Performance Evaluation 68 (2011) 938–954 Contents lists available at ScienceDirect Performance Evaluation journal homepage: www.elsevier.com/locate/peva A refined EM algorithm for PH distributions Hiroyuki Okamura a,∗ , Tadashi Dohi a , Kishor S. Trivedi b a Department of Information Engineering, Graduate School of Engineering, Hiroshima University, 1–4–1 Kagamiyama, Higashi-Hiroshima 739–8527, Japan b Department of Electrical and Computer Engineering, Duke University, Hudson Hall, Durham 27707, USA article info Article history: Received 24 December 2009 Received in revised form 22 March 2011 Accepted 11 April 2011 Available online 1 May 2011 Keywords: PH distribution PH fitting EM algorithm Canonical form Uniformization abstract This paper proposes an improved computation method of maximum likelihood (ML) estimation for phase-type (PH) distributions with a number of phases. We focus on the EM (expectation-maximization) algorithm proposed by Asmussen et al. [27] and refine it in terms of time complexity. Two ideas behind our method are a uniformization- based procedure for computing a convolution integral of the matrix exponential and an improvement of the forward–backward algorithm using time intervals. Compared with the differential-equation-based EM algorithm discussed in Asmussen et al. [27], our approach succeeds in the reduction of computation time for the PH fitting with a moderate to large number of phases. In addition to the improvement of time complexity, this paper discusses how to estimate the canonical form by applying the EM algorithm. In numerical experiments, we examine computation times of the proposed and differential-equation- based EM algorithms. Furthermore, the proposed EM algorithm is also compared with the existing PH fitting methods in terms of computation time and fitting accuracy. © 2011 Elsevier B.V. All rights reserved. 1. Introduction A phase-type (PH) distribution is defined as the probability distribution of the time to absorption in a finite state Markov chain with one or more transient states and one absorbing state. PH distributions include several important distributions such as the Erlang, the hypoexponential and the hyperexponential distributions. An advantage of PH distributions is their mathematical tractability. Since PH distributions are easily combined with Markov models, they are often utilized in modeling queueing systems, insurance risk and dependability/performance of technical systems [1–4]. PH distributions can approximate arbitrary continuous probability distributions with arbitrary precision. Thus the PH approximation is popular as an approximation to a general distribution in quantitative dependability/performance evaluation. For example, Markov regenerative stochastic Petri nets (MRSPNs) and their associated Markov regenerative analysis are often used in the performance evaluation of communication systems [5–7]. Since the MRSPNs involve general distributions as well as exponential distributions, PH distributions can be applied as a method of approximate solution of MRSPNs. For this reason, some MRSPN tools incorporate the PH approximation [8–10]. A significant problem in PH fitting is the estimation/approximation of parameters. In general, three kinds of criteria are applied to the parameter estimation of PH distributions: moment matching, difference in probability densities and Kullback–Leibler (KL) divergence [11]. Table 1 presents a classification of PH estimation/approximation methods based on the three criteria. The second column in Table 1 represents the class of PH distributions that is used in the parameter estimation. To the best of our knowledge, moment matching was the first method adopted for PH fitting. Marie [12] proposed a simple method by using the first two moments to fit either Coxian or Erlang distributions. The method was extended by ∗ Corresponding author. Tel.: +81 82 424 7697; fax: +81 82 422 7025. E-mail address: okamu@rel.hiroshima-u.ac.jp (H. Okamura). 0166-5316/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.peva.2011.04.001