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Subexponential asymptotics of asymptotically block-Toeplitz and upper block-Hessenberg Markov chains

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  • Hiroyuki Masuyama

    (Tokyo Metropolitan University)

Abstract

This paper studies the subexponential asymptotics of the stationary distribution vector of an asymptotically block-Toeplitz and upper block-Hessenberg (atUBH) Markov chain in discrete time. The atUBH Markov chain is a kind of the upper block-Hessenberg (UBH) one and is a generalization of the M/G/1-type one. The atUBH Markov chain typically arises from semi-Markovian retrial queues, as the queue-length process, its embedded process, or appropriately time-scaled versions of these processes. In this paper, we present subexponential and locally subexponential asymptotic formulas for the stationary distribution vector. We then extend the locally subexponential asymptotic formula to a continuous-time version of the atUBH Markov chain by uniformization and change of time scale. This extension expands the applicability of the locally subexponential asymptotic formula.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:queues:v:102:y:2022:i:1:d:10.1007_s11134-022-09857-5
    DOI: 10.1007/s11134-022-09857-5
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    References listed on IDEAS

    as
    1. Hiroyuki Masuyama, 2019. "Correction to: 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 201-202, June.
    2. Kim, Bara & Kim, Jeongsim, 2012. "A note on the subexponential asymptotics of the stationary distribution of M/G/1 type Markov chains," European Journal of Operational Research, Elsevier, vol. 220(1), pages 132-134.
    3. 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.
    4. Masuyama, Hiroyuki, 2011. "Subexponential asymptotics of the stationary distributions of M/G/1-type Markov chains," European Journal of Operational Research, Elsevier, vol. 213(3), pages 509-516, September.
    5. Hiroyuki Masuyama, 2016. "A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications," Annals of Operations Research, Springer, vol. 247(1), pages 65-95, December.
    6. Tetsuya Takine, 2004. "Geometric and Subexponential Asymptotics of Markov Chains of M / G /1 Type," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 624-648, August.
    7. 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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