Category: Regional Rail

Frequency is Relative

Five years ago, I wrote a blog post about frequency-ridership spirals, mentioning as a side comment that the impact of mass transit frequency on ridership can be lumped together with the trip time. I’d like to develop this point here, and talk about how it affects various kinds of public transportation, including intercity trains.

The rule of thumb I’ve advocated for in ETA reports (for example, on commuter rail) is that the maximum headway should be no more than half the trip time. Untimed transfers reset the clock, since passengers have to wait another time every time they make such a transfer; timed transfers do not, but are rare enough that local public transportation doesn’t usually need to consider them in service planning. Intercity transportation can follow the same rule of thumb, but can also get away with worse frequency since passengers time themselves to arrive shortly before the train does; in particular, hourly trains between cities that are three hours apart are frequent enough that increasing service is valuable only insofar as it provides more capacity, and is unlikely to lead to higher ridership through shorter waits.

Wait and transfer penalties

In the literature on modeling public transportation ridership, it is universal that passengers prefer spending time on board a moving vehicle to waiting for a vehicle, walking to the station, or walking between platforms. This preference is expressed as a factor, called the waiting or transfer penalty. different models have different levels for these penalties; passengers also likely have different penalties depending on circumstances, such as familiarity with the route or how much luggage they’re carrying.

The papers I’ve seen have penalties ranging from 1.75 (in the MTA’s model) to 3 (the higher end cited in Lago-Mayworm-McEnroe). I usually model with 2, in Teulings-Ossokina-de Groot. The factor of 2 has the advantage of consistency with an assumption that passengers don’t have a wait penalty but do assume a worst-case scenario for waits, so that the generalized travel time is equal to the maximum headway plus the in-vehicle travel time.

Update 4-19: I was just alerted to a new study by Yap-Wong-Cats using London ridership, finding an out-of-vehicle penalty factor of 1.94 pre-pandemic and 1.92 post-pandemic.

The impact of frequency relative to trip time

If the elasticity of ridership with respect to the generalized travel time, summing the headway and in-vehicle travel time but not walking time to and from the station, is e, then we can compute the elasticity of ridership with respect to frequency as a fraction of e. If the current headway is a proportion r of the in-vehicle trip time, that is to say a fraction r/(r+1) of the generalized travel time, then the elasticity is er/(r+1).

In Lago-Mayworm-McEnroe, the value of e appears to be 0.8. This means that if r = 0.5, the elasticity of ridership with respect to frequency is 0.267. The paper doesn’t quantify elasticities relative to different levels of r but only relative to absolute frequencies, but 0.267 is within the range it finds for different frequencies, dipping to 0.22 for high-frequency lines. Other papers have different figures of e, often higher in the long run as passengers adjust, but those go up to around 1 or, reasoning backward from a VTPI report, a little higher.

Of notes:

  1. The value of r is not constant across different uses of the same line. A commuter traveling from near the outer end of a subway line to city center faces much lower r than a traveler going a short distance, within city center or within a large outlying neighborhood. In particular, r is generally lower for commutes than for non-commute trips, which is why the latter are more sensitive to frequency.
  2. Systems that rely on extensive transfers can have very high values of r with short in-vehicle trips. New York averages 13.5 minutes per unlinked subway trip, with many trips facing an effective off-peak headway of 10 or even 12 minutes, at which point e is high enough that increasing off-peak frequency could pay for itself through higher paying ridership (see analysis in a blog post and an ETA report). This, again, depends on the type of trip – commuters may pick an apartment or a job based on ease of travel, reducing the need to transfer, but their non-commute trips are usually a collection of irregular trips to various destinations and are likelier to involve a transfer.
  3. The cost of higher frequency depends on mode (it’s higher on buses than trains) and time of day (it’s very low on off-peak trains until it matches peak frequency). Together with points 1 and 2, this argues in favor of raising the off-peak frequency on urban and inner suburban trains, potentially to the point of matching peak frequency. On longer-range commuter trains, the impact of frequency on ridership is lower, and thus the marginal cost may be such that a ratio of peak to off-peak service larger than 1 is desirable.

Intercity trains

The papers I’m citing aim to fit elasticity factors to observed ridership on local and regional public transportation. Intercity rail has its own set of models, with different assumptions. Frequency again matters, but because passengers time themselves to arrive at the departing station shortly before the train leaves, its impact is reduced.

I don’t know the elasticity of intercity rail ridership with respect to frequency; Cascetta-Coppola have the elasticity of ridership with respect to in-vehicle trip time as about -2, while Börjesson has it at -1.5 for business travel and -1 for non-business travel with a rough rule of thumb trying to approximate the impact of frequency. At the level of the sanity check, the low frequency of TGV services is not visible in TGV ridership between the provinces and Paris, compared with Japan (which charges higher fares) and as far as I can tell from a few data points Germany. TGV ridership between the provinces is bad, but that involves trains with service gaps that are much larger than the trip times, reaching six hours between Marseille and Lyon. In contrast, those three-hour gaps in service between Paris and cities three hours away by TGV don’t seem to impact ridership visibly.

What this means is that intercity trains do need a certain baseline frequency. The German system of a train every two hours on every city pair is wise, in light of the typical intercity rail travel distances in a large country with slow trains. Higher frequency is warranted if the cities are bigger and therefore require more service, or if they are closer together in time through either a short geographical distance or higher speeds. New York and Philadelphia are about 1:10 apart by rail, and high-speed rail could cut this to about 45 minutes; half-hourly frequencies in the current situation are sufficient that more service would have a second-order effect, and even with high-speed rail, a train every 15 minutes is more than enough for all purposes except capacity (the current offer is 3-4 trains an hour with irregular spacing). Frequency is freedom, but this depends on trip times; what works for four-station subway trips is not what works for trips between cities 140 km apart, let alone 360 km, and vice versa.

Local and Intercity Rail are Complements

An argument in my comments section is reminding me of a discussion by American transit advocates 15 years ago, I think by The Overhead Wire, about the tension between funding local transit and high-speed rail. I forget who it was – probably Jeff Wood himself – pointing out that the argument in 2008-9 about whether the priority was local transit or intercity rail didn’t make much sense. There are separate questions of how to allocate funding for intercity transportation and how to do the same for local transportation, and in both cases the same group of activists can push for a more favorable rail : car funding ratio. Jeff was talking about this in the sense of political activism; the purpose of this post is to explain the same concept from the point of view of public transportation connectivity and network effects. This is not an obvious observation, judging by how many people argue to the contrary – years ago I had a debate with Noah Smith about this, in which he said the US shouldn’t build high-speed rail like the Shinkansen before building urban rail systems like those of Japanese cities (see my side here and here).

I’ve written about related issues before, namely, in 2022 when I recommended that countries invest against type. For example, France with its TGV-centric investment strategy should invest in connecting regional lines, whereas Germany with its hourly regional train connections should invest in completing its high-speed rail network. It’s also worthwhile to reread what I wrote about Metcalfe’s law for high-speed rail in 2020, here and here. Metcalfe’s law is an abstract rule about how the value of a network with n nodes is proportional to n^2, and Odlyzko-Tilly argue strongly that it is wrong and in fact the value is n\log n; my post just looks at specific high-speed rail connections rather than trying to abstract it out, but the point is that in the presence of an initial network, even weaker-looking extensions can be worth it because of the connections to more nodes. Finally, this builds on what I said five days ago about subway-intercity rail connections.

The combined point is that whenever two forms of local, regional, or intercity public transportation connect, investments in one strengthen the case for investments in the other.

In some edge cases, those investments can even be the same thing. I’ve been arguing for maybe 12 years that MBTA electrification complements Northeast Corridor high-speed rail investment, because running fast electric multiple units (EMUs) on the Providence Line and its branches instead of slow diesel locomotive-hauled trains means intercity trains wouldn’t get stuck behind commuter trains. Similarly, I blogged five years ago, and have been doing much more serious analysis recently with Devin Wilkins, that coordinating commuter rail and intercity rail schedules on the New Haven Line would produce very large speed gains, on the order of 40-45 minutes, for both intercity and commuter trains.

But those are edge cases, borne of exceptionally poor management and operations by Amtrak and the commuter railroads in the Northeast. Usually, investments clearly are mostly about one thing and not another – building a subway line is not an intercity rail project, and building greenfield high-speed rail is not a local or regional rail project.

And yet, they remain complements. The time savings that better operations and maintenance can produce on the New Haven Line are also present on other commuter lines in New York, for example on the LIRR (see also here, here, and here); they don’t speed up intercity trains, but do mean that people originating in the suburbs have much faster effective trips to where they’d take intercity rail. The same is true for physical investments in concrete: the North-South Rail Link in Boston and a Penn Station-Grand Central connection in New York both make it easier for passengers to connect to intercity trains, in addition to benefits for local and regional travel, and conversely, fast intercity trains strengthen the case for these two projects since they’d connect passengers to better intercity service.

Concretely, let’s take two New York-area commuter lines, of which one will definitely never have to interface with intercity rail and one probably will not either. The definitely line is the Morristown Line: right now it enters New York via the same North River Tunnels as all other trains from points west, intercity or regional, but the plan for the Gateway Tunnel is to segregate service so that the Morris and Essex Lines use the new tunnel and the Northeast Corridor intercity and commuter trains use the old tunnel, and so in the future they are not planned to interact. The probably line is the LIRR Main Line, which currently doesn’t interface with intercity trains as I explain in my post about the LIRR and Northeast Corridor, and which should keep not interfacing, but there are Amtrak plans to send a few daily intercities onto it.

Currently, the trip time from Morristown to New York is around 1:09 off-peak, with some peak-only express trains doing it in 1:01. With better operations and maintenance, it should take 0:47. The upshot is that passengers traveling from Morristown to Boston today have to do the trip in 1:09 plus 3:42-3:49 (Acela) or 4:15-4:35 (Regional). The commuter rail improvements, which other than Gateway and about one unfunded tie-in do not involve significant investment in concrete, turn the 4:51 plus transfer time trip to 4:29 plus transfer time – say 5 hours with the transfer, since the intercities run hourly and the transfers are untimed and, given the number of different branches coming in from New Jersey, cannot be timed. High-speed rail, say doing New York-Boston in 2 hours flat (which involves an I-95 bypass from New Haven to Kingston but no other significant deviations from the right-of-way), would make it 2:47 with a transfer time capped at 10 minutes, so maximum 2:57. In effect, these two investments combine to give people from Morristown an effective 41% reduction in trip time to Boston, which increases trip generation by a factor of 2.87. Of course, far more people from Morristown are interested in traveling to New York than to Boston, but the point is that in the presence of cheap interventions to rationalize and speed up commuter rail, intercity rail looks better.

