Extremes of the time-average of stationary Gaussian processes

We study the exact asymptotics of , as u-->[infinity], where and {Z(t):t>=0} is a centered stationary Gaussian process with covariance function satisfying some regularity conditions. As an application, we analyze the probability of buffer emptiness in a Gaussian fluid queueing system and the collision probability of differentiable Gaussian processes with stationary increments. Additionally, we find estimates for analogues of Piterbarg-Prisyazhnyuk constants, that appear in the form of the considered asymptotics.