General description
Queuing Theory is an application of mathematics and probability theory that arises directly from a practical situation, viz the development and dissipation of queues. Whilst mathematics can go a long way to analysing such situations, some complex systems defy analytical treament, and Simulation is concerned with ways in which practical situations can be 'mimicked' mathematically. This course introduces these two areas.

Aims
To introduce the student to the basic theory of queues and to the technique of simulation.

Learning outcomes
On completion of this module, a student should be able to:

• explain the basic theory of random arrival modelling;
• set up and solve the equations governing a simple queue;
• formulate a given real situation as a queueing model;
• describe the failings of simple models and what modified models are trying to cover;
• design algorithms to simulate from a given distribution using different methods;
• design a queue simulation and interpret the results of running it.

Syllabus
1. INTRODUCTION: Random events. Queues and their characteristics. Markovian arrival processes. Queue model notation.
2. THE SIMPLE QUEUE: Simple Markovian (M/M/1) Queue; basic theory; steady state solutions. Waiting times and their districution. The output process.
3. 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
4. NON-MARKOVIAN QUEUES: The method of stages; Erlangian sitributions. Pollaczek-Khintchine methods. Priority queues.
5. INTRODUCTION TO SIMULATION: Random numbers. Inverse transform theorem. Composition methods. Acceptance-rejection techniques. Simulating a queue.
6. PROJECT: Detailed study of a specific queueing situation to be completed by the end of the semester, with most of the work being in the second half.