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 syst em operating within the laboratory, with R-D(t), the reliability funct ion of the same system operating subject to environmental effects. Wit hin the laboratory, all component lifetimes are independent and identi cally distributed according to G(alpha + 1, lambda), a gamma distribut ion with index alpha + 1 and scale lambda. 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 e ta which models the effect of the common environment, all component li fetimes are independent and identically distributed according to G(alp ha + 1, lambda eta). When alpha is a non-negative integer we prove tha t for R-D(t) to underestimate (resp. overestimate) R-I(t) for all t su fficiently close to zero, it is necessary and sufficient that E(eta((n -m+1)(alpha+1))) > 1 (resp. E(eta((n-m+1)(alpha+1)))< 1). In the case in which n = 2, m = 1 and alpha = 0 we obtain a special case of a resu lt of Currit and Singpurwalla (J. Appl. Prob. 26 (1988), 763-771). As an application, we obtain a necessary and sufficient condition under w hich R-D(t) initially underestimates (or overestimates) R-I(t) when et a follows a gamma or an inverse Gaussian distribution.