We consider the supermarket model in the usual Markovian setting where jobs arrive at rate for some , with n parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among randomly selected service queues. We show that when and , 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 . Previous papers, for example, (Stoch. Syst. 8 (2018) 265–292) have considered the regime 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 . In particular, we consider the three canonical regimes (a) ; (b) and, (c) . In all three regimes, we show, by establishing suitable functional limit theorems, that (under conditions on ) fluctuations of the state process about its near equilibrium are of order 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 , our work gives alternative proofs for the universality results established in (Stoch. Syst. 8 (2018) 265–292).
Research of SB is supported in part by NSF Grants DMS-1613072, DMS-1606839, DMS-2113662 and ARO Grant W911NF-17-1-0010.
Research of AB is supported in part by the National Science Foundation (DMS-1814894 and DMS-1853968).
Research of MD is supported by the NSF Grant DMS-1613072 and NIH R01 grant HG009125-01.
AB is grateful for the support from Nelder Fellowship from Imperial College, London, where part of this research was completed.
We thank the two anonymous referees for a very careful review of the manuscript that led to numerous improvements in the work.
"Near equilibrium fluctuations for supermarket models with growing choices." Ann. Appl. Probab. 32 (3) 2083 - 2138, June 2022. https://doi.org/10.1214/21-AAP1729