ST406-15 Applied Stochastic Processes with Advanced Topics

22/23
Department
Statistics
Level
Sam Olesker-Taylor
Credit value
15
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

Introductory description

This module runs in Term 1 and is available for students on a course where it is a listed option and as an Unusual Option to students who have completed the prerequisite modules.

The ideas presented in this module have a vast range of applications, for example routing algorithms in telecommunications (queues), assessment of apparent spatial order in astronomical data (stochastic geometry), description of outbreaks of disease (epidemics). We will only be able to introduce each area - indeed each area could easily be the subject of a course on its own! But the introduction will provide you with a good base to follow up where and when required. (For example: a MORSE graduate found that their firm was asking them to address problems in queuing theory, for which ST333 provided the basis.) We will discuss these and other applications and show how the ideas of stochastic process theory help in formulating and solving relevant questions.

Students will be given selected advanced research material for independent study and examination.

Pre-requisites: ST202 Stochastic Processes

Module aims

To provide an introduction to concepts and techniques which are fundamental in modern applied probability theory and operations research:
Models for queues, point processes, and epidemics.
Notions of equilibrium, threshold behaviour, and description of structure.

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1: Continuous time Markov Chains.
Terms used in the analysis of continuous-time Markov chains: Markov property, transition probability function, standing assumptions, Chapman-Kolmogorov equations, Q-matrix, Kolmogorov forward and backward differential equations, equilibrium distribution. The simplest case: finite state-space Markov chains. The "switcher" example. Exact transition densities for processes on a small number of states. The strong Markov property.
2: Linear Birth-Death processes.
Poisson (counting) process: construction, ideas of independent increments, superposition, counts and thinning. Pure birth process, linear birth-death process, birth-death-immigration process: construction using "microscopic model", derivation of extinction and equilibrium probabilities. Generalized birth-death processes.
3: Queuing theory.
The Markov single-server (M/M/1) queue. The concept of detailed balance. Measures of effectiveness. Multiserver (M/M/cl/c2) queues. Erlang's formula. Queues with general service-time distribution (M/G/l) and their embedded Markov chains. Little's formula, Pollaczek-Khintchine formula.
4: Other Markov properties.
Stopping times. Strong Markov property. Holding theorem.
5: Epidemics.
Deterministic Epidemic model. Stochastic model without removals. Stochastic model with removals.

Learning outcomes

By the end of the module, students should be able to:

• Be able to formulate continuous-time Markov chain models for applied problems.
• Be able to use basic theory to gain quick answers to important questions (for example, what is the equilibrium distribution for a specific reversible Markov chain?).
• Be able to solve for the transition probabilities for Markov chains on a finite state space.
• Understand how to use Markov chains in the modelling and analysis of queues and epidemics.

TBC

TBC

Study time

Type Required Optional
Lectures 30 sessions of 1 hour (20%) 2 sessions of 1 hour
Seminars 5 sessions of 1 hour (3%)
Private study 115 hours (77%)
Total 150 hours
Private study description

Study of advanced topic, completion of non-credit bearing coursework, weekly revision of lecture notes and materials, wider reading, practice exercises and preparing for examination.

Costs

No further costs have been identified for this module.

You must pass all assessment components to pass the module.

Students can register for this module without taking any assessment.

Assessment group B4
Weighting Study time
In-person Examination 100%

The examination will contain one compulsory question on the advanced topic and four additional questions of which the best marks of TWO questions will be used to calculate your grade.

• Students may use a calculator
Assessment group R2
Weighting Study time
In-person Examination - Resit 100%

The examination will contain one compulsory question on the advanced topic and four additional questions of which the best marks of TWO questions will be used to calculate your grade.

• Students may use a calculator
• Cambridge Statistical Tables (blue)
Feedback on assessment

Opportunities will be provided to submit non-credit bearing coursework for which feedback will be provided in the following problem class.
Solutions and cohort level feedback will be provided for the examination.

Anti-requisite modules

If you take this module, you cannot also take:

• ST333-15 Applied Stochastic Processes

Courses

This module is Optional for:

• TMAA-G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of TMAA-G1PD Postgraduate Taught Interdisciplinary Mathematics (Diploma plus MSc)
• Year 1 of TMAA-G1P0 Postgraduate Taught Mathematics
• Year 1 of TMAA-G1PC Postgraduate Taught Mathematics (Diploma plus MSc)
• Year 1 of TSTA-G4P1 Postgraduate Taught Statistics
• USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• Year 3 of G300 Mathematics, Operational Research, Statistics and Economics
• Year 4 of G300 Mathematics, Operational Research, Statistics and Economics

This module is Core option list A for:

• Year 3 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
• Year 4 of USTA-G1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

This module is Option list A for:

• Year 4 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• Year 5 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
• USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
• Year 3 of G1G3 Mathematics and Statistics (BSc MMathStat)
• Year 4 of G1G3 Mathematics and Statistics (BSc MMathStat)
• USTA-G1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
• Year 4 of G1G4 Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
• Year 5 of G1G4 Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

This module is Option list B for:

• Year 4 of USTA-G304 Undergraduate Data Science (MSci)
• Year 4 of UCSA-G4G3 Undergraduate Discrete Mathematics
• Year 3 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
• Year 3 of G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int
• Year 4 of G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int

This module is Option list D for:

• USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• Year 4 of G30C Master of Maths, Op.Res, Stats & Economics (Operational Research and Statistics Stream)
• Year 4 of G30C Master of Maths, Op.Res, Stats & Economics (Operational Research and Statistics Stream)
• Year 5 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated

This module is Option list E for:

• Year 4 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• Year 5 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated

This module is Option list F for:

• Year 3 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
• Year 3 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
• Year 4 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)

Assessments dates for Statistics modules, including coursework and examinations, can be found in the Statistics Assessment Handbook.