Computational and Mathematical Methods in Medicine
Volume 2012 (2012), Article ID 290360, 6 pages
Research Article

A Mathematical Modelling Approach for Systems Where the Servers Are Almost Always Busy

1Clinical Operational Research Unit, University College London, 4 Taviton Street, London WC1H 0BT, UK
2Psychology, College of Life and Environmental Sciences, University of Exeter, Perry Road, Exeter EX4 4QG, UK

Received 15 November 2011; Accepted 23 January 2012

Academic Editor: John Hotchkiss

Copyright © 2012 Christina Pagel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The design and implementation of new configurations of mental health services to meet local needs is a challenging problem. In the UK, services for common mental health disorders such as anxiety and depression are an example of a system running near or at capacity, in that it is extremely rare for the queue size for any given mode of treatment to fall to zero. In this paper we describe a mathematical model that can be applied in such circumstances. The model provides a simple way of estimating the mean and variance of the number of patients that would be treated within a given period of time given a particular configuration of services as defined by the number of appointments allocated to different modes of treatment and the referral patterns to and between different modes of treatment. The model has been used by service planners to explore the impact of different options on throughput, clinical outcomes, queue sizes, and waiting times. We also discuss the potential for using the model in conjunction with optimisation techniques to inform service design and its applicability to other contexts.