ABC Tools

Airplane Boarding, Explained

After reading this you will know why back-to-front boarding is slow, why random boarding beats it, and how to read the aisle interference that drives every boarding time you measure in the simulator.

What the model is

Boarding a plane looks like a queue problem, but it is really a problem about how many people can do useful work at the same time. The useful work is stowing a bag and sitting down. The bottleneck is a single narrow aisle. Only one person can occupy each aisle cell, so two passengers cannot stow bags at the same spot at the same time.

The simulator gives you a cabin of thirty rows, six seats per row (three on each side), and one aisle. Passengers enter at the front in an order set by the chosen method. Each passenger walks until the person ahead blocks the way, stops at their assigned row, spends a fixed time stowing a bag, and sits down. If someone is already seated between them and their window seat, a seat shuffle happens: the seated passengers stand, step into the aisle, let the new arrival pass, then sit again.

The hook is the result airlines keep ignoring. Board back-to-front, the tidy way, and the cabin serialises: one block queues behind a handful of rows while the rest of the plane stands empty. Board at random and the load spreads along the aisle, so more people stow bags at once. Random finishes faster. The numbers below show by how much.

When this model helps and when it misleads

Use it to build intuition about parallel work under a shared bottleneck. The same idea appears in the Queue Simulator and the Zipper Merge Simulator: throughput depends on how you feed work into a constrained channel, not on how neat the input looks.

Do not read exact minutes off it as if they were airline data. The model fixes the aisle width at one, assumes every passenger walks at the same speed, and uses a single bag-stowing time drawn from one distribution. Real cabins have wheelchairs, families, overhead bins that fill up, and passengers who freeze. The model is honest about ranking methods against each other. It is not a schedule.

Do not compare two runs of the same method and conclude one method is faster. Each run is one random draw. To compare methods you must average many silent trials, which is exactly what the compare button does.

The mechanism behind the times

Total boarding time is set almost entirely by aisle interference. Write the cabin as a line of aisle cells. A passenger heading for row r occupies aisle cells 1 through r on the way in, and blocks cell r for the whole stow time. Two passengers can stow in parallel only if their rows are far enough apart that neither stands in the other's aisle path during the stow.

T \approx t_{\text{stow}} \cdot \left\lceil \frac{N}{P} \right\rceil + t_{\text{walk}}

Here T is total boarding time, t_{\text{stow}} is the time one passenger spends stowing a bag, N is the number of passengers, P is the average number of passengers stowing in parallel, t_{\text{walk}} is the fixed walking overhead to clear the aisle, and \lceil \cdot \rceil is the ceiling. The whole game is raising P. Tidy orders lower it; spread-out orders raise it.

Two effects push P down. First, clustering: if the next twenty passengers all want the back six rows, they queue single file and P collapses toward 1. Second, seat shuffles: boarding a window passenger after an aisle passenger in the same row forces the aisle passenger to stand, which freezes an aisle cell and blocks everyone behind. WILMA removes the second effect entirely by boarding window seats first. Steffen removes both by spacing rows apart and ordering by seat.

A worked example on the demo defaults

Random versus back-to-front, 180 passengers

Run the demo defaults: 30 rows, 6 seats, so N = 180 passengers, a mean stow time of about 6 seconds, and strict order following. The compare button runs silent trials and reports mean boarding times. Take representative means from those trials and reason about them with the formula above.

  1. Back-to-front blocks. Passengers arrive in tight clumps aimed at the same few rows, so effective parallelism is low, roughly P \approx 4. Then T \approx 6 \cdot \lceil 180 / 4 \rceil = 6 \cdot 45 = 270 seconds plus walking, near 1140 seconds in trials.
  2. Random. Rows are scattered, so several passengers stow far apart at once, roughly P \approx 6. Then T \approx 6 \cdot \lceil 180 / 6 \rceil = 6 \cdot 30 = 180 seconds plus walking, near 960 seconds in trials.
  3. WILMA. No seat shuffles and good spread give P \approx 8, so T \approx 6 \cdot \lceil 180 / 8 \rceil = 6 \cdot 23 = 138 seconds plus walking, near 720 seconds.
  4. Steffen perfect. Alternate rows, one side, window first: nearly everyone stows at once, P \approx 15 or more, so T \approx 6 \cdot \lceil 180 / 15 \rceil = 6 \cdot 12 = 72 seconds plus walking, near 420 seconds.

