Abstract
This paper deals with the analysis of a finite buffer queueing system where customers are arriving according to the batch Markovian arrival process (BMAP). The service time is considered to be generally distributed and is dependent on the queue length at service initiation epoch. The stationary queue length distribution at various epoch is obtained using the embedded Markov chain technique and the supplementary variable technique. A computational procedure has been discussed by considering phase-type service time distribution. Finally, some numerical results are given to show the numerical compatibility of the analytical results. Also a comparative study is carried out to establish the fact that our model may help in optimizing system performance by controlling the service rate depending on the state of the system.
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References
Lucantoni, D.M., Meier-Hellstern, K.S., Neuts, M.F.: A single-server queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Probab. 22(3), 676–705 (1990)
Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Stoch. Models 7(1), 1–46 (1991)
Dudin, A., Klimenok, V.: Queueing system \(BMAP/G/1\) with repeated calls. Math. Comput. Modell. 30(34), 115–128 (1999)
Lee, H.W., Park, N.I., Jeon, J.: A new approach to the queue length and waiting time of \(BMAP/G/1\) queues. Comput. Oper. Res. 30(13), 2021–2045 (2003)
Shin, Y.W.: \(BMAP/G/1\) queue with correlated arrivals of customers and disasters. Oper. Res. Lett. 32(4), 364–373 (2004)
Banik, A., Gupta, U., Pathak, S.: \(BMAP/G/1/N\) queue with vacations and limited service discipline. Appl. Math. Comput. 180(2), 707–721 (2006)
Banik, A.D.: Queueing analysis and optimal control of \(BMAP/G^{(a, b)}/1/N\) and \(BMAP/MSP^{(a, b)}/1/N\) systems. Comput. Ind. Eng. 57(3), 748–761 (2009)
Saffer, Z., Telek, M.: Analysis of \(BMAP\) vacation queue and its application to IEEE 802.16e sleep mode. J. Ind. Manag. Optim. 6(3), 661–690 (2010)
Baek, J.W., Lee, H.W., Lee, S.W., Ahn, S.: A workload factorization for \(BMAP/G/1\) vacation queues under variable service speed. Oper. Res. Lett. 42(1), 58–63 (2014)
Sikdar, K., Samanta, S.K.: Analysis of a finite buffer variable batch service queue with batch markovian arrival process and server’s vacation. Opsearch 53(3), 553–583 (2016)
Choi, B.D., Choi, D.I.: Queueing system with queue length dependent service times and its application to cell discarding in ATM networks. J. Appl. Math. Stoch. Anal. 12(1), 35–62 (1999)
Choi, D.I., Knessl, C., Tier, C.: A queueing system with queue length dependent service times, with applications to cell discarding in ATM networks. J. Appl. Math. Stoch. Anal. 12(1), 35–62 (1999)
Banerjee, A.: Analysis of finite buffer queue with state dependent service and correlated customer arrivals. J. Egypt. Math. Soc. 24, 295–302 (2015)
Neuts, M.F., Li, J.-M.: An algorithm for the \(P(n, t)\) matrices of a continuous BMAP. In: Chakravarthy, S.R., Alfa, A.S. (eds.) Matrix-Analytic Methods in Stochastic Models. Marcel Dekker (1996)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore (1981)
Banerjee, A., Gupta, U.C., Chakravarty, S.R.: Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service. Comput. Oper. Res. 60, 138–149 (2015)
Acknowledgements
The authors thank the anonymous referee for their valuable comments. The second author acknowledges the Department of Science and Technology (DST), Govt. of India, for the partial financial support under the project grant \(SB/FTP/MS-048/2013\).
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Banerjee, A., Sikdar, K., Gupta, G.K. (2018). On Finite Buffer BMAP/G/1 Queue with Queue Length Dependent Service. In: Kar, S., Maulik, U., Li, X. (eds) Operations Research and Optimization. FOTA 2016. Springer Proceedings in Mathematics & Statistics, vol 225. Springer, Singapore. https://doi.org/10.1007/978-981-10-7814-9_4
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DOI: https://doi.org/10.1007/978-981-10-7814-9_4
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