On Stochastic Recursive Equations and Infinite Server Queues
The purpose of this paper is to investigate some performance measures of the
discrete time G/G/$\infty$ 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 these class of recursive
equations allow to solve not only the G/PH/$\infty$ queue but also a network
of such queues. We finally investigate the process of residual activity time
in a G/G/$\infty$ queue under general stationary ergodic assumptions, obtain
the unique stationary solution and establish coupling convergence to it from
any initial state.
Philippe Nain
Last modified: Tue Mar 15 14:27:23 MET 2005