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 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.
|Name||Memorandum / Department of Applied Mathematics|
|Publisher||Department of Applied Mathematics, University of Twente|
- Geometric product form
- Invariant Measure
- Continuous-time Markov process
- Linear combination