Abstract
We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron–Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111–1118, 2007) on the M/G/1 retrial queue.
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Artalejo, J.R.: A classified bibliography of research on retrial queues: progress in 1990–1999. Top 7, 187–211 (1999)
Artalejo, J.R.: Accessible bibliography on retrial queues. Math. Comput. Model. 30, 1–6 (1999)
Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems. Springer, Berlin (2008)
Baiocchi, A., Blefari-Melazzi, N.: Analysis of the loss probability of the MAP/G/1/K queue Part II: approximations and numerical results. Commun. Stat.-Stoch. Models 10(4), 895–925 (1994)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Falin, G.I.: A survey of retrial queues. Queueing Syst. 7, 127–168 (1990)
Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman & Hall, London (1997)
Falkenberg, E.: On the asymptotic behavior of the stationary distribution of the Markov chains of M/G/1-type. Commun. Stat.-Stoch. Models 10(1), 75–97 (1994)
Kim, J., Kim, J., Kim, B.: Regularly varying tail of the waiting time distribution in M/G/1 retrial queue (2010). doi:10.1007/s11134-010-9180-3
Kim, J., Kim, B., Ko, S.-S.: Tail asymptotics for the queue size distribution in an M/G/1 retrial queue. J. Appl. Probab. 44, 1111–1118 (2007)
Kulkarni, V.G.: Modeling and Analysis of Stochastic Systems. Chapman & Hall, London (1995)
Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat.-Stoch. Models 7, 1–46 (1991)
Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Applications. Dekker, New York (1989)
Nobel, R.D., Tijms, H.C.: Waiting-time probabilities in the M/G/1 retrial queue. Stat. Neerlandica 60, 73–78 (2006)
Seneta, E.: Nonnegative Matrices and Markov Chains. Springer, Berlin (1981)
Shang, W., Liu, L., Li, Q.: Tail asymptotics for the queue length in an M/G/1 retrial queue. Queueing Syst. 52, 193–198 (2006)
Yang, T., Templeton, J.G.C.: A survey on retrial queues. Queueing Syst. 2, 201–233 (1982)
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This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment) and the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2008-314-C00031).
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Kim, B., Kim, J. & Kim, J. Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue. Queueing Syst 66, 79–94 (2010). https://doi.org/10.1007/s11134-010-9179-9
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DOI: https://doi.org/10.1007/s11134-010-9179-9