Long-range dependence of stationary processes in single-server queues

The stationary processes of waiting times 1 Introduction {W(n)}(n=1,2,...) in a GI/G1 queue and queue sizes at successive departure epochs {Q(n)}(n=1,2,...) 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 1/1(5 - kappa(S)), i.e. sup{h: lim sup/n ->infinity var(W(1)+center dot center dot center dot+W(n)/n(2h) = infinity} = 5-kappa(s)/2. If this assumption does not hold but the sequence of serial correlation coefficients {p(n)} of the stationary process {W(n)} behaves asymptotically as cn(-alpha) for some finite positive c and alpha epsilon (0,1), where alpha = kappa(S) - 3, then {W(n)} has Hurst index 1/2(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 1/2(5 - kappa(S)).