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

TL;DR

This paper characterizes the invariant measures of homogeneous continuous-time Markov processes in the quarter-plane, showing they are either single geometric distributions or countable coupled combinations, with a complete geometric characterization.

Contribution

It provides a complete geometric characterization of all invariant measures formed by countable combinations of geometric distributions in the quarter-plane.

Findings

Invariant measure cannot be a finite sum of geometric distributions unless it is a single distribution.

Countable linear combinations of geometric terms form invariant measures only if terms are pairwise-coupled.

Complete geometric characterization of invariant measures in the quarter-plane.

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.