Fair enough, with your permission we could discuss the exact formulation in chat so that we could narrow down the definition a little. I guess if we need map from $\mathbb{R}$ to $\text{Perm}(S)$, we'd have a countable subset of $\mathbb{R}$ that aren't trivial elements in $\text{Perm}(S)$, what makes this tricky is that the embedding of "interesting transitions" in real time effect what the interesting transitions are and vice versa, deterministically (no Markov theory here!). The closest theoretical relative would be timed automata in CS, even then.