Stochastic and discrete mathematical models

Entry requirements: Basic knowledge in probability theory, statistics and programming

Credits: 4

Semester: 2

Course: Core

Language of the course: English

Objectives

After completing the course you should be able to: analyze given problems concerning properties of wide sense stationary stochastic processes; analyze given problems in filtering and modelling of stochastic processes; understand the classical theory for queuing systems.

Contents

Basics about continuous and discrete time stochastic processes, especially wide sense stationary processes. Definitions of probability distribution and density functions, statistical mean, mean power, variance, autocorrelation function, power spectral density, Gaussian processes and white noise. Linear filtering of stochastic processes, ergodicity, estimation of statistical properties from measurements. Estimation theory: linear estimation, orthogonality conditions. Prediction and Wiener filtering. Model based signal processing: Linear signal models, AR-models. Spectral estimation. Application of the above to simple computer science applications. Basic terminology of queuing theory, Kendall’s notation and Little’s theorem. Discrete and continuous time Markov chains, birth-death processes, and the Poisson process. Markovian waiting systems with one or more servers, and systems with infinite as well as finite buffers and finite user populations (M/M/1). Systems with general service distributions (M/G/1): the method of stages, Pollaczek-Khinchin mean-value formula and systems with priority and interrupted service. Loss systems according to Erlang, Engset and Bernoulli. Open and closed queuing networks, Jacksonian networks.

Format

Lectures and lab sessions

Assessment

The grade is consisted of the oral examination - 40% and lab reports - 60%.