Explicit solution for the stationary distribution of a discrete-time finite buffer queue

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July 2016, 12(3): 1121-1133. doi: 10.3934/jimo.2016.12.1121

Explicit solution for the stationary distribution of a discrete-time finite buffer queue

Bara Kim ^1, and Jeongsim Kim ^2,

1.	Department of Mathematics, Korea University, 145, Anam-ro, Seongbuk-gu, Seoul, 02841, South Korea
2.	Department of Mathematics Education, Chungbuk National University, 1, Chungdae-ro, Seowon-gu, Cheongju, Chungbuk, 28644, South Korea

Received October 2013 Revised February 2015 Published September 2015

Abstract
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We consider a discrete-time single server queue with finite buffer. The customers arrive according to a discrete-time batch Markovian arrival process with geometrically distributed batch sizes and the service time is one time slot. For this queueing system, we obtain an exact closed-form expression for the stationary queue length distribution. The expression is in a form of mixed matrix-geometric solution.

Keywords: Discrete-time queue, stationary distribution, quadratic matrix equation, mixed matrix-geometric solution., discrete-time batch markovian arrival process.

Mathematics Subject Classification: Primary: 60K25; Secondary: 60J1.

Citation: Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121

References:

[1]	N. Akar, N. C. Oǧuz and K. Sohraby, Matrix-geometric solutions of M/G/1-type Markov chains: A unifying generalized state-space approach, IEEE Journal on Selected Areas in Communications, 16 (1998), 626-639. doi: 10.1109/49.700901.
[2]	C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belgian J. Oper. Res. Statist. Comput. Sci., 32 (1993), 3-23.
[3]	C. Blondia and O. Casals, Performance analysis of statistical multiplexing of VBR sources, Proc. IEEE INFOCOM, Vol. 2, Florence, Italy, 1992, 828-838. doi: 10.1109/INFCOM.1992.263492.
[4]	C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach, Performance Evaluation, 16 (1992), 5-20. doi: 10.1016/0166-5316(92)90064-N.
[5]	M. L. Chaudhry and U. C. Gupta, Queue length distributions at various epochs in discrete-time D-MAP/G/1/N queue and their numerical evaluations, Information and Management Science, 14 (2003), 67-83.
[6]	C. Herrmann, The complete analysis of the discrete time finite DBMAP/G/1/N queue, Performance Evaluation, 43 (2001), 95-121. doi: 10.1016/S0166-5316(00)00037-7.
[7]	G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM series on Statistics and Applied Probability, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9780898719734.
[8]	D. Moltchanov, Y. Koucheryavy and J. Harju, Non-preemptive $\sum$$_i D$-$BMAP_i$/D/1/Kqueuing system modeling the frame transmission process over wireless channels, in 19th International Teletraffic Congress (ITC19): Performance Challenges for Efficient Next Generation Networks, Beijing, China, 2005, 1335-1344.
[9]	M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981.
[10]	M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, 1989.
[11]	J. A. Silvester, N. L. S. Fonseca and S. S. Wang, D-BMAP models for performance evaluation of ATM networks, in Performance Modelling and Evaluation of ATM Networks, 1995, 325-346. doi: 10.1007/978-0-387-34881-0_17.
[12]	S. S. Wang and J. A. Silvester, A discrete-time performance model for integrated services in ATM multiplexers, in Proc. IEEE GLOBECOM, Houston, Vol. 2, IEEE, 1993, 757-761. doi: 10.1109/GLOCOM.1993.318182.
[13]	J.-A. Zhao, B. Li, C.-W. Kok and I. Ahmad, MPEG-4 video transmission over wireless networks: A link level performance study, Wireless Networks, 10 (2004), 133-146. doi: 10.1023/B:WINE.0000013078.74259.13.

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References:

[1]	N. Akar, N. C. Oǧuz and K. Sohraby, Matrix-geometric solutions of M/G/1-type Markov chains: A unifying generalized state-space approach, IEEE Journal on Selected Areas in Communications, 16 (1998), 626-639. doi: 10.1109/49.700901.
[2]	C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belgian J. Oper. Res. Statist. Comput. Sci., 32 (1993), 3-23.
[3]	C. Blondia and O. Casals, Performance analysis of statistical multiplexing of VBR sources, Proc. IEEE INFOCOM, Vol. 2, Florence, Italy, 1992, 828-838. doi: 10.1109/INFCOM.1992.263492.
[4]	C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach, Performance Evaluation, 16 (1992), 5-20. doi: 10.1016/0166-5316(92)90064-N.
[5]	M. L. Chaudhry and U. C. Gupta, Queue length distributions at various epochs in discrete-time D-MAP/G/1/N queue and their numerical evaluations, Information and Management Science, 14 (2003), 67-83.
[6]	C. Herrmann, The complete analysis of the discrete time finite DBMAP/G/1/N queue, Performance Evaluation, 43 (2001), 95-121. doi: 10.1016/S0166-5316(00)00037-7.
[7]	G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM series on Statistics and Applied Probability, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9780898719734.
[8]	D. Moltchanov, Y. Koucheryavy and J. Harju, Non-preemptive $\sum$$_i D$-$BMAP_i$/D/1/Kqueuing system modeling the frame transmission process over wireless channels, in 19th International Teletraffic Congress (ITC19): Performance Challenges for Efficient Next Generation Networks, Beijing, China, 2005, 1335-1344.
[9]	M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981.
[10]	M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, 1989.
[11]	J. A. Silvester, N. L. S. Fonseca and S. S. Wang, D-BMAP models for performance evaluation of ATM networks, in Performance Modelling and Evaluation of ATM Networks, 1995, 325-346. doi: 10.1007/978-0-387-34881-0_17.
[12]	S. S. Wang and J. A. Silvester, A discrete-time performance model for integrated services in ATM multiplexers, in Proc. IEEE GLOBECOM, Houston, Vol. 2, IEEE, 1993, 757-761. doi: 10.1109/GLOCOM.1993.318182.
[13]	J.-A. Zhao, B. Li, C.-W. Kok and I. Ahmad, MPEG-4 video transmission over wireless networks: A link level performance study, Wireless Networks, 10 (2004), 133-146. doi: 10.1023/B:WINE.0000013078.74259.13.

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