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 < ? S < 4, where ? 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 ½(5?? S ), i.e. $${\rm sup} \left\{h : {\rm lim sup}_{n\to\infty}\, \frac{{\rm var}(W_1+\cdots+W_n)}{n^{2h}} = \infty \right\} = \frac{5-\kappa_S} {2}\,.$$ If this assumption does not hold but the sequence of serial correlation coefficients {? n } of the stationary process {W n } behaves asymptotically as cn ?? for some finite positive c and ? ? (0,1), where ? = ? S ? 3, then {W n } has Hurst index ½(5?? 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 ½(5? S )