The NTSB Wants American Trains to Be Less Safe

In 2017, an Amtrak Cascades train derailed outside Seattle. The train driver sped on a curve and the heavy locomotive derailed, dragging the trains with it, as had happened in 2013 in New York and in 2015 in Philadelphia. The primary culprit was the tardy installation of automatic train protection (“positive train control”), which would have prevented overspeed: the Philadelphia accident happened shortly before that section of track was scheduled to get PTC, the New York accident happened on a line with weaker protection against running red signals but not against overspeed, and the Seattle accident happened on a line not-yet equipped with PTC but with ongoing installation.

Despite the similarity, the National Transportation Safety Board’s recommendations, just released after a year and a half’s investigation, are different: it demands that Amtrak withdraw the passenger cars used. Those cars are tilting trains manufactured by Talgo, running on a waiver from federal regulations that have since changed. They are not what led to the derailment, and evidently the crash was less deadly than the New York crash (which killed 4) as well as the Philadelphia one (which killed 8). And yet, the NTSB wants these coaches removed where it said no such thing about the coaches used at previous accidents.

Rolling stock that is designed to avoid derailments at curves should focus on getting its center of gravity under control. Diesel locomotives have high centers of gravity, limiting their speed on curves. The standard measure for how fast a train can go on curves is called cant deficiency, and de facto has the conversion rate of 150 mm = 1 m/s^2 of lateral acceleration in the plane of the tracks (which may be banked, so the acceleration in the horizontal plane is larger). Top-of-the-line tilting trains in Europe can safely take curves at 300 mm cant deficiency, at which point the limiting factor isn’t passenger safety, but the complexity of maintaining the equipment’s tilt system. The Amtrak Talgos are less powerful – their tilt system allows only about 180 mm, but their center of gravity is so low that they could go much faster without derailing, just with a lot of passenger discomfort over the centrifugal force.

In contrast, the heavy diesel locomotive is limited to 6″ of cant deficiency, as its center of gravity is so high that going much faster could risk derailment, as indeed happened. Electric trains have no such problem – the Acelas do 7″ every day, and could have done 9″, but as with the Talgos the limit is passenger comfort and not train toppling. In fact, some non-tilting trains in Europe run about as fast on curves as the Amtrak Cascades service does, and FRA regulations from 2010 even suggest that modern non-tilting trains could do about 6″ of cant deficiency.

I bring all of this up to explain that the Talgos on their own, with a typical European locomotive, would not have derailed. Moreover, after the derailment, they stayed upright, unlike the Amtrak coaches in Philadelphia or the Metro-North ones in New York. The reason people died is that the train fell from a bridge. As far as factors that are controllable by the coach builder go, the Talgos performed well.

What’s more, the regulators who compared European and American designs for rolling stock have come to the same conclusion. This underlay the tests Caltrain used to argue in front of the FRA for its waiver to get rolling not compliant with the FRA’s then-current regulations demanding heavy trains designed around static buff strength; this would later underlie the FRA’s own rule change, now permitting lightly-modified European imports on American tracks.

So why is the NTSB so dead set against them? In three words: not invented here. The Talgos were designed and built in Europe. They are designed around European ideas of crash avoidance. Trains here have buff strength requirements too (and are too heavy as a result), but they’re much laxer than those of last generation’s American regulations, because at the end of the day lighter trains are no less safe than American tanks on rails. Lighter trains, designed to brake more quickly and not to derail in the first place, underlie the superior train safety of Europe to that of the United States – and Europe is downright dangerous compared with Japan, whose ultralight trains kill passengers in crashes at maybe 1/15th the per-passenger-km rate of American ones.

Foreign ideas threaten Americans, especially incurious Americans. Americans are not used to not being in the center of the world. They have many reactions to the fact that when it comes to transportation, the global centers of innovation are elsewhere. Some believe private cars are just superior. Others do not but still deny the need for trains, for example elements in tech media who cover Elon Musk as if he were God. And yet others assert that the US actually has amazing rail service on a variety of specious grounds – freight, safety, unique history. All of these are excuses made to avoid learning from foreigners.

The United States should treat its mainline passenger rail as a tabula rasa. Its ridership is a rounding error by European and Asian standards, and this is in large part due to the failure of management to modernize over the last few generations. An American regulator, investigator, or manager who neither knows how to adapt best industry practices, none of which is domestic, nor has any interest in learning, has no business working in the industry. Moreover, to reinforce the need to learn from the best, the NTSB should begin an exchange program with European and East Asian regulators; exchange programs among railroads in Western Europe, Russia, and Japan already exist, but to my knowledge American railroads do not participate in them.

The lessons from the Cascades crash are the same as those from the New York and Philadelphia crash, and drawing different lessons is prima facie evidence of irrational prejudice against foreign design elements that in fact work better than domestic ones. The people at the NTSB who authored the recommendation to get rid of the Talgos have just shown themselves to be incurious about practices in parts of the world with better rail safety. Thus, they should all be immediately dismissed from their jobs and replaced by people who are more informed. Any day they keep their jobs is a signal that the NTSB will be an obstacle to any program to make American passenger rail not only faster, more convenient, and more reliable, but also safer for passengers and employees.

