Reliability Evaluation of Engineering Systems: Concepts and Techniques Roy Billinton PhD, DSc, FEIC, FRSC, FIEEE, PE c. J. MacKenzie Professor of Electrical Engineering University of Saskatchewan and Ronald N Allan PhD, FSRS, MIEEE, MIEE, CEng Senior Lecturer in Electrical Power Systems University of Manchester Institute of Science and Technology Springer Science+Business Media, LLC
Longman Scientific & Technical Longman Group UK Limited, Longman House, Bumt Mill, Harlow, Essex CM20 2JE, England and Associated Companies throughout the world. Co-published in the United States by Plenum Press Springer Science+Business Media New York 1983 Originally published by Plenum Press in 1983 Softcover reprint ofthe hardcover 1 st edition 1983 AII rights reserved; no part of this publicat ion may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publishers. First published in Great Britain by Pitman Publishing Limited 1983 Reprinted 1985 (twice) Reprinted by Longman Scientific & Technical 1987 Library of Congress Cataloging in Publication Data Billinton, Roy. Reliability evaluation of engineering systems. lncludes bibliographical references and index. 1. Reliability (Engineering) I. Alian, Ronald N. (Ronald Norman) II. Title. TA169.B54 1983 620'.00452 82-18578 ISBN 978-1-4615-7730-0 ISBN 978-1-4615-7728-7 (ebook) DOI 10.1007/978-1-4615-7728-7
Contents Preface ix 1 Introduction 1 2 Basic probability theory 5 2.1 Probability concepts 5 2.2 Permutations and combinations 7 2.2.1 General concepts 7 2.2.2 Permutations 7 2.2.3 Combinations 9 2.2.4 Comparison of permutations and combinations 11 2.2.5 Application in probability evaluation 11 2.3 Practical engineering concepts 13 2.4 Venn diagrams 15 2.5 Rules for combining probabilities 16 2.5.1 Rule I-independent events 16 2.5.2 Rule 2-mutually exclusive events 16 2.5.3 Rule 3-complementary events 17 2.5.4 Rule 4-conditional events 17 2.5.5 Rule 5-simuItaneous occurrence of events 18 2.5.6 Rule 6--occurrence of at least one of two events 20 2.5.7 Rule 7 -application of conditional probability 22 2.6 Probability distributions 25 2.6.1 Random variables 25 2.6.2 Density and distribution functions 26 2.6.3 Mathematical expectation 31 2.6.4 Variance and standard deviation 33 2.7 Conclusions 34 Problems 35 3 Application of the binomial distribution 36 3.1 Binomial distribution concepts 36 3.2 Properties of the binomial distribution 37 3.2.1 General characteristics 37 3.2.2 Binomial coefficients 41 iii
iv Contents 3.2.3 Expected value and standard deviation 42 3.3 Engineering applications 44 3.3.1 Restricting the assessment 44 3.3.2 Implication of economics 45 3.3.3 Effect of redundancy 46 3.3.4 Effect of partial output (derated) states 51 3.3.5 Effect of unavailability 55 3.3.6 Effect of one unit in reserve 55 3.3.7 Non-identical capacities 57 3.3.8 Non-identical unavailabilities 59 3.4 Conclusions 60 Problems 61 4 Network modelling and evaluation of simple systems 62 4.1 Network modelling concepts 62 4.2 Series systems 63 4.3 Parallel systems 66 4.4 Series-parallel systems 70 4.5 Partially redundant systems 73 4.6 Standby redundant systems 75 4.6.1 Redundancy concepts 75 4.6.2 Perfect switching 76 4.6.3 Imperfect switching 76 4.6.4 Standby redundancy calculations 78 4.7 Conclusions 79 Problems 79 5 Network modelling and evaluation of complex systems 81 5.1 Modelling and evaluation concepts 81 5.2 Conditional probability approach 82 5.3 Cut set method 84 5.3.1 Cut set concepts 84 5.3.2 Application of cut sets 85 5.3.3 Approximate evaluation 87 5.3.4 Deducing the minimal cut sets 89 5.4 Application and comparison of previous techniques 91 5.5 Tie set method 94 5.6 Connection matrix techniques 96 5.7 Event trees 100 5.7.1 General concepts 100 5.7.2 Continuously operated systems 101 5.7.3 Cut and tie set deduction 106 5.7.4 Standby and sequential logic systems 107 5.8 Fault trees 113
