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Continuous-time Markov chains with alarms (ACTMCs) allow for alarm events that can be non-exponentially distributed. Within parametric ACTMCs, the parameters of alarm-event distributions are not given explicitly and can be the subject of parameter synthesis. In this line, an algorithm is presented that solves the ε-optimal parameter synthesis problem for parametric ACTMCs with long-run average optimization objectives. The approach provided in this article is based on a reduction of the problem to finding long-run average optimal policies in semi-Markov decision processes (semi-MDPs) and sufficient discretization of the parameter (i.e., action) space. Since the set of actions in the discretized semi-MDP can be very large, a straightforward approach based on an explicit action-space construction fails to solve even simple instances of the problem. The presented algorithm uses an enhanced policy iteration on symbolic representations of the action space. Soundness of the algorithm is established for parametric ACTMCs with alarm-event distributions that satisfy four mild assumptions, fulfilled by many kinds of distributions. Exemplifying proofs for the satisfaction of these requirements are provided for Dirac, uniform, exponential, Erlang, and Weibull distributions in particular. An experimental implementation shows that the symbolic technique substantially improves the efficiency of the synthesis algorithm and allows us to solve instances of realistic size.

Mean-payoff Optimization in Continuous-time Markov Chains with Parametric Alarms