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Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

David Gamarnik  and
Petar Momčilović
David Gamarnik*
Affiliation:
Massachusetts Institute of Technology
Petar Momčilović*
Affiliation:
University of Michigan
*
∗ Postal address: Operations Research Center and Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Email address: gamarnik@mit.edu
∗∗ Postal address: EECS Department, University of Michigan, Ann Arbor, MI 48109, USA. Email address: petar@eecs.umich.edu
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Abstract

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We consider a multiserver queue in the Halfin-Whitt regime: as the number of servers n grows without a bound, the utilization approaches 1 from below at the rate Assuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

Keywords

Multiserver queueheavy-traffic approximationHalfin-Whitt (QED) regime

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 548 - 577
Copyright
Copyright © Applied Probability Trust 2008 

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