Deciding the Value 1 Problem for Reachability in 1-Clock Decision Stochastic Timed Automata

We consider reachability objectives on an extension of stochastic timed automata (STA) with nondeterminism. Decision stochastic timed automata (DSTA) are Markov decision processes based on timed automata where delays are chosen randomly and choices between enabled edges are nondeterministic. Given a reachability objective, the value 1 problem asks whether a target can be reached with probability arbitrary close to 1. Simple examples show that the value can be 1 and yet no strategy ensures reaching the target with probability 1. In this paper, we prove that, the value 1 problem is decidable for single clock DSTA by non-trivial reduction to a simple almost-sure reachability problem on a finite Markov decision process. The ε-optimal strategies are involved: although the precise probability distributions do not change the winning nature of a state, they impact in a crucial way the timings at which ε-optimal strategies must change their decisions.