Abstract
The present paper contains a specification of the EM algorithm in order to fit an empirical counting process, observed at discrete times, to a Markovian arrival process. The given data are the numbers of observed events in disjoint time intervals. The underlying phase process is not observable. An exact numerical procedure to compute the E and M steps is given.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Albert, A.: Estimating the infinitesimal generator of a continuous time, finite state Markov process. Ann. Math. Stat. 33, 727–753 (1962)
Asmussen, S.: Phase-type distributions and related point processes: Fitting and recent advances. In: Chakravarthy, Alfa (eds.) Matrix-analytic methods in stochastic models. Lect. Notes Pure Appl. Math, vol. 183, pp. 137–149. Marcel Dekker, NY (1997)
Asmussen, S., Koole, G.: Marked point processes as limits of Markovian arrival streams. J. Appl. Probab. 30(2), 365–372 (1993)
Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23(4), 419–441 (1996)
Bladt, M., Gonzalez, A., Lauritzen, S.: The estimation of phase-type related functionals using Markov chain Monte Carlo methods. Scandinavian Actuarial Journal 2003(4), 280–300 (2003)
Bladt, M., Soerensen, M.: Statistical inference for discretely observed Markov jump processes. Journal of the Royal Statistical Society: Series B 67(3), 395–410 (2005)
Breuer, L.: An EM Algorithm for Batch Markovian Arrival Processes and its Comparison to a Simpler Estimation Procedure. Annals of Operations Research 112, 123–138 (2002)
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)
Buchholz, P., Kriege, J.: A Heuristic Approach for Fitting MAPs to Moments and Joint Moments. In: 2009 Sixth International Conference on the Quantitative Evaluation of Systems, pp. 53–62. IEEE Computer Society, Los Alamitos (2009)
Carbonell, F., Jimenez, J.C., Pedroso, L.M.: Computing multiple integrals involving matrix exponentials. Journal of Computational and Applied Mathematics 213, 300–305 (2007)
Casale, G., Zhang, Z., Smirni, E.: KPC-Toolbox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes. In: International Conference on Quantitative Evaluation of Systems, pp. 83–92. IEEE Comp. Soc., Los Alamitos (2008)
Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. Discussion. J. R. Stat. Soc., Ser. B 39, 1–38 (1977)
Fearnhead, P., Sherlock, C.: An exact Gibbs sampler for the Markov–modulated Poisson process. J. R. Statist. Soc. B 68(5), 767–784 (2006)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Horvath, G., Telek, M., Buchholz, P.: A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation. In: QEST 2005: Proc Sec. Inter. Conf. Quant. Eval. Syst., p. 124. IEEE Comp. Soc., Los Alamitos (2005)
Jewell, N.P.: Mixtures of exponential distributions. Ann. Stat. 10, 479–484 (1982)
Kent, J.: The complex Bingham distribution and shape analysis. J.R. Statist. Soc. Series B 56, 285–289 (1994)
Klemm, A., Lindemann, C., Lohmann, M.: Modeling IP traffic using the batch Markovian arrival process. In: Perform. Eval., pp. 149–173. Elsevier Science Publishers B. V., Amsterdam (2003)
Kume, A., Wood, A.T.A.: On the normalising constant of the Bingham distribution. Statistics and Probability Letters 77, 832–837 (2007)
Leroux, B.G., Puterman, M.L.: Maximum-Penalized-Likelihood Estimation for Independent and Markov-Dependent Mixture Models. Biometrics 48, 545–558 (1992)
Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat., Stochastic Models 7(1), 1–46 (1991)
Lucantoni, D.M.: The BMAP/G/1 Queue: A Tutorial. In: Donatiello, L., Nelson, R. (eds.) SIGMETRICS 1993 and Performance 1993. LNCS, vol. 729, pp. 330–358. Springer, Heidelberg (1993)
McLachlan, G.J., Krishnan, T.: The EM algorithm and extensions. John Wiley & Sons, New York (1997)
Meng, X.-L., van Dyk, D.: The EM algorithm - an old folk-song sung to a fast new tune. J. R. Stat. Soc., Ser. B 59(3), 511–567 (1997)
Neuts, M.F.: A versatile Markovian point process. J. Appl. Probab. 16, 764–774 (1979)
Nijenhuis, A., Herbert, S.W.: Combinatorial Algorithms. Academic Press, London (1978)
Ryden, T.: An EM algorithm for estimation in Markov-modulated Poisson processes. Comput. Stat. Data Anal. 21(4), 431–447 (1996)
Okamura, H., Dohi, T., Trivedi, S.K.: Markovian arrival process parameter estimation with group data. IEEE/ACM Trans. Netw. 17(4), 1326–1339 (2009)
Ryden, T.: Estimating the order of continuous phase-type distributions and Markovmodulated Poisson processes. Commun. Stat., Stochastic Models 13(3), 417–433 (1997)
Sundberg, R.: Maximum Likelihood Theory for Incomplete Data from an Exponential Family. Scand. J. Statist. 1, 49–58 (1974)
Van Loan, C.F.: Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control 23, 395–404 (1978)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Breuer, L., Kume, A. (2010). An EM Algorithm for Markovian Arrival Processes Observed at Discrete Times. In: Müller-Clostermann, B., Echtle, K., Rathgeb, E.P. (eds) Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance. MMB&DFT 2010. Lecture Notes in Computer Science, vol 5987. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12104-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-12104-3_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12103-6
Online ISBN: 978-3-642-12104-3
eBook Packages: Computer ScienceComputer Science (R0)