Mathematical Theory of Reliability

Cover......Page 1
S Title......Page 2
Editors & Published Texts......Page 3
Title: Mathematical Theory of Reliability......Page 6
TA169.B37 1996 620\'.00452\'015118—dc20......Page 7
Contents......Page 8
Preface to the Classics Edition......Page 12
Preface......Page 14
Acknowledgments......Page 18
1. HISTORICAL BACKGROUND OF THE MATHEMATICAL THEORY OF RELIABILITY......Page 20
2. DEFINITIONS OF RELIABILITY......Page 24
1. INTRODUCTION......Page 28
2. TYPICAL FAILURE LAWS......Page 31
3. THE EXPONENTIAL AS THE FAiLURE LAW OF COMPLEX EQUIPMENT......Page 37
Definitions......Page 41
Comparison with the exponential distribution......Page 45
5. PRESERVATION OF MONOTONE FAILURE RATE......Page 54
6. ADDITIONAL INEQUALITIES......Page 58
7. GENERAL FAILURE RATES......Page 60
1. INTRODUCTION......Page 65
2. RENEWAL THEORY......Page 67
Mean time to failure with replacement......Page 80
Replacement of redundant structures......Page 82
Exponential bounds on operating characteristics......Page 84
4. COMPARISON OF AGE AND BLOCK REPLACEMENT POLICIES......Page 86
5. RANDOM REPLACEMENT......Page 91
6. REPAIR OF A SiNGLE UNIT......Page 93
The distribution of the number of failures......Page 94
The distribution of unit down time......Page 98
The expected unit \"on\" time......Page 99
Interval reliability......Page 101
1. INTRODUCTION......Page 103
2.1 Age replacement......Page 104
2.2 Block replacement......Page 114
2.3 Periodic replacement with minimal repair at failure......Page 115
2.4 Sequential replacement over a finite time span......Page 117
3. INSPECTION POLiCIES......Page 126
3.1 Minimizing expected cost until detection of failure......Page 127
3.2 Minimizing expected cost assuming renewal at detection of failure......Page 133
3.3 Opportunistic replacement of a single part in the presence of several monitored parts......Page 136
1. INTRODUCTION......Page 138
2. MARKOV CHAINS AND SEMI-MARKOY PROCESSES......Page 140
Discrete time Markov chains......Page 141
Classification of states of a Markov chain......Page 142
Absorbing Markov chains......Page 144
Ergodic Markov chains......Page 146
Semi-Markov processes......Page 150
Moments of first-passage distributions for semi-Markov processes......Page 151
Limit theorems......Page 154
3. REPAIRMAN PROBLEMS......Page 158
Birth-and-death processes......Page 159
Equal numbers of machines and repairmen—no spares......Page 164
More general models......Page 165
General failure and exponential-repair distributions......Page 168
4. MARGINAL CHECKING......Page 170
Formal model......Page 171
IFR Markor chains......Page 173
Optimal marginal checking intervals......Page 174
5. OPTIMAL MAINTENANCE POLICIES UNDER MARKOV1AN DETERIORATION......Page 175
Optimal maintenance policies with probability bounds on failure......Page 177
1. INTRODUCTION......Page 181
Introduction......Page 182
General solution of redundancy allocation problem......Page 184
Procedure 1 (single cost factor)......Page 185
Procedure I (multiple cost factors)......Page 186
3. APPLICATION TO PARALLEL REDUNDANCY MODEL......Page 189
Explicit formulas for procedure 2......Page 192
4. APPLICATION TO STANDBY REDUNDANCY MODEL......Page 194
Exponential failure distributions......Page 195
Procedure 3 (Kettelle algorithm for single cost factor)......Page 199
6. OPTIMAL REDUNDANCY ASSUMING TWO TYPES OF FAILURE......Page 204
Maximizing system reliability......Page 207
Maximizing expected system life......Page 211
Exponential failure distribution......Page 213
Uniform failure distribution......Page 214
1. INTRODUCTION......Page 217
2. ACHIEViNG RELiABLE RELAY CiRCUITS......Page 218
Structure function......Page 223
Reliability......Page 226
Path and cut bounds on structure reliability......Page 227
4. S-SHAPED RELIABILITY FUNCTIONS FOR MONOTON1C STRUCTURES......Page 230
S. k-OUT-OF-n STRUCTURES......Page 237
Introduction......Page 240
Basic theorem for structures of identical components......Page 241
Applications......Page 242
Bounds on structure failure rate for structures of nonidentical components......Page 244
APPENDIX 1  Total Positivity......Page 248
VARIATION DIMINISHING PROPERTY......Page 250
APPENDIX 2  Test for Increasing Failure Rate......Page 253
UNBIASEDNESS OF TEST......Page 254
ASYMPTOT1C RELATIVE EFF1CIENCY......Page 255
APPENDiX 3  Tables Giving Bounds on Distributions with Monotone Failure Rate......Page 257
References......Page 262
Author Index......Page 270