Asymptotics of the area under the graph of a Lévy-driven workload process

Let (Qt)t∈R be the stationary workload process of a Lévy-driven queue, where the driving Lévy process is light-tailed. For various functions T(u), we analyze P(∫0T(u)Qsds>u) for u large. For T(u)=o(u) the asymptotics resemble those of the steady-state workload being larger than u/T(u). If T(u) is proportional to u they look like e−αu for some α>0. Interestingly, the asymptotics are still valid when u=o(T(u)), T(u)=o(u), and T(u)=βu for β suitably small.