# zbMATH — the first resource for mathematics

Repairable systems reliability. Modeling, inference, misconceptions and their causes. (English) Zbl 0543.62083
Lecture Notes in Statistics (Marcel Dekker), Vol. 7. New York - Basel: Marcel Dekker, Inc. XIV, 223 p. $45.00 (U.S. & Canada);$ 54.00; SFr. 119.00 (all other countries) (1984).
In the first paragraph of Chapter 1 the authors define: ”A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method, other than replacement of the entire system.” One of the intents of the monograph is to emphasize that in spite of the fact that most ”real world” systems are repairable, only a small fraction of the reliability literature deals with repairable systems. Possible reasons for ignoring repairable systems in the literature are discussed. Another intent of the monograph is to describe in a semiformal way some available statistical analysis of repairable systems failure data. Still another purpose of the monograph is to list and discuss various probabilistic models of repairable systems including models for reliability growth, software reliability and for repairable systems cost.
The first chapter, called ”Scope”, emphasizes the lack of statistical analysis and probabilistic modeling of repairable systems in the literature. The chapter also describes what is not going to be treated in the monograph. In particular, the first system failures of systems with repairable subsystems are not covered in detail in the monograph. Also, in spite of the fact that the monograph emphasizes point processes, the point process approach to nonrepairable systems and components is not treated.
Chapter 2 introduces some terminology, notation and some basic models for stochastic point processes applicable to repairable systems. The confusion which arose from calling two different functions by the same name is highlighted. An example is given in which the ”failure rate” increases and simultaneously decreases. Some basic probabilistic modeling of repairable systems (such as homogeneous Poisson process, nonhomogeneous Poisson process, superimposed renewal process and branching Poisson process) and their shortcomings are listed in Chapters 3 and 4. The well known conflict between the complicated ”real world” and the simplified mathematical model is discussed in detail.
Chapters 5, 6 and 7 review the literature on the statistical analysis of repairable systems data, reliability growth and repairable systems cost models. The discussion again is semiformal and many details are omitted; however references for complete details are given throughout.
In the preface Chapter 8 is called the heart of the monograph. It deals with misconceptions which are rampant in the reliability field. For example, it is emphasized that a linear plot on linear paper does not imply an underlying homogeneous Poisson process. Similarly the confusion between ”reliability with repair” and ”repairable system” is noted down. Further in Chapter 8, some misconceptions due to terminology and notation are given. It is shown that ”failure rate” is used to mean so many things that it has become meaningless. Also many works use ”intensity”, ”hazard rate”, ”rate of occurrence of failures”, ”peril rate”, ”event rate” and other variations of the above, to mean the same or different quantities. Other misconceptions arise from the various meanings of ”times between failures”, of ”wearout” and ”deterioration”, of ”burn-in” and ”improvement” and so on. This chapter can bring to frustration many readers. The lesson is that one should make sure what a term means in a particular paper or a book, although it may mean something else somewhere else. Chapter 8 is the one that I enjoyed the most in the monograph and I think that every reliability practitioner should read it. Chapter 9, the last chapter, lists some research topics for future work and describes some military standards and handbooks.
Overall it is a charming monograph. It is easy to read and it contains a large amount of information. The Table of Contents consists of 187 items for a book of 223 pages. That is, each subsection is a little more than one page long, a fact which makes the reading even easier. Every reliability practitioner should be aware of the main messages of the monograph (that is, that confusion is wide spread in the practical reliability literature and that very little work has been done in the area of repairable systems).
Reviewer: M.Shaked

##### MSC:
 62N05 Reliability and life testing 90B25 Reliability, availability, maintenance, inspection in operations research 62-02 Research exposition (monographs, survey articles) pertaining to statistics 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)