  We consider a stationary queueing process $Q_X$ fed by a centered Gaussian
process $X$ with stationary increments and variance function satisfying
classical regularity conditions. A criterion when, for a given function $f$,
$\mathbb P (Q_{X}(t) > f(t)\, \text{ i.o.})$ equals 0 or 1 is provided.
Furthermore, an Erd\"os-R\'ev\'esz type law of the iterated logarithm is proven
for the last passage time $\xi (t) = \sup\{s:0\le s\le t, Q_{X}(s)\ge f(s)\}$.
Both of these findings extend previously known results that were only available
for the case when $X$ is a fractional Brownian motion.
