Failure Rate Modelling for Reliability and Risk (Springer Series in Reliability Engineering)

296 Pages·2008·2.1 MB·English

by  Maxim Finkelstein

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

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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
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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