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Decay property of a class of d-dimensional Markov processes

Yanyun Li

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 $λ_{\mathcal{C}}$ is obtained by using a new method. It is proved that the process is $λ_{\mathcal{C}}$-transient. The corresponding $λ_{\mathcal{C}}$-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.

Decay property of a class of d-dimensional Markov processes

Abstract

In this paper, we consider the decay property of a special class of -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 is obtained by using a new method. It is proved that the process is -transient. The corresponding -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.
Paper Structure (1 section, 10 theorems, 31 equations)

This paper contains 1 section, 10 theorems, 31 equations.

Table of Contents

  1. Acknowledgement

Key Result

Lemma 2.1

Let $Q=(q_{\textbf{ij}}:\textbf{i},\textbf{j}\in \mathbf{Z}_+^d)$ be the $q$-matrix given in $(eq1.2)$, $P(t)=(p_{\textbf{ij}}(t):\textbf{i},\textbf{j}\in \mathbf{Z}_+^d)$ and $\Phi(\lambda)=(\phi_{\textbf{ij}}(\lambda):\textbf{i},\textbf{j}\in \mathbf{Z}_+^d)$ be the $Q$-function and $Q$-resolvent or in resolvent version where $F_{\textbf{i}}(t,\textbf{x})=\sum\limits_{\textbf{j}\in \mathcal{C}

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 4 more