We consider a fluid queueing system with infinite storage capacity and constant output rate offered a superposition of N identical On/Off sources, where the ratio of input to output rate is small. The On and/or Off periods have heavy tailed distributions with infinite variance, giving rise to long range dependence in the arrival process. In the limit of a large number of sources and high load, it is shown that the tail of the stationary queue content distribution is Weibullian, implying much larger queue contents than in the classical case of exponential tails. Noting that similar results were recently found by I. Norros for a storage system input by a Fractional Brownian Motion, we then show how the two models are related, thus providing a further physical motivation for the Fractional Brownian Motion model.
