When it comes to potties, “separate but equal” may not be equal enough – NY’s City Council decided that public venues like arenas, nightclubs and theaters must provide a female-to-male restroom ratio of two to one. This provides an entertaining context to discuss queuing theory.
We all know that women often have to queue to use the restroom while men usually do not. To devise the appropriate solution, it is important to know why. Queuing theory provides useful guidance.
In short, a queue will form when the load on the system exceeds its capacity, where the load is the arrival rate multiplied by the time to process each request. Think of load as the desired number of flushes per minute and the capacity as the feasible number of flushes per minute. This raises several possibilities for why the queue in the women’s room is longer:
(1) The arrival rate of women is higher, either because there are more women than men at a particular venue (doubtful at Madison Square Garden’s Monster Truck Smash) or because women need to use the restroom more often (absolutely true in some families);
(2) The arrival rate of women is more variable. Hard to imagine this is so unless the women’s basketball team bus arrives or women have a greater tendency to use the restroom in packs.
(3) Women have a longer processing time. Data could in theory be collected on this, though discretion would be appropriate;
(4) Women have a more variable processing time. Again, many theories, little hard evidence;
(5) Less capacity per restroom – How many people can flush simulataneously in a restroom? On a per square meter basis, urinals are very efficient.
When it comes to restrooms we probably do not want to consider options that would either strive to (i) decrease the mean or variance of their arrival rate or (ii) change their restroom behavior to decrease the mean or variance of their “processing times”. That leaves cause (5) above as the solution – increase capacity.
A simple solution to increase capacity is to pool capacity - unisex restrooms. This, of course, is a non-starter in some cultures. The obvious alternative is to add more women’s restrooms (i.e., more possible flushes per unit time), which is exactly what NY City has legislated.
This leaves open two questions – why do we need legislation to fix this? (i.e., why doesn’t the market work here) and how do we get men to leave the seat down?
NY Times, July 18, 2008
A ‘Women Only’ Restroom Renovation Tips the Balance at Grand Central