Continuous-time block-monotone Markov chains and their block-augmented truncations

TL;DR

This paper studies continuous-time block-monotone Markov chains, introduces block-augmented truncations, and proves convergence and approximation bounds for their stationary distributions, with applications to queueing systems.

Contribution

It introduces block monotonicity and block-wise dominance for continuous-time Markov chains, and shows that block-augmented truncations effectively approximate the stationary distribution.

Findings

Stationary distributions of truncated chains converge to the original's.

LC-block-augmented truncation provides the best approximation.

Provides computable bounds for approximation error.

Abstract

This paper considers continuous-time block-monotone Markov chains (BMMCs) and their block-augmented truncations. We first introduce the block monotonicity and block-wise dominance relation for continuous-time Markov chains, and then provide some fundamental results on the two notions. Using these results, we show that the stationary distribution vectors obtained by the block-augmented truncation converge to the stationary distribution vector of the original BMMC. We also show that the last-column-block-augmented truncation (LC-block-augmented truncation) provides the best (in a certain sense) approximation to the stationary distribution vector of a BMMC among all the block-augmented truncations. Furthermore, we present computable upper bounds for the total variation distance between the stationary distribution vectors of a Markov chain and its LC-block-augmented truncation, under the assumption that the original Markov chain itself may not be block-monotone but is block-wise dominated by a BMMC with exponential ergodicity. Finally, we apply the obtained bounds to a queue with a batch Markovian arrival process and state-dependent departure rates.