# #TIL: **A mathematical route to bus efficiency**

By Anish Michael

Many times we find ourselves in the bus stop, waiting for hours on end only for not one but multiple buses arriving one after the other headed on the same route — a phenomenon known as *bus bunching*. Researchers and governments have endlessly tried to understand and solve this problem. Interestingly though, it has a mathematical reason as well as a solution.

The time buses spend serving passengers at a bus-stop is related to the amount of time between consecutive bus arrival. When a bus runs late, its headway increases resulting in more passengers needing to be served at its next stop. The problem tends to compound itself since a delayed bus will have more passengers waiting for it, resulting in longer boarding times, delaying the bus further. This cycle continues until multiple buses eventually catch up and bunch.

To solve this problem, researchers use the *Markov Chain* — a mathematical theory predicting long-term behavior, to calculate how long a bus needs to wait at each stop. It is based on the estimated arrival time of the next bus. This theory has been tested by Georgia Tech University on three high transit routes. Buses are all connected through GPS which signals the driver the exact time to depart from each stop, improving efficiency. Passengers are thus happy as service is more regular and often at equal intervals.

While in reality, buses seldom run like clockwork, a simple mathematical calculation can ease a long wait for our next bus.

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