Infinitely Many Doctors? Hmmm...
6th March 2016
I am currently in the process of putting together a literature review on one of the research topics that were presented to us in January
and February. The topic I have chosen to look at is Infinite Server Queuing Theory and looking at its applications within Health Care
Services. The topic was presented by . Although the model does not
include the effects of hitting the limit of servers available, it does give an indication of when the pressure is applied and the magnitude
of that pressure, which can then be used to work out how many servers one needs.
In a paper by Massey and Whitt, the theorem was generalised to a network of queues, each with infinitely many servers. The main difference
being that $Q_i(t)$ at point $i$ in the network has mean dependent on its aggregated arrival rate (that is, from both outside and elsewhere in
the network).
A few of the STOR-i crew queuing up. Note that this wouldn't have happened if there were infinitely many servers!
You may still be wondering what this has to do with Health Care. Well, one particular example comes from a paper by N Izady and D Worthington.
Their idea was to model A&E as a network of queues with different resources, but each with infinitely many servers (Doctors or Nurses). Then,
they used a so called Square Root Staffing Law to find the true number of servers $s_i(t)$:
$$s_i(t) = \bigg\lceil m_i(t) + \beta\sqrt{m_i(t)} \bigg\rceil$$
where $\beta$ is a quality of service parameter. Once this had been generated, the true system was simulated using these staffing levels to see
whether or not the 98% discharge within 4 hours (set by the government) was fulfilled. If not, the value of $\beta$ was changed until it was
met. This approach allowed a spreading out of pressure and waiting times over the day whilst reducing the man hours of the Doctors and Nurses.
I hope this post has shown that even something that sounds absurd, like assuming infinitely many doctors in a model, can still be of use in the
health service. I hope that in the next post or two, I will be able to give another example of where this has been useful.
References
[1] ,
Eick S., Massey W. and Whitt W., Operations Research, Vol.41, 31-42
(1993)
[2] ,
Eick S., Massey W. and Whitt W., Management Science, Vol.39(2), 241-252
(1993)
[3] ,
Massey W. and Whitt W., Queueing Systems, Vol.13(1), 183-250
(1993)
[4] -->
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