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Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

Author

Listed:
  • Toshihisa Ozawa

    (Komazawa University)

  • Masahiro Kobayashi

    (Tokai University)

Abstract

We consider a discrete-time two-dimensional process $$\{(X_{1,n},X_{2,n})\}$$ { ( X 1 , n , X 2 , n ) } on $$\mathbb {Z}_+^2$$ Z + 2 with a supplemental process $$\{J_n\}$$ { J n } on a finite set, where the individual processes $$\{X_{1,n}\}$$ { X 1 , n } and $$\{X_{2,n}\}$$ { X 2 , n } are both skip-free. We assume that the joint process $$\{\varvec{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$$ { Y n } = { ( X 1 , n , X 2 , n , J n ) } is Markovian and that the transition probabilities of the two-dimensional process $$\{(X_{1,n},X_{2,n})\}$$ { ( X 1 , n , X 2 , n ) } are modulated depending on the state of the supplemental process $$\{J_n\}$$ { J n } . This modulation is space homogeneous except for the boundaries of $$\mathbb {Z}_+^2$$ Z + 2 . We call this process a discrete-time two-dimensional quasi-birth-and-death process. Under several conditions, we obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.

Suggested Citation

  • Toshihisa Ozawa & Masahiro Kobayashi, 2018. "Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process," Queueing Systems: Theory and Applications, Springer, vol. 90(3), pages 351-403, December.
    • Handle: RePEc:spr:queues:v:90:y:2018:i:3:d:10.1007_s11134-018-9586-x
    DOI: 10.1007/s11134-018-9586-x
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    References listed on IDEAS

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    1. Masakiyo Miyazawa, 2011. "Light tail asymptotics in multidimensional reflecting processes for queueing networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 233-299, December.
    2. Masakiyo Miyazawa, 2009. "Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 547-575, August.
    3. Masakiyo Miyazawa, 2011. "Rejoinder on: Light tail asymptotics in multidimensional reflecting processes for queueing networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 313-316, December.
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    Citations

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    Cited by:

    1. Valeriy Naumov, 2024. "A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes," Mathematics, MDPI, vol. 12(3), pages 1-15, January.
    2. Ioannis Dimitriou, 2022. "Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane," Annals of Operations Research, Springer, vol. 310(2), pages 355-387, March.
    3. Toshihisa Ozawa, 2021. "Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 125-161, February.
    4. Toshihisa Ozawa, 2022. "Tail asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 227-267, October.
    5. Yiqiang Q. Zhao, 2022. "The kernel method tail asymptotics analytic approach for stationary probabilities of two-dimensional queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 100(1), pages 95-131, February.

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