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On the correlation structure of a Lévy-driven queue. (English) Zbl 1154.60348
Summary: We consider a single-server queue with Lévy input and, in particular, its workload process $$(Q_t)_{t\geq 0}$$, with a focus on the 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), we first determine its transform $$\int _{0}^{\infty }r(t)$$e$$^{-\vartheta t}$$d$$t$$. This expression allows us to prove that $$r(\cdot )$$ is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that $$r(\cdot )$$ 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 large $$t$$, for the cases of light-tailed and heavy-tailed Lévy inputs.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60G51 Processes with independent increments; Lévy processes 90B05 Inventory, storage, reservoirs
##### Keywords:
Lévy process; reflection; correlation; complete monotonicity
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