Near equilibrium fluctuations for supermarket models with growing choices

Abstract

We consider the supermarket model in the usual Markovian setting where jobs arrive at rate $\mathit{n}{\mathit{\lambda }}_{\mathit{n}}$ for some ${\mathit{\lambda }}_{\mathit{n}}>0$, with n parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among ${\mathit{d}}_{\mathit{n}}\le \mathit{n}$ randomly selected service queues. We show that when ${\mathit{d}}_{\mathit{n}}\to \infty$ and ${\mathit{\lambda }}_{\mathit{n}}\to \mathit{\lambda }\in (0,\infty )$, under natural conditions on the initial queues, the state occupancy process converges in probability, in a suitable path space, to the unique solution of an infinite system of constrained ordinary differential equations parametrized by λ. Our main interest is in the study of fluctuations of the state process about its near equilibrium state in the critical regime, namely when ${\mathit{\lambda }}_{\mathit{n}}\to 1$. Previous papers, for example, (Stoch. Syst. 8 (2018) 265–292) have considered the regime $\frac{{\mathit{d}}_{\mathit{n}}}{\sqrt{\mathit{n}}log\mathit{n}}\to \infty$ while the objective of the current work is to develop diffusion approximations for the state occupancy process that allow for all possible rates of growth of ${\mathit{d}}_{\mathit{n}}$. In particular, we consider the three canonical regimes (a) ${\mathit{d}}_{\mathit{n}}/\sqrt{\mathit{n}}\to 0$; (b) ${\mathit{d}}_{\mathit{n}}/\sqrt{\mathit{n}}\to \mathit{c}\in (0,\infty )$ and, (c) ${\mathit{d}}_{\mathit{n}}/\sqrt{\mathit{n}}\to \infty$. In all three regimes, we show, by establishing suitable functional limit theorems, that (under conditions on ${\mathit{\lambda }}_{\mathit{n}}$) fluctuations of the state process about its near equilibrium are of order ${\mathit{n}}^{-1/2}$ and are governed asymptotically by a one-dimensional Brownian motion. The forms of the limit processes in the three regimes are quite different; in the first case, we get a linear diffusion; in the second case, we get a diffusion with an exponential drift; and in the third case we obtain a reflected diffusion in a half space. In the special case ${\mathit{d}}_{\mathit{n}}/(\sqrt{\mathit{n}}log\mathit{n})\to \infty$, our work gives alternative proofs for the universality results established in (Stoch. Syst. 8 (2018) 265–292).