Necessary conditions for the compensation approach for a random walk in the quarter-plane

Yanting Chen (), Richard J. Boucherie () and Jasper Goseling ()
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Yanting Chen: Hunan University
Richard J. Boucherie: University of Twente
Jasper Goseling: University of Twente

Queueing Systems: Theory and Applications, 2020, vol. 94, issue 3, No 4, 257-277

Abstract: 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 countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

Keywords: Compensation approach; Random walk; Quarter-plane; Invariant measure; Geometric term; Algebraic curve; Pairwise-coupled; 60J10; 60G50 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s11134-019-09622-1

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