A Novel Approach for Phase-Type Fitting with the EM Algorithm

A. Thummler, P. Buchholz, M. Telek
2006 IEEE Transactions on Dependable and Secure Computing  
The representation of general distributions or measured data by phase-type distributions is an important and non-trivial task in analytical modeling. Although a large number of different methods for fitting parameters of phase-type distributions to data traces exist, many approaches lack efficiency and numerical stability. In this paper, a novel approach is presented that fits a restricted class of phase-type distributions, namely mixtures of Erlang distributions, to trace data. For the
more » ... r fitting an algorithm of the expectation maximization type is developed. The paper shows that these choices result in a very efficient and numerically stable approach which yields phase-type approximations for a wide range of data traces that are as good or better than approximations computed with other less efficient and less stable fitting methods. To illustrate the effectiveness of the proposed fitting algorithm, we present comparative results for our approach and two other methods using six benchmark traces and two real traffic traces as well as quantitative results from queueing analysis. Keywords: Performance and dependability assessment/analytical and numerical techniques, design of tools for performance/dependability assessment, traffic modeling, hyper-Erlang distributions. but it seems to be more appropriate to develop an EM algorithm tailored to specific types of PH distributions. Based on earlier work from [10], El Abdouni Khayari et al. developed an EM algorithm in [8] to fit the parameters of a hyperexponential distribution to values of a data trace. The resulting approach is extremely efficient and yields good fitting results for heavytailed distributions with monotonically decreasing density functions. However, the use of hyperexponential distributions restricts the class of distributions, which can be represented. In fact, hyperexponential distributions cannot adequately capture general distributions with increasing and decreasing densities or with a coefficient of variation less than one. Since the fitting of parameters of a PH distribution is in general a non-linear optimization problem, apart from the EM algorithm also other optimization algorithms can be applied. Furthermore, the fitting algorithm is rather stable due to the specific structure of the density function, which yields a fast and reliable convergence of the EM method. Additionally, the -3fitting of the first three moments using a polynomial of degree 5 is introduced and it is shown how moment fitting can be integrated in the proposed EM algorithm that fits the empirical distribution function. Apart from the efficiency of the approach, the quality of the approximation for a given number of phases is important. We tested the approach on a set of six benchmark traces [3] and compared it with general PH-fitting [1] and fitting of acyclic PH distributions [14] . As expected, G-FIT is significantly faster than the other two approaches. Additionally, we were able to reach with an identical number of states a similar or better fitting quality than with the other two approaches on almost all examples. This result was not expected, because hyper-Erlang distributions of a given order are in general less flexible than acyclic or general PH distributions of the same order. The practical applicability of G-FIT is demonstrated by fitting a call center trace [21] and a large traffic trace, which was recorded at the Web proxy server at the University of Dortmund in March 2005. The presented EM algorithm is implemented in the software package G-FIT, which is available for download on the Web [12]. The paper is organized as follows. Section 2 introduces the considered class of hyper-Erlang distributions, it studies its relationship to general phase-type distributions, and it introduces the fitting of the first three moments of a hyper-Erlang distribution. Section 3 develops a specialized EM algorithm for fitting the continuous parameters of a hyper-Erlang distribution and Section 4 presents an approach for finding optimal settings of the discrete parameters of the distribution. Experimental results obtained from fitting synthetically generated benchmark traces and two real traffic traces as well as quantitative results from queueing analysis are presented in Section 5. Finally, concluding remarks are given. Hyper-Erlang Distributions and its Properties Hyper-Erlang Distributions We consider a mixture of M mutually independent Erlang distributions weighted with the (initial) probabilities α 1 ,...,α M with α m ≥ 0 and α 1 +α 2 +...+α M = 1. The number of phases of
doi:10.1109/tdsc.2006.27 fatcat:hfyjru2ljvh2fnfqzfyp2akmz4