On a processor sharing queue that models balking

نویسندگان

Qiang Zhen

Johan van Leeuwaarden

Charles Knessl

چکیده

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 − bn. If bn 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/1PS queue. We also discuss a perturbation method for deriving the asymptotics, which should apply to more general balking functions. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL 60607-7045, USA. Email: [email protected] Department of Mathematics and Computer Science, Eindhoven University of Technology, Room HG 9.13, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Email: [email protected] Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL 60607-7045, USA. Email: [email protected] Acknowledgements: Knessl was partly supported by NSF grant DMS 05-03745 and NSA grant H 98230-08-1-0102. Van Leeuwaarden was supported by a VENI grant from The Netherlands Organization for Scientific Research (NWO).

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