The same is true from the other direction, from the LIRR Main Line. The two busiest suburban stations in the United States are on 2000s and 10s numbers Ronkonkoma and Hicksville, each with about 10,000 weekday boardings. Ronkonkoma-Penn Station is 1:18 and Hicksville-Penn Station is 0:42 off-peak; a few peak express trains per day do the trip a few minutes faster from Ronkonkoma by skipping Hicksville, but the fastest looks like 1:15. If the schedule is rationalized, Ronkonkoma is about 0:57 from New York and Hicksville 0:31, on trains making more stops than today. I don’t have to-the-minute New York-Washington schedules with high-speed rail yet, but I suspect 1:50 plus or minus 10 minutes is about right, down from 2:53-3:01 on the Acela and 3:17-3:38 on the Regional. So the current timetable for Ronkonkoma-Washington is, with a half-hour transfer time, around 4:45 today and 2:57 in the future, which is a 38% reduction in time and a factor of 2.59 increase in the propensity to travel. From Hicksville, the corresponding reduction is from 4:09 to 2:31, a 39% reduction and a factor of 2.72 increase in trip generation. Again, Long Islanders are far more interested in traveling to Manhattan than to Washington, but a factor of 2.59-2.72 increase in trip generation is nothing to scoff at.

The issue here is that once the cheap upgrades are done, the expensive ones start making more sense – and this is true for both intercity and regional trains. The New York-Boston timetable assumes an I-95 bypass between New Haven and Kingston, saving trains around 24 minutes, at a cost of maybe $5 billion; those 24 minutes matter more when they cut the trip time from 2:24 to 2:00 than when the current trip time is about 3:45 and the capacity on the line is so limited any increase in underlying demand has to go to higher fares, not more throughput. For suburban travelers, the gains are smaller, but still, going from 5:00 to 4:36 matters less than going from 3:21 to 2:57.

Conversely, the expensive upgrades for regional trains – by which I mean multi-billion dollars tunnels, not $300 million junction grade separations like Hunter or the few tens of millions of dollars on upgrading the junction and railyard at Summit – work better in a better-operated system. Electronics before concrete, not instead of concrete – in fact, good operations (i.e. good electronics) create more demand for megaprojects.

At no point are these really in competition, not just because flashy commuter rail projects complement intercity rail through mutual feeding, but also because the benefits for non-connecting passengers are so different that different funding mechanisms make sense. The North-South Rail Link has some benefits to intercity travel, as part of the same program with high-speed rail on the Northeast Corridor, and as such, it could be studied as part of the same program, if there is enough money in the budget for it, which there is not. Conversely, it has very strong local benefits, ideal for a funding partnership between the federal government and Massachusetts; similarly, New York commuter rail improvements are ideal for a funding partnership between the federal government, New York State, New Jersey, and very occasionally Connecticut.

In contrast, intercity rail benefits people who are far away from where construction is done: extensive bypasses built in Connecticut would create a small number of jobs in Connecticut temporarily, but the bigger benefits would accrue not just to residents of the state (through better New Haven-Boston and perhaps New Haven-New York trip times) but mostly to residents of neighboring states traveling through Connecticut. This is why there’s generally more national coordination of intercity rail planning than of regional rail planning: the German federal government, too, partly funds S-Bahn projects in major German cities, but isn’t involved in planning S21 S15 or the second S-Bahn trunk in Munich, whereas it is very involved in decisions on building high-speed rail lines. The situation in France is similar – the state is involved in decisions on LGVs and on Parisian transit but not on provincial transit, though it helps fund the latter; despite the similarity in the broad outlines of the funding structure, the outcomes are different, which should mean that the differences between France and Germany do not boil down to funding mechanisms or to inherent competition between intercity rail funds and regional rail funds.

Intercity Trains and Long Island

Amtrak wants to extend three daily Northeast Corridor trains to Long Island. It’s a bad idea – for one, if the timetable can accommodate three daily trains, it can accommodate an hourly train – but beyond the frequency point, this is for fairly deep reasons, and it took me years of studying timetabling on the corridor to understand why. In short, the timetabling introduces too many points of failure, and meanwhile, the alternative of sending all trains that arrive in New York from Philadelphia and Washington onward to New Haven is appealing. To be clear, there are benefits to the Long Island routing, they’re just smaller than the operational costs; there’s a reason this post is notably not tagged “incompetence.”

How to connect the Northeast Corridor with Long Island

Map of the Northeast Corridor and LIRR Main Line; the Northeast Corridor is depicted with extensive bypasses in Connecticut and small curve modifications in New Jersey and Pennsylvania

The Northeast Corridor has asymmetric demand on its two halves. North of New York, it connects the city with Boston. But south of New York, it connects to both Philadelphia and Washington. As a result, the line can always expect to have more traffic south of New York than north of it; today, this difference is magnified by the lower average speed of the northern half, due to the slowness of the line in Connecticut. Today, many trains terminate in New York and don’t run farther north; in the last 20 years, Amtrak has also gone back and forth on whether some trains should divert north at New Haven and run to Springfield or whether such service should only be provided with shuttle trains with a timed connection. Extending service to Long Island is one way to resolve the asymmetry of demand.

Such an extension would stop at the major stattions on the LIRR Main Line. The most important is Jamaica, with a connection to JFK; then, in the suburbs, it would be interesting to stop at least at Mineola and Hicksville and probably also go as far as Ronkonkoma, the end of the line depicted on the map. Amtrak’s proposed service makes exactly these stops plus one, Deer Park between Hicksville and Ronkonkoma.

The entire Main Line is electrified, but with third rail, not catenary. The trains for it therefore would need to be dual-voltage. This requires a dedicated fleet, but it’s not too hard to procure – it’s easier to go from AC to DC than in the opposite direction, and Amtrak and the LIRR already have dual-mode diesel locomotives with third rail shoes, so they could ask for shoes on catenary electric locomotives (or on EMUs).

The main benefit of doing this, as opposed to short-turning surplus Northeast Corridor trains in New York, is that it provides direct service to Long Island. In theory, this provides access to the 2.9 million people living on Long Island. In practice, the shed is somewhat smaller, because people living near LIRR branches that are not the Main Line would be connecting by train anyway and then the difference between connecting at Jamaica and connecting at Penn Station is not material; that said, Ronkonkoma has a large parking lot accessible from all of Suffolk County, and between it and significant parts of Nassau County near the Main Line, this is still 2 million people. There aren’t many destinations on Long Island, which has atypically little job sprawl for an American suburb, but 2 million originating passengers plus people boarding at Jamaica plus people going to Jamaica for JFK is a significant benefit. (How significant I can’t tell you – the tools I have for ridership estimation aren’t granular enough to detect the LIRR-Amtrak transfer penalty at Penn Station.)

My early Northeast Corridor ideas did include such service, for the above reasons. However, there are two serious drawbacks, detailed below.

Timetabling considerations

Under current plans, there is little interaction between the LIRR and the Northeast Corridor. There are two separate routes into Penn Station from the east, one via 32nd Street (“southern tunnels”) and one via 33rd (“northern tunnels”), each a two-track line with one track in each direction. The North River Tunnels, connecting Penn Station with New Jersey and the rest of the United States, face the southern tunnels; the Gateway tunnels under construction to double trans-Hudson capacity are not planned to pair with the northern tunnels, but rather to connect to stub-end tracks facing 31st Street. For this reason, Amtrak always or almost always enters Penn Station from the east using the southern tunnels; the northern tunnels do have some station tracks that connect to them and still allow through-service to the west, but the moves through the station interlocking are more complex and more constrained.

As seen on the map, east of Penn Station, the Northeast Corridor is to the north of the LIRR. Thus, Amtrak has to transition from being south of the LIRR to being north of it. This used to be done at-grade, with conflict with same-direction trains (but not opposite-direction ones); it has since been grade-separated, at excessive cost. With much LIRR service diverted to Grand Central via the East Side Access tunnel, current traffic can be divided so that LIRR Main Line service exclusively uses the northern tunnels and Northeast Corridor (Amtrak or commuter rail under the soon to open Penn Station Access project) service exclusively uses the southern tunnels; the one LIRR branch not going through Jamaica, the Port Washington Branch, can use the southern tunnels as if it is a Penn Station Access branch. This is not too far from how current service is organized anyway, with the LIRR preferring the northern (high-numbered) tracks at Penn Station, Amtrak the middle ones, and New Jersey Transit the southern ones with the stub end:

The status quo, including any modification thereto that keeps the LIRR (except the Port Washington Branch) separate from the Northeast Corridor, means that all timetabling complexity on the LIRR is localized to the LIRR. LIRR timetabling has to deal with all of the following issues today:

  • There are many different branches, all of which want to go to Manhattan rather than to Brooklyn, and to a large extent they also want to go on the express tracks between Jamaica and Manhattan rather than the local tracks.
  • There are two Manhattan terminals and no place to transfer between trains to different ones except Jamaica; an infill station at Sunnyside Yards, permitting trains from the LIRR going to Grand Central to exchange passengers with Penn Station Access trains, would be helpful, but does not currently exist.
  • The outer Port Jefferson Branch is unelectrified and single-track and yet has fairly high ridership, so that isolating it with shuttle trains is infeasible except in the extreme short run pending electrification.
  • All junctions east of Jamaica are flat.
  • The Main Line has three tracks east of Floral Park, the third recently opened at very high cost, purely for peak-direction express trains, but cannot easily schedule express trains in both directions.

There are solutions to all of these problems, involving timetable simplification, reduction of express patterns with time saved through much reduced schedule padding, and targeted infrastructure interventions such as electrifying and double-tracking the entire Port Jefferson Branch.

