块结构马尔可夫链的末列块增广截断的误差界
Error bounds for last-column-block-augmented truncations of block-structured Markov chains
AI总结：
本文研究连续时间块结构马尔可夫链的末列块增广截断误差，推导一般$f$-调制漂移条件下的上界，建立指数遍历情形的可计算界并提出一般情形的转化方法，将其应用于水平依赖拟生灭过程，还给出平稳分布的可计算扰动界。
摘要：
本文讨论连续时间块结构马尔可夫链（block-structured Markov chains，简称BSMCs）的末列块增广西北角截断（简称LC-block-augmented截断）的误差估计。我们首先在BSMC满足一般$f$-调制漂移条件的假设下，推导BSMC与其LC-block-augmented截断的时间平均泛函之间绝对差的上界。随后，我们针对BSMC指数遍历的特殊情况建立了可计算的界。为推导一般情形下的此类可计算界，我们提出一种将BSMC转化为指数遍历的方法。我们还将所得界应用于水平依赖拟生灭过程（level-dependent quasi-birth-and-death processes，简称LD-QBDs），并通过M/M/$s$重试排队（LD-QBDs的典型例子）的数值结果讨论界的性质。最后，我们给出了BSMC平稳分布向量的可计算扰动界。
英文摘要：
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.
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