Suppose we have an $M/M/1$ queue with arrival rate $\lambda$ and service rate $\mu$. Let $S_i$ and $D_i$ denote the arrival time and departure time of the $i$-th customer, and fix $S_1$ as $0$.
Given that $n$ customers finish their services during time interval $[0, T]$ (i.e., $D_n \leq T$), what can we say about the (joint) distribution of $D_i,\ i = 2,\dots,n$? What is the expected total waiting time (i.e., $D_1+\cdots+D_n-S_2-\cdots-S_n$) of these $n$ customers?
From Stochastic Processes by Ross (Chapter 2), I know that $S_i\sim\operatorname{Uniform}([S_{i-1},T]), i = 2,\cdots,n$, if the given condition is '$n$ customers arrive during $[0, T]$'. However, I am not sure if we have a similar result in the above case.
I have been thinking about this problem for a long time. I will be very appreciative if anyone could help. Thanks in advance!