Queueing Systems

, Volume 53, Issue 1–2, pp 31–51

Sojourn time asymptotics in processor-sharing queues

  • Sem Borst
  • Rudesindo Núñez-Queija
  • Bert Zwart
Article

Abstract

Over the past few decades, the Processor-Sharing (PS) discipline has attracted a great deal of attention in the queueing literature. While the PS paradigm emerged in the sixties as an idealization of round-robin scheduling in time-shared computer systems, it has recently captured renewed interest as a useful concept for modeling the flow-level performance of bandwidth-sharing protocols in communication networks. In contrast to the simple geometric queue length distribution, the sojourn time lacks such a nice closed-form characterization, even for exponential service requirements. In case of heavy-tailed service requirements however, there exists a simple asymptotic equivalence between the sojourn time and the service requirement distribution, which is commonly referred to as a reduced service rate approximation. In the present survey paper, we give an overview of several methods that have been developed to obtain such an asymptotic equivalence under various distributional assumptions. We outline the differences and similarities between the various approaches, discuss some connections, and present necessary and sufficient conditions for an asymptotic equivalence to hold. We also consider the generalization of the reduced service rate approximation to several extensions of the M/G/1 PS queue. In addition, we identify a relationship between the reduced service rate approximation and a queue length distribution with a geometrically decaying tail, and extend it to so-called bandwidth-sharing networks. The state-of-the-art with regard to sojourn time asymptotics in PS queues with light-tailed service requirements is also briefly described. Last, we reflect on some possible avenues for further research.

Keywords

Processor sharing Bandwidth-sharing networks Large deviations Tail asymptotics Reduced service rate approximation Heavy-tailed distributions Light-tailed distributions 

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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Sem Borst
    • 1
    • 2
    • 3
  • Rudesindo Núñez-Queija
    • 1
    • 2
  • Bert Zwart
    • 1
    • 2
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Department of Mathematics & Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.Bell Laboratories, Lucent TechnologiesMurray HillUSA

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