Veckodisposition
-
- CONTENT
The course is directed to describe some key structures which can be used in modeling stochastic processes. The course starts with a repetition on Poisson processes and continuous-time Markov chains. The main topics which follow are renewal theory, queueing theory and Brownian motion. The course includes both analysis of these and related processes and presentation of methods for their simulation.
Regenerative processes are such that the process starts over with the same stochastic properties at some random regeneration times. Typical examples are recurrent Markov processes, renewal processes, and processes describing functions of queuing systems. Brownian motion (Wiener process) plays a fundamental role in stochastic processes. It is a continuous Gaussian process with independent increments possessing variances proportional to time. Brownian motion is effectively used for the construction of wide classes of more complicated stochastic processes, for example stationary processes, diffusion processes, general processes with independent increments, and processes which are solutions of so-called stochastic differential equations.
- TEACHING
Lectures: Matteo Sfragara, matteo.sfragara "at" math.su.se
Computer labs: Taariq Fahran Nazar, taariq.nazar "at" math.su.se
Course literature: S.M. Ross, Introduction to Probability Models, 11th edition, Academic Press, 2014, (available at http://www.sciencedirect.com/science/book/9780124079489).
- EXAMINATION
Examination Form
The examination consists of two parts: a written exam on campus for the theoretical part and a report of lab exercises.
The written exam will consist of 5 problems, each covering one of the main subjects of the theoretical part of the course (ie, (i) Poisson processes, (ii) renewal theory, (iii) queueing theory, (iv) simulation and (v) Brownian motion). Students are not allowed to bring any material, including calculators, notes or cheat-sheets.
Dates for the written exam:
Exam: Tuesday, 30th May, 14:00-19:00.
Re-Exam: Tuesday, 15th August.Approved lab reports, which are performed in pairs or individually, are required for passing the course. The reports must be in pdf format and submitted electronically here on the course page. The code used to solve the assignment must be displayed in the report. If a pair works on the same report, both students must submit the same pdf with both their names on it. The programming language used in the course is R and the programming environment is R-studio. The submission deadline for the laboratory reports is Monday, May 22, 23:59. You will receive feedback within a week after submission and will then have the opportunity to submit a corrected version if your report has not been approved. If you miss the deadline, you can submit the report for late submission. The deadline for both the corrected versions and the late submissions is Monday, June 5, 23:59. Note that these reports will be corrected after the exam. There will be a second round for submitting reports, with submission deadline on Monday, July 31, 23:59. If you fail to submit before this deadline, the next chance will be during the course in spring 2024 with a possibly different assignment.
Note: In recent times, students have been suspended from studies for a certain period of time due to plagiarism. Due to this, we would like to remind everyone that it is not allowed to copy text or computer code (which is not given in the lab instructions) from previously submitted reports, and that all reports are checked for similarity to previously submitted reports in Urkund.
Do not forget to register for the exam at least 10 days in advance!
Examination Rules at the Department of Mathematics.
More info also at https://www.su.se/english/education/student-support/examination-services.
Grading criteria
In order to pass the course you have to score sufficiently on the computer lab report and you have to score at least 3 out of 12 points for each question and 30 out of 60 points in total at the written exam. Conditioned on passing the exam, the grade for the course is a non-decreasing function of the points scored at the written exam.
After the exams have been graded, students can apply for return of their exams by using the following form: https://survey.su.se/Survey/44514/en.- SCHEDULE
The lectures will be given in Room 8, Albano house 2, floor 2. The computer labs will be in the computer Room 33, Albano house 4, floor 2.
The preliminary schedule is given at
https://cloud.timeedit.net/su/web/stud1/ri16eXQu017Z5bQv37093ZZ6y2Y770QQ5Y62Y0gQ607.html.
Lectures:
09:15-10:00, exercise solutions
10:15-11:00, lecture
11:15-12:00, lecture -
Thursday, March 23, 09:15-12:00.
Subject: Repetition of Poisson processes and continuous time Markov chains.
Literature: Sections 5.1-5.3 and 6.1-6.5.
Exercises for next session: Chapter 5: 3, 8, 36, 40, 45, 49, 60.
-
Monday, March 27, 09:15-12:00.
Exercises to discuss: Chapter 5: 3, 8, 36, 40, 45, 49, 60.
Subject: Generalizations of Poisson processes.
Literature: Section 5.4.
Exercises the next session: Chapter 5: 46, 78, 81b (assume that the result of 81a is given), 95.
