In New York, the MTA has consistent guidelines for how frequently to run each subway route, based on crowding levels. The standards are based on crowding levels at the point of maximum crowding on each numbered or lettered route. Each line is designed to have the same maximum crowding, with different systemwide levels for peak and off-peak crowding. While this approach is fair, and on the surface reasonable, it is a poor fit for New York’s highly branched system, and in my view contributes to some of the common failings of the subway.

Today, the off-peak guidelines call for matching frequency to demand, so that at the most crowded, the average train on each route has 25% more passengers than seats. Before the 2010 service cuts, the guidelines had the average train occupied to exact seating capacity. At the peak, the peak crowding guidelines are denser: 110 passengers on cars on the numbered lines, 145 on shorter (60’/18 m) cars on the lettered lines, 175 on longer (75’/23 m) cars on the lettered lines. There’s a minimum frequency of a train every 10 minutes during the day, and a maximum frequency at the peak depending on track capacity. When the MTA says certain lines, such as the 4/5/6, are operating above capacity, what it means is that at maximum track capacity, trains are still more crowded than the guideline.

In reality, guideline loads are frequently exceeded. Before the 2010 service cuts, many off-peak trains still had standees, often many standees. Today, some off-peak trains are considerably fuller than 25% above seated capacity. In this post, I’d like to give an explanation, and tie this into a common hazard of riding the subway in New York: trains sitting in the tunnels, as the conductor plays the announcement, “we are delayed because of train traffic ahead of us.”

The key takeaway from the system is that frequency at each time of day is calculated separately for each numbered or lettered route. Even when routes spend extensive distance interlined, as the 2/3 and 4/5 do, their frequencies are calculated separately. As of December 2014, we have the following headways, in minutes:

Line AM peak Noon off-peak PM peak 1 3 6 4 2 6:30 7:30 6:45 3 6 8:30 6:45 4 4:30 7:30 4:24 5 5 8:30 5:45 6 2:30 4 3:18 7 2:30 5 2:30 A 4:45 10 4:45 B 8:45 10 9:15 C 9:15 10 10 D 6:15 10 6:45 E 4 7:30 4 F 4:45 7:30 5 G 6:30 10 10 J/Z 5 10 5 L 4:30 6 4 M 8:45 10 9:25 N 7:15 10 7:30 Q 7:15 10 7:45 R 7:30 10 7:30

Consider now the shared segments between the various lines. The 4 comes every 4.5 minutes in the morning peak, and the 5 every 5 minutes. There is no way to maintain even spacing on both lines with these headways: they share tracks for an extensive portion of their trip. Instead, the dispatchers move trains around to make sure that headways are as even as possible on both the shared trunk segments and the branches, but something has to give. In 45 minutes, there are ten 4s and nine 5s. Usually, on trunk lines with two branches, trains alternate, but here, it’s not possible to have a perfect alternation in which each 4 is followed by a 5 and each 5 is followed by a 4. There is bound to be a succession of two 4s: the second 4 is going to be less crowded than the guideline, and the following 5 is going to be more crowded.

It gets worse when we consider the extensive reverse-branching, especially on the lettered lines. For example, on its northbound journey, the Q initially does not share tracks with any line; then it shares tracks with the B, into Downtown Brooklyn; then it crosses into Manhattan sharing tracks with the N; then it again shares tracks with no other route, running express in Manhattan while the N runs local; then it shares tracks with the N and R into Queens; and then finally it shares tracks with the N in Queens. It is difficult to impossible to plan a schedule that ensures smooth operations like this, even off-peak, especially when the frequency is so variable.

Concretely, consider what happens when the Q enters Manhattan behind an N. Adequate separation between trains is usually 2 minutes – occasionally less, but the schedule is not robust to even slight changes then. To be able to go to Queens ahead of the N, the Q has to gain 4 minutes running express in Manhattan while the N runs local. Unfortunately, the Q’s express jaunt only skips 4 stations in Manhattan, and usually the off-peak stop penalty is only about 45 seconds, so the Q only gains 3 minutes on the N. Thus, the N has to be delayed at Herald Square for a minute, possibly delaying an R behind it, or the Q has to be delayed 3 minutes to stay behind the N.

In practice, it’s possible to schedule around this problem when schedules are robust. Off-peak, the N, Q, and R all come every 10 minutes, which makes it possible to schedule the northbound Q to always enter Manhattan ahead of the N rather than right behind it. Off-peak, the services they share tracks with – the B, D, and M – all come every 10 minutes as well. The extensive reverse branching still makes the schedule less robust than it can be, but it is at least possible to schedule non-conflicting moves. (That said, the M shares tracks with the much more frequent F.) At the peak, things are much harder: while the N, Q, and R have very similar headways, the D is considerably more frequent, and the B and M considerably less frequent.

