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Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms

Author

Listed:
  • Yanting Chen

    (Hunan University
    University of Twente)

  • Richard J. Boucherie

    (University of Twente)

  • Jasper Goseling

    (University of Twente)

Abstract

We consider homogeneous random walks in the quarter-plane. The necessary conditions which characterize random walks of which the invariant measure is a sum of geometric terms are provided in Chen et al. ( arXiv:1304.3316 , 2013, Probab Eng Informational Sci 29(02):233–251, 2015). Based on these results, we first develop an algorithm to check whether the invariant measure of a given random walk is a sum of geometric terms. We also provide the explicit form of the invariant measure if it is a sum of geometric terms. Second, for random walks of which the invariant measure is not a sum of geometric terms, we provide an approximation scheme to obtain error bounds for the performance measures. Our results can be applied to the analysis of two-node queueing systems. We demonstrate this by applying our results to a tandem queue with server slow-down.

Suggested Citation

  • Yanting Chen & Richard J. Boucherie & Jasper Goseling, 2016. "Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 21-48, October.
    • Handle: RePEc:spr:queues:v:84:y:2016:i:1:d:10.1007_s11134-016-9483-0
    DOI: 10.1007/s11134-016-9483-0
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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. Nico M. Dijk, 2011. "Error Bounds and Comparison Results: The Markov Reward Approach For Queueing Networks," International Series in Operations Research & Management Science, in: Richard J. Boucherie & Nico M. Dijk (ed.), Queueing Networks, chapter 9, pages 397-459, Springer.
    4. 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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    Cited by:

    1. Yanting Chen & Richard J. Boucherie & Jasper Goseling, 2020. "Necessary conditions for the compensation approach for a random walk in the quarter-plane," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 257-277, April.

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