Let $S_{k}$ be the sum of $k$ independent exponential random variables of rate $k \mu$. Compute the moment generating function of $S_{k}$.

Consider, for each fixed $k$ and for $0<\lambda<\mu$, an $M / G / 1$ queue with arrival rate $\lambda$ and with service times distributed as $S_{k}$. Assume that the queue is empty at time 0 and write $T_{k}$ for the earliest time at which a customer departs leaving the queue empty. Show that, as $k \rightarrow \infty, T_{k}$ converges in distribution to a random variable $T$ whose moment generating function $M_{T}(\theta)$ satisfies

$$\log \left(1-\frac{\theta}{\lambda}\right)+\log M_{T}(\theta)=\left(\frac{\theta-\lambda}{\mu}\right)\left(1-M_{T}(\theta)\right)$$

Hence obtain the mean value of $T$.

For what service-time distribution would the empty-to-empty time correspond exactly to $T$ ?