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Necessary conditions for the compensation approach for a random walk in the quarter-plane

Author

Listed:
  • Yanting Chen

    (Hunan University
    University of Twente)

  • Richard J. Boucherie

    (University of Twente)

  • Jasper Goseling

    (University of Twente)

Abstract

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:queues:v:94:y:2020:i:3:d:10.1007_s11134-019-09622-1
    DOI: 10.1007/s11134-019-09622-1
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    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.
    Full references (including those not matched with items on IDEAS)

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