Positive and Negative Interactions

This is a theoretical post about a practical matter that arises whenever multiple variables interact. Two variables x and y, both correlated positively a dependent variable z, are said to positively interact if when x is larger, the effect of y on z gets larger and vice versa, and to negatively interact if when x is larger, the effect of y on z gets smaller. If z is transit ridership, let alone any of the direct benefits of good transit (good job access, environmental protection, public health, etc.), then it is affected by a slew of variables concerning service provision, infrastructure, and urban design, and they interact in complex ways.

I have not found literature on this interaction, which does not mean that this literature does not exist. The papers I’ve seen about correlates of bus ridership look at it one variable at a time, and yet they are suggestive of positive as well as negative interactions. More broadly, there are interactions between different types of service.

Positive interactions tend to involve network effects. These include the interaction between transit and transit-oriented development, as well as that between different aspects of rail modernization. Whenever there is positive interaction between variables, half-measures tend to flop; some are a reverse 80/20 situation, i.e. 80% of the cost yields 20% of the benefits. In some cases, compromises are impossible without making service useless. In others, some starter service is still viable, but in its presence, the case for expansion becomes especially strong, which can lead to a natural virtuous cycle.

Negative interactions occur when different improvements substitute for one another. One straightforward example is bus stops and frequency: frequency and the quality of bus shelter both impact bus ridership, but have a negative interaction, in that at higher frequency, the inconvenience coming from not having bus shelter is less important. In some cases, negative interactions can even lead to either/or logic, in which, in the presence of one improvement, another may no longer be worth the economic or political cost. In others it’s still useful to pursue multiple improvements, but the negative interaction implies the benefits are not as great as one might assume in isolation, and transit planners and advocates must keep this in mind and not overpromise.

Door-to-door trip times

The door-to-door trip time includes walking distance to and from the station, waiting time, transferring time, and in-vehicle time. Each of these components affects ridership in that longer trips reduce people’s propensity to choose public transport.

There is strong positive interaction between variables affecting the trip time. This is not directly attested in the literature that I know of, but it is a consequence of any ridership model that lumps the different components of trip time into one. If public transportation runs faster, that is if the in-vehicle time is reduced, then the share of the other components of the trip time rises, which means that the importance of frequency for reducing wait time is increased. Thus, speed and frequency have a positive interaction.

However, at the same time, there is a subtle negative interaction between speed and service provision on buses. The reason is that bus operating expenses are largely a linear function of overall service-hours, since costs are dominated by driver wages, and even maintenance is in practice a function of service-hours and not just service-km, since low speeds come from engine-stressing stop-and-go traffic conditions. In this case, increasing the speed of buses automatically means increasing their frequency, as the same resources are plugged into more service-km. In that case, the impact of a further increase in service is actually decreased: by speeding up the buses, the transit agency has reduced the share of the door-to-door trip time that is either in-vehicle or waiting at a stop, and thus further reductions in wait time are less valuable.

In the literature, the fact that investing in one portion of the trip makes its share of the overall trip length smaller and thus reduces the impact of further investments is seen in research into ridership-frequency elasticity. My standard references on this – Lago-Mayworm-McEnroe and Totten-Levinson – cite lit reviews in which the elasticity is far higher when frequency is low than when it is high, about 1 in the lowest-frequency cases and 0.3 in the highest-frequency ones. When frequency is very low, for example hourly, the elasticity is so high that adding service increases ridership proportionally; when frequency is a bus every few minutes, the impact of service increase on ridership is much smaller.

I’ve focused on in-vehicle time and waiting time, but the other two components are sometimes within the control of the transit agency as well, especially on rapid transit. Station design can reduce transfer time by providing clear, short passageways between platforms; it can also reduce access time by including more exits, for example at both ends of the platform rather than just at one end or in the middle. As such design positively interacts with other improvements to speed, it makes sense to bundle investments into more exits and better transfers with programs that add train service and speed up the trains.

Network effects

There is positive interaction between different transit services that work together in a network. In the presence of a north-south line through a city, the case for east-west transportation strengthens, and vice versa. This is not a new insight – Metcalfe’s law predicts usage patterns of communications technologies and social networks. The same effect equally holds for fixed infrastructure such as rail, and explains historical growth patterns. The first intercity steam railway opened in 1830, but the fastest phase of growth of the British rail network, the Railway Mania, occurred in the late 1840s, after main lines such as the London and Birmingham had already been established. 150 years later, the first TGV would start running in 1981, but the network’s biggest spurt of growth in terms of both route-km and passenger numbers occurred in the 1990s.

