Table of Contents
Foreword vii
1 Markov reliability and availability analysis vii
1.1 Introduction 1
1.2 Discrete-time, discrete-state Markov processes 2
1.2.1 The conceptual model 2
1.2.2 State probabilities 5
1.2.3 Multi-step transition probabilities 7
1.2.4 Solution of the fundamental equation 9
1.2.5 Steady state probabilities for ergodic systems 19
1.2.6 First passage probabilities 20
1.3 Continuous time, discrete-state Markov processes 24
1.3.1 The conceptual model 24
1.3.2 Solution of the fundamental equation 30
1.3.3 Failure intensity 34
1.3.4 Average time of occupancy of a given state 36
1.3.5 System availability 37
1.3.6 System reliability 38
2 Monte Carlo simulations for reliability and availability analysis
2.1 Introduction 59
2.2 Monte Carlo simulation for system engineering 60
2.3 Monte Carlo simulation for system unreliability and unavailability estimation
2.3.1 Indirect and direct Monte Carlo simulation 66
3 Markov Chain Monte Carlo for applications to reliability and availability analysis
3.1 Introduction 71
3.2 The Metropolis-Hastings algorithm 73
3.2.1 Application to the estimation of the failure rate of a deteriorating component 74
3.3 The Gibbs sampler 78
3.3.1 Application to the estimation of a rare failures process 80
3.4 The reversible-jump MCMC algorithm 83
3.4.1 Application to the estimation of the failure rate of a component subject to degradation or improvement 88
3.4.2 Application to the estimation of the parameters of a deterioration process due to fatigue 95
3.5 Bayesian updating 103
3.6 Practical issues in implementing MCMC algorithms 108
3.6.1 Choice of the kinetics K(.|.) 108
3.6.2 Burn-in period109
3.6.3 Number of iterations 109
3.6.4 Initial conditions 110
3.6.5 Other algorithms 110
4 Basics of genetic algorithms with application to system reliability and availability optimization
4.1 Introduction 115
4.2 Genetic Algorithms at a glance 117
4.3 The standard Genetic Algorithm 121
4.4 Affine transforming the chromosome fitness 124
4.5 More sophisticated breeding procedures 131
4.6 Efficiency of breeding procedures 134
4.6.1 The figures of merit 134
4.6.2 The test functions 138
4.6.3 Results 144
4.7 Inducement of species and niches 151
4.7.1 Isolation by distance 151
4.7.2 Spatial mating 152
4.7.3 Sharing 153
4.8 Multi-objective optimization 155
4.9 Application of genetic algorithms to RAMS 158
4.10 Examples 163
4.10.1 Multi-objective optimization of system design: a simple application 163
4.10.2 Multi-objective optimization of the inspection policy of a nuclear safety system 169
4.11 Discussion 180
5 Dependent failures
5.1 Introduction 187
5.2 General classification 188
5.3 Identification of dependent failures and protection from their occurrence 191
5.4 Definition of dependent failures 192
5.5 Methods for dependent-failure analysis 194
5.5.1 Examples of explicit methods 194
5.5.2 An example of an implicit method for modeling dependent failures 205
5.6 A methodological framework for common cause failures analysis 208
5.6.1 System logic model development 208
5.6.2 Identification of common cause component groups 208
5.6.3 Common cause failure modeling and data analysis 212
6 Importance measures
6.1 Introduction 235
6.2 Birnbaum's measure 238
6.2.1 Relation with the system structure function 239
6.3 Criticality importance 243
6.4 Fussell-Vesely importance measure 245
6.5 Risk Achievement Worth and Risk Reduction Worth 249
6.5.1 Risk Achievement Worth 249
6.5.2 Risk Reduction Worth 249
6.6 Observations and limitations of importance measures 252
6.7 Generalized risk importance measure 257
6.8 Importance measures for multiple basic events 259
6.8.1 Risk Achievement Worth 259
6.8.2 Birnbaum importance measure 261
6.8.3 Fussell-Vesely importance 262
6.8.4 Risk Reduction Worth 263
6.9 Relationship of importance measures to system risk changes 264
6.10 The Differential Importance Measure (DIM) 265
6.11 Importance measures for multi-state systems 277
6.11.1 Introduction 277
6.11.2 The model of a multi-state system 278
6.11.3 Importance measures for multi-state systems 279
6.11.4 Importance measures based on limitations on the performance of multi-state components 281
6.11.5 Comparison of importance measures for multi-state systems 288
7 Basic concepts of uncertainty and sensitivity analysis
7.1 Introduction 295
7.2 Local and global uncertainty analysis 297
7.3 Approximated analytical methods: the method of moments 300
7.4 Discrete methods 302
7.4.1 Sensitivity on the nominal range 302
7.4.2 Event and probability tree 303
7.4.3 Discrete probability method 305
7.5 Monte Carlo method 306
7.6 Linear regression method 307
7.7 The variance decomposition method 310
7.8 Sobol indexes and Fourier Amplitude Sensitivity Test 323
7.9 Model structure uncertainty 325
7.9.1 The alternative models approach 325
7.9.2 Adjustment factor approach 326
