  Let {X(t)}_{t\ge0} be a locally bounded and infinitely divisible stochastic
process, with no Gaussian component, that is self-similar with index H>0.
  Pick constants \gamma >H and c>0. Let \nu be the L\'evy measure on
R^{[0,\infty)} of X, and suppose that R(u)\equiv\nu({y\inR^{[0,\infty)}:supt\ge
0y(t)/(1+ct^{\gamma})>u}) is suitably ``heavy tailed'' as u\to\infty (e.g.,
subexponential with positive decrease). For the ``storage process'' Y(t)\equiv
sup_{s\ge t}(X(s)-X(t)-c(s-t)^{\gamma}), we show that
P{sup_{s\in[0,t(u)]}Y(s)>u}\sim P{Y(\hat t(u))>u} as u\to\infty, when 0\le \hat
t(u)\le t(u) do not grow too fast with u [e.g., t(u)=o(u^{1/\gamma})].
