A Lévy process $\left\{ X_{t}\right\} $with values in $\mathbb{R}^{+}$ is termed a subordinator if it is a.s. increasing as a function of $t$, i.e. the map $t\mapsto X_{t}(\omega)$ is increasing for every $\omega$ in a set of probability 1. Furthermore, $\left\{ X_{t}\right\} $ is said to be an $\alpha$-stable subordinator, with $\alpha\in(0,1)$, if and only if, for some $c>0$ and $\beta>0$, its Laplace exponent can be written as:
$$ \mathbb{E}\left[\exp\left(\lambda X_{t}\right)\right]=\exp\left\{ -t\left(c\lambda^{\alpha}+\beta\lambda\right)\right\} . $$
So far, my question is: why may we need a subordinator to be $\alpha$-stable? Which is the main tool that $\alpha$-stability brings about?