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Bounding Mean First Passage Times in Population Continuous-Time Markov Chains

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Quantitative Evaluation of Systems (QEST 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12289))

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Abstract

We consider the problem of bounding mean first passage times and reachability probabilities for the class of population continuous-time Markov chains, which capture stochastic interactions between groups of identical agents. The quantitative analysis of such models is notoriously difficult since typically neither state-based numerical approaches nor methods based on stochastic sampling give efficient and accurate results. Here, we propose a novel approach that leverages techniques from martingale theory and stochastic processes to generate constraints on the statistical moments of first passage time distributions. These constraints induce a semi-definite program that can be used to compute exact bounds on reachability probabilities and mean first passage times without numerically solving the transient probability distribution of the process or sampling from it. We showcase the method on some test examples and tailor it to models exhibiting multimodality, a class of particularly challenging scenarios from biology.

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Acknowledgements

We would like to thank Andreas Karrenbauer for helpful comments on the usage of SDP solvers and Gerrit Großmann for the valuable comments on this manuscript. This work is supported by the DFG project “MULTIMODE”, and partially supported by the italian PRIN project “SEDUCE” n. 2017TWRCNB.

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Backenköhler, M., Bortolussi, L., Wolf, V. (2020). Bounding Mean First Passage Times in Population Continuous-Time Markov Chains. In: Gribaudo, M., Jansen, D.N., Remke, A. (eds) Quantitative Evaluation of Systems. QEST 2020. Lecture Notes in Computer Science(), vol 12289. Springer, Cham. https://doi.org/10.1007/978-3-030-59854-9_13

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