Contents v 5.9 Multi-failure modes 116 5.10 Conclusions 120 Problems 121 6 Probability distributions in reliability evaluation 124 6.1 Distribution concepts 124 6.2 Terminology of distributions 125 6.3 General reliability functions 128 6.4 Evaluation of the reliability functions 130 6.5 Shape of reliability functions 134 6.6 The Poisson distribution 136 6.6.1 General concepts 136 6.6.2 Derivation of the Poisson distribution 136 6.6.3 Relationship with the binomial distribution 140 6.7 The normal distribution 143 6.7.1 General concepts 143 6.7.2 Probability density function 143 6.7.3 Evaluation of probabilities 146 6.8 The exponential distribution 149 6.8.1 General concepts 149 6.8.2 Reliability functions 150 6.8.3 A posteriori failure probability 152 6.8.4 Mean value and standard deviation 154 6.9 The Weibull distribution 156 6.10 The gamma distribution 159 6.11 The Rayleigh distribution 161 6.12 The lognormal distribution 162 6.13 The rectangular (or uniform) distribution 164 6.14 Summary of reliability functions 165 6.15 Conclusions 167 Problems 167 7 System reliability evaluation using probability distributions 170 7.1 Introduction 170 7.2 Series systems 171 7.3 Parallel systems 173 7.4 Partially redundant systems 175 7.5 Mean time to failure 178 7.6 Standby systems 179 7.6.1 General concepts 179 7.6.2 Perfect switching 181 7.6.3 Imperfect switching 183 7.6.4 Effect of spare components 185 7.6.5 Non-identical components 188
vi Contents 7.6.6 Failures in the standby mode 190 7.7 Wearout and component reliability 195 7.8 Maintenance and component reliability 198 7.9 Conclusions 202 Problems 203 8 Disel'ete Markov chains 206 8.1 Introduction 206 8.2 General modelling concepts 207 8.3 Stochastic transitional probability matrix 211 8.4 Time dependent probability evaluation 212 8.5 Limiting state probability evaluation 214 8.6 Absorbing states 216 8.7 Application of discrete Markov techniques 218 8.8 Conclusions 222 Problems 223 9 Continuous Markov processes 225 9.1 Introduction 225 9.2 General modelling concepts 225 9.2.1 Transition rate concepts 225 9.2.2 Evaluating time dependent probabilities 227 9.2.3 Evaluating limiting state probabilities 230 9.3 State space diagrams 231 9.3.1 General concepts 231 9.3.2 Single repairable component 232 9.3.3 Two repairable components 233 9.3.4 Three component system 234 9.3.5 Larger number of components 236 9.3.6 Standby redundant systems 236 9.3.7 Mission-orientated systems 237 9.4 Stochastic transitional probability matrix 237 9.5 Evaluating limiting state probabilities 238 9.5.1 Single repairable component 238 9.5.2 Two identical repairable components 239 9.6 Evaluating time dependent state probabilities 241 9.6.1 Differential equations method 241 9.6.2 Matrix multiplication method 242 9.7 Reliability evaluation in repairable systems 243 9.8 Mean time to failure 244 9.8.1 Evaluation concepts 244 9.8.2 Stochastic transitional probability matrix method 245 9.9 Application of techniques to complex systems 247 9.10 Conclusions 250 Problems 251
Contents vii 10 Frequency and duration techniques 253 10.1 Introduction 253 10.2 Frequency and duration concepts 253 10.3 Application to multi-state problems 256 10.3.1 Two component repairable system 256 10.3.2 State probabilities 257 10.3.3 Frequency of encountering individual states 258 10.3.4 Mean duration of individual states 259 10.3.5 Cycle time between individual states 260 10.3.6 Frequency of encountering cumulated states 261 10.3.7 Recursive evaluation of cumulative frequency 264 10.3.8 Mean duration of cumulated states 266 10.4 Frequency balance approach 267 10.5 Two stage repair and installation process 269 10.5.1 General concepts 269 10.5.2 One component system-no spare available 270 10.5.3 One component system-one spare available 272 10.5.4 One component system-two spares available 274 10.5.5 Two component system-one spare available 275 10.5.6 Limiting number of spares 276 10.5.7 Application of the techniques 277 10.6 Conclusions 280 Problems 281 11 Approximate system reliability evaluation 282 11.1 Introduction 282 11.2 Series systems 282 11.3 Parallel systems 285 11.3.1 Two component system 285 11.3.2 Systems with more than two components 287 11.4 Network reduction techniques 288 11.5 Minimal cut set/failure modes approach 289 11.6 Inclusion of scheduled maintenance 291 11. 7 Common mode failures 294 11. 7.1 General concepts 294 11.7.2 Modelling and evaluation techniques 296 11.8 Conclusions 301 Problems 301 12 Systems with non-exponential distnbutions 302 12.1 Introduction 302 12.2 Method of stages 303 12.3 Stages in series 305 12.4 Stages in parallel 306 12.5 Series stages in series with two parallel stages 307
