In this presentation I'll consider a single-server queue with Levy input, and in particular its workload process (Q_t), focusing on its 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), I first show how powerful results for spectrally positive Levy processes can be used to find its Laplace transform. This expression allows us to prove that r(.) is positive, decreasing, and convex --- properties that are likely to hold, but far from straightforward to prove directly from first principles. Relying on the machinery of completely monotone functions, however, a rather elegant proof can be given.

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 Levy input.

Joint work with Abdelghafour Es-Saghouani