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On a processor sharing queue that models balking

Author

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  • Qiang Zhen
  • Johan Leeuwaarden
  • Charles Knessl

Abstract

We consider the processor sharing M/M/1-PS queue which also models balking. A customer that arrives and sees n others in the system “balks” (i.e., decides not to enter) with probability 1−b n . If b n is inversely proportional to n + 1, we obtain explicit expressions for a tagged customer’s sojourn time distribution. We consider both the conditional distribution, conditioned on the number of other customers present when the tagged customer arrives, as well as the unconditional distribution. We then evaluate the results in various asymptotic limits. These include large time (tail behavior) and/or large n, lightly loaded systems where the arrival rate λ → 0, and heavily loaded systems where λ → ∞. We find that the asymptotic structure for the problem with balking is much different from the standard M/M/1-PS queue. We also discuss a perturbation method for deriving the asymptotics, which should apply to more general balking functions. Copyright The Author(s) 2010

Suggested Citation

  • Qiang Zhen & Johan Leeuwaarden & Charles Knessl, 2010. "On a processor sharing queue that models balking," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(3), pages 453-476, December.
    • Handle: RePEc:spr:mathme:v:72:y:2010:i:3:p:453-476
    DOI: 10.1007/s00186-010-0328-z
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    References listed on IDEAS

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    1. H. Christian Gromoll & Philippe Robert & Bert Zwart, 2008. "Fluid Limits for Processor-Sharing Queues with Impatience," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 375-402, May.
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