Commercial airlines often prioritize boarding for passengers traveling with small children, or for those who need extra assistance—in other words, those likely to be slower to stow their bags and take their seats—before starting to board the faster passengers. It's counter-intuitive, but it turns out that letting slower passengers board first actually results in a more efficient process and less time before takeoff, according to a new paper in Physical Review E.

Physicists have been puzzling over this particular optimization problem for several years now. While passengers all have reserved seats, they arrive at the gate in arbitrary order, and over the years, airlines have tried any number of boarding strategies to make the process as efficient and timely as possible. Flight delays have a ripple effect on the complex interconnected network of air travel and often result in extra costs and disgruntled passengers.

Back in 2011, Jason Steffen, now a physicist at the University of Nevada, Las Vegas, became intrigued by the problem and applied the same optimization routine used to solve the famous traveling salesman problem to airline boarding strategies. Steffen fully expected that boarding from the back to the front would be the most efficient strategy and was surprised when his results showed that strategy was actually the least efficient. The most efficient, aka the "Steffen method," has the passengers board in a series of waves. "Adjacent passengers in line will be seated two rows apart from each other," Steffen wrote at The Conversation in 2014. "The first wave of passengers would be, in order, 30A, 28A, 26A, 24A, and so on, starting from the back."

Field tests bore out the results, showing that Steffen's method was almost twice as fast as boarding back-to-front or rotating blocks of rows and 20-30 percent faster than random boarding. The key is parallelism, according to Steffen: the ideal scenario is having more than one person sitting down at the same time. "The more parallel you can make the boarding process, the faster it will go," he told Ars. "It's not about structuring things as much as it is about finding the best way to facilitate multiple people sitting down at the same time."

Steffen used a standard agent-based model using particles to represent individual agents. This latest study takes a different approach, modeling the boarding process using Lorentzian geometry—the mathematical foundation of Einstein's general theory of relativity. Co-author Sveinung Erland of Western Norway University and colleagues from Latvia and Israel exploited the well-known connection between microscopic dynamics of interacting particles and macroscopic properties and applied it to the boarding process. In this case, the microscopic interacting particles are the passengers waiting in line to board, and the macroscopic property is how long it takes all the passengers to settle into their assigned seats.

"The ability of a passenger to delay other passengers depends on their queue positions and row designations," the authors wrote. "This is equivalent to the causal relationship between two events in space-time, whereas two passengers are timelike separated if one is blocking the other and space like if both can be seated simultaneously."

Erland et al. treated the boarding process as an iterative two-step process. The passengers move until they either reach their assigned rows or are blocked by other passengers in the aisle, and the second step is how long passengers stand next to their designated rows to stow luggage and sit down.

The passengers form a one-dimensional line to fit into a matrix of seats. The researchers predicted passenger speed based on where each person was in line, which row they were seated in, and how long it took to clear the aisle. The model calculates whether passengers will eventually run into one another based on how far apart they are sitting and how far apart they are standing in line. Seated close together but standing far apart in line (a space-like separation) means there will be no interference; seated far apart, but standing close together (a time separation) is more likely to lead to interference.

The researchers ended up with another counter-intuitive result: it's actually 28 percent more efficient to let slower passengers board first. Furthermore, "This is a universal result," the authors wrote, "valid for any combination of the parameters that characterize the problem." Those parameters are the percentage of slow passengers, the ratio between aisle-clearing times of fast and slow passengers, and the density of passengers in the aisle.

Once again, it comes down to maximizing parallelism. As boarding progresses, those at the tail-end of the slow group will still be getting settled as the first influx of fast people begins boarding. For instance, three or four fast people might take their seats in the time it takes a single slow person near the back of the aircraft to sit down. Having all the fast people board first actually minimizes parallelism: the last fast passenger is already seated before the first slow passenger gets settled.

To illustrate, Steffen cites the well-known analogy of trying to pack rocks and sand in a jar. Put the sand in first and there won't be much room left for the rocks. Put the larger rocks in first, and you can then pour in plenty of sand to fill in all the gaps around the rocks. "That's the lesson of this [latest] result," said Steffen. "If you're going to pour a bunch of passengers into a vessel like this, and you're dividing them up into slow people versus fast people, it's better to get the slow people out of the way first and then let the fast people trickle in."

Of course, even the best models are a simplification of an actual boarding process, where many other factors must be considered, including human behavior. First-class and frequent flyer passengers, for instance, are accustomed to preferential boarding. In an era where more people are not checking bags because they want to avoid extra charges, competition for overhead bin space can be fierce—and those who board earlier are more likely to snag precious bin space. Such studies are nevertheless useful, Steffen maintains. "It gives you a quantifiable result to consider when crafting policy," he said. "And it's counter-intuitive information, which makes it even more valuable because it shows where your intuition can lead you astray."

DOI: Physical Review E, 2020. 10.1103/PhysRevE.100.062313 (About DOIs).