Jan Kriege

Multi-class Markovian arrival processes and their parameter fitting

Markovian arrival processes are a powerful class of stochastic processes to represent stochastic workloads that include autocorrelation in performance or dependability modeling. However, fitting the parameters of a Markovian arrival process to given measurement data is non-trivial and most known methods focus on a single class case, where all events are of the same type and only the sequence of interarrival times is of interest. In this paper, we propose a method to fit data to a multi-class Markovian arrival process, where arrivals can be partitioned into a finite set of classes. This allows us to use a Markovian arrival process to represent workloads where interarrival times are correlated across customer classes and to achieve models of greater accuracy. The fitting approach performs in several consecutive steps and applies a single non-linear optimization step and several non-negative least squares computations.

Input Modeling with Phase-Type Distributions and Markov Models

Containing a summary of several recent results on Markov-based input modeling in a coherent notation, this book introduces and compares algorithms for parameter fitting and gives an overview of available software tools in the area. Due to progress made in recent years with respect to new algorithms to generate PH distributions and Markovian arrival processes from measured data, the models outlined are useful alternatives to other distributions or stochastic processes used for input modeling. Graduate students and researchers in applied probability, operations research and computer science along with practitioners using simulation or analytical models for performance analysis and capacity planning will find the unified notation and up-to-date results presented useful. Input modeling is the key step in model based system analysis to adequately describe the load of a system using stochastic models. The goal of input modeling is to find a stochastic model to describe a sequence ofmeasurements from a real system to model for example the inter-arrival times of packets in a computer network or failure times of components in a manufacturing plant. Typical application areas are performance and dependability analysis of computer systems, communication networks, logistics or manufacturing systems but also the analysis of biological or chemical reaction networks and similar problems. Often the measured values have a high variability and are correlated. Its been known for a long time that Markov based models like phase type distributions or Markovian arrival processes are very general and allow one to capture even complex behaviors. However, the parameterization of these models results often in a complex and non-linear optimization problem. Only recently, several new results about the modeling capabilities of Markov based models and algorithms to fit the parameters of those models have been published.

A Heuristic Approach for Fitting MAPs to Moments and Joint Moments

Fitting of the parameters of a Phase Type (PH) Distribution or a Markovian Arrival Process (MAP) according to some quantities of measured data streams is still a challenge. This paper presents a new approach which computes in two steps for a set of moments and joint moments for an Acyclic PH distribution that is expanded into a MAP. In contrast to other known approaches, parameters are computed to minimize the weighted squared difference between the measured moments and the moments of the resulting PH Distribution or MAP. The proposed approach is very flexible and allows one to generate a MAP of a predefined order to approximate a given set of moments and joint moments. It is shown that the approximation is often sufficiently accurate even with MAPs of a moderate size. However, we also show that the practical applicability of the approach is limited since the exact determination of higher order moments from traces requires an extremely high effort.

ProFiDo - The Processes Fitting Toolkit Dortmund

This paper describes the Java-based toolkit ProFiDo which integrates several tools for fitting input models. Currently supported are command line tools for fitting probability distributions, ARIMA processes and Markovian arrival processes. The toolkit provides a graphical user interface which allows for the specification of workflows that describe the different steps of data preprocessing, parameter fitting and result visualization. The basis for the interoperability of the different tools is an XML based interchange format for the specification of various types of processes. An XML based configuration file supports the extension of the toolkit by integrating additional fitting methods or analysis approaches.

An empirical comparison of MAP fitting algorithms

The paper presents an empirical comparison of different methods to fit the parameters of a MAP according to the quantities derived from three different real traces. The results indicate that for two of the three traces an adequate fitting with low order MAPs is possible whereas almost all approaches failed for the third trace. Apart form this the question for the best approach for fitting MAPs is still open although there seems to be a tendency that the most costly EM algorithms provide the best fitting results.

A Framework for Simulation Models of Service-Oriented Architectures

Service-Oriented Architectures (SOA) are one of the main paradigms for future software systems. Since these software systems are composed of a large number of different components it is non trivial to assure an adequate Quality of Service (QoS) of the overall system and performance analysis becomes an important issue. To consider performance issues early in the development process, a model based approach becomes necessary which has to be embedded into the development process of SOA to avoid overhead and assure consistency. In particular the specification of the software system should be used as a base for the resulting performance model. However, since common specification techniques for SOA are very high level, many details have to be added to come to an executable simulation model which is often needed for a detailed analysis of performance or dependability. This paper presents an approach which combines an extended version of process chains to describe the SOA components and some quantitative specifications at the higher levels. For the modelling of the detailed architecture and protocols the simulation tool OMNeT++ is used. Both modelling levels are combined resulting in an executable simulation model for the whole architecture.

Correlated phase-type distributed random numbers as input models for simulations

The adequate modeling of correlated input processes is an important step in building simulation models. Modeling independent identically distributed data is well established in simulation whereas the integration of correlation is still a challenge. In this paper, ARTA processes which have been used several times for describing correlated input processes in simulation are extended by using ARMA instead of AR processes to realize the correlation and Acyclic Phase Type distributions to model the marginal distribution. For this new process type a fitting algorithm is presented. By means of some real network traces it is shown that the extended model allows a better fitting of the marginal distribution as well as the correlation structure and results in a compact process description that can be used in simulation models.

A comparison of Markovian arrival and ARMA/ARTA Processes for the modeling of correlated input processes

The adequate modeling of input processes often requires that correlation is taken into account and is a key issue in building realistic simulation models. In analytical modeling Markovian Arrival Processes (MAPs) are commonly used to describe correlated arrivals, whereas for simulation often ARMA/ARTA-based models are in use. Determining the parameters for the latter input models is well-known whereas good fitting methods for MAPs have been developed only in recent years. Since MAPs may as well be used in simulation models, it is natural to compare them with ARMA/ARTA models according to their expressiveness and modeling capabilities for dependent sequences. In this paper we experimentally compare MAPs and ARMA/ARTA-based models.

Markov Modeling of Availability and Unavailability Data

Markov models are often used in performance and dependability analysis and allow a precise and numerically stable computation of many result measures including those which result from rare events. It is, however, known that simple exponential distributions, which are the base of Markov modeling, cannot adequately describe the duration of availability or unavailability intervals of components in a distributed system. Commonly used in modeling those durations are Weibull, log-normal or Pareto distributions that can also capture a possibly heavy tailed behavior but cannot be analyzed analytically or numerically. An alternative to applying the mentioned distributions in modeling availability or unavailability intervals are phase type distributions and Markovian arrival processes that still result in a Markov model. Based on experiments for a large number of publically available availability traces, we show that phase type distributions are a flexible alternative to other commonly known distributions and even more that Markov models can be easily extended to capture also correlation in the length of availability or unavailability intervals.

Simulating process chain models with OMNeT

This paper presents an approach to simulate complex hierarchical process chains resulting from large logistics networks in OMNeT++, a discrete event simulation environment designed for communication networks. For this purpose OMNeT++ has been integrated as a new simulation engine into the ProC/B toolset which is designed for the analysis and optimization of large logistics networks. The paper highlights the main steps of the automatic transformation of a hierarchical process chain model into a hierarchical model in OMNeT++. Furthermore it shows how the transformation has been validated and how detailed performance figures can be evaluated with OMNeT++.