Baccelli, François; Makowski, Armand M. Dynamic, transient and stationary behavior of the M/GI/1 queue via martingales. (English) Zbl 0686.60097 Ann. Probab. 17, No. 4, 1691-1699 (1989). From the introduction: An exponential martingale is associated with the Markov chain that describes the number of customers at departure epochs in the M/GI/1 queue. Basic regularity properties of this martingale and standard arguments from renewal theory are shown to provide a unified probabilistic framework for deriving three well-known results for the M/GI/1 queue, to wit, the generating function of the number of customers served in a busy period, the Pollaczek-Khintchine formula, and the transient generating function of the number of customers at departure epochs. The main ingredient of the approach is a relationship between the law of the embedded Markov chain and the law of the forward recurrence time of a discrete-time renewal process associated with this chain. Reviewer: E.van Doorn Cited in 2 ReviewsCited in 7 Documents MSC: 60K25 Queueing theory (aspects of probability theory) 60G42 Martingales with discrete parameter 60K05 Renewal theory 60E10 Characteristic functions; other transforms 60F05 Central limit and other weak theorems 60G17 Sample path properties 60G40 Stopping times; optimal stopping problems; gambling theory 60J05 Discrete-time Markov processes on general state spaces Keywords:exponential martingale; regularity properties; Pollaczek-Khintchine formula; law of the embedded Markov; discrete-time renewal process PDF BibTeX XML Cite \textit{F. Baccelli} and \textit{A. M. Makowski}, Ann. Probab. 17, No. 4, 1691--1699 (1989; Zbl 0686.60097) Full Text: DOI OpenURL