Mathematical Models for Systems Reliability

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  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2008-05-09
  • Publisher: Chapman & Hall/

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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.

Table of Contents

Preliminariesp. 1
The Poisson process and distributionp. 1
Waiting time distributions for a Poisson processp. 6
Statistical estimation theoryp. 8
Basic ingredientsp. 8
Methods of estimationp. 9
Consistencyp. 11
Sufficiencyp. 12
Rao-Blackwell improved estimatorp. 13
Complete statisticp. 14
Confidence intervalsp. 14
Order statisticsp. 16
Generating a Poisson processp. 18
Nonhomogeneous Poisson processp. 19
Three important discrete distributionsp. 22
Problems and commentsp. 24
Statistical life length distributionsp. 39
Stochastic life length modelsp. 39
Constant risk parametersp. 39
Time-dependent risk parametersp. 41
Generalizationsp. 42
Models based on the hazard ratep. 45
IFR and DFRp. 48
General remarks on large systemsp. 50
Problems and commentsp. 53
Reliability of various arrangements of unitsp. 63
Series and parallel arrangementsp. 63
Series systemsp. 63
Parallel systemsp. 64
The k out of n systemp. 66
Series-parallel and parallel-series systemsp. 67
Various arrangements of switchesp. 70
Series arrangementp. 71
Parallel arrangementp. 72
Series-parallel arrangementp. 72
Parallel-series arrangementp. 72
Simplificationsp. 73
Examplep. 74
Standby redundancyp. 76
Problems and commentsp. 77
Reliability of a one-unit repairable systemp. 91
Exponential times to failure and repairp. 91
Generalizationsp. 97
Problems and commentsp. 98
Reliability of a two-unit repairable systemp. 101
Steady-state analysisp. 101
Time-dependent analysis via Laplace transformp. 105
Laplace transform methodp. 105
A numerical examplep. 111
On Model 2(c)p. 113
Problems and Commentsp. 114
Continuous-time Markov chainsp. 117
The general casep. 117
Definition and notationp. 117
The transition probabilitiesp. 119
Computation of the matrix P(t)p. 120
A numerical example (continued)p. 122
Multiplicity of rootsp. 126
Steady-state analysisp. 127
Reliability of three-unit repairable systemsp. 128
Steady-state analysisp. 128
Steady-state results for the n-unit repairable systemp. 130
Example 1 - Case 3(e)p. 131
Example 2p. 131
Example 3p. 131
Example 4p. 132
Pure birth and death processesp. 133
Example 1p. 133
Example 2p. 133
Example 3p. 134
Example 4p. 134
Some statistical considerationsp. 135
Estimating the ratesp. 136
Estimation in a parametric structurep. 137
Problems and commentsp. 138
First passage time for systems reliabilityp. 143
Two-unit repairable systemsp. 143
Case 2(a) of Section 5.1p. 143
Case 2(b) of Section 5.1p. 148
Repairable systems with three (or more) unitsp. 150
Three unitsp. 150
Mean first passage timesp. 152
Other initial statesp. 154
Examplesp. 157
Repair time follows a general distributionp. 160
First passage timep. 160
Examplesp. 164
Steady-state probabilitiesp. 165
Problems and commentsp. 167
Embedded Markov chains and systems reliabilityp. 173
Computations of steady-state probabilitiesp. 173
Example 1: One-unit repairable systemp. 174
Example 2: Two-unit repairable systemp. 175
Example 3: n-unit repairable systemp. 177
Example 4: One out of n repairable systemsp. 183
Example 5: Periodic maintenancep. 184
Example 6: Section 7.3 revisitedp. 189
Example 7: One-unit repairable system with prescribed on-off cyclep. 192
Mean first passage timesp. 194
Example 1: A two-unit repairable systemp. 194
Example 2: General repair distributionp. 195
Example 3: Three-unit repairable systemp. 195
Computations based on s[subscript jk]p. 197
Problems and commentsp. 200
Integral equations in reliability theoryp. 207
Introductionp. 207
Example 1: Renewal processp. 208
Some basic factsp. 208
Some asymptotic resultsp. 210
More basic factsp. 212
Example 2: One-unit repairable systemp. 213
Example 3: Preventive replacements or maintenancep. 215
Example 4: Two-unit repairable systemp. 218
Example 5: One out of n repairable systemsp. 219
Example 6: Section 7.3 revisitedp. 220
Example 7: First passage time distributionp. 223
Problems and commentsp. 224
Referencesp. 247
Indexp. 251
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