Why can irregular departures increase expected wait?

Equal average gaps between departures do not necessarily produce equal average waits for passengers. People arriving independently across time are more likely to land inside a long gap than a short one, because the long gap occupies more of the schedule.

The fictional schedule repeats every eight minutes. Regular departures at 0, 4 and 8 create two four-minute gaps. Irregular departures at 0, 2 and 8 create gaps of two and six minutes. Both have a mean gap of four minutes: their two gaps add to eight. The departure at 8 is the next cycle’s boundary.

In the arrival model, 0.5, 1.5 and every following half-minute position through 7.5 are equally likely. Each passenger boards the next departure. Regular waits across these eight arrival positions sum to sixteen minutes, for a mean wait of two. Irregular waits sum to twenty, giving a mean wait of two and a half minutes.

Six of the eight possible arrivals fall within the irregular six-minute gap. Passengers therefore sample gaps by time exposure, not by giving each of the two gaps equal probability. That difference explains the longer expected wait, meaning the average under the stated arrival model. Passengers timing their arrivals to departures could change the result. The illustration is a repeating mathematical schedule, not measured evidence about a real service.