Consider a M/G/1 queueing system, when the server frees up, it chooses a random job to run next. We want to find the varance of response time $T = T_Q + S$, where $T_Q$ is waiting time and $S$ is service time.
For example, if a customer arrives and finds $k$ customers in the queue and 1 customer in service. Then, he or she has to waited until the server becomes free. We assume $m$ customers arrives during the service. Then, with probability $\frac{1}{k+m}$, he or she will be the next person to be served.
I am thinking, maybe we can find the laplace transform of $T_Q$ and then derive the $\tilde{T}(s)$. But I have no idea about how to iterate this process.