by Steven E. Rigdon and Asit P. Basu

© 2000 John Wiley and Sons, New York, NY

Last Updated: April 7, 2000

Preface

Reliability plays a key role in developing quality products and in enhancing
competitiveness. For most products, customers see reliability as one of the
most important quality characteristics. In the last several decades, there has
been much research on the theory and applications of reliability. Most of this
literature, however, is directed toward nonrepairable systems, that is,
systems that fail only once and are then discarded. This book covers the
reliability of repairable systems only, and is intended to
give a thorough presentation of

-- Probabilistic models for the reliability of repairable systems

-- Statistical methods, including graphical methods, for analyzing data
from repairable systems.

The first part of this book looks much like a book on stochastic processes,
although only selected topics from that subject are presented. This part of
the book is a reasonably thorough presentation of stochastic point processes.
The second part of the book deals with analyzing data from repairable systems,
and includes graphical methods, point estimation, interval estimation,
hypothesis tests, goodness-of-fit tests, and reliability prediction.

This book is written for reliability engineers, quality engineers, statisticians, quality managers, and anyone who is involved in producing reliable systems. It should be useful as a reference for practioners and researchers in the field. In addition, the book may also serve as a textbook for an upper level or beginning graduate level course on reliability. For this reason we have included numerous exercises, many with real data. Readers with a background consisting of a calculus-based course on probability and statistics should be able to follow much of the book, although some of the proofs will be difficult to follow. Readers with a background that includes random variables (discrete and continuous), marginal and joint probability distributions, expectation, point estimation, confidence intervals, and hypothesis testing should be able to follow nearly all of the book. Some of the more advanced developments, such as the derivation of maximum likelihood estimates, the derivation of Bayesian estimates, and proofs of some of the theorems are optional.

Chapter 1 begins with a discussion of several terms that are often used (and abused) in reliability. The distinction between repairable systems and nonrepairable systems, and the corresponding sets of terms and notation, is also discussed in this chapter. Chapter 2 discusses the Poisson process, including the nonhomogeneous Poisson process, and gives some of its properties. Chapter 3 discusses other probabilistic models that can be applied to the reliability of repairable systems. These models include the renewal process, as well as some other more specialized models. Chapters 4 and 5 discuss the analysis of data from repairable systems. Chapter 4 deals with analyzing data from a single repairable system, and Chapter 5 deals with several systems.

We would like to thank all the students who have read drafts of this book. Michael Gough and Randall Holden deserve special thanks for their careful reading of the manuscript in its latter stages. We also want to thank the staff at John Wiley and Sons, especially Stephen Quigley, Andrew Prince, and Heather Haselkorn, for their hard work and encouragement along the way. Finally, we wish to thank our wives, Pat and Sandra, for their patience and understanding during the years that it took to complete this project.

Steven E. Rigdon, Edwardsville, Illinois

Asit P. Basu, Columbia, Missouri

A unique, practical guide for industry professionals who need to improve product quality and reliability in repairable systems. Owing to its vital role in product quality, reliability has been intensely studied in recent decades. Most of this research, however, addresses systems that are nonrepairable and therefore discarded upon failure. Statistical Methods for the Reliability of Repairable Systems fills the gap in the field, focusing exclusively on an important yet long-neglected area of reliability. Written by two highly recognized members of the reliability and statistics community, this new work offers a unique, systematic treatment of probabilistic models used for repairable systems as well as the statistical methods for analyzing data generated from them.

Liberally supplemented with examples as well as exercises boasting real data, the book clearly explains the difference between repairable and nonrepairable systems and helps readers develop an understanding of stochastic point processes. Data analysis methods are discussed for both single and multiple systems and include graphical methods, point estimation, interval estimation, hypothesis tests, goodness-of-fit tests, and reliability prediction. Complete with extensive graphs, tables, and references, Statistical Methods for the Reliability of Repairable Systems is an excellent working resource for industry professionals involved in producing reliable systems and a handy reference for practitioners and researchers in the field.

Chapter 1. Terminology and Notation for Repairable Systems

1.2 Nonrepairable Systems

1.2.2 The Weibull Distribution

1.2.3 The Gamma Distribution

1.4 Overview of Models

1.5 Exercises

2.2 The Homogeneous Poisson Process

2.3.2 Time Truncated Case

3.2 The Piecewise Exponential Model

3.3 Modulated Processes

3.4 The Branching Poisson Process

3.5 Imperfect Repair Models

3.6 Exercises

4.1.2 Total Time on Test (TTT) Plots

4.2.2 Kernel Estimates

4.2.3 An Estimate Assuming a Convex Intensity Function

4.2.4 Examples

4.4 Inference for the Homogeneous Poisson Process

4.5 Inference for the Power Law Process: Failure Truncated Case

4.5.2 Interval Estimation and Tests of Hypotheses

4.5.3 Estimation of the Intensity Function

4.5.4 Goodness-of-Fit Tests

4.6.2 Interval Estimation and Tests of Hypotheses

4.6.3 Estimation of the Intensity Function

4.6.4 Goodness-of-Fit Tests

4.8 Bayesian Estimation

4.8.2 Bayesian Inference for Predicting the Number of Failures from the HPP

4.8.3 Bayesian Inference for the Parameters of the Power Law Process

4.8.4 Bayesian Inference for Predicting the Number of Failures

k 4.9.2 Hypothesis Tests for the Modulated Power Law Process

4.9.3 Confidence Intervals for Parameters

4.9.4 An Example

4.11 Standards

4.11.2 MIL-HDBK-781 and MIL-STD-781

4.11.3 ANSI/IEC/ASQ 61164

4.13 Exercises

5.1.2 Interval Estimation for q

5.1.3 Hypothesis Testing for q

5.2.2

5.3.2 Hierarchical Bayes Models

5.5 Testing for the Equality of Growth Parameters in the Power Law Process

5.5.2 Testing Equality of b’s for

5.7 Parametric Empirical Bayes Models for the PLP

5.8 Exercises

A.2 Critical Values for the

A.3 Confidence Limits for the Mean of a Poisson Distribution Given an Observation of

A.4 Factors for Obtaining a Confidence Interval for the Intensity at the Time of the Last Failure for a Failure Truncated Power Law Process

A.5 Factors for Obtaining a Confidence Interval for the Intensity at the Time of the Last Failure for a Time Truncated Power Law Process

A.6 Critical Values for the Cramer-von Mises Goodness-of-Fit Test

A.7 Critical Values for Lillierfors’ Goodness of Fit Test

Index

Errata

p. 121, last two displayed equations. In the denominators, the subscripts
on the chi-square percentile should be a
and 1-a, respectively.

p. 127, third displayed equation. The definition for *R_i* be for *i=*1,2,…,*n-*1.