The Washington DC Metro recently opened the Silver Line to travel to Dulles Airport (end of the line). It only runs “local” (all stops) trains. That got me thinking of the following problem.

A subway system with 25 stops is trying to decide whether to run all locals or locals with some express trains mixed in. A local train takes 4 minutes to go from leaving one stop to leaving the next one after unloading and loading passengers. An express train takes 2 minutes to go from passing one local-only stop to passing the next local-only stop. It also takes the same extra 2 minutes at an express stop to unload and load passengers. At any time there are 12 trains operating on the system. Passengers are assumed to arrive at each station in groups of 24 every four minutes with one person bound for each of the 24 other stations. If your goal is to minimize the average travel time of the patrons, should you run any express trains, and if so how many of the 12 trains should be express trains and how often should they stop? To simplify the solution space, all express trains have the same set of stops and the stop are uniformly spaced, that is, there are the same number of non-stops between adjacent express stations and the time between expresses is fixed as is the time between locals, and further the endpoints are always express stations. The tracks have adequate dual sections to allow express trains to pass local ones. And stations can serve up to four trains at a time with passengers magically able to transfer between any trains that stop at a given station without extra time to change platforms. The system is a single fee cost, that is, you pay to enter and may ride as many or as few trains as like before you exit.
L. Flynn
P.S. Since the Silver Line is part of and connects to the whole system, it could be the case that an express to get you to and from the first transfer station and Dulles would be beneficial overall in reducing the travel time of passengers going beyond the single line portion of the system. That is a problem for another day.