  Let $\{X(t):t\in[0,\infty)\}$ be a centered Gaussian process with stationary
increments and variance function $\sigma^2_X(t)$. We study the exact
asymptotics of ${\mathbb{P}}(\sup_{t\in[0,T]}X(t)>u)$ as $u\to\infty$, where
$T$ is an independent of $\{X(t)\}$ non-negative Weibullian random variable. As
an illustration, we work out the asymptotics of the supremum distribution of
fractional Laplace motion.
