1. Notices of the AMS, September 1998

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

3. 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

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

5. The SOC (Self-Organized Criticality) view
Links

6. Average
Queue
Links
Flow
“phase transition”
capacity

7. 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

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

9. 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)

10. 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)

11. 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

12. 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

13. 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,…

14. 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