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A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains

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  • Hiroyuki Masuyama

    (Kyoto University)

Abstract

This paper proposes a new algorithm for computing the stationary distribution vector in continuous-time upper block-Hessenberg Markov chains. To this end, we consider the last-block-column-linearly-augmented (LBCL-augmented) truncation of the (infinitesimal) generator of the upper block-Hessenberg Markov chain. The LBCL-augmented truncation is a linearly augmented truncation such that the augmentation distribution has its probability mass only on the last block column. We first derive an upper bound for the total variation distance between the respective stationary distribution vectors of the original generator and its LBCL-augmented truncation. Based on the upper bound, we then establish a series of linear fractional programming (LFP) problems to obtain augmentation distribution vectors such that the bound converges to zero. Using the optimal solutions of the LFP problems, we construct a matrix-infinite-product (MIP) form of the original (i.e., not approximate) stationary distribution vector and develop a sequential update algorithm for computing the MIP form. Finally, we demonstrate the applicability of our algorithm to BMAP/M/ $$\infty $$ ∞ queues and M/M/s retrial queues.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:queues:v:92:y:2019:i:1:d:10.1007_s11134-019-09599-x
    DOI: 10.1007/s11134-019-09599-x
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    References listed on IDEAS

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    1. Yang Woo Shin, 2009. "Fundamental Matrix Of Transient Qbd Generator With Finite States And Level Dependent Transitions," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 26(05), pages 697-714.
    2. 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.
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    Cited by:

    1. 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.
    2. 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.

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