


Politecnico di Torino  
Academic Year 2016/17  
02KRRPE, 02KRRBG, 02KRRND Stochastic processes 

Master of sciencelevel of the Bologna process in Nanotechnologies For Icts  Torino/Grenoble/Losanna Master of sciencelevel of the Bologna process in Communications And Computer Networks Engineering  Torino Master of sciencelevel of the Bologna process in Energy And Nuclear Engineering  Torino 





Esclusioni: 01NNM 
Subject fundamentals
The course is taught in English.
The purpose of this course is to review the basic concepts of probability theory and to introduce the theory of the stochastic processes which are especially relevant in queueing systems, telematics networks and in telecommunication and software engineering. The most relevant stochastic processes in discrete and continuous time and space are described, together with a comprehensive list of examples of application. 
Expected learning outcomes
The knowledge obtained through this course consists on the basic theory of the stochastic processes useful to describe and solve problems characterized by nondeterministic time, At the end the course the students will be able to describe and design simple probabilistic models in queues, network or software reliability problems, and to solve them both through analytical and numerical methods, computing useful quantities like stationary distributions of systems, distribution of waiting times, mean times to reach absorbing states. They will be also able to understand which one of the presented processes is more appropriate in the analysis they have to perform, and the meaning of the values assumed by the mathematical objects presented during the lectures. The ability to apply the gained knowledge will be verified through class exercises. 
Prerequisites / Assumed knowledge
A basic knowledge of calculus and a first course in probability theory are the prerequisites for this course.

Contents
• Recap of basic probability: probability spaces, random variables and notable distributions, expectations, conditioning (12 hours)
• Poisson process: equivalent definitions, generalizations (nonhomogeneous, compound, mixed) (12 hours) • Renewal processes: equilibrium distributions and stationary processes (10 hours) • Discrete time Markov chains: transition matrices, classification of states, stationarity and ergodicity, time reversibility, techniques for aggregation of states, branching processes (12 hours) • Continuous time Markov chains: transitions, birth and death processes, stationarity(10 hours) • Brownian motions: definitions, properties, applications (4 hours) 
Delivery modes
Lectures are held with the support of slides. Exercises are presented and solved in the class. Technical discussions during class lectures will also help to assess the acquired level of knowledge and ability at the different stages of the course.

Texts, readings, handouts and other learning resources
Sheldon N. Ross Stochastic processes, ed John Wiley, any edition.
Slides of the lectures, execises and examples of written exams, both with solutions, are available in the website of the course. 
Assessment and grading criteria
The exam consists of a written examination and a facultative oral examination. The written exaqmination (2 hours) consists of 5 exercises, 4 of them similar to those presented during the lectures, where it is required to model some practical problems and to compute useful quantities like stationary distributions or expected times to reach specific states, and one on the theoretical aspects of processes. The oral exam is possible under request for those students that in the written exam get a positive mark (greater or equal to 18/30).