Using a primitive model in which high-speed rail ridership is proportional to the product of city populations, and insensitive to trip length, the United States’ strongest potential line is naturally the Northeast Corridor, between Boston and Washington. However, direct extensions of the line toward Virginia and points south are extremely strong per the same model and, depending on construction costs, may have even higher return on investment than the initial line, as 180 km of Washington-Richmond construction produce 540 km of New York-Richmond passenger revenue. In some places, the extra link may make all the difference, such as extending New York-Buffalo high-speed rail to Toronto; what looks like a basic starter system may be cost-ineffective without the extra link.

Network effects produce positive interactions not just between different high-speed rail lines, but also between transit services at lower levels. Rail service to a particular suburb has positive interaction with connecting bus service, for which the train station acts as an anchor; in some cases, such as the Zurich model for suburban transit planning, these are so intertwined that they are planned together, with timed transfers.

Network effects do not go on forever. There are diminishing returns – in the case of rail, once the biggest cities have been connected, new lines duplicate service or connect to more marginal nodes. However, this effect points out to a growth curve in which the first application has a long lead time, but the next few additions are much easier to justify. This is frustrating since the initial service is hard to chop into small manageable low-risk pieces and may be canceled entirely, as has happened repeatedly to American high-speed rail lines. And yet, getting over the initial hurdle is necessary as well as worth it once subsequent investments pan out.

Either-or improvements

In the introduction, I gave the example of negative interaction between bus shelter amenities and frequency: it’s good to have shelter as well as shorter waits, but if waits are shorter, the impact of shelter is lessened. There are a number of other negative interactions in transit. While it is good to both increase bus frequency and install shelter at every stop, some negative interactions lead to either-or logic, in which once one improvement is made, others are no longer so useful.

Fare payment systems exhibit negative interactions between various positive features. The way fare payment works in Germany and Switzerland – paper tickets, incentives for monthly passes to reduce transaction costs, proof of payment – is efficient. But the same can be said about the smartcard system in Singapore, EZ-Link. EZ-Link works so rapidly that passengers can board buses fast, which reduces (but does not eliminate) the advantage of proof-of-payment on buses. It also drives transaction costs down to the point of not making a monthly pass imperative, so Singapore has no season passes, and it too works.

Interior circulation displays negative interactions as well. There are different aspects of rolling stock design that optimize for fast boarding and disembarking of passengers, which is of critical importance on the busiest rail lines, even more than interior capacity. Trains so designed have a single level, many doors (four pairs per 20-meter car in Tokyo), interiors designed for ample standing space, and level boarding. Each of these factors interacts negatively with the others, and in cities other than Tokyo, regional trains like this are overkill, so instead designers balance circulation with seated capacity. Berlin has three door pairs per car and seats facing front and back, Zurich has double-deckers with two pairs of triple-wide doors and has been quite tardy in adopting level boarding, Paris has single-level cars with four door pairs and crammed seats obstructing passageways (on the RER B) and bespoke double-deckers with three pairs of triple-wide doors (on the RER A).

Finally, speed treatments on scheduled regional and intercity trains may have negative interactions. The Swiss principle of running trains as fast as necessary implies that once various upgrades have cut a route’s trip time to that required for vigorous network connections – for example, one hour or just a few minutes less between two nodes with timed transfers – further improvements in speed are less valuable. Turning a 1:02 connection into a 56-minute one is far more useful than further turning a 56-minute service into a 50-minute trip. This means that the various programs required to boost speed have negative interactions when straddling the boundary of an even clockface interval, such as just less than an hour, and therefore only the cheapest ones required to make the connections should receive investment.

Conclusion

Good transit advocates should always keep the complexities that affect transportation in mind. Negative interactions between different investments have important implications for activism as well as management, and the same is true for positive interactions.

When variables interact negatively, it is often useful to put a service in the good enough basket and move on. In some cases, further improvements are even cost-ineffective, or require unduly compromising other priorities. Even when such improvements remain useful, the fact that they hit diminishing returns means advocates and planners should be careful not to overpromise. Cutting a two-hour intercity rail trip to an hour is great; cutting a 40-minute trip to a 20-minute one may seem like a game changer, but really isn’t given the importance of access and egress times, so it’s usually better to redeploy resources elsewhere.

Conversely, when variables interact positively, transit service finds itself in an 80% of the cost for 20% of the benefits situation. In such case, compromises are almost always bad, and advocates have to be insistent on getting everything exactly right, or else the system will fail. Sometimes a phased approach can still work, but then subsequent phases become extremely valuable, and it is useful to plan for them in advance; other times, no reasonable intermediate phase exists, and it is on activists to convince governments to spend large quantities of upfront money.

Transportation is a world of tradeoffs, in which benefits are balanced against not just financial costs but also costs in political capital, inconvenience during construction, and even activist energy. Positive and negative interactions have different implications to how people who want to see better public transport should allocate resources; one case encourages insisting on grand plans, another encourages compromise.

