The Limit of Stationary Distributions of Many-Server Queues in the Halfin-Whitt Regime

TL;DR

This paper proves the convergence of stationary distributions for many-server queues in the Halfin-Whitt regime under general service time distributions, resolving a long-standing open problem and characterizing the limit as an infinite-dimensional Markov process.

Contribution

It introduces a new representation for the queue state and establishes convergence for general service distributions, extending previous results beyond exponential cases.

Findings

Proves convergence of stationary distributions in the Halfin-Whitt regime.

Characterizes the limit as a solution to a stochastic PDE.

Handles general service distributions, not just exponential.

Abstract

We consider the so-called GI/GI/N queue, in which a stream of jobs with independent and identically distributed service times arrive as a renewal process to a common queue that is served by $N$ identical parallel servers in a first-come-first-serve manner. We introduce a new representation for the state of the system and, under suitable conditions on the service and interarrival distributions, establish convergence of the corresponding sequence of centered and scaled stationary distributions in the so-called Halfin-Whitt asymptotic regime. In particular, this resolves an open question posed by Halfin and Whitt in 1981. We also characterize the limit as the stationary distribution of an infinite-dimensional two-component Markov process that is the unique solution to a certain stochastic partial differential equation. Previous results were essentially restricted to exponential service distributions or service distributions with finite support, for which the corresponding limit process admits a reduced finite-dimensional Markovian representation. We develop a different approach to deal with the general case when the Markovian representation of the limit is truly infinite-dimensional. This approach is more broadly applicable to a larger class of networks.