The Annals of Applied Probability

The waiting time distribution for the random order service $M/M/1$ queue

L. Flatto

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The $M/M/1$ queue is considered in the case in which customers are served in random order. A formula is obtained for the distribution of the waiting time w in the stationary state. The formula is used to show that $P9w > t) \sim \alpha t^{-5/6} \exp (-\beta t - \gamma t^{1/3})$ as $t \to \infty$, with the constants $\alpha, \beta$, and $\gamma$ expressed as functions of the traffic intensity $\rho$. The distribution of w for the random order discipline is compared to that of the first in, first out discipline.

Article information

Ann. Appl. Probab., Volume 7, Number 2 (1997), 382-409.

First available in Project Euclid: 14 October 2002

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 90B22: Queues and service [See also 60K25, 68M20]
Secondary: 30C20: Conformal mappings of special domains 30D20: Entire functions, general theory 44R10

$M/M/1$ queue random order service discipline waiting time distribution Little's law


Flatto, L. The waiting time distribution for the random order service $M/M/1$ queue. Ann. Appl. Probab. 7 (1997), no. 2, 382--409. doi:10.1214/aoap/1034625337.

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