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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References
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures (Cambridge Univ. Press, Cambridge, 1998).
P. Burke, The output of a queueing system, Oper. Res. 4 (1956) 699–704.
J.W. Cohen, The Single Server Queue, 2nd ed. (North-Holland, Amsterdam, 1982).
P. Flajolet and F. Guillemin, The formal theory of birth-and-death processes, lattice path combinatorics, and continued fractions, Adv. in Appl. Probab. 32 (2000) 750–778.
R. Graham, D. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed. (Addison-Wesley, Reading, MA, 1994).
F. Guillemin and D. Pinchon, On the area swept under the occupation process of an M/M/1 queue in a busy period, Queueing Systems 29(2–4) (1998) 383–398.
E. Reich, Waiting times when queues are in tandem, Ann. Math. Statist. 28 (1957) 527–530.
P. Robert, Réseaux et files d’attente: Méthodes probabilistes, in: Mathématiques & Applications, Vol. 35 (Springer, Paris, 2000).
A. Schwartz and A. Weiss, Large Deviations for Performance Analysis. Queues, Communications, and Computing (Chapman & Hall, London, 1995).
R. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62 (Cambridge Univ. Press, Cambridge, 1999).
L. Takács, Introduction to the Theory of Queues, University Texts in the Mathematical Sciences (Oxford Univ. Press, Oxford, 1962).
L. Takács, Queueing Methods in the Theory of Random Graphs, Probability and Stochastics Series (CRC Press, Boca Raton, FL, 1995) 45–78.
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Draief, M., Mairesse, J. Services within a Busy Period of an M/M/1 Queue and Dyck Paths. Queueing Syst 49, 73–84 (2005). https://doi.org/10.1007/s11134-004-5556-6
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DOI: https://doi.org/10.1007/s11134-004-5556-6