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Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

Part of: Hamilton-Jacobi theories, including dynamic programming Operations research and management science Special processes Limit theorems

Published online by Cambridge University Press:  17 March 2021

Ari Arapostathis ,
Guodong Pang  and
Yi Zheng
Ari Arapostathis*
Affiliation:
The University of Texas at Austin
Guodong Pang*
Affiliation:
Pennsylvania State University
Yi Zheng*
Affiliation:
Pennsylvania State University
*
*Postal address: Department of Electrical and Computer Engineering, The University of Texas at Austin, 2501 Speedway, EERC 7.824, Austin, TX78712. Email address: [email protected]
**Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, College of Engineering, Pennsylvania State University, University Park, PA16802.
**Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, College of Engineering, Pennsylvania State University, University Park, PA16802.
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Abstract

We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.

Keywords

Markov-modulated processbirth–death processexponential ergodicitysteady-state approximationsfast regime switching

MSC classification

Primary: 60K25: Queueing theory 90B20: Traffic problems
Secondary: 90B36: Scheduling theory, stochastic 49L20: Dynamic programming method 60F17: Functional limit theorems; invariance principles
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 1 - 29
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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