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.
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Acknowledgements
The authors thank Anton A. Stoorvogel for useful discussions. Yanting Chen acknowledges support through the NSFC Grant 71701066, the Fundamental Research Funds for the Central Universities and a CSC scholarship [No. 2008613008]. This work is partly supported by the Netherlands Organization for Scientific Research (NWO) Grant 612.001.107.
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Chen, Y., Boucherie, R.J. & Goseling, J. Necessary conditions for the compensation approach for a random walk in the quarter-plane. Queueing Syst 94, 257–277 (2020). https://doi.org/10.1007/s11134-019-09622-1
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DOI: https://doi.org/10.1007/s11134-019-09622-1
Keywords
- Compensation approach
- Random walk
- Quarter-plane
- Invariant measure
- Geometric term
- Algebraic curve
- Pairwise-coupled