Cumulative broadband network traffic is often thought to be well
modeled by fractional Brownian motion (FBM). However, some traffic measurements
do not show an agreement with the Gaussian marginal distribution assumption. We
show that if connection rates are modest relative to heavy tailed connection
length distribution tails, then stable Lévy motion is a sensible
approximation to cumulative traffic over a time period. If connection rates are
large relative to heavy tailed connection length distribution tails, then FBM
is the appropriate approximation. The results are framed as limit theorems for
a sequence of cumulative input processes whose connection rates are varying in
such a way as to remove or induce long range dependence.