The waiting time paradox: why is my bus always late? (2018)
Everyday experiences & humor
- Many describe always seeing the bus or train just leaving, or the bus arriving only once they give up and start walking.
- “Lighting a cigarette makes the bus arrive” is a widely recognized running joke; some treat it as an almost deterministic law.
- Several doctor/patient and TV show jokes spin off from the “if it hurts, don’t do it” attitude toward watching the station.
Intuition for the waiting time paradox
- Key assumption: passengers arrive at random times, not synchronized to the timetable.
- Because long gaps contain more time, you’re more likely to arrive during a long interval than a short one.
- With perfectly punctual buses, average wait is half the headway; with random (Poisson-like) headways, average experienced gap doubles, so the average wait approaches the nominal interval.
- Some confusion appears around notation (N vs 2N vs wait time); others clarify: average span riders experience is 2N, so average wait is N.
Related paradoxes and probability debates
- Analogies:
- Class-size paradox: the “average student” sees larger classes than the average class size.
- “Your friends have more friends than you” phenomenon.
- Bitcoin block times: memoryless exponential waiting.
- Large subthread debates the claim: “if you win the lottery jackpot, you win less than the average lottery winner.”
- One side emphasizes conditional expectation: given that you win, you tend to be in drawings with more co‑winners.
- Others argue about what “average winner” and “average prize” mean (per draw vs per winner, conditioning on at least one winner, etc.).
- Consensus: the paradox is real at the level of conditional expectations, but precise statements depend heavily on definitions.
Transit reliability, operations, and bus bunching
- Many note that in practice, apps and GPS-based predictions can be wildly inaccurate, sometimes no better than random.
- Others report fairly accurate real-time systems where knowing “4 minutes away” matters more than schedule adherence.
- Bus bunching is discussed: late buses get more passengers and fall further behind; following buses catch up and form clumps.
- Proposed mitigations: multi-door boarding, off-bus or app ticketing, random inspections, mandatory vs optional stops, dispatch control, and self-coordinating headway control (including approaches that hold buses at control points).