Distribution-valued heavy-traffic limits for the $G/GI/\infty$ queue

Details Distribution-valued heavy-traffic limits for the $G/GI/\infty$ queue Distribution-valued heavy-traffic limits for the $G/GI/\infty$ queue

2009-02-13

arxiv.org/abs/0902.2236v1

Mathematics

Probability

32 pages

Scientific paper

We study the $G/GI/\infty$ queue from two different perspectives in the same heavy-traffic regime. First, we represent the dynamics of the system using a measure-valued process that keeps track of the age of each customer in the system. Using the continuous-mapping approach together with the martingale functional central limit theorem, we obtain fluid and diffusion limits for this process in a space of distribution-valued processes. Next, we study a measure-valued process that keeps track of the residual service time of each customer in the system. In this case, using the functional central limit theorem and the random time change theorem together with the continuous-mapping approach, we again obtain fluid and diffusion limits in our space of distribution-valued processes. In both cases, we find that our diffusion limits may be characterized as distribution-valued Ornstein-Uhlenbeck processes. Further, these diffusion limits can be analyzed using standard results from the theory of Markov processes.

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