Question

CASE POTTY PARITY

Here is a question you might not have thought about much, unless you are a woman: “What is the right number of toilets to have in a bathroom?” We all must use toilets every day, and we often need to use public toilets. So somebody in charge of building codes needs to determine if there should be a standard for restroom construction in public buildings.

Before 2005, New York City architectural codes stipulated that there should be an equal amount of space given to the men’s and women’s rooms. For example, if the women’s room has 25 square meters, then the men’s room should have 25 square meters. Sounds fair, right? Not so fast. Women’s rooms only have toilets, whereas men’s rooms can have toilets and urinals. Urinals take less space than a toilet, so it is possible that a men’s room with the same area as a women’s room actually has more “flushing capacity.” For example, with equal space dedicated to the two restrooms, a building might have five toilets in the women’s room and three toilets and three urinals in the men’s room.

In 2005, New York City changed the requirements. In what has been come to be called “potty parity,” New York City stipulated that there should be at least a 2:1 ratio of flushing units between the women’s room and the men’s room.

The issue of potty parity is not just for New Yorkers. In China, petitions have been filed with the Ministry of Housing and Urban-Rural Development and other departments to account for the extra time women need on each bathroom visit—89 seconds instead of 39 seconds for men. Maybe as a result of their efforts, the World Expo in Shanghai in 2010 used a ratio of 2.5 to one with its new bathrooms.

1.	To what extent does the equation for the waiting time in a queue help explain why queues for the women’s room might be longer than queues for the men’s room before 2005 in New York City?
2.	Say women on average take twice as long in the restroom (not including waiting time) than men. Is the 2:1 ratio for flushing capacity the right ratio?
3.	Besides adding flushing capacity, what can be done to reduce waiting times for restrooms?

Answer 1

1. In queuing theory the simplest equation is the single server queue. Let’s take that as an example. The arrival rate of people wanting to use a washroom can be denoted by lambda. The rate at which people finish using could be denoted by mu. Then we know that the waiting time in the queue is given by

Wq = lambda / mu(mu – lambda)

From this we can clearly see that as mu increases the waiting time will reduce. This is because mu is in the denominator and the relationship between waiting time and mu will be inversely related.

Now mu is a function of how long it takes to use the washroom. This is usually denoted by

mu = time period / time taken

This means if time taken increases, the value of mu will reduce. If mu reduces, the waiting time in the queue will increase. Naturally since it is established that women take longer time to use the washroom, their waiting time will increase in the queue. We are assuming that the arrival rate, lambda, is equal for both men and women.

2. In this case let us consider the arrival rate of both men and women to be 0.5 per minute. This means every 2 minutes there is a person (men or women) that comes to use the washroom. Now let’s use our information given as 89 seconds for women and 39 seconds for men. We need to determine the utilization factor to understand how busy the washroom is. The service rate for women will be 60/89 = 0.67 per minute and for men 60/0.39 = 1.53 per minute.

The utilization factor rho is calculated as

Rho = lambda / mu

For women it will be

Rho = 0.5 /0.67 = 0.74 = 74% utilization

For men it will be

Rho = 0.5 / 1.53 = 0.32 = 32% utilization

We can clearly see that the utilization of men’s washroom is less than half of women’s. Thus a 2:1 ratio is still not fair. The ratio between the two should be at least 74:32, which is 2.3:1.

3. Waiting time is a function of arrival and service rate. Thus in order to reduce the time without adding more servers (adding more flushing capacity) we either need to increase the service rate (how quickly women can finish using the washroom or reduce the arrival rate (slow down the rate at which people come to the washroom). Reducing the arrival rate is beyond our control. However, we could probably improve the layout of the washrooms and their locations such that they are more efficient. This could lead to reducing the usage time and bring down the overall waiting time in the queue.