Abstract
In this paper, we show that the discrete GI/G/1 system with Bernoulli retrials can be analyzed as a level-dependent QBD process with infinite blocks; these blocks are finite when both the inter-arrival and service times have finite supports. The resulting QBD has a special structure which makes it convenient to analyze by the Matrix-analytic method (MAM). By representing both the inter-arrival and service times using a Markov chain based approach we are able to use the tools for phase type distributions in our model. Secondly, the resulting phase type distributions have additional structures which we exploit in the development of the algorithmic approach. The final working model approximates the level-dependent Markov chain with a level independent Markov chain that has a large set of boundaries. This allows us to use the modified matrix-geometric method to analyze the problem. A key task is selecting the level at which this level independence should begin. A procedure for this selection process is presented and then the distribution of the number of jobs in the orbit is obtained. Numerical examples are presented to demonstrate how this method works.
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References
Alfa, A.S. (2004). “Markov Chain Representations of Discrete Distributions Applied to Queueing models.” Computers & Operations Research 31, 2365–2385.
Artalejo, J.R. (1999). “Accessible Bibliography on Retrial Queues.” Mathematical & Computer Modelling 30, 1–6.
Choi, B.D. and J.W. Kim. (1997). “Discrete-Time Geo1, Geo2/G/1 Retrial Queueing System with Two Types of Calls.” Computers & Mathematics with Applications 33, 79–88.
Dafermos, S. and M.F. Neuts. (1971). “A Single Server Queue in Discrete Time.” Cahiers du Centre Recherche Opérationelle 13, 23–40.
Diamond, J.E. and A.S. Alfa. (1998). “The MAP/PH/1 Retrial Queue.” Comm. Statist. Stochastic Models 14, 1151–1177.
Falin, G.I. (1983). “Calculation of Probabilistic Characteristics of a Multi-Channel Queue with Repeated Calls.” Vestnik Mosk. Univ. Ser. 15, Vychisl. Mat. Kibernet 3, 66–69.
Falin, G.I. (1990). “A Survey on Retrial Queues.” Queueing Systems—Theory and Applications 7, 127–168.
Falin, G.I. and Templeton. (1997). Retrial Queues, Chapman & Hall, London.
Greenberg, B.S. and R.W. Wolff. (1987). “An Upper Bound on the Performance of Queues with Returning Customers.” Journal of Applied Probability 24, 466–475.
Kulkarni, V.G. and Liang, H.M. (1997). “Retrial Queues Revisited.” In J.H. Dshalalow (ed.), Frontiers in Queueing, CRC Press, pp. 19–34.
Latouche, G. (1987). “A Note on Two Matrices Occurring in the Solution of Quasi-Birth-and-Death Processes.” Comm. Statist. Stochastic Models 3, 251–257.
Latouche, G. and V. Ramaswami. (1999). Introduction to Matrix Analytic Methods in Stochastic Modelling, ASA SIAM Series on Statistics and Applied Probability, ASA, Alexandria, Virginia.
Li, H. and T. Yang. (1998). “Geo/G/1 Discrete-Time Retrial Queue with Bernoulli Schedule.” European Journal of Operational Research 111, 629–649.
Li, H. and T. Yang. (1999). “Steady-State Queue Size Distribution of Discrete-Time PH/Geo/1 Retrial Queue.” Mathematical & Computer Modelling 30, 51–63.
Neuts, M.F. and B.M. Rao. (1990). “Numerical Investigation of a Multi-Server Retrial Model.” Queueing Systems—Theory and Applications 7, 169–190.
Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models, The Johns Hopkins University Press.
Neuts, M.F. (1989). Structured Stochastic Matrices of the M/G/1 Type and Their Applications, Marcel Dekker Inc., New York.
Ramaswami, V. and P.G. Taylor. (1996). “Some Properties of the Rate Operators in Level Dependent Quasi-Birth-and-Death Processes with a Countable Number of Phases.” Comm. Statist. Stochastic Models 12, 143–164.
Stephanov, S. (1988). “Optimal Calculation of Characteristics of Models with Repeated Calls.” In M. Bonatti (ed.), Teletraffic Science—Proceedings of the 12th ITC, Torino, North-Holland Studies in Telecommunication, Vol. 12, pp. 1427–1433.
Yang, T. and H. Li. (1995). “On the Steady-State Queue Size Distribution of the Discrete-Time Geo/G/1 Queue with Repeated Customers.” Queueing Systems—Theory and Applications 21, 199–215.
Yang, T. and J.G.C. Templeton. (1987). “A Survey on Retrial Queues.” Queueing Systems—Theory and Applications 2, 201–233.
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Alfa, A.S. Discrete-time analysis of the GI/G/1 system with Bernoulli retrials: An algorithmic approach. Ann Oper Res 141, 51–66 (2006). https://doi.org/10.1007/s10479-006-5293-9
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DOI: https://doi.org/10.1007/s10479-006-5293-9