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

Abstract

The stationary processes of waiting times {W n }n = 1,2,... in a GI/G/1 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 \u3c κ S \u3c 4, where κ 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 1/2(5-κ S ), i.e. sup {h : lim sup n→∞\, var(W1+⋯+Wn)/n 2h = ∞} = 5 - κS}/2 If this assumption does not hold but the sequence of serial correlation coefficients {ρ n } of the stationary process {W n } behaves asymptotically as cn -α for some finite positive c and α ∈ (0,1), where α = κ S - 3, then {W n } has Hurst index 1/2(5-κ 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(5κ S ). © Springer Science+Business Media, LLC 2007