I have annoying commenters. They nitpick what I say and point out errors in my thinking – or if there are no errors, they take it beyond where I thought it could be taken and find new ways of looking at it. After I wrote about frequency relative to trip length last week, Colin Parker pointed out on the Fediverse that this can be simplified into thinking about frequency not in units of time (trains or buses per hour), but in units of distance (trains or buses per km of route). This post is dedicated to developing this idea on various kinds of transit service, including buses and trains.

The key unit throughout, as Colin points out, is the number of buses available per route, the assumption being that the average trip length is proportional to the average route length. However, this is not a perfect assumption, because then the introduction of network effects changes things – generally in the direction of shorter average trip length, as passengers are likelier to transfer, in turn forcing agencies to run more vehicles on a given route to remain useful. Conversely, timed transfers permit running fewer vehicles, or by the same token more routes with the same resources – but the network had better have a strong node to connect to after a series of vehicle changes, more like the Swiss rail network than like a small American city’s bus network.

Frequency and resources

On a bus network with even frequency across all routes, the following formula governs frequency, as I discussed six years ago:

Daily service hours * average speed per hour = daily trips * network length

When Eric and I proposed our Brooklyn bus redesign, we were working with a service-hour budget of about 10,800 per weekday; status quo as of 2017-8 was 11 km/h, 550 km, and thus 216 daily trips (108 per direction), averaging around a bus per 11 minutes during the daytime, while we were proposing speed up treatments and a redesign to change these figures to 15 km/h, 355 km, and thus 456 daily trips (228 per direction). The six-minute service ideal over 16 hours requires 188 trips per direction; the difference between 188 and 228 is due to higher frequencies on the busiest routes, which need the capacity.

To express this in units of length, we essentially eliminate time from the above dimensional analysis. Daily service hours is a dimensionless quantity: 10,800 hours per weekday means 450 buses circulating at a given time on average, in practice about 570 during the daytime but not many more than 100 buses circulating overnight. If there are 570 buses circulating at a given time, then a 550 km network will average a bus every 1.9 km and a 355 km one will average a bus every 1.25 km. With pre-corona New York bus trips averaging 3.4 km unlinked, a bus every 1.9 km means the maximum headway is a little higher than half the trip time, and a bus every 1.25 km means the maximum headway is a little higher than one third the trip time, independently of speed.

This calculation already illustrates one consequence of looking at frequency in units of distance and not time: your city probably needs to aggressively prune its bus network to limit the wait times relative to overall trip times.

Route length and trip length

On an isolated bus or train route, serving an idealized geography with a destination at its center and isotropic origins along the line, the average trip length is exactly one quarter of the route length. The frequency of service in units of distance should therefore be one eighth of the route length, requiring 16 vehicles to run service plus spares and turnarounds. This is around 18-20 vehicles in isolation, though bear in mind, the 10,800 service hours/day figure for Brooklyn buses above is only for revenue service, and thus already incorporates the margin for turnaround times and deadheads.

Colin points out that where he lives, in Atlanta, bus routes usually have around four vehicles circulating per route at a given time, rather than 16. With the above assumptions, this means that the average wait is twice the average trip time, which goes a long way to explaining why Atlanta’s bus service quality is so poor.

But then, different assumptions of how people travel can reduce the number 16:

1. If destinations are isotropic, then the average trip length rises from one quarter of the route length to 3/8 of the route length, and then the frequency should be 1.5/8 of the route length, which requires 11 vehicles in revenue service.
2. If origins are not isotropic, then the average trip length can rise or fall, depending on whether they are likelier to be farther out or closer in. A natural density gradient means origins are disproportionately closer-in, but then in a city with a natural density gradient and only four buses to spare per route, the route is likely to be cut well short of the end of the built-up area. If the end of the route is chosen to be a high-density anchor, then the origins relative to the route itself may be disproportionately farther out. In the limiting case, in which the average trip is half the route length, only eight buses are needed to circulate.

To be clear, this is for a two-tailed route; a one-tailed route, connecting city center at one end to outlying areas at the other, needs half the bus service, but then a city needs twice as many such routes for its network.

The impact of transfers

Transfers can either reduce the required amount of service for it to be worth running or increase it, depending on type. The general rule is that untimed transfers occurring at many points along the line reduce the average unlinked trip and therefore force the city to run more service, while timed transfers occurring at a central node lengthen the effective trip relative to the wait time and therefore permit the city to run less service. In practice, this describes both how existing bus practices work in North America, and even why the Swiss rail network is so enamored with timed connections.

