FROM YOUR chair in the departure lounge you can already see the plane, tanks topped up, ready to go. All that stands between you and a speedy departure are your fellow passengers.
“Ladies and gentlemen, we are now ready to begin boarding…”
The surge begins, and within a few minutes you are all hopelessly backed up in the bridge onto the plane. It’s enough to give you air rage. What’s so hard about getting into a seat?
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You may not realise it, but it’s a question with cosmic significance. While shuffling on board a plane feels like a distinctly Earth-bound experience, there’s more to it than that. In what appears to be the first out-of-physics application for Einstein’s theory of relativity, it seems the answer to faster boarding lies in a trip through space-time. And, you’ll be glad to hear, this approach shows there really are ways to improve the experience.
A few seconds’ thought will give you some of the options airlines could consider. One is random boarding: everyone for themselves. This is not as chaotic as it sounds, but nor is it edifying, and seems unlikely to be the most efficient approach.
At the other extreme you could be ultra-organised and specify the exact order in which you want everyone to queue up. In such a rigidly marshalled scheme, however, you’d spend so much time ordering people about that it would be no better than a free-for-all. And any airline that tried to be that bossy would quickly find itself without any passengers to worry about. After all, one person’s orderly queuing protocol is another’s intolerable control-freakery.
In between lie myriad options: calling passengers by rows, half-rows, blocks of rows, back-of-plane-to-front-of-queue, and “outside in” – window seats first, then middle, then aisle – and so on. They all sound reasonable enough, but there’s scant evidence to prove that one is better than another. Isn’t there a more scientific way to work this out?
Enter Eitan Bachmat and his colleagues at Ben-Gurion University of the Negev in Israel. Bachmat wasn’t initially aiming to solve the problem of “enplaning”, as airlines rather grandly call it. He was studying the performance of digital storage systems such as PC hard drives, looking at how to read and write data most efficiently. Say you send your hard drive a big bunch of read/write requests. What you want to know is the quickest way to carry out those tasks. How do you find the fewest drive rotations needed to do the job?
Boarding for bozos
One day a colleague who saw this work said, “Hey, that looks a lot like airplane boarding,” and so Bachmat and his colleagues decided to explore this parallel route. “We thought we could do this and take stuff back to the hard drive problem,” he says.
The first step, they realised, was to see what actually holds things up when you’re waiting in the aisle of the plane. While that bozo up ahead of you is hunting for his iPod and then trying to jam that stupidly big bag into the locker before taking his seat, everyone behind is either standing waiting to get past – or, if they have already reached their row, doing pretty much the same thing as the bozo. Once those passengers have sat down, everyone moves on a bit, and the process repeats itself. At each stage, some passengers remain blocked by those in front: the blocked wait their turn, then become blockers, and so on.
What everyone wants is to find a way to minimise the time from the start of boarding until the last person sits down and buckles up. This is easy enough in simple approximations, but as soon as your model includes variables like distance between rows, width of passengers’ bodies, time taken to stow luggage and so on, it quickly becomes intractable.
Fortunately, physics has something to say about problems like this. Odd as it may seem, it occurred to Bachmat and his colleagues that the way passengers fill up a plane looks like relativity’s description of how things move through the four dimensions of space-time under the influence of gravity.
According to relativity, an object in “freefall” follows the trajectory that ages it the most. Throw a stone into the air: it will trace out an arc and return to the ground. Attach a stopwatch to the stone and you would see that, out of all the trajectories through space-time the stone might have taken under the influence of gravity, the arc of its flight is the one in which the most time passes. In other words, the stone maximises the amount of time (in its own reference frame) that passes as it flies up and then falls back to Earth.
Back in the departure lounge, the key is to identify the trajectory of the passenger who will sit down last – the maximum time span of the boarding process. “Once you realise you have a maximisation problem in both cases, suddenly it clicks,” says Bachmat. “Everything falls together, it’s a great moment.”
OK, so herding holidaymakers onto a jet is tied to the fabric of the cosmos. But does that help? How do you actually solve the problem of speeding things up?
The first thing Bachmat and his colleagues did was to set up their own version of space-time. Unlike the more familiar four dimensions of our universe’s space-time – three of space and one of time – this had just two: passengers’ row numbers and their places in the boarding queue when it first forms. In effect, each person trying to board the plane becomes a point in the passenger “space-time”.
Space-time seating
Having set up their new space-time, the researchers needed to find the right “metric”. This is, effectively, a formula for working out distances. The metric most familiar to us is the length of the straight line between points, but for mathematicians there are many other, equally good ways to define distance. And in passenger “space-time” there’s a crucial difference: the metric tells you not just where passengers are relative to each other in space, but also in time. So you can use the metric to determine whether passengers are in each other’s future or past. Bachmat and his colleagues chose their metric so that the past-future relationship between passenger “space-time” points is precisely the same as the blocking relationship on a real aircraft: in passenger “space-time”, if someone is blocking you, they are in your future.
