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.
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Acknowledgments
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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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Zhen, Q., van Leeuwaarden, J.S.H. & Knessl, C. On a processor sharing queue that models balking. Math Meth Oper Res 72, 453–476 (2010). https://doi.org/10.1007/s00186-010-0328-z
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DOI: https://doi.org/10.1007/s00186-010-0328-z