We explore formal approximation techniques for Markov chains based on state–space reduction that aim at improving the scalability of the analysis, while providing formal bounds on the approximation error. We first present a comprehensive survey of existing state-reduction techniques based on clustering or truncation. Then, we extend existing frameworks for aggregation-based analysis of Markov chains by allowing them to handle chains with an arbitrary structure of the underlying state space – including continuous-time models – and improve upon existing bounds on the approximation error. Finally, we introduce a new hybrid scheme that utilises both aggregation and truncation of the state space and provides the best available approach for approximating continuous-time models. We conclude with a broad and detailed comparative evaluation of existing and new approximation techniques and investigate how different methods handle various Markov models. The results also show that the introduced hybrid scheme significantly outperforms existing approaches and provides a speedup of the analysis up to a factor of 30 with the corresponding approximation error bounded within 0.1%.

我们探索基于状态空间约简的马尔可夫链形式近似方法，旨在提高分析的可扩展性，同时提供近似误差的形式边界。我们首先对基于聚类或截断的现有状态约简技术进行全面概述。然后，我们扩展了现有框架，使之能够对基于马尔可夫链的聚合进行分析，方法是使它们处理具有潜在状态空间的任意结构的链（包括连续时间模型），并改善近似误差的现有界限。最后，我们介绍了一种新的混合方案，该方案同时利用状态空间的聚合和截断，并为近似连续时间模型提供了最佳的可用方法。我们以现有和新的近似技术的广泛而详细的比较评估作为结束，并研究不同的方法如何处理各种马尔可夫模型。结果还表明，引入的混合方案明显优于现有方法，并提供了高达30倍的分析速度，相应的近似误差限制在0.1％以内。