Amtrak is Blocking MBTA Electrification

Ten years ago, Amtrak began putting out its outrageously expensive proposals for high-speed rail on the Northeast Corridor. Already then, when it asked for $10 billion to barely speed up trains, there was a glaring problem with coordination: Amtrak wanted hundreds of millions of dollars to three-track the Providence Line so that its trains could overtake the MBTA’s commuter trains between Providence and Boston, even though the same benefit could be obtained for cheaper by building strategic overtakes and electrifying the MBTA so that its trains would run faster. Unfortunately, Amtrak has not only displayed no interest in coordinating better service with the MBTA this way, but has just actively blocked the MBTA.

The issue at hand is MBTA electrification: the MBTA runs an exclusively diesel fleet. These trains are slow, polluting, and unreliable. Lately they have had breakdowns every few thousand kilometers, whereas electric trains routinely last multiple hundreds of thousands of kilometers between breakdowns. The current scheduled trip time between Boston and Providence is about 1:10 on the MBTA with a total of nine stops, whereas Amtrak’s southbound trains do the same trip in 35 minutes with three stops, leading to a large schedule difference between the trains, requiring overtaking. Fortunately, modern electric multiple units, or EMUs, could make the same stops as the MBTA in about 45 minutes, close enough to Amtrak that Amtrak could speed up its trains without conflict.

The MBTA would benefit from electrification without any reference to Amtrak. Connecting Boston and Providence in 45 minutes rather than 70 has large benefits for suburban and regional travelers, and the improved reliability means trains can follow the schedule with fewer unexpected surprises. The line is already wired thanks to Amtrak’s investment in the 1990s, and all that is required is wiring a few siding and yard tracks that Amtrak did not electrify as it does not itself use them. With the diesel locomotives falling apart, the MBTA has begun to seriously consider electrifying.

Unfortunately, the MBTA has made some questionable decisions, chief of which is its attempt to procure electric locomotives rather than self-propelled EMUs. The MBTA’s reasoning is that EMUs require high platforms, which cost about $10 million per station, which is a small but nonzero amount of money on the Providence Line. As a result, it neglected any solution involving buying new EMUs or even leasing them from other railroads for a pilot project. It’s only looking at electric locomotives, whose travel time benefits are about half as large as those of EMUs.

And yet, Amtrak is blocking even the half-measures. The MBTA sought to lease electric locomotives from Amtrak, which uses them on its own trains; Amtrak quoted an unreasonably high monthly price designed to get the MBTA to lose interest. As of last week, the MBTA put the plan to lease electric locomotives for its electrification pilot on hold. No plans for purchase of rolling stock are currently active, as the MBTA is worried about lead time (read: having to actually write down and execute a contract) and does not know how to buy lightly-modified European products on the open market.

As far as Amtrak is concerned, speeding up the MBTA is not really relevant. Yes, such a speedup would improve Amtrak’s own scheduling, removing a few minutes from the Northeast Corridor’s travel time that would otherwise cost hundreds of millions of dollars. But Amtrak has time and time again displayed little interest in running fast trains. All it wants is money, and if it can ask for money without having to show anything for it, then all the better. Coordinating schedules with other railroads is hard, and only improves the experience of the passengers and not Amtrak’s managers.

The MBTA’s decisionmaking is understandable, in contrast, but still questionable. It was worth asking; the MBTA is no worse for having received an unreasonable offer. However, it is imperative that the MBTA understand that it must be more proactive and less hesitant. It must electrify, and commit a real budget to it rather than a pilot. This means immediately raising the platforms on the stations of the Providence Line that do not yet have level boarding and buying (not leasing) modern EMUs, capable of running fast schedules.

Even with some infill, there are only seven low-platform stations on the mainline and two on the branch to Stoughton, none in a constrained location where construction is difficult. This is at most a $100 million project, excluding the trains themselves. The MBTA could operate the Providence Line with seven to eight trainsets providing service every 15 minutes and the Stoughton Line with another four providing the same. Procuring trains for such service would cost another $250 million or so, but the MBTA needs to buy new rolling stock anyway as its diesel locomotives are past the end of their useful lives, and buying EMUs would pay for itself through higher ridership and lower operating expenses coming from much faster trips.

The MBTA is fortunately salvageable. It has a serious problem in that the state leadership is indecisive and noncommittal and prefers a solution that can be aborted cheaply to one that provides the best long-term financial and social return on investment. However, it is seriously looking in the right direction, consisting of better equipment providing higher-quality service to all passengers.

Amtrak is unfortunately not salvageable. An intercity railroad whose reaction to a commuter railroad’s attempt to improve service for both systems is to overcharge it on rolling stock proves that it is ignorant of, indifferent to, and incurious about modern rail operations. A chain of managers from the person who made the decision to offer the MBTA bad lease terms upward must be removed from their positions if there is any hope for improved intercity rail in the Northeastern United States.