However, Amtrak service throws multiple wrenches in this system. First, it requires a vigorous all-day express service between New York and Hicksville if not Ronkonkoma. Between Floral Park and Hicksville, there are three tracks. Right now the local demand is weak, but this is only because there is little local service, and instead the schedule encourages passengers to drive to Hicksville or Mineola and park there. Any stable timetable has to provide much stronger local service, and this means express trains have to awkwardly use the middle track as a single track. This isn’t impossible – it’s about 15 km of fast tracks with only one intermediate station, Mineola – but it’s constraining. Then the constraint propagates east of Hicksville, where there are only two tracks, and so those express trains have to share tracks with the locals and be timetabled not to conflict.

And second, all these additional conflict points would be transmitted to the entire Northeast Corridor. A delay in Deer Park would propagate to Philadelphia and Washington. Even without delays, the timetabling of the trains in New Jersey would be affected by constraints on Long Island; then the New Jersey timetabling constraints would be transmitted east to Connecticut and Massachusetts. All of this is doable, but at the price of worse schedule padding. I suspect that this is why the proposed Amtrak trip time for New York-Ronkonkoma is 1:25, where off-peak LIRR trains do it in 1:18 making all eight local stops between Ronkonkoma and Hicksville, Mineola, Jamaica, and Woodside. With low padding, which can only be done with more separated out timetables, they could do it in 1:08, making four more net stops.

Trains to New Haven

The other reason I’ve come to believe Northeast Corridor trains shouldn’t go to Jamaica and Long Island is that more trains need to go to Stamford and New Haven. This is for a number of different reasons.

The impact of higher average speed

The higher the average speed of the train, the more significant Boston-Philadelphia and Boston-Washington ridership is. This, in turn, reduces the difference in ridership north and south of New York somewhat, to the point that closer to one train in three doesn’t need to go to Boston than one train in two.

Springfield

Hartford and Springfield can expect significant ridership to New York if there is better service. Right now the line is unelectrified and runs haphazard schedules, but it could be electrified and trains could run through; moreover, any improvement to the New York-Boston line automatically also means New York-Springfield trains get faster, producing more ridership.

New Haven-New York trips

If we break my gravity model of ridership not into larger combined statistical areas but into smaller metropolitan statistical areas, separating out New Haven and Stamford from New York, then we see significant trips between Connecticut and New York. The model, which is purely intercity, at this point projects only 15% less traffic density in the Stamford-New York section than in the New York-Trenton section, counting the impact of Springfield and higher average speed as well.

Commutes from north of New York

There is some reason to believe that there will be much more ridership into New York from the nearby points – New Haven, Stamford, Newark, Trenton (if it has a stop), and Philadelphia – than the model predicts. The model doesn’t take commute trips into account; thus, it projects about 7.78 million annual trips between New York and either Stamford or New Haven, where in fact the New Haven Line was getting 125,000 weekday passengers and 39 million annual passengers in the 2010s, mostly from Connecticut and not Westchester County suburbs. Commute trips, in turn, accrete fairly symmetrically around the main city, reducing the difference in ridership between New York-Philadelphia and New York-New Haven, even though Philadelphia is the much larger city.

Combining everything

With largely symmetric ridership around New York in the core, it’s best to schedule the Northeast Corridor with the same number of trains immediately north and immediately south of it. At New Haven, trains should branch. The gravity model projects a 3:1 ratio between the ridership to Boston and to Springfield. Thus, if there are eight trains per hour between New Haven and Washington, then six should go to Boston and two to Springfield; this is not even that aggressive of an assumption, it’s just hard to timetable without additional bypasses. If there are six trains per hour south of New Haven, which is more delicate to timetable but can be done with much less concrete, then two should still go to Springfield, and they’ll be less full but over this short a section it’s probably worth it, given how important frequency is (hourly vs. half-hourly) for trips that are on the order of an hour and a half to New York.

Serving Metro-North Fordham Station

In the Bronx, the Metro-North Harlem Line runs north-south, west of the 2/5 subway lines on White Plains Road and east of the 4 on Jerome Avenue and the B/D on Grand Concourse. It makes multiple stops, all served rather infrequently, currently about every half hour, with some hourly midday gaps, at premium fare. The north-south bus lines most directly parallel to the line, the Bx15 and Bx41, ranked #20 and #24 respectively in ridership on the New York City Transit system in 2019, though both have lost considerable ground since the pandemic. Overall, there is serious if not extremely high demand for service at those stations. There is already a fair amount of reverse-peak ridership: while those half-hourly frequencies can’t compete with the subway for service to Manhattan, they are the only non-car option for reverse-peak service to White Plains, and Fordham gets additional frequencies as well as some trains to Stamford. A city report from 2011 says that Fordham has 51 inbound passengers and 3,055 outbound ones boarding per weekday on Metro-North. Figuring out how to improve service in the Bronx requires a paradigm shift in how commuter rail is conceived in North America. Fordham’s reverse-peak service is a genuinely hard scheduling question, which we’re having to wrestle with as we’re proposing a (much) faster and smoother set of timetables for Northeast Corridor trains. Together, they make for a nontrivial exercise in tradeoffs on a busy commuter line, in which all options leave something to be desired.

Harlem Line local service

The bulk of demand from Fordham is local service, mostly toward Manhattan. The area is a bedroom community: within 1 km of the station at Park and Fordham there are 35,338 employed residents and only 22,515 jobs as of 2019; the largest destination is Manhattan (12,734 commuters), followed by the Bronx (7,744), then the rest of the city (8,069 across three boroughs), and only then Westchester (2,207), Long Island (1,660), and Connecticut (220). But to an extent, the station’s shed is larger for reverse-commute service, because people can connect from the Bx12 bus, which ranked second in the city behind the M15 in both 2019 and 2022; in contrast, Manhattan-bound commuters are taking the subway if they live well east or west of the station along Fordham. Nonetheless, the dominance of commutes to city destinations means that the most important service is to the rest of the city.

Indeed, the nearest subway stations have high subway ridership. The city report linked in the lede cites ridership of 11,521 on the B/D and another 12,560 on the 4 every weekday, as of 2012; both stations saw declines by 2019. The West Bronx’s hilly terrain makes these stations imperfect substitutes for each other and for the Metro-North station, despite the overlap in the walk sheds – along Fordham, Park is 600 meters from the Concourse and 950 from Jerome. Nonetheless, “roughly the same as either of the Fordham stations on the subway” should be a first-order estimate for the ridership potential; better Metro-North service would provide a much faster option to East Harlem and Grand Central, but conversely require an awkward transfer to get to points south, which predominate among destinations of workers from the area, who tilt working-class and therefore peak in Midtown South and not in the 50s:

Adding up all the Bronx stations on the line – Wakefield, Woodlawn, Williams Bridge, Botanical Garden, Fordham, Tremont, Melrose – we get 170,049 employed residents (as always, as of 2019), of whom 62,837 work in Manhattan. The line is overall in a subway desert, close to the 4 and B/D but along hills, and not so close to the 2/5 to the east; several tens of thousands of boardings are plausible if service is improved. For comparison, the combination of Westchester, Putnam, and Dutchess Counties has 115,185 Manhattan-bound commuters, split across the Harlem Line, the Hudson Line, and the inner New Haven Line. The Bronx is thus likely to take a majority of Manhattan-bound ridership on the Harlem Line, though not an overwhelming one.

To serve all this latent demand, it is obligatory to run more local service. A minimum service frequency of a train every 10 minutes is required. The current outbound schedule is 20 minutes from Grand Central to Fordham, and about four of those are a slowdown in the Grand Central throat that can be waived (the current speed limit is 10 miles per hour for the last mile; the infrastructure can largely fit trains running three times that fast almost all the way to the bumpers). Lower frequency than this would not really make use of the line’s speed.

Moreover, using the track chart as of 2015 and current (derated) M7 technical performance, the technical trip time is 18 minutes, over which 20 minutes is not too onerously padded; but removing the derating and the four gratuitous minutes crawling into and out of Grand Central, this is about 14 minutes, with some pad. The speed zones can be further increased by using modern values of cant and cant deficiency on curves, but the difference isn’t very large, only 40 seconds, since this is a section with frequent stops. It’s fast, and to reinforce this, even higher frequency may be warranted, a train every 7.5 or 6 or even 5 minutes.

There is room on the tracks for all of this. The issue is that this requires dedicating the local tracks on the Harlem Line in the Bronx to local service, instead of having trains pass the platforms without stopping. This, in turn, requires slowing down some trains from Westchester to make more local stops. Current peak traffic on the Harlem Line is 15 trains per hour, of which 14 run past Mount Vernon West, the current northern limit of the four-track section, and 13 don’t stop in the Bronx at all. The line has three tracks through past Crestwood, and the stations are set up with room for a fourth track, but a full 10 trains per hour, including one that stops in the Bronx, run past Crestwood. In theory it’s possible to run 12 trains per hour to Mount Vernon West making all stops, and 12 trains past it skipping Bronx stops; this slows down the express trains from White Plains, which currently skip seven stops south of White Plains to Mount Vernon West inclusive, but higher speeds in the Bronx, speeding up the Grand Central throat, higher frequency, and lower schedule padding would together lead to improvements in trip times. However, this introduces a new set of problems, for which we need to consider the New Haven Line too.

Harlem Line express service and the New Haven Line

Currently, the New Haven Line runs 20 trains per hour into Grand Central at the peak. This number will go down after Penn Station Access opens in 2027, but not massively; a split of 6-8 trains per hour into Penn Station and 16-18 into Grand Central, with the new service mildly increasing total throughput, is reasonable.

