Abstract
Approximating general distributions by phase-type (PH) distributions is a popular technique in queueing analysis, since the Markovian property of PH distributions often allows analytical tractability. This paper proposes an algorithm for mapping a general distribution G to a PH distribution where the goal is to find a PH distribution which matches the first three moments of G. Since efficiency of the algorithm is of primary importance, we first define a particular subset of the PH distributions, which we refer to as EC distributions. The class of EC distributions has very few free parameters, which narrows down the search space, making the algorithm efficient – In fact we provide a closed-form solution for the parameters of the EC distribution. Our solution is general in that it applies to any distribution whose first three moments can be matched by a PH distribution. Also, our resulting EC distribution requires a nearly minimal number of phases, always within one of the minimal number of phases required by any acyclic PH distribution. Lastly, we discuss numerical stability of our solution.
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References
Aldous, D., Shepp, L.: The least variable phase type distribution is Erlang. Communications in Statistics – Stotchastic Models 3, 467–473 (1987)
Altiok, T.: On the phase-type approximations of general distributions. IIE Transactions 17, 110–116 (1985)
Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance Evaluation 32, 245–279 (1998)
Franke, H., Jann, J., Moreira, J., Pattnaik, P., Jette, M.: An evaluation of parallel job scheduling for ASCI blue-pacific. In: Proceedings of Supercomputing 1999, pp. 679–691 (November 1999)
Harchol-Balter, M., Li, C., Osogami, T., Scheller-Wolf, A., Squillante, M.S.: Analysis of task assignment with cycle stealing under central queue. In: Proceedings of ICDCS 2003, pp. 628–637 (May 2003)
Horváth, A., Telek, M.: Approximating heavy tailed behavior with phase type distributions. In: Advances in Matrix-Analytic Methods for Stochastic Models, pp. 191–214. Notable Publications (July 2000)
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)
Johnson, M.A.: Selecting parameters of phase distributions: Combining nonlinear programming, heuristics, and Erlang distributions. ORSA Journal on Computing 5, 69–83 (1993)
Johnson, M.A., Taaffe, M.F.: An investigation of phase-distribution momentmatching algorithms for use in queueing models. Queueing Systems 8, 129–147 (1991)
Johnson, M.A., Taaffe, M.R.: Matching moments to phase distributions: Mixtures of Erlang distributions of common order. Communications in Statistics – Stochastic Models 5, 711–743 (1989)
Johnson, M.A., Taaffe, M.R.: Matching moments to phase distributions: Density function shapes. Communications in Statistics – Stochastic Models 6, 283–306 (1990)
Johnson, M.A., Taaffe, M.R.: Matching moments to phase distributions: Nonlinear programming approaches. Communications in Statistics – Stochastic Models 6, 259–281 (1990)
Karlin, S., Studden, W.: Tchebycheff Systems: With Applications in Analysis and Statistics. John Wiley and Sons, Chichester (1966)
Khayari, R.E.A., Sadre, R., Haverkort, B.: Fitting world-wide web request traces with the EM-algorithm. Performance Evalutation 52, 175–191 (2003)
Marie, R.: Calculating equilibrium probabilities for λ(n)/c k /1/n queues. In: Proceedings of Performance 1980, pp. 117–125 (1980)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore (1981)
Osogami, T., Harchol-Balter, M.: A closed-form solution for mapping general distributions to minimal PH distributions. Technical Report CMU-CS-03-114, School of Computer Science, Carnegie Mellon University (2003)
Osogami, T., Harchol-Balter, M.: Necessary and sufficient conditions for representing general distributions by Coxians. In: Kemper, P., Sanders, W.H. (eds.) TOOLS 2003. LNCS, vol. 2794, pp. 182–199. Springer, Heidelberg (2003)
Osogami, T., Harchol-Balter, M., Scheller-Wolf, A.: Analysis of cycle stealing with switching cost. In: Proceedings of Sigmetrics 2003, pp. 184–195 (June 2003)
Riska, A., Diev, V., Smirni, E.: Efficient fitting of long-tailed data sets into PH distributions. Performance Evaluation (2003) (to appear)
Sauer, C., Chandy, K.: Approximate analysis of central server models. IBM Journal of Research and Development 19, 301–313 (1975)
Schmickler, L.: Meda: Mixed Erlang distributions as phase-type representations of empirical distribution functions. Communications in Statistics – Stochastic Models 8, 131–156 (1992)
Squillante, M.: Matrix-analytic methods in stochastic parallel-server scheduling models. In: Advances in Matrix-Analytic Methods for Stochastic Models. Notable Publications (July 1998)
Starobinski, D., Sidi, M.: Modeling and analysis of power-tail distributions via classical teletraffic methods. Queueing Systems 36, 243–267 (2000)
Telek, M., Heindl, A.: Matching moments for acyclic discrete and continuous phase-type distributions of second order. International Journal of Simulation 3, 47–57 (2003)
Whitt, W.: Approximating a point process by a renewal process: Two basic methods. Operations Research 30, 125–147 (1982)
Zhang, Y., Franke, H., Moreira, J., Sivasubramaniam, A.: An integrated approach to parallel scheduling using gang-scheduling, backfilling, and migration. IEEE Transactions on Parallel and Distributed Systems 14, 236–247 (2003)
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Osogami, T., Harchol-Balter, M. (2003). 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. TOOLS 2003. Lecture Notes in Computer Science, vol 2794. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45232-4_13
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DOI: https://doi.org/10.1007/978-3-540-45232-4_13
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