To the point of untimed transfers, their benefit is that there can be very many of them. On an idealized grid – let’s call it Toronto, or maybe Vancouver, or maybe Chicago – every grid corner is a transfer point between an east-west and a north-south route, and passengers can get from anywhere to anywhere. But then they have to wait multiple times; in transit usage statistics, this is seen in low average unlinked trip lengths. New York, as mentioned above, averages 3.4 km bus trips, with a network heavily based on bus-subway transfers; Chicago averages a not much higher 3.9 km. This can sort of work for New York with its okay if not great relative frequency, and I think also for Chicago; Vancouver proper (not so much its suburbs) and Toronto have especially strong all-day frequencies. But weaker transit networks can’t do this – the transfers can still exist but are too onerous. For example, Los Angeles has about the same total bus resources as Chicago but has to spread them across a much larger network, with longer average trip times to boot, and is not meaningfully competitive. The untimed grid, then, is a good feature for transit cities, which have the resources and demand to support the required frequencies.

Not for nothing, rapid transit networks love untimed transfers, and often actively prefer to spread them across multiple stations, to avoid overwhelming the transfer corridors. Subways are only built on routes that are strong enough to have many vehicles circulating, to the point that all but the shortest trips have low ratios of wait to in-vehicle times. They are also usually radial, aiming to get passengers to connect between any pair of stations with just one transfer; Berlin, Paris, and New York are among the main exceptions. These features make untimed transfers tolerable, in ways they aren’t on weaker systems; not for nothing, a city with enough resources for a 100 km bus network and nothing else does not mimic a 100 km subway network.

Timed transfers have the opposite effect as untimed transfers. By definition, a timed transfer means the wait is designed to be very short, ideally zero. At this point, the unlinked trip length ceases to be meaningful – the quantity that should be compared with frequency is the entire trip with all timed transfers included. In particular, lower frequencies may be justifiable, because passengers travel to much more than just the single bus or rail route.

This can be seen in small-city American bus networks, or some night bus networks, albeit not with good quality. It can be seen much more so on transfer-based rail networks like Switzerland’s. The idealized timed transfer network comprises many routes all converging on one node where they are timed to arrive and depart simultaneously, with very short transfers; this is called a knot in German transit planning and a pulse in American transit planning. American networks like this typically run a single bus circulating on each one-tailed route; the average wait works out to be four times longer than the average unlinked trip, and still twice as long as a transfer trip, which helps explain why ridership on such networks is a rounding error, and this system is only used for last-resort transportation in small cities where transit is little more than a soup kitchen or on night bus networks that are hardly more ridden. It would be better to redo such networks, pruning weaker routes to run more service on stronger ones, at least two per one-tailed route and ideally more.

But then the Swiss rail network is very effective, even though it’s based on a similar principle: there’s no way to fill more frequent trains than one every hour to many outlying towns, and even what are midsize cities by Swiss standards can’t support more than a train every half hour, so that many routes have a service offer of two to four vehicles circulating at a given time. However, on this network, the timed transfers are more complex than the idealized pulse – there are many knots with pulses, and they work to connect people to much bigger destinations than could be done with sporadic one-seat rides. A succession of timed connections can get one from a small town in eastern Switzerland to St. Gallen, then Zurich, then Basel, stretching the effective trip to hours, and making the hourly base frequency relatively tolerable. The key feature is that the timed transfers work because while individual links are weak enough to need them, there are some major nodes that they can connect to, often far away from the towns that make the most use of the knot system.

This year, there have been some positive signs about things changing in New York on subway construction – and yet, I’m uncertain about them. There are some signs that construction costs for Second Avenue Subway Phase 2 are coming under control. The New York Post broke in January that the MTA is eying smaller station designs, to reduce costs, to the tun of $300 million; an article released a few hours ago adds that there may be another $600 million in potential savings. So, in theory, costs are going down, and they’re going down as the MTA implements something we’ve been screaming about at the Transit Costs Project, so we should be happy.

And yet, I’m uncertain – not negative, but still somewhat pessimistic about whether this portends an era in which New York can finally build more subways. The main reason isn’t even some mistrust in the MTA at this point – the reduction in station footprints is a genuinely good thing, and to the extent it’s incomplete, it’s because it’s a longstanding project with older designs. Rather, it’s a combination of what this means for future projects, and how it interacts with federal funding. In brief, federal funding is at the level of the project rather than agency, and this makes it hard for cost savings to be plugged into the most straightforward benefit – namely, being able to build more on the same budget.

How is the money being saved?

