Springer Series in Reliability Engineering Series Editor Professor Hoang Pham Department of Industrial and Systems Engineering Rutgers, The State University of New Jersey 96 Frelinghuysen Road Piscataway, NJ 08854-8018 USA Other titles in this series The Universal Generating Function in The Maintenance Management Framework Reliability Analysis and Optimization Adolfo Crespo Márquez Gregory Levitin Human Reliability and Error in Trans- Warranty Management and Product portation Systems Manufacture B.S. Dhillon D.N.P. Murthy and Wallace R. Blischke Complex System Maintenance Handbook Maintenance Theory of Reliability D.N.P. Murthy and Khairy A.H. Kobbacy Toshio Nakagawa Recent Advances in Reliability and Quality System Software Reliability in Design Hoang Pham Hoang Pham Reliability and Optimal Maintenance Product Reliability Hongzhou Wang and Hoang Pham D.N.P. Murthy, Marvin Rausand and Trond Østerås Applied Reliability and Quality B.S. Dhillon Mining Equipment Reliability, Maintain- ability, and Safety Shock and Damage Models in Reliability B.S. Dhillon Theory Toshio Nakagawa Advanced Reliability Models and Maintenance Policies Risk Management Toshio Nakagawa Terje Aven and Jan Erik Vinnem Justifying the Dependability of Computer- Satisfying Safety Goals by Probabilistic based Systems Risk Assessment Pierre-Jacques Courtois Hiromitsu Kumamoto Reliability and Risk Issues in Large Scale Offshore Risk Assessment (2nd Edition) Safety-critical Digital Control Systems Jan Erik Vinnem Poong Hyun Seong Maxim Finkelstein Failure Rate Modelling for Reliability and Risk 123 Maxim Finkelstein, PhD, DSc Department of Mathematical Statistics University of the Free State Bloemfontein South Africa and Max Planck Institute for Demographic Research Rostock Germany ISBN 978-1-84800-985-1 e-ISBN 978-1-84800-986-8 DOI 10.1007978-1-84800-986-8 Springer Series in Reliability Engineering ISSN 1614-7839 A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008939573 © 2008 Springer-Verlag London Limited Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: deblik, Berlin, Germany Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com To my wife Olga Preface In the early 1970s, after obtaining a degree in mathematical physics, I started working as a researcher in the Department of Reliability of the Saint Petersburg Elektropribor Institute. Founded in 1958, it was the first reliability department in the former Soviet Union. At first, for various reasons, I did not feel a strong incli- nation towards the topic. Everything changed when two books were placed on my desk: Barlow and Proshcan (1965) and Gnedenko et al. (1964). On the one hand, they showed how mathematical methods could be applied to various reliability engineering problems; on the other hand, these books described reliability theory as an interesting field in applied mathematics/probability and statistics. And this was the turning point for me. I found myself interested–and still am after more than 30 years of working in this field. This book is about reliability and reliability-related stochastics. It focuses on failure rate modelling in reliability analysis and other disciplines with similar set- tings. Various applications of risk analysis in engineering and biological systems are considered in the last three chapters. Although the emphasis is on the failure rate, one cannot describe this topic without considering other reliability measures. The mean remaining lifetime is the first in this list, and we pay considerable atten- tion to describing and discussing its properties. The presentation combines classical results and recent results of other authors with our research over the last 10 to15 years. The recent excellent encyclopaedic books by Lai and Xie (2006) and Marshall and Olkin (2007) give a broad picture of the modern mathematical reliability theory and also present an up-to-date source of references. Along with the classical text by Barlow and Proschan (1975), the excellent textbook by Rausand and Hoyland (2004) and a mathematically oriented reliability monograph by Aven and Jensen (1999), these books can be considered as complementary or further reading. I hope that our text will be useful for reliabil- ity researchers and practitioners and to graduate students in reliability or applied probability. I acknowledge the support of the University of the Free State, the National Re- search Foundation (South Africa) and the Max Planck Institute for Demographic Research (Germany). I thank those with whom I had the pleasure of working and (or) discussing reli- ability-related problems: Frank Beichelt, Ji Cha, Pieter van Gelder, Waltraud viii Preface Kahle, Michail Nikulin, Jan van Noortwijk, Michail Revjakov, Michail Rosenhaus, Fabio Spizzichino, Jef Teugels, Igor Ushakov, James Vaupel, Daan de Waal, Ter- tius de Wet, Anatoly Yashin, Vladimir Zarudnij. Chapters 6 and 7 are written in co-authorship with my daughter Veronica Esaulova on the basis of her PhD thesis (Esaulova, 2006). Many thanks to her for this valuable contribution. I would like to express my gratitude and appreciation to my colleagues in the department of mathematical statistics of the University of the Free State. Annual visits (since 2003) to the Max Planck Institute for Demographic Research (Ger- many) also contributed significantly to this project, especially to Chapter 10, which is devoted to demographic and biological applications. Special thanks to Justin Harvey and Lieketseng Masenyetse for numerous sug- gestions for improving the presentation of this book. Finally, I am indebted to Simon Rees, Anthony Doyle and