Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-30T20:16:29.111Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of system reliability functions under laboratory and common operating environments

Part of: Multivariate analysis Distribution theory Design 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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

[1] Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
[2] Chhikara, R. S. and Folks, J. L. (1989) The Inverse Gaussian Distribution: Theory, Methodology, and Applications. Marcel Dekker, New York.Google Scholar
[3] Currit, A. and Singpurwalla, N. D. (1988) On the reliability of a system of components sharing a common environment. J. Appl. Prob. 26, 763–771.Google Scholar
[4] Egeland, T. (1992) Mixing dependence and system reliability. Scand. J. Statist. 19, 173–183.Google Scholar
[5] Gross, A. J. and Lee, M.-L. (1989) Properties of conditionally independently generated gamma distributions. Prob. Eng. Inf. Sci. 3, 289–297.Google Scholar
[6] Gupta, R. D. and Richards, D. St. P. (1992) Multivariate Liouville distributions, III. J. Multivar. Anal. 43, 29–57.Google Scholar
[7] Herstein, I. N. (1964) Topics in Algebra. Xerox College Pub, Lexington, MA.Google Scholar
[8] Lefevre, C. and Malice, M.-P. (1989) On a system of components with joint lifetimes distributed as a mixture of independent exponential laws. J. Appl. Prob. 26, 202–208.Google Scholar
[9] Lindley, D. V. and Singpurwalla, N. D. (1986) Multivariate distributions for the life lengths of components of a system sharing a common environment. J Appl. Prob. 23, 418–431.Google Scholar
[10] Roman, S. (1984) The Umbral Calculus. Academic Press, New York.Google Scholar
[11] Stanley, R. P. (1986) Enumerative Combinatorics. Vol. 1. Wadsworth, Monterey, CA.Google Scholar
[12] Whitmore, G. A. and Lee, M.-L. T. (1991) A multivariate survival distribution generated by an inverse Gaussian mixture of exponentials. Technometrics 33, 39–50.Google Scholar
[13] Woodham, S.-A. (1993) Modeling environmental impact on fault-tolerant systems using mixed reliability functions. Ph.D. dissertation. University of Virginia.Google Scholar
[14] Youngren, M. A. (1991) Dependence in target element detection induced by the environment. Naval Res. Logist. 38, 567–577.Google Scholar