The New York Post is relying on an MTA presentation from January that defends the cost structure but talks about how to reduce station costs through reducing back-of-house space. Phase 1 of Second Avenue Subway built two deep-mined intermediate stations, at 72nd and 86th Streets; the platforms are 610′ (187 meters) long, and there are no serious prospects of ever running longer trains since the line is an extension of older lines, but the station caverns are, respectively, 398 and 295 meters long, where the norm in the European comparison cases we’ve seen is that the station dig is 3-15% longer than the platforms, not twice as long.

Both stations have extensive back-of-house space, which New York City Transit demanded so that each department using the station would have its own space; 72nd also has a crossover inherited from older designs that would have permitted some trains from the south to terminate there on a third station track, which was later removed from scope to reduce costs. (The terminal station, the cut-and-cover 96th Street, is a 485 meter long dig, but that’s an artifact of block-level geology: the northern end had to go as far as 99th in order to connect with an older tunnel built in the 1970s, whereas the southern end had to go as far as 92nd because the underground geology changes abruptly there and it was easier to start boring at 92nd than at 96th.)

The plan for Phase 2 initially included much longer digs than the platform, for the same reason. However, it has since changed, and now the digs are substantially reduced. The MTA’s presentation looks like the overage at 125th Street is reduced from somewhat more than 100% in the 2004 plan to about 40-50% in the 2023 plan, and the overage at 106th and 116th is somewhat less than that, maybe 30-40%. While the 125th Street station dig is still as deep as in prior plans, the deep-mined station will also extend less far up, reducing the extent of space required to be outfit with systems.

The MTA could shrink the stations’ footprint further and save more money, but it’s fairly late in the design, and thus the opportunity to take full benefit of this improvement is for future projects. If this establishes precedent for future station construction, then it’s an unmixed blessing.

Money is saved. So what?

The broader issue is that the savings from shrinking the stations’ footprint – totaling potentially $1 billion out of a budget of $7 billion – don’t have much to go. The rub is that the project already has a Full Funding Grant Agreement. If the MTA manages to do it for less, then the most obvious, and most pessimistic, answer to where the money goes is “preventing future overruns.” The savings, in the worst case, then transfer waste from one basket, namely oversize stations, to other baskets, which could be future conflict with contractors, last-minute design changes, or betterments to the neighborhood.

That said, there are plans to spend it on something useful. But the problem is that this is limited by the scope of the project itself. Second Avenue Subway in its current iteration dates to the 1990s, and is reusing some infrastructure from the 1970s. The intention in the 1990s was to do the entire thing, or at least Phases 1 and 2, together, and the project was only chopped into four phases due to high costs. There was design work done 20 years ago or more, and the environmental impact statement is roughly that old.

I suspect the reason the cost saving from shrinking the stations is $1 billion and not much more – we estimated that Phase 1’s cost would have been halved if the stations had been only as long as the platforms – is that the designs are already spoken for, with 20 years of optimization involved. Thus the change is reducing the hard costs of construction, but not the soft costs. It will not surprise me if a postmortem will reveal an elevated ratio of soft to hard costs, purely because the cost savings are happening at such a late stage; in this case, and only in this case, it is important to forgive a high ratio of soft to hard costs, since it portends that future designs will be cheaper, and future cost savings larger. Normally, a high soft-to-hard cost ratio suggests red tape and waste involving consultants, but in this one case, it would suggest something else; I highlight this so that watchdogs for government waste, including the New York Post, realize what is going on and avoid hitting the project if it turns out to indeed have a high soft-to-hard cost ratio as I expect.

Current plans include potentially continuing the tunnel boring under 125th Street. Governor Hochul expressed some interest in a subway extension under 125th Street, extending Phase 2 from 125th/Lexington to the Hudson, with stops at the intersections with the subway lines, at Lenox (2/3), St. Nicholas (A/B/C/D), and Broadway (1 and also potential commuter rail). Such an extension was long on the wishlist of New York-area railfans, and an operations planner mentioned it to me as a future desire more than 10 years ago, unprompted. But there is no way to just reallocate $1 billion to this line; that’s not how federal funding works. At best, it will be possible to continue boring the tunnel all the way to the west, and leave the systems and stations to a future project.

My pessimism is that the cost figure given for the 125th Street extension is $7.6 billion, around $3.3 billion per kilometer. Even taking into account future inflation, it’s costlier than Second Avenue Subway Phases 1 and 2. Now, this is an early number, one that hasn’t really made it into any plans. I hope that the current cost savings are then plugged into the 125th Street extension plan, and that this project is pursued seriously at a much lower cost figure; since all stations would have to be complex digs underneath older north-south subway lines, the benefits of shrinking station footprint are especially large. But I worry that this will not happen; I’ve had hopes dashed before – for example, FRA reform did not lead American commuter rail agencies to start buying alt-compliant vehicles. We’ll see what happens if there’s more detailed work on the 125th Street extension proposal.

