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 πT(u)(u):=P(∫T(u)0Q(r)dr>u) is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of πT(u)(u) are given for the case that T(u) grows slower than u√, and then logarithmic asymptotics for (i) T(u)=Tu√ (relying on sample-path large deviations), and (ii) u√=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.

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