SCS

Abstract

In this paper we consider a single-server queue with Lévy input, and in particular its workload process $(Q_t)_{t\ge 0}$, focusing on its correlation structure. With the correlation function defined as $r(t):= {\mathbb C}{\rm ov}(Q_0,Q_t)/{\mathbb V}{\rm ar}\, Q_0$ (assuming the workload process is in stationarity at time 0), we first study its transform $\int_0^\infty r(t) e^{-\vartheta t}{\rm d}t$, both for the case that the Lévy process has positive jumps, and that it has negative jumps. These expressions allow us to prove that $r(\cdot)$ is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of $r(t)$ for $t$ large. We then focus on techniques to estimate $r(t)$ by simulation. Naive simulation techniques require roughly $(r(t))^{-2}$ runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (required number of runs being roughly $(r(t))^{-1}$). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm. We present a set of simulation experiments, underscoring the superior performance of our techniques.