Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms

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Abstract

We consider homogeneous random walks in the quarter-plane. The necessary conditions which characterize random walks of which the invariant measure is a sum of geometric terms are provided in Chen et al. (arXiv:1304.3316, 2013, Probab Eng Informational Sci 29(02):233–251, 2015). Based on these results, we first develop an algorithm to check whether the invariant measure of a given random walk is a sum of geometric terms. We also provide the explicit form of the invariant measure if it is a sum of geometric terms. Second, for random walks of which the invariant measure is not a sum of geometric terms, we provide an approximation scheme to obtain error bounds for the performance measures. Our results can be applied to the analysis of two-node queueing systems. We demonstrate this by applying our results to a tandem queue with server slow-down.
Original languageUndefined
Pages (from-to)21-48
Number of pages28
JournalQueueing systems
Volume84
Issue number1
DOIs
Publication statusPublished - Oct 2016

Keywords

  • MSC-60G50
  • MSC-60J10
  • EWI-27032
  • Error bounds
  • Quarter-plane
  • IR-101439
  • Performance measure
  • Tandem queue
  • Geometric terms
  • METIS-318453
  • Random walk

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