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arXiv · 0707.4124

Services within a busy period of an M/M/1 queue and Dyck paths

Moez Draief·Jean Mairesse
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Abstract

We analyze the service times of customers in a stable M/M/1 queue in equilibrium depending on their position in a busy period. We give the law of the service of a customer at the beginning, at the end, or in the middle of the busy period. It enables as a by-product to prove that the process of instants of beginning of services is not Poisson. We then proceed to a more precise analysis. We consider a family of polynomial generating series associated with Dyck paths of length 2n and we show that they provide the correlation function of the successive services in a busy period with (n+1) customers.

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Moez Draief, Jean Mairesse. 2007-07-27. Services within a busy period of an M/M/1 queue and Dyck paths. https://doi.org/10.1007/s11134-004-5556-6

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