On the maximum queue length in the supermarket model

Details On the maximum queue length in the supermarket model On the maximum queue length in the supermarket model

2006-05-24

arxiv.org/abs/math/0605639v1

Annals of Probability 2006, Vol. 34, No. 2, 493-527

Mathematics

Probability

Published at http://dx.doi.org/10.1214/00911790500000710 in the Annals of Probability (http://www.imstat.org/aop/) by the Inst

Scientific paper

10.1214/00911790500000710

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 $d\geq2$ 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 $n\to\infty$ the maximum queue length takes at most two values, which are $\ln\ln n/\ln d+O(1)$.

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