NMAK15019U CANCELLED Phase-type Distributions: Theory and Applications

This course deals with stochastic modelling with examples from insurance risk, queueing theory and reliability theory. Necessary tools from renewal theory, random walks and Markov processes are introduced. In particular ladder height methods and Wiener–hopf techniques will be of importance.

A distribution on the non-negative reals is a matrix-exponential (ME) distribution if the density can  be written as a matrix-exponential of a fixed matrix multiplied with the argument. This expression is pre- and post-multiplied by constant vectors to give a scalar expression.

A phase–type (PH) distribution is defined as the distribution of the time to the  first exit from a set of transient states of a Markov jump process. A PH distribution is also ME.

These two distribution classes will play an important role in the course. Included in the class PH are convolutions and mixtures of exponential distributions. The class of ME distributions is dense in the class of distributions on the positive reals, and the distributional assumption will in many cases provide exact, even explicit, solutions to many complex stochastic models.

This is for example the case in renewal theory, ladder height distributions, ruin probabilities, the severity of ruin, or some waiting time distributions in queues, where we obtain explicit formulas. In reliability theory, the PH distributions naturally describe the distribution of lifetimes which may be seen to go through different stages prior to failure.

The assumption of PH distributions being involved enables us to use probabilistic reasoning, which is a powerful technique that often avoids tedious calculations, and in some cases provides solutions which cannot readily be obtained by analytic methods. A probabilistic method uses sample path arguments rather than transition probabilities or transform methods, which would be the preferred methods in an analytic approach.