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Birth and death processes in interactive random environments

Author

Listed:
  • Guodong Pang

    (Rice University)

  • Andrey Sarantsev

    (University of Nevada)

  • Yuri Suhov

    (University of Cambridge
    Pennsylvania State University)

Abstract

This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results, a coupling method turns out to be useful.

Suggested Citation

  • Guodong Pang & Andrey Sarantsev & Yuri Suhov, 2022. "Birth and death processes in interactive random environments," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 269-307, October.
    • Handle: RePEc:spr:queues:v:102:y:2022:i:1:d:10.1007_s11134-022-09855-7
    DOI: 10.1007/s11134-022-09855-7
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    References listed on IDEAS

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    1. Zeifman, A.I., 1995. "Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 157-173, September.
    2. Sarantsev, Andrey, 2021. "Sub-exponential rate of convergence to equilibrium for processes on the half-line," Statistics & Probability Letters, Elsevier, vol. 175(C).
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