## The Annals of Applied Probability

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 14 October 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1034625337

**Digital Object Identifier**

doi:10.1214/aoap/1034625337

**Mathematical Reviews number (MathSciNet)**

MR1442319

**Zentralblatt MATH identifier**

0883.60086

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1034625337