Error Bounds for Last-Block-Column-Augmented Truncations of Block-Structured Markov Chains

Hiroyuki Masuyama

Published 2016-01-14Version 1

This paper considers the estimation of the difference between the time-averaged functionals of a continuous-time block-structured Markov chain (BSMC) and its last-blockcolumn-augmented northwest-corner truncation (called the LBC-augmented truncation, for short). The stationary probability vectors of a BSMC and its LBC-augmented truncation can be connected through the deviation matrix of the BSMC, which is a solution to a certain Poisson equation. Combining this fact with Dynkin's formula, we derive error bounds for the time-averaged functional obtained by LBC-augmented truncation, under the assumption that the BSMC satisfies the general f-modulated drift condition. We also establish computable bounds for a special case where the BSMC is exponentially ergodic. To derive such computable bounds for the general case, we propose a method that reduces BSMCs to be exponential ergodic. Furthermore, we focus on the level-dependent quasi-birth-and-death process (LD-QBD), which is a typical example of BSMCs connected with queueing theory. As an application of our results, we consider a retrial queueing model and provide some numerical examples. Finally, we provide some remarks on the perturbation analysis of the stationary probability vectors BSMCs.