A functional central limit theorem for the M/GI/. queue

In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/.. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space S. of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a S.-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes.