Asymptotic analysis of the heterogeneous machine interference problem with random environments

Abstract

This paper presents a queueing model that can be used to analyze the asymptotic behavior of the machine interference problem composed of N heterogeneous machines and n operatives. Each machine and the repair facility are assumed to operate in independent random environments governed by ergodic Markov chains. The running and repair times of a machine are assumed to be exponentially distributed random variables with parameters depending on the index of the machine and the state of the corresponding random environment. Assuming that the repair rates are much larger than the corresponding failure rates (i.e., “fast” service), it is shown that the time until the number of stopped machines reaches a certain level converges weakly, under appropriate norming, to an exponentially distributed random variable. Furthermore, some numerical examples that compare the approximate and exact characteristics are presented.