Gómez Corral, Antonio and Ramalhoto, M.F.
(1999)
*The stationary distribution of a Markovian process arising in the theory of multiserver retrial queueing systems.*
Mathematical and Computer Modelling, 30
.
pp. 141-158.
ISSN 0895-7177

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Official URL: http://www.sciencedirect.com/science/article/pii/S0895717799001387

## Abstract

In this paper, we introduce a bivariate Markov process {X(t), t greater than or equal to 0} = {(C(t), Q(t)), t greater than or equal to 0} whose state space is a lattice semistrip E = {0, 1, 2, 3} x Z(+). The process {X(t), t greater than or equal to 0} can be seen as the joint process of the number of servers and waiting positions occupied, and the number of customers in orbit of a generalized Markovian multiserver queue with repeated attempts and state dependent intensities. Using a simple approach, we derive closed form expressions for the stationary distribution of {X(t), t greater than or equal to 0} when a sufficient condition is satisfied. The stationary analysis of the M/M/2/2 + 1 and M/M/3/3 queues with linear retrial rates is studied as a particular case in this process.

Item Type: | Article |
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Uncontrolled Keywords: | multiserver queue; repeated attempt; stationary distribution; closed form formulae |

Subjects: | Sciences > Mathematics > Stochastic processes |

ID Code: | 15876 |

References: | T. Yang and J.G.C. Templeton, A survey on retrial queues, Queue&g Systems 2, 201-233 (1987). G.1. Falin, A survey of retrial queues, Queueing Systems 7, 127-168 (1990). V.G. Kulkarni and H.M. Liang, Retrial queues revisited, In Frontiers in Queueing. Models and Applications in Science and Engineering, (Edited by J.H. Dshalalow), pp. 19-34, CRC Press (1996). G. Fayolle, A simple telephone exchange with delayed feedbacks, In Teletnzfic Analysis and Computer Performance Evaluation, (Edited by O.J. Boxma, J.W. Cohen and H.C. Tijms), pp. 245-253, Elsevier Science, (1986). M. Martin and J.R.. ArtaIejo, Analysis of an M/G/l queue with two types of impatient units, Advances in Applied Prubability 27, 840-861 (1995). J.R. Artalejo and A. G6mez-Corral, Steady state solution of a single-server queue with linear repeated requests, Journal of Applied Probability 34, 223-233 (1997). J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecommunication Review 18, 49-100 (1957). G.L. Jonin and J.J. Sedol, Telephone systems with repeated calls, In Proceedings of the tih International Teletmfic Congress, Munich, pp. 435/l-435/5, (1970). T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts,Journal of Applied P&ability 24, 486494 (1987). C.E.M. Pearce, Extended continued fractions, recurrence relations and two-dimensional Markov processes,Advances in Applied Probability 21, 357-375 (1989). M.F. Neuts and M.F. Ramalhoto, A service model in which the server is required to search for customers,Journal of Applied Probability 21, 157-166 (1984). RI. Wilkinson, Theories for toll traffic engineering in the U.S.A., The Bell System Technic& Journal 35,421-507 (1956). G.I. Falin, On the accuracy of a numerical method of calculation of characteristics of systems with repeated calls, Elektrosvyaz 8, 35-36 (1983). B.S. Greenberg and R.W. Wolff, An upper bound on the performance of queues with returning customers,Journal of Applied Probability 24, 466-475 (1987). M.F. Neuts and B.M. Rae, Numerical investigation of a multiserver retrial model, Queueing Systems 7,169-189 (1990). S.N. Stepanov, Numerical Methods of Calculation for Systems with Repeated Calls, (in Russian) Nauka,(1983). S.N. Stepanov, Increasing the efficiency of numerical methods for models with repeated calls, Problems of Information Thnsmission 22, 313-326 (1986). G.I. Falin and J.G.C. Templeton, Retrial Queues, Chapman and Hall, (1997). G.I. Falin, Probabilistic model for investigation of load of subscriber’s lines with waiting places, In Probability Theory, Stochastic Processes and finctional Analysis, (in Russian) Moscow State University, Moscow,64-66(1985). G.I. Falin, Single-line repeated orders queueing systems, Mathematische Operationsforschung und Statistik, Optimization 17, 649467 (1986). M.F. Ramalhoto and A. G6mez-Corral, Some decomposition formulae for the M/M/T/~ + d queue with constant retrial rate, Communicntions in Statistics-Stochastic Models 14, 123-146 (1998). J.R. Artalejo, Stationary analysis of the characteristics of the M/M/2 queue with constant repeated attempts,Opsearch 33, 83-95 (1996). Y.C. Kim, On M/M/3/3 retrial queueing systems, Honam Mathematical Journal 17, 141-147 (1995). S. Asmussen, Applied Probability and Queues, John Wiley and Sons, (1987). G.I. E&n, Heavy traffic analysis of 8 random walk on a lattice semi-strip, Communications in Statistics-Stochastic Modek 11, 395-409 (1995). D. Gross and C. Harris, Fundamentals of Queueing Theory, John Wiley and Sons, (1985). G-1. FaIin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls,Advances in Applied Probability 16, 447-448 (1984). M. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The John Hopkins University Press, (1981). |

Deposited On: | 10 Jul 2012 08:01 |

Last Modified: | 06 Feb 2014 10:33 |

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