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Giuliano Casale, Eddy Z. Zhang, Evgenia Giuliano Casale, Eddy Z. Zhang, Evgenia SmirniSmirni

{casale, eddy, esmirni}@cs.wm.edu{casale, eddy, esmirni}@cs.wm.edu

Speaker:Speaker:Giuliano CasaleGiuliano Casale

Numerical Methods for Structured Markov Chains, Dagstuhl Numerical Methods for Structured Markov Chains, Dagstuhl SeminarSeminar

November 11-14, 2007 November 11-14, 2007

College of William & Mary Department of Computer Science Williamsburg, 23187-8795, Virginia, US

Interarrival Times Interarrival Times Characterization and Fitting Characterization and Fitting

for Markovian Traffic for Markovian Traffic AnalysisAnalysis

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OutlineOutline MotivationsMotivations Review of MAP Fitting Algorithms Review of MAP Fitting Algorithms

from fitting counts to interarrival times (IAT) from fitting counts to interarrival times (IAT) fittingfitting

observations on eigenvalue-based methodsobservations on eigenvalue-based methods Jordan characterization of MAP moments Jordan characterization of MAP moments

and autocorrelationsand autocorrelations analysis of small MAPsanalysis of small MAPs

Composition of large MAPsComposition of large MAPs MAP fitting using higher-order MAP fitting using higher-order

correlationscorrelations

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MotivationMotivation MAP/MMPP Model ParameterizationMAP/MMPP Model Parameterization

Markovian models of network trafficMarkovian models of network traffic MAP closed queueing networks (see slides E. MAP closed queueing networks (see slides E.

Smirni)Smirni) MAP fitting is not fully understoodMAP fitting is not fully understood E.g., some questions:E.g., some questions:

Fit the counting process or the interarrival Fit the counting process or the interarrival process?process?

How many moments? Which correlation coeffs?How many moments? Which correlation coeffs? How fitting decisions affect queueing How fitting decisions affect queueing

prediction?prediction? Is nonlinear optimization appropriate?Is nonlinear optimization appropriate?

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MMPP Counting Process MMPP Counting Process FittingFitting

Measuring counts in networks can be often Measuring counts in networks can be often easier than measuring interarrival timeseasier than measuring interarrival times

S. Li & C.L. Hwang, 1992, 1993: S. Li & C.L. Hwang, 1992, 1993: Circulant matrices to impose MMPP power Circulant matrices to impose MMPP power

spectrumspectrum A.T. Andersen & B.F. Nielsen, 1998: A.T. Andersen & B.F. Nielsen, 1998:

Superposition of MMPP(2)s (Kronecker sum)Superposition of MMPP(2)s (Kronecker sum) Matching of the Hurst parameterMatching of the Hurst parameter Degrees of freedom for optional least-square Degrees of freedom for optional least-square

fitting of the interarrival time (IAT) fitting of the interarrival time (IAT) autocorrelations (ACF)autocorrelations (ACF)

Good accuracy on the Bellcore Aug89/Oct89 Good accuracy on the Bellcore Aug89/Oct89 tracestraces

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MAP Counting Process MAP Counting Process FittingFitting

A. Horváth & M. Telek, 2002: A. Horváth & M. Telek, 2002: Multifractal traffic model, e.g., Riedi et al., Multifractal traffic model, e.g., Riedi et al.,

19991999 Traffic analysis based on Haar wavelet Traffic analysis based on Haar wavelet

transformtransform Each MAP(2) describes variability in the Haar Each MAP(2) describes variability in the Haar

wavelet coefficients at a specific time scalewavelet coefficients at a specific time scale Almost optimal fitting of the BC-Aug89 traceAlmost optimal fitting of the BC-Aug89 trace

Further improvements may not be easy:Further improvements may not be easy: Higher-order moments of counts hard to Higher-order moments of counts hard to

manipulatemanipulate

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MAP Interarrival ProcessMAP Interarrival Process Two-phase fitting fitting of PH-type Two-phase fitting fitting of PH-type

distribution followed by fitting of IAT distribution followed by fitting of IAT ACFACF