“You can determine whether passengers are in each other’s future or past”
Having called their virtual passengers to this mathematical boarding gate, Bachmat and his colleagues can calculate the overall boarding time by finding the equivalent of a free-fall trajectory through that passenger “space-time”. The maths isn’t simple but it gives you formulae for calculating boarding time under different boarding policies. Crucially, it also lets you see the effect of varying parameters such as legroom and luggage-stowing time (Journal of Physics A, DOI: 10.1088/0305-4470/39/29/L01).
So what did they discover? “What turns out to be critical is the notion of congestion,” says Bachmat. The team incorporated this into their model as a variable k, which depends on legroom, passenger width and number of people per row. “It tells you how crammed passengers are.”
The team found that policies that work well for small k – the rarefied world of first class – become terrible for the high-k crush back in economy. Where congestion is low, furthest-row-first works well, but this becomes progressively worse as congestion increases. The people destined for seats 68A and 68B, for example, could be standing alongside row 67, waiting for the woman in 68C to stuff her bag into the overhead locker. While they wait, they block the people heading for row 67 from reaching their locker, and so on. So even with a furthest-row-first policy, high congestion means that the queue will be all the way out of the plane door within minutes, with not one other person able to get to their own row.
For really high congestion, random boarding soon becomes the better option. For the moderate-squeeze conditions that are the reality for most of us, the best balance is a high level of row randomness combined with a touch of back-to-front.
“For moderate-squeeze conditions the best balance is a high level of row randomness combined with a touch of back-to-front”
Unsurprisingly, you can improve things significantly if you can persuade people to arrange themselves in window-middle-aisle order for each row. Teaming that with a bit of back-to-front helps, but row order makes relatively little difference compared with the order within each row. “Unless you can really force passengers [to do what you want], playing with the row orders isn’t going to get you very far,” says Bachmat.
What also emerged is that boarding time is proportional to the square root of the number of passengers. Considering how complex things become as the numbers increase, that’s actually not bad, but it still has significant implications for today’s ever-bigger airplanes, like Airbus’s 555-seat superjumbo, the A380.
Airbus has clearly anticipated the problem – even without the aid of relativity. “You can get people in on two different levels at once,” says spokesman Justin Dubon. “It’s tantamount to loading two planes at once.” In tests on real aircraft, all 555 passengers could be seated in just 22 minutes, compared with 21 minutes for the 471 passengers on a 747.
So are Bachmat and his colleagues sledgehammering a trivial problem by using relativity? Hendrik Van Landeghem of the University of Ghent in Belgium doesn’t think so. Working with Ghent colleague Annelies Besuselinck, Van Landeghem carried out a simulation-based study in 2002 (see “All aboard”), and he is impressed by Bachmat’s further step. “I actually think it was fairly original that they attempted such a thing. If you are able to validate simulations by mathematical formulation it strengthens both approaches.” Maybe he is pleased because the results tally well with his own. “It has validated what we did, which means that we did a good job,” he says.
Airlines themselves remain oddly uninterested in any of this work, however. Van Landeghem did manage to establish a collaboration with Belgian airline Sabena, but the airline went bankrupt before they could take it forward. There has been no further interest from any airline, Van Landeghem says. Neither has Bachmat been offered a chance to try out his new application for relativity. “I would be very happy if an airline that does unassigned boarding would work with us,” he says.
Aside from honing their boarding policies, what else could airlines do? One thing stands out a mile: “Widen the aisle, to make it one-and-a-half person to allow bypassing, that would have the single greatest impact,” says Van Landeghem. Bachmat agrees: “If you had several entrances and more aisles, that would help.”
You are unlikely to see airlines ripping out seats to make more space any time soon, but there are things passengers can do to ease things along. “Keep your distance from your peers to lower the congestion parameter,” says Bachmat. And obviously, take less carry-on luggage with you. “Be disciplined,” he says.
So next time you’re surrounded by dithering travellers as you board your plane, take comfort from the fact that you’re all acting out some very clever physics. “I think it’s exciting,” says Bachmat. “It’s kind of nice that you can tell people that they compute a relativistic calculation with their feet.”
All aboard
Airlines have never put much serious effort into getting people onto planes efficiently, even though a lousy “turn time” – the period a plane spends on the ground between flights – hits them smack in the wallet. In 1998, however, Seattle-based Boeing heard rumours that airlines who were potential customers for its new 757-300 jet, which had 40 more seats than its predecessor, were worried about the extra capacity inflating turn times. The company set Scott Marelli, one of its mathematical analysts, the task of doing a few computer simulations to see how different strategies affected boarding times. Marelli’s study revealed that boarding the new craft could be just as speedy, given the right tweaks to boarding policy.
Four years later, however, Hendrik Van Landeghem and Annelies Besuselinck at the University of Ghent in Belgium published a paper calling Marelli’s conclusions “rather optimistic”. The Ghent researchers thought he had underestimated the problem of passengers blocking each other, so they set out to do another analysis. Among various surprises, they revealed that a random approach – allowing people to board the plane in any order they want – performs pretty well. Only 9 out of 46 other strategies did better; one of the best approaches was a half-row boarding pattern combined with a little randomness and some back-to-front. No airline, however, has jumped on that complex answer as a solution to the boarding shuffle.
