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

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• 作者：Hiroyuki Masuyama ^{masuyama@sys.i.kyoto-u.ac.jp}
• 关键词：60J27 ; 60J22
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：514
• 期：Complete
• 页码：105-150
• 全文大小：672 K
• 卷排序：514

文摘

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.