APP MTH 7056 - Random Processes

North Terrace Campus - Semester 2 - 2019

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.

  • General Course Information
    Course Details
    Course Code APP MTH 7056
    Course Random Processes
    Coordinating Unit 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
    Assessment ongoing assessment, exam
    Course Staff

    Course Coordinator: Dr Andrew Black

    Course Timetable

    The full timetable of all activities for this course can be accessed from Course Planner.

  • Learning Outcomes
    Course Learning Outcomes
    1.     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)
    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
    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
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    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
  • Learning Resources
    Required Resources
    Recommended Resources
    Students 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 Learning
    Assignments, 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 Modes
    The 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 Summary
    Lecture Schedule

    Week 1           Introduction and Review of Discrete-time Markov chains

    Week 2           Modelling      
                          Understanding the formulation of CTMCs

    Week 3           Transient Behaviour            
                          Kolmogorov DEs

    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.
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

    1. Assessment must encourage and reinforce learning.
    2. Assessment must enable robust and fair judgements about student performance.
    3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
    4. Assessment must maintain academic standards.

    Assessment Summary
    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 Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    Assessment task
    Set Due
    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%
    Homework 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.
    Course Grading

    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.

  • Student Feedback

    The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

    SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy ( course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

  • Student Support
  • Policies & Guidelines
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