Semi-Markov-Modulated Infinite-Server Queues: Approximations by Time-Scaling

This article studies an infinite-server queue in a semi-Markov environment: the queue’s input rate is modulated by a semi-Markovian background process, and the service times are assumed to be exponentially distributed. The primary objective of this article is to propose approximations for the queue-length distribution, based on time-scaling arguments. The analysis starts with an explicit analysis of the cases in which the transition times of the modulating semi-Markov process are either all deterministic or all exponential. We use these results to obtain approximations under time-scalings; both a quasi-stationary regime (in which time is slowed down) and a fluid-scaling regime (in which time is sped up) are considered. Notably, in the latter regime, the limiting distribution of the number of customers present is Poisson, irrespective of the distribution of the transition times. The accuracy of the resulting approximations is illustrated by several numerical experiments, that moreover give an indication of the speed of convergence in both regimes, for various distributions of the transition times. The last section derives conditions under which the distribution of the number of customers present is Poisson (in an exact sense, i.e., not in a limiting regime).