It is our goal to introduce the most basic mathematical and simulative tools to face problems where the evolution of a system is random, like for waiting times, first crossing times of threshold levels, or the number of failures in a given time of electronic devices. At the end of the course the students will be able to define and analyze simple stochastic models in queuing theory, network or software reliability, and from different areas of engineering. They will be able to solve them both through analytical and simulations, 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 and analysis of simple case studies.

A basic knowledge of calculus and a first course in probability theory are the prerequisites for this course. A minimum knowledge of basic probability will be assumed as previously acquired at the beginning of the course.

• Complements of basic probability: notable distributions, moment generating functions,conditional expectations, mixtures, order statistics – 6 hours. • Poisson process: equivalent definitions, generalizations (non-homogeneous, compound, mixed) and brief presentation of Renewal 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 - 9 hours • Brownian motions: definitions and main properties - 3 hours. • Simulation with Matlab and analysis of some case studies (times to hit fixed thresholds for different kind of processes, probabilities of reaching absorbing states for analitically untractable processes, simulations of queueing systems) – 20 hours

In the first part of the course the lectures are held with the support of slides. Exercises are presented and solved in the class as well. In the second part of the course the lessons will mainly consist in process simulation with Mathlab, under the guidance of the teacher. Technical discussions during class lectures will also help to assess the acquired level of knowledge and ability at the different stages of the course. The theoretical lectures and the process simulations will be performed in class or online or blended, depending on sanitary emergency conditions.

Sheldon N. Ross Stochastic processes, ed John Wiley, any edition. Slides of the lectures, exercises and examples of written exams, both with solutions, are available in the website of the course.

Exam: Written test; Optional oral exam; Practical lab skills test;

The exam consists of a written examination (open books) and a facultative oral examination. The exam’s aim is to test the student's ability to apply the methods of analysis of processes described during the course. The written examination (2 hours) consists of 4 exercises, 3 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. One of the exercises consists on production of a Mathlab script that allows to simulate a process and to numerically provide useful quantities of interest in a practical engineering problem. The length of the written exam is of two hours, and during the test it is allowed the use of textbooks, student notes or formularies provided by the teacher during the year. 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), and concerns the theoretical results presented during lectures. After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.

Exam: Optional oral exam; Individual essay; Paper-based written test with video surveillance of the teaching staff;

The exam consists of a written examination (open books: it is allowed the use of textbooks, student notes or formularies provided by the teacher during the year), the individual writing of a report related to the simulation of processes in Matlab, and a facultative oral examination. The exam’s aim is to test the student's ability to apply the methods of analysis of processes described during the course. The written examination (1.5 hours) consists of 3 exercises 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. The individual report will be assigned in advance, and it will deal with a practical problem which requires the simulation of a process. The problem will be assigned individually at the end of the exam bookings and within one week the student must send back a short report in which is described how it has been modeled/solved, and must contain a description of the related simulation script. The oral exam is possible under request for those students who in the written exam get a positive mark (greater or equal to 18/30), and concerns the theoretical results presented during lectures. After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points

Exam: Written test; Optional oral exam; Paper-based written test with video surveillance of the teaching staff;

In case of onsite exams there will be no differences with respect to the grading criteria described for the online exams