Iryna Felko

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.

Transformation of Acyclic Phase Type Distributions for Correlation Fitting

In this paper similarity transformations for Acyclic Phase Type Distributions (APHs) are considered, and representations maximizing the first joint moment that can be reached when the distribution is expanded into a Markovian Arrival Process (MAP) are investigated. For the acyclic case the optimal representation corresponds to a hyperexponential representation, which is optimal among all possible representations that can be reached by similarity transformations. The parameterization aspect for the possible transformation of APHs into a hyperexponential form is revealed, together with corresponding transformation rules. For the case when APHs cannot be transformed into a hyperexponential representation a heuristic optimization method is presented to obtain good representations, while transformation methods to increase the first joint moment by adding additional phases are derived.

PH-graphs for analyzing shortest path problems with correlated traveling times

This paper presents a new approach to model weighted graphs with correlated weights at the edges. Such models are important to describe many real world problems like routing in computer networks or finding shortest paths in traffic models under realistic assumptions. Edge weights are modeled by phase type distributions (PHDs), a versatile class of distributions based on continuous time Markov chains (CTMCs). Correlations between edge weights are introduced by adding dependencies between the PHDs of adjacent edges using transfer matrices. The new model class, denoted as PH graphs (PHGs), allows one to formulate many shortest path problems as the computation of an optimal policy in a continuous time Markov decision process (CTMDP). The basic model class is defined, methods to parameterize the required PHDs and transfer matrices based on measured data are introduced and the formulation of basic shortest path problems as solutions of CTMDPs with the corresponding solution algorithms are also provided. Numerical examples for some typical stochastic shortest path problems demonstrate the usability of the new approach.

Phase-Type Distributions

Continuous-time Markov chainsContinuous-time Markov chain (CTMCs)CTMC seealso Continuous-time Markov chain Markov chain seealso Continuous-time Markov chain are a class of stochastic processes with a discrete state space in which the time between transitions follows an exponential distribution. In this section, we first provide the basic definitions for CTMCs and notations associated with this model. We then proceed with an explanation of the basic concepts for phase-type distributions (PHDs) and the analysis of such models. For theoretical details about CTMCs and related stochastic processes we refer to the literature [151].

Modeling Human Decisions in Performance and Dependability Models

Many systems are driven partially by human operators who decide about basic operations that influence system behavior. Therefore the performance and dependability depend on the technical system and the human operator. Performance and dependability models usually include a detailed model of the technical infrastructure but the human decision maker is only roughly modeled by simple probabilities or delays. However, in psychology much more sophisticated models of human decision making exist. For tasks with two choices usually diffusion models are applied. These models include information about the process of human decision making based on perception or memory retrieval and take into account the time pressure under which decisions have to be made. In this paper we combine these diffusion models with Markov models for performance and dependability analysis. By using a discretization approach for the diffusion model the combined model is a Markov chain which can be analyzed with standard means. The approach allows one to integrate detailed models of human two-way decisions in performance and dependability models.

Markovian Arrival Processes

PHDs can be extended to describe correlated inter-event times. The resulting models are denoted as Markovian Arrival Processes (MAPs) and have been introduced in the pioneering work of Neuts [124]. MAPs are a very flexible and general class of stochastic processes. In this chapter we first introduce the general model and its analysis, then the specific case of MAPs with only two states is considered because it allows one to derive some analytical results and canonical representations. The last section extends the model class to stochastic processes generating different event types.

Stochastic Models Including PH Distributions and MAPs

PHDs and MAPs are used to define inter-event times at various levels and in different model types. Originally, phase-type representations of inter-event times are used in models that are mapped on Markov processes and are solved numerically. However, this is only one application area.

Parameter Fitting of MAPs

Fitting the parameters of a MAP is much more complex than the parameter fitting for PHDs. The major reasons for the complexity of the fitting problem are missing canonical representations for MAPs and the necessity to consider long traces to adequately capture the correlation.

Parameter Fitting for Phase Type Distributions

An important step when developing models that will be subject to a numerical or simulative analysis is the definition of input data for e.g. inter-event or service times which is denoted as input modeling. Usually, one has some observations measured in a real system, called trace, and tries to estimate (fit) the parameters of a distribution, such that the distribution captures characteristics of the given data. In this book we consider Markov processes as models for the data.