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Asymptotic methods in reliability theory: a review

Published online by Cambridge University Press:  01 July 2016

I. B. Gertsbakh
I. B. Gertsbakh*
Affiliation:
Ben Gurion University of the Negev
*
∗Postal address: Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beersheva 84 105, Israel.
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Abstract

Section 1 of this paper reviews some works related to reliability evaluation of systems without repair. The assumption that element failure rates are low enables one to obtain an expression for the main term in the asymptotic representation of system reliability function. Section 2 is devoted to repairable systems. The main index of interest in reliability is the time to the first system failure. A typical situation in reliability is that the repair time is much smaller than the element lifetime. This ‘fast repair' property leads to an interesting phenomenon, that for many renewable systems the time to system failure converges in distribution, under appropriate norming, to an exponential random variable. Some basic theorems explaining this fact are presented and a series of typical examples is considered. Special attention is paid to reviewing the works describing the exponentiality phenomenon in birth-and-death processes. Some important aspects of computing the normalizing constants are considered, among them the role played by the so-called ‘main event'. Section 2 also reviews various bounds on the deviation from exponentiality. Section 3 gives brief comments on some works and techniques related to asymptotic reliability analysis. In particular, attention is paid to the works presenting upper and lower bounds on the reliability function.

A considerable part of this review is based on sources originally published in Russian.

Keywords

RELIABILITYEXPONENTIALITYFAST REPAIRSMALL PARAMETERMAIN EVENTREPAIRABLE SYSTEMSASYMPTOTICS
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 1 , March 1984 , pp. 147 - 175
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

This research was carried out during the author's visit to the University of Delaware in 1981–82 and was supported in part by the National Science Foundation under Grant No. ENG-7908351 and the Air Force Office of Scientific Research under Grant No. AFOSR-77–3236.

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