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Dynamic, transient and stationary behavior of the M/GI/1 queue via martingales. (English) Zbl 0686.60097

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

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
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