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The kernel method tail asymptotics analytic approach for stationary probabilities of two-dimensional queueing systems

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  • Yiqiang Q. Zhao

    (Carleton University)

Abstract

Many queueing systems can be modelled as two-dimensional random walks with reflective boundaries, discrete, continuous or mixed. Stationary probabilities are one of the most sought after statistical quantities in queueing analysis. However, explicit expressions are only available for a very limited number of models. Therefore, tail asymptotic properties become more important, since they provide insightful information on the structure of the tail probabilities, and often lead to approximations, performance bounds, algorithms, among possible other applications. In this survey, we provide key ideas of a kernel method, developed from the classical kernel method in analytic combinatorics, for studying so-called exact tail asymptotic properties in stationary probabilities for this type of random walk.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:queues:v:100:y:2022:i:1:d:10.1007_s11134-021-09727-6
    DOI: 10.1007/s11134-021-09727-6
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    References listed on IDEAS

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    1. 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.
    2. Jiashan Tang & Yiqiang Zhao, 2008. "Stationary tail asymptotics of a tandem queue with feedback," Annals of Operations Research, Springer, vol. 160(1), pages 173-189, April.
    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. Wendi Li & Yuanyuan Liu & Yiqiang Q. Zhao, 2019. "Exact tail asymptotics for fluid models driven by an M/M/c queue," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 319-346, April.
    5. Yang Song & Zaiming Liu & Yiqiang Q. Zhao, 2016. "Exact tail asymptotics: revisit of a retrial queue with two input streams and two orbits," Annals of Operations Research, Springer, vol. 247(1), pages 97-120, December.
    6. Liming Liu & Masakiyo Miyazawa & Yiqiang Zhao, 2008. "Geometric decay in level-expanding QBD models," Annals of Operations Research, Springer, vol. 160(1), pages 83-98, April.
    7. 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.
    8. 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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