Miklós Telek

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

The representation of general distributions or measured data by phase-type distributions is an important and nontrivial 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. This 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 queuing analysis

PhFit: a general phase-type fitting tool

PhFit, a new phase-type fitting tool is presented in this paper. PhFit allows for approximating distributions or set of samples not only by continuous but by discrete phase-type distributions as well. The implemented algorithms separate the fitting of the body and the tail part of the distribution which results in satisfactory fitting also for heavy-tail distributions. Moreover, PhFit allows the user to choose the distance measure according to which the fitting is performed. The tool is equipped with a graphical interface that visualizes the goodness of the provided approximation from several aspects.

MATCHING THREE MOMENTS WITH MINIMAL ACYCLIC PHASE TYPE DISTRIBUTIONS

Abstract A number of approximate analysis techniques are based on matching moments of continuous time phase type (PH) distributions. This paper presents an explicit method to compose minimal order continuous time acyclic phase type (APH) distributions with a given first three moments. To this end we also evaluate the bounds for the first three moments of order n APH distributions (APH(n)). The investigations of these properties are based on a basic transformation, which extends the APH(n − 1) class with an additional phase in order to describe the APH(n) class.

Acyclic discrete phase type distributions: properties and a parameter estimation algorithm

This paper provides a detailed study on discrete phase type (DPH) distributions and its acyclic subclass referred to as acyclic-DPH (ADPH). Previously not considered similarities and differences between DPH and continuous phase type (CPH) distributions are investigated and minimal representations, called canonical forms, for the subclass of ADPH distributions are provided. We investigate the consequences of the recent result about the minimal coefficient of variation of the DPH class [The minimal coefficient of variation of discrete phase type distributions, in: Proceedings of the Third International Conference on Matrix-analytic Methods in Stochastic Models, July 2000] and show that below a given order (that is a function of the expected value) the minimal coefficient of variation of the DPH class is always less than the minimal coefficient of variation of the CPH class. Since all the previously introduced Phase Type fitting methods were designed for fitting over the CPH class we provide a DPH fitting method for the first time. The implementation of the DPH fitting algorithm is found to be simple and stable. The algorithm is tested over a benchmark consisting of 10 different continuous distributions. The error resulted when a continuous distribution sampled in discrete points is fitted by a DPH is also considered.

A benchmark for ph estimation algorithms: results for acyclic-ph

Distribution functions that can be expressed as exponential polynomials have useful computational properties in applied stochastic modeling and have gained widespread acceptance in recent years. Nevertheless, the implementation of efficient numerical procedures for estimating the distribution parameters remains an open problem that limits the use of this class of distributions in applications. The difficulty of the fitting problem is largely related to the non-linearity of the model and to the number of the parameters to be estimated Many attempts have been presented in the literature. However, the lack of accepted and standardized test examples makes it difficult to establish a comparative merit among the various approaches. This paper proposes a benchmark based on the workshop on Fitting phase type distributions, organized by S. Asmussen in February 1991. It also presents the results obtained by applying the Maximum Likelihood (ML) estimation procedure to the canonical representation of Acyclic Phase Ty...

A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation

This paper proposes a two-step Markov arrival process (MAP) fitting approach, where the first step is the phase type fitting of the inter-arrival time and the second step is the approximation of the first n lag correlation values. Depending on the description of the arrival process to approximate various phase type fitting methods can be applied for the first step. In the second step the approximation of the lag correlation values is computed through a non-linear optimization problem. Numerical examples demonstrate the abilities and the limits of the fitting method.

Optimal software rejuvenation for tolerating soft failures

Abstract In recent studies, the phenomenon of software “aging” has come to light which causes performance of a software to degrade with time. Software rejuvenation is a fault tolerance technique which counteracts aging. In this paper, we address the problem of determining the optimal time to rejuvenate a server type software which experiences “soft failures” (witnessed in telecommunication systems) because of aging. The service rate of the software gradually decreases with time and settles to a very low value. Since the performability in this state is unacceptable, it is necessary to “renew” the software to its peak performance level. We develop Markov decision models for such a system for two different queuing policies. For each policy, we define the look-ahead- n cost functions and prove results on the convergence of these functions to the optimal minimal cost function. We also prove simple rules to determine optimal times to rejuvenate for a realistic cost criterion. Finally, the results are illustrated numerically and the effectiveness of the MDP model is compared with that of the simple rules.

PhFit: A General Phase-Type Fitting Tool

PhFit, a new Phase-type fitting tool is presented in this paper. The presented tool is novel from several aspects. It allows for approximating distributions or set of samples not only by continuous but by discrete Phase-type distributions as well. The implemented algorithms separate the fitting of the body and the tail part of the distribution which results in satisfactory fitting also for heavy-tail distributions. Moreover, PhFit allows the user to choose the distance measure according to which the fitting is performed.

RECENT DEVELOPMENTS IN NON-MARKOVIAN STOCHASTIC PETRI NETS

Analytical modeling plays a crucial role in the analysis and design of computer systems. Stochastic Petri Nets represent a powerful paradigm, widely used for such modeling in the context of dependability, performance and performability. Many structural and stochastic extensions have been proposed in recent years to increase their modeling power, or their capability to handle large systems. This paper reviews recent developments by providing the theoretical background and the possible areas of application. Markovian Petri Nets are first considered together with very well established extensions known as Generalized Stochastic Petri Nets and Stochastic Reward Nets. Key ideas for coping with large state spaces are then discussed. The challenging area of non-Markovian Petri nets is considered, and the related analysis techniques are surveyed together with the detailed elaboration of an example. Finally new models based on Continuous or Fluid Stochastic Petri Nets are briefly discussed.

A Markovian canonical form of second-order matrix-exponential processes

Besides the fact that - by definition - matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that - in two dimensions - the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP(2)[reverse not equivalent]MAP(2)[reverse not equivalent]MEP(2). For higher orders, these equivalences do not hold. The second-order equivalence is established via a novel canonical form for the (correlated) processes. An explicit moment/correlation-matching procedure to construct the canonical form from the first three moments of the interarrival time distribution and the lag-1 correlation coefficient shows how these compact processes may conveniently serve as input models for arrival/service processes in applications.