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Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane

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  • Ioannis Dimitriou

    (University of Patras)

Abstract

In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series approximation method, and the theory of Riemann boundary value problems. We also obtain explicit expressions for the first moments of the stationary distribution under some symmetry assumptions. An application in the modelling of a priority retrial system with coupled orbit queues is also presented. Using a queueing network example, we numerically validated the theoretical findings.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:310:y:2022:i:2:d:10.1007_s10479-020-03676-8
    DOI: 10.1007/s10479-020-03676-8
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    References listed on IDEAS

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    1. D. Anderson & J. Blom & M. Mandjes & H. Thorsdottir & K. Turck, 2016. "A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 153-168, March.
    2. 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.
    3. 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.
    4. Blom, Joke & De Turck, Koen & Mandjes, Michel, 2017. "Refined large deviations asymptotics for Markov-modulated infinite-server systems," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1036-1044.
    5. 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.
    6. Sigrún Andradóttir & Hayriye Ayhan & Douglas G. Down, 2001. "Server Assignment Policies for Maximizing the Steady-State Throughput of Finite Queueing Systems," Management Science, INFORMS, vol. 47(10), pages 1421-1439, October.
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    Cited by:

    1. Xianyue Shi & Liwei Liu, 2023. "Equilibrium Joining Strategies in the Retrial Queue with Two Classes of Customers and Delayed Vacations," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-27, June.

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