Mathematical models for systems reliability

Benjamin Epstein, Ishay Weissman1420080822, 9781420080827, 9781420080834

Evolved from the lectures of a recognized pioneer in developing the theory of reliability, Mathematical Models for Systems Reliability provides a rigorous treatment of the required probability background for understanding reliability theory.
This classroom-tested text begins by discussing the Poisson process and its associated probability laws. It then uses a number of stochastic models to provide a framework for life length distributions and presents formal rules for computing the reliability of nonrepairable systems that possess commonly occurring structures. The next two chapters explore the stochastic behavior over time of one- and two-unit repairable systems. After covering general continuous-time Markov chains, pure birth and death processes, and transitions and rates diagrams, the authors consider first passage-time problems in the context of systems reliability. The final chapters show how certain techniques can be applied to a variety of reliability problems.
Illustrating the models and methods with a host of examples, this book offers a sound introduction to mathematical probabilistic models and lucidly explores how they are used in systems reliability problems.

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Table of contents :
c0822fm.pdf……Page 1
MATHEMATICAL MODELS FOR SYSTEMS RELIABILITY……Page 2
Preface……Page 5
Contents……Page 8
Table of Contents……Page 0
1.1 The Poisson process and distribution……Page 13
1.2 Waiting time distributions for a Poisson process……Page 18
1.3.1 Basic ingredients……Page 20
1.3.2 Methods of estimation……Page 21
1.3.3 Consistency……Page 23
1.3.4 Sufficiency……Page 24
1.3.5 Rao-Blackwell improved estimator……Page 25
1.3.7 Confidence intervals……Page 26
1.3.8 Order statistics……Page 28
1.4 Generating a Poisson process……Page 30
1.5 Nonhomogeneous Poisson process……Page 31
1.6 Three important discrete distributions……Page 34
Problems for Section 1.1……Page 36
Problems for Section 1.2……Page 38
Problems for Section 1.3……Page 39
Problems for Section 1.4……Page 41
Problems for Section 1.5……Page 44
Problems for Section 1.6……Page 45
2.1.1 Constant risk parameters……Page 50
2.1.2 Time-dependent risk parameters……Page 52
2.1.3 Generalizations……Page 53
2.2 Models based on the hazard rate……Page 56
2.2.1 IFR and DFR……Page 59
2.3 General remarks on large systems……Page 61
Problems for Section 2.1……Page 64
Problems for Section 2.2……Page 68
Problems for Section 2.3……Page 73
3.1.1 Series systems……Page 74
3.1.2 Parallel systems……Page 75
3.1.3 The k out of n system……Page 77
3.2 Series-parallel and parallel-series systems……Page 78
3.3 Various arrangements of switches……Page 81
3.3.1 Series arrangement……Page 82
3.3.4 Parallel-series arrangement……Page 83
3.3.5 Simplifications……Page 84
3.3.6 Example……Page 85
3.4 Standby redundancy……Page 87
Problems for Section 3.1……Page 88
Problems for Section 3.2……Page 95
Problems for Section 3.3……Page 96
Problems for Section 3.4……Page 97
4.1 Exponential times to failure and repair……Page 102
4.2 Generalizations……Page 108
Problems for Section 4.1……Page 109
Problems for Section 4.2……Page 110
5.1 Steady-state analysis……Page 111
5.2.1 Laplace transform method……Page 115
5.2.2 A numerical example……Page 121
5.3 On Model 2(c)……Page 123
Problems for Section 5.1……Page 124
Problems for Section 5.2……Page 125
6.1.1 Definition and notation……Page 126
6.1.2 The transition probabilities……Page 128
6.1.3 Computation of the matrix P(t)……Page 129
6.1.4 A numerical example (continued)……Page 131
6.1.5 Multiplicity of roots……Page 135
6.1.6 Steady-state analysis……Page 136
6.2.1 Steady-state analysis……Page 137
6.3 Steady-state results for the n-unit repairable system……Page 139
6.3.3 Example 3……Page 140
6.3.4 Example 4……Page 141
6.4.2 Example 2……Page 142
6.4.4 Example 4……Page 143
6.5 Some statistical considerations……Page 144
6.5.1 Estimating the rates……Page 145
6.5.2 Estimation in a parametric structure……Page 146
Problems for Section 6.1……Page 147
Problems for Section 6.2……Page 148
Problems for Section 6.3……Page 149
Problem for Section 6.5……Page 150
7.1.1 Case 2(a) of Section 5.1……Page 151
7.1.2 Case 2(b) of Section 5.1……Page 156
7.2.1 Three units……Page 158
7.2.2 Mean first passage times……Page 160
7.2.3 Other initial states……Page 162
7.2.4 Examples……Page 165
7.3.1 First passage time……Page 168
7.3.2 Examples……Page 172
7.3.3 Steady-state probabilities……Page 173
Problems for Section 7.1……Page 175
Problems for Section 7.2……Page 176
Problems for Section 7.3……Page 178
8.1 Computations of steady-state probabilities……Page 180
8.1.1 Example 1: One-unit repairable system……Page 181
8.1.2 Example 2: Two-unit repairable system……Page 182
8.1.3 Example 3: n-unit repairable system……Page 184
8.1.4 Example 4: One out of n repairable systems……Page 190
8.1.5 Example 5: Periodic maintenance……Page 191
8.1.6 Example 6: Section 7.3 revisited……Page 196
8.1.7 Example 7: One-unit repairable system with prescribed on-off cycle……Page 199
8.2.1 Example 1: A two-unit repairable system……Page 201
8.2.3 Example 3: Three-unit repairable system……Page 202
8.2.4 Computations based on sjk……Page 204
Problems for Section 8.1……Page 207
Problems for Section 8.2……Page 211
9.1 Introduction……Page 214
9.2.1 Some basic facts……Page 215
9.2.2 Some asymptotic results……Page 217
9.2.3 More basic facts……Page 219
9.3 Example 2: One-unit repairable system……Page 220
9.4 Example 3: Preventive replacements or maintenance……Page 222
9.5 Example 4: Two-unit repairable system……Page 225
9.6 Example 5: One out of n repairable systems……Page 226
9.7 Example 6: Section 7.3 revisited……Page 227
9.8 Example 7: First passage time distribution……Page 230
Problems for Subsection 9.2.2……Page 231
Problems for Subsection 9.2.3……Page 234
Problems for Section 9.3……Page 236
Problems for Section 9.4……Page 239
Problems for Section 9.5……Page 244
Problems for Section 9.6……Page 245
Problems for Section 9.7……Page 246
Problems for Section 9.8……Page 250
References……Page 253