NMAK15019U Phase-type distributions: Theory and applications (AAM)

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.

The class of phase–type (PH) distributions, defined as the distribution of a first exit time from a transient set of a Markov jump process, will play an important role in the course. Included in the class PH are convolutions and mixtures of exponential distributions. The class of PH distributions is dense in the class of distributions on the positive reals, and the PH 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.