Delivery Type	Delivery length / details
Lecture	18 X 1 HOUR
Seminars / Tutorials	6 X 1 HOUR
Practical	8 X 2 HOUR
Workload Breakdown	(EVERY 10 CREDITS CARRIES A NOTIONAL STUDENT WORKLOAD OF 100 HOURS) Lectures and tutorials 24 hours Worksheets (6 x 5 hours) 30 hours Practical classes 16 hours Practical report writing 16 hours Private study 112 hours Examination 2 hours LECTURES AND TUTORIALS

Assessment Type	Assessment length / details	Proportion
Semester Assessment	PRACTICAL REPORTS	25%
Semester Exam	2 Hours CONVENTIONAL 2 HOUR EXAMINATION	75%
Supplementary Assessment	2 Hours CONVENTIONAL 2 HOUR EXAMINATION	100%

Learning Outcomes

On successful completion of this module students should be able to:

1. formulate and solve simple problems in random arrival modelling;

2. set up and solve the equations governing a simple queue;

3. formulate a given real situation as a queueing model;

4. appreciate the failings of simple models and what modified models are trying to cover;

5. design algorithms to simulate from a given distribution using different methods;

6. design a queue simulation and interpret the results of running it.

7. evaluate and use the reliability function, hazard function, mean time to failure and reliable lifetime of lifetime distributions in common use;

8. evaluate the reliability of systems of independent components;

9. calculate and interpret simple bounds on reliability;

10. explain and use the notation used in fault trees and be aware of their uses;

11. in life testing, appropriately use censoring and acceleration; and estimate exponential and Weibull parameters in such life tests.

Aims

IMAPS has for some years been able to offer a very limited range of modules for final year students. With the imminent arrival of more staff, it is now proposed to rectify this by introducing Level 3 modules that can be extended to cater for Level M students also. This module introduces the important area of stochastic Operational Research, covering Queueing and Reliability Theory, and also the technique of Simulation; this latter topic greatly extends the area of application and is a technique that has much wider application than the two particular areas covered. It is intended to offer this module in alternate years.

Brief description

Queueing Theory is a classic application of probability that falls within the umbrella of stochastic operational research, motivated by practical problems. The module covers the basic models of Queueing Theory and the formulation of real scenarios in their terms.
The module also introduces students to the properties of lifetime distributions, to study the reliability of systems of components and gain an appreciation of how high levels of reliability and safety may be achieved in practice. The fitting of models to data is also covered.
The statistical technique of Simulation has extensive applications both in Queueing and Reliability. It also greatly extends the range of problems that can be studied and solved; the module introduces the simulation of queues as an example of its scope.

Content

Random events
Modelling of Markovian arrival processes. Queues and their characteristics. Queue model notation. The Simple Queue
Simple Markovian (M/M/1) Queue; basic theory; steady state solutions. Waiting times and their distribution. The output process.
Generalised Markovian systems
Queues where arrival and service rates are dependent on system size. Examples will include limited waiting rooms, multiple server queues, self-service queues, telephone exchange design, machine minding, server fatigue, etc.
Non-Markovian Queues
The method of stages; Erlangian distributions. Pollaczek Khintchine theory. Priority queues.
Introduction to Simulation
Random numbers. Inverse transform theorem. Composition and Acceptance-rejection techniques. Simulating a queue.
Statistical Failure Models:
Reliability and hazard functions, mean time to failure, reliable lifetime; distributions (Exponential, Weibull, Gamma, Gumbel, Log Normal) competing risks; simple bounds on reliability.
Fitting Models to Reliability Data
Life tests: type 1 and 2 censoring, progressive censoring; accelerated life tests. Kaplan-Meier estimator. Maximum likelihood estimation for exponential and Weibull with censoring; reliability function, reliable lifetime. Arrhenius and power law models.
Systems Reliability
Series, parallel, k out of n systems. Path and cut sets, monotonic systems, modules. Bounds on system reliability. Fault trees.

Skills Type	Skills details
Application of Number	Throughout the module
Communication	Students will be required to communicate their findings and results in writing (in practical classes) and to submit worksheets (for tutorials).
Improving own Learning and Performance	N/A
Information Technology	Use of spreadsheets and Statistical Packages.
Personal Development and Career planning	Students will be exposed to an area of application that they have not previously encountered.
Problem solving	All situations considered are problem-based to a greater or lesser degree.
Research skills	Students will be encouraged to consult various books and journals for examples of application.
Subject Specific Skills	Modelling of practical situations in stochastic terms
Team work	Students will be encouraged to work together in practical classes

Course Delivery

Assessment

Learning Outcomes

Aims

Brief description

Content

Module Skills

Reading List

Notes