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February 2020 Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter
Sayan Banerjee, Debankur Mukherjee
Ann. Appl. Probab. 30(1): 80-144 (February 2020). DOI: 10.1214/19-AAP1496

Abstract

Consider a system of $N$ parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate $\lambda (N)$. When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (Math. Oper. Res. 43 (2018) 867–886) identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin–Whitt regime, where $(N-\lambda (N))/\sqrt{N}\to \beta >0$ as $N\to \infty$. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is nontrivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein’s method, Braverman (2018) established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (Ann. Appl. Probab. 29 (2018) 1262–1309) analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion.

However, the analysis of the bulk behavior of the stationary distribution, namely, the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits different qualitative behavior, depending on the value of the heavy-traffic parameter $\beta$. Moreover, we obtain precise asymptotic laws of the centered and scaled steady-state distribution, as $\beta $ tends to 0 and $\infty$. Of particular interest, we also establish a certain intermittency phenomena in the $\beta \to \infty$ regime and a surprising distributional convergence result in the $\beta \to 0$ regime.

Citation

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Sayan Banerjee. Debankur Mukherjee. "Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter." Ann. Appl. Probab. 30 (1) 80 - 144, February 2020. https://doi.org/10.1214/19-AAP1496

Information

Received: 1 September 2018; Revised: 1 February 2019; Published: February 2020
First available in Project Euclid: 25 February 2020

zbMATH: 07200524
MathSciNet: MR4068307
Digital Object Identifier: 10.1214/19-AAP1496

Subjects:
Primary: 60J60 , 60K25
Secondary: 60H20 , 60K05

Keywords: diffusion limit , Halfin–Whitt regime , Join the shortest queue , Local time , nonelliptic diffusion , regenerative processes , steady-state analysis

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 1 • February 2020
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