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

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Stochastic Processes in Communication Sciences

In this paper we consider a single-server queue with Lévy input, and in particular its workload process $(Q_t)_{tge 0}$, focusing on its correlation structure. With the correlation function defined as $r(t):= {mathbb C}{ m ov}(Q_0,Q_t)/{mathbb V}{ m ar}, Q_0$ (assuming the workload process is in stationarity at time 0), we first study its transform $int_0infty r(t) e{- artheta t}{ m d}t$, both for the case that the Lévy process has positive jumps, and that it has negative jumps. These expressions allow us to prove that $r(ot)$ 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 $(r(t))$ 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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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