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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 98-10708

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

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One View of Network

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Network Flows

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

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Brief History Shannon-Hartley (classical channel capacity) Shannon-Hartley (classical channel capacity) C=B log 2 (1+SNR) C=B log 2 (1+SNR) Leland, Taqqu, Willinger, Wilson, Paxon, … Leland, Taqqu, Willinger, Wilson, Paxon, … Self-similar traffic Self-similar traffic Cao, Cleveland, Lin, Sun, Ramanan Cao, Cleveland, Lin, Sun, Ramanan Poisson in limit Poisson in limit

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

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

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

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

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

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

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Poisson Distribution is mean Thin Tail

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

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Real versus Poisson

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

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Pareto Distribution

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

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

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Multicast Possibilities

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

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

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Unicast Flow Time

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Future Directions More detailed network More detailed network Bulk properties Bulk properties Online tool Online tool

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