1. Maurizio Naldi Università di Roma “Tor Vergata” POPULARITY DISTRIBUTIONS AND INTERNET TRAFFIC MODELLING Workshop “Statistica e Telecomunicazioni”, Roma 2-3 Luglio 2001 Università Di Roma “Tor Vergata” Dip. Informatica Sistemi Produzione

2. WHAT’S A POPULARITY MODEL Popularity models describe the way users distribute their preferences among a set of objects. They are represented under the form of either a frequency-rank plot (suitable for highly preferred objects) or a frequency-count plot (suitable for the less preferred objects.

3. EXAMPLES OF FREQUENCY-RANK AND FREQUENCY-COUNT PLOTS A frequency-rank plot No. of preferences vs. rank A frequency-count plot No. of preferences vs. no. of objects that have those preferences

4. SOME POPULARITY MODELS (FREQUENCY-RANK LAWS) Zipf Simon Yule

5. RELATIONSHIP TO PARETO’S MODEL If the objects in a set of N are ranked by size according to Zipf’s law Then the number of objects having a size greater or equal to is The probability distribution function is therefore i.e. of the Pareto type

6. APPLICATIONS Present Cache algorithm design Address cache table dimensioning Optimization of Video-on-Demand servers’ architecture Possible Any communications context where the user has a wide choice

7. TRAFFIC MONITORING POINTS UsersSites Web proxy observation point Some-to-All Web server observation point All-to-One UsersSites

8. OBSERVED REQUEST DISTRIBUTIONS

11. OBSERVED DISTRIBUTIONS OF USERS AMONG SITES

12. THE 20/80 (10/90) RULE The proportion of requests for the top documents is overestimated Fixed proportion rules are false

13. GENERAL COMMENTS When fitted by linear regression via Zipf’s law the estimated parameter typically lies in the 0.6-0.85 range All log-log frequency-rank plots exhibit an initial concavity (top objects’ preferences are overestimated) All log-log frequency-count plots exhibit final (count vs. frequency) spreading

14. OPEN ISSUES Search for better models (solving the initial concavity problem) Search for parameter estimation methds other than linear regression Definition of proper goodness-of-fit tests