Parameter Fitting for Phase Type Distributions

  • Peter Buchholz
  • Jan Kriege
  • Iryna Felko
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


An important step when developing models that will be subject to a numerical or simulative analysis is the definition of input data for e.g. inter-event or service times which is denoted as input modeling. Usually, one has some observations measured in a real system, called trace, and tries to estimate (fit) the parameters of a distribution, such that the distribution captures characteristics of the given data. In this book we consider Markov processes as models for the data.


  1. 6.
    Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM-algorithm. Scand. J. Stat. 23(4), 419–441 (1996)zbMATHGoogle Scholar
  2. 7.
    Atkinson, K.A.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)zbMATHGoogle Scholar
  3. 15.
    Bilmes, J.: A gentle tutorial on the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models. Technical Report TR-97-021, University of Berkeley (1997)Google Scholar
  4. 18.
    Bobbio, A., Cumani, A.: ML estimation of the parameters of a PH distribution in triangular canonical form. In: Balbo, G., Serazzi, G. (eds.) Computer Performance Evaluation, pp. 33–46. Elsevier, Amsterdam (1992)Google Scholar
  5. 19.
    Bobbio, A., Telek, M.: Parameter estimation of phase type distributions. Technical Report R.T.423, Instituto Elettrotechnico Nazional Galileo Ferraris (1997)Google Scholar
  6. 20.
    Bobbio, A., Horváth, A., Scarpa, M., Telek, M.: Acyclic discrete phase type distributions: properties and a parameter estimation algorithm. Perform. Eval. 54(1), 1–32 (2003)CrossRefGoogle Scholar
  7. 21.
    Bobbio, A., Horváth, A., Telek, M.: Matching three moments with minimal acyclic phase type distributions. Stoch. Model. 21(2–3), 303–326 (2005)CrossRefzbMATHGoogle Scholar
  8. 23.
    Bodrog, L., Heindl, A., Horváth, G., Telek, M., Horváth, A.: Current results and open questions on PH and MAP characterization. In: Bini, D., Meini, B., Ramaswami, V., Remiche, M., Taylor, P. (eds.) Numerical Methods for Structured Markov Chains, No. 07461 in Dagstuhl Seminar Proceedings (2008)Google Scholar
  9. 31.
    Buchholz, P.: An EM-algorithm for MAP fitting from real traffic data. In: Kemper, P., Sanders, W.H. (eds.) Computer Performance Evaluation/TOOLS. Lecture Notes in Computer Science, vol. 2794, pp. 218–236. Springer, New York (2003)Google Scholar
  10. 32.
    Buchholz, P., Kriege, J.: A heuristic approach for fitting MAPs to moments and joint moments. In: Proceedings of the 6th International Conference on the Quantitative Evaluation of Systems (QEST), pp. 53–62. IEEE Computer Society, Budapest (2009)Google Scholar
  11. 35.
    Buchholz, P., Panchenko, A.: An EM algorithm for fitting of real traffic traces to PH-distribution. In: Proceedings of the International Conference on Parallel Computing in Electrical Engineering, PARELEC, pp. 283–288. IEEE Computer Society, Dresden (2004)Google Scholar
  12. 44.
    Collection of availability traces.
  13. 47.
    Cumani, A.: On the canonical representation of homogeneous Markov processes modeling failure-time distributions. Micorelectron. Reliab. 22(3), 583–602 (1982)CrossRefMathSciNetGoogle Scholar
  14. 51.
    Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39(1), 1–38 (1977)zbMATHMathSciNetGoogle Scholar
  15. 54.
    Fackrell, M.: Characterization of matrix-exponential distributions. Ph.D. thesis, School of Applied Mathematics, The University of Adelaide (2003)Google Scholar
  16. 56.
    Failure trace archive.
  17. 58.
    Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Perform. Eval. 31, 245–279 (1998)CrossRefGoogle Scholar
  18. 60.
    Fox, B.L., Glynn, P.W.: Computing Poisson probabilities. Commun. ACM. 31(4), 440–445 (1988)CrossRefMathSciNetGoogle Scholar
  19. 74.
    Horváth, A., Telek, M.: Approximating heavy tailed behavior with phase type distributions. In: Proceedings of the 3rd International Conference on Matrix-Analytic Methods in Stochastic Models. Leuven, Belgium (2000)Google Scholar
  20. 76.
    Horváth, A., Telek, M.: Markovian modeling of real data traffic: Heuristic phase type and MAP fitting of heavy tailed and fractal like samples. In: Calzarossa, M.C., Tucci, S. (eds.) Proceedings of the Performance 2002. Lecture Notes in Computer Science, vol. 2459, pp. 405–434. Springer, Berlin (2002)Google Scholar
  21. 77.
    Horváth, A., Telek, M.: Matching more than three moments with acyclic phase type distributions. Stoch. Model. 23, 167–194 (2007)CrossRefzbMATHGoogle Scholar
  22. 80.
    Horváth, A., Rácz, S., Telek, M.: Moments characterization of order 3 matrix exponential distributions. In: Al-Begain, K., Fiems, D., Horváth, G. (eds.) Proceedings of the Analytical and Stochastic Modeling Techniques and Applications (ASMTA). Lecture Notes in Computer Science, vol. 5513, pp. 174–188. Springer, Berlin (2009)CrossRefGoogle Scholar
