# American Institute of Mathematical Sciences

• Previous Article
Some characterizations of the approximate solutions to generalized vector equilibrium problems
• JIMO Home
• This Issue
• Next Article
An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity
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

 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.
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
 [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
 [1] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [2] Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166 [3] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [4] Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375 [5] Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016 [6] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [7] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435 [8] Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012 [9] Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012 [10] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029 [11] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [12] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [13] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [14] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [15] Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269 [16] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [17] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [18] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [19] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [20] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

2019 Impact Factor: 1.366