# The supermarket model with arrival rate tending to 1

#### by Malwina Luczak

Institution: University of Sheffield
Date: Wed 16th November 2011
Time: 3:00 PM
Location: Room 213, Richard Berry Building

Abstract: There are $n$ queues, each with a single server. Customers
arrive in a Poisson process at rate $\lambda n$, where $0 < \lambda = \lambda (n) < 1$. Upon arrival each customer selects $d = d(n) \ge 2$
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 will briefly review the results of Luczak and McDiarmid (2006) in the
case where $\lambda$ and $d$ are constants independent of $n$.

We will then investigate the speed of convergence to equilibrium and the
maximum length of a queue in the equilibrium distribution when $\lambda (n) \to 1$ and $d(n) \to \infty$ as $n \to \infty$. This is joint work
with Graham Brightwell.