I believe that this system is one of the factors contributing to uneven frequency in New York, with all of the problems it entails: crowding levels in excess of guidelines, trains held in the tunnel, unpredictable wait times at stations. Although the principle underlying the crowding guidelines is sound, and I would recommend it in cities without much subway branching, in New York it fails to maintain predictable crowding levels, and introduces unnecessary problems elsewhere.

Instead of planning schedules around consistent maximum crowding, the MTA should consider planning schedules around predictable alternation of services on shared trunk lines. This means that, as far as practical, all lettered lines except the J/Z and the L should have the same frequency, and in addition the 2/3/4/5 should also have the same frequency. The 7 and L, which do not share their track or route with anything else, would maintain the present system. The J/Z, which have limited track sharing with other lines (only the M), could do so as well. The 1 and 6 do not share tracks with other lines, but run local alongside the express 2/3 and 4/5. Potentially, they could run at exactly twice the frequency of the 2/3/4/5, with scheduled timed local/express transfers; however, while this may work for the 6, it would give the 1 too much service, as there is much more demand for express than local service on the line.

To deal with demand mismatches, for example between the E/F and the other lettered lines, there are several approaches, each with its own positives and negatives:

– When the mismatch in demand is not large, the frequencies could be made the same, without too much trouble. The N/Q/R could all run the same frequency. More controversially, so could the 2/3/4/5: there would be more peak crowding on the East Side than on the West Side, but, to be honest, at the peak the 4 and 5 are beyond capacity anyway, so they already are more crowded.

– Some services could run at exactly twice the frequency of other services. This leads to uneven headways on the trunks, but maintains even headways on branches. For example, the A’s peak frequency is very close to exactly twice that of the C, so as they share tracks through Lower Manhattan and Downtown Brooklyn, they could alternate A-C-A-empty slot.

– Services that share tracks extensively could have drastic changes in frequency to each route, preserving trunk frequency. This should be investigated for the E/F on Queens Boulevard: current off-peak frequency is 8 trains per hour each, so cutting the E to 6 and beefing the F to 12 is a possibility.

– Service patterns could be changed, starting from the assumption that every lettered service runs every 10 minutes off-peak and (say) 6-7 minutes at the peak. If some corridors are underserved with just two services with such frequency, then those corridors could be beefed with a third route: for example, the Queens Boulevard express tracks could be supplanted with a service that runs the F route in Jamaica but then enters Manhattan via 53rd Street, like the E, and then continues either via 8th Avenue like the E or 6th Avenue like the M. Already, some peak E trains originate at Jamaica-179th like the F, rather than the usual terminus of Jamaica Center, which is limited to a capacity of 12 trains per hour.

– The service patterns could be drastically redrawn to remove reverse branching. I worked this out with Threestationsquare in comments on this post, leading to a more elegant local/express pattern but eliminating or complicating several important transfers. In particular, the Broadway Line’s N/Q/R trains could be made independent of the Sixth Avenue trains in both Queens and Brooklyn, allowing their frequencies to be tailored to demand without holding trains in tunnels to make frequencies even.

For the lettered lines, I have some affinity for the fourth solution, which at least in principle is based on a service plan from start to finish, rather than on first drawing a map and then figuring out frequency. But it has two glaring drawbacks: it involves more branching than is practiced today, since busy lines would get three services rather than two, making the schedule less robust to delays; and it is so intertwined with crowding levels that every major service change is likely to lead to complete overhaul of the subway map, as entire routes are added and removed based on demand. The second drawback has a silver lining; the first one does not.

I emphasize that this is more a problem of reverse branching than of conventional branching. The peak crowding on all lines in New York, with the exception of the non-branched 7 and 1, occurs in the Manhattan core. Thus, if routes with different colors never shared tracks, it would not be hard to designate a frequency for each trunk route at each time of day, without leading to large mismatches between service and demand. In contrast, reverse branching imposes schedule dependencies between many routes, to the point that all lettered routes except the L have to have the same frequency, up to integer multiples, to avoid conflicts between trains.

The highly branched service pattern in New York leads to a situation in which there is no perfect solution to train scheduling. But the MTA’s current approach is the wrong one, certainly on the details but probably also in its core. It comes from a good place, but it does not work for the system New York has, and the planners should at least consider alternatives, and discuss them publicly. If the right way turns out to add or remove routes in a way that makes it easier to schedule trains, then this should involve extensive public discussion of proposed service maps and plans, with costs and benefits to each community openly acknowledged. It is not good transit to maintain the current scheduling system just because it’s how things have always worked.