  This paper studies the joint tail asymptotics of extrema of the
multi-dimensional Gaussian process over random intervals defined as
  $$ P(u):=\mathbb{P}\left\{\cap_{i=1}^n \left(\sup_{t\in[0,\mathcal{T}_i]} (
X_{i}(t) +c_i t )>a_i u \right)\right\}, \ \ \ u\to\infty, $$ where $X_i(t),
t\ge0$, $i=1,2,\cdots,n,$ are independent centered Gaussian processes with
stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \cdots,
\mathcal{T}_n)$ is a regularly varying random vector with positive components,
which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$,
$a_i>0$, $i=1,2,\cdots,n$. Our result shows that the structure of the
asymptotics of $P(u)$ is determined by the signs of the drifts $c_i$'s. We also
discuss a relevant multi-dimensional regenerative model and derive the
corresponding ruin probability.
