Mathematical Models for Systems Reliability

Evolved from the lectures of a recognized pioneer in developing the theory of reliability, Mathematical Models for Systems Reliabilityprovides 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.

Preliminaries	p. 1
The Poisson process and distribution	p. 1
Waiting time distributions for a Poisson process	p. 6
Statistical estimation theory	p. 8
Basic ingredients	p. 8
Methods of estimation	p. 9
Consistency	p. 11
Sufficiency	p. 12
Rao-Blackwell improved estimator	p. 13
Complete statistic	p. 14
Confidence intervals	p. 14
Order statistics	p. 16
Generating a Poisson process	p. 18
Nonhomogeneous Poisson process	p. 19
Three important discrete distributions	p. 22
Problems and comments	p. 24
Statistical life length distributions	p. 39
Stochastic life length models	p. 39
Constant risk parameters	p. 39
Time-dependent risk parameters	p. 41
Generalizations	p. 42
Models based on the hazard rate	p. 45
IFR and DFR	p. 48
General remarks on large systems	p. 50
Problems and comments	p. 53
Reliability of various arrangements of units	p. 63
Series and parallel arrangements	p. 63
Series systems	p. 63
Parallel systems	p. 64
The k out of n system	p. 66
Series-parallel and parallel-series systems	p. 67
Various arrangements of switches	p. 70
Series arrangement	p. 71
Parallel arrangement	p. 72
Series-parallel arrangement	p. 72
Parallel-series arrangement	p. 72
Simplifications	p. 73
Example	p. 74
Standby redundancy	p. 76
Problems and comments	p. 77
Reliability of a one-unit repairable system	p. 91
Exponential times to failure and repair	p. 91
Generalizations	p. 97
Problems and comments	p. 98
Reliability of a two-unit repairable system	p. 101
Steady-state analysis	p. 101
Time-dependent analysis via Laplace transform	p. 105
Laplace transform method	p. 105
A numerical example	p. 111
On Model 2(c)	p. 113
Problems and Comments	p. 114
Continuous-time Markov chains	p. 117
The general case	p. 117
Definition and notation	p. 117
The transition probabilities	p. 119
Computation of the matrix P(t)	p. 120
A numerical example (continued)	p. 122
Multiplicity of roots	p. 126
Steady-state analysis	p. 127
Reliability of three-unit repairable systems	p. 128
Steady-state analysis	p. 128
Steady-state results for the n-unit repairable system	p. 130
Example 1 - Case 3(e)	p. 131
Example 2	p. 131
Example 3	p. 131
Example 4	p. 132
Pure birth and death processes	p. 133
Example 1	p. 133
Example 2	p. 133
Example 3	p. 134
Example 4	p. 134
Some statistical considerations	p. 135
Estimating the rates	p. 136
Estimation in a parametric structure	p. 137
Problems and comments	p. 138
First passage time for systems reliability	p. 143
Two-unit repairable systems	p. 143
Case 2(a) of Section 5.1	p. 143
Case 2(b) of Section 5.1	p. 148
Repairable systems with three (or more) units	p. 150
Three units	p. 150
Mean first passage times	p. 152
Other initial states	p. 154
Examples	p. 157
Repair time follows a general distribution	p. 160
First passage time	p. 160
Examples	p. 164
Steady-state probabilities	p. 165
Problems and comments	p. 167
Embedded Markov chains and systems reliability	p. 173
Computations of steady-state probabilities	p. 173
Example 1: One-unit repairable system	p. 174
Example 2: Two-unit repairable system	p. 175
Example 3: n-unit repairable system	p. 177
Example 4: One out of n repairable systems	p. 183
Example 5: Periodic maintenance	p. 184
Example 6: Section 7.3 revisited	p. 189
Example 7: One-unit repairable system with prescribed on-off cycle	p. 192
Mean first passage times	p. 194
Example 1: A two-unit repairable system	p. 194
Example 2: General repair distribution	p. 195
Example 3: Three-unit repairable system	p. 195
Computations based on s[subscript jk]	p. 197
Problems and comments	p. 200
Integral equations in reliability theory	p. 207
Introduction	p. 207
Example 1: Renewal process	p. 208
Some basic facts	p. 208
Some asymptotic results	p. 210
More basic facts	p. 212
Example 2: One-unit repairable system	p. 213
Example 3: Preventive replacements or maintenance	p. 215
Example 4: Two-unit repairable system	p. 218
Example 5: One out of n repairable systems	p. 219
Example 6: Section 7.3 revisited	p. 220
Example 7: First passage time distribution	p. 223
Problems and comments	p. 224
References	p. 247
Index	p. 251
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