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arXiv · 1511.04669

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

Hiroyuki Masuyama
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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.

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Hiroyuki Masuyama. 2016-11-19. Continuous-time block-monotone Markov chains and their block-augmented truncations. https://doi.org/10.1016/j.laa.2016.10.014

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