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On partially homogeneous nearest-neighbour random walks in the quarter plane and their application in the analysis of two-dimensional queues with limited state-dependency

Author

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  • Ioannis Dimitriou

    (University of Patras)

Abstract

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such random walks are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that stationary behaviour is investigated by solving a finite system of linear equations, two matrix functional equations, and a functional equation with the aid of the theory of Riemann (–Hilbert) boundary value problems. This work is strongly motivated by emerging applications in flow level performance of wireless networks that give rise in queueing models with scalable service capacity, as well as in queue-based random access protocols, where the network’s parameters are functions of the queue lengths. A simple numerical illustration, along with some details on the numerical implementation are also presented.

Suggested Citation

  • Ioannis Dimitriou, 2021. "On partially homogeneous nearest-neighbour random walks in the quarter plane and their application in the analysis of two-dimensional queues with limited state-dependency," Queueing Systems: Theory and Applications, Springer, vol. 98(1), pages 95-143, June.
    • Handle: RePEc:spr:queues:v:98:y:2021:i:1:d:10.1007_s11134-021-09705-y
    DOI: 10.1007/s11134-021-09705-y
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    References listed on IDEAS

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    1. Rama Cont & Adrien de Larrard, 2013. "Price Dynamics in a Markovian Limit Order Market," Post-Print hal-00552252, HAL.
    2. Seva Shneer & Alexander Stolyar, 2020. "Stability conditions for a decentralised medium access algorithm: single- and multi-hop networks," Queueing Systems: Theory and Applications, Springer, vol. 94(1), pages 109-128, February.
    3. Choi, Doo Il & Kim, Tae-Sung & Lee, Sangmin, 2007. "Analysis of a queueing system with a general service scheduling function, with applications to telecommunication network traffic control," European Journal of Operational Research, Elsevier, vol. 178(2), pages 463-471, April.
    4. Ivo J. B. F. Adan & Onno J. Boxma & Stella Kapodistria & Vidyadhar G. Kulkarni, 2016. "The shorter queue polling model," Annals of Operations Research, Springer, vol. 241(1), pages 167-200, June.
    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. J. W. Cohen, 1998. "Analysis of the asymmetrical shortest two-server queueing model," International Journal of Stochastic Analysis, Hindawi, vol. 11, pages 1-48, January.
    7. Boxma, O. J. & Down, D. G., 1997. "Dynamic server assignment in a two-queue model," European Journal of Operational Research, Elsevier, vol. 103(3), pages 595-609, December.
    8. 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.
    9. H. Christian Gromoll & Philippe Robert & Bert Zwart, 2008. "Fluid Limits for Processor-Sharing Queues with Impatience," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 375-402, May.
    10. Borst, S.C. & den Hollander, F. & Nardi, F.R. & Sfragara, M., 2020. "Transition time asymptotics of queue-based activation protocols in random-access networks," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7483-7517.
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