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Abstract

Phase Type Distributions (PHDs) and Markovian Arrival Processes (MAPs) are versatile models for the modeling of timing behavior in stochastic models. The parameterization of the models according to measured traces is often done using Expectation Maximization (EM) algorithms, which have long runtimes when applied to realistic datasets. In this paper, new versions of EM algorithms are presented that use only an aggregated version of the trace. Experiments show that these realizations of EM algorithms are much more efficient than available EM algorithms working on the complete trace and the fitting quality remains more or less the same.

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References

  1. Artalejo, J.R., Gomez-Corral, A., He, Q.M.: Markovian arrivals in stochastic modelling: a survey and some new results. Sort: Statistics and Operations Research Transactions 34(2), 101–156 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM-algorithm. Scand. J. Stat. 23(4), 419–441 (1996)

    MATH  Google Scholar 

  3. Breuer, L.: An EM algorithm for batch Markovian arrival processes and its comparison to a simpler estimation procedure. Annals OR 112(1-4), 123–138 (2002)

    Article  MathSciNet  Google Scholar 

  4. Buchholz, P.: An EM-algorithm for MAP fitting from real traffic data. In: Kemper, P., Sanders, W.H. (eds.) TOOLS 2003. LNCS, vol. 2794, pp. 218–236. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  5. Buchholz, P., Felko, I., Kriege, J.: Input Modeling with Phase Type Distributions and Markov Models. Springer (to appear)

    Google Scholar 

  6. Buchholz, P., Kriege, J.: A heuristic approach for fitting MAPs to moments and joint moments. In: QEST, pp. 53–62 (2009)

    Google Scholar 

  7. Bux, W., Herzog, U.: The phase concept: approximation of measured data and performance analysis. In: Computer Performance, pp. 23–38 (1977)

    Google Scholar 

  8. Casale, G., Zhang, E.Z., Smirni, E.: KPC-toolbox: Simple yet effective trace fitting using Markovian arrival processes. In: QEST, pp. 83–92 (2008)

    Google Scholar 

  9. Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance Evaluation (1998)

    Google Scholar 

  10. Gerhardt, I., Nelson, B.L.: On capturing dependence in point processes: Matching moments and other techniques. Technical report, Northwestern Univ. (2009)

    Google Scholar 

  11. Heindl, A., Telek, M.: Output models of MAP/PH/1(/K) queues for an efficient network decomposition. Perform. Eval. 49(1/4), 321–339 (2002)

    Article  Google Scholar 

  12. Horváth, A., Telek, M.: Matching more than three moments with acyclic phase type distributions. Stochastic Models 23, 167–194 (2007)

    Article  MathSciNet  Google Scholar 

  13. Horváth, G., Telek, M., Buchholz, P.: A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag-correlation. In: QEST, pp. 124–133. IEEE CS Press (2005)

    Google Scholar 

  14. Johnson, M.A., Taaffe, M.R.: Matching moments to phase distributions: Nonlinear programming approaches. Stochastic Models 2(6), 259–281 (1990)

    Article  MathSciNet  Google Scholar 

  15. El Abdouni Khayari, R., Sadre, R., Haverkort, B.: Fitting world-wide web request traces with the EM-algorithm. Performance Evaluation 52, 175–191 (2003)

    Article  Google Scholar 

  16. Klemm, A., Lindemann, C., Lohmann, M.: Modeling IP traffic using the batch Markovian arrival process. Perform. Eval. 54(2), 149–173 (2003)

    Article  Google Scholar 

  17. Kriege, J., Buchholz, P.: An Empirical Comparison of MAP Fitting Algorithms. In: Müller-Clostermann, B., Echtle, K., Rathgeb, E.P. (eds.) MMB & DFT 2010. LNCS, vol. 5987, pp. 259–273. Springer, Heidelberg (2010)

    Google Scholar 

  18. Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V.: On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Trans. Netw. (1994)

    Google Scholar 

  19. McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. John Wiley and Sons (1997)

    Google Scholar 

  20. Neuts, M.F.: A versatile Markovian point process. Jour. of Appl. Probability (1979)

    Google Scholar 

  21. Neuts, M.F.: Matrix-geometric solutions in stochastic models. Johns Hopkins University Press (1981)

    Google Scholar 

  22. O’Cinneide, C.A.: Phase-type distributions: open problems and a few properties. Stochastic Models 15(4), 731–757 (1999)

    Article  MathSciNet  Google Scholar 

  23. Okamura, H., Dohi, T., Trivedi, K.S.: Markovian arrival process parameter estimation with group data. IEEE/ACM Trans. Netw. 17(4), 1326–1339 (2009)

    Article  Google Scholar 

  24. Okamura, H., Dohi, T., Trivedi, K.S.: A refined EM algorithm for PH distributions. Perform. Eval. 68(10), 938–954 (2011)

    Article  Google Scholar 

  25. Okamura, H., Dohi, T., Trivedi, K.S.: Improvement of expectation-maximization algorithm for phase-type distributions with grouped and truncated data. Applied Stochastic Models in Business and Industry 29(2), 141–156 (2012)

    Article  MathSciNet  Google Scholar 

  26. Panchenko, A., Thümmler, A.: Efficient phase-type fitting with aggregated traffic traces. Perform. Eval. 64(7-8), 629–645 (2007)

    Article  Google Scholar 

  27. Paxson, V., Floyd, S.: Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Netw. 3(3), 226–244 (1995)

    Article  Google Scholar 

  28. Stewart, W.J.: Introduction to the numerical solution of Markov chains. Princeton University Press (1994)

    Google Scholar 

  29. Thümmler, A., Buchholz, P., Telek, M.: A novel approach for phase-type fitting with the EM algorithm. IEEE Trans. Dep. Sec. Comput. 3(3), 245–258 (2006)

    Article  Google Scholar 

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Kriege, J., Buchholz, P. (2014). PH and MAP Fitting with Aggregated Traffic Traces. In: Fischbach, K., Krieger, U.R. (eds) Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance. MMB&DFT 2014. Lecture Notes in Computer Science, vol 8376. Springer, Cham. https://doi.org/10.1007/978-3-319-05359-2_1

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  • DOI: https://doi.org/10.1007/978-3-319-05359-2_1

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-05358-5

  • Online ISBN: 978-3-319-05359-2

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