Stochastic Models
| Code | School | Level | Credits | Semesters |
| MATH3039 | School of Mathematical Sciences | 3 | 20 | Autumn China |
- Code
- MATH3039
- School
- School of Mathematical Sciences
- Level
- 3
- Credits
- 20
- Semesters
- Autumn China
Summary
In this module the ideas of discrete-time Markov chains, introduced in the module MATH2039, are extended to include more general discrete-state space stochastic processes evolving in continuous time and applied to a range of stochastic models for situations occurring in the natural sciences and industry. The module begins with an introduction to Poisson processes and birth-and-death processes. This is followed by more extensive studies of epidemic models and queueing models, and introductions to component and system reliability. The course finishes with a brief introduction to Stochastic Differential Equations. Students will gain experience of classical stochastic models arising in a wide variety of practical situations. In more detail, the course includes:
• homogeneous Poisson processes and their elementary properties;
• birth-and-death processes - forward and backward equations, extinction probability;
• epidemic processes - chain-binomial models, parameter estimation, deterministic and stochastic general epidemic, threshold behaviour, carrier-borne epidemics;
• queueing processes - equilibrium behaviour of single server queues;
• queues with priorities;
• component reliability and replacement schemes;
• system reliability;
• Stochastic differential equations and Ito's lemma.
Target Students
Single Honours students from the School of Mathematical Sciences.Requisites: MATH2039 Probability Models and Methods
Classes
- One 2-hour lecture each week for 12 weeks
- Two 1-hour lectures each week for 12 weeks
Two one-hour and one two-hour classes per week timetabled centrally, some of which may be used for examples classes and/or problem classes.
Assessment
- 100% Exam 1 (3-hour): 3 hour written examination
Assessed by end of autumn semester
Educational Aims
The purpose of this module is to broaden the students' knowledge and experience of stochastic processes by studying a wide range of stochastic models that have application in the natural sciences and in industry.This course builds upon the ideas of stochastic processes introduced in the module MATH2039. Students will acquire knowledge and skills of the analysis and application of stochastic models.Learning Outcomes
A student who completes this module successfully should be able to:
L1 - derive and apply basic properties of homogeneous Poisson processes;
L2 - state and apply basic properties of continuous-time Markov chains (e.g. definition of chain; holding times and jump probabilities; Chapman-Kolmogorov equations );
L3 - derive forward and backward equations and equilibrium distributions for continuous-time Markov chains;
L4 - state and apply basic properties of chain-binomial and carrier-borne epidemic models (e.g. definitions of Reed-Frost and Greenwood models; calculation of epidemic chain probabilities; calculation of final size distributions);
L4 - state and apply basic results and methods for analysing queues (e.g. method of stages; Little's Theorem; mean busy period)
L5 - state and apply basic results and methods for system reliability (e.g. reliability function; parallel and series systems; new-better-than-used distributions; calculation of optimal block replacement schemes);
L6 - state and apply Ito's Lemma (e.g. for calculating expectations and integrals);
L7 - solve simple stochastic differential equations.