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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

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 & 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.  doi: 10.1109/49.700901.  Google Scholar

[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.   Google Scholar

[3]

C. Blondia and O. Casals, Performance analysis of statistical multiplexing of VBR sources,, Proc. IEEE INFOCOM, (1992), 828.  doi: 10.1109/INFCOM.1992.263492.  Google Scholar

[4]

C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach,, Performance Evaluation, 16 (1992), 5.  doi: 10.1016/0166-5316(92)90064-N.  Google Scholar

[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.   Google Scholar

[6]

C. Herrmann, The complete analysis of the discrete time finite DBMAP/G/1/N queue,, Performance Evaluation, 43 (2001), 95.  doi: 10.1016/S0166-5316(00)00037-7.  Google Scholar

[7]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling,, ASA-SIAM series on Statistics and Applied Probability, (1999).  doi: 10.1137/1.9780898719734.  Google Scholar

[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, (2005), 1335.   Google Scholar

[9]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach,, The Johns Hopkins University Press, (1981).   Google Scholar

[10]

M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications,, Marcel Dekker, (1989).   Google Scholar

[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.  doi: 10.1007/978-0-387-34881-0_17.  Google Scholar

[12]

S. S. Wang and J. A. Silvester, A discrete-time performance model for integrated services in ATM multiplexers,, in Proc. IEEE GLOBECOM, (1993), 757.  doi: 10.1109/GLOCOM.1993.318182.  Google Scholar

[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.  doi: 10.1023/B:WINE.0000013078.74259.13.  Google Scholar

show all references

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.  doi: 10.1109/49.700901.  Google Scholar

[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.   Google Scholar

[3]

C. Blondia and O. Casals, Performance analysis of statistical multiplexing of VBR sources,, Proc. IEEE INFOCOM, (1992), 828.  doi: 10.1109/INFCOM.1992.263492.  Google Scholar

[4]

C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach,, Performance Evaluation, 16 (1992), 5.  doi: 10.1016/0166-5316(92)90064-N.  Google Scholar

[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.   Google Scholar

[6]

C. Herrmann, The complete analysis of the discrete time finite DBMAP/G/1/N queue,, Performance Evaluation, 43 (2001), 95.  doi: 10.1016/S0166-5316(00)00037-7.  Google Scholar

[7]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling,, ASA-SIAM series on Statistics and Applied Probability, (1999).  doi: 10.1137/1.9780898719734.  Google Scholar

[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, (2005), 1335.   Google Scholar

[9]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach,, The Johns Hopkins University Press, (1981).   Google Scholar

[10]

M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications,, Marcel Dekker, (1989).   Google Scholar

[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.  doi: 10.1007/978-0-387-34881-0_17.  Google Scholar

[12]

S. S. Wang and J. A. Silvester, A discrete-time performance model for integrated services in ATM multiplexers,, in Proc. IEEE GLOBECOM, (1993), 757.  doi: 10.1109/GLOCOM.1993.318182.  Google Scholar

[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.  doi: 10.1023/B:WINE.0000013078.74259.13.  Google Scholar

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