Abstract
In this paper similarity transformations for Acyclic Phase Type Distributions (APHs) are considered, and representations maximizing the first joint moment that can be reached when the distribution is expanded into a Markovian Arrival Process (MAP) are investigated. For the acyclic case the optimal representation corresponds to a hyperexponential representation, which is optimal among all possible representations that can be reached by similarity transformations. The parameterization aspect for the possible transformation of APHs into a hyperexponential form is revealed, together with corresponding transformation rules. For the case when APHs cannot be transformed into a hyperexponential representation a heuristic optimization method is presented to obtain good representations, while transformation methods to increase the first joint moment by adding additional phases are derived.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bladt, M., Nielson, B.F.: On the construction of bivariate exponential distributions with an arbitrary correlation coefficient. Stochastic Models 26, 295–308 (2010)
Buchholz, P., Felko, I., Kriege, J.: Online Companion to the Paper Transformation of Acyclic Phase Type Distributions for Correlation Fitting (2013), http://www4.cs.tu-dortmund.de/download/kriege/publications/asmta2013_online_companion.pdf
Buchholz, P., Kriege, J.: Equivalence Transformations for Acyclic Phase Type Distributions. Technical Report 827, Dep. of Informatics, TU Dortmund (2009)
Buchholz, P., Kriege, J.: A heuristic approach for fitting MAPs to moments and joint moments. In: QEST, pp. 53–62 (2009)
Buchholz, P., Panchenko, A.: A two-step EM algorithm for MAP fitting. In: Aykanat, C., Dayar, T., Körpeoğlu, İ. (eds.) ISCIS 2004. LNCS, vol. 3280, pp. 217–227. Springer, Heidelberg (2004)
Buchholz, P., Telek, M.: Stochastic Petri nets with matrix exponentially distributed firing times. Performance Evaluation 67(12), 1373–1385 (2010)
Corana, A., Marchesi, M., Martini, C., Ridella, S.: Minimizing multimodal functions of continuous variables with the ”simulated annealing” algorithm. ACM Trans. Math. Softw. 13(3), 262–280 (1987)
Cumani, A.: On the canonical representation of homogeneous Markov processes modeling failure-time distributions. Micorelectronics and Reliability 22(3), 583–602 (1982)
Fackrell, M.: Modelling healthcare systems with phase-type distributions. Health Care Management Science 12(1), 11–26 (2009)
Horváth, A., Telek, M.: PhFit: A general phase-type fitting tool. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, pp. 82–91. Springer, Heidelberg (2002)
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 (2005)
Iversen, V.B., Nielsen, B.F.: Some properties of Coxian distributions with applications. In: Modeling Techniques and Tools for Performance Analysis, pp. 61–66. Elsevier (1986)
Klemm, A., Lindemann, C., Lohmann, M.: Modeling IP traffic using the batch Markovian arrival process. Perform. Eval. 54(2), 149–173 (2003)
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)
Lipsky, L.: Queueing Theory: A Linear Algebraic Approach. Springer (2010)
Mészáros, A., Telek, M.: A two-phase MAP fitting method with APH interarrival time distribution. In: Winter Simulation Conference. ACM (2012)
Neuts, M.F.: A versatile Markovian point process. Journ. of Applied Probability 16 (1979)
Neuts, M.F.: Matrix-geometric solutions in stochastic models. Johns Hopkins University Press (1981)
Paxson, V., Floyd, S.: Wide-area traffic: The failure of Poisson modeling. IEEE/ACM Transactions on Networking 3, 226–244 (1995)
Telek, M., Horváth, G.: A minimal representation of Markov arrival processes and a moments matching method. Perform. Eval. 64(9-12), 1153–1168 (2007)
Telek, M., et al.: Butools - program packages, webspn.hit.bme.hu/~butools/
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Buchholz, P., Felko, I., Kriege, J. (2013). Transformation of Acyclic Phase Type Distributions for Correlation Fitting. In: Dudin, A., De Turck, K. (eds) Analytical and Stochastic Modeling Techniques and Applications. ASMTA 2013. Lecture Notes in Computer Science, vol 7984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39408-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-39408-9_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39407-2
Online ISBN: 978-3-642-39408-9
eBook Packages: Computer ScienceComputer Science (R0)