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Stationary distributions and convergence for M/M/1 queues in interactive random environment

Author

Listed:
  • Guodong Pang

    (The Pennsylvania State University)

  • Andrey Sarantsev

    (University of Nevada in Reno)

  • Yana Belopolskaya

    (Saint Petersburg State University of Architecture and Civil Engineering)

  • Yuri Suhov

    (University of Cambridge
    The Pennsylvania State University)

Abstract

A Markovian single-server queue is studied in an interactive random environment. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depend on the queue length. We consider in detail two types of Markov random environments: a pure jump process and a reflected jump diffusion. In both cases, the joint dynamics are constructed so that the stationary distribution can be explicitly found in a simple form (weighted geometric). We also derive an explicit estimate for the exponential rate of convergence to the stationary distribution via coupling.

Suggested Citation

  • Guodong Pang & Andrey Sarantsev & Yana Belopolskaya & Yuri Suhov, 2020. "Stationary distributions and convergence for M/M/1 queues in interactive random environment," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 357-392, April.
    • Handle: RePEc:spr:queues:v:94:y:2020:i:3:d:10.1007_s11134-019-09644-9
    DOI: 10.1007/s11134-019-09644-9
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    References listed on IDEAS

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    Cited by:

    1. Sarantsev, Andrey, 2020. "Convergence rate to equilibrium in Wasserstein distance for reflected jump–diffusions," Statistics & Probability Letters, Elsevier, vol. 165(C).

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