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

We consider the invariant measure of a homogeneous continuoustime 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 geometric distribution. Second, we show that a countable linear combination 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 distributions that may yield an invariant measure for a homogeneous continuous-time Markov process in the quarter-plane. keywords: Invariant measure, Continuous-time Markov process, Quarterplane, Geometric product form, Random walk. ∗P.O. Box 217, 7500 AE Enschede, The Netherlands. Email:{Y.Chen, R.J.Boucherie, J.Goseling}@utwente.nl