Efficient Implementations of the EM-Algorithm for Transient Markovian Arrival Processes

  • Mindaugas Bražėnas
  • Gábor Horváth
  • Miklós Telek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9845)

Abstract

There are real life applications (e.g., requests of http sessions in web browsing) with a finite number of events and correlated inter-arrival times. Terminating point processes can be used to model such behavior. Transient Markov arrival processes (TMAPs) are computationally appealing terminating point processes which are terminating versions of Markov arrival processes.

In this work we propose algorithms for creating a TMAP based on empirical measurement data and compare various (series/parallel, CPU/GPU) implementations of the EM method for TMAP fitting.

References

  1. 1.
    Daley, D.J., Vere-Jones, D.: Finite point processes. In: Daley, D.J., Vere-Jones, D. (eds.) An Introduction to the Theory of Point Processes, pp. 111–156. Springer, New York (2003)Google Scholar
  2. 2.
    Anderson, T.W., Goodman, L.A.: Statistical inference about Markov chains. Ann. Math. Stat. 28, 89–110 (1957)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23, 419–441 (1996)MATHGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Hautphenne, S., Latouche, G.: The Markovian binary tree applied to demography. J. Math. Biol. 64, 1109–1135 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hautphenne, S., Telek, M.: Extension of some MAP results to transient MAPs and Markovian binary trees. Perform. Eval. 70(9), 607–622 (2013)CrossRefGoogle Scholar
  7. 7.
    Horváth, G., Okamura, H.: A fast EM algorithm for fitting marked Markovian arrival processes with a new special structure. In: Balsamo, M.S., Knottenbelt, W.J., Marin, A. (eds.) EPEW 2013. LNCS, vol. 8168, pp. 119–133. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Latouche, G., Remiche, M.-A., Taylor, P.: Transient Markov arrival processes. Ann. Appl. Probab. 13, 628–640 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Okamura, H., Dohi, T.: Faster maximum likelihood estimation algorithms for Markovian arrival processes. In: Sixth International Conference on the Quantitative Evaluation of Systems, QEST 2009, pp. 73–82. IEEE (2009)Google Scholar
  10. 10.
    Thümmler, A., Buchholz, P., Telek, M.: A novel approach for phase-type fitting with the EM algorithm. IEEE Trans. Dependable Secure Comput. 3(3), 245–258 (2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mindaugas Bražėnas
    • 1
  • Gábor Horváth
    • 2
  • Miklós Telek
    • 2
    • 3
  1. 1.Department of Applied MathematicsKaunas University of TechnologyKaunasLithuania
  2. 2.Department of Networked Systems and ServicesBudapest University of Technology and EconomicsBudapestHungary
  3. 3.MTA-BME Information Systems Research GroupBudapestHungary

Personalised recommendations