A functional central limit theorem for the M/GI/$\infty$ queue

Details A functional central limit theorem for the M/GI/$\infty$ queue A functional central limit theorem for the M/GI/$\infty$ queue

2006-08-10

arxiv.org/abs/math/0608258v3

Annals of Applied Probability 2008, Vol. 18, No. 6, 2156-2178

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP518 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP518

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/$\infty$. 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 $\mathcal{S}^{\prime}$ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a $\mathcal{S}^{\prime}$-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes.

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