In chapter one some distributions which have been used for modelling within classical queueing theory are considered. This introductory chapter is primarily intended as an illustration of the generality of PH-distributions.

PH-distributions are distributions for the sojourn time in the transient states of a Markov chain in continuous or discrete time with a single absorbing state. In chapter two a comprehensive treatment of phase type distributions and related distributions are given. The main results known to date for these distributions are described and a few new ideas concerning canonical form representations of PH-distributions are presented.

The problem of parameter estimation in PH-distributions is considered in chapter 3. It is shown that there exist several candidate algorithms for this task. It is also shown that none of these algorithms are generally applicable. Some ideas for improving the methods are presented among which some have been implemented and to some extend tested.

Chapters 4 and 5 are dedicated to the treatment of some multiple access system models. In chapter 4 the time-in service protection model is discussed. This scheme is used for the protection of a type of customers without waiting possibilities against overload from another type of customers which share finite or infinite waiting space. The analysis of this model is new and a number of figures for three cases regarding the waiting space are given. Finally in chapter 5 some models for the analysis of local area networks are presented. The main model discussed is based on another work in this area by Coyle and Liu. Their model is extended to the case with two inhomogeneous traffic sources (i.e. voice and data) with different performance demands; but no figures or numerical results are presented. Also a model for the analysis of a token ring system is sketched.

The conclusion of the thesis is that modelling of complex systems with phase type distributions and analysis with the related matrix geometric methods is possible and to be recommended. Many physical models which could hardly be analyzed by other techniques are within the range of numerical solution with the use of these tools. Because it is possible to estimate distribution parameters from real life data also end-to-end performance analysis of systems is possible.

With the more complex modelling the models obtained is somewhat limited in scope. But the restricted generality which partially follows from the levels of details included should be compared to the generality of simulation models which usually would be the alternative modelling tool. The latter tools usually have less generality.

Of course modelling with more comprehensive tools is more consuming, both in the terms of knowledge and in the terms of computer power. Thus cost considerations should be involved before the decision is taken to use these tools for new design issues in telecommunications and related areas. Nevertheless, it is the opinion of the author that there is a promising field for this theory in the modelling of many new areas. An example of this is the new ATD -technology as discussed in the Cost-214 project and to be considered in future Race and Cost projects.

Finn Kuno Christensen