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April 2019 Ergodicity of an SPDE associated with a many-server queue
Reza Aghajani, Kavita Ramanan
Ann. Appl. Probab. 29(2): 994-1045 (April 2019). DOI: 10.1214/18-AAP1419

Abstract

We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by $N$ identical servers in a first-come-first-serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as $1-\beta N^{-1/2}+o(N^{-1/2})$ for some $\beta>0$. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued Itô equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [Aghajani and Ramanan (2016)] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in [Oper. Res. 29 (1981) 567–588]. The methods introduced here are more generally applicable for the analysis of a broader class of networks.

Citation

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Reza Aghajani. Kavita Ramanan. "Ergodicity of an SPDE associated with a many-server queue." Ann. Appl. Probab. 29 (2) 994 - 1045, April 2019. https://doi.org/10.1214/18-AAP1419

Information

Received: 1 October 2016; Revised: 1 December 2017; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047443
MathSciNet: MR3910022
Digital Object Identifier: 10.1214/18-AAP1419

Subjects:
Primary: 60G10, 60H15, 60K25
Secondary: 90B22

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 2 • April 2019
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