Unit information: Introduction to Queueing Networks in 2012/13

Unit name	Introduction to Queueing Networks
Unit code	MATH35800
Credit points	10
Level of study	H/6
Teaching block(s)	Teaching Block 1A (weeks 1 - 6)
Unit director	Dr. Ayalvadi Ganesh
Open unit status	Not open
Pre-requisites	MATH21400
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Queues are a fact of life - banks, supermarkets, registration at university - especially in the UK where we wait patiently in line for service! The modelling and evaluation of individual queueing systems (in terms of quantities such as customer arrival patterns, service demands, scheduling priorities for different customer classes, queue size and waiting times) has been a rich source of theory and application in applied probability and operational research. More recently, networks of linked queueing systems have gained wide popularity for modelling and performance-evaluation purposes in telecommunications, computer technology and manufacturing. Much of the success of these queueing models can be attributed to their flexible modelling capabilities and to the simple Jackson product-form expressions that are often available to describe steady-state distributions. The course will introduce relevant concepts in the context of a single server queue and look at simple parallel and tandem systems, before going on to develop models and performance criteria applicable to more general networks.

Aims

To introduce stochastic models for the description and analysis of simple queueing systems and queueing networks.

Syllabus

1. Introduction: Markov chains, Exponential distributions, Poisson processes.

2. Continuous time Markov processes: models, full balance equations, equilibrium distributions.

3. Reversibility: Detailed balance, birth and death processes, Kelly's lemma.

4. Queues: Single-server and infinite server queues; Jackson networks, traffic equations.

5. Arrival/Departure theorems: Distributions seen by arriving and departing customers, PASTA property

6. Performance measures: Little's theorem. Service disciplines, multiple queues.

7. Mean-field queueing models.

Relation to Other Units

The units Information Theory, Financial Mathematics, Queuing Networks and Complex Networks apply probabilistic methods to problems arising in various fields.

Intended Learning Outcomes

Students who successfully complete this unit should be able to:

• identify the transition rates for simple Markov processes from an informal description of the system;

• construct Markov process models of simple queueing networks, specified in terms of the transition rates, and understand the basic properties of such models;

• define the concepts of reversed and reversible Markov processes and use them to construct equilibrium distributions for simple queueing networks;

• compute the distribution of the queue size as seen in equilibrium by arrivals and departures;

• use Little's theorem to compute appropriate performance measures for simple systems.

Transferable Skills:

The ability to translate practical problems into mathematics and the construction of appropriate probabilistic models.

Teaching Information

Lectures and weekly problem sheets, from which work will be set and marked, with outline solutions handed out a fortnight later.

Assessment Information

The assessment mark for Introduction to Queuing Networks is calculated from a 1½-hour written examination in April, consisting of THREE questions. The candidate's best TWO answers will be used for assessment. Calculators are not allowed.

Reading and References

Main text:

• F P Kelly, Reversibility and stochastic networks, Wiley, 1979.

Other recommended texts:

• I Mitrani, Modelling of computer and communication systems, Cambridge University Press, 1998.

• J Walrand, An introduction to queuing networks, Prentice-Hall International, 1988.