June 2022 Near equilibrium fluctuations for supermarket models with growing choices
Shankar Bhamidi, Amarjit Budhiraja, Miheer Dewaskar
Author Affiliations +
Ann. Appl. Probab. 32(3): 2083-2138 (June 2022). DOI: 10.1214/21-AAP1729


We consider the supermarket model in the usual Markovian setting where jobs arrive at rate nλn for some λn>0, with n parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among dnn randomly selected service queues. We show that when dn and λnλ(0,), 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 λn1. Previous papers, for example, (Stoch. Syst. 8 (2018) 265–292) have considered the regime dnnlogn 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 dn. In particular, we consider the three canonical regimes (a) dn/n0; (b) dn/nc(0,) and, (c) dn/n. In all three regimes, we show, by establishing suitable functional limit theorems, that (under conditions on λn) fluctuations of the state process about its near equilibrium are of order n1/2 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 dn/(nlogn), our work gives alternative proofs for the universality results established in (Stoch. Syst. 8 (2018) 265–292).

Funding Statement

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.


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Shankar Bhamidi. Amarjit Budhiraja. Miheer Dewaskar. "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


Received: 1 June 2020; Revised: 1 April 2021; Published: June 2022
First available in Project Euclid: 29 May 2022

MathSciNet: MR4430009
zbMATH: 1496.90022
Digital Object Identifier: 10.1214/21-AAP1729

Primary: 60C05 , 60F17 , 90B15 , 90B22

Keywords: diffusion approximations , fluid limits , Functional limit theorems , Halfin–Whitt , heavy traffic , join-the-shortest-queue , load balancing , Power of Choice , Reflected diffusions , Skorohod problem

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 3 • June 2022
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