Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 27I / a

Independence and the moment-generating function of an exponential distribution giveFor each fixed $\theta$, taking logarithms and using $-\log (1-x)\sim x$ yields ${\varphi }_{S_k}(\theta )\to e^{\theta /\mu }$. The limiting service time is the constant $S=1/\mu$. Its moment-generating function is exactly this limit.