We consider a stationary queueing process QX 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, P(QX(t) > f(t) i.o.) equals 0 or 1 is provided. Furthermore, an Erdös–Révész type law of the iterated logarithm is proven for the last passage time ξ(t) = sup{s : 0 ≤ s ≤ t, QX(s) ≥ f(s)}. Both of these findings extend previously known results that were only available for the case when X is a fractional Brownian motion.