Error bounds for last-column-block-augmented truncations of block-structured Markov chains

TL;DR

This paper develops error bounds for a specific truncation method of block-structured Markov chains, providing practical estimates for their accuracy and applications to queueing models.

Contribution

It introduces new error bounds for LC-block-augmented truncations of BSMCs, including computable bounds for exponentially ergodic cases and a reduction method for general cases.

Findings

Derived upper bounds for the difference in time-averaged functionals.

Established computable bounds for exponentially ergodic BSMCs.

Applied bounds to LD-QBDs and queueing models, demonstrating effectiveness.

Abstract

This paper discusses the error estimation of the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of block-structured Markov chains (BSMCs) in continuous time. We first derive upper bounds for the absolute difference between the time-averaged functionals of a BSMC and its LC-block-augmented truncation, under the assumption that the BSMC satisfies the general $f$-modulated drift condition. We then 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 exponentially ergodic. We also apply the obtained bounds to level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the properties of the bounds through the numerical results on an M/M/$s$ retrial queue, which is a representative example of LD-QBDs. Finally, we present computable perturbation bounds for the stationary distribution vectors of BSMCs.