Poisson convergence, in large deviations, for the superposition of independent point processes

For a small buffer queueing system fed by many flows of a large class of traffic processes we show the single server queue and associated sample paths behave as if fed by marked Poisson traffic in a large deviations limit.

The timescale of events of interest tends to zero, so we study the log moment generating function as time tends to zero. The associated rate function depends only on the mean arrival rate and the moment generating function of the arrivals. These results are useful in estimating drop probabilities while studying the effect of small buffers on communication protocols.