# Maths without equations: Dr Tom Monks insights into patient flow from queuing theory

Previously I have written about using a detailed computer model to ask ‘what-if’ an emergency department could be run differently.   Hidden away in complex models like these are important rules of thumb that tell us how to efficiently manage patient flow.

Here I am going to illustrate one of these rules of thumb– the resourcing consequences of making wards more specialised. It is a really important concept. I am going to use mathematical queuing theory to do this, but before you run away in fear I promise that there will be no equations. If you stick with it you might even learn something.

To begin with we need a simple context. Consider patients who have been treated in an emergency department, but now must be admitted to a ward for a short stay where they receive some additional assessment and monitoring.   Our fictional ward has a fixed number of beds, so sometimes the patient has to wait on a trolley in the emergency department until there is a free bed.   We could set up the system up into two ways: a single large ‘aeroplane hangar’ ward that all patients stay in or three smaller specialised wards that receive patients with specific conditions.

Queuing theory tells us that in order to achieve the same waiting times the specialised approach always needs more beds than the aeroplane hangar approach.   The chart below illustrates this in our fictional hospital. To achieve a patient waiting time of 0.2 days for our aeroplane hangar ward we need 15 beds. But with three separate specialised wards we need 20 beds.

Why do the two systems need a different numbers of beds?

This video  illustrates the answer. In it we can see stickmen arrive in an unpredictable order to three specialised wards. They then leave in an unpredictable order. It is precisely because of this unpredictability that our poor stickmen end up queuing for their specialised ward while another ward has free beds. Our single general ward is what mathematicians would call a pooled system. Pooling of resources is very efficient and can for example help manage variability in admissions and discharges. In this case the pooled system would accommodate three of the queuing patients. This rule of thumb holds for bigger and smaller hospitals as well as different types of system. For example, surgeons lists for operating theatres, or the shopping queue at Tesco. A word of warning though. This is just a rule of thumb the specific reduction in waiting times will vary for a number of reasons.

Are there any missing details in our ward model?

Yes. Specialised wards might be faster than general wards: a patient may have a shorter stay on their ward due to the specialised care they receive. So it may not be as bad as it seems. But beware; they need to be A LOT faster. In our example the three specialised wards need to reduce average length of stay by more than 60%.

Of course, I am not arguing that we should not have specialisation in healthcare systems. Far from it. There are critical patient benefits of doing so. However, those managing a hospital need to balance the need for specialisation with the need to keep the system moving. In Wessex we are working with a partner organisations on a variety of modelling projects from the detailed to the simple.

That wasn’t so painful was it?