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Comparison of system reliability functions under laboratory and common operating environments

Part of:Distribution theoryMultivariate analysisDesign of experiments

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 R 1(t), the reliability function of a redundant m-of-n system operating within the laboratory, with R D(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 R D(t) to underestimate (resp. overestimate) R 1(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 R D(t) initially understimates (or overestimates) R 1(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 designs62E10: Characterization and structure theory
Secondary:62H10: Distribution of statistics62E15: Exact distribution theory

Information

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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