Queueing Theory and Ideal Hospital Occupancy

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1 Queueing Theory and Ideal Hospital Occupancy Peter Taylor Department of Mathematics and Statistics The University of Melbourne

2 Hospital Occupancy A statement to think about. Queuing theory developed by Erlang nearly 100 years ago tells us that systems are most efficient when they operate at 85% capacity. This applies to queues at the local bank waiting for the teller or at ticket booths at the MCG". Access block and overcrowding in emergency departments. Australasian College for Emergency Medicine, April 2004, Page 5.

3 The Statement Again Queuing theory developed by Erlang nearly 100 years ago tells us that systems are most efficient when they operate at 85% capacity. This applies to queues at the local bank waiting for the teller or at ticket booths at the MCG". Why?????? Where does the 85% figure come from? What does most efficient mean?

4 Why?????? The following comes some work that we did jointly with the Royal Melbourne Hospital. The analysis is based on data collected between 1 January 2001 and 18 April A total of 199,480 records appear. WARD 1 Triage Desk Assessment and Treatment WARD 2 WARD 3 WARD 4 DNW Discharge/DOA

5 ATS The Australian Triage Scale Category Access Performance Level 1 Immediate 100% 2 10 minute 80% 3 30 minute 75% 4 60 minute 70% minute 70%

6 ATS The Canadian Triage Scale Category Access Performance Level 1 Immediate 98% 2 15 minute 95% 3 30 minute 90% 4 60 minute 85% minute 80%

7 Queue Size Percentage (%) Bed Treatment Size of Queue

8 Waiting Time Percentage (%) Treatment: Waiting Time Bed: Admission Delay >=24 Hours

9 Bagust, Place and Posnett The source of the 85% figure appears to be the paper A. Bagust, M. Place and J.W. Posnett. Dynamics of bed use in accommodating emergency admissions: stochastic simulation model. British Medical Journal, 319 (1999) This paper, written by three health economists, reported the results of a discrete-event simulation implemented on a Microsoft Excel 5 spreadsheet.

10 Bagust, Place and Posnett They modelled new arrivals using an empirically derived normal distribution adjusted for seasonal and day of the week variations, and used empirically derived length-of-stay distributions.

11 Bagust, Place and Posnett For each repetition of the simulation, the authors recorded The proportion of new arrivals for emergency admission who cannot be accommodated owing to lack of available beds, the proportion of days in a year when at least one patient requiring immediate admission cannot be accommodated and the bed occupancy rates achieved. (my itemisation).

12 Bagust, Place and Posnett Their results indicated Risks are discernable when average bed occupancy rates exceed about 85%, and an acute hospital can expect regular bed shortages and periodic bed crises if average occupancy rises to 90% or more.

13 Bagust, Place and Posnett and their conclusions were There are limits to the occupancy rates that can be achieved safely without considerable risk to patients and to the efficient delivery of emergency care. Spare bed capacity is therefore essential for the effective management of emergency admissions, and its costs should be borne by purchasers as an essential element of an acute hospital service.

14 Bagust, Place and Posnett None of this would be surprising to people who work in queueing theory - we all know that we cannot expect full availability and full utilisation at the same time. What is surprising is that the paper was published in a very well-regarded journal in 1999, this primitive paper has had a big effect on health services in the UK, and it has been re-quoted uncritically in situations beyond the scope of the original study (witness the ACEM statement).

15 Our response It is true that the origin of queueing theory can be traced to Erlang, who published his first paper 100 years ago. However, no professional queueing theorist would support the ACEM statement quoted above. C.A. Bain, P.G. Taylor, G. McDonnell and A. Georgiou Myths of ideal hospital occupancy. Medical Journal of Australia, 192 (2010)

16 Our response We also said There is also a more fundamental problem with making general statements that relate blocking probabilities with steady-state mean occupancy. Both these measures are outputs of queueing models... For a specific model, it may be possible to state a relationship between them. However, a better way to look at the situation is to express both these performance measures in terms of its inputs, namely, the arrival and service processes and the number of servers (beds).

17 A Better Approach We also suggested that our medical readers should look at the paper F. Goronescu F., S.I. McClean, and P.H. Millard A Queueing Model for Bed-Occupancy Management and Planning of Hospitals, The Journal of the Operational Research Society (2002) 53, This paper discussed a method for optimising the number of beds in a hospital unit. It showed that the optimal number of beds depends on the relative value placed upon blocked patients and empty beds.

18 A Question Did we do the right thing? A crude simulation that people thought they could follow was more effective in changing the understanding of system behaviour than undergraduate-level analysis. Ultimately, real-life resource allocation decisions are made in a political context. For example, we were congratulated by a number of people who think running hospitals at 100% utilisation is optimal. Will our contribution serve a useful purpose in this context?