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Simulation-based computation of the workload correlation function in a Lévy-driven queue. (English) Zbl 1213.60147
Summary: We consider a single-server queue with Lévy input, and, in particular, its workload process $$(Q_t)_{t\geq 0}$$, focusing on its correlation structure. With the correlation function defined as $$r(t):= \mathrm{cov}(Q_{0}, Q_t) / \mathrm{var} \, Q_{0}$$ (assuming that the workload process is stationary at time 0), we first study its transform $$\int _{0}^{\infty }r(t)\mathrm{e}^{-\vartheta t}\,dt$$, both for when the Lévy process has positive jumps and when 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 large $$t$$. 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 (the required number of runs being roughly $$(r(t))^{-1})$$. If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60G51 Processes with independent increments; Lévy processes
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