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

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. ( http://arxiv.org/abs/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.