Mathematics – Probability
Scientific paper
2003-12-17
Mathematics
Probability
A new preprint math.PR/0403538, has been written as a combined version of the present preprint and the preprint math.PR/031233
Scientific paper
We consider N single server infinite buffer queues with service rate beta. Customers arrive at rate N times alpha,choose L queues uniformly, and join the shortest one. The stability condition is alpha strictly less than beta. We study in equilibrium the sequence of the fraction of queues of length at least k, in the large N limit. We prove a functional central limit theorem on an infinite-dimensional Hilbert space with its weak topology, with limit a stationary Ornstein-Uhlenbeck process. We use ergodicity and justify the inversion of limits of long times and large sizes N by a compactness-uniqueness method. The main tool for proving tightness of the ill-known invariant laws and ergodicity of the limit is a global exponential stability result for the nonlinear dynamical system obtained in the functional law of large numbers limit.
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