The supermarket model with arrival rate tending to one

In the supermarket model, there are $n$ queues, each with a single server. Customers arrive in a Poisson process with arrival rate $\lambda n$, where $\lambda = \lambda (n) \in (0,1)$. Upon arrival, a customer selects $d=d(n)$ servers uniformly at random, and joins the queue of a least-loaded server amongst those chosen. Service times are independent exponentially distributed random variables with mean~1. In this paper, we analyse the behaviour of the supermarket model in a regime where $\lambda(n)$ tends to~1, and $d(n)$ tends to infinity, as $n \to \infty$. For suitable triples $(n,d,\lambda)$, we identify a subset ${\cal N}$ of the state space where the process remains for a long time in equilibrium. We further show that the process is rapidly mixing when started in ${\cal N}$, and give bounds on the speed of mixing for more general initial conditions.