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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
Software:
longmemo
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