An Analytical Model for An Analytical Model for Network Flow AnalysisNetwork Flow Analysis

Ernesto Gomez, Yasha Karant, Keith SchubertErnesto Gomez, Yasha Karant, Keith SchubertInstitute for Applied SupercomputingInstitute for Applied Supercomputing

Department of Computer ScienceDepartment of Computer ScienceCSU San BernardinoCSU San Bernardino

The authors gratefully acknowledge the support of the The authors gratefully acknowledge the support of the NSF under award CISE 98-10708NSF under award CISE 98-10708

OutlineOutline

Networks and FlowsNetworks and FlowsHistoryHistoryStatistical MechanicsStatistical MechanicsSelf-similar trafficSelf-similar trafficTraffic creation and destructionTraffic creation and destructionMaster Equation and traffic flowMaster Equation and traffic flow

One View of NetworkOne View of Network

Network FlowsNetwork Flows

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Networks and FlowsNetworks and FlowsHistoryHistoryStatistical MechanicsStatistical MechanicsSelf-similar trafficSelf-similar trafficTraffic creation and destructionTraffic creation and destructionMaster Equation and traffic flowMaster Equation and traffic flow

Brief HistoryBrief History

Shannon-Hartley (classical channel Shannon-Hartley (classical channel capacity)capacity)C=B logC=B log22(1+SNR)(1+SNR)

Leland, Taqqu, Willinger, Wilson, Paxon, …Leland, Taqqu, Willinger, Wilson, Paxon, …Self-similar trafficSelf-similar traffic

Cao, Cleveland, Lin, Sun, RamananCao, Cleveland, Lin, Sun, RamananPoisson in limitPoisson in limit

Stochastic vs. AnalyticStochastic vs. Analytic

Stochastic best tools currentlyStochastic best tools currentlyOpnet, NSOpnet, NSProblemsProblems

limiting caseslimiting casesImproving estimatesImproving estimates

Analytic (closed form equations)Analytic (closed form equations)Handles problems of stochasticHandles problems of stochastic Insight into structureInsight into structureFluid modelsFluid modelsStatistical MechanicsStatistical Mechanics

OutlineOutline

Networks and FlowsNetworks and FlowsHistoryHistoryStatistical MechanicsStatistical MechanicsSelf-similar trafficSelf-similar trafficTraffic creation and destructionTraffic creation and destructionMaster Equation and traffic flowMaster Equation and traffic flow

OverviewOverview

Large number of entitiesLarge number of entitiesBulk propertiesBulk properties

Equilibrium or non-equilibrium propertiesEquilibrium or non-equilibrium propertiesTime-dependenceTime-dependence

Conservation over ensemble averagesConservation over ensemble averagesCan handle classical and quantum flowsCan handle classical and quantum flows

Density Matrix FormalismDensity Matrix Formalism

Each component Each component Label by stateLabel by state

n = node source and destinationn = node source and destinationf = flow indexf = flow indexc = flow characteristicsc = flow characteristicst = time stept = time step

tcfn ,,,

Density Matrix IIDensity Matrix II

Probability of a flow Probability of a flow

Element in Density Matrix isElement in Density Matrix is

Averaged PropertiesAveraged Properties

tcfn ,,,Pr

tcfntcfntcfn ,,,,,,,,,Pr

StcfntcfnStcfnxqxq ,,,,,,,,,Pr

OutlineOutline

Networks and FlowsNetworks and FlowsHistoryHistoryStatistical MechanicsStatistical MechanicsSelf-similar trafficSelf-similar trafficTraffic creation and destructionTraffic creation and destructionMaster Equation and traffic flowMaster Equation and traffic flow

Poisson DistributionPoisson Distribution

!

,t

etp

t

is mean

Thin Tail

Problem with PoissonProblem with Poisson

BurstBurstExtended period above the meanExtended period above the meanVariety of timescalesVariety of timescales

Long-range dependenceLong-range dependencePoisson or Markovian arrivalsPoisson or Markovian arrivals

Characteristic burst lengthCharacteristic burst lengthSmoothed by averaging over timeSmoothed by averaging over time

Real distribution is self-similar or multifractalReal distribution is self-similar or multifractalProven for EthernetProven for Ethernet

Real versus PoissonReal versus Poisson

Pareto DistributionPareto Distribution

Shape parameter (Shape parameter ())Smaller means heavier tailSmaller 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) tt≥k≥k

1 tktp

Pareto DistributionPareto Distribution

1 tktp

OutlineOutline

Networks and FlowsNetworks and FlowsHistoryHistoryStatistical MechanicsStatistical MechanicsSelf-similar trafficSelf-similar trafficTraffic creation and destructionTraffic creation and destructionMaster Equation and traffic flowMaster Equation and traffic flow

Flow OriginationFlow Origination

UnicastUnicastOne sourceOne sourceOne destinationOne destinationMany segmentsMany segments

MulticastMulticastOne sourceOne sourceMany destinationsMany destinations

Multicast PossibilitiesMulticast Possibilities

OutlineOutline

Networks and FlowsNetworks and FlowsHistoryHistoryStatistical MechanicsStatistical MechanicsSelf-similar trafficSelf-similar trafficTraffic creation and destructionTraffic creation and destructionMaster Equation and traffic flowMaster Equation and traffic flow

Probability in Density MatrixProbability in Density Matrix

Tr = eTr = eHtHt (H is energy function)(H is energy function) Tr= (1+t/tTr= (1+t/tnsns))-1-1 Cauchy Boundary conditions Cauchy Boundary conditions

hypersurface of flow spacehypersurface of flow space Ill behavedIll behaved Gaussian quadrature, Monte Carlo, Pade ApproximationGaussian quadrature, Monte Carlo, Pade Approximation

tptcfnTrtcfn

tptcfnTrtcfndt

tcfnd

outinout

ininoutin

,,,,,,

,,,,,,,,,Pr

Unicast Flow TimeUnicast Flow Time

Future DirectionsFuture Directions

More detailed networkMore detailed networkBulk propertiesBulk propertiesOnline toolOnline tool