• Train Length:  xL = 510 ft (10-car consist of R142 vehicles, each 51 ft long)
• Cruise Velocity: vC = 50 mph (73.3 fpsps)
• Platform Velocity: vP = 30 mph (44.0 fps)
• Normal Acceleration: aS = 2.5 mphps (3.7 fpsps)
• Service Braking Deceleration: dS = 3.0 mphps (4.4 fpsps)
• Feathered Stop Decleration: dF = 1.5 mphps (2.2 fpsps)
• Dwell Time: tD = 20 seconds

Our Express vs Local puzzle posed the question, "What are the real payoffs attributable to express service?" There can be no doubt that for transit passengers, especially commuters, any payoff must take the form of reduced Trip Time.  Getting from Station A to Station B -- that's all that matters. Quicker the better.  The solution might seem to be Trip Time savings for passengers on express trains, amounting to about a minute for every bypassed station.  Oh, but there is more to the subject.

The graph below is called a "string chart."  This one shows four theoretical subway stations with local and express trains that have been modeled using the Base Case parameters developed above.  Various times and distances are indicated on the graph.

1. Three stations (1, 3, 4) have local train service only.
2. One station (2) has both local and express service.
3. By assumption, the distance between stations xS = 1.5 miles.
4. Both train services are assumed to be running on five-minute headways, tH = 5 min.
5. Average wait for a local train at any station ("boarding delay") tBD =  tH / 2.
6. Average wait for an express train at an express station ("boarding delay") tBD = tH / 2.
7. Arrival times at station 2 are staggered, with the local train arriving a minute ahead of the express train.
8. Passengers changing trains from local to express will have to wait about a minute.
9. Passengers changing trains from express to local will have to wait about four minutes.
10. Average wait ("cross-platform delay") tXP = (1 + 4) / 2 = 2.5 min (tH / 2).
11. Trip Time between local stations tS = tSS + (xS - xSS) / vC = 168 sec (2.8 min).
12. Express trains bypass local stations, saving one minute of Trip Time per station.
13. Trip Time between express stations bypassing n local stations tS = tSS + (n xS - xSS) / vC 

• Category 1 Passengers, with routes between two express stations will surely choose to ride express trains and enjoy a reduction in overall Trip Time by about one minute per bypassed station.
• Category 2 Passengers, traveling between local stations, will not be able to make use of express train service unless two or more of the enroute stations having express service, which would then put those passengers in Category 5.
• Category 3 Passengers must board a local train and, for destinations at express stations, with routes that include two or more express stations, can choose to change trains enroute to bypass local stations.
• Category 4 Passengers, departing at an express station, with routes that include two or more express stations, can choose to board an express train and bypass local stations, but must then change trains to reach destinations at local station.
• Category 5 Passengers must board a local train and, with routes that include two or more express stations, can choose to change trains and bypass local stations, but then must change trains again to reach destinations at local stations.

Consider a Category 1 or 4 Passenger arriving at the departure platform just as the express train pulls out of the station.  Suppose also that a local train is just pulling in.  The choices would be to leave right away on the local train or to wait for as long as tBD = tH to board an express train

Whenever there is a choice that requires changing train enroute, cross-platform delays impact Trip Time and reduce the benefits of express service.  Category 3 and 4 Passengers will suffer tBD = tH / 2 plus a cross-platform delay tXP = tH / 2. 

Category 5 Passengers will suffer two cross-platform delays for a total of tH.   Thus, for example, with a headway tH = 5 min, Category 5 Passengers will wind up riding an express train that must bypasses seven local stations just to break even.

The savings in Trip Time attributable to express train service will depend on the number n of local stations bypassed between successive express stations.

To justify the investment, there must be some other real payoffs attributable to express service beyond those for passengers. How about another kind of speed: flow-rate?  Not miles-per-hour as experienced by passengers but speed measured as passengers-per-hour achieved by stations.  A transportation system is not just about getting people from A to B quickly but getting the most people loaded at A and ultimately unloaded at B -- and C and D -- whatever the Trip Time over each individual route.

Two clues for this speculation: [1] that, as described above, there are so few local stations situated between express stations ready to be bypassed by express train service (n = 1.7) and [2] that express stations often serve sites that produce and absorb high volumes of patrons (Grand Central Terminal, Pennsylvania Station, Rockefeller Center, Times Square). Having both express and local service, an express station will handle twice the volume of passengers.  Never mind that some express train passengers cannot access local stations for their destinations. 

Exclamation Point Alert: Today there is an alternative for handling twice the volume!  Modern transit systems do not need to budget for all those construction expenditures to double each rail line to provide express trackway, including all those requisite tunnels and all those extra platforms at each express station, along with extra street-level entrances, extra stairways and escalators for circulation between concourses and platforms. 

A comparable benefit in system flow-rate would accrue to all stations by running more trains with closer headways.  Oh right, the New York Subway system applies 'legacy technologies' dating back a century or more.  Closer headways require modern systems, specifically automatic train control.

Accordingly, let us take as our solution for the Express vs Local puzzle...