Traffic Flow

[marcmagus]

Consider a mixed-sex set of people who occupy the same space for an extended period of time. Assume there are three restrooms attached to the space, one labeled Men, one Women, and one Men/Women. Each can accomodate a single person at a time. Assume that all the people involved obey the restrictions, and that everybody involved has a non-ambiguous identity of either Male or Female.

If you are going to occupy a bathroom for a relatively long time, should you use the single-sex bathroom for your own sex, or the mixed-sex bathroom? Your goal is to minimize the impact on the other occupants of the space in terms of their ability to use a bathroom with minimal wait.

Feel free to make the simplifying assumptions that the sex mix is approximately even and that there isn't a sex bias to frequency or duration of visits to the bathroom. Also feel free to assume that people will preferentially use their single-sex bathroom if it is available, and go to the mixed-sex bathroom only as an "overflow". Do these assumptions change your results? Can you get a more favorable result by relaxing them (changing the behavior of the other people)?

Yes, I think about these things...

Tangent: Why do we specify gender on single-person bathrooms, anyway? I can accept cultural mores against people of different sex being in the same bathroom at the same time, but I can't see what possible use there is when the occupancy limit is one.

Comments

[kiddens.livejournal.com]

Written from the perspective of a man and using your assumptions about equal demand and duration:

Regardless of which bathroom I choose, we can simultaneously also satisfy the demand of one woman - so remove the lady's room from the equation.

If I use the men's room, we can satisfy additional demand of one person regardless of gender. If I use the neutral room, we can only satisfy the demand of an additional man. Therefore the gender neutral bathroom is preferable for demand satisfaction.

If you can't use the assumptions about equal demand and duration, then the problem changes. You need to know something about the distribution of requests and the service times and the problem becomes much harder since it may become more favorable to leave the option for two females rather than be able to serve either gender.

[kiddens.livejournal.com]

Whoops, I meant the men's room is better. I had the logic all in my head and got the label wrong.

[marcmagus]

Yes, but consider the high-demand situation.

Hmmm...I had this whole write-up about how the women's queue would move at 3x the rate of the men's queue in the situation that the men's room is occupied. Halfway through I realized I was wrong: if you assume a single queue which dispatches the frontmost person permitted to use a room as it becomes available, you'll briefly see higher throughput of women. Then the queue will become stacked at the front with men, and the mixed-sex room will become a de-facto men's room if the distribution is roughly equal.

If you occupy the mixed-sex room, queues move normally at reduced capacity.

So it's pretty much irrelevant which room you occupy if there's a lot of contention for the resources (assuming a queuing behavior rather than a mob . . . if someone random is selected as each room becomes available, then occupying the men's room creates a preferential situation for women in line over men--again assuming high contention for the shared resources). If it mattered it's an interesting question: are we optimizing for average wait (in which case it doesn't matter) or for worst expected wait (in which case it does) or for something else?

Another thought: if we assume demand and duration are equal (or don't show a sex bias), but allow for the possibility that another person will have an unusually high duration, it's preferable to choose the single-sex room. If you occupy the mixed-sex room and any other person enters for a long duration, one gender will block until one of you finishes. If you occupy the single-sex room, there are two possibilities, roughly equally weighted: someone of your sex enters for a long duration, and your sex blocks until one of you finishes or someone of the opposite sex enters for a long duration, and everybody can use the remaining room with reduced capacity.

I'm trying to think of a situation in which the optimal choice is to occupy the mixed-sex room. I thought I had one, but, as stated, I was wrong.