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 web page

Module aims

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.

Indicative reading list

View reading list on Talis Aspire

Subject specific skills

TBC

Transferable skills

TBC