A fitting method with generalized Erlang distributions

Abstract We present a fitting technique that fits trace data into a generalized Erlang distribution class using an EM method. A generalized Erlang (GEr) distribution can be made by convolution of the third order ME distributions similar to the formulation of an Erlang distribution with exponential distributions. We give a sufficient condition for the representation to make a probability density function and we implement a fitting algorithm into a GEr distribution set by solving a nonlinear optimization problem with the EM algorithm. The effectiveness of the proposed fitting algorithm is presented by applying fitting methods to sets of synthetic data and measurement data. We present comparative numerical simulation results of our approach and other methods.

[1]  L. Schmickler.  Meda: mixed erlang distributions as phase-type representations of empirical distribution functions. , 1992 .

[2]  Gábor Horváth,et al.  A minimal representation of Markov arrival processes and a moments matching method , 2007, Perform. Evaluation.

[3]  Mogens Bladt,et al.  Point processes with finite-dimensional conditional probabilities , 1999 .

[4]  Luca Benvenuti,et al.  A tutorial on the positive realization problem , 2004, IEEE Transactions on Automatic Control.

[5]  A. Cumani.  On the canonical representation of homogeneous markov processes modelling failure - time distributions , 1982 .

[6]  Colm Art O'Cinneide,et al.  Phase-type distributions and invariant polytopes , 1991, Advances in Applied Probability.

[7]  Miklós Telek,et al.  Moment characterization of matrix exponential and Markovian arrival processes , 2008, Ann. Oper. Res..

[8]  Ren Asmussen,et al.  Fitting Phase-type Distributions via the EM Algorithm , 1996 .

[9]  Qi-Ming He,et al.  On matrix exponential distributions , 2007, Advances in Applied Probability.

[10]  Peter Buchholz,et al.  A Novel Approach for Phase-Type Fitting with the EM Algorithm , 2006, IEEE Transactions on Dependable and Secure Computing.

[11]  Mogens Bladt,et al.  Renewal Theory and Queueing Algorithms for Matrix-Exponential Distributions , 1996 .

[12]  William G. Marchal,et al.  Distribution Estimation Using Laplace Transforms , 1998, INFORMS J. Comput..

[13]  Marcel F. Neuts,et al.  Matrix‐Exponential Distributions: Calculus and Interpretations via Flows , 2003 .

[14]  Jeff A. Bilmes,et al.  A gentle tutorial of the em algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models , 1998 .

[15]  Nigel G. Bean,et al.  Characterization of Matrix-Exponential Distributions , 2008 .

[16]  Christian Commault,et al.  Phase-type distributions and representations: Some results and open problems for system theory , 2003 .

[17]  Mark Fackrell,et al.  Fitting with Matrix-Exponential Distributions , 2005 .

[18]  Bruno Sericola,et al.  Second order fluid models with general boundary behaviour , 2008, Ann. Oper. Res..