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Transient behavior of the Halfin-Whitt diffusion

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  • van Leeuwaarden, Johan S.H.
  • Knessl, Charles

Abstract

We consider the heavy-traffic approximation to the GI/M/s queueing system in the Halfin-Whitt regime, where both the number of servers s and the arrival rate [lambda] grow large (taking the service rate as unity), with and [beta] some constant. In this asymptotic regime, the queue length process can be approximated by a diffusion process that behaves like a Brownian motion with drift above zero and like an Ornstein-Uhlenbeck process below zero. We analyze the transient behavior of this hybrid diffusion process, including the transient density, approach to equilibrium, and spectral properties. The transient behavior is shown to depend on whether [beta] is smaller or larger than the critical value [beta]*[approximate]1.85722, which confirms the recent result of Gamarnik and Goldberg (2008) [8].

Suggested Citation

  • van Leeuwaarden, Johan S.H. & Knessl, Charles, 2011. "Transient behavior of the Halfin-Whitt diffusion," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1524-1545, July.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:7:p:1524-1545
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    References listed on IDEAS

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    1. Shlomo Halfin & Ward Whitt, 1981. "Heavy-Traffic Limits for Queues with Many Exponential Servers," Operations Research, INFORMS, vol. 29(3), pages 567-588, June.
    2. Sem Borst & Avi Mandelbaum & Martin I. Reiman, 2004. "Dimensioning Large Call Centers," Operations Research, INFORMS, vol. 52(1), pages 17-34, February.
    3. Noah Gans & Ger Koole & Avishai Mandelbaum, 2003. "Telephone Call Centers: Tutorial, Review, and Research Prospects," Manufacturing & Service Operations Management, INFORMS, vol. 5(2), pages 79-141, September.
    4. O. Garnet & A. Mandelbaum & M. Reiman, 2002. "Designing a Call Center with Impatient Customers," Manufacturing & Service Operations Management, INFORMS, vol. 4(3), pages 208-227, October.
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    Cited by:

    1. Josh Reed & Bo Zhang, 2017. "Managing capacity and inventory jointly for multi-server make-to-stock queues," Queueing Systems: Theory and Applications, Springer, vol. 86(1), pages 61-94, June.

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