2004/03/31 by C. D. Graham, Carl Graham, Graham, Carl
Business, Management and Accounting · Economics, Econometrics and Finance · Mathematics · #Advanced Queuing Theory Analysis #Markov Chains and Monte Carlo Methods #Stochastic processes and financial applications #math.PR #msc:37A30 #msc:60K25 #msc:60K35
paper · pdf · doi:10.48550/arxiv.math/0403538
This preprint is a combined version of the two preprints math.PR/0312334 and math.PR/0312335. It is recommended to read this combined version instead of the two precedent ones
arxiv created 2004/03/31 · arxiv updated 2009/12/01
We consider N single server infinite buffer queues with service rate β. Customers arrive at rate Nα, choose L queues uniformly, and join the shortest. We study the processes RN for large N, where RNt(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data RN0 satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the RN in equilibrium under the stability condition α< β, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits for large times and for large sizes N by a compactness-uniqueness method. We deduce a posteriori the CLT for RN0 under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit.