On stochastic recursive equations and infinite server queues

The purpose of this paper is to investigate some performance measures of the discrete time infinite server queue under a general arrival process. We assume, more precisely, that at each time unit a batch with a random size may arrive, where the sequence of batch sizes need not be i.i.d. All we request is that it would be stationary ergodic and that the service duration has a phase type distribution. Our goal is to obtain explicit expressions for the first two moments of number of customers in steady state. We obtain this by computing the first two moments of some generic stochastic recursive equations that our system satisfies. We then show that this class of recursive equations allow to solve not only the G/PH/∞ queue but also a network of such queues. We finally investigate the process of residual activity time in a G/G/∞ queue under general stationary ergodic assumptions, obtain the unique stationary solution and establish coupling convergence to it from any initial state.