Stochastic Models in Reliability and Maintenance

by Shunji Osaki

• ISBN13: 9783540248088
• ISBN10: 3540248080
• Format: Nonspecific Binding
• Copyright: 2012-11-02
• Publisher: Springer Nature

Table of Contents

Renewal Processes and Their Computational Aspects	p. 1
Introduction	p. 2
Basic Renewal Theory	p. 3
Continuous renewal theory	p. 3
Discrete renewal theory	p. 6
Some Useful Properties of the Renewal Function	p. 7
Specific examples	p. 7
Asymptotic properties	p. 8
Analytical Approximation Methods	p. 9
Phase renewal processes	p. 9
Gamma approximations	p. 10
Methods based on equilibrium distribution	p. 13
Bounds	p. 14
Numerical Methods	p. 16
Laplace inversion technique	p. 17
Cubic spline algorithm	p. 18
Discritization algorithm	p. 19
Approximation by rational functions	p. 20
Concluding Remarks	p. 23
Stochastic Orders in Reliability Theory	p. 31
Introduction	p. 31
Definitions and Basic Properties	p. 32
Stochastic orders generated from univariate functions	p. 33
Conditional stochastic orders	p. 37
Bivariate characterization of stochastic orders	p. 41
Applications in Reliability Theory	p. 42
Notions of aging	p. 42
Useful stochastic inequalities in reliability theory	p. 47
Stochastic comparisons of system reliabilities	p. 48
Redundancy improvement	p. 51
Stochastic comparisons of maintenance policies	p. 53
Replacements upon failures	p. 53
Age replacement	p. 54
Block replacement	p. 55
Minimal repair	p. 56
Minimal repair with block replacement	p. 56
Stochastic comparison of different maintenance policies	p. 57
TP2 Functions	p. 63
Classical Maintenance Models	p. 65
Introduction	p. 65
Block Replacement	p. 67
Age Replacement	p. 71
Order Replacement	p. 75
Inspection Strategies	p. 79
Conclusions	p. 81
A Review of Delay Time Analysis for Modelling Plant Main-tenance	p. 89
Introduction	p. 89
Maintenance Practice	p. 93
The Delay Time Concept	p. 94
Basic Delay Time Maintenance Model: Complex Plant	p. 96
Basic Maintenance Model: Component Tracking	p. 97
Relaxation of Assumptions	p. 98
Non-perfect Inspection	p. 98
Non-steady-state Condition	p. 99
Non-homogeneous Defect Arrival Rate ¿	p. 100
Condition-dependent Cost and Downtime for Repair	p. 102
Case Experience Using Subjective Data: Case Experience	p. 103
Revision of Subjectively Estimated Delay Time Distribution	p. 106
Correction for Sampling Bias	p. 106
Subjective Estimation of the Delay Time Distribution Directly	p. 107
Objective Estimation of Delay Time Parameters	p. 107
Case Experience Using Objective Data: HPP of Defect Arrival	p. 110
Discussion of Further Developments in Delay Time Modelling	p. 115
Conclusions	p. 116
Imperfect Preventive Maintenance Models	p. 125
Introduction	p. 125
Sequential Imperfect Preventive Maintenance	p. 126
Introduction	p. 126
Model A - age	p. 127
Model B - failure rate	p. 128
Numerical examples	p. 129
Shock Model with Imperfect Preventive Maintenance	p. 131
Introduction	p. 131
Model and expected cost	p. 132
Optimal policies	p. 135
Conclusions	p. 139
Generalized Renewal Processes and General Repair Models	p. 145
Background and Motivation	p. 145
Generalized Renewal Processes	p. 149
p-Renewal Processes in Discrete Time	p. 153
Monotonicity and Asymptotic Properties of the p-Renewal Density	p. 155
On the g-Renewal Function	p. 156
A General Repair Model	p. 159
Two-Unit Redundant Models	p. 165
Introduction	p. 165
Two-Unit Standby System	p. 167
Model and assumptions	p. 167
First-passage time distributions	p. 169
Expected numbers of visits to state	p. 170
Transition probabilities	p. 171
Preventive Maintenance of Two-Unit Systems	p. 173
Model and analysis	p. 173
Optimum preventive maintenance policies	p. 175
Replacement of a two-unit parallel system	p. 178
Other Two-Unit Systems	p. 179
Two-unit parallel system	p. 179
Two-unit priority standby system	p. 181
Two-unit standby system with imperfect switchover	p. 182
Other models	p. 184
Optimal Maintenance Problems for Markovian Deteriorat-ing Systems	p. 193
