Abstract
We consider a BMAP/G/1 type queueing model with gated service and duration of vacations depending on how many times in turn the system was empty at the previous vacation completion moments. We compute stationary distributions of the queue length at the embedded moments (vacation completions) and at arbitrary time as well as of a customer waiting time. The results of our analysis can be useful for determining strategy of adaptive choosing duration of sleep periods, e.g., in mobile networks where power consumption is an important issue.
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Abbas, C. B., Gonzlez, R., Cardenas, N., & Villalba, L. J. G. (2008). A proposal of a wireless sensor network routing protocol. Telecommunication Systems, 38, 61–68.
Baek, S., Son, J. J., & Choi, B. D. (2011). Performance analysis of Push-To-Talk over IEEE 802.16e with sleep mode and idle mode. Telecommunication Systems, 47(3–4), 291–302.
Chakravarthy, S. R. (2001). The batch Markovian arrival process: A review and future work. In A. Krishnamoorthy, et al. (Eds.), Advances in probability theory and stochastic process: Proc (pp. 21–49). Branchburg, NJ: Notable Publications.
Cinlar, E. (1975). Introduction to stochastic process. Englewood Cliffs, NJ: Prentice-Hall.
Heyman, D. P., & Lucantoni, D. (2003). Modelling multiple IP traffic streams with rate limits. IEEE/ACM Transactions on Networking, 11, 948–958.
Hurni, P., Braun, T., & Anwander, M. (2010). Evaluation of WiseMAC and extensions on wireless sensor nodes. Telecommunication Systems, 43, 49–58.
Hwang, E., Kim, K. J., Lyakhov, A., & Choi, B. D. (2012). Performance analysis of bandwidth requests under unicast, multicast and broadcast pollings in IEEE 802.16d/e. Telecommunication Systems, 50(1), 15–30.
Kesten, H., & Runnenburg, J Th. (1956). Priority in waiting line problems. Amsterdam: Mathematisch Centrum.
Khan, B. M., & Ali, F. H. (2013). Collision Free Mobility Adaptive (CFMA) MAC for wireless sensor networks. Telecommunication Systems, 52(4), 2459–2474.
Klemm, A., Lindermann, C., & Lohmann, M. (2003). Modelling IP traffic using the batch Markovian arrival process. Performance Evaluation, 54, 149–173.
Lucantoni, D. M. (1991). New results on the single server queue with a batch Markovian arrival process. Communications in Statistics-Stochastic Models, 7, 1–46.
Lucantoni, D. M., & Neuts, M. F. (1994). Some steady-state distribution for the MAP/SM/1 queue. Communications in Statistics-Stochastic Models, 10, 575–598.
Mann, C. R., Baldwin, R. O., Kharoufeh, J. P., & Mullins, B. E. (2007). A trajectory-based selective broadcast query protocol for large-scale, high-density wireless sensor networks. Telecommunication Systems, 35, 67–86.
Neuts, M. F. (1989). Structured stochastic matrices of M/G/1 type and their applications. New York: Marcel Dekker.
Saffer, Z., & Telek, M. (2010). Analysis of BMAP vacation queue and its application to IEEE 802.16e sleep mode. Journal of Industrial and Management Optimization, 6(3), 661–690.
Takagi, H. (1991). Queueing analysis: A foundation of performance evaluation (Vol. 1). New York, NY: North-Holland.
Turck, K. D., Vuyst, S. D., Fiems, D., & Wittevrongel, S. (2008). Performance analysis of the IEEE 802.16e sleep mode for correlated downlink traffic. Telecommunication Systems, 39, 145–156.
van Danzig, D. (1955). Chaines de Markof dans les ensembles abstraits et applications aux processus avec regions absorbantes et au probleme des boucles. Ann. de l’Inst. H. Poincare, 14(fasc. 3), 145–199.
Vishnevskii, V. M., Lakontsev, D. V., Semenova, O. V., & Shpilev, S. A. (2006). Polling model for investigation of the broadband wireless networks. Automation and Remote Control, 67, 1974–1995.
Vishnevsky, V., Dudin, A., Klimenok, V., Semenova, O., & Shpilev, S. (2012). Approximate method to study M/G/1-type polling system with adaptive polling mechanism. Quality Technology & Quantitative Management, 9, 211–228.
Vishnevsky, V. M., Dudin, A. N., Semenova, O. V., & Klimenok, V. I. (2011). Performance analysis of the BMAP/G/1 queue with gated servicing and adaptive vacations. Performance Evaluation, 68, 446–462.
Vishnevsky, V., Semenova, O. Dudin, A., Klimenok, V. (2009). Queueing model with gated service and adaptive vacations. In IEEE International Conference on Communications Workshops, ICC-2009 14–18 June, Dresden, Germany.
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The research is supported by the Russian Foundation for Basic Research (grant no. 14-07-90015) and the Belarusian Republican Foundation for Fundamental Research (grant no. F14R-126).
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Dudin, A.N., Vishnevsky, V.M. & Sinjugina, J.V. Analysis of the BMAP/G/1 queue with gated service and adaptive vacations duration. Telecommun Syst 61, 403–415 (2016). https://doi.org/10.1007/s11235-014-9946-8
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DOI: https://doi.org/10.1007/s11235-014-9946-8