
Forget the plane for a minute. Boarding is a scheduling problem. There is one narrow lane. Only one person can use each bit of it at a time. And 72 jobs each need the lane for a moment, then leave it forever by sitting down. Programmers meet this shape every day: one disk head, one lock that every thread wants, one checkout counter.
In 2006 the mathematician Eitan Bachmat worked out what boarding time actually is. It is the length of the longest waiting chain. You wait for the man ahead of you, he waits for the woman stowing her bag, she waited for someone else before that. The plane is boarded when the longest such chain has finished. Nothing else matters. (He proved it by mapping the cabin onto the geometry of spacetime, the same math as relativity, which is its own kind of joke about air travel.)
The three pictures show the three shapes a chain can take. Back-to-front links all 72 people into one long chain, so everything happens one at a time. A random order does much better, and the reason is lovely: when you shuffle people, the longest run of them that happen to block each other is only about 2√N people long. Mathematicians proved that in 1977. For 72 passengers it means a chain of about 17, and that is the whole reason chaos beats the airlines. Steffen's order goes further. It is built so people standing next to each other in the queue never need the same bit of plane, so chains barely form at all.
This trick already runs the world. Just not at the gate.
Your airline's queue and your hard drive's queue are the same queue. One of them has been optimised since the 1970s.
The order airlines believe in came fourth. The order most of them actually run came last. The difference is bags: Steffen's order keeps 2.0 going into bins at the same moment, block boarding manages 1.3.
And the order decides your evening. Seat 8D stands still for 0:11 under Steffen, 1:13 boarding back-to-front, and in block groups it sits down at 4:32, after the whole plane would have finished in take 4.
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