### Simulation-based computation of the workload correlation function in a Lévy-driven queue

Mandjes, M; Glynn, P *(Stanford; Amsterdam)*

Wednesday 23 June 2010, 09:00-09.50

Seminar Room 1, Newton Institute

#### 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.

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