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

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.

This version has corrected minor errors related to the domain of "sup" in several error bounds

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• Error bounds for augmented truncations of discrete-time block-monotone Markov chains under subgeometric drift conditions
• A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains
• Stationary distributions of continuous-time Markov chains: a review of theory and truncation-based approximations
• Simple error bounds for the QBD approximation of a special class of two dimensional reflecting random walks