The invariant measure of homogeneous Markov processes in the quarter-plane: Representation in geometric terms

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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.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages15
Publication statusPublished - Dec 2011

Publication series

NameMemorandum / Department of Applied Mathematics
PublisherDepartment of Applied Mathematics, University of Twente
ISSN (Print)1874-4850


  • Geometric product form
  • Invariant Measure
  • EWI-20980
  • Continuous-time Markov process
  • Linear combination
  • MSC-60J27
  • IR-78962
  • Quarter-plane

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