In this paper we study the distribution of the supremum over interval [0, T] of a centered Gaussian process with stationary increments with a general negative drift function. This problem is related to the distribution of the buffer content in a transient Gaussian fluid queue Q(T) at time T, provided that at time 0 the buffer is empty. The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion. We give asymptotic results for ℙ(Q(T) > x) and ℙ(sup0≤t≤T Q(t) > x) as x → ∞.
