Overflow behavior in queues with many long-tailed inputs

We consider a fluid queue fed by the superposition of n homogeneous on-off sources with generally distributed on and off periods. The buffer space B a nd link rate C are scaled by n, so that we get nb and nc, respectively. The n we let n grow large. In this regime, the overflow probability decays expo nentially in the number of sources n. We specifically examine the scenario where b is also large. We obtain explicit asymptotics for the case where th e on periods have a subexponential distribution, e.g., Pareto, Lognormal, o r Weibull. The results show a sharp dichotomy in the qualitative behavior, depending o n the shape of the function nu (t) := -logP(A* > t) for large t, A* represe nting the residual on period. If nu(.) is regularly varying of index 0 (e.g ., Pareto, Lognormal), then, during the path to overflow, the input rate wi ll only slightly exceed the link rate. Consequently, the buffer will fill ' slowly', and the typical time to overflow will be 'more than linear' in the buffer size. In contrast, if nu(.) is regularly varying of index strictly between 0 and 1 (e.g., Weibull), then the input rate will significantly exc eed the link rate, and the time to overflow is roughly proportional to the buffer size. In both cases there is a substantial fraction of the sources that remain in the on state during the entire path to overflow, while the others contribu te at their mean rates. These observations lead to approximations for the o verflow probability. The approximations may be extended to the case of hete rogeneous sources. The results provide further insight into the so-called r educed-load approximation.