Today, New Haven Line locals stop at Fordham, and nowhere else in the Bronx. This is inherited from the trackage rights agreement between the New York Central and the New York, New Haven and Hartford Railroad, allowing the latter to make only one stop in the Bronx on the former’s territory; it used to be Woodlawn, the branch point, but has been moved to Fordham, which has busier reverse-peak traffic. The two railroads merged in 1969, and all service is currently run by Metro-North, but the practice persists. This is not necessarily stupid: the New Haven locals are long – Stamford is 53 km from Grand Central, 50% farther than White Plains – and a system in which the New Haven trains are more express than the Harlem trains is not by itself stupid, depending on other system constraints. Unfortunately, this setup introduces all manners of constraints into the system:

  1. Fordham is a local-only station, and thus New Haven locals have to use the local tracks, using awkwardly-placed switches to get between the express and the local tracks. In fact, all stations up to and including Woodlawn are local-only; the first station with platforms on the express tracks is Wakefield, just north of the split between the two lines.
  2. If there are 12 Harlem Line trains per hour expressing through the Bronx, then the New Haven Line is limited to about 12 trains per hour as well unless the local trains make all the Bronx local stops.
  3. The Hudson Line has a flat junction with the Harlem Line at Mott Haven Junction, which means that any regular schedule has to have gaps to let Hudson Line trains pass; current peak Hudson Line traffic is 11 trains per hour, but it was 14 before corona.

This leads to a number of different options, each problematic in its own way.

Maximum separation

In this schedule, all Harlem Line trains run local, and all New Haven Line trains run express in the Bronx. This is the easiest to timetable – the junction between the two lines, unlike Mott Haven, is grade-separated. This also requires splitting the Hudson Line between local and express tracks, so delays will still propagate in any situation unless the Hudson Line is moved to the Empire Connection (6-8 trains per hour can stay on the current route); but in a future with Penn Station Access West, building such service, it does allow for neat separation of the routes, and I usually crayon it this way in very high-investment scenarios with multiple through-running tunnels.

But in the near future, it is a massive slowdown for Harlem Line riders who currently have express service from White Plains to Manhattan. The current peak timetable has a 15-minute difference between local and express trains on this section; this figure is padded but not massively so, and conversely, higher speeds on curves increase the express train speed premium.

It also severs the connection between the Bronx and the New Haven Line, unless passengers take east-west buses to the Penn Station Access stations. It is possible to add infill at Wakefield on the New Haven Line: this is north of the junction with the Harlem Line, but barely, so the separation between the lines is short, and a transfer station is feasible. But it wouldn’t be a cross-platform transfer, and so Fordham-Stamford service would still be degraded.

Locals run local

The New Haven Line locals can make local stops in the Bronx. It’s a slowdown of a few minutes for those trains – the current outbound timetable is 18 minutes from Grand Central to Fordham, two minutes faster than on Harlem Line locals while skipping Melrose and Tremont. Overall, it’s a slowdown of around six minutes; the current speed zones are 60 and 75 mph, and while raising the speed limits increases the extent of the slowdown (most of the track geometry is good for 160 km/h), getting new trainsets with better acceleration performance decreases it, and overall it’s likely a wash.

    From there, the service pattern follows. New Haven Line locals to Grand Central have little reason to run more frequently than every 10 minutes at peak – the local stations are 30-60 minutes out of Grand Central today, and this is massively padded, but with the timetables Devin produced, fixing the Grand Central throat, New Rochelle would still be 20 minutes out on a local train (stopping at Fordham only as is the case today) and Stamford would be 45 minutes out. What’s more, there will be some local trains to Penn Station starting in three years, boosting the effective frequency to a train every five minutes, with a choice of Manhattan station if express trains can be made to stop at New Rochelle with a timed connection.

    Now, if there are six local trains per hour in the Bronx going to Stamford, then the Harlem Line locals only take six trains per hour of their own, and then 12 trains should run express from Wakefield to Harlem. What they do to the north depends. The simplest option is to have all of them make all stops, which costs White Plains 7-8 minutes relative to the express stopping pattern. But if the line can be four-tracked to Crestwood, then half the trains can run local to White Plains and half can run nonstop between Wakefield or Mount Vernon West and White Plains. Two local stations, Scarsdale and Hartsdale, are in two-track territory, but timetabling a local to follow an express when both run every 10 minutes and there are only two local stops’ worth of speed difference is not hard.

    The New Haven Line, meanwhile, gets 12 express trains per hour. Those match 12 express Harlem Line trains per hour, and then there’s no more room on the express tracks; Hudson Line trains have to use the local tracks and somehow find slots for the northbound trains to cross both express tracks at-grade.

    Status quo

    The status quo balances Bronx-Stamford connectivity with speed. Bear in mind, the New Haven Line today has truly massive timetable padding, to the point that making trains make all six stops in the Bronx and not just Fordham would still leave New Rochelle locals faster than they are today if the other speedup treatments were put into place. But the status quo would allow New Rochelle, Larchmont, Rye, and Greenwich to take maximum advantage of the speedup, which is good. The problem with it is that it forces New Haven Line locals to take slots from both the express tracks and the local tracks in the Bronx.

    In this situation, the New Haven Line still runs six locals and 12 express trains to Grand Central per hour. The Harlem Line is reduced to six express trains through the Bronx and has to run 12 trains local. Transfers at Wakefield allow people in suburbs south of White Plains to get on a faster train, but this in effect reduces the effective peak-of-peak throughput from the suburbs to Manhattan to just six trains per hour.

    Express Fordham station

    Rebuilding Fordham as an express station means there’s no longer any need to figure out which trains stop there and which don’t: all would. Then the New Haven Line would run express and the Harlem would either run local or run a mix.

    The problem is that Fordham is in a constrained location, where such a rebuild is hard:

    The line is below-grade, with a tunnel from Fordham Road to 189th Street. The platforms are short and narrow, and partly overlap the tunnel. Any conversion has to involve two island platforms north of the tunnel, where there is room but only if the right-of-way is expanded a little, at the expense of some parkland, and possibly a lane of Webster Avenue. The cost would not be pretty, independently of the inability of the MTA to build anything on a reasonable budget.

    That said, the timetables on the Northeast Corridor require some infrastructure intervention to smooth things, like grade-separating some junctions for hundreds of millions of dollars each. New Rochelle, which has only a local platform southbound, should almost certainly be rebuilt as a full express stop. So rebuilding Fordham is not out of the question, even if the cost is high (which it is).

    In this situation, all New Haven Line trains should use the express tracks. Thus, as in the status quo alternative, the Harlem Line gets six express trains, the other trains having to run local. Potentially, there may be a schedule in which the New Haven Line runs 16 trains to Grand Central and eight to Penn Station, and then the Harlem Line can get eight local and eight express trains; but then the local trains have to be carrying the load well into Westchester, and four-tracking the line to Crestwood is likely obligatory.

    The New Haven Line and intercity trains

    The above analysis elides one important factor: intercity trains. The current practice in the United States is a three-way separation of urban rail, suburban commuter rail, and intercity rail, with fares designed to discourage riders from taking trains that are not in their sector. However, in much the same way the best industry practice is to charge mode-neutral fares within cities, it is also valuable to charge mode-neutral fares between them. In other words, it’s useful to look at the impact of permitting people with valid commuter rail tickets to take intercity trains, without seat reservations.

    To be clear, this means that at rush hour, there are going to be standees on busy commuter routes including Stamford-New York and Trenton-New York. But it’s not necessarily bad. The intercity trip time in our timetables between Stamford and New York is around 29 minutes without high-speed bypasses; the standing time would be less than some subway riders endure today – in the morning rush hour the E train departs Jamaica Center full and takes 34 minutes to get to its first Manhattan stop. And then there’s the issue of capacity: commuter trains on the New Haven Line are eight cars long, intercity trains can be straightforwardly expanded to 16 cars by lengthening the platforms at a very small number of stations.

    And if the intercity trains mop up some of the express commuter rail traffic, then the required service on the New Haven Line at rush hour greatly decreases. An intercity train, twice as long as a commuter train (albeit with somewhat fewer seats and less standing space per car), could plausibly displace so much commuter traffic that the peak traffic on the line could be reduced, say to 18 trains per hour from today’s 20. Moreover, the reduction would be disproportionately at longer distance: passengers west of Stamford would not have any replacement intercity train unless they backtracked, but passengers east of Stamford would likely switch. This way, the required New Haven Line traffic shrinks to 12 local trains per hour and six express trains; half the locals run to Penn Station and half to Grand Central, and all express trains run to Grand Central.

    In that situation, we can rerun the scenarios for what to do about Fordham; the situation generally improves, since less commuter traffic is required. The maximum separation scenario finally permits actual separation – the Hudson Line would run on the express tracks into Grand Central and have to cross the southbound Harlem Line locals at Mott Haven Junction, with predictable gaps between trains. The locals-run-local scenario gives Harlem Line express trains more wiggle room to slot between New Haven Line express trains. The status quo option lets the Harlem Line run six local trains and 12 express trains, though that likely underserves the Bronx. Converting Fordham to an express stop straightforwardly works with zero, six, or 12 express Harlem Line trains per hour.

    Or maybe not. It’s fine to assume that letting passengers get on a train that does Stamford-Penn Station in 29 minutes and New Haven-Penn Station in 57 for the price of a commuter pass is going to remove passengers from the express commuter trains and put them onto longer intercity trains. But by the same token, the massive speed improvements to the other stops could lead to an increase in peak demand. The current trip time to Stamford is 1:12 on a local train; cutting that to 45 minutes means so much faster trips to the suburbs in between that ridership could increase to the point of requiring even more service. I’m not convinced on this – the modal split for peak commutes to Manhattan is already very high (Metro-North claimed 80% in the 2000s), and these suburbs are incredibly NIMBY. But it’s worth considering. At the very least, more local service is easier to add to the timetable than more express service – locals to Grand Central don’t share tracks with intercities at all, and even locals to Penn Station only do on a controllable low-speed section in Queens.

    New York-New Haven Trains in an Hour

    Devin Wilkins and I are still working on coming up with a coordinated timetable on the Northeast Corridor, north to south. Devin just shared with me the code she was running on both routes from New Haven to New York – to Grand Central and to Penn Station – and, taking into account the quality of the right-of-way and tunnels but not timetable padding and conservative curve speeds – it looks like intercity trains would do it in about an hour. The current code produces around 57 minutes with 7% timetable pad if I’m getting the Penn Station throat and tunnel slowdowns right – but that’s an if; but at this point, I’m confident about the figure of “about an hour” on the current right-of-way.

