  Let $\{D(s), s \geq 0\}$ be a non-decreasing L\'evy process. The
first-hitting time process $\{E(t) t \geq 0\}$ (which is sometimes referred to
as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$ is a
process which has arisen in many applications. Of particular interest is the
mean first-hitting time $U(t)=\mathbb{E}E(t)$. This function characterizes all
finite-dimensional distributions of the process $E$. The function $U$ can be
calculated by inverting the Laplace transform of the function
$\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}$, where $\phi$ is the
L\'evy exponent of the subordinator $D$. In this paper, we give two methods for
computing numerically the inverse of this Laplace transform. The first is based
on the Bromwich integral and the second is based on the Post-Widder inversion
formula. The software written to support this work is available from the
authors and we illustrate its use at the end of the paper.
