# The supermarket model with arrival rate tending to 1

*Stochastic Processes and Financial Mathematics*

*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.