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

The purpose of the teaching is to introduce the theory of Markov Chains and to develop the skills necessary to model by means of such processes dynamic systems that evolve randomly, and solve problems related to them.

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.

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.

Students with disabilities or specific learning disorders (SLDs) are reminded that in order to request adaptations in the exam, they must first enter their certification on the University website at servizionline.unige.it in the “Students” section. The documentation will be verified by the University's Services for the Inclusion of Students with Disabilities and DSA Sector.

Subsequently, significantly in advance (at least 7 days) of the examination date, an e-mail must be sent to the teacher with whom the examination will be taken, including in the knowledge copy both the School's Teacher Referent for the Inclusion of Students with Disabilities and with DSA (sergio.didomizio@unige.it) and the above-mentioned Sector. The e-mail should specify:

- the name of the teaching

- the date of the roll call

- the student's last name, first name and roll number

- the compensatory tools and dispensatory measures deemed functional and required.

The contact person will confirm to the teacher that the applicant has the right to request adaptations in the examination and that these adaptations must be agreed upon with the teacher. The lecturer will respond by informing whether it is possible to use the requested adaptations.

Requests should be sent at least 7 days before the date of the call in order to allow the lecturer to assess the content. In particular, in the case of intending to make use of concept maps for the exam (which must be much more concise than the maps used for study) if the submission does not meet the deadline there will be no technical time to make any adjustments.

For more information regarding the request for services and adaptations see the document: Guidelines for requesting services, compensatory tools and/or dispensatory measures and specific aids.

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.