Machine interference problem with a random environment

This paper is concerned with a queueing model to analyse the asymptotic behaviour of the machine interference problem with N machines and a single operative. The running and repair times of each machine are supposed to be exponentially distributed random variables with parameter depending on the state of a varying environment. Assuming that the repair rate is much greater than the failure rate (‘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 illustrate the problem in question in the field of textile winding.