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October, 1989 Dynamic, Transient and Stationary Behavior of the $M/GI/1$ Queue Via Martingales
Francois Baccelli, Armand M. Makowski
Ann. Probab. 17(4): 1691-1699 (October, 1989). DOI: 10.1214/aop/1176991182

Abstract

An exponential martingale is associated with the Markov chain of the number of customers in the $M/GI/1$ queue. With the help of arguments from renewal theory, this martingale provides a unified probabilistic framework for deriving several well-known generating functions for the $M/GI/1$ queue, such as the Pollaczek-Khintchine formula, the transient generating function of the number of customers at departure epochs and the generating function of the number of customers served in a busy period.

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Francois Baccelli. Armand M. Makowski. "Dynamic, Transient and Stationary Behavior of the $M/GI/1$ Queue Via Martingales." Ann. Probab. 17 (4) 1691 - 1699, October, 1989. https://doi.org/10.1214/aop/1176991182

Information

Published: October, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0686.60097
MathSciNet: MR1048954
Digital Object Identifier: 10.1214/aop/1176991182

Subjects:
Primary: 60E10
Secondary: 60F05 , 60G17 , 60G40 , 60G42 , 60J05 , 60K05 , 60K25

Keywords: busy period , Doob's optional sampling theorem , generating functions , Martingales , Pollaczek-Khintchine formula , Queueing theory , renewal theory

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 4 • October, 1989
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