## The Annals of Applied Probability

### Ergodicity of an SPDE associated with a many-server queue

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 994-1045.

Dates
Revised: December 2017
First available in Project Euclid: 24 January 2019

https://projecteuclid.org/euclid.aoap/1548298935

Digital Object Identifier
doi:10.1214/18-AAP1419

Mathematical Reviews number (MathSciNet)
MR3910022

Zentralblatt MATH identifier
07047443

#### Citation

Aghajani, Reza; Ramanan, Kavita. Ergodicity of an SPDE associated with a many-server queue. Ann. Appl. Probab. 29 (2019), no. 2, 994--1045. doi:10.1214/18-AAP1419. https://projecteuclid.org/euclid.aoap/1548298935

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