Adaptive formal approximations of Markov chains

Abstract

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%.