    I bring this up to give updates on how the more accurate coding is changing the timetable compared to previous estimates, but also to talk about what this means for future investment priorities.

    First, the curve radii I was assuming in posts I was writing last decade were consistently too optimistic. I wrote three months ago about how even within the highest speed zone in southern Rhode Island, there’s a curve with radius 1,746 meters (1 degree in American parlance), which corresponds to about 215 km/h with aggressive cant and cant deficiency. At this point we’ve found numbers coming straight from Amtrak, Metro-North, and MBTA, letting us cobble together speed zones for the entire system.

    But second, conversely, I was being too conservative with how I was setting speed zones. My principle was that the tightest curve on a section sets the entire speed limit; when writing commuter rail timetables, I would usually have each interstation segment be a uniform speed zone, varying from this practice only when the interstation was atypically long and had long straight sections with a tight curve between them. When writing intercity timetables, I’d simplify by having the typical curves on a line set the speed limit and then have a handful of lower speed limits for tighter curves; for example, most curves on the New Haven Line are 873 meters, permitting 153 km/h with aggressive high-speed rail cant and cant deficiency, and 157 km/h with aggressive limits for slower trains, which can run at slightly higher cant deficiency, but those sections are punctuated by some sharper curves with lower limits. Devin, using better code than me, instead lets a train accelerate to higher speed on straight sections and then decelerate as soon as it needs to. Usually such aggressive driving is not preferred, and is used only when recovering from delays – but the timetable is already padded somewhat, so it might as well be padded relative to the fastest technical speed.

    The upshot of all of this is that the speed gains from just being able to run at the maximum speed permitted by the right-of-way are massive. The trip time today is 1:37 on the fastest trains between New York and New Haven. Commuter trains take 2:10, making all stops from New Haven to Stamford and then running nonstop between Stamford and Manhattan; in our model, with a top speed of 150 km/h, high-performance regional trains like the FLIRT, Talent 3, or Mireo should do the trip in about 1:15-1:20, and while we didn’t model the current rolling stock, my suspicion is that it should be around six minutes longer. The small difference in trip time is partly because Penn Station’s approach is a few kilometers longer than Grand Central’s and the curves in Queens and on the Hell Gate Bridge are tight.

    What this means is that the highest priority should be getting trains down to this speed. In the Swiss electronics-before-concrete schema, the benefits of electronics on the Northeast Corridor are massive; concrete has considerable benefits as well, especially on sections where the current right-of-way constrains not just speed but also reliability and capacity, like New Haven-Kingston, but the benefits of electronics are so large that it’s imperative to make targeted investments to allow for such clean schedules.

    Those investments do include concrete, to be clear. But it’s concrete that aims to make the trains flow more smoothly, in support of a repetitive schedule with few variations in train stopping patterns, so that the trains can be timetabled in advance not to conflict. At this point, I believe that grade-separating the interlocking at New Rochelle, popularly called Shell Interlocking and technically called CP 216, is essential and must be prioritized over anything else between the city limits of New York and New Haven Union Station. Currently, there’s very high peak traffic through the interlocking, with a flat junction between trains to Penn Station and trains to Grand Central.

    On the electronics side, the timetables must become more regular. There are currently 20 peak trains per hour on the New Haven Line into Grand Central; of those, four go to branches and 16 are on the main line, and among the 16, there are 13 different stopping patterns, on top of the intercity trains. It is not possible to timetable so many different trains on a complex system and be sure that everything is conflict-free, and as a result, delays abound, to which the response is to pad the schedules. But since the padded schedules still have conflicts, there is a ratchet of slowdowns and padding, to the point that a delayed train can recover 20 minutes on less than the entire line. Instead, every train should either be a local train to Stamford or an express train beyond Stamford, and there should only be a single express pattern on the inner line, which today is nonstop between Harlem and Stamford and in the future should include a stop at New Rochelle; this means that, not taking intercity trains into account, the main line should have at most four stopping patterns (local vs. express, and Penn Station vs. Grand Central), and probably just three, since express commuter trains should be going to Grand Central and not Penn Station, as passengers from Stamford to Penn Station can just ride intercity trains.

    Also on the electronics side, the way the line is maintained currently is inefficient to an extent measured in orders of magnitude and not factors or percents. Track inspection is manual; Metro-North finally bought a track geometry machine but uses it extremely unproductively, with one report saying it gets one tenth as much work done as intended. Normally these machines can do about a track-mile in an overnight work window, which means the entire four-track line can be regraded and fixed in less than a year of overnights, but they apparently can’t achieve that. Whatever they’re doing isn’t working; the annual spending on track renewal in Connecticut is what Germany spends on once-in-a-generation renewal. The endless renewal work includes a plethora of ever-shifting slow zones, and at no point is the entire system from New York to New Haven clear for trains, even on weekdays. The excessively complex schedule, on tracks that constantly shift due to segment-by-segment daytime repairs, is turning a trip that should be doable on current rolling stock in perhaps 1:23 into one that takes 2:10.

    The billions of dollars in the Bipartisan Infrastructure Law that are dedicated to the Northeast Corridor and have not yet been spent can reduce trip times further. But the baseline should be that the bucket of money is a few hundreds of millions smaller and the base case trip time from New York to New Haven is an hour and not 1:37; this is what the system should be compared with.

    Reverse- and Through-Commute Trends

    I poked around some comparable data for commuting around New York for 2007 and 2019 the other day, using OnTheMap. The motivation is that I’d made two graphics of through-commutes in the region, one in 2017 (see link here, I can’t find the original article anymore) and one this year for the ETA report (see here, go to section 2B). The nicer second graphic was made by Kara Fischer, not by me, but also has about twice the volume of through-commutes, partly due to a switch in source to the more precise OnTheMap, partly due to growth. It’s the issue of growth I’d like to go over in this post.

    In all cases, I’m going to compare data from 2007 and 2019. This is because these years were both business cycle peaks, and this is the best way to compare data from different years. The topline result is that commutes of all kinds are up – the US had economic growth in 2007-19 and New York participated in it – but cross-regional commutes grew much more than commutes to Manhattan. New Jersey especially grew as a residential place, thanks to its faster housing growth, to the point that by 2019, commute volumes from the state to Manhattan matched those of all east-of-Hudson suburbs combined. The analysis counts all jobs, including secondary jobs.

    For the purposes of the tables below, Long Island comprises Nassau and Suffolk Counties, and Metro-North territory comprises Westchester, Putnam, and Duchess Counties and all of Connecticut.

    2007 data

    From\ToManhattanBrooklynQueensBronxStaten IslandLong IslandMetro-NorthNew Jersey
    Manhattan449,30830,71622,02817,7461,97417,57420,28129,031
    Brooklyn385,943299,05676,49916,1219,28840,84717,17525,887
    Queens328,78589,982216,98819,2274,350107,63421,73718,555
    Bronx184,59435,99429,81897,3972,33720,20041,31615,467
    Staten Island59,57230,2417,2232,32649,6797,5143,65517,919
    Long Island163,98845,12177,33712,7245,103926,91232,80612,557
    Metro-North124,95212,60614,22824,1311,96229,3441,897,39215,413
    New Jersey245,37323,45517,49611,0228,10917,46022,0733,523,860

    2019 data

    From\ToManhattanBrooklynQueensBronxStaten IslandLong IslandMetro-NorthNew Jersey
    Manhattan570,32156,01944,06331,9474,00020,67822,14635,243
    Brooklyn486,757429,234119,58826,19217,07343,41018,30133,119
    Queens384,186134,063308,90336,3397,640121,19425,21622,863
    Bronx224,58362,37758,124135,2884,36426,17245,34717,387
    Staten Island59,77840,99413,9715,21856,9539,8773,51019,442
    Long Island191,23959,241102,93923,2468,132971,19340,13014,724
    Metro-North153,48221,28323,49837,1473,17940,5861,874,61820,819
    New Jersey345,55140,39729,52317,46714,13423,43929,7553,614,386

    Growth

    From\ToManhattanBrooklynQueensBronxStaten IslandLong IslandMetro-NorthNew Jersey
    Manhattan26.93%82.38%100.03%80.02%102.63%17.66%9.20%21.40%
    Brooklyn26.12%43.53%56.33%62.47%83.82%6.27%6.56%27.94%
    Queens16.85%48.99%42.36%89.00%75.63%12.60%16.00%23.22%
    Bronx21.66%73.30%94.93%38.90%86.74%29.56%9.76%12.41%
    Staten Island0.35%35.56%93.42%124.33%14.64%31.45%-3.97%8.50%
    Long Island16.62%31.29%33.10%82.69%59.36%4.78%22.33%17.26%
    Metro-North22.83%68.83%65.15%53.94%62.03%38.31%-1.20%35.07%
    New Jersey40.83%72.23%68.74%58.47%74.30%34.24%34.80%2.57%

    Some patterns

    Commutes to Manhattan are up 24.37% over the entire period. This is actually higher than the rise in all commutes in the table combined, because of the weight of intra-suburban commutes (internal to New Jersey, Metro-North territory, or Long Island), which stagnated over this period. However, the rise in all commutes that are not to Manhattan and are also not internal to one of the three suburban zones is much greater, 41.11%.

    This 41.11% growth was uneven over this period. Every group of commuters to the suburbs did worse than this. On net, commutes to New Jersey, Metro-North territory, and Long Island, each excluding internal commutes, grew 21.34%, 15.95%, and 18.62%, all underperforming commutes to Manhattan. Some subgroups did somewhat better – commutes from New Jersey and Metro-North to the rest of suburbia grew healthily (they’re the top four among the cells describing commutes to the suburbs) – but overall, this isn’t really about suburban job growth, which lagged in this period.

    In contrast, commutes to the Outer Boroughs grew at a collective rate of 50.31%. All four intra-borough numbers (five if we include Manhattan) did worse than this; rather, people commuted between Outer Boroughs at skyrocketing rates in this period, and many suburbanites started commuting to the Outer Boroughs too. Among these, the cis-Manhattan commutes – Long Island to Brooklyn and Queens, and Metro-North territory to the Bronx – grew less rapidly (31.29%, 33.1%, 53.94% respectively), while the trans-Manhattan commutes grew very rapidly, New Jersey-Brooklyn growing 72.23%.

