Comparison of system reliability functions under laboratory and common operating environments

We compare R_I(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_I(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_I(t) when η follows a gamma or an inverse Gaussian distribution.