Mathematical models for systems reliability / Benjamin Epstein, Ishay Weissman.

Format:

Book

Author/Creator:

Epstein, Benjamin, 1918-

Contributor:

Weissman, Ishay, 1940-

Anne and Joseph Trachtman Memorial Book Fund.

Language:

English

Subjects (All):

Reliability (Engineering)--Mathematics.

Reliability (Engineering).

System failures (Engineering)--Mathematical models.

System failures (Engineering).

Physical Description:

253 pages : illustrations ; 25 cm

Place of Publication:

Boca Raton : CRC Press, [2008]

Summary:

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.

Contents:

1.1 The Poisson process and distribution 1

1.2 Waiting time distributions for a Poisson process 6

1.3 Statistical estimation theory 8

1.3.1 Basic ingredients 8

1.3.2 Methods of estimation 9

1.3.3 Consistency 11

1.3.4 Sufficiency 12

1.3.5 Rao-Blackwell improved estimator 13

1.3.6 Complete statistic 14

1.3.7 Confidence intervals 14

1.3.8 Order statistics 16

1.4 Generating a Poisson process 18

1.5 Nonhomogeneous Poisson process 19

1.6 Three important discrete distributions 22

1.7 Problems and comments 24

2 Statistical life length distributions 39

2.1 Stochastic life length models 39

2.1.1 Constant risk parameters 39

2.1.2 Time-dependent risk parameters 41

2.1.3 Generalizations 42

2.2 Models based on the hazard rate 45

2.2.1 IFR and DFR 48

2.3 General remarks on large systems 50

2.4 Problems and comments 53

3 Reliability of various arrangements of units 63

3.1 Series and parallel arrangements 63

3.1.1 Series systems 63

3.1.2 Parallel systems 64

3.1.3 The k out of n system 66

3.2 Series-parallel and parallel-series systems 67

3.3 Various arrangements of switches 70

3.3.1 Series arrangement 71

3.3.2 Parallel arrangement 72

3.3.3 Series-parallel arrangement 72

3.3.4 Parallel-series arrangement 72

3.3.5 Simplifications 73

3.4 Standby redundancy 76

3.5 Problems and comments 77

4 Reliability of a one-unit repairable system 91

4.1 Exponential times to failure and repair 91

4.2 Generalizations 97

4.3 Problems and comments 98

5 Reliability of a two-unit repairable system 101

5.1 Steady-state analysis 101

5.2 Time-dependent analysis via Laplace transform 105

5.2.1 Laplace transform method 105

5.2.2 A numerical example 111

5.3 On Model 2(c) 113

5.4 Problems and Comments 114

6 Continuous-time Markov chains 117

6.1 The general case 117

6.1.1 Definition and notation 117

6.1.2 The transition probabilities 119

6.1.3 Computation of the matrix P(t) 120

6.1.4 A numerical example (continued) 122

6.1.5 Multiplicity of roots 126

6.1.6 Steady-state analysis 127

6.2 Reliability of three-unit repairable systems 128

6.2.1 Steady-state analysis 128

6.3 Steady-state results for the n-unit repairable system 130

6.3.1 Example 1 - Case 3(e) 131

6.3.3 Example 3 131

6.3.4 Example 4 132

6.4 Pure birth and death processes 133

6.4.3 Example 3 134

6.4.4 Example 4 134

6.5 Some statistical considerations 135

6.5.1 Estimating the rates 136

6.5.2 Estimation in a parametric structure 137

6.6 Problems and comments 138

7 First passage time for systems reliability 143

7.1 Two-unit repairable systems 143

7.1.1 Case 2(a) of Section 5.1 143

7.1.2 Case 2(b) of Section 5.1 148

7.2 Repairable systems with three (or more) units 150

7.2.1 Three units 150

7.2.2 Mean first passage times 152

7.2.3 Other initial states 154

7.3 Repair time follows a general distribution 160

7.3.1 First passage time 160

7.3.3 Steady-state probabilities 165

7.4 Problems and comments 167

8 Embedded Markov chains and systems reliability 173

8.1 Computations of steady-state probabilities 173

8.1.1 Example 1: One-unit repairable system 174

8.1.2 Example 2: Two-unit repairable system 175

8.1.3 Example 3: n-unit repairable system 177

8.1.4 Example 4: One out of n repairable systems 183

8.1.5 Example 5: Periodic maintenance 184

8.1.6 Example 6: Section 7.3 revisited 189

8.1.7 Example 7: One-unit repairable system with prescribed on-off cycle 192

8.2 Mean first passage times 194

8.2.1 Example 1: A two-unit repairable system 194

8.2.2 Example 2: General repair distribution 195

8.2.3 Example 3: Three-unit repairable system 195

8.2.4 Computations based on s[subscript jk] 197

8.3 Problems and comments 200

9 Integral equations in reliability theory 207

9.2 Example 1: Renewal process 208

9.2.1 Some basic facts 208

9.2.2 Some asymptotic results 210

9.2.3 More basic facts 212

9.3 Example 2: One-unit repairable system 213

9.4 Example 3: Preventive replacements or maintenance 215

9.5 Example 4: Two-unit repairable system 218

9.6 Example 5: One out of n repairable systems 219

9.7 Example 6: Section 7.3 revisited 220

9.8 Example 7: First passage time distribution 223

9.9 Problems and comments 224.

Notes:

"A Chapman & Hall book."

Includes bibliographical references (pages 247-250) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

9781420080827

1420080822

OCLC:

213375865