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THE INVARIANT MEASURE OF RANDOM WALKS IN THE QUARTER-PLANE: REPRESENTATION IN GEOMETRIC TERMS

Published online by Cambridge University Press:  26 January 2015

Yanting Chen ,
Richard J. Boucherie  and
Jasper Goseling
Yanting Chen
Affiliation:
Stochastic Operations Research, University of Twente The Netherlands E-mails: y.chen@utwente.nl, r.j.boucherie@utwente.nl
Richard J. Boucherie
Affiliation:
Stochastic Operations Research, University of Twente The Netherlands E-mails: y.chen@utwente.nl, r.j.boucherie@utwente.nl
Jasper Goseling
Affiliation:
Department of Intelligent Systems, Delft University of Technology The Netherlands;Stochastic Operations Research, University of Twente The Netherlands E-mail: j.goseling@utwente.nl
Corresponding
E-mail address:
y.chen@utwente.nl
r.j.boucherie@utwente.nl
j.goseling@utwente.nl
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Abstract

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these linear combinations such that the resulting measure may yield an invariant measure of a random walk. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space and further show that the geometric terms in an invariant measure must have a pairwise-coupled structure. Finally, we show that at least one of the coefficients in the linear combination must be negative.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 2 , April 2015 , pp. 233 - 251
Copyright
Copyright © Cambridge University Press 2015 

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