The Importance of Radial Urban Rail

Most urban rail networks consist predominantly of radial lines, connecting city center with outlying areas. However, a fair minority attempt a different typology, reminiscent of a grid. This does not work so well in practice, and I want to bring it up in the context of both my current domicile and my previous one, which have non-radial metro systems, for different reasons. The problem with non-radial metro networks is that many trips require multiple transfers, and even single-transfer trips often place the connection far out of the way. The average effective speed in such cases is often lower than that of a slow bike.

Berlin

Berlin’s S-Bahn network is fully radial, consisting of two radial trunks and the Ringbahn. However, its U-Bahn network fails to be radial in three distinct ways. First, the two north-south trunks, U6 and U8, are parallel. Second, the east-west U5 only goes from Alexanderplatz eastward, although an under-construction extension will extend it slightly to the west and connect with U6. And third, the postwar lines were constructed in a Cold War context around a temporary city center.

The problem can be summarized with the following map, in which the green dot is where I live and the three red dots are places I’ve gone to recently:

A larger 13 MB image can be seen here.

My trips to the two southern red dots, both gaming events, are not too onerous, using the Ringbahn. But my trip to the northern red dot isn’t so easy; besides being circuitous, the Ringbahn is currently shut down for maintenance for a segment in between. Now, there’s redundancy, I can still get there, but it requires a three-seat ride involving U7, U8, and U2, and what’s more, the U8/U2 transfer at Alexanderplatz is long.

The route U7 takes is a Cold War relic. During the division of Berlin, city center was in East Berlin, forcing West Berlin to build a new city center, currently called City West. East Berlin got the S-Bahn, which West Berliners eventually began to boycott even when it served the West, and West Berlin got nearly the entire U-Bahn, with the exception of just U5 and the eastern parts of U2. U6 and U8 served West Berlin going from the north to the south without stopping in the Walled center. To provide entirely Western routes, West Berlin built two new lines, U7 and U9.

The decisions made about U7 and U9 routing were then about the context of the Wall. U9 is a straight north-south trunk passing through City West, with connections to every West Berlin U-Bahn line with the exception of the low-ridership U4 stub. U7, originally part of the U6 mainline south of their junction, was extended northwest to become a new trunk line, and it too is designed to connect to the U-Bahn network of West Berlin alone rather than to the combined one. Thus, there is a reasonable U7-U1 connection at Möckernbrücke, but instead of continuing due west to connect with U2, U7 detours southwest and only connects with U2 at Bismarck Strasse, too far to the west to be of use for people connecting between the two lines’ eastern legs. In the Cold War, this worked, in the sense that the parts of U2 on the Western side of the Wall are either still convenient with the Bismarck Strasse transfer or very close to U1. In the context of reunification, this doesn’t work so well.

Here is a map of passenger rail traffic by interstation segment:

Note that U9 goes strong throughout its run; while the area it serves is no longer the CBD, it is still a strong high-end retail center, defined around Kurfürstendamm. However, U7’s traffic actually peaks in Neukölln and Kreuzberg and is lower west of the junction with U6 at Mehringdamm. Moreover, coming from Spandau at the other end, U7 loses traffic as it crosses the Ringbahn (as do many lines coming from the west) rather than gaining it. The shift in the center of the city has rendered U7 a mixed radial-circumferential line.

Paris

Like Berlin, Paris has a radial mainline rail network and a more complicated dedicated urban rail system.

The Metro defies easy categorization. It has the characteristics of a grid, with Lines 1, 3, 8, 9, and 10 running east-west, and Lines 4, 5, 7, 12, and 13 running north-south. It also has the characteristics of two overlain radial networks, one consisting of Lines 1, 4, 7, 11, and 14 around Chatelet, and one consisting of Lines 3, 7, 8, 9, 12, 13, and 14 around the Opera and Saint-Lazare. Some support for the latter view is that the weakest lines are M5 and M10, passing respectively too far east and south of city center.

Out of 14 Paris Metro lines, only two can connect to every other line in the network, M4 and M9, and the latter has an unadvertised connection from Saint-Augustin to Saint-Lazare, without which no transfer to M12 or M14 is possible.

This is not quite as dire as it may seem at first glance. So many stations are transfers, and stations are spaced so close together, that even though M1 runs parallel to M3 and M10, nearly all intra muros M1 stations still have two-seat rides to M3 and M10 via a different line.

Nevertheless, some parts of the city are poorly connected as a result. Eastern Paris’s connections to the Left Bank are not good. Some are more circuitous than they need to be, such as between a dorm I stayed at near Porte de Vincennes in 2010 for a conference and the site of the conference itself at Jussieu. Some are three-seat rides. Most involve changing trains at Chatelet, which adds 5-7 minutes to the trip in walking time alone.

In the case of Berlin, explaining how it got this way requires going into Cold War history. In that of Paris, a city with continuous development, this is just a matter of uncoordinated layers of planning. The plan from the 1890s produced M1-6, shaped as a hex symbol inside a circle; the lack of connection between M1 and M3 was not thought a problem then, and the remaining lack of connectivity if one originates in the suburbs was never a planning priority either.

