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

Abstract

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 parameter 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.

A Fast EM Algorithm for Fitting Marked Markovian Arrival Processes with a New Special Structure

“…It turned out that fitting with sub-classes of PH distributions like hyper-exponential [10], hyperErlang [20], or acyclic PH distributions [17] is more efficient than fitting with the general PH class [2].…”

“…Thummler et al [20] presented a heuristic method that tries to predict promising combinations of shape parameters by doing a few iterations of the EM algorithm which is called the progressive preselection. This method first considers all vectors r ∈ H N .…”

Efficient Implementations of the EM-Algorithm for Transient Markovian Arrival Processes

“…However, computing the matrix exponential terms is numerically demanding. To reduce the computational complexity we apply the same structural restriction as in [10], [9] and [7], thus we introduce a special TMAP structure composed of a number of Erlang distributed branches. When a given branch is selected the inter-arrival time is Erlang distributed defined by the parameters (rate and order) of the selected Erlang branch, and after each arrival event a sub-stochastic transition probability matrix determines which Erlang branch to choose for the next inter-arrival given the branch generating the current arrival (see Figure 2).…”

“…The EM method has been used successfully for parameter estimation of several models with background Markov chains, e.g., for PH distributions [3], for PH distributions with structural restriction [10], for MAPs [4], for MAPs with structural restrictions [9,7]. The experiences from these previous research results indicate that the inherent redundancy of the stochastic models with background Markov chains makes the parameter estimation of the general models inefficient.…”

“…The experiences from these previous research results indicate that the inherent redundancy of the stochastic models with background Markov chains makes the parameter estimation of the general models inefficient. In this work we avoid the implementation of the EM based estimation of general TMAPs and immediately apply a similar structural restriction as the one which turned out to be efficient in case of PH distributions [10] and MAPs [9,7]. The formulas of the EM method for TMAP fitting show similarities with the ones for MAP fitting in [7], but there are introcate details associated with the handling of background process termination which require a non-trivial reconsideration of the expectation and the maximization steps of the method.…”