Funding projects vs. funding agencies

The current way federal funding works for public transportation in the United States is that the government funds specific capital projects. The MTA can ask for funding for big-ticket items like Second Avenue Subway Phase 1, Phase 2, a future 125th Street subway, the Interborough Express, or any similar such line. It can also ask for a rolling program of improvements, for example installing elevators at stations to make them accessible in line with the Americans with Disabilities Act. But the federal government does not make it easy to move money between such projects.

Phase 2 is just one project, but imagine that there are three subway lines funded concurrently: Phase 2, a subway extension under Nostrand, and a subway extension under Utica. If the MTA finds 25% cost savings, it can’t easily flex the money to a fourth line, say under 125th. It would need to start the planning process early, which is so cumbersome and expensive that it wouldn’t do so for a project it wasn’t certain it wanted to do; there is no shelf of subway extensions that are approved and are just waiting for money.

This makes the incentives for cost savings uncertain: cost savings could be used to establish the agency’s bona fides with a distrustful public, or to establish a warchest guarding against future cost overruns, or to trial new ways of working that could lead to bigger cost savings in the future, but the most straightforward benefit of cost savings – building more infrastructure on the same budget – is not generally available. For Phase 2, the best that can happen is, again, continuing boring the tunnel to the western end of 125th Street, which could be connected with the current project because the geology under the street is the same from Second Avenue to Broadway so might as well future-proof it.

And unfortunately, in the United States, the current examples of funding agencies rather than projects have been lacking. The programs in Los Angeles and Seattle, funded by sales taxes, in effect fund the agencies. There are long lists of projects in both metro areas that are funded from them, but they come from the same pool of money in each region. The situation in Los Angeles is that there’s a decided priority list, with money allocated through the end of the 2050s; if Los Angeles figures out how to cut costs significantly, all of the opening dates will be moved closer and additional lines could be planned and built with the money (the planning and environmental review process takes years, not decades, so by 2050 they will have reviews for the additional subways they could build).

And yet, the same process that’s produced lush agency-level funding in both regions has also led to bad prioritization. New York may have the world’s highest construction costs, but at least what it’s building is what it should be building: Second Avenue Subway Phase 2 is indeed the highest priority right now, and among the next priorities, the Interborough Express and 125th Street are solid choices, according to most area rail advocates two of the top five, and potentially even the top two (the other three are Nostrand, Utica, and an elevated extension of the Astoria Line to LaGuardia). In contrast, Los Angeles is prioritizing the wrong projects. The same ballot proposition process there that produces agency-level funding also requires the agency to bribe local actors who care little for public transportation or for ideological politics with lines to their own subregions of the county, not because they or their constituents will ride the trains, but because they will be able to tell their constituents “I managed to force the county to give us infrastructure money.” This way, each region of the county gets a light rail extension, no matter how lightly-ridden, while the core of the system receives little investment: while the busiest bus corridor in Los Angeles, Wilshire, is getting a subway in the D (Purple) Line Extension, the next two busiest, Vermont and Western, are not getting any rail through the 2050s, despite calls from advocates to built a line on Vermont to turn the B (Red) Line into a north-south line rather than a branch with the Purple Line.

Incentives for the future

The MTA is clearly capable of saving money. The question is now how to incentivize doing more of it. First of all, I urge New York-area advocates to pursue the 125th Street extension, and demand that the cost savings identified for Phase 2 apply to it too. The savings may even potentially be relevant to the Interborough Express, though with at-grade and above-ground stations, the impact is greatly reduced. The Phase 2 savings are reactive; applying them to future lines is proactive.

Second, I urge both the MTA and advocates to look for cost savings in areas where it is easier to flex money – namely, ADA accessibility. Being able to make a station accessible not for the current budget of about $70 million per station with contingency but the $25 million of Boston or $10 million of Madrid would enable New York to have an all-accessible subway system not in the 2050s but in the early 2030s.

Finally, at the federal level, it is useful to figure out how to fund agencies with a positive track record and not just specific projects. Potentially, agencies could be encouraged to submit wishlists of future projects that may be cleared in case money becomes available on short notice; this is useful not just in the case of cost savings, but also in the case of an unexpected infrastructure stimulus – neither the scope of the 2009 stimulus in the early Obama era nor that of 2021 under Biden had been telegraphed until shortly before, and so agencies have not always been able to take maximum advantage of the additional funds.