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Comparison of system reliability functions under laboratory and common operating environments

Part of: Multivariate analysis Design of experiments Distribution theory

Published online by Cambridge University Press:  14 July 2016

Sara-Anne Woodham  and
Donald St. P. Richards
Sara-Anne Woodham*
Affiliation:
AT&T Network Systems
Donald St. P. Richards*
Affiliation:
University of Virginia
*
∗Postal address: AT&T Network Systems, 480 Red Hill Road, Middletown, NJ 07748, USA.
∗∗Postal address: Division of Statistics, University of Virginia, Charlottesville, VA 22903, USA.
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Abstract

We compare R1(t), the reliability function of a redundant m-of-n system operating within the laboratory, with RD(t), the reliability function of the same system operating subject to environmental effects. Within the laboratory, all component lifetimes are independent and identically distributed according to G(α + 1, λ), a gamma distribution with index α + 1 and scale λ. Outside the laboratory, we adopt the model of Lindley and Singpurwalla (J. Appl. Prob. 23 (1986), 418-431) and assume that, conditional on a positive random variable η which models the effect of the common environment, all component lifetimes are independent and identically distributed according to G(α + 1, λη). When α is a non-negative integer we prove that for RD(t) to underestimate (resp. overestimate) R1(t) for all t sufficiently close to zero, it is necessary and sufficient that E(η(n-m + 1)(α+1)) > 1 (resp. E(η(n-m + 1)(α+1)) < 1). In the case in which n = 2, m= 1 and α = 0 we obtain a special case of a result of Currit and Singpurwalla (J. Appl. Prob. 26 (1988), 763-771). As an application, we obtain a necessary and sufficient condition under which RD(t) initially understimates (or overestimates) R1(t) when η follows a gamma or an inverse Gaussian distribution.

Keywords

EXPONENTIAL DISTRIBUTIONGAMMA DISTRIBUTIONGENERALIZED GAMMA DISTRIBUTIONINVERSE GAUSSIAN DISTRIBUTIONMIXTURES OF DISTRIBUTIONSm-OF-n SYSTEMSYSTEM RELIABILITYUMBRAL CALCULUSWEIBULL DISTRIBUTION

MSC classification

Primary: 62K10: Block designs 62E10: Characterization and structure theory
Secondary: 62H10: Distribution of statistics 62E15: Exact distribution theory
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 536 - 545
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Research supported in part by National Science Foundation grant DMS-9401322.

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