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Long-range dependence of stationary processes in single-server queues. (English) Zbl 1178.60063
Summary: The stationary processes of waiting times $$\{W_{n}\}_{n = 1,2,\ldots}$$ in a $$GI / G /1$$ queue and queue sizes at successive departure epochs $$\{Q_{n}\}_{n = 1,2,\ldots}$$ in an $$M / G /1$$ queue are long-range dependent when $$3 < \kappa_{S} < 4$$, where $$\kappa_{S}$$ is the moment index of the independent identically distributed (i.i.d.) sequence of service times. When the tail of the service time is regularly varying at infinity the stationary long-range dependent process $$\{W_{n}\}$$ has Hurst index $$\frac 12(5 - \kappa_{S})$$, i.e. $\sup \left\{h : \limsup_{n\to\infty} \frac{\text{var}(W_1+\cdots+W_n)}{n^{2h}} = \infty \right\} = \frac{5-\kappa_S}{2}.$ If this assumption does not hold but the sequence of serial correlation coefficients $$\{\rho_{n}\}$$ of the stationary process $$\{W_{n}\}$$ behaves asymptotically as $$cn^{- \alpha}$$ for some finite positive $$c$$ and $$\alpha \in (0,1)$$, where $$\alpha = \kappa_{S} - 3$$, then $$\{W_{n}\}$$ has Hurst index $$(5 - \kappa_{S})$$. If this condition also holds for the sequence of serial correlation coefficients $$\{r_{n}\}$$ of the stationary process $$\{Q_{n}\}$$ then it also has Hurst index $$(5\kappa_{S})$$
##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
longmemo
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