Notices of the AMS, September 1998

Poisson Measured Internet traffic Standard Poisson models don’t capture long-range correlations. “bursty” on all time scales

Fractal Measured Internet traffic Fractional Gaussian (fractal) noise models measurements well. Hurst parameter H is an aggregate measure of long-range correlations. “bursty” on all time scales

The “physics” of the Internet “Physicists use chaos to calm the web,” (Physics World, 2001) www.networkphysics.com Large literature in physics journals and recently in Science, Nature, etc…

The SOC (Self-Organized Criticality) view Links

Average Queue Links Flow “phase transition” capacity

Lattice without congestion control (?!?) “Critical” phase transition at max capacity At criticality: self-similar fluctuations, long tailed queues and latencies, 1/f time series, etc Flow capacity Average Queue

Alternative “edge of chaos” models Self-similarity due to chaos and independent of higher-layer characteristics

Why SOC/EOC/… models fail No “critical” traffic rate Self-similar scaling at all different rates TCP can be unstable and perhaps chaotic, but does not generate self-similar scaling Self-similar scaling occurs in all forms of traffic (TCP and nonTCP) Measured traffic is not consistent with these models Fractal and scale-free topology models are equally specious (for different reasons)

A network based explanation Underlying cause: If connections arrive randomly (in time) and if their size (# packets) have high variability (i.e. are heavy-tailed with infinite variance) then the aggregate traffic is perforce self-similar Evidence Coherent and mathematically rigorous framework Alternative measurements (e.g. TCP connections, IP flows) Alternative analysis (e.g. heavy-tailed property)

Web servers Heavy tailed web traffic p  s- Typical web traffic Heavy tailed web traffic  > 1.0 log(freq > size) p  s- log(file size) Is streamed out on the net. Creating fractal Gaussian internet traffic (Willinger,…) Web servers

creating long-range correlations with Is streamed onto the Internet Fat tail web traffic time creating long-range correlations with Is streamed onto the Internet

Heavy tails in networks? Heavy tails and divergent length scales are everywhere in networks. There is a large literature since 1994: Leland, Taqqu, Willinger, Wilson Paxson, Floyd Crovella, Bestavros Harchol-Balter,…

Piece of a consistent, rigorous theory with supporting measurements Typical web traffic Heavy tailed web traffic  > 1.0 log(freq > size) p  s- log(file size) Is streamed out on the net. Piece of a consistent, rigorous theory with supporting measurements Web servers