See related post: Expected Wait Time given Multiple Trains
Is it worth it to run and rush to the train station? It might payoff if running there gets us on an earlier train. But it might not payoff, as we simply end up waiting a longer time for the next train. Does it make a difference what time of the day it is?
To answer this question, we need just two parameters: the frequency of the arrival event, and the extent to which the rushing shortens the time getting onto the site. Let's use the train example for easier understanding. Suppose that the train comes every 10 minutes. As we saw on the previous post, the average wait time until the first train is 5 minutes. Now, suppose that we run to the train station and get there 3 minute earlier than if we had walked.
It turns out that this answer is much simpler than the calculations from the last post. The average wait time is still 5 minutes. However, the probability that we end up catching an earlier train now is 3/10. We can easily see that we'll be able to catch an earlier train, if a train comes within those first 3 minute that we wouldn't have been there otherwise. Given 10-minute frequencies, that probability is simply 30%. This helps us answer the question of whether or not rushing to the arrival site will benefit. Using the variables defined earlier, the probability of benefit is simply (shortened time) / (frequency). In the event that (shortened time) > (frequency), that probability simply caps at 100% certainty.
That simply results also helps us answer whether time of the day affects this decision. Since during rush-hours the trains come more frequent (lower frequencies), rushing to the station will be more likely to be beneficial then. It makes intuitive sense, because in the same 3 minutes or whatever the shortened time may be, there will be a higher chance that a train comes during that period then.
