Problem Assume that passengers arrive at a bus station as a Poisson process with rate λ. The only bus departs after a deterministic time T. Let S' be the combined waiting time for all passengers. Compute ES'.
I do know about other ways to solve this problem, but I WANT TO SALVAGE MY SOLUTION:
Passenger $m$ waiting time: $$S'_{m}=T-S_m=T-\sum_{k=1}^{m}X_k$$
$$E(S'_{m})=T-\mu m$$
All waiting time: $$S'=\sum_{m=1}^{N(t)} S'_{m}=\sum_{m=1}^{N(t)}T-S_m$$
And now I want find the expected value:
$$ES'=\sum_{m=1}^{EN(t)} ES'_{m}$$
The problem is of course $EN(t)\not\in\mathbb{N}$.
Ignoring $EN(t)\not\in\mathbb{N}$ I get:
$$ES'=\frac{\lambda T^2+T}{2}$$
Actual answer:
$$ES'=\frac{\lambda T^2}{2}$$
The answer I get ignoring $EN(t)\not\in\mathbb{N}$ is very simmilar to the actual solution. Is there a way to salvage this solution?