On the tail asymptotics of the area swept under the Brownian storage graph

In this paper, the area swept under the workload graph is analyzed: with {Q(t): t=0} denoting the stationary workload process, the asymptotic behavior of p_{T(u)} (u):=P(\int^T(u)_0 Q(r)dr > u) is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of pT(u)(u) are given for the case that T(u) grows slower than vu, and then logarithmic asymptotics for (i) T(u)=Tvu (relying on sample-path large deviations), and (ii) vu=o(T(u)) but T(u)=o(u). Finally, the Laplace transform of the residual busy period are given in terms of the Airy function.