Queueing Systems

, 58:261 | Cite as

Corrected asymptotics for a multi-server queue in the Halfin-Whitt regime

  • A. J. E. M. Janssen
  • J. S. H. van Leeuwaarden
  • B. Zwart
Open Access


To investigate the quality of heavy-traffic approximations for queues with many servers, we consider the steady-state number of waiting customers in an M/D/s queue as s→∞. In the Halfin-Whitt regime, it is well known that this random variable converges to the supremum of a Gaussian random walk. This paper develops methods that yield more accurate results in terms of series expansions and inequalities for the probability of an empty queue, and the mean and variance of the queue length distribution. This quantifies the relationship between the limiting system and the queue with a small or moderate number of servers. The main idea is to view the M/D/s queue through the prism of the Gaussian random walk: as for the standard Gaussian random walk, we provide scalable series expansions involving terms that include the Riemann zeta function.


M/D/s queue Halfin-Whitt scaling Gaussian random walk All-time maximum Riemann zeta function Lerch’s transcendent Spitzer’s identity Queues in heavy traffic Lambert W Function Corrected diffusion approximation 

Mathematics Subject Classification (2000)

11M06 30B40 60G50 60G51 65B15 


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Copyright information

© The Author(s) 2008

Authors and Affiliations

  • A. J. E. M. Janssen
    • 1
  • J. S. H. van Leeuwaarden
    • 2
  • B. Zwart
    • 3
  1. 1.Digital Signal Processing GroupPhilips ResearchEindhovenThe Netherlands
  2. 2.Mathematics and Computer Science DepartmentEindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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