-
Thursday, March 30, 09:15-12:00.
Exercises to discuss: Chapter 5: 46, 78, 81b (assume that the result of 81a is given), 95.
Subject: Renewal theory.
Literature: Sections 7.1-7.3.
Exercises for next session: Chapter 7: 1, 3, 5, 6a, 9, 12.
-
Monday, April 3, 09:15-12:00.
Exercises to discuss: Chapter 7: 1, 3, 5, 6a, 9, 12.
Subject: Renewal reward theory.
Literature: Section 7.4.
Exercises for next session: Chapter 7: 15, 16, 19, 20 (convergence is almost surely), 26.
-
Thursday, April 6, 09:15-12:00.
Exercises to discuss: Chapter 7: 15, 16, 19, 20 (convergence is almost surely), 26.
Subject: Regenerative and semi-Markov processes and the inspection paradox.
Literature: Sections 7.5-7.7.
Exercises for next session: Chapter 7: 22, 31, 38, 46, 47. For 46 assume that jumps are independent of waiting times, while for 47 allow for dependence.
-
Thursday, April 13, 09:15-12:00.
Exercises to discuss: Chapter 7: 22, 31, 38, 46, 47. For 46 assume that jumps are independent of waiting times, while for 47 allow for dependence.
Subject: Queueing theory: exponential models.
Literature: Sections 8.1, 8.3.1-3 and 8.9.1-2.
Exercises for next session: Chapter 8: 1, 6, 8 (do part c before part b), 12a-b.
-
Monday, April 17, 09:15-12:00.
Exercises to discuss: Chapter 8: 1, 6, 8, 12a-b.
Subject: Queueing theory: PASTA and general workloads.
Literature: Sections 8.2 and 8.5.
Exercises for next session: Chapter 8: 23 (for questions c,d,e, express the answer in P_S's, where S is the state, but you do not have to compute the P_S's), 28, 36, 37, 40.
-
Thursday, April 20, 09:00-12:00.
!!! We start at 09:00 !!!
The session consists of 1 hour of discussing exercises from Chapter 8 and 2 hours of lectures on Chapter 11.
Exercises to Discuss: Chapter 8: 23 (for questions c,d,e, express the answer in P_S's, where S is the state, but you do not have to compute the P_S's), 28, 36, 37, 40.
Subject: Simulation. Variance reduction techniques.
Literature: Sections 11.1-11.4. Section 11.6 (excluding 11.6.3).
Exercises for session 10: Chapter 11: 1, 5, 7, 8, 13, 30-33. For 8 use Wikipedia for Stirling's approximation.
-
Monday, April 24, 09:00-12:00.
!!! We start at 09:00 !!!
The session consists of 3 hours of lectures on Chapter 11.
Subject: Simulation of stochastic processes and MCMC methods.
Literature: Sections 11.5.0-1 and Section 4.9. See also the extra material in the reading guide.
Exercises for session 10: Chapter 11: 17, 23, 24. For 23 assume that the intensity is strictly positive everywhere.
-
Thursday, April 27, 09:15-12:00.
The entire session is dedicated to discussing the exercises on Chapter 11 with Tariq Nazar, in preparation for the computer assignment.
Exercises to discuss: Chapter 11: 1, 5, 7, 8, 13, 30-33, 17, 23, 24.
-
Thursday, May 4, 09:15-12:00.
Computer lab.
-
Monday, May 8, 09:15-12:00.
Computer lab.
-
Thursday, May 11, 09:15-12:00.
!!! This session will be in Room 26, Albano house 4, floor 2 !!!
Subject: Brownian motion.
Literature: Sections 10.1-10.3
Exercises for next session: Chapter 10: 1-4, 6.
-
Monday, May 15, 09:15-12:00.
Exercises to discuss: Chapter 10: 1-4, 6.
Subject: Variations on Brownian motion, Gaussian processes and stationary processes.
Literature: Sections 10.5, 10.7 and 10.8.
Exercises for next session: Chapter 10: 7, 9, 10, 32, 35. Suggested extra: 11, 33, 34. Note that answers might contain some integrals that are very hard to evaluate. It is okay to provide answers as not evaluated integrals.
-
Monday, May 22, 09:15-12:00.
Exercises to discuss: Chapter 10: 7, 9, 10, 32, 35. Extra: 11, 33, 34.
Subject: Repetition.
-
Tuesday, May 30, 14.00-19.00: Exam.
-
Tuesday, August 15: Re-exam.