  For the stationary storage process $\{Q(t), t\ge0\}$, with $ Q(t)=\sup_{ s
\ge t}\left(X(s)-X(t)-c(s-t)^\beta\right), $ where $\{X(t),t\ge 0\}$ is a
centered Gaussian process with stationary increments, $c>0$ and $\beta>0$ is
chosen such that $Q(t)$ is finite a.s., we derive exact asymptotics of
$\mathbb{P}\left(\sup_{t\in [0,T_u]} Q(t)>u\right)$ and
$\mathbb{P}\left(\inf_{t\in [0,T_u]} Q(t)>u\right)$, as $u\rightarrow\infty$.
As a by-product we find conditions under which strong Piterbarg property holds.
