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With all the insanity on terra firma, you’re forgiven if you missed the 30,000-foot brilliance of this year’s Crystal Cabin Awards—the Only International Award For Excellence in Aircraft Interior InnovationTM. Ladies and gents, it was marvelous: heated floor panels, rentable beds for economy fliers, a rollout mattress for those in business, and more.
One finalist that saw its hopes for a trophy go down the drain is Zodiac Aerospace’s Durinal. As the name (sort of) indicates, this innovation would replace one standard lavatory with two urinals. Because in an age where flying is more misery than miracle, anything that can make the experience better—like shortening bathrooms lines—is worth a shot.
“The innovation will benefit female passengers too, at least indirectly: when the men have their own urinal facility, the queue for the rest of the cabin will be reduced,” the press release announcing the award finalists read. “Things may be a little cleaner too.” Zing, dudes.
But will giving half the population a yellow-tinged fast track really save everybody time? On one hand, stand-up pee-ers wouldn’t be clogging up the lines to those sit-down commodes. On the other, the Durinal removes an outlet for sit-down urinators who share their reduced number of toilets with women, as well as with men who aren’t headed to the lav to pee.
Fortunately, mathematicians have created an entire field dedicated to striking the balance between one gender’s convenience and another’s irritation.
OK, the science of waiting in line—technically known as queuing theory—actually began about a century ago, helping old-timey switchboard operators efficiently manage influxes of telephone calls. But today, it has all sorts of applications. “It is basically an abstract way of dealing with any system where there is a waiting phenomenon,” says Wouter Rogiest, a mathematician at the University of Ghent in Belgium.
The two fundamental variables in queueing are arrival time (how quickly the line forms) and service time (how long it takes each person to do the thing they were waiting to do). The former is hard to predict. Mostly, people get in line whenever they've got all their groceries, or want to get a cup of coffee, or need to hit the head (usually about 15 minutes after the coffee stop). For mathematicians, modeling such random arrivals requires turning to Markov models, the same ones they use to estimate other inevitable but unpredictable processes, like radioactive decay.
Conversely, service time is often quite predictable. A network router can ship a data packet in a fixed period; a cashier needs only so much time to ring you up; or in an example salient to the issue at hand, most mammals bigger than rats need roughly 21 seconds to clear their bladders. Humanity’s penchant for clothing and privacy has added time to its whiz clock, and led to a significant disparity in service times for males and females going number one. Rogiest has actually done the math: On average, men take one minute to pee. Women take about a minute and a half. As you might have guessed, men’s ability to remain standing and mostly dressed while urinating explains some of the difference. But women don’t just have to undo more clothing and more complicated clothing—ever try to bundle a romper?—they also have to open and close doors, as well as wipe the seatie, sweetie.
However, the biggest reason women waste more of their lives queuing for the bathroom is that the world has a surfeit of facilities that serve men. Urinals take up much less space than individual stalls. Diving into the literature on bathroom design, Rogiest and a colleague found that the average men’s room has 20 to 30 percent more toilets than the average women’s room. So, those long lines at the women’s room aren’t just from clothing logistics and extra cleanliness. Women just have fewer places to pee.
Rogiest actually tested how this disparity in bathroom service times and available toilets cost women more time. To set a baseline, he and his co-author Kurt Van Hautegem created a model where men and women had the same number of toilets and arrived in line at 10-second intervals. The only variable changed between the two sexes was in service times. With a 60-second service time, they found that men waited an average of just 1.52 seconds. Women, with a minute and a half service time, waited about a minute.
Then they started playing with bathroom layouts: changing the ratio of urinals to squatters, giving women more toilets than men, and toggling whether the bathrooms were segregated or not. They also upped the intensity of the arrival times to mimic a concert or movie letting out. Their ideal outcome would be short, and mostly equal wait times for both sexes. They found a pretty steady pattern: “It basically comes down to there is a bottleneck whenever you separate the sexes,” he says. When the bathrooms had no fixed gender, however, the sex-specific bottleneck disappeared, and everyone spent about 2 minutes, 10 seconds in line.
But they actually got their best results when they offered that fast lane for stand-up pee-ple. Let’s see if you can figure out the ideal ratio, in the context of an airplane-based word problem: You have 300 people on a five-hour flight. Everyone will need to pee twice, half of them taking 60 seconds, half taking 90 seconds. One-third will have to poop once, which (let’s be optimistic) takes five minutes. If this plane comes standard with six single-occupant sit-downers, how many would you replace with Durinals to reach the optimum pee times for everyone?
Savvy readers might have figured out a shortcut to all of this: Call Zodiac Aerospace and ask them. Too bad 1) this is cheating, and 2) Zodiac did not respond to my emails. To calculate an exact answer, you’d probably need sophisticated software using Markov randomization sequences to simulate how quickly people line up and then clear out from each “operation.”
Luckily we have common sense and a scientist here to help. First, common sense. If you replace half of the commodes with urinals at a two-for-one discount, you’re going to wind up with mostly empty urinals and three very overburdened commodes. That’s unfair to half the plane’s passengers, with added discomfort for men who must advertise when they’re going to be sitting a spell.
Working from there, Rogiest and his co-author actually found the optimal ratio was to put no more than half the number of urinals to commodes. In their concert scenario, a ratio of 14 commodes to eight urinals gave women a wait time of one minute, 27 seconds, and men just under a minute—each sex was waiting nearly equal to its service time. So on this six-toilet plane, replacing one or even two of the cabinets with Durinals should improve bathroom wait times for everyone.
However, as a dutiful scientist, Rogiest warns against applying his ratio on spec. “In any queueing scenario, you quickly you end up with difficult mathematical conditions that are difficult to model,” he says. For instance, people on an airplane can often spy the bathroom line from their seats. “That means there might be four people in line," he adds, "but there are eight or nine others sitting down who also want to use the restroom.” Rogiest says he would like to run mathematical simulations on how a Durinal would affect airplane bathroom usage before throwing his academic reputation behind a back-of-the-tissue calculation.
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