Queues and Air Ambulances

When I need a succinct introduction to Operational Research, I generally refer to the application of queue models.  The two-minute lesson goes like this:
(1) Think of the number of cashiers in your local supermarket.  What's the best number?
(2) It can't be too many - they wouldn't have work to do; it can't be too few - then the queues would be over-long; so there is a right number, and hence some mathematics to find that number.
(3) But that number depends on knowledge of how many customers will come at different times of the day and days of the week, and seasons of the year.
(4) And it must fit in with the shift pattern of staff at the store
(5) So what started as a simple question of finding the right number becomes part of a problem of using data from the past to make forecasts, and managing the employees in the store.
(6) O.R. people deal with problems like that

It's much easier than trying to explain about linear programming, or game theory, or Black Scholes volatility.

But even that explanation hides a great deal of complexity.  Many supermarket queues are themselves complex systems, with separate queues for those with one basket, those with less than 10 items, the self-service queues ....  And generally, queue models work best when the servers and customers are not human beings but things.  Human beings have personalities and psychology.  So queues with human beings are more sophisticated and more complex than the ones one meets in theoretical courses.  Despite these complications, queue models are extremely useful.

One area where O.R. has used queue models to good effect is in the emergency services.  Call for police, fire brigade or ambulance, and you become a customer in a queueing system.  The dispatcher selects a server (or more than one) to send to you, unless there is none conveniently available for your need.  So you may wait for a server to become free, or your service may start as soon as your need has been assessed - although you may not actually see the server for some time, as "service time" includes the travel time to the point of need.  This queue system is one with multiple servers, the possibility of service by more than one server, and the possibility of pre-emption.  (Pre-emption means that the dispatcher may decide that another emergency is more pressing than yours and take away the server from you.)  And the dispatcher has the potential to change the availability of servers by moving them around as those in one geographical area become unavailable or free.  Good dispatchers know all this and handle their resources without solving too many mathematical models of queues - but in many circumstances the initial planning of the emergency service has been done with such models.

Last week, there was a news story about emergency vehicles for which queue modelling is relevant.  In the United Kingdom, there are about 30 air ambulances, mostly helicopters, which provide rapid transport of patients from their point of need to a hospital.  As my house is close to the hospital in Exeter, we frequently see one of the two air ambulances that fly in our county of Devon from our garden.  I am not sufficiently familiar with the air ambulances from the neighbouring counties to be able to recognise them, but I know that there are times when "Our" ambulances will cross the border and vice versa, so the Dorset ambulance sometimes comes to Exeter, and - rarely -  that from Cornwall flies here, though that generally only visits the west and the north of Devon.  So, essentially, the emergency service in Devon might be modelled as a queue with two servers.  Or, you could model the emergency service in Dorset, Devon and Cornwall as a queue with four servers.  However, as soon as you allow the latter, you need to consider the counties to the east of Dorset, who can also call on the Dorset helicopter.  So maybe you need to consider the whole country, and model a queue with 30 servers?  No, not really, because the response time would become excessive; a helicopter from London would not respond to an emergency in Devon, nor would one of ours respond to an emergency in Manchester.

Hence most modelling of air ambulances is going to be based on the county service, and not extend to the whole of England or Britain.  At least that is one way of looking at the news story which emerged last week. 

That story concerned the deployment of a dedicated Children's Air Ambulance whose specific role is to transfer children in a specialised vehicle between hospitals anywhere in the country.  Such inter-hospital transfers, for adults and children, are already part of the tasks undertaken by the county air ambulances.  Not all the county organisers are happy with the concept of such a dedicated "server".  They will say that the response time of a helicopter based in the centre of the UK will be too great for the needs of most counties, and therefore they will continue to treat their system as having "their" vehicles, and none from outside.  The proposers will argue that nationwide, the queue for air ambulances will have one more server, and that can only be a good thing - more servers reduce the waiting time (lesson number 1 in queue models). 

So it looks as if we have a clash of two ways of modelling the emergency service provided.  Is it one based on service by local helicopters, in Devon's case two?  Or is it one provided by helicopters which might come from elsewhere in the country, with a longer response time?  Can O.R. help?  Yes, but there is the psychology of the proponents of the new service and the established administrators of the existing one to consider!  And they each seem committed to their conceptual models of the air ambulance emergency service. 

And another thought about a dedicated service for a particular category of "customer".  Queue systems exist which offer dedicated servers to particular classes of customer.   The executive lift which only the holders of a particular key can use.  The private hospital operation only available to those who can pay for it.  Parking spaces for staff only.  Whenever customers are divided into different types, the queue modeller needs to calculate the optimal number of servers providing that service, and normal service.  It is easy (exercise for the reader) to create a queueing system where the division of customers into two classes and dividing the servers between them means that the quality of service for each customer deteriorates.   (hint, you only need two servers to create such an example)

O.R. modellers of the activities of air ambulances, please step forward (and watch out for the rotor blades)!


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