On a Processor Sharing Queue That Models Balking

Details On a Processor Sharing Queue That Models Balking On a Processor Sharing Queue That Models Balking

2009-05-22

arxiv.org/abs/0905.3700v1

Math. Meth. Oper. Res. 72 (2010) 453-472

Mathematics

Probability

36 pages

Scientific paper

10.1007/s00186-010-0328-z

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 $\lambda\to 0$, and heavily loaded systems where $\lambda\to\infty$. 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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