Estimation of availability function using Student distribution

We consider the asymptotic behavior of the empirical availability function, which is an important reliability characteristic of technical, communication, information, or transport systems. Our main focus is the case of violation of the classical assumptions of homogeneity of failure flow or the existence of the expectation of failure-free performance or repair duration. Both the classical situation dealing with samples with nonrandom size and the situation where the number of available observations is unknown beforehand, that is, the sample size is random, are considered. In a special situation where the sample size has a negative binomial distribution, an analogue of the law of large numbers is proved for random sums of not necessarily identically distributed random variables describing conditions for the convergence of the distributions of negative binomial random sums to generalized gamma distributions. Thus, a simple limit scheme is proposed, within which generalized gamma distributions turn out to be limit laws. As a corollary, conditions are obtained for the convergence of the distributions of geometric random sums of independent nonidentically distributed random variables to the Weibull distribution.