Why random arrivals find longer gaps
A bus rider usually does not arrive at the stop at the moment a bus pulls away. They arrive at some arbitrary point in the schedule, so the gap they encounter is more likely to be a long one. This is a form of length-biased sampling: when you sample a stretch of time at random, longer stretches have more chances to contain your sampling point.
Imagine a route with one five-minute gap and one twenty-minute gap. Together they span 25 minutes. A random arrival has a 20-in-25 chance of landing in the longer gap—not because the bus service chose that gap more often, but because it occupies more of the clock. Once there, the rider may wait anywhere from almost nothing to nearly twenty minutes.
Why the average wait can surprise you
The average gap between buses is not the same as the average time a randomly arriving passenger waits. Under a simple model in which arrivals are evenly spread within each gap, the wait within a gap averages half its length. That means long gaps inflate waiting disproportionately: a gap twice as long is both twice as likely to catch a random passenger and offers a longer possible wait.
For example, if buses arrive exactly every ten minutes, a random arrival waits five minutes on average. But if the same average service interval comes from alternating short and long gaps, passengers can wait longer on average. The total time between buses may be unchanged; its uneven distribution changes what riders experience.
The practical consequence
This is why a timetable or an average frequency can give an overly reassuring picture of service. Regular spacing matters as much as the average interval: bunching buses together can leave a long empty stretch behind them, and that stretch captures a disproportionate share of riders arriving at random. For someone who misses a bus, the gap—not the route’s average—sets the wait.