The ranking is the point. Random beats back-to-front by about 15 percent here, and Steffen roughly halves it. Your exact seconds will differ because stow times are random, but the order of the methods is stable across trials.

Back-to-front is near the slow end and random beats it. Steffen's spaced, seat-ordered scheme is fastest.

Explore the parallelism yourself

As the number of passengers stowing at once (parallelism P) rises from 1 to 15, boarding time for 180 passengers with a 6 second stow time falls sharply then flattens: at P=1 it is about 1080 s, at P=4 about 270 s, at P=6 about 180 s, at P=15 about 72 s, before adding a fixed 700 s walking overhead. The curve is a hyperbola, so early gains are large and later gains are small.

Reading the results

Watch two things while a run plays. First, how much of the aisle is doing work. In back-to-front you see a dense plug near the back and an empty aisle everywhere else. In WILMA and Steffen you see stow events spread along the whole cabin. That spread is P made visible.

Second, watch for seat shuffles. Each shuffle is a passenger standing to let someone reach a window. In front-to-back and random you see many; in WILMA you see essentially none because windows load first. A single shuffle can stall the entire tail of the queue, which is why removing shuffles alone (WILMA) buys most of the gain a real airline can capture.

Turn the order-strictness slider down and rerun WILMA. Boarding time rises but stays well below back-to-front. That is the honest version of WILMA: even when groups board together and break the window-first rule, it beats the tidy blocks.

Common mistakes

The first mistake is trusting one run. Stow times are random draws, so a single Steffen run can lose to a lucky random run. Average at least 20 trials before ranking anything.

The second mistake is confusing tidy with fast. Back-to-front and front-to-back both look organised and both serialise the work. Organisation of the passengers is not the goal. Parallelism of the stowing is.

The third mistake is expecting Steffen perfect to survive real life. It requires every passenger to arrive in an exact position, alone, with no family or gate-check chaos. The simulator shows it as a theoretical floor. WILMA is the number to quote when someone asks what actually helps.

The fourth mistake is ignoring the stow time. If you set stow time near zero, all methods converge because there is no work to parallelise. The method only matters when bags take real time, which is the realistic case.

Related tools

The shared-bottleneck logic here reappears across the site. The Traffic Jam Simulator shows how a single slowdown propagates backward through a line, the same wave you see when one seat shuffle stalls the aisle. The Crowd Evacuation Simulator covers the mirror-image problem of emptying a space through one exit, including the paradox that an obstacle can speed things up. For the counterintuitive-network family, Braess's Paradox shows adding capacity making everyone slower. If you want the pure queueing theory under all of this, start with the City Traffic Grid.

Frequently asked questions

Why is back-to-front slower than random?

Back-to-front sends everyone to the same few rows at once, so they queue single file and only a handful stow bags in parallel. Random scatters passengers along the cabin, so more stow at the same time. In the demo, random finishes near 960 seconds against back-to-front's 1140.

Is Steffen boarding actually used?

Rarely, because it requires each passenger to line up in an exact slot, alone. Field trials found it roughly halved boarding time, which matches the simulator's ranking, but airlines cannot enforce the order with families and groups.

What does WILMA stand for?

Window, middle, aisle. Passengers board all window seats first, then all middle seats, then all aisle seats. This removes seat shuffles entirely, which is where most of the realistic saving comes from.

Does the number of overhead bags matter?

Yes, through the stow time. Longer stow times widen the gap between methods because there is more parallelisable work. At very short stow times every method converges, since the aisle bottleneck barely bites.

Why does one seat shuffle slow so many people?

A shuffle forces a seated passenger to stand in the aisle, freezing an aisle cell. Everyone behind that cell waits. Because the aisle is one lane wide, a single frozen cell blocks the entire tail of the queue.

Try it yourself in the tool

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