viii Contents 12.6 Time dependent and limiting state probabilities 308 12.7 Conclusions 312 Problem 313 13 Epilogue 314 Appendix I-Rules of Boolean algebra 31S Appendix 2-The normal distribution function 316 Appendix 3-Elementary matrix algebra 317 A3.1 Concepts of matrices 317 A3.2 Square matrix 317 A3.3 Column matrix (or vector) 317 A3.4 Row matrix (or vector) 318 A3.S Transposed matrix 318 A3.6 Diagonal matrix 318 A3.7 Identity (or unit) matrix 318 A3.8 Symmetric matrix 319 A3.9 Determinant of a matrix 319 A3.10 Co-factors 319 A3.11 Evaluation of determinants 319 A3.12 Addition of matrices 320 A3.13 Subtraction of matrices 321 A3.14 Multiplication of matrices 321 A3.1S Multiplication by a constant 322 A3.16 Inverse of a matrix 322 A3.17 Solution of simultaneous equations 323 A3.18 Cramer's rule for solving simultaneous equations Appendix 4-Differential equations and Laplace transforms 324 326 A4.1 Differential equations 326 A4.2 Laplace transforms 326 A4.3 Solving differential equations using Laplace transforms 328 Appendix S-Confidence levels and limits 331 AS.l Introduction 331 AS.2 Unavailability at selected confidence levels 331 AS.3 Failure rate at selected confidence levels 334 AS.4 Conclusions 338 Problems 338 References 339 Solutions 342 Index 34S
Preface This book has evolved from our deep interest and involvement in the development and application of reliability evaluation techniques. Its scope is not limited to anyone engineering discipline as the concepts and basic techniques for reliability evaluation have no disciplinary boundaries and are applicable in most, if not all, engineering applications. We firmly believe that reliability evaluation is an important and integral feature of the planning, design and operation of all engineering systems; from the smallest and most simple to the largest and most complex. Also, we believe that all engineers involved with such systems should be aware of, and appreciate, not only the benefits which can accrue from reliability assessment, but also how such assessments can be made. Our primary objective has been to compile a book which provides practising engineers and engineering graduates who have little or no background in probability theory or statistics, with the concepts and basic techniques for evaluating the reliability of engineering systems. It is hoped that the material presented will enable them to reach quickly a level of self-confidence which will permit them to assimilate, understand and appreciate the more detailed applications and additional material which is available in the journals and publications associated with their own discipline. We have attempted to structure the book in such a way that only one new concept or technique is introduced and applied at a time. We have also made frequent use of numerical examples to consolidate the concepts and techniques. We believe that this structure will permit the reader to become confident with the application and understanding of reliability evaluation and enable the book to be used either as a self-tutorial text or as the text for a formally taught reliability evaluation course at undergraduate and postgraduate level. It would not have been possible for us to have written this book without our involvement and close association with many individuals and organizations: the students who have been on our respective (post) graduate research programmes, our colleagues on IEEE, CEA and lee committees and at our respective universities, and the engineers in the various industries with which we have been involved. Several typists have helped in the preparation of the manuscripts and ix
x Preface we would like to express our appreciation to them as a group. Finally, but by no means least, we would like to thank our respective wives, Joyce and Diane, for their perseverance and constant encouragement and for their assistance in reading and checking the manuscript. August 1982 Roy Billinton Ron Allan