How many toilets do you need in a Theatre? #MoreLoos #LittlesLaw

Joanna Lumley.
Image source: Irish Independent.

I read with interest an article by Sinéad Ryan in yesterday's Irish Independent: "Biggest drama in theatre is getting to loos". She had attended an event in a community hall where there were "just two cubicles for a room holding 300 people". As a man I have had rarely needed to queue to use the loo in places such as theatres, arenas, or public places - there is a definite anatomical advantage to being male when it comes to time for a pee. "Joanna Lumley won’t take shortage of ladies’ loos sitting down" writes Rebecca Nicholson in The Guardian newspaper - the lovely Joanna has launched a "More Loos" campaign and says that the "ladies are about to storm the men’s loos. They can’t manage to have a drink and a waz at half time"! In her article, Sinéad Ryan uses our own Abbey Theatre which has been criticised recently for its lack of loos for women. 

So how many loos are needed to satisfy demand?

Prof John Little.
Image source: MIT Sloan.

Fortunately, there is a formula for working these things out. In 1961, Professor John Little of MIT published what became to be known as Little's Law. This law illustrates the mathematical relationship between throughput, work-in-progress (WIP), and cycle time. Let's use Dublin's Abbey Theatre as an example to work out how many toilets for women are needed during a 20 minute interval in the middle of a show.

The capacity of the Abbey Theatre is currently 492 seats - let's assume for simplicity that at a typical performance that 50% of the attendance are female. This means that there would be 246 women at each performance. This is our WIP (demand for toilets). The throughput is the length of the interval time - in this case 20 minutes. Finally, let's assume that on average, each woman spends 3 minutes in the toilet cubicle, and that there are 10 cubicles available (I don't know exact figure as I have never been in the ladies' toilets in the Abbey!).

In summary:

    WIP (Demand for toilets) = 246 women

    Throughput time = 20 minutes

    Work content = 3 minutes

First - we need to calculate the cycle time:

        Cycle time = Throughput = 20 = 0.08 minutes

                    WIP      246

Next - calculate the number of toilets required :

        # toilets required = Work content =  3  = 37.5 toilets

                         Cycle Time    0.08

There are not enough toilets to deal with demand since as 37.5 (say 38) are required. Given that the work content (the time taken to use the loo) cannot realistically be shortened, nor WIP (demand for the loo) be reduced, then what are the options?

1. More toilets - 38 to cover demand
2. A longer interval:

     (New) cycle time = Work content =  3  = 0.3

                         # toilets     10

Throughput time = WIP x Cycle time = 246 x 0.03 = 73.8 minutes. The interval would need to be 74 minutes long to ensure that all demand was met.