1 Giuliano Casale, Eddy Z. Zhang, Evgenia Smirni {casale, eddy, Speaker: Giuliano Casale Numerical Methods for Structured Markov Chains, Dagstuhl Seminar November 11-14, 2007 College of William & Mary Department of Computer Science Williamsburg, , Virginia, US Interarrival Times Characterization and Fitting for Markovian Traffic Analysis

2 Outline Motivations Motivations Review of MAP Fitting Algorithms Review of MAP Fitting Algorithms from fitting counts to interarrival times (IAT) fitting from fitting counts to interarrival times (IAT) fitting observations on eigenvalue-based methods observations on eigenvalue-based methods Jordan characterization of MAP moments and autocorrelations Jordan characterization of MAP moments and autocorrelations analysis of small MAPs analysis of small MAPs Composition of large MAPs Composition of large MAPs MAP fitting using higher-order correlations MAP fitting using higher-order correlations

3 Motivation MAP/MMPP Model Parameterization MAP/MMPP Model Parameterization Markovian models of network traffic Markovian models of network traffic MAP closed queueing networks (see slides E. Smirni) ‏ MAP closed queueing networks (see slides E. Smirni) ‏ MAP fitting is not fully understood MAP fitting is not fully understood E.g., some questions: E.g., some questions: Fit the counting process or the interarrival process? Fit the counting process or the interarrival process? How many moments? Which correlation coeffs? How many moments? Which correlation coeffs? How fitting decisions affect queueing prediction? How fitting decisions affect queueing prediction? Is nonlinear optimization appropriate? Is nonlinear optimization appropriate?

4 MMPP Counting Process Fitting Measuring counts in networks can be often easier than measuring interarrival times Measuring counts in networks can be often easier than measuring interarrival times S. Li & C.L. Hwang, 1992, 1993: S. Li & C.L. Hwang, 1992, 1993: Circulant matrices to impose MMPP power spectrum Circulant matrices to impose MMPP power spectrum 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 parameter Matching of the Hurst parameter Degrees of freedom for optional least-square fitting of the interarrival time (IAT) autocorrelations (ACF) ‏ Degrees of freedom for optional least-square fitting of the interarrival time (IAT) autocorrelations (ACF) ‏ Good accuracy on the Bellcore Aug89/Oct89 traces Good accuracy on the Bellcore Aug89/Oct89 traces

5 MAP Counting Process Fitting A. Horváth & M. Telek, 2002: A. Horváth & M. Telek, 2002: Multifractal traffic model, e.g., Riedi et al., 1999 Multifractal traffic model, e.g., Riedi et al., 1999 Traffic analysis based on Haar wavelet transform Traffic analysis based on Haar wavelet transform Each MAP(2) describes variability in the Haar wavelet coefficients at a specific time scale Each MAP(2) describes variability in the Haar wavelet coefficients at a specific time scale Almost optimal fitting of the BC-Aug89 trace Almost 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 manipulate Higher-order moments of counts hard to manipulate

6 MAP Interarrival Process Two-phase fitting fitting of PH-type distribution followed by fitting of IAT ACF Two-phase fitting fitting of PH-type distribution followed by fitting of IAT ACF Feasible manipulation of higher-order moments Feasible manipulation of higher-order moments P. Buchholz et al., 2003, 2004: P. Buchholz et al., 2003, 2004: Expectation Maximization (EM) algorithms Expectation Maximization (EM) algorithms Support for two-phase fitting Support for two-phase fitting Scalability of EM rapidly increasing (Panchenko & Thümmler, 2007) ‏ Scalability of EM rapidly increasing (Panchenko & Thümmler, 2007) ‏

7 MAP 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 ACF Two-phase least-square fitting of PH distrib. and ACF Optimization variables are the MAP transition rates, i.e., the O(n 2 ) entries of the D 0 and D 1 matrices Optimization variables are the MAP transition rates, i.e., the O(n 2 ) entries of the D 0 and D 1 matrices Simple to understand and implement Simple to understand and implement Least-squares can be numerically difficult: Least-squares can be numerically difficult: small magnitude of transition rates compared to tolerance small magnitude of transition rates compared to tolerance infeasibility due to inappropriate choice of step size infeasibility due to inappropriate choice of step size

8 Our observations Observation 1: eigenvalues give direct control to the nonlinear solver on ACF decay and CDF tail Observation 1: eigenvalues give direct control to the nonlinear solver on ACF decay and CDF tail Observation 2: lack of general Jordan analysis of IAT moments and autocorrelations Observation 2: lack of general Jordan analysis of IAT moments and autocorrelations Observation 3: eigenvalue-based least-squares tends to be numerically well-behaved Observation 3: eigenvalue-based least-squares tends to be numerically well-behaved Observation 4: inverse eigenvalue problems often prohibitive, how do we determine D 0 and D 1 ? Observation 4: inverse eigenvalue problems often prohibitive, how do we determine D 0 and D 1 ?  Superposition does not help for IAT process

