Given that the time to hit state $s_j$ starting from $s_i$ in a Markov process is $\tau$, what is the probability that in a simulation of the system, the actual hitting time is $\tau + \delta$, for some $\delta \in \mathbb{R}$?
EDIT:
Expected hitting time for a Markov process (assuming the transition matrix has been constructed) already has a closed form expression. It should be the case that there should also exist a closed-form expression for the probability of being $\delta$ far from the expected time.