Checkout lines and genderfree bathrooms

Yes, the two things mentioned in the title have something in common.

A Long Line for a Shorter Wait -- June 23 New York Times.

Whole Foods stores in New York City have moved from having a line for each checkout register (which is for the most part standard in American food stores) to a single line for the whole store. This means that the line looks longer but customers get through it faster. A few of the commenters at the NYT article have pointed out that you don't actually get through the line faster with this system; however, the probability of waiting a very long time is reduced. And that's really what the store wants to minimize. When you go grocery shopping, you understand that you're going to have to wait in line.

In general, if you wait in a line and there's a line on either side of you, the chances are one in three that your line will be faster than both of the lines adjacent to you. So there's a two in three chance that you'll regret your choice and think "damn, I should have gotten in that one!" -- and that's if you can't see any lines other than those two. I suspect that what a grocery store actually wants to minimize is a combination of average waiting time, some sort of "maximum" waiting time (maybe the 95th percentile?), and the number of people who feel like they got screwed over.

The article claims that the waits are much longer in NYC than elsewhere, though. If this is true, why? My guess is the following. Let's say your store's checkout people can serve 5 people per minute. Then in any given minute, the line only gets longer if more than 5 people come in. If on average four people come to the checkout per minute, then the line will only get longer in 22% of minutes (the minutes when six or more people get in line); it'll stay the same length in 16% of minutes (those when five people get in line); it'll shrink in 62% of minutes. So the line doesn't have much of a chance to get long. If on average 4.8 people come to the checkout per minute (96% of your store's capacity), these probabilities are 35%, 17%, 48% respectively; suddenly it's easier for the line to get long.

If on average 5 people come per minute, the line is equally likely to grow or shrink in any given minute. And if the store's understaffed (someone's sick, maybe?) then forget it -- the line is more likely to grow than to shrink and will probably get out of control. Perhaps NYC grocery stores are slightly understaffed relative to grocery stores elsewhere but this translates into big differences for line length.


Incidentally, a related problem comes to mind. There are a large number of small business establishments (restaurants, coffee shops, etc.) which have two bathrooms, each of which has a single toilet in it. In many cases these two bathrooms are marked "men" and "women". This means one has to wait longer than if there were two bathrooms, both of which were marked "bathroom". One might argue, though, that since men on average take less time in the bathroom than women, a system such as this would actually slow things down for men.

I've seen more and more people taking this into their own hands by using whatever bathroom is free in such establishments. The usual protocol seems to be to try the bathroom marked with one's gender first, and then try the other one if the "right" one is occupied.

And let's not forget that some people face a real quandary trying to decide which bathroom to use! safe2pee.org -- bathrooms for everyone has listings and maps of bathrooms which are safe for such people, either because they are single-occupancy or because they are explicitly genderfree.