There is another way

Paris may have Europe’s largest rail network, and Berlin may have the fourth largest, but they are in this sense atypical. London’s radial Underground network provides better connectivity, and the radial typology is increasingly dominant as Chinese metro networks follow the Soviet model, which is even more strictly radial than the British one.

The distance between my apartment and the northernmost red dot on the Berlin map is 8 kilometers. I looked at similar 8-kilometer trips within Inner London, picking two different starting points at Brixton and Bromley-by-Bow. Each 8-kilometer trip passing through or near Central London had a viable one- or two-seat ride, with the exception of Brixton-Canary Wharf, which is a three-seat ride with a cross-platform interchange at Stockwell.

There are a lot of defects in the London Underground network, and the two starting points I picked are somewhat cherrypicked to avoid them. Every pair of London Underground main lines intersects with a transfer, except the Metropolitan line and the Charing Cross branch of the Northern line, which have a missed connection at Euston and Euston Square; my two starting points are on neither of these two lines. Moreover, going northwest, there are some suburban missed connections between branches. This is on top of serious problems with capacity coming from the trains’ small size.

However, it’s notable that nobody is reproducing the small profile of the Tube networks. What cities around the world are reproducing is the radial network design, in which most trips within the urban core are reasonably direct. Going forward, Paris may even consider building connections between Metro lines to make its network more radial, for example extending M11 to the west. Berlin, likewise, should look to invest in radial S-Bahn trunks, following the busiest corridors connecting more areas to and beyond Mitte, where it’s already building extensive office space.

Optimization

This post may be of general interest to people looking at optimization as a concept; it’s something I wish I’d understood when I taught calculus for economics. The transportation context is network optimization – there is a contrast between the sort of continuous optimization of stop spacing and the discrete optimization of integrated timed transfers.

Minimum and maximum problems: short background

One of the most fundamental results students learn in first-semester calculus is that minimum and maximum points for a function occur when the derivative is zero – that is, when the graph of the function is flat. In the graph below, compare the three horizontal tangent lines in red with the two non-horizontal ones:

A nonzero derivative – that is, a tangent line slanting up or down – implies that the point is neither a local minimum nor a local maximum, because on one side of the point the value of the function is higher and on the other it is lower. Only when the derivative is zero and the tangent line is flat can we get a local extreme point.

Of course, a local extreme point does not have to be a global one. In the graph above, there are three local extreme points, two local maxima and one local minimum, but only the local maximum on the left is also a global maximum since it is higher than the local maximum on the right, and the local minimum is not a global minimum because the very left edge of the graph dips lower. In real-world optimization problems, the global optimum is one of the local ones, rather than an edge case like the global minimum of the above graph.

First-semester calculus classes love giving simplified min/max problems. This class of problems is really one of two or three serious calc 1 exercises; the other class is graphing a function, and the potential third is some integrals, at universities that teach the basics of integration in calc 1 (like Columbia and unlike UBC, which does so in calc 2). There’s a wealth of functions that are both interesting from a real-world perspective and doable by a first-semester calc student, for example maximizing the volume of some shape with prescribed surface area.

My formulas for stop spacing come from one of these functions. The overall travel time is a function of walking time, which increases as stops get farther apart, and in-vehicle time, which decreases as stops get farther apart. A certain stop spacing produces the minimum overall trip time; this is precisely the global minimum of the travel time function, which is ultimately of the form f(x) = ax + b/x where a and b are empirical parameters depending on walking speed and other relevant variables.

Continuous optimization

The fundamental fact of continuous optimization, one I wish I’d learned in time to teach it to students, is that at the optimum the derivative is zero, and therefore making a small mistake in the value of the optimum is not a big problem.

What does “mistake” mean in this context? It does not mean literally getting the computation wrong. There is no excuse for that. Rather, it means choosing a value that’s slightly suboptimal for ancillary reasons – perhaps small discontinuities in the shape of the network, perhaps political considerations.

Paul Krugman brings this concept up in the context of wages. The theory of efficiency wages asserts that firms often pay workers above the bare minimum required to get any workers at all, in order to get higher-quality workers and incentivize them to work harder. In this theory, the wage level is set to maximize employer productivity net of wages. At the employer’s optimum the derivative of profit is by definition zero, so a small change in wages has little impact to the employer. However, to the workers, any wage increase is good, as their objective function is literally their wage rather than profits. They may engage in industrial action to raise wages, or push for favorable regulations like a high minimum wage, and these will have a limited effect on profits.

In the context of transit, this has the obvious implication to wages – it’s fine to set them somewhat above market rate since the agency will get better workers this way. But there are additional implications to other continuous variables.

