  Various empirical and theoretical studies indicate that cumulative network
traffic is a Gaussian process. However, depending on whether the intensity at
which sessions are initiated is large or small relative to the session duration
tail, Mikosch et al. (Ann Appl Probab, 12:23-68, 2002) and Kaj and Taqqu
(Progress Probab, 60:383-427, 2008) have shown that traffic at large time
scales can be approximated by either fractional Brownian motion (fBm) or stable
Levy motion. We study distributional properties of cumulative traffic that
consists of a finite number of independent streams and give an explanation of
why Gaussian examples abound in practice but not stable Levy motion. We offer
an explanation of how much vertical aggregation is needed for the Gaussian
approximation to hold. Our results are expressed as limit theorems for a
sequence of cumulative traffic processes whose session initiation intensities
satisfy growth rates similar to those used in Mikosch et al. (Ann Appl Probab,
12:23-68, 2002).
