Crossing a road through a Poisson process Am currently working on a Stochastic Poisson process on my project. I have thought and settled on the below scenario which I think is appropriate. However, solving it am not getting what I expect. 
I want to cross a road at a spot where cars pass according to a Poisson process with a rate of lambda. I will begin to cross as soon as I see there will be no cars passing for the next time C units. I have taken N=Number of cars that pass before I cross and T=Time I begin to cross the road.
I want to determine the E(N) and Also E(T). What I know is that to find the E(T), I will have to condition on N
 A: I am not sure, what you are trying to model here. Maybe if you give some more details, I may be able to provide a better answer.
From what you are saying, it seem to me like you are doing the "gambler's fallacy" here, just for Poisson processes instead of a binomial process.
Now the gambler's fallacy arises, because a predictor is used to predict a value, that can by definition of the underlying process be shown to be independent from each other. For example if I throw a fair die, the probability of the die showing a 6 is by definition of the underlying binomial process independent of the number of 6 that I have seen before (or rather have not seen before). If I assume the probability I will see a 6, because I have not seen a 6 on the die for a long time, I will overestimate it.
The same is true for a Poisson process. By definition of a Poisson process, the expected number of cars in the next time interval (e.g. time it takes to cross the road), is by definition of a Poisson process independent of the number of cars that have passed before. Therefore, if you use the number of cars that have passed before to predict anything, you can only succumb to the gambler's fallacy and you must overestimate your chances, just as a gambler looking at the lack of 6es in the run before must overestimate his chances.
So for the question about when to cross the road, if the cars really follow a Poisson process? Just pick any time. The Poisson process guarantees that the probability of at least one car in the next time interval is independent of the number of previous cars. Therefore, my chances will be the same for all T and I should not worry about prediction and just see to get across as fast as possible.
A: You might have a check at this post:
Poisson Process,
which dealt with $E(T)$.
Concerning $E(N)$, the situation is similar, you still need to condition upon the first passage, that is $$E(N)= E(E(N|J_1)),$$ where $J_1$='when the first car will pass'. Then
$$E(N)= E(N|J_1>a)P(J_1>C)+ \int_0^C E(N|J_1=u)f_{J_1}(u) du$$
$$= 0+ \int_0^C (1+ E(N)) \lambda e^{-\lambda u} du .$$
It remains to solve the equation to find $E(N).$
LiKao, in your answer  $E(T), E(N)$ do not depend at all on the constant $C$, while if it increases also $E(T), E(N)$ should increase. 
