1. An Analytical Model for Network Flow Analysis
Ernesto Gomez, Yasha Karant, Keith Schubert
Institute for Applied Supercomputing
Department of Computer Science
CSU San Bernardino
The authors gratefully acknowledge the support of the NSF under award CISE

2. Outline Networks and Flows History Statistical Mechanics
Self-similar traffic
Traffic creation and destruction
Master Equation and traffic flow

3. One View of Network

4. Network Flows

5. Outline Networks and Flows History Statistical Mechanics
Self-similar traffic
Traffic creation and destruction
Master Equation and traffic flow

6. Brief History Shannon-Hartley (classical channel capacity)
C=B log2(1+SNR)
Leland, Taqqu, Willinger, Wilson, Paxon, …
Self-similar traffic
Cao, Cleveland, Lin, Sun, Ramanan
Poisson in limit

7. Stochastic vs. Analytic
Stochastic best tools currently
Opnet, NS
Problems
limiting cases
Improving estimates
Analytic (closed form equations)
Handles problems of stochastic
Insight into structure
Fluid models
Statistical Mechanics

8. Outline Networks and Flows History Statistical Mechanics
Self-similar traffic
Traffic creation and destruction
Master Equation and traffic flow

9. Overview Large number of entities
Bulk properties
Equilibrium or non-equilibrium properties
Time-dependence
Conservation over ensemble averages
Can handle classical and quantum flows

10. Density Matrix Formalism
Each component
Label by state
n = node source and destination
f = flow index
c = flow characteristics
t = time step

11. Density Matrix II Probability of a flow Element in Density Matrix is
Averaged Properties

12. Outline Networks and Flows History Statistical Mechanics
Self-similar traffic
Traffic creation and destruction
Master Equation and traffic flow

13. Poisson Distribution
 is mean
Thin Tail

14. Problem with Poisson Burst Long-range dependence
Extended period above the mean
Variety of timescales
Long-range dependence
Poisson or Markovian arrivals
Characteristic burst length
Smoothed by averaging over time
Real distribution is self-similar or multifractal
Proven for Ethernet

15. Real versus Poisson

16. Pareto Distribution Shape parameter () Location parameter (k)
Smaller means heavier tail
Infinite varience when 2 ≥ 
Infinite mean when 1 ≥ 
Location parameter (k)
t≥k

17. Pareto Distribution

18. Outline Networks and Flows History Statistical Mechanics
Self-similar traffic
Traffic creation and destruction
Master Equation and traffic flow

19. Flow Origination Unicast Multicast One source One destination
Many segments
Multicast
Many destinations

20. Multicast Possibilities

21. Outline Networks and Flows History Statistical Mechanics
Self-similar traffic
Traffic creation and destruction
Master Equation and traffic flow

22. Probability in Density Matrix
Tr = eHt (H is energy function)
Tr= (1+t/tns)-1
Cauchy Boundary conditions
hypersurface of flow space
Ill behaved
Gaussian quadrature, Monte Carlo, Pade Approximation

23. Unicast Flow Time

24. Future Directions
More detailed network
Bulk properties
Online tool