  We investigate the tail asymptotics of the supremum of X(t)+Y(t)-ct, where
X={X(t),t\geq 0} and Y={Y(t),t\geq 0} are two independent stochastic processes.
We assume that the process Y has subexponential characteristics and that the
process X is more regular in a certain sense than Y. A key issue examined in
earlier studies is under what conditions the process X contributes to large
values of the supremum only through its average behavior. The present paper
studies various scenarios where the latter is not the case, and the process X
shows some form of ``atypical'' behavior as well. In particular, we consider a
fluid model fed by a Gaussian process X and an (integrated) On-Off process Y.
We show that, depending on the model parameters, the Gaussian process may
contribute to the tail asymptotics by its moderate deviations, large
deviations, or oscillatory behavior.
