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February 2013 SPDE limits of many-server queues
Haya Kaspi, Kavita Ramanan
Ann. Appl. Probab. 23(1): 145-229 (February 2013). DOI: 10.1214/11-AAP821


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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Haya Kaspi. Kavita Ramanan. "SPDE limits of many-server queues." Ann. Appl. Probab. 23 (1) 145 - 229, February 2013.


Published: February 2013
First available in Project Euclid: 25 January 2013

zbMATH: 1271.60098
MathSciNet: MR3059233
Digital Object Identifier: 10.1214/11-AAP821

Primary: 60F17, 60H15, 60K25
Secondary: 68M20, 90B22

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.23 • No. 1 • February 2013
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