Comparison of system reliability functions under laboratory and common operating environments

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.