The Annals of Applied Probability

Ergodicity of an SPDE associated with a many-server queue

Reza Aghajani and Kavita Ramanan

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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

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

Received: October 2016
Revised: December 2017
First available in Project Euclid: 24 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G10: Stationary processes 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 90B22: Queues and service [See also 60K25, 68M20]

Stochastic partial differential equations Itô process stationary distribution ergodicity asymptotic coupling asymptotic equivalent coupling GI/GI/N queue Halfin–Whitt regime diffusion approximations


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.

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