# An extension of the inverse method to probabilistic timed automata

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## Abstract

Probabilistic timed automata can be used to model systems in which probabilistic and timing behaviour coexist. Verification of probabilistic timed automata models is generally performed with regard to a single reference valuation *π*_{0} of the timing parameters. Given such a parameter valuation, we present a method for obtaining automatically a constraint *K*_{0} on timing parameters for which the reachability probabilities (1) remain invariant and (2) are equal to the reachability probabilities for the reference valuation. The method relies on parametric analysis of a non-probabilistic version of the probabilistic timed automata model using the “inverse method”. The method presents the following advantages. First, since *K*_{0} corresponds to a dense domain around *π*_{0} on which the system behaves uniformly, it gives us a measure of *robustness* of the system. Second, it allows us to obtain a valuation satisfying *K*_{0} which is *as small as possible* while preserving reachability probabilities, thus making the probabilistic analysis of the system easier and faster in practice. We provide examples of the application of our technique to models of randomized protocols, and introduce an extension of the method allowing the generation of a “probabilistic cartography” of a system.

### Keywords

Probabilistic model checking Parametric timed automata## Notes

### Acknowledgements

We are grateful to the anonymous referees for their helpful comments. Étienne André and Laurent Fribourg have been partially supported by the Agence Nationale de la Recherche, grant ANR-06-ARFU-005, and by Institute Farman (project SIMOP). Jeremy Sproston is supported in part by the project AMALFI—Advanced Methodologies for the AnaLysis and management of the Future Internet (Università di Torino/Compagnia di San Paolo).

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