Overflow Behavior in Queues with Many Long-Tailed Inputs

We consider a fluid queue fed by a superposition of n homogeneous on-off sources with generally distributed on- and off-periods. We scale buffer space B and link rate C by n, such that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n; we specifically examine the situation in which also b is large. We explicitly compute asymptotics for the case in which the on-periods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull. We provide a detailed interpretation of our results. Crucial is the shape of the function v(t) :=-log P(A* > t) for large t, A* being the residual on-period. If v(dot) is slowly varying (e.g., Pareto, Lognormal), then, during the trajectory to overflow, the input rate will 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.