MARKOV PROCESSES | FMS180 |

**Aim**

Markov chains and processes are a class of models which, apart from a rich mathematical structure, also has applications in many disciplines, such as telecommunications and production (queue and inventory theory), reliability analysis, financial mathematics (e.g., hidden Markov models), automatic control, and image processing (Markov fields).

The aim of this course is to give the student the basic concepts and methods for Poisson processes, discrete Markov chains and processes, and also the ability to apply them. The course presents examples of applications in different fields, in order to facilitate the use of the knowledge in other courses where Markov models appear.

*Knowledge and understanding*

For a passing grade the student must

- be able to explain the Markov property and the intensity concept, as well as the concepts of recurrence, communication, stationary distribution, and how they relate to each other,
- perform calculations of stationary distributions and absorption times for discrete Markov chains and processes,
- explain the suitability of the Poisson process as a model for rare events and perform calculations of probabilities using the properties of the Poisson process in one and several dimensions.

*Skills and abilities*

For a passing grade the student must

- be able to construct a model graph for a Markov chain or process describing a given system, and use the model for studying the system,
- in connection with problem solving, show ability to integrate knowledge from the different parts of the course,
- read and interpret easier literature with elements of Markov models and their applications.

*Judgement and approach*

For a passing grade the student must

- identify problems that can be solved using Markov models, and choose an appropriate method,
- use knowledge of Markov models in other courses, as transfer concepts, tools, and knowledge between different courses where Markov models are used.

**Contents**

Markov chains: model graphs, Markov property, transition probabilities, persistent and transient states, positive and null persistent states, communication, existence and uniqueness of stationary distribution, and calculation thereof, absorption times.

Poisson process: Law of small numbers, counting processes, event distance, non-homogeneous processes, diluting and super positioning, processes on general spaces.

Markov processes: transition intensities, time dynamic, existence and uniqueness of stationary distribution, and calculation thereof, birth-death processes, absorption times.

Introduction to renewal theory and regenerative processes.

**Literature**

Lindgren, G. & Rydén, T.: Markovprocesser. KFS, 2002.