Yanting Chen, Richard J. Boucherie, Jasper Goseling
Published 2011-12-13, updated 2014-02-24Version 3
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.
Comments: The 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
Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms
The Invariant Measure of Random Walks in the Quarter-plane: Representation in Geometric Terms
On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation
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