APP MTH 4116 - Random Processes - Honours

North Terrace Campus - Semester 2 - 2016

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, mean; 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 4116
    Course Random Processes - Honours
    Coordinating Unit Mathematical Sciences
    Term Semester 2
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 hours per week
    Available for Study Abroad and Exchange
    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, mean; 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 Staff

    Course Coordinator: Dr Giang Nguyen

    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)
    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
    Intercultural and ethical competency
    • adept at operating in other cultures
    • comfortable with different nationalities and social contexts
    • able to determine and contribute to desirable social outcomes
    • demonstrated by study abroad or with an understanding of indigenous knowledges
    all
    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
    all
  • Learning Resources
    Required Resources
    None.
    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.
    Workload

    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 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%
    Submission
    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:

    M11 (Honours Mark Scheme)
    GradeGrade reflects following criteria for allocation of gradeReported on Official Transcript
    Fail A mark between 1-49 F
    Third Class A mark between 50-59 3
    Second Class Div B A mark between 60-69 2B
    Second Class Div A A mark between 70-79 2A
    First Class A mark between 80-100 1
    Result Pending An interim result RP
    Continuing Continuing CN

    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 (http://www.adelaide.edu.au/policies/101/) 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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