On a processor sharing queue that models balking |
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Authors: | Qiang Zhen Johan S H van Leeuwaarden Charles Knessl |
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Institution: | 1. Department of Mathematics and Statistics, University of North Florida, 1 UNF DR, Bldg 14/2731, Jacksonville, FL, 32224-7699, USA 2. Department of Mathematics and Computer Science, Eindhoven University of Technology, Room HG 9.13, P.O. Box 513, 5600?MB, Eindhoven, The Netherlands 3. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL, 60607-7045, USA
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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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Keywords: | |
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