the Springer staff for their editorial work. University of the Free State Maxim Finkelstein South Africa July 2008 Contents 1 Introduction.......................................................................................................1 1.1 Aim and Scope of the Book.......................................................................1 1.2 Brief Overview..........................................................................................5 2 Failure Rate and Mean Remaining Lifetime..................................................9 2.1 Failure Rate Basics..................................................................................10 2.2 Mean Remaining Lifetime Basics............................................................13 2.3 Lifetime Distributions and Their Failure Rates.......................................19 2.3.1 Exponential Distribution...............................................................19 2.3.2 Gamma Distribution.....................................................................20 2.3.3 Exponential Distribution with a Resilience Parameter.................22 2.3.4 Weibull Distribution.....................................................................23 2.3.5 Pareto Distribution........................................................................24 2.3.6 Lognormal Distribution................................................................25 2.3.7 Truncated Normal Distribution.....................................................26 2.3.8 Inverse Gaussian Distribution......................................................27 2.3.9 Gompertz and Makeham–Gompertz Distributions.......................27 2.4 Shape of the Failure Rate and the MRL Function....................................28 2.4.1 Some Definitions and Notation....................................................28 2.4.2 Glaser’s Approach........................................................................30 2.4.3 Limiting Behaviour of the Failure Rate and the MRL Function...36 2.5 Reversed Failure Rate..............................................................................39 2.5.1 Definitions....................................................................................39 2.5.2 Waiting Time................................................................................42 2.6 Chapter Summary....................................................................................43 3 More on Exponential Representation...........................................................45 3.1 Exponential Representation in Random Environment.............................45 3.1.1 Conditional Exponential Representation......................................45 3.1.2 Unconditional Exponential Representation..................................47 3.1.3 Examples......................................................................................48 3.2 Bivariate Failure Rates and Exponential Representation.........................52 x Contents 3.2.1 Bivariate Failure Rates.................................................................52 3.2.2 Exponential Representation of Bivariate Distributions................54 3.3 Competing Risks and Bivariate Ageing...................................................59 3.3.1 Exponential Representation for Competing Risks........................59 3.3.2 Ageing in Competing Risks Setting.............................................60 3.4 Chapter Summary....................................................................................65 4 Point Processes and Minimal Repair............................................................67 4.1 Introduction – Imperfect Repair...............................................................67 4.2 Characterization of Point Processes.........................................................70 4.3 Point Processes for Repairable Systems..................................................72 4.3.1 Poisson Process............................................................................72 4.3.2 Renewal Process...........................................................................73 4.3.3 Geometric Process........................................................................76 4.3.4 Modulated Renewal-type Processes.............................................79 4.4 Minimal Repair........................................................................................81 4.4.1 Definition and Interpretation........................................................81 4.4.2 Information-based Minimal Repair..............................................83 4.5 Brown–Proschan Model..........................................................................84 4.6 Performance Quality of Repairable Systems...........................................85 4.6.1 Perfect Restoration of Quality......................................................86 4.6.2 Imperfect Restoration of Quality..................................................88 4.7 Minimal Repair in Heterogeneous Populations.......................................89 4.8 Chapter Summary....................................................................................92 5 Virtual Age and Imperfect Repair................................................................93 5.1 Introduction – Virtual Age.......................................................................93 5.2 Virtual Age for Non-repairable Objects...................................................95 5.2.1 Statistical Virtual Age..................................................................95 5.2.2 Recalculated Virtual Age..............................................................98 