Decay property of a class of d-dimensional Markov processes

TL;DR

This paper investigates the decay properties of a specific class of d-dimensional Markov processes, deriving the decay parameter exactly, proving transience, and providing invariant measures with an illustrative example.

Contribution

The paper introduces a new method to precisely compute the decay parameter for a class of Markov processes and establishes their transience and invariant measures.

Findings

Exact decay parameter $\lambda_{\mathcal{C}}$ obtained

Proved the process is $\lambda_{\mathcal{C}}$-transient

Presented invariant measures and an example

Abstract

In this paper, we consider the decay property of a special class of $d$-dimensional Markov processes, which can be viewed as a stopped network with the external customer being blocked to empty nodes. The exact value of the decay parameter $\lambda_{\mathcal{C}}$ is obtained by using a new method. It is proved that the process is $\lambda_{\mathcal{C}}$-transient. The corresponding $\lambda_{\mathcal{C}}$-invariant measures and quasi-distributions are also presented. Finally, an example on auto quick repair service network is presented to illustrate the results obtained in this paper.