With stop spacing specifically, the street network isn’t perfectly continuous. There are more important and less important streets. Getting transit stops to align with major streets is important, even if it forces the stop spacing to be somewhat different from the optimum. The same is true of ensuring that whenever two transit lines intersect, there is a transfer between them. This is the reason my bus redesign for Brooklyn together with Eric Goldwyn involved drawing the map before optimizing route spacing – the difference between 400 and 600 meters between bus stops is not that important. For the same reason, my prescription for Chicago, and generally other American cities with half-mile grids of arterial roads, is a bus stop every 400 meters, to align with the grid distance while still hewing close to the optimum, which is about 500.

When I talked about stop consolidation with a planner at New York City Transit who worked on the Staten Island express bus redesign, the planner explained the philosophy to me: “get rid of every other stop.” In the context of redesigning a single route, this is an excellent idea as well: the process of adding and removing bus stops in New York is not easy, so minimizing the net change by deleting stops at regular intervals so as to space the remaining stops close to the optimum is a good idea.

The world of public transit is full of these tradeoffs with continuous variables. It’s not just wages and interstations. Fares are another continuous variable, involving particular tensions as different political factions have different objective functions, such as revenue, social rate of return, and social rate of return for the working class alone. Frequency is a continuous variable too in isolation. Top speed for a regional train is in effect a continuous variable. All of these have different optimization processes, and in all cases, it’s fine to slightly deviate from the strict optimum to fulfill a different goal.

Discrete optimization

Whereas continuous optimization deals with flat tangent lines, discrete optimization may deal with delicate situations in which small changes have catastrophic consequences. These include connections between different lines, clockface scheduling, and issues of integration between different services in general.

An example that I discussed in the early days of this blog, and again in a position paper I just wrote to some New Hampshire politicians, is the Lowell Line, connecting Boston with Lowell, a distance of 41 km. The line is quite straight, and were it electrified and maintained better, trains could run at 160 km/h between stops with few slowdowns. The current stop spacing is such that the one-way trip time would be just less than half an hour. The issue is that it matters a great deal whether the trip time is 25 or 27 minutes. A 25-minute trip allows a 5-minute turnaround, so that half-hourly service requires just two trainsets. A 27-minute trip with half-hourly service requires three trainsets, each spending 27 minutes carrying passengers and 18 minutes depreciating at the terminal.

A small deterioration in trip time can literally raise costs by 50%. It gets to the point that extending the line another 50 kilometers north to Manchester, New Hampshire improves operations, because the Lowell-Manchester trip time is around 27-28 minutes, so the extension can turn a low-efficiency 27-minute trip into a high-efficiency 55-minute trip, providing half-hourly service with four trainsets.

In theory, frequency is a continuous variable. However, in the range relevant to regional rail, it is discrete, in fractions of an hour. Passengers can memorize a half-hourly schedule: “the inbound train leaves my stop at :10 and :40.” They cannot and will not memorize a schedule with 32-minute frequency, and needing to constantly consult a trip planner will degrade their travel experience significantly. Not even smartphone apps can square this circle. It’s telling that the smartphone revolution of the last decade has not been accompanied with rapid increase in ridership on transit lines without clockface schedules, such as those of the United States – if anything, ridership has grown faster in the clockface world, such as Germany and Switzerland.

Transit networks involving timed connections are another case of discrete optimization in which all parts of the network must work together, and small changes can make the network fall apart. If a train is late by a few minutes and its passengers miss their connection, the short delay turns into a long one for them. As a result, conscientious schedule planners make sure to write timetables with some contingency time to recover from delays; in Switzerland this is 7%, so in practice, out of every 15 minutes, one minute is contingency, typically spent waiting at a major station.

But this gets even more delicate, because different aspects of the transit network impact how reliable the schedule is. If it’s a bus, it matters how much traffic there is on the line. Buses in traffic not reliable enough for tight connections, so optimizing the network means giving buses dedicated lanes wherever there may be traffic congestion. Even though it’s a form of optimization, and even though there’s a measure of difficulty coming from political opposition by drivers, it is necessary to overrule the opposition, unlike in continuous cases such as wages and fares.

Infrastructure planning for rail has the same issues of discrete optimization. It is necessary to design complex junctions to minimize the ability of one late train to delay other trains. This can take the form of flying junctions or reducing interlining; in Switzerland there are also examples of pocket tracks at flat junctions where trains can wait without delaying other trains behind them. Then, the decision of how much to upgrade track speed, and even how many intermediate stations to allow on a line, has to come from the schedule, in similar vein to the Lowell Line’s borderline trip time.

Continuous and discrete optimization

Many variables relevant to transit are in theory continuous, such as trip time, frequency, stop spacing, wages, and fares. However, some of these have discontinuities in practice. Stop spacing on a real-world city street network must respect the hierarchy of more and less important destinations. Frequency and trip times are discrete variables except at the highest intensity of service, perhaps every 7.5 minutes or better; 11-minute frequency is worse to the passenger who has to memorize a difficult schedule than either 10- or 12-minute frequency.

