The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms

Yanting Chen; Richard J. Boucherie; Jasper Goseling

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arXiv · 1112.2890

The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms

Yanting Chen·Richard J. Boucherie·Jasper Goseling

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Abstract

We consider the invariant measure of a homogeneous continuous- time Markov process in the quarter-plane. The basic solutions of the global balance equation are the geometric distributions. We first show that the invariant measure can not be a finite linear combination of basic geometric distributions, unless it consists of a single basic geo- metric distribution. Second, we show that a countable linear combina- tion of geometric terms can be an invariant measure only if it consists of pairwise-coupled terms. As a consequence, we obtain a complete characterization of all countable linear combinations of geometric dis- tributions that may yield an invariant measure for a homogeneous continuous-time Markov process in the quarter-plane.

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Yanting Chen, Richard J. Boucherie, Jasper Goseling. 2014-02-24. The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms. https://arxiv.org/abs/1112.2890

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