This module consists of two distinct parts, the first dealing with Queueing Theory and Stock Control and the second with Game Theory. The Queueing Theory section develops methodology introduced in MA0261 in order to analyse more realistic queueing systems. Embedded Markov Chain Methods are introduced for solving systems with either general arrival distributions or general service patterns. Some case studies and cost analyses are undertaken. The Stock Control section is an introduction to one of the most important management decision processes. Efficient control of stock levels is an essential feature in the financial well-being of any company. Again the emphasis is on the solution of real-life problems, with mini-case studies and cost analyses playing a major role. The Game Theory section will introduce students to the mathematical study of multiple interactive agent decision making. This part of the module provides an introduction to Game Theory through notions such as Nash Equilibria, Evolutionary Game Theory and Bayesian Games. Students will learn Game Theory in an active way through role playing and student-led activities.

This module provides a background in Queueing Theory, Stock Control and Game Theory which is more than adequate for the several MSc courses available in other UK universities. Careers in this subject area are plentiful, interesting and well-paid.

Prerequisite Modules: MA0261 Operational Research

• set up differential-difference equations for the method of phases as applicable to Erlang service times.
• use generating function techniques to provide steady-state solutions of such equations.
• set up steady state equations for M/G/1 and GI/M/1 queues using the imbedded Markov Chain technique.
• undertake simple cost analyses relating to the above queues.
• understand the assumptions behind Cyclical Review and Re-order Level stock control systems.
• recognise the cost implications relevant to different stock control models.
• formulate and analyse simple models of Stock control situations in both deterministic and stochastic forms.
• represent games in both Extensive Form and Normal Form.
• understand concepts linked to mixed strategies.
• compute dominant strategies.
• identify best responses.
• define and compute Nash Equilibria.
• understand and carry out backward induction.
• understand concepts of subgame perfection.
• compute equilibria and best responses in repeated games.
• understand concepts of Evolutionary Game Theory
• compute evolutionary Stable Strategies
• represent incomplete information games
• understand concepts of Bayesian Nash Equilibria and rationalizability

44 – 50 minute lectures

10 – 50 minute exercise classes.

Both parts of the course will be presented in parallel (2.5 hours per week for each part, for a total of 5 hours per week)

Students are also expected to undertake at least 100 hours private study including preparation of worked solutions for tutorial classes.

Problem solving

Logical thinking

Mathematical formulation of real-life situations

Formative assessment is carried out by means of regular exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during classes.  There is no summative coursework assessment for this module

Summative assessment is by means of the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from six equally weighted questions.

• Queueing Theory:
  • Derivation of the Erlang service time distribution. 
  • Setting up differential-difference equations for queueing systems with Erlang service time distributions. 
  • The method of phases. 
  • Use of p.g.f.'s in solution of queueing equations. 
  • Derivation of summary measures of queues. 
  • Cost Analyses. 
  • The imbedded Markov Chain technique. 
  • M/G/1 and GI/M/1 queues. 
  • Case studies.

• Stock Control:
  • Deterministic models involving cyclical review and re-order level methods. 
  • Costing in inventory control. 
  • Power-demand and Time-dependent demand patterns. 
  • Constrained systems. 
  • Simple stochastic models.

• Game Theory:
  • Representations and basic concepts. 
  • Analysing behaviour in static settings. 
  • Further analysis of extensive form games. 
  • Repeated games. 
  • Evolutionary Game Theory.  
  • Games with incomplete information

Cox, D. R., and Smith, W. L., Queues, Chapman and Hall

Taha H A., Operations Research : An Introduction, MacMillan

Watson J., Strategy: An Introduction to Game Theory, Norton