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.
Citation
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
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