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IMA Journal of Management Mathematics Advance Access originally published online on June 27, 2008
IMA Journal of Management Mathematics 2009 20(1):39-49; doi:10.1093/imaman/dpn007
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© The authors 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Predicting overflow in an emergency department

L. Au{dagger}

Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia

G. B. Byrnes

Centre for Molecular, Environmental, Genetic and Analytic Epidemiology, Department of Public Health, The University of Melbourne, Victoria, Australia

C. A. Bain

Directorate Office, Western and Central Melbourne Integrated Cancer Service, Victoria, Australia

M. Fackrell

Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia

C. Brand

Clinical Epidemiology and Health Service Evaluation Unit, Melbourne Health, Victoria, Australia

D. A. Campbell

Department of Medicine, Southern Clinical School, Monash University, Victoria, Australia

P. G. Taylor

Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia

{dagger} Email: l.au{at}ms.unimelb.edu.au

Received on 9 October 2007. Accepted on 4 February 2008.

Ambulance bypass occurs when the emergency department (ED) of a hospital becomes so busy that ambulances are requested to take their patients elsewhere, except in life-threatening cases. It is a major concern for hospitals in Victoria, Australia, and throughout most of the western world, not only from the point of view of patient safety but also financially—hospitals lose substantial performance bonuses if they go on ambulance bypass too often in a given period. We show that the main cause of ambulance bypass is the inability to move patients from the ED to a ward. In order to predict the onset of ambulance bypass, the ED is modelled as a queue for treatment followed by a queue for a ward bed. The queues are assumed to behave as inhomogeneous Poisson arrival processes. We calculate the probability of reaching some designated capacity C within time t, given the current time and number of patients waiting.

Keywords: access block; ambulance bypass; continuous-time Markov chain; emergency department; Laplace transform


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