Feasible manipulation of higher-order Feasible manipulation of higher-order moments moments

P. Buchholz et al., 2003, 2004:P. Buchholz et al., 2003, 2004: Expectation Maximization (EM) algorithmsExpectation Maximization (EM) algorithms Support for two-phase fittingSupport for two-phase fitting Scalability of EM rapidly increasing Scalability of EM rapidly increasing

((Panchenko & Thümmler, 2007Panchenko & Thümmler, 2007))

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MAP Interarrival ProcessMAP Interarrival Process Moment and ACF Analytical Fitting:Moment and ACF Analytical Fitting:

Results only for MMPP(2), MAP(2), MAP(3)Results only for MMPP(2), MAP(2), MAP(3) G. Horváth, M. Telek & P. Buchholz, 2005: G. Horváth, M. Telek & P. Buchholz, 2005:

Two-phase least-square fitting of PH distrib. and Two-phase least-square fitting of PH distrib. and ACFACF

Optimization variables are the MAP transition Optimization variables are the MAP transition rates, i.e., the O(nrates, i.e., the O(n22) entries of the D) entries of the D

00 and D and D11

matricesmatrices Simple to understand and implementSimple to understand and implement

Least-squares can be numerically difficult: Least-squares can be numerically difficult: small magnitude of transition rates compared to small magnitude of transition rates compared to

tolerancetolerance infeasibility due to inappropriate choice of step size infeasibility due to inappropriate choice of step size

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Our observationsOur observations Observation 1Observation 1: eigenvalues give direct control : eigenvalues give direct control

to the nonlinear solver on ACF decay and CDF to the nonlinear solver on ACF decay and CDF tailtail

Observation 2Observation 2: lack of general Jordan analysis : lack of general Jordan analysis of IAT moments and autocorrelationsof IAT moments and autocorrelations

Observation 3Observation 3: eigenvalue-based least-squares : eigenvalue-based least-squares tends to be numerically well-behavedtends to be numerically well-behaved

Observation 4Observation 4: inverse eigenvalue problems : inverse eigenvalue problems often prohibitive, how do we determine Doften prohibitive, how do we determine D

00 and and DD11?? Superposition does not help for IAT processSuperposition does not help for IAT process

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Our contributionsOur contributions

A general Jordan analysis of MAP A general Jordan analysis of MAP moments and autocorrelationsmoments and autocorrelations

Using this characterization we analyze Using this characterization we analyze the IAT process in small MAPsthe IAT process in small MAPs

We find a compositional approach to We find a compositional approach to define the IAT process in large MAPs define the IAT process in large MAPs using small MAPsusing small MAPs

Main resultMain result: A least-squares that can : A least-squares that can fit IAT moments and correlations fit IAT moments and correlations of of any orderany order

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Why statistics of “any Why statistics of “any order”?order”?

Literature evaluates up to second-Literature evaluates up to second-order propertiesorder properties Higher-order correlations are neglected, Higher-order correlations are neglected,

but....but....

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MAP Jordan Analysis MAP Jordan Analysis Definition: MAP moments Definition: MAP moments

Definition: MAP autocorrelationsDefinition: MAP autocorrelations

Moments and correlations depend on matrix Moments and correlations depend on matrix powerspowers Eigenvalues explicited by the Cayley-Hamilton Eigenvalues explicited by the Cayley-Hamilton

theoremtheorem

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MAP Jordan Analysis – MAP Jordan Analysis – Cont'dCont'd

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MAP(3) Characterization MAP(3) Characterization ExampleExample

Define MAPs with given oscillatory Define MAPs with given oscillatory ACFACF Generalization of Circulant MAPs to IAT Generalization of Circulant MAPs to IAT

processprocessMAP DefinitionMAP Definition CharacterizationCharacterization

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MAP(3) Characterization MAP(3) Characterization ExampleExample

