Solution
= Solution
Independence and the <moment-generating function> of an <exponential distribution> give
$$
\phi_{S_k}(\theta)=\left(\frac{k\mu}{k\mu-\theta}\right)^k,\qquad\theta<k\mu.
$$
For each fixed $\theta$, taking logarithms and using $-\log(1-x)\sim x$ yields $\phi_{S_k}(\theta)\to e^{\theta/\mu}$. \b[The limiting service time is the constant $S=1/\mu$.] Its moment-generating function is exactly this limit.