October 2021 Many-server asymptotics for join-the-shortest-queue: Large deviations and rare events
Amarjit Budhiraja, Eric Friedlander, Ruoyu Wu
Author Affiliations +
Ann. Appl. Probab. 31(5): 2376-2419 (October 2021). DOI: 10.1214/20-AAP1650


The join-the-shortest-queue routing policy is studied in an asymptotic regime where the number of processors n scales with the arrival rate. A large deviation principle (LDP) for the occupancy process is established, as n, in a suitable infinite-dimensional path space. Model features that present technical challenges include, Markovian dynamics with discontinuous statistics, a diminishing rate property of the transition probability rates, and an infinite-dimensional state space. The difficulty is in the proof of the Laplace lower bound which requires establishing the uniqueness of solutions of certain infinite-dimensional systems of controlled ordinary differential equations. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of long queues. In particular, denoting by Ejn(T) the event that there is at least one queue with j or more jobs at some time instant over [0,T], we show that, in the critical case, for large n and T, P(Ejn(T))exp[n(j2)24T].

Funding Statement

The first author was supported in part by the National Science Foundation Grants DMS-1814894 and DMS-1853968.


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Amarjit Budhiraja. Eric Friedlander. Ruoyu Wu. "Many-server asymptotics for join-the-shortest-queue: Large deviations and rare events." Ann. Appl. Probab. 31 (5) 2376 - 2419, October 2021. https://doi.org/10.1214/20-AAP1650


Received: 1 April 2019; Revised: 1 March 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

MathSciNet: MR4332700
zbMATH: 1477.60053
Digital Object Identifier: 10.1214/20-AAP1650

Primary: 34H05 , 60F10 , 60J75 , 90B15 , 91B70

Keywords: calculus of variations , diminishing rates , discontinuous statistics , golden ratio , infinite-dimensional Skorokhod problem , JSQ , jump-Markov processes in infinite dimensions , large deviations , load balancing

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 5 • October 2021
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