On Internet Traffic Dynamics and Internet Topology I

High Variability Phenomena

Topics Covered

Motivation

A Working Definition

Some Illustrative Examples

Probability density functions

Cumulative Distribution Function

Complementary CDFs

Slide 9

Why “Heavy Tails” Matter …

Some First Properties

Some Simple Constructions

Key Mathematical Properties of Scaling Distributions

Linear Aggregation:
Classical Central Limit Theorem

Linear Aggregation:
Non-classical Central Limit Theorem

Maximization:
Maximum Domain of Attraction

Intuition for “Mild” vs. “Wild”

Weighted Mixture

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Resilience to Ambiguity

On the Ubiquity of Heavy Tails

Heavy Tails and Statistics

“Curve-fitting” approach

“Curve-fitting” by example

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The “truth” about “curve-fitting”

“Borrowing strength” approach

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“Borrowing strength” (example 1)

Fitting lognormal: n=20,000

Fitting lognormal: n=40,000

Fitting lognormal: n=80,000

Fitting lognormal: n=160,000

Fitting lognormal: All data

The case against lognormal

Randomizing observations

Matching “mild” distributions

Lognormal or scaling distribution

Slide 43

“Borrowing strength” (example 2)

Fitting Pareto: n=20,000

Fitting Pareto: n=40,000

Fitting Pareto: n=80,000

Fitting Pareto: n=160,000

Fitting Pareto: All data

The case for scaling distributions

Looking ahead …

Some Words of Caution …

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A Word of Wisdom …