Sheldon Ross's Introduction to Probability Models, exercise 5.44.b: cars pass a certain street location according to a Poisson process with rate $\lambda$. A woman who wants to cross the street at that location waits until she can see that no cars will come by in the next $\tilde t$ time units. I'm trying to find her expected waiting time.
I am given a hint to condition on the time of the first car, which I'll call $T_1$. The process is noted $\{N(t) \;|\; t \geq 0\}$. Her waiting time can be defined as $S := \min_{t\geq0}\{N(t+\tilde t) - N(t) = 0\} \geq 0\quad.$
Clearly, $\mathbb E[S\;|\;T_1 \geq \tilde t] = \mathbb E[0] = 0\quad.$ If this approach is good, how should I proceed for $T_1 < \tilde t$?