New York supplies a great example showcasing how bad it can be to slavishly hew to some optimal interstation and not consider the street network. The Lexington Avenue Line has a stop every 9 blocks from 33rd Street to 96th, offset with just 8 blocks between 51st and 59th and 10 between 86th and 96th. In particular, on the Upper East Side it skips the 72nd and 79th Street arterials and serves the less important 68th and 77th Streets instead. As a result, east-west buses on the two arterials cross Lexington without a transfer.

Just east of Lex, there is also a great example of optimization on Second Avenue Subway. The stops on Second Avenue are at 72nd, 86th, and 96th, skipping 79th. It turns out that skipping 79th is correct – the optimum for the subway is to the meter the planned stop spacing for the line between 125th and Houston Streets, so it’s okay to have slightly non-uniform stop spacing to make sure to hit the important east-west streets.

Frequency and trip times are subject to the Swiss maxim, run trains as fast as necessary, not as fast as possible. Hitting trip times equal to an integer or half-integer number of hours minus a turnaround time has great value, but small further speedups do not. Passengers still benefit from the speedup, but the other benefits of higher speed to the network, such as better connections and lower crew costs, are no longer present.

The most general rule here is really that continuous optimization tolerates small errors, whereas discrete optimization does not. Therefore, it’s useful to do both kinds of optimization in isolation, and then modify the continuous variable somewhat based on the needs of the discrete one. If you calculate and find that the optimal frequency for your bus or train is once every 16 minutes, you should round it to 15, based on the discrete optimization rule that the frequency should be a divisor of the hour to allow for clockface timetable. If you calculate and find that the optimal bus stop spacing is 45% of the distance between two successive arterial streets, you should round it to 50% so that every arterial gets a bus stop.

Getting continuous optimization right remains important. If the optimal stop spacing is 500 meters and the current one is 200 meters, the network is so far from the local maximum of passenger utility that the derivative is large and stop consolidation has strong enough positive effects to justify overruling any political opposition. However, it is subsequently fine to veer from the optimum based on discrete considerations, including political ones if removing every 1.7th bus stop is harder than removing every other stop. Close to the local maximum or minimum, small changes really are not that important.

Construction Costs in the Nordic Countries

I write a lot about stereotypes in the context of construction costs. Countries with a reputation for corruption, such as Spain, South Korea, Greece, and Italy, often build subways very cheaply. Germany, for all its stereotype of efficiency, has high costs and some dysfunctional decisionmaking in what to build. Singapore, the self-styled most efficient government, pays its transport minister more than a million dollars per year to make excuses for why it has such high construction costs.

In the Nordic countries, the stereotype is correct: those countries have transparent, clean governments, and also build infrastructure cheaply.

Subway tunnels

All four mainland Nordic capitals have recent or ongoing metro expansion projects:

Stockholm just opened Citybanan, a regional rail connection including 6 km of tunnel with two deep stations in Central Stockholm and a 1.4 km bridge. The total cost was 16.8 billion SEK in 2007 terms, which in today’s PPP terms is about $330 million per km. It’s expensive for a suburban subway but not for regional rail.

Copenhagen is currently wrapping up construction on the fully underground, driverless City Circle Line. It is a circular but not circumferential line through city center. With repeated schedule slips, the budget is now 24.8 billion DKK, or $3.4 billion in PPP terms, which is $220 million per km.

Stockholm is expanding its metro in three directions. The fully underground extensions are together 19 km and 22.4 billion SEK, which in PPP terms is $130 million per km.

Helsinki has just opened an expansion of its metro westward to Espoo. This is a 13.5 km, 8-station fully underground line with a water crossing. After cost overruns, the current cost estimate is €1,186 million, which is in PPP terms $115 million per km.

Oslo recently opened a short connection, called Lørenbanen. It’s 1.6 km long and includes a single new station, for a total of NOK 1.33 billion, including 150 million for modernization of an existing connecting line. In PPP terms this is just $90 million per km in today’s money.

Other rail infrastructure

Sweden is investing heavily in mainline rail modernization. This includes a planned high-speed rail network connecting the country’s three biggest cities, which are spaced far apart and not on a line, requiring the total system to be 740 km long. The cost projection as of 2015 is 125 billion SEK, which in PPP terms is $14 million per km; I do not know if it is in 2015 prices or expected year of construction prices. This cost figure is comparable to that of Madrid-Barcelona and about half the at-grade norm for Europe.

Sweden is simultaneously investing in its mainline network, rather than neglecting it in favor of just HSR the way France is. A document from 2009 lists some of these on p. 38 based on the national plan of 2010-21, which did not include HSR. Of note, two full double-track projects are coming it at about $10 million per km or slightly more. In contrast, in Berlin, suburban S-Bahn double-tracking is around twice as expensive per the list on PDF-pp. 73-77 of the official wishlist.

In Denmark, a recent double-tracking project cost 675 million DKK for 20 km, or $4.6 million per km, even cheaper than in Sweden. