OVERVIEW

Teaching introduces the student to the Markov chains theory, both discrete-time and continuous, with emphasis on Poisson Processes and queuing theory, developing the essential skills to model and analyze concrete problems of stochastic evolution.

Teaching contributes to the achievement of Goals 4 and 5 of Sustainable Development of the UN 2030 Agenda.

AIMS AND CONTENT

LEARNING OUTCOMES

We want to introduce Markov chains and other simple stochastic processes in order to model and solve real problems of stochastic evolution.

AIMS AND LEARNING OUTCOMES

The goal is to have the student learn the language of Markov chains, so that he will be able to build an accurate model starting from real problems of stochastic evolutions taking values in a finite or countable set (the set of states). 

At the end of the course the student will have to:
- know the general theory of Markov chains, both at discrete and continuous time,          

- classify the states and determine the invariant laws with respect to the evolutions of the system,                                                          

- suitably model real situations of the queueing theory in the language of Markov chains, and be able to study the efficacy of the model. 

PREREQUISITES

Probability.

You can find more details on Aulaweb.

TEACHING METHODS

Teaching is done the traditional way, with lectures held at the blackboard. Expect 2 theory classes per week (4 hours) and 1 of exercises (2 hours).
At the end of the course there will be a guided full-text exercise so as to give students the opportunity to understand their degree of readiness and to clarify together possible doubts. 

Attendance is not mandatory but strongly recommended.

SYLLABUS/CONTENT

Discrete time Markov chains. Definition. Classification of states. Transience and recurrence criteria. Probability of absorbtion in recurrent classes. Invariant laws. Limit Theorems. Convergence to equilibrium. 

Contnuous time Markov chains. Hitting time. Chapman-Kolmogorov equations. Invariant laws. Jumps chain. Born and death chains. Poisson processes.

Queueing theory.

RECOMMENDED READING/BIBLIOGRAPHY

P. Baldi, Calcolo delle Probabilità e Statistica Matematica

W. Feller, An introduction to Probability Theory and its Applications

S. Karlin, H.M. Taylor, A First Course in Stochastic Processes.

S. Karlin, H.M. Taylor, A Second Course in Stochastic Processes.

S.M. Ross, Introduction to Probability Models.

G. Grimmett, D. Stirzaker, (2001). Probability and Random Processes. 

J.R. Norris. Markov Chains.

P. Brémaud. Markov Chains: Gibbs Fields, Montecarlo Simulation, and Queues.

Notes

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar https://corsi.unige.it/corsi/11900/studenti-orario 

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written test + oral test. 

To participate in the written test you must register on the UNIGE site.

The written test consists of 2 exercises, one on the discrete part and the other one on the continuous part, each with 6 questions.
The duration of the test is 3 hours and access to the course notes (including exercises done in the classroom) and handouts is allowed.  Reporting a grade greater than or equal to 18 gives access to the oral test.

The oral test will consist of exposition of theoretical arguments, proofs and exercises. In addition, students in the master's degree in mathematics will also be asked for a proof (of the teacher's choice) from among those not given in class but pointed out by the lecturer (on Aulaweb).

The oral examination can be taken immediately after the written test or even in subsequent exam sessions during the academic year in progress. The written test will be considered valid until the September roll call of the academic year in which it was taken. After that date, the student must retake the written test.

There will be 2 rounds available for the winter session (January-February) and 3 rounds for the summer session (June, July and September). A round around mid-December is also planned for Smid students. No special rounds will be granted outside of those specified in the course regulations, except for students who have not completed exams within set time period.

ASSESSMENT METHODS

The oral examination is aimed at assessing the general understanding of the course topics and it is required that the student knows how to properly expose the concepts seen in the course, to show the main results and to solve the exercises.

The written grade is the starting point for the final score, and it is modified depending on the oral performance, either positively or negatively.

FURTHER INFORMATION

Contact the lecturer for further information not included in the teaching sheet.

Compensatory and dispensatory measures

Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it