A Basic Optimal Replacement Problem for a Discrete Time Markovian Deteriorating System	p. 193
Some conditions on transition probabilities and cost structure	p. 194
Formulation by Markovian decision process (MDP)	p. 194
Optimality of control limit rule	p. 195
An Optimal Inspection and Replacement Problem	p. 195
Transition probability	p. 196
Formulation by semi-Markov decision process (SMDP)	p. 196
Structure of optimal inspection and replacement policy	p. 197
An Optimal Inspection and Replacement Policy with Incom-plete Information	p. 199
Some notations and conditions	p. 200
Formulation by partially observable Markov decision process (POMDP)	p. 200
Some properties of TP2 order	p. 202
Some properties of optimal function	p. 204
Structure of optimal inspection and replacement policy 206
An Optimal Inspection and Replacement Problem of a Con-tinuous Time Markovian Deteriorating System	p. 207
A continuous time Markovian deteriorating system	p. 207
Transition probability	p. 208
Formulation by semi-Markov decision process	p. 209
Structure of optimal policy	p. 209
An Optimal Maintenance Problem for a Queueing System	p. 211
Model description	p. 211
Formulation by semi-Markov decision process	p. 214
Properties of value function	p. 214
Structure of optimal policy	p. 215
Transient Analysis of Semi-Markov Reliability Models - A Tutorial Review with Emphasis on Discrete-Parameter Ap- proaches	p. 219
Introduction	p. 219
Modelling Framework	p. 220
Dependability Measures	p. 222
Methods of Analysis	p. 226
Continuous-parameter models	p. 226
Discrete-parameter models	p. 233
Equations for the Dependability Measures	p. 234
Numerical Solution Techniques	p. 239
Solving the integral equations	p. 239
Discrete-parameter approximations	p. 240
Recent Developments, Conclusions and Further Work	p. 242
Software Reliability Models	p. 253
Introduction	p. 253
Definitions and Software Reliability Model	p. 254
Software Reliability Growth Modeling	p. 256
Imperfect Debugging Modeling	p. 260
Imperfect debugging model with perfect correction rate	p. 262
Imperfect debugging model for introduced faults	p. 263
Software Availability Modeling	p. 265
Model description	p. 265
Software availability measures	p. 266
Application of Software Reliability Assessment	p. 268
Optimal software release problem	p. 269
Maintenance cost model	p. 269
Maintenance cost model with reliability re-quirement	p. 271
Statistical software testing-progress control	p. 272
Optimal testing-effort allocation problem	p. 274
Reliability Models in Data Communication Systems	p. 281
Introduction	p. 282
SW ARQ Model with Intermittent Faults	p. 283
Intermittent faults	p. 283
ARQ policy	p. 285
Optimal retransmission number	p. 287
Numerical examples and remarks	p. 288
SR ARQ Model with Retransmission Number	p. 288
Model and analysis	p. 289
Optimal policy	p. 291
Numerical examples and remarks	p. 293
Hybrid ARQ Models with Response Time	p. 294
Type-I hybrid ARQ	p. 295
Type-II hybrid ARQ	p. 296
Comparison of type-I and type-II hybrid ARQs	p. 299
Numerical examples and remarks	p. 300
Quick Monte Carlo Methods in Stochastic Systems and Re-liability	p. 307
Introduction	p. 307
The Problem with Direct Simulation	p. 308
Importance Sampling	p. 309
The Optimal Change of Measure	p. 310
Remarks	p. 310
Preliminary definitions	p. 311
The recursive approach	p. 312
Exact calculation of ¿(x)	p. 314
Cases of Application of the Recursive Approach	p. 314
System Model	p. 316
Regenerative Simulation	p. 318
Failure Biasing Methods	p. 319
Simple failure biasing (SFB)	p. 319
Balanced failure biasing (BFB)	p. 320
Bias2 failure biasing	p. 321
Failure distance biasing (FDB)	p. 322
Balanced 1 failure biasing (B1FB)	p. 322
Balanced 2 failure biasing (B2FB)	p. 323
Bounded relative error and failure biasing	p. 323
Unreliability Estimation	p. 323
One-component system	p. 323
General case	p. 324
Example	p. 326
Analytical-Statistical Methods	p. 326
Concluding Remarks	p. 329
Index	p. 335
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