  23. 84.
    The internet traffic archive.
  24. 87.
    Johnson, M.: Selecting parameters of phase distributions: combining nonlinear programming, heuristics, and Erlang distributions. INFORMS J. Comput. 5(1), 69–83 (1993)CrossRefzbMATHGoogle Scholar
  25. 89.
    Johnson, M., Taaffe, M.: Matching moments to phase distributions: nonlinear programming approaches. Stoch. Model. 2(6), 259–281 (1990)CrossRefMathSciNetGoogle Scholar
  26. 90.
    Jordan, M.I., Jacobs, R.A.: Hierarchical mixtures of experts and the EM algorithm. Neural Comput. 6(2), 181–214 (1994)CrossRefGoogle Scholar
  27. 94.
    Khayari, R.E.A., Sadre, R., Haverkort, B.: Fitting world-wide web request traces with the EM-algorithm. Perform. Eval. 52, 175–191 (2003)CrossRefGoogle Scholar
  28. 97.
    Klemm, A., Lindemann, C., Lohmann, M.: Modeling IP traffic using the batch Markovian arrival process. Perform. Eval. 54(2), 149–173 (2003)CrossRefGoogle Scholar
  29. 100.
    Krijnen, W.P.: Convergence of the sequence of parameters generated by alternating least squares algorithms. Comput. Stat. Data Anal. 51, 481–489 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 107.
    Lawson, C.L., Hanson, B.J.: Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs (1974)zbMATHGoogle Scholar
  31. 110.
    van de Liefvoort, A.: The moment problem for continuous distributions. Technical Report WP-CM-1990-02, University of Missouri, Kansas City (1990)Google Scholar
  32. 117.
    McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. Wiley, Hoboken (1997)zbMATHGoogle Scholar
  33. 134.
    Okamura, H., Dohi, T., Trivedi, K.S.: A refined EM algorithm for PH distributions. Perform. Eval. 68(10), 938–954 (2011)CrossRefGoogle Scholar
  34. 135.
    Okamura, H., Dohi, T., Trivedi, K.S.: Improvement of expectation-maximization algorithm for phase-type distributions with grouped and truncated data. Appl. Stoch. Model. Bus. Ind. 29(2), 141–156 (2012)CrossRefMathSciNetGoogle Scholar
  35. 136.
    Olsson, M.: The EMpht-programme. Technical Report, Chalmers University of Technology (1998)Google Scholar
  36. 137.
    Osogami, T., Harchol-Balter, M.: A closed-form solution for mapping general distributions to minimal PH distributions. In: Kemper, P., Sanders, W.H. (eds.) Computer Performance Evaluation. Modelling Techniques and Tools. Lecture Notes in Computer Science, vol. 2794, pp. 200–217. Springer, Berlin (2003)CrossRefGoogle Scholar
  37. 138.
    Osogami, T., Harchol-Balter, M.: Necessary and sufficient conditions for representing general distributions by Coxians. In: Kemper, P., Sanders, W.H. (eds.) Computer Performance Evaluation. Modelling Techniques and Tools. Lecture Notes in Computer Science, vol. 2794, pp. 182–199. Springer, Berlin (2003)CrossRefGoogle Scholar
  38. 139.
    Panchenko, A., Thümmler, A.: Efficient phase-type fitting with aggregated traffic traces. Perform. Eval. 64(7–8), 629–645 (2007)CrossRefGoogle Scholar
  39. 142.
  40. 146.
    Riska, A., Diev, V., Smirni, E.: An EM-based technique for approximating long-tailed data sets with PH distributions. Perform. Eval. 55, 147–164 (2004)CrossRefGoogle Scholar
  41. 148.
    Sauer, C.H., Chandy, K.M.: Computer Systems Performance Modeling. Prentice Hall, Englewood Cliffs (1981)Google Scholar
  42. 149.
    Schmickler, L.: MEDA: mixed Erlang distributions as phase-type representations of empirical distribution functions. Stoch. Model. 8(1), 131–156 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 151.
    Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  44. 153.
    Takahashi, Y.: Asymptotic exponentiality of the tail of the waiting-time distribution in a PH/PH/c queue. Adv. Appl. Probab. 13(3), 619–630 (1981)CrossRefzbMATHGoogle Scholar
  45. 154.
    Telek, M., Heindl, A.: Matching moments for acyclic discrete and continuous phase-type distributions of second order. Int. J. Simulat. Syst. Sci. Tech. 3(3–4), 47–57 (2002). [Special Issue on: Analytical and Stochastic Modelling Techniques]Google Scholar
  46. 155.
    Telek, M., Horváth, G.: A minimal representation of Markov arrival processes and a moments matching method. Perform. Eval. 64(9–12), 1153–1168 (2007)CrossRefGoogle Scholar
  47. 156.
    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)CrossRefGoogle Scholar
  48. 159.
    Vehicular mobility trace of the city of Cologne, Germany.
  49. 160.
    Wu, C.F.J.: On the convergence properties of the EM algorithm. Ann. Stat. 11(1), 95–103 (1983)CrossRefzbMATHGoogle Scholar

Copyright information

© Peter Buchholz, Jan Kriege, Iryna Felko 2014

Authors and Affiliations

  • Peter Buchholz
    • 1
  • Jan Kriege
    • 1
  • Iryna Felko
    • 1
  1. 1.Department of Computer ScienceTechnical University of DortmundDortmundGermany

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