APP MTH 7056 - Random Processes
North Terrace Campus - Semester 2 - 2018
General Course Information
Course Code APP MTH 7056 Course Random Processes Coordinating Unit School of Mathematical Sciences Term Semester 2 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Available for Study Abroad and Exchange Y Prerequisites MATHS 1012 Assumed Knowledge Knowledge of Markov chains, such as would be obtained from MATHS 2103 Course Description This course introduces students to the fundamental concepts of random processes, particularly continuous-time Markov chains, and related structures. These are the essential building blocks of any random system, be it a telecommunications network, a hospital waiting list or a transport system. They also arise in many other environments, where you wish to capture the development of some element of random behaviour over time, such as the state of the surrounding environment.
Topics covered are: Continuous-time Markov-chains: definition and basic properties, transient behaviour, the stationary distribution, hitting probabilities and expected hitting times, reversibility; Queueing Networks: Kendall's notation, Jackson networks, Loss Networks: truncated reversible processes, circuit-switched networks, reduced load approximations. Basic Queueing Theory: arrival processes, service time distributions, Little's Law; Point Processes: Poisson process, properties and generalisations; Renewal Processes: preliminaries, renewal function, renewal theory and applications, stationary and delayed renewal processes.
Course Coordinator: Professor Joshua Ross
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning Outcomes1. demonstrate understanding of the mathematical basis of continuous-time Markov chains
2. demonstrate the ability to formulate continuous-time Markov chain models for relevant practical systems
3. demonstrate the ability to apply the theory developed in the course to problems of an
appropriate level of difficulty
4. develop an appreciation of the role of random processes in system modelling
5. demonstrate skills in communicating mathematics orally and in writing
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s) Deep discipline knowledge
- informed and infused by cutting edge research, scaffolded throughout their program of studies
- acquired from personal interaction with research active educators, from year 1
- accredited or validated against national or international standards (for relevant programs)
all Critical thinking and problem solving
- steeped in research methods and rigor
- based on empirical evidence and the scientific approach to knowledge development
- demonstrated through appropriate and relevant assessment
all Teamwork and communication skills
- developed from, with, and via the SGDE
- honed through assessment and practice throughout the program of studies
- encouraged and valued in all aspects of learning
all Career and leadership readiness
- technology savvy
- professional and, where relevant, fully accredited
- forward thinking and well informed
- tested and validated by work based experiences
1,3 Self-awareness and emotional intelligence
- a capacity for self-reflection and a willingness to engage in self-appraisal
- open to objective and constructive feedback from supervisors and peers
- able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
Recommended ResourcesStudents may wish to consult any of the following books, available in the Library.
1. Introduction to Probability Models, (currently the 10th edition), Sheldon Ross, Academic Press, 2009
2. Introduction to Stochastic Models (2nd edition), R. Goodman, Dover Publications, 2006
Online LearningAssignments, tutorial exercises, handouts, video recordings of lectures and course announcements will be posted on MyUni.
Please don't hesitate to email the lecturer should anything be missing.
Learning & Teaching Activities
Learning & Teaching ModesThe lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures, but lectures will be recorded to help with occasional absences and for revision purposes. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload hours Lectures 30 90 Tutorials 5 20 Assignments 5 25 Project 1 20 Total 155
Learning Activities SummaryLecture Schedule
Week 1 Introduction and Review of Discrete-time Markov chains
Week 2 Modelling
Understanding the formulation of CTMCs
Week 3 Transient Behaviour
Week 4 Equilibrium Behaviour
Global Balance equations and characterisation
Week 5 Hitting Times and Reversibility
Hitting probabilities, expected hitting times and reversible Markov chains
Week 6 Queueing Networks
Burke’s Theorem, Jackson Networks, the theory of truncation of reversible Markov chains and application to queueing networks
Week 7 Reduced Load Approximations
Erlang Fixed Point Method
Week 8 Observed distributions
PASTA, Waiting time distributions, Little’s Law, Pollaczek-Khinchin
Week 9 Point processes
Background and Markovian Arrival Processes
Week 10 Renewal Theory
Riemann-Stieltjes Integration, Laplace Stieltjes Transform, the Convolution Theorem
Week 11 Renewal Theory
Convergence of random variables, the counting and waiting time processes, the renewal function
Week 12 Renewal Theory
Generalised renewal equation, the Basic, Blackwell’s and Elementrary Renewal Theorems, forward and backward recurrence times, Delayed and Stationary renewal processes
Tutorials in Weeks 3, 5, 7, 9, 11 cover the material of the previous few weeks.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 70% All Homework assignments Formative and summative Weeks 2, 4, 6, 8, 12 15% All Project Formative and summative Week 10 15% All
Assessment Related RequirementsAn aggregate score of 50% is required to pass the course.
Assessment task Set Due Weighting Assignment 1 Week 1 Week 2 3% Assignment 2 Week 3 Week 4 3% Assignment 3 Week 5 Week 6 3% Assignment 4 Week 7 Week 8 3% Assignment 5 Week 11 Week 12 3% Project Week 3 Week 10 15%
SubmissionHomework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible.
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description FNS Fail No Submission F 1-49 Fail P 50-64 Pass C 65-74 Credit D 75-84 Distinction HD 85-100 High Distinction CN Continuing NFE No Formal Examination RP Result Pending
Further details of the grades/results can be obtained from Examinations.
Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.
Final results for this course will be made available through Access Adelaide.
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