9 Our contributions A general Jordan analysis of MAP moments and autocorrelations A general Jordan analysis of MAP moments and autocorrelations Using this characterization we analyze the IAT process in small MAPs Using this characterization we analyze the IAT process in small MAPs We find a compositional approach to define the IAT process in large MAPs using small MAPs We find a compositional approach to define the IAT process in large MAPs using small MAPs Main result: A least-squares that can fit IAT moments and correlations of any order Main result: A least-squares that can fit IAT moments and correlations of any order

10 Why statistics of “any order”? Literature evaluates up to second-order properties Literature evaluates up to second-order properties Higher-order correlations are neglected, but.... Higher-order correlations are neglected, but....

11 MAP Jordan Analysis Definition: MAP moments Definition: MAP moments Definition: MAP autocorrelations Definition: MAP autocorrelations Moments and correlations depend on matrix powers Moments and correlations depend on matrix powers Eigenvalues explicited by the Cayley-Hamilton theorem Eigenvalues explicited by the Cayley-Hamilton theorem

12 MAP Jordan Analysis – Cont'd

13 MAP(3) Characterization Example Define MAPs with given oscillatory ACF Define MAPs with given oscillatory ACF Generalization of Circulant MAPs to IAT process Generalization of Circulant MAPs to IAT process MAP Definition Characterization

14 MAP(3) Characterization Example SCV=4.87 p1=0 p2=0.0286

15 Composition of Large Processes Idea: use Kronecker product to overcome inverse eigenvalue problem in eigenvalue-based fittings Idea: use Kronecker product to overcome inverse eigenvalue problem in eigenvalue-based fittings Kronecker product composition (KPC) Kronecker product composition (KPC) One of the two D 0 matrices must be diagonal One of the two D 0 matrices must be diagonal No loss of generality No loss of generality Prevents negative (infeasible) off-diagonal entries in the D 0 matrix of the KPC Prevents negative (infeasible) off-diagonal entries in the D 0 matrix of the KPC

16 Jordan Analysis of KPC process Eigenvalues and projectors Eigenvalues and projectors Moments and autocorrelations Moments and autocorrelations

17 KPC Example To the best of our knowledge, never shown in the literature a MAP with lag-1 acf > 0.5 To the best of our knowledge, never shown in the literature a MAP with lag-1 acf > 0.5 Does it exist? Does it exist? MAP(2) must have lag-1 acf <0.5 MAP(2) must have lag-1 acf <0.5 Not found in random MAP(3) and MAP(4) Not found in random MAP(3) and MAP(4) Answer: yes it exists, it can be defined by KPC Answer: yes it exists, it can be defined by KPC A simple MAP(2): A simple MAP(2):

18 KPC Example – Cont'd What happens if we compose with KPC the MAP(2) with a PH renewal process? What happens if we compose with KPC the MAP(2) with a PH renewal process? Composition with a hypoexponential process Composition with a hypoexponential process

19 Jordan Analysis of KPC process IAT Joint Moments Joint moments, e.g.,G. Horváth & M. Telek, 2007 Joint moments, e.g.,G. Horváth & M. Telek, 2007 Admits characterization similar to moments/acf Admits characterization similar to moments/acf Joint moments in KPC process Joint moments in KPC process Conclusion: KPC can fit moments of any order Conclusion: KPC can fit moments of any order

20 Two-Phase Least Squares We determine J small MAPs to be composed by KPC in order to best fit a trace We determine J small MAPs to be composed by KPC in order to best fit a trace Lessons learned from Jordan analysis: Lessons learned from Jordan analysis: first fit ACF and SCV, then moments first fit ACF and SCV, then moments Phase 2: Fit moments Phase 1: ACV+SCV eigenvalue-based mean and bispectrum

21 Results: BC-Aug89 quality of fitting - MAP(16) ‏

22 Results: Seagate-Web quality of fitting - MAP(16) ‏

23 Results: BC-Aug89 queueing prediction - MAP(16) ‏

24 Results: Seagate Web queueing prediction - MAP(16) ‏

25 Conclusion Jordan characterization allows: Jordan characterization allows: analysis of simple MAP processes analysis of simple MAP processes least-square fitting that is numerically well-behaved least-square fitting that is numerically well-behaved Joint IAT moments required for accurate queueing prediction of real workloads Joint IAT moments required for accurate queueing prediction of real workloads even bispectrum fitting leaves room for improvement even bispectrum fitting leaves room for improvement KPC indispensable for definition of large processes KPC indispensable for definition of large processes Future work Future work Fitting traces with strong oscillatory patterns (e.g., MPEG traces) ‏ Fitting traces with strong oscillatory patterns (e.g., MPEG traces) ‏ Comparison with circulant MAPs approach Comparison with circulant MAPs approach