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

For a small buffer queueing system fed by a large class of traffic processes we show that the single server queue and the associated sample paths behave as if fed by marked Poisson traffic; in a large deviations limit, for a system fed by many flows. The timescale upon which events occur in this setting tends to zero, hence we study a limit on the log moment generating function as time tends to zero. Additionally, we prove the associated rate function depends only on the mean arrival rate and the moment generating function of the size of each arrival. These results are useful in estimating drop probabilities while studying the effect of small buffers on communication protocols, for example, the Transmission Control Protocol (TCP).