The Annals of Applied Probability

SPDE limits of many-server queues

Haya Kaspi and Kavita Ramanan

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This paper studies a queueing system in which customers with independent and identically distributed service times arrive to a queue with many servers and enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, and a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Itô diffusion with a constant diffusion coefficient that is insensitive to the service distribution beyond its mean. In addition, the limit of the sequence of (centered and scaled) age processes is shown to be a diffusion taking values in a Hilbert space and is characterized as the unique solution of a stochastic partial differential equation that is coupled with the Itô diffusion describing the limiting number in system. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.

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Ann. Appl. Probab., Volume 23, Number 1 (2013), 145-229.

First available in Project Euclid: 25 January 2013

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F17: Functional limit theorems; invariance principles 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 90B22: Queues and service [See also 60K25, 68M20] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Many-server queues GI/G/N queue fluid limits diffusion limits stochastic partial differential equations measure-valued processes Itô diffusion Halfin–Whitt regime


Kaspi, Haya; Ramanan, Kavita. SPDE limits of many-server queues. Ann. Appl. Probab. 23 (2013), no. 1, 145--229. doi:10.1214/11-AAP821.

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