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## Course Outline: 0432 L 998 VL Markov chain modeling of communication systems

Introduction

• 2 SWS, block lecture on fixed dates (see below)
• Presenter: Dr. Andreas Willig, University of Canterbury
• Pre-requisites:

• BSc module "Kommunikationsnetze" or similar knowledge is required
• Solid grasp of elementary probability theory and probability distributions, and matrix algebra.
• Experience in a programming language (e.g. MATLAB, Python)

• Limitation of entry: 15 students
Contents
The goal of this lecture is to introduce students to the theory and practice of Markov chain modeling in communication systems. Markov chains are a class of stochastic processes that is applicable to modeling a wide range of phenomena in communications and networking, for example queueing in routers, performance of retransmission protocols, performance of medium access control protocols and many more. At the same time, many Markov models are still tractable enough to be evaluated analytically or numerically. The basic theory of Markov chains will be developed, a range of modeling examples from the domain of communication systems and networks will be discussed, and students will also be given opportunity to develop and evaluate own models.
Prerequisites
BSc module „Kommunikationsnetze” or similar knowledge is required.
Lecture topics
• Review of probability theory and transforms
• Introduction to stochastic processes

• Discrete-time Markov chains

• Definition and basic properties

• Classification of states

• Hitting times, stopping times, Potentials
• Long-term behaviour
• Foster/Lyapunov stability theory
• Modeling examples

• Continuous-time Markov chains

• Matrix exponentials and Q-Matrices
• Definition and basic properties
• Classification of states
• Hitting times, hitting probabilities
• Long-term behaviour
• Modeling examples

• Simple Markovian Queueing Systems

• Definition, Markov Chain Model, Performance Measures

• Modeling examples
• The M/G/1 system
• If time permits: Matrix-geometric solutions
• If time permits: ODE approximations and mean-field theory
Timetable
Due to the limited availability of the lecturer the course will be given as a block course. More specifically, the times will be:

Wed, Dec 3: 1 - 6 pm
Wed, Dec 10: 1 - 6 pm
Sat, Dec 13: 10 am - 3 pm
Wed, Dec 17: 1 - 6 pm
Wed, Jan 7: 1 - 6 pm, Deadline for assignment

which gives 25 contact hours in total. Each of these lecture days will be split in two parts. The first part will be a classical lecture, the second part will be a mixture of a tutorial (in which a range of example problems and modeling examples is presented by the lecturer) and own work on small modeling examples.

The oral exams will be held on (or around) Wed, Jan 14. I will make individual appointments with students.
Assessment
• Problem sheets (20% weight): students will be given three to four problem sheets, to which they have to submit solutions. The problem sheets will focus on computational skills and small modeling examples. The solutions will be graded.
• One assignment (30% weight): In the assignment a Markovian model for a given problem from communication network needs to be developed and evaluated numerically. Students will have to deliver a report which explains their model in detail, explains the code developed for numerical evaluation and discusses / interprets the numerical results.
• Oral examination (50% weight): To be held at the end of the lecture. The exam will focus solely on the contents of the lecture and on small modeling examples, not on the assignment.
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