    New Jersey had especially high growth rates as an origin. Not counting intra-state commutes, commutes as an origin grew 45% (Long Island: 25.75%; Metro-North territory: 34.75%), due to the relatively high rate of housing construction in the state. By 2019, commutes from New Jersey to Manhattan grew to be about equal in volumes to commutes from the two east-of-Hudson suburban regions combined.

    Overall, trans-Hudson through-commutes – those between New Jersey and anywhere in the table except Manhattan and Staten Island – grew from 179,385 to 249,493, 39% in total, with New Jersey growing much faster as an origin than a destination for such commutes (53.63% vs. 23.93%); through-commutes between the Bronx or Metro-North territory and Brooklyn grew 56.48%, reaching 128,153 people, with Brooklyn growing 72.13% as a destination for such commutes and 33.63% as an origin.

    What this means for commuter rail

    Increasingly, through-running isn’t about unlocking new markets, although I think that better through-service is bound to increase the size of the overall commute volume. Rather, it’s about serving commutes that exist, or at least did on the eve of the pandemic. About half of the through-commutes are to Brooklyn, the Bronx, or Queens; the other half are to the suburbs (largely to New Jersey).

    The comparison must be with reverse-commutes. Those are also traditionally ignored by commuter rail, but Metro-North made a serious effort to accommodate the high-end ones from the city to edge cities including White Plains, Greenwich, and Stamford, where consequently transit commuters outearn drivers in workplace geography. The LIRR, which long ran its Main Line one-way at rush hour to maintain express service on the two-track line, sold the third track project as opening new reverse-commutes. But none of these markets is growing much, and the only cis-Manhattan one that’s large is Queens to Long Island, which has an extremely diffuse job geography. In contrast, the larger and faster-growing through-markets are ignored.

    Short (cis-Manhattan) trips are growing healthily too. They are eclipsed by some through-commutes, but Long Island to Queens and Brooklyn and Metro-North territory to the Bronx all grew very fast, and at least for the first two, the work destinations are fairly clustered near the LIRR (but the Bronx jobs are not at all clustered near Metro-North).

    The fast job growth in all four Outer Boroughs means that it’s better to think of commuter rail as linking the suburbs with the city than just linking the suburbs with Penn Station or Grand Central. There isn’t much suburban job growth, but New Jersey has residential growth (the other two suburban regions don’t), and the city has job growth, with increasing complexity as more job centers emerge outside Manhattan and as people travel between them and not just to Manhattan.

    Commuter Suburbs and Express and Local Trains

    At both TransitMatters and my ongoing Northeast Corridor high-speed rail timetabling project, one question we face is how to mix local and express trains on the same line. I blogged about this years ago, but that was from first principles and this is from a much better position of using Devin Wilkins’ code and analysis of recent research on rail timetabling to evaluate alternatives.

    Most of this post is going to be about the Worcester Line in Boston, which we used as a test case; thus, the following two sections cover how to modernize the line, which will be covered in greater detail in an upcoming TransitMatters report, and you can skip them if you genuinely don’t care about Boston. But much of the analysis generalizes, especially when it relates to the issue of American commuter suburbs and their land use. This land use makes neat express patterns hard to justify in most cases, and the outcome in historic American planning has often been irregular patterns, which in postwar suburban New York led to regularizing around zonal express trains, designed to be usable only by rush hour suburb-to-city commuters and nobody else. Nonetheless, it’s still possible to run coherent timetables that make suburb-to-city commutes convenient while also making other trips viable – it just requires running fewer express trains in most cases.

    The Worcester Line’s current situation

    The Worcester Line connects Boston and Worcester. It is 71 km long and double-track and has 17 stops on the way. There is a planned infill station within Boston at West Station in Allston, and one to four potential infill locations on the way (Newton Corner is the most interesting; the other three are US 20 and the poorly-named Plantation Street in Worcester and Parsons Street in Faneuil). On the way, it passes largely through commuter suburbs of Boston, with one intermediate city, Framingham, station #12 out of Boston, at km-point 34.4, dividing the line into an inner and outer zone. Atypically for a Boston commuter line, seven stations in the inner zone not only don’t have level boarding, but also don’t even have a railcar length’s worth of high platform for wheelchair accessibility (called “mini-highs” in Boston).

    All trains are pulled by diesel locomotives. Currently, off-peak and on weekends, all trains make all stops, running roughly hourly. At rush hour, trains either run local between Framingham and Boston, or express between Worcester and Boston, the latter trains running nonstop between the last station in Boston (Boston Landing, #3) and West Natick (#11) and running local beyond; each of the two patterns ran roughly half-hourly before corona, but currently runs roughly every 45 minutes.

    Exceptionally, reverse-peak and some midday trains do not stop at the Newton stations (#4-6), where not only are the platforms low but also they only serve one track, and so the peak trains use the track with Newton platforms and the reverse-peak trains use the track without; they switch to the usual right-hand running farther west, the line running infrequently enough it can be scheduled. However, a project to build high platforms on both tracks at these stations is currently in design, and all future modernization assumes it will be completed by then; the current pattern is so atypical that what should generalize is the timetable after completion, not the current one.

    Worcester Line modernization and timetabling

    Modernization of the Worcester Line means, at a minimum, high platforms at all stations and electrification. This is the starting point of everything that follows; while the North American rail network has practically no electrification measured by route-length, the electrified share measured by ridership is fairly high (nearly all ridership in metro New York is on electric lines, for one).

    The combination of those two, plus the improvements in reliability that would follow permitting less timetable padding, would make trip times much faster. Where today, locals to Framingham take 58 minutes and expresses to Worcester take 1:26, EMUs would do these trips in 35 and 46 minutes respectively even with infill stations, or maybe slightly more with schedule padding. This would induce higher ridership, requiring higher frequency – not to mention that at stations 15-20 minutes out of Boston, which the Newton stations will be if this is implemented, increasing frequency from a train every 30 minutes to a train every 15 or ideally less would increase ridership in and of itself.

    Then, there’s planning for intercity trains beyond Worcester, to Springfield, which is called East-West Rail in Massachusetts. The plans have gotten some funding, but it isn’t enough, and the current plans are still measured in diesel trains per day and not electric trains per hour. But for future planning, we look at space for faster trains, running even faster than Boston-Worcester express trains. Internally in meetings, Devin has come to calling the three patterns local (current locals), local-express (current expresses, so named because they run local between Framingham and Worcester), and Heart-to-Hub (trains running express between Framingham and Worcester, named after a daily express train that got a lot of love from Worcester boosters but not much ridership). The Heart-to-Hub’s ridership was low and therefore its main use is to speed up Boston-Springfield trips. We express frequency in trains per hour at rush hour in both directions, in the order above: 4/4/0 means the local and local-express patterns run every 15 minutes and there are no Heart-to-Hubs, 4/2/2 means the locals run every 15 minutes and the other two run every 30 minutes, etc.

    To boost frequency to 4/4/0 or 4/2/2, even with very fast EMU acceleration rates, requires additional infrastructure. The options are to rebuild Framingham from an at-grade two-track station to an elevated four-track station, so that locals could terminate while local-express and Heart-to-Hub trains continue west, and to add a third track in Wellesley (current stops #7-9, the overtake done toward the east). An ongoing plan to triple-track both Wellesley and Natick is budgeted at $400 million, including four station rebuilds; even net of the rebuilds, it’s expensive, and being able to build a shorter triple-track section would save a lot. (Another option is a modified 4/0/4, with the locals running all the way to Worcester; close to 100% of the riders from the local outer-zone stations would transfer at Framingham.)

    Anything beyond eight trains per hour requires too much extra infrastructure – at a minimum, both quad-tracking Framingham and triple-tracking Wellesley, and even then the timetable would be fragile. A coherent 4/4/2 pattern would even require an additional passing track around Southborough (stop #14): the issue is that local-express and Heart-to-Hub trains have narrow windows to depart between pairs of local trains, so if there are four local trains per hour and six express trains, then two pairs of express trains have to be closely spaced, forcing an overtake on the outer section even though the speed difference between them is small.

    Thankfully, the Worcester Line can live with eight peak trains per hour indefinitely: it’s a doubling over current frequency, and modernization stands to raise ridership by more than that but largely off-peak, as the modal split at rush hour is healthy.

    Short-turn trains

    Eight trains per hour on the Worcester Line, four running local to Framingham and four running express to Framingham and continuing on to Worcester, is solid. But are there other ways to shove more trains in? This is where the compromises that lead to irregular express patterns become apparent.

    The first possibility is to add short-turn trains: those are trains that run short of the outer terminal. Technically, Framingham locals can be thought of as short-turns, but it’s perhaps better to think of Framingham as the outer terminal of local trains, and then conceive of local trains turning short of that. Newton is a good candidate for short-turns: it is on the dense side for an American suburb, it’s close to Boston, and there’s a place for trains to short-turn off the track via a disused connection to Riverside, where the Green Line D branch terminates. But this still doesn’t work, for a subtle reason that generalizes.

    The generalization is that instead of the three names for future Worcester Line trains, we will have more patterns, so let’s refer to them by letters. Local trains are L; local trains that turn short are M. Express trains are X (local-express) and Y (Heart-to-Hub), but in the most general case, it’s fine to think in terms of just L, M, and X, since on the inner zone, X and Y make the same stops.

