Here's the introduction:
In Boston, the Red Line is a subway that runs north-south from Cambridge to Boston. When I was working in Cambridge I took the Red Line from Kendall Square to South Station and caught the commuter rail to Needham. During rush hour Red Line trains run every 7--8 minutes, on average.
When I arrived at the station, I could estimate the time until the next train based on the number of passengers on the platform. If there were only a few people, I inferred that I just missed a train and expected to wait about 7 minutes. If there were more passengers, I expected the train to arrive sooner. But if there were a large number of passengers, I suspected that trains were not running on schedule, so I would go back to the street level and get a taxi.
While I was waiting for trains, I thought about how Bayesian estimation could help predict my wait time and decide when I should give up and take a taxi. This chapter presents the analysis I came up with.
Sadly, this problem has been overtaken by history: the Red Line now provides real-time estimates for the arrival of the next train. But I think the analysis is interesting, and still applies for subway systems that don't provide estimates.
One interesting tidbit:
As it turns out, the average time between trains, as seen by a random passenger, is substantially higher than the true average.
Why? Because a passenger is more like to arrive during a large interval than a small one. Consider a simple example: suppose that the time between trains is either 5 minutes or 10 minutes with equal probability. In that case the average time between trains is 7.5 minutes.
But a passenger is more likely to arrive during a 10 minute gap than a 5 minute gap; in fact, twice as likely. If we surveyed arriving passengers, we would find that 2/3 of them arrived during a 10 minute gap, and only 1/3 during a 5 minute gap. So the average time between trains, as seen by an arriving passenger, is 8.33 minutes.
This kind of observer bias appears in many contexts. Students think that classes are bigger than they are, because more of them are in the big classes. Airline passengers think that planes are fuller than they are, because more of them are on full flights.
In each case, values from the actual distribution are oversampled in proportion to their value. In the Red Line example, a gap that is twice as big is twice as likely to be observed.
The data for the Red Line are close to this example. The actual time between trains is 7.6 minutes (based on 45 trains that arrived at Kendall square between 4pm and 6pm so far this week). The average gap as seen by random passengers is 8.3 minutes.
Interestingly, the Red Line schedule reports that trains run every 9 minutes during peak times. This is close to the average seen by passengers, but higher than the true average. I wonder if they are deliberately reporting the mean as seen by passengers in order to forestall complaints.
You can read the rest of the chapter here. One of the figures there didn't render very well. Here is a prettier version:
This figure shows the predictive distribution of wait times if you arrive and find 15 passengers on the platform. Since we don't know the passenger arrival rate, we have to estimate it. Each possible arrival rate yields one of the light blue lines; the dark blue line is the weighted mixture of the light blue lines.
So in this scenario, you expect the next train in 5 minutes or less, with 80% confidence.
UPDATE 10 May 2013: I got the following note from email@example.com, confirming that their reported gap between trains is deliberately conservative:
Thank you for writing to let us know about the Red Line case study in your book, and thank you for your question. You’re right that the scheduled time between trains listed on the subway schedule card for rush hour is greater than what you observed at Kendall Square. It’s meant as a slightly conservative simplification of the actual frequency of trains, which varies by time throughout rush hour – to provide maximum capacity during the very peak of rush hour when ridership is normally highest – as well as by location along the Red Line during those different times, since when trains begin to leave more frequently from Alewife it takes time for that frequency to “travel” down the line. So yes it is meant to be slightly conservative for that reason. We hope this information answers your question.