The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms
arXiv:1112.2890
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.
arXiv commentThe author has submitted a better version of this paper. However, the author did not know the new version should replace the old version. Therefore, there is another paper in arXiv with the same topic