On the correlation structure of a Lévy-driven queue

In this paper we consider a single-server queue with Lévy input, and in particular its workload process (Q(t)), for t > 0, with a focus on the correlation structure. With the correlation function defined as r(t) := Cov(Q(0),Q(t))/Var Q(0) (assuming that the workload process is in stationarity at time 0), we first determine its transform. This expression allows us to prove that r(.) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that r(.) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of r(t), for t large, for the cases of light-tailed and heavy-tailed Lévy input.