A queue that eats itself
Why Do Buses Come in Threes?
Buses set off evenly spaced and end up bunched together, and it happens for a reason no driver can fix.
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Step 1 of 5
Passengers arrive at a steady trickle.
People turn up at a stop at roughly a constant rate, so the number waiting depends on how long it has been since the last bus went through.
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Step 2 of 5
A late bus has more people to pick up.
A bigger queue takes longer to board, so being late directly causes being later. The delay feeds itself instead of correcting.
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Step 3 of 5
The bus behind has the opposite problem.
It arrives at stops that were just cleared, so it picks up almost nobody and sails through. It gains on the one in front with every stop.
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Step 4 of 5
So they meet, and then travel together.
Once the second bus catches the first they stay together, because they now share every queue between them. A pair of buses is stable in a way a single one is not.
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Step 5 of 5
The fix is to hold buses back on purpose.
Operators make an early bus wait at a stop rather than run ahead. Deliberately wasting time is the only thing that keeps the gaps even.
The same effect shows up in lifts, supermarket tills and cars on a motorway. Anywhere a queue slows the server, the delays clump.
The short version
Buses bunch because falling behind means bigger queues, which means falling further behind, while the bus behind catches up.
Try it yourself
Watch a busy bus stop for twenty minutes. The buses do not arrive evenly, and they never will.