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Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/ $$\infty $$ ∞ and BMAP/M/c+M queues

Author

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  • Tetsuya Takine

    (Osaka University)

Abstract

This paper considers a special class of continuous-time Markov chains of level-dependent M/G/1-type, where block matrices representing downward jumps in the infinitesimal generator are nonsingular. This special class naturally arises in the analysis of BMAP/M/ $$\infty $$ ∞ queues and BMAP/M/c queues with exponential impatience times (BMAP/M/c+M). We first formulate the boundary probability vector in terms of a solution of a system of infinitely many linear inequalities. We then reveal that in the above special class, this infinite system is regarded as a nested sequence of simplices, and we identify their vertices. Based on these results, we develop a simple yet efficient computational algorithm for the stationary distribution conditioned that the level is not greater than a predefined N. Note that for a large N, the conditional distribution will provide a good approximation to the stationary distribution. Some numerical examples for BMAP/M/ $$\infty $$ ∞ and BMAP/M/c+M queues are shown.

Suggested Citation

  • Tetsuya Takine, 2016. "Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/ $$\infty $$ ∞ and BMAP/M/c+M queues," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 49-77, October.
    • Handle: RePEc:spr:queues:v:84:y:2016:i:1:d:10.1007_s11134-016-9482-1
    DOI: 10.1007/s11134-016-9482-1
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    Cited by:

    1. Tetsuya Takine, 2022. "On level-dependent QBD processes with explosive state space," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 353-355, April.
    2. Hiroyuki Masuyama, 2022. "Subexponential asymptotics of asymptotically block-Toeplitz and upper block-Hessenberg Markov chains," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 175-217, October.
    3. Yera Mora, Yoel Gustavo & Lillo Rodríguez, Rosa Elvira & Ramírez-Cobo, Pepa, 2017. "Findings about the two-state BMMPP for modeling point processes in reliability and queueing systems," DES - Working Papers. Statistics and Econometrics. WS 24622, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Alexander Dudin & Olga Dudina & Sergei Dudin & Konstantin Samouylov, 2021. "Analysis of Multi-Server Queue with Self-Sustained Servers," Mathematics, MDPI, vol. 9(17), pages 1-18, September.
    5. Hiroyuki Masuyama, 2019. "A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 173-200, June.

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