Geometric sum of exponential variables
= Geometric sum of exponential variables
If $N$ is geometric on $\{1,2,\ldots\}$ with parameter $p$, independently of unit-rate exponential variables $S_i$, then
$$
\sum_{i=1}^NS_i
$$
is exponential with rate $p$. Its moment-generating function is $p/(p-\theta)$ for $\theta<p$.