SCV=4.87p1=0p2=0.0286

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Composition of Large Composition of Large ProcessesProcesses

Idea:Idea: use Kronecker product to overcome use Kronecker product to overcome inverse eigenvalue problem in eigenvalue-inverse eigenvalue problem in eigenvalue-based fittingsbased fittings

Kronecker product composition (KPC)Kronecker product composition (KPC)

One of the two DOne of the two D00 matrices must be diagonal matrices must be diagonal

No loss of generalityNo loss of generality Prevents negative (infeasible) off-diagonal Prevents negative (infeasible) off-diagonal

entries inentries inthe Dthe D

0 0 matrix of the KPC matrix of the KPC

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Jordan Analysis of KPC Jordan Analysis of KPC processprocess

Eigenvalues and projectorsEigenvalues and projectors

Moments and autocorrelationsMoments and autocorrelations

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KPC ExampleKPC Example

To the best of our knowledge, never To the best of our knowledge, never shown in the literature a MAP with shown in the literature a MAP with lag-1 acf > 0.5lag-1 acf > 0.5

Does it exist? Does it exist? MAP(2) must have lag-1 acf <0.5MAP(2) must have lag-1 acf <0.5 Not found in 100.000 random MAP(3) Not found in 100.000 random MAP(3)

and MAP(4) and MAP(4) Answer: yes it exists, it can be Answer: yes it exists, it can be

defined by KPCdefined by KPC A simple MAP(2):A simple MAP(2):

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KPC Example – Cont'dKPC Example – Cont'd

What happens if we compose with What happens if we compose with KPC the MAP(2) with a PH renewal KPC the MAP(2) with a PH renewal process?process?

Composition with a Composition with a hypoexponential processhypoexponential process

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Jordan Analysis of KPC Jordan Analysis of KPC processprocess

IAT Joint MomentsIAT Joint Moments Joint moments, e.g.,G. Horváth & M. Telek, Joint moments, e.g.,G. Horváth & M. Telek,

20072007

Admits characterization similar to Admits characterization similar to moments/acfmoments/acf

Joint moments in KPC processJoint moments in KPC process

Conclusion:Conclusion: KPC can fit moments of any order KPC can fit moments of any order

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Two-Phase Least SquaresTwo-Phase Least Squares We determine We determine J J small MAPs to be small MAPs to be

composed by KPC in order to best fit composed by KPC in order to best fit a tracea trace

Lessons learned from Jordan analysis:Lessons learned from Jordan analysis: first fit ACF and SCV, then momentsfirst fit ACF and SCV, then moments

Phase 2:Phase 2: Fit moments Fit momentsPhase 1:Phase 1: ACV+SCVACV+SCV

eigenvalue-eigenvalue-basedbased mean and mean and

bispectrumbispectrum

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Results: BC-Aug89Results: BC-Aug89quality of fitting - MAP(16)quality of fitting - MAP(16)

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Results: Seagate-WebResults: Seagate-Webquality of fitting - MAP(16)quality of fitting - MAP(16)

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Results: BC-Aug89Results: BC-Aug89queueing prediction - MAP(16)queueing prediction - MAP(16)

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Results: Seagate WebResults: Seagate Webqueueing prediction - MAP(16)queueing prediction - MAP(16)

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ConclusionConclusion Jordan characterization allows:Jordan characterization allows:

analysis of simple MAP processes analysis of simple MAP processes least-square fitting that is numerically well-least-square fitting that is numerically well-

behavedbehaved Joint IAT moments required for accurate Joint IAT moments required for accurate

queueing prediction of real workloadsqueueing prediction of real workloads even bispectrum fitting leaves room for even bispectrum fitting leaves room for

improvementimprovement KPC indispensable for definition of large KPC indispensable for definition of large

processesprocesses Future workFuture work

Fitting traces with strong oscillatory patterns Fitting traces with strong oscillatory patterns (e.g., MPEG traces)(e.g., MPEG traces)

Comparison with circulant MAPs approachComparison with circulant MAPs approach