Artalejo, Jesús R. and Economou, A. and López Herrero, María Jesús
(2007)
*Algorithmic approximations for the busy period distribution of the M/M/c retrial queue.*
European journal of operational research, 176
(3).
pp. 1687-1702.
ISSN 0377-2217

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Official URL: http://www.sciencedirect.com/science/article/pii/S0377221705008684

## Abstract

In this paper we deal with the main multiserver retrialqueue of M/M/c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busyperioddistribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busyperiod moments. The corresponding distributions for the total number of customers served during a busyperiod are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/M/cretrialqueue.

Item Type: | Article |
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Additional Information: | The authors thank the support received from the research project MTM2005-01248. A. Economou was supported by the University of Athens grant ELKE/70/4/6415 and by the Greek Ministry of Education and European Union program PYTHAGORAS/2004. |

Uncontrolled Keywords: | Queueing; M/M/c retrial queue; Generalized truncation models; Busy period; First-step analysis;Numerical inversion; Algorithmic probability |

Subjects: | Sciences > Mathematics > Operations research |

ID Code: | 15455 |

References: | Anisimov, V.A., Artalejo, J.R., 2001. Analysis of Markov multiserver retrial queues with negative arrivals. Queueing Systems 39, 157–182. Artalejo, J.R., Falin, G., 2002. Standard and retrial queueing systems: A comparative analysis. Revista Matematica Complutense 15, 101–129. Artalejo, J.R., Lopez-Herrero, M.J., 2000. On the busy period of the M/G/1 retrial queue. Naval Research Logistics 47, 115–127. Artalejo, J.R., Pozo, M., 2002. Numerical calculation of the stationary distribution of the main multiserver retrial queue. Annals of Operations Research 116, 41–56. Artalejo, J.R., Economou, A., Lopez-Herrero, M.J., in press. Algorithmic analysis of the maximum queue length in a busy period for the M/M/c retrial queue. INFORMS Journal on Computing. Breuer, L., Dudin, A.N., Klimenok, V.I., 2002. A retrial BMAP/PH/N system. Queueing Systems 40, 433–457. Bright, L., Taylor, P.G., 1995. Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Communications in Statistics—Stochastic Models 11, 497–525. Chakravarthy, S.R., Dudin, A.N., 2002. Multiserver retrial queue with BMAP arriving and group services. Queueing Systems 42, 5–31. Choo, Q.H., Conolly, B., 1979. New results in the theory of repeated orders queueing systems. Journal of Applied Probability 16, 631– 640. Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls. Moscow University Computational Mathematics and Cybernetics 1, 43–49. Falin, G.I., Templeton, J.G.C., 1997. Retrial Queues. Monographs on Statistics and Applied Probability, vol. 75. Chapman and Hall, London. Gomez-Corral, A., 2006. A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Annals of Operations Research 141, 177–207. Latouche, G., Ramaswami, V., 1999. Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia. Lopez-Herrero, M.J., Neuts, M.F., 2002. The distribution of the maximum orbit size of anM/G/1 retrial queue during the busy period. In: Artalejo, J.R., Krishnamorthy, A. (Eds.), Advances in Stochastic Modelling. Notable Publications Inc., pp. 219–231. Neuts, M.F., 1981. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD (Reprinted: Dover, New York, 1994). Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model. Queueing Systems 7, 169–190. Shin, Y.W., Kim, Y.C., 2000. Stochastic comparisons of Markovian retrial queues. Journal of the Korean Statistical Society 29, 473–488. Tijms, H.C., 2003. A First Course in Stochastic Models. Wiley, Chichester. Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA. The Bell System Technical Journal 35, 421–514. |

Deposited On: | 01 Jun 2012 10:47 |

Last Modified: | 06 Feb 2014 10:24 |

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