SIAM Review, Volume 63, Issue 1, Page 3-64, January 2021.
Computing the stationary distributions of a continuous-time Markov chain (CTMC) involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too large, and the equations cannot be solved analytically or numerically. Several approximation schemes overcome this issue by truncating the state space to a manageable size. In this review, we first give a comprehensive theoretical account of the stationary distributions and their relation to the long-term behaviour of CTMCs that is readily accessible to non-experts and free of irreducibility assumptions made in standard texts. We then review truncation-based approximation schemes for CTMCs with infinite state spaces paying particular attention to the schemes' convergence and the errors they introduce, and we illustrate their performance with an example of a stochastic reaction network of relevance in biology and chemistry. We conclude by elaborating on computational trade-offs associated with error control and several open questions.

中文翻译：

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连续时间马尔可夫链的平稳分布：理论回顾和基于截断的近似

SIAM 评论，第 63 卷，第 1 期，第 3-64 页，2021 年 1 月。
计算连续时间马尔可夫链 (CTMC) 的平稳分布涉及求解一组线性方程。在大多数感兴趣的情况下，方程的数量是无限的或太多，并且无法解析或数值求解方程。几种近似方案通过将状态空间截断到可管理的大小来克服这个问题。在这篇综述中，我们首先对平稳分布及其与 CTMC 的长期行为的关系进行了全面的理论解释，非专家可以轻松获得并且没有标准文本中的不可约假设。然后，我们回顾了具有无限状态空间的 CTMC 的基于截断的近似方案，特别注意方案的收敛性和它们引入的误差，我们通过一个与生物学和化学相关的随机反应网络的例子来说明它们的性能。最后，我们详细阐述了与错误控制相关的计算权衡和几个悬而未决的问题。