ST33315 Applied Stochastic Processes
Introductory description
This module runs 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.
Prerequisites: ST202 Stochastic Processes or ST227 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.

Continuous time Markov Chains.
Terms used in the analysis of continuoustime Markov chains: Markov property, transition probability function, standing assumptions, ChapmanKolmogorov equations, Qmatrix, Kolmogorov forward and backward differential equations, equilibrium distribution. The simplest case: finite statespace Markov chains. The "switcher" example. Exact transition densities for processes on a small number of states. The strong Markov property. 
Linear BirthDeath processes.
Poisson (counting) process: construction, ideas of independent increments, superposition, counts and thinning. Pure birth process, linear birthdeath process, birthdeathimmigration process: construction using "microscopic model", derivation of extinction and equilibrium probabilities. Generalized birthdeath processes. 
Queuing theory.
The Markov singleserver (M/M/1) queue. The concept of detailed balance. Measures of effectiveness. Multiserver (M/M/cl/c2) queues. Erlang's formula. Queues with general servicetime distribution (M/G/l) and their embedded Markov chains. Little's formula, PollaczekKhintchine formula. 
Other Markov properties.
Stopping times. Strong Markov property. Holding theorem. 
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 continuoustime 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

Demonstrate facility with rigorous probabilistic methods.

Evaluate, select and apply appropriate mathematical and/or probabilist techniques.

Demonstrate knowledge of and facility with formal probability concepts, both explicitly and by applying them to the solution of mathematical problems.

Create structured and coherent arguments communicating them in written form.

Construct logical arguments with clear identification of assumptions and conclusions.

Reason critically, carefully, and logically.
Transferable skills

Problem solving: Use rational and logical reasoning to deduce appropriate and wellreasoned conclusions. Retain an open mind, optimistic of finding solutions, thinking laterally and creatively to look beyond the obvious. Know how to learn from failure.

Self awareness: Reflect on learning, seeking feedback on and evaluating personal practices, strengths and opportunities for personal growth.

Communication: Present arguments, knowledge and ideas, in a range of formats.

Professionalism: Prepared to operate autonomously. Aware of how to be efficient and resilient. Manage priorities and time. Selfmotivated, setting and achieving goals, prioritising tasks.
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 noncredit 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  

Inperson Examination  100%  
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.

Assessment group R3
Weighting  Study time  

Inperson Examination  Resit  100%  
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.

Feedback on assessment
Opportunities will be provided to submit noncredit bearing coursework for which feedback will be provided in the following problem class.
Solutions and cohort level feedback will be provided for the examination.
Antirequisite modules
If you take this module, you cannot also take:
 ST40615 Applied Stochastic Processes with Advanced Topics
Courses
This module is Optional for:
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics

UCSAG4G1 Undergraduate Discrete Mathematics
 Year 3 of G4G1 Discrete Mathematics
 Year 3 of G4G1 Discrete Mathematics
 Year 3 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 4 of UCSAG4G4 Undergraduate Discrete Mathematics (with Intercalated Year)
 Year 4 of UCSAG4G2 Undergraduate Discrete Mathematics with Intercalated Year

USTAG300 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 Unusual option for:
 Year 2 of TMAAG1PC Postgraduate Taught Mathematics (Diploma plus MSc)
This module is Core option list A for:
 Year 3 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 4 of USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
This module is Option list A for:

USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 3 of G1G3 Mathematics and Statistics (BSc MMathStat)
 Year 4 of G1G3 Mathematics and Statistics (BSc MMathStat)

USTAG1G4 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)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 3 of GG14 Mathematics and Statistics
 Year 3 of GG14 Mathematics and Statistics
 Year 4 of USTAGG17 Undergraduate Mathematics and Statistics (with Intercalated Year)

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 4 of USTAY603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)
This module is Option list B for:

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 4 of G105 Mathematics (MMath) with Intercalated Year
 Year 5 of G105 Mathematics (MMath) with Intercalated Year
 Year 3 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

USTAG301 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

UMAAG100 Undergraduate Mathematics (BSc)
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 3 of G100 Mathematics
 Year 3 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)

UMAAG106 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 UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Option list C for:

USTAG302 Undergraduate Data Science
 Year 3 of G302 Data Science
 Year 3 of G302 Data Science
 Year 3 of USTAG304 Undergraduate Data Science (MSci)
 Year 4 of USTAG303 Undergraduate Data Science (with Intercalated Year)
This module is Option list F for:
 Year 3 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

USTAG301 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)