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
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.
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.
By the end of the module, students should be able to:
View reading list on Talis Aspire
TBC
TBC
| 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 |
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.
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.
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| In-person Examination | 100% | No | |
|
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.
|
|||
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| In-person Examination - Resit | 100% | No | |
|
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.
|
|||
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.
If you take this module, you cannot also take:
This module is Optional for:
This module is Core option list A for:
This module is Option list A for:
This module is Option list B for:
This module is Option list D for:
This module is Option list E for:
This module is Option list F for: