Birth and death processes in interactive random environments

Guodong Pang (), Andrey Sarantsev () and Yuri Suhov ()
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Guodong Pang: Rice University
Andrey Sarantsev: University of Nevada
Yuri Suhov: University of Cambridge

Queueing Systems: Theory and Applications, 2022, vol. 102, issue 1, No 12, 269-307

Abstract: 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.

Keywords: Birth–death processes; Interactive random environment; Markov jump process (jump)Diffusion process; Invariant measures; Product-form formula; Exponential/polynomial convergence rate to stationarity; 60H10; 60J60; 60K25; 90B22 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11134-022-09855-7

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