ST333 Applied Stochastic Processes
Please note that all lectures for Statistics modules taught in the 2022-23 academic year will be delivered on campus, and that the information below relates only to the hybrid teaching methods utilised in 2021-22 as a response to Coronavirus. We will update the Additional Information (linked on the right side of this page) prior to the start of the 2022/23 academic year.
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ST333-15 Applied Stochastic Processes
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.
Pre-requisites: ST202 Stochastic Processes
Results from this module may be partly used to determine of exemption eligibility in the Institute and Faculty of Actuaries (IFoA) modules CS2.
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:
- formulate continuous-time Markov chain models for applied problems.
- use basic theory to gain quick answers to important questions (for example, what is the equilibrium distribution for a specific reversible Markov chain?).
- 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.
Indicative reading list
View reading list on Talis Aspire
Subject specific skills
TBC
Transferable skills
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
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 B2
Weighting | Study time | Eligible for self-certification | |
---|---|---|---|
On-campus Examination | 100% | No | |
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade. ~Platforms - Moodle
|
Assessment group R1
Weighting | Study time | Eligible for self-certification | |
---|---|---|---|
In-person Examination - Resit | 100% | No | |
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade. ~Platforms - Moodle
|
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:
- ST406-15 Applied Stochastic Processes with Advanced Topics
Courses
This module is Optional for:
- Year 3 of UCSA-G4G1 Undergraduate Discrete Mathematics
- Year 3 of UCSA-G4G3 Undergraduate Discrete Mathematics
- Year 4 of UCSA-G4G2 Undergraduate Discrete Mathematics with Intercalated Year
-
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)
This module is Option list A for:
-
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)
- Year 5 of USTA-G1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
- Year 3 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
- Year 4 of USTA-GG17 Undergraduate Mathematics and Statistics (with Intercalated Year)
- Year 3 of USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics
- Year 4 of USTA-Y603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)
This module is Option list B for:
- Year 3 of USTA-G304 Undergraduate Data Science (MSci)
- Year 4 of USTA-G303 Undergraduate Data Science (with Intercalated Year)
-
UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
- Year 3 of G105 Mathematics (MMath) with Intercalated Year
- Year 5 of G105 Mathematics (MMath) with Intercalated Year
- 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
- Year 3 of UMAA-G100 Undergraduate Mathematics (BSc)
-
UMAA-G103 Undergraduate Mathematics (MMath)
- Year 3 of G100 Mathematics
- Year 3 of G103 Mathematics (MMath)
- Year 4 of G103 Mathematics (MMath)
-
UMAA-G106 Undergraduate Mathematics (MMath) with Study in Europe
- Year 3 of G106 Mathematics (MMath) with Study in Europe
- Year 4 of G106 Mathematics (MMath) with Study in Europe
- Year 4 of UMAA-G101 Undergraduate Mathematics with Intercalated Year
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)