5.2.3 Information-based Virtual Age...................................................102 5.2.4 Virtual Age in a Series System...................................................105 5.3 Age Reduction Models for Repairable Systems....................................107 5.3.1 G-renewal Process......................................................................107 5.3.2 ‘Sliding’ Along the Failure Rate Curve......................................109 5.4 Ageing and Monotonicity Properties.....................................................115 5.5 Renewal Equations................................................................................123 5.6 Failure Rate Reduction Models.............................................................125 5.7 Imperfect Repair via Direct Degradation ..............................................127 5.8 Chapter Summary..................................................................................130 6 Mixture Failure Rate Modelling..................................................................133 6.1 Introduction – Random Failure Rate......................................................133 6.2 Failure Rate of Discrete Mixtures..........................................................138 6.3 Conditional Characteristics and Simplest Models.................................139 6.3.1 Additive Model...........................................................................141 6.3.2 Multiplicative Model..................................................................143 Contents xi 6.4 Laplace Transform and Inverse Problem...............................................144 6.5 Mixture Failure Rate Ordering...............................................................149 6.5.1 Comparison with Unconditional Characteristic..........................149 6.5.2 Likelihood Ordering of Mixing Distributions............................152 6.5.3 Mixing Distributions with Different Variances..........................157 6.6 Bounds for the Mixture Failure Rate.....................................................159 6.7 Further Examples and Applications.......................................................163 6.7.1 Shocks in Heterogeneous Populations........................................163 6.7.2 Random Scales and Random Usage...........................................164 6.7.3 Random Change Point................................................................165 6.7.4 MRL of Mixtures........................................................................167 6.8 Chapter Summary..................................................................................168 7 Limiting Behaviour of Mixture Failure Rates............................................171 7.1 Introduction............................................................................................171 7.2 Discrete Mixtures...................................................................................172 7.3 Survival Models.....................................................................................175 7.4 Main Asymptotic Results.......................................................................177 7.5 Specific Models.....................................................................................179 7.5.1 Multiplicative Model..................................................................179 7.5.2 Accelerated Life Model..............................................................182 7.5.3 Proportional Hazards and Other Possible Models......................183 7.6 Asymptotic Mixture Failure Rates for Multivariate Frailty...................184 7.6.1 Introduction................................................................................184 7.6.2 Competing Risks for Mixtures...................................................185 7.6.3 Limiting Behaviour for Competing Risks..................................187 7.6.4 Bivariate Frailty Model..............................................................189 7.7 Sketches of the Proofs............................................................................192 7.8 Chapter Summary..................................................................................196 8 ‘Constructing’ the Failure Rate...................................................................197 8.1 Terminating Poisson and Renewal Processes........................................197 8.2 Weaker Criteria of Failure.....................................................................201 8.2.1 Fatal and Non-fatal Shocks.........................................................201 8.2.2 Fatal and Non-fatal Failures.......................................................205 8.3 Failure Rate for Spatial Survival............................................................207 8.3.1 Obstacles with Fixed Coordinates..............................................207 8.3.2 Crossing the Line Process...........................................................210 8.4 Multiple Availability on Demand..........................................................213 8.4.1 Introduction................................................................................213 8.4.2 Simple Criterion of Failure.........................................................215 8.4.3 Two Consecutive Non-serviced Demands..................................218 8.4.4 Other Weaker Criteria of Failure................................................221 8.5 Acceptable Risk and Thinning of the Poisson Process..........................222 8.6 Chapter Summary..................................................................................223