2003/12/17 by Carl Graham, Graham, Carl
Business, Management and Accounting · Decision Sciences · Mathematics · #Advanced Queuing Theory Analysis #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR #msc:60K25 #msc:60K35
paper · pdf · doi:10.48550/arxiv.math/0312334
A new preprint math.PR/0403538, has been written as a combined version of the present preprint and the preprint math.PR/0312335. It is recommended to read the new combined version instead of the two others
arxiv created 2004/03/31 · arxiv updated 2009/12/01
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.