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 $\{Q(t) : t\ge0\}$ denoting the stationary workload process, the asymptotic behavior of $$\pi_{T(u)}(u):={\mathbb{P}}\biggl(\int_0^ {T(u)}Q(r)\,\mathrm{d}r>u\biggr)$$ is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of $\pi_{T(u)}(u)$ are given for the case that $T(u)$ grows slower than $\sqrt{u}$, and then logarithmic asymptotics for (i) $T(u)=T\sqrt{u}$ (relying on sample-path large deviations), and (ii) $\sqrt{u}=\mathrm{o}(T(u))$ but $T(u)=\mathrm{o}(u)$. Finally, the Laplace transform of the residual busy period are given in terms of the Airy function.