    The issue is that X and Y trains still have to fit between L trains or between L and M trains. Under the 4/2/2 option, with Framingham overtakes and no short-turns, outbound departures look roughly as follows:

    :00 X
    :02 L
    :15 Y
    :17 L
    (Repeat every half hour)

    L is allowed to take at most 11 more minutes to get to the overtake point than X/Y, otherwise X/Y have to be slowed down. Under our current assumptions, this is the exact difference. So there’s no space for additional trains unless they turn short, enough that by the time they’d be overtaken mid-line, they’re on the spur to Riverside. This leads to the following principle: if short-turn locals are added to a line with local and express trains, the express train must run behind a short-turn local and not a full-line local. Concretely, where would M short-turns fit? It would look roughly as follows:

    :00 X
    :02 L
    :07 M
    :15 Y
    :17 L
    :22 M
    (Repeat every half hour)

    This shoves more frequency on the line, which is good for residents of the stations served by M, neutral to slightly bad for everyone else, and costly. The issue is that on the section where M trains run, the operating costs are those of eight local trains per hour, but the maximum gap is not 7.5 but 10 minutes. Moreover, the trains arriving after the longer gap are L trains and not M trains. Thus, the L trains would end up considerably more crowded than the M trains – all passengers traveling beyond the short-turn would be on L and so would two-thirds of the passengers traveling on the short-turn section. Ideally, if there are programmed irregular gaps on a line with short-turn trains, the short-turns should arrive after the longer gap and not after the shorter gap; if L departs :02 and M departs :12, then L takes just one-third of the passengers on the shared section and M takes two-thirds, which manages capacity better. But the presence of X and Y makes this impossible, because X and Y have to run behind M and ahead of L and still have a fairly long gap from M to avoid having to overtake.

    Irregular express patterns

    Okay, so short-turn trains are not a good way to add capacity to a line with a mix of local and express trains. This leads to the next step: slowing down some express trains, reducing the speed difference between locals and expresses, and compensating by running more express trains serving different stations.

    On the Worcester Line, this means running trains in the pattern Y-X-L, with X making more stops to slow it down a little. But this then raises the question of which intermediate stops get to run express.

    The New York City Subway standard of having express stops at regular intervals every three to six stations doesn’t work on commuter rail. The subway is designed around four-track trunk lines with cross-platform transfers at express stations, and those work because the trains are very frequent (or, off-peak, used to be). What works for a four-track system that runs local trains every five minutes doesn’t work for a two-track one that runs them every 15. On a commuter rail network, if a station only gets local trains, there is not going to be an opportunity to transfer, not at the scale we’re talking about.

    The simplest answer to who gets the express stations is “the busiest stations.” This is a valid answer. Here is the RER B, between Massy-Palaiseau and Cité-Universitaire, the last station within Paris, where all trains make all stops. Off-peak, the southbound stopping patterns include S trains to Robinson on a branch, K trains making all stops to Massy, and P trains running express to Massy and local beyond, each every 15 minutes; at peak, the S is as before, the K skips one stop, there’s an L pattern skipping stops and running local from Massy to Orsay, and the P runs the fastest and is express to Orsay and only local past that, each every 12 minutes.

    StationRidershipSP (base)K (rush)L (rush)P (rush)
    Cité-U7,531,642*****
    Gentilly2,415,990 ****
    Laplace3,707,718****
    Arcueil-Cachan3,634,116****
    Bagneux2,241,461****
    Bourg-la-Reine4,446,499*****
    Parc de Sceaux574,699*
    La Croix de Berny3,161,602****
    Antony6,304,424****
    Fontaine-Michalon731,635*
    Les Baconnets1,860,108 **
    Massy-Verrières607,314*
    Massy-Palaiseau9,141,486****

    There are no overtake locations on the RER B; because of this, the shared trunk to Bourg-la-Reine has to run local at rush hour. The express patterns still run on a regular clockface schedule every 12 minutes at rush hour and every 15 off-peak, but they aren’t neat, due to the density of traffic. The stations that always get served are the busiest ones – Bourg-la-Reine, La Croix de Berny, Antony, Massy – and are an order of magnitude busier than the stations that get skipped the most. This is not an artifact of service – at the distance of those stations from Paris, a train every 12 or 15 minutes is enough for ridership not to depend too much on frequency. There really is much higher demand at Bourg-la-Reine and Massy than at the minor stations.

    In contrast, here is ridership per station between Boston Landing and Framingham, on weekdays:

    Boston Landing: 479
    Newtonville: 429
    West Newton: 243
    Auburndale: 203
    Wellesley Farms: 285
    Wellesley Hills: 322
    Wellesley Square: 591
    Natick: 697
    West Natick: 914
    Framingham: 995

    So, the first thing to notice is that Framingham, the busiest station, has 995 weekday boardings, which is maybe 290,000 per year, which would be by far the least busy on the RER B (the least busy on the entire RATP-run part of the RER is La Hacquinière, on the outer tail of the RER B, with 419,294/year). But also, the spread is much smaller than on the RER B, a factor of about 5 rather than 16. The spread in potential demand is actually larger, since Auburndale and West Newton are close enough to Boston that the hourly frequencies hurt a lot, whereas Framingham and West Natick both are farther away and get much more rush-hour frequency at any case.

    A line like the Worcester Line should only even be running express trains in the first place to speed up some outer-zone trips, and with the expectation that Framingham could develop to something bigger. But that brings in the issue of land use in American suburbs.

    Land use and commuter rail ridership

    I encourage people to compare land use near American and Parisian commuter rail stations. Here is Bourg-la-Reine:

    And here is Massy:

    These are town centers. Massy is a postwar suburb developed around the train station, with town center development near the station and plenty of later urban renewal as the area got a TGV station. There’s visibly more stuff near Massy or Bourg-la-Reine than near the minor stations on the RER B, in the same way there is more stuff near a major transfer point on most subways than near a station on one line on a tail. They anchor express service.

    The land use near American commuter rail stations works differently. It is lower-density, of course, but more importantly, it is uniformly low-density. Density isn’t especially oriented near the train stations. Occasionally there is walkable retail from the train, but it’s not consistent, and there are no clusters of mid-rise buildings with retail and some local office jobs. American suburbanization of residences may follow the train, with gray near the lines and green between them, but suburbanization of jobs never does, instead following highways.

    In this context, there’s no real distinguishing feature that allows some stations to get more express service than others. Riders get to the station by car; the fares and schedules don’t allow for integration with suburban buses, and there is no reason for anyone in these suburbs to rely on bikes when all local destinations are auto-centric. The car has a fairly long range within suburbia, and thus riders drive to a better park-and-ride or kiss-and-ride; the busiest suburban American train station, Ronkonkoma, with around 10,000 weekday riders pre-corona, is a parking lot with practically nothing else near it.

    What this means is that express trains often generate their own demand, as passengers start driving to them, neglecting other stations. It creates a fiction of lopsided demand with similar ratios between busier and less busy stations as on the RER B, with no underlying reason for it; the Worcester Line has no such ratio, but the LIRR Main Line does, largely due to the park-and-ride effect. Once planners accept that everyone needs an express trains, schedules evolve to be ever more irregular and less reliable, in the search of the perfect express. Caltrain even came up with the push model, in which the scheduler’s job is to push passengers to park-and-rides with open spots, and otherwise there would be no reason to run trains other than the fastest express trains.

    This, in turn, guarantees bad service – these irregular patterns repel riders who are not city-bound commuters, and the frequency is never good enough to sustain such patterns off-peak. In the most extreme cases, it can even backfire: the LIRR’s split between Grand Central and Penn Station frequencies has made it so that so far, the East Side Access project has generated zero new ridership. It’s sometimes possible to salvage something: in the case of Framingham and Worcester, both cities have skeletons of bus networks, and coordinated planning could ensure that the buses would be timed with the train and have free transfers. But at most stations, it’s pointless to try to turn them into distinguished nodes beyond the usual for a local train station.

    The upshot is that the only way to run coherent timetables is to focus on local trains. Express trains are for express stops, and few places in the suburban United States are worth the effort; Framingham and Worcester are two of those few because of their town center development, but nothing else on the line is.

    In that sense, when I harp on the need for high platforms and electrification, it’s not just because these are good practices in and of themselves, but also because they’re necessary for making local service work. Otherwise, the stop penalty kills you: Boston-Worcester is 1:38 on all-local off-peak trains today, which is an average speed of 43 km/h. And nothing except local stretches works in the context of continuous, isotropic suburban density.

    We Gave a Talk About New York Commuter Rail Modernization

    Blair Lorenzo and I gave the talk yesterday, as advertised. The slide deck was much more in her style than in mine – more pictures, fewer words – so it may not be exactly clear what we said.

    Beyond the written report itself (now up in web form, not just a PDF), we talked about some low-hanging fruit. What we’re asking for is not a lot of money – the total capital cost of electrification and high platforms everywhere and the surface bottlenecks we talk about like Hunter Junction is around $6 billion, of which $800 million for Portal Bridge need to happen regardless of anything else; Penn Reconstruction is $7 billion and the eminently cancelable Penn Expansion is $17 billion. However, it is a lot of coordination, of different agencies, of capital and operations, and so on. So it’s useful to talk about how to, in a way, fail gracefully – that is, how to propose something that, if it’s reduced to a pilot program, will still be useful.

    The absolute wrong thing to do in a pilot program situation is to just do small things all over, like adding a few midday trains. That would achieve little. There is already alternation between hourly and half-hourly commuter trains in most of the New York region; this doesn’t do much when the subway or a subway + suburban bus combination runs every 10-12 minutes (and should be running every six). The same can be said for CityTicket, which incrementally reduces fares on commuter rail within New York City but doesn’t integrate fares with the subway and therefore produces little ridership increase.

    Instead, the right thing to do is focus on one strong corridor. We propose this for phase 1, turning New Brunswick-Stamford or New Brunswick-New Rochelle into a through-line running every 10 minutes all day, as soon as Penn Station Access opens. But there are other alternatives that I think fall into the low-hanging fruit category.

    One is the junction fixes, like Hunter as mentioned above (estimated at $300 million), or similar-complexity Shell in New Rochelle, which is most likely necessary for any decent intercity rail upgrade on the Northeast Corridor. It costs money, but not a lot of it by the standards of what’s being funded through federal grants, including BIL money for the Northeast Corridor, which is relevant to both Hunter and Shell.

    The other is Queens bus redesign. I hope that as our program at Marron grows, we’ll be able to work on a Queens bus redesign that assumes that it’s possible to connect to the LIRR with fare integration and high frequency; buses would not need to all divert to Flushing or Jamaica, but could run straight north-south, leaving the east-west Manhattan-bound traffic to faster, more efficient trains.

    I’m Giving a Talk in New York About Commuter Rail

    At the Effective Transit Alliance, we’re about to unveil a report explaining how to modernize New York’s commuter rail system (update 10-31: see link to PDF here). The individual elements should not surprise regular readers of this blog, but we go into more detail about things I haven’t written before about peakiness, and combine everything together to propose some early action items.

    To that effect, we will present this in person on Wednesday November 1st, at 1 pm. The event will take place at Marron, in Room 1201 of 370 Jay Street; due to NYU access control, signing up is mandatory using this form, but it can be done anytime until the morning of (or even later, but security will be grumpy). At the minimum, Blair Lorenzo and I will talk about commuter rail and what to do to improve it and take questions from the audience; we intend to be there for two hours, but people can break afterward and still talk, potentially.

    Setting Speed Zones

    At the Boston meetup two days ago, I was asked about what tools I use to generate timetables, for example for my New York commuter rail posts. The answer is that I use speed zones and then run this code on them – but then the question is how to figure out speed zones. I hope that this sequence of steps will help advocates who are interested in rail modernization.

    Generating curve radii

    The most difficult element to fix on mainline rail is the right-of-way geometry. Most other things that can restrict a train’s speed can be fixed with more modern maintenance, but right-of-way geometry doesn’t change without physical construction, often in constrained areas – if they weren’t constrained, the curves would have been built wider in the first place.

    The best case scenario is that there exist track maps with exact curve radii. I have these for large chunks of the Northeast Corridor, but not all. For example, here is Metro-North (with thanks to The Korot). Curves on such maps are denoted as circles or bumps deviating from a line, with the direction of the circle indicating the direction of the curve’s curvature. On this and other American maps, the radius is listed in degrees, and the cant (see the section below) in inches.

    To convert the radius from degrees to more usual units, set one degree to be 1,746 meters, and note that degrees measure curvature and not radius, so a two degree curve has half the radius of a one degree curve. More precisely, the formula is that degrees measure the change in azimuth over 100 feet; 100 feet are 30.48 meters, and converting 30.48 from degrees to radians gives 1,746.37536… meters.

    For example, on the Metro-North chart, let’s look at Harrison, New York. It’s on PDF-p. 24 of the chart; Harrison is sandwiched between two curves with opposite orientations, with the platforms on tangent (uncurved) track. The curve just west of Harrison has radius 1° 58′ 30″, which is 884.24069… meters; the curve just east has radius 2° 2′ 15″, which is 857.11674… meters.

    Converting curve radii to speeds

    The formula for the speed of a train, in SI units, is

    \mbox{speed}^{2} = \mbox{radius} \times \mbox{lateral acceleration}

    I wrote about lateral acceleration, cant, and cant deficiency two and a half years ago. In short, lateral acceleration, in m/s^2, is the centrifugal force coming from the action of the train rounding the curve at speed. For the purposes of the formula, it is measured in the horizontal plane. To reduce the centrifugal force felt by the passengers (for comfort and safety) as well as that felt by the train body (for safety and maintenance costs), the tracks will typically be banked so that the inner rail is lower than the outer rail, which is called cant or superelevation, and is written in units of distance, such as mm or inches.

    The speed of a train on canted track is typically higher than the perfect balancing speed, where the force of gravity counteracts that of centrifugal force; thus, on a fast train there is a residual force pointing to the outside of the train, which can be written down as lateral acceleration in the plane of the tracks (in m/s^2), but is more typically written down in the same units as superelevation, representing the additional superelevation required for the speed to perfectly balance, which is called cant deficiency or underbalance.

    The conversion rate between cant (or cant deficiency) is the track gauge measured between the middle of the two rails, divided by the gravitational constant (9.8 m/s^2). Track gauge is typically given as inner rail to inner rail; standard gauge is 1,435 mm inner rail to inner rail. The relevant quantity to superelevation calculations is a few cm more; on standard gauge, it’s taken to be about 1,470-1,500 mm, so the conversion rate is 1 m/s^2 = 150 mm of cant or cant deficiency. The two quantities, cant and cant deficiency, are additive.

    The American track charts that I have specify the actual cant. However, the values tend to be too conservative. Again with the example of Harrison, the slightly wider western curve has 5″ cant and the slightly tighter eastern curve has 4.125″ cant. Regulations for maximum cant depend on the country and maintenance standards. The absolute maximum cant I am aware of on any standard-gauge railway is 200 mm on the Tokaido and Tohoku Shinkansen. The reasons not to raise cant further include maintenance difficulties and the risk of a train running at lower speed or even stopping on the track. On lines that are not captive to just high-speed trains, the highest cant I am aware of is 180 mm, in Germany, and this is rare; 160 mm is more common. The American limit is 7″, but frequent inspections are required at that point to ensure that the tracks don’t get bent out of shape to produce higher cant; 8″ is a do-not-exceed level, and in practice track irregularities may lead to exceeding it if there isn’t regular track maintenance.

    In practice, raising the cant is usually easy – it can be done with a track geometry machine automatically. However, in one case, it is not: that of S-curves, which have exactly the shape implied by the letter S. Our example of Harrison has two reverse curves in close proximity, but is not an S-curve, as there are hundreds of meters of tangent track between the two curves. Other places do have S-curves, and there, the maximum cant must be lowered somewhat; regulations vary on this, but in Europe, the maximum change in cant is 30-55 mm per second depending on the country (there’s a secondary regulation on mm per meter, but on the Northeast Corridor, the binding rule is mm/s, not mm/m). For example, if we take 45 mm/s, and 180 mm of cant, then it takes four seconds to reverse a curve; note that it is four and not eight, because half of the increase in cant, called a superelevation spiral, is within the curve. Harrison’s current curves impose a hard limit of about 150 km/h, at which point the hundreds of meters of tangent track make it trivially easy to have full superelevation. However, other places, most infamously among Northeastern railfans Elizabeth, there is an actual S-curve, forcing lower cant and lower speeds.

    Finally, the maximum cant deficiency depends on the track, the train, and the regulations. Traditionally, American regulations limited most passenger trains to 3″ of cant deficiency, which is ridiculously conservative; in the 2000s, a waiver allowing 5″ in some cases was derided as the “magic high-speed rail waiver” because it applied not just to higher-speed tracks but also to lower-speed lines that had through-service to higher-speed tracks. Since then, FRA regulations have changed, and now the practical limit in the US, with extensive testing, is 6″, or 150 mm cant deficiency, at most speeds. European limits tend to be around 130-150 mm; high-speed trains are at the lower end of this range unless they are tilting trains, which nearly all trains are not. Cant deficiency, like cant, requires its own superelevation spiral on S-curves, but the limits are in practice looser than for cant, and in some cases trains can change superelevation abruptly, with no spiral, for example on switches.

    The upshot is that in the typical case, the most aggressive assumption should be 180 mm cant, 150 mm cant deficiency, for a lateral acceleration in the horizontal plane of 2.2 m/s^2. Most lines will not have this pair of aggressive assumptions: 180 mm is only viable when it’s guaranteed that trains will not stop on a canted curve, which is a reasonable assumption on a reliable high-speed line and even on a German high-speed line. Moreover, if there is any freight on the line, superelevation must fall drastically: slower trains would be at cant excess, and freight trains have high center of mass (diesel locomotives and double-stacked containers both have higher center of mass than electric passenger trains) and therefore have tight cant excess limits. Aggressive assumptions are viable on the Northeast Corridor and on controllable commuter lines with no or almost no freight, such as the LIRR, but not everywhere else.

    Finding curve radii

    In some cases, curve radii are spelled out in a chart. In others, they are not, and must be figured out. Our program’s schedule writer, Devin Wilkins, tells me she has just found a track chart for SEPTA, but otherwise, I have no such charts south of New York. There, the dirty, imperfect method of estimating curve radii must be used.

    For that, I use Google Earth. Nowadays, Google Earth Pro is free, and comes with a circle tool. In theory, I can fidget with the radius of a circle until I find that it approximates the arc of a curve well. This requires paying special attention to how the drawn curve compares with not just the broad outline of the curve but also the exact arc of each track or even each rail: the drawn curve should be at the same relative position to the rails, such as following one rail of one track, or right in the center of one track, or right between the track centers, and so on.

    In cases of uncertainty, it’s also possible to use Google Earth line tools, which state the azimuth of each line. If I can find the exact start and end points of each curve, and the azimuths of the tangents on both side, then I can draw the chord with a line tool, verifying that its azimuth is the exact arithmetic mean of the azimuths of the two tangents; if it is not the mean, then either I made an error (more likely) or the curve is not a perfect circle (possible but less likely). The radius of the curve is approximately the length of the chord times 180/pi divided by change in azimuth; more precisely, the radius is

    \mbox{chord length}/(2 \times \arcsin(\mbox{difference in azimuth})/2).

    This method is error-prone, especially for short, sharp curves. Computing the start and end points of the curve will always have errors, and if the change in azimuth is small, then these will lead to large errors. The circle tool method suffers from the same drawback: it’s easier to use it to estimate the radius of a curve with 60 degree change in azimuth than that of one with 10 degree change.

    Over time I’ve gotten this method down to the point that my errors from what I later find with track charts such as that of Metro-North are fairly small, and not very biased in the larger-radius direction. But it takes time and practice and ideally you should avoid it for short, sharp curves.

    Update 10-28: Ari Ofsevit has a third method, using chord lines.

    Other speed limits

    Speed limits on intercity trains mostly come from curves. But there are other things to keep in mind, not all of which are fixable:

    • Tunnels increase air resistance to the point that unless the tunnels are constructed with large enough radius to have a lot of free air (modern tunnels are, legacy ones aren’t), or unless the trains are pressurized, the speed limit has to be lower just to avoid popping passengers’ ears at entry and exit.
    • Switches generally have low speed limits – they have tight curves and no superelevation – making all complex junctions and major stations slow.
    • Terminal stations have another set of speed limits coming from the bumper tracks. American limits are very conservative – 10 mph where a ramp down from 40 or 50 km/h is more normal in Europe – but even 50 km/h is not 200 km/h.
    • Some pieces of infrastructure are so shoddy that they limit the dynamical axle load of the train, which is derived from both static axle load, which is a function of train mass, and speed. As usual, American limits on this are conservative, assuming high static axle load and fixing a low speed on some very old bridges, instead of permitting lighter trains to run faster. But it is sometimes a real problem.