everyday
did you know random arrivals make long bus gaps find you?
you are more likely to land inside a long gap than a short one.
5 → 8.2 min
even gaps → uneven gaps
both example services average a ten-minute gap; passengers arrive independently at random.
- 2-minute gap
- 18-minute gap
- 8.2-minute wait
the idea
a bus service can average one bus every ten minutes while a randomly arriving passenger waits more than five minutes on average. uneven gaps matter because long gaps occupy more of the clock.
how it works
inside the two-minute gap, the average remaining wait is one minute. inside the eighteen-minute gap, it is nine. weight those by 10% and 90%, and the overall average wait becomes 8.2 minutes.
go deeper
whose average are we taking?
a bus service can average one bus every ten minutes while a randomly arriving passenger waits more than five minutes on average. uneven gaps matter because long gaps occupy more of the clock.
suppose gaps alternate between two and eighteen minutes. the operator averages those two gaps and gets ten. a passenger arriving at a random time has a 90% chance of landing in the eighteen-minute gap.
weight each gap by the time it occupies
inside the two-minute gap, the average remaining wait is one minute. inside the eighteen-minute gap, it is nine. weight those by 10% and 90%, and the overall average wait becomes 8.2 minutes.
perfect ten-minute spacing would give a five-minute average wait. the same average service frequency can therefore produce a very different experience.
how does variability enter the answer?
for random arrivals independent of the service, average waiting time is the mean squared gap divided by twice the mean gap. long gaps get extra weight.
average wait = E(gap²) / [2 × E(gap)] = (2² + 18²) / [2 × (2 + 18)] = 8.2 minutes
when does this calculation not apply?
- someone checking a live timetable and timing their arrival is not arriving independently at a random time.
- the same bias appears when sampling intervals, queues, and lifetimes: a random observation tends to encounter things that last longer.
sources & further reading
concepts: inspection paradox · weighted averages · sampling bias
2 minute read