Fluid queues and regular variation

This paper considers a fluid queueing system, fed by $N$ independent sources that alternate between silence and activity periods. We assume that the distribution of the activity periods of one or more sources is a regularly varying function of index $zeta$. We show that its fat tail gives rise to an even fatter tail of the buffer content distribution, viz., one that is regularly varying of index $zeta +1$. In the special case that $zeta in (-2,-1)$, which implies long-range dependence of the input process, the buffer content does not even have a finite first moment. As a queueing-theoretic by-product of the analysis of the case of $N$ identical sources, with $N rightarrow infty$, we show that the busy period of an M/G/$infty$ queue is regularly varying of index $zeta$ iff the service time distribution is regularly varying of index $zeta$.