Waiting Time Paradox or Why Is My Bus Always Late?
The Waiting Time Paradox reveals that bus wait times can exceed expectations due to arrival variability. Simulations and Seattle data confirm theoretical predictions, aiding public transit scheduling and commuter understanding.
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The Waiting Time Paradox illustrates a common experience for public transit users: the perception that buses are often late, leading to longer average wait times than expected. While one might assume that if buses arrive every 10 minutes, the average wait time would be around 5 minutes, the reality is different. Due to variability in bus arrival times, the average wait time can actually be as long as 10 minutes. This phenomenon is linked to the inspection paradox, where the probability of observing longer wait times increases because passengers tend to arrive at random times. The post explores this paradox through simulations and real data from Seattle's bus system, confirming that the average wait time aligns with the predicted outcomes of the paradox. The analysis shows that when bus arrivals follow a Poisson process, the expected wait time for passengers remains consistent, regardless of previous wait times. The findings suggest that real-world bus arrival patterns often reflect these theoretical models, providing insight into commuter experiences.
- The Waiting Time Paradox suggests average wait times can exceed expected values due to variability in bus arrivals.
- Simulations confirm that average wait times can align with theoretical predictions of the paradox.
- The inspection paradox explains why longer wait times are more frequently experienced by passengers.
- Real-world data from Seattle supports the notion that bus arrivals often follow a Poisson process.
- Understanding these patterns can help improve public transit scheduling and commuter expectations.
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- Many commenters discuss the phenomenon of buses being late and how this affects passenger wait times, often humorously noting personal experiences.
- Several comments reference related mathematical concepts and theories, such as queueing theory and renewal theory, to explain the paradox.
- Some users share anecdotes about their experiences with bus schedules and the unpredictability of arrival times, including the common belief that lighting a cigarette can influence bus arrival.
- There is a discussion about the implications of the paradox in other contexts, such as lottery winnings and Bitcoin block times, highlighting its broader relevance.
- Several commenters express a desire for better tools or apps to predict bus arrival times more accurately.
24 commentsThe further behind the previous bus a bus is, the more people will arrive at the bus stop. The more people there are at the stop, the longer the bus has to spend picking them all up and selling them tickets etc. Therefore the delayed bus will tend to experience more delay. The bus behind them will have less people to pick up, so it will spend a shorter time at stops and tend to catch up with the first bus, so the two busses are dragged towards each other.
Average Jackpot prize is JackpotPool/Average winners.
Average Jackpot prize given you win is JackpotPool/(1+Average winners).
The number of expected other winners on the date you win is the same as the average number of winners. Your winning ticket doesn't affect the average number of winners.
This is similar to the classroom paradox where there are more winners when the prize is poorly split, so the average observed jackpot prize is less than the average jackpot prize averaged over events.
Who is arriving in the first part of the sentence? At first I thought he meant the bus arrival, thus N = 10, and 2N would be 20. But then he says
>The average wait time is also close to 10 minutes, just as the waiting time paradox predicted.
10 isn't 20 so ???
That is, they sound like similar questions, but they are not. How long can one random person expect to wait at a stop is different from how long a population will wait at a given spot. In large because a person can only arrive at a single time in the waiting interval, but more passengers become less likely the closer to departure time.
(I realize I didn't word all of this as a question, but I am not asserting I'm correct here. Genuinely curious if I understand correctly.)
As the article notes, it's the same reason the average coin in a sequence of tosses will be in a longer run than the average run length.
someone comes to a subway station at (uniform) random times between 6p and 8p; he notices that the first train he observes arriving at the same station is 3 times more often inbound than outbound. He also knows that time intervals between the trains going in the opposite directions are fixed and equal — say, always 6 minutes, so the only random event here is when this person arrives at the platform. Explain how this is possible.
Past discussion: https://news.ycombinator.com/item?id=18321062
edit: interesting post with a different ending than I imagined.
Been thinking about using some statistical methods to give me some better estimates of the busses I take to work. Like "given that it's $today, which is a Tuesday, it's 17:30, and the display says the bus is 7 minutes delayed, how long til it will actually come?
Some discussion then: https://news.ycombinator.com/item?id=18321062
OK sometimes the app / bus location system fucks up but most of the time or their are unexpected road road work or traffic accident that suddently forces it to be slower but most of the time it is pretty much accurate.
I really enjoy having 10-20 lines that produces the expectation value directly by simulation; that's a fast way for me to understand the underlying values.
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