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Department of Mathematical Sciences

# Seminar Archives

"There are $n$ queues, each with a single server. Customers arrive in a Poisson process at rate $\lambda n$, where $0<\lambda <1$. Upon arrival each customer selects $d \geq 2$ servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as $n \rightarrow \infty$ the maximum queue length takes at most two values, which are $\ln\ln n/ \ln d +O(1)$. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as $n \to \infty$. We quantify this convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order $n^{-1}$; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit propagation of chaos: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most $n^{-1}$. This is joint work with Colin McDiarmid."