Why You Keep Catching the Long Wait

Two buses almost together. Then a long empty gap. Why does your arrival seem to land in the long one? Our original repeating toy schedule alternates 2-minute and 10-minute gaps. Passenger arrivals are uniformly random in time and independent of the buses; everyone can board the next bus. The average gap is 6 minutes, but the mean remaining wait is 4 minutes 20 seconds. Even 6/6 spacing has the same frequency and a 3-minute average wait. This is not a real timetable, a claim about every journey, or a model of why buses bunch. Timed arrivals using a reliable schedule can behave differently. Background: Karl Sigman, Columbia University, renewal-reward theory and the inspection paradox: https://www.columbia.edu/~ks20/4106-2... https://www.columbia.edu/~ks20/4106-2... Original AI-assisted miniature animation. Locally synthesized narration; original procedural music and cartoon sound effects. Small Rules: simple rules, surprising outcomes.