Heavy Traffic Limits for Queues with Many Deterministic Servers

  • Authors:

  • Predrag Jelenković;Avishai Mandelbaum;Petar Momčilović

  • Affiliations:

  • Department of Electrical Engineering, Columbia University, New York, NY 10027, USA predrag@ee.columbia.edu;Faculty of Industrial Engineering and Management, Technion – Israel Institute of Technology, Haifa 32000, Israel avim@tx.technion.ac.il;Department of Electrical Engineering, Columbia University, New York, NY 10027, USA petar@ee.columbia.edu

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 2004

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Abstract

Consider a sequence of stationary GI/D/N queues indexed by N↑∞, with servers' utilization 1−β/$\sqrt{N}$, β0. For such queues we show that the scaled waiting times $\sqrt{N}$WN converge to the (finite) supremum of a Gaussian random walk with drift −β. This further implies a corresponding limit for the number of customers in the system, an easily computable non-degenerate limiting delay probability in terms of Spitzer's random-walk identities, and $\sqrt{N}$ rate of convergence for the latter limit. Our asymptotic regime is important for rational dimensioning of large-scale service systems, for example telephone- or internet-based, since it achieves, simultaneously, arbitrarily high service-quality and utilization-efficiency.