2.2 Element with 3 States 334
4.2.3 Element with 4 States 335
4.3 Lz-Transform for Serial Topology System 336
4.3.1 Sub-System 1 (SS1s) 336
4.3.2 Sub-System 2 (SS2s) 336
4.3.3 Serial Topology System (STS) 337
4.3.4 Serial Topology System with Energy Storage (STS-ESs) 337
4.4 Lz-Transform for Combined Topology System 338
4.4.1 Sub-System 1 (SS1c) 338
4.4.2 Sub-System 2 (SS2) 339
4.4.3 Sub-System 3 (SS3) 339
4.4.4 Combined Topology System (CTS) 340
4.4.5 Combined Topology System with Energy Storage (CTS-ESc) 340
5 Availability and Mean Power Performance Calculation 341
6 Conclusion 345
References 346
18 Patient Diagnostic State Evolution During Hospitalization: Developing a Model for Measuring Clinical Diagnostic Dynamics 348
Abstract 348
1 Introduction 348
2 Some Basic Definitions 350
3 The Distance Between Two Diagnoses as a Measure of Their Dissimilarity 350
4 Some Facts Concerning the Similarities and Differences Between Two Consecutives Diagnoses in Hospital 352
5 How to Measure Divergence Between Two Sets of Diagnoses 353
6 Example of Segregation Index Calculations 354
7 Applying Proposed Measures to Patients Which Passed Only One Ward 356
8 Brief Summary and Future Research 358
References 359
19 Automated Development of the Markovian Chains to Assess the Availability and Performance of Multi-state Multiprocessor System 360
Abstract 360
1 Introduction 361
1.1 Motivation and Approach 361
1.2 Related Works Analysis 361
2 Description of the Multi-state Multiprocessor System 363
3 Technique for Automated Development of the Markov Model of the Multi-state Multiprocessor System 364
3.1 Procedures for Behaviour Description of the Multi-state Multiprocessor System 365
3.2 Basic Events 365
3.3 Components of Vector States 365
3.4 Parameters of the Markov Model 366
3.5 Structural-Automated Model for Multi-state Multiprocessor System 366
4 Availability and Performance Analysis. Markov Model Development 372
4.1 Simulation Results 375
5 Conclusion 377
References 378