On the time of first level crossing and inverse Gaussian distribution

We propose a new approximation for the distribution of the time of the first level $u$ crossing by the random process $\text{homV}s-cs$, where $\text{homV}s$, $s>0$, is compound renewal process and $c>0$. It is competitive with respect to existing approximations, particularly in the region around the critical point $c=\text{cS}$ which separates processes with positive and negative drifts. This approximation is tightly related to inverse Gaussian distributions.