Consider a queue with Poisson arrivals of rate $\lambda$ in which each arrival $i$, independent of other arrivals, remains in the system for a time $X_i$, where {$X_i;i\geq1$} is a set of IID rv's with some given distribution function $F(x)$.
You may assume that the number of arrivals in any interval $(t, t+\epsilon)$ that are still in the system at some later time interval $\tau \geq t+\epsilon$ is statistically independent of the number of arrivals in that same time interval $(t, t+\epsilon)$ that have departed from the system by time $\tau$.
Let $N(\tau)$ be the number of customers in the system at time $\tau$. Find Pr{$N(\tau) = n$}.
The only idea I have right now to approach this question is to consider the arrival time of each person as Wi and sum it for all for the condition Wi+Xi <= t. As you can tell, it's an extremely loosely defined answer and I am struggling with formalizing it or even figuring out whether it is correct