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

• 成果类型: Article
• 署名作者: Aghajani, Reza; Ramanan, Kavita
• 署名单位: University of California System; University of California San Diego; Brown University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2019.1021
• 发表日期: 2020
• 页码: 1016-1055
• 关键词: heavy-traffic limits

摘要

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-served 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.