Functional central limit theorems for a large network in which customers join the shortest of several queues

摘要

We consider N single server infinite buffer queues with service rate beta. Customers arrive at rate Nalpha, choose L queues uniformly, and join the shortest. We study the processes t is an element of R+ (bar right arrow) R-t(N) = (R-t(N)(k))(kis an element ofN) for large N, where R-t(N)(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. 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 R-0(N) 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 R-N in equilibrium under the stability condition alphainfinity) lim (t-->infinity)= lim(t-->infinity)lim(N-->infinity) by a compactness-uniqueness method. We deduce a posteriori the CLT for R-0(N) 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.