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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 the regime where \(\lambda (n) = 1 - n^{-\alpha }\) and \(d(n) = \lfloor n^\beta \rfloor \), where \(\alpha \) and \(\beta \) are fixed numbers in (0, 1]. For suitable pairs \((\alpha , \beta )\), our results imply that, in equilibrium, with probability tending to 1 as \(n \rightarrow \infty \), the proportion of queues with length equal to \(k = \lceil \alpha /\beta \rceil \) is at least \(1-2n^{-\alpha + (k-1)\beta }\), and there are no longer queues. We further show that the process is rapidly mixing when started in a good state, and give bounds on the speed of mixing for more general initial conditions.

The Supermarket Model with Bounded Queue Lengths in Equilibrium