# Queuing Theory

My previous blog about queuing ‘Queuing:there has to be a better way.’ received an unexpected response from my son.  As a University researcher studying something to do with physics and mathematical modelling, he suggested that I might have mentioned queuing theory in the blog.  I have to say this was very remiss of me.  I asked him to briefly clarify my understanding.

The theory is used extensively by organisations that deliver queuing to their customers as part of their experience.  Theme Parks, banks and supermarkets employ mathematical models to optimise the most efficient way for their customers to be processed.  After a brief resume of the following we decided to apply this to a supermarket queue:

Expected average queue length  E(m)= (2ρ- ρ2)/ 2  (1- ρ)

Expected average total time  E(v) = 2- ρ / 2 μ  (1- ρ)

Expected average waiting time  E(w) = ρ / 2 μ  (1- ρ)

Expected average waiting time  E(w) = E(v) – 1/μ

λ = Arrival Rate

μ = Service Rate

ρ = λ / μ

Sam used QT to choose the line he was to follow, whereas I used my own unique queuing criteria.  My criteria was simply ‘Which queue would be most fun to join?’, including sub-criteria such as: queue-ers who looked funny, potential for the cashier to smile, banter, amusing purchases and vegetables that looked like Prince Charles.  Sam used these various indexes and a calculator.

Admittedly he did get served marginally quicker but only because my fellow queue-ers couldn’t agree which member of the royal family bore most resemblance to the sweet potato I had offered up for the ‘vegetable that looked like Prince Charles’.  I had to disagree entirely with the Security Guard who was brought in to make the final judgement, there was no way it was a Lady Di lookalike.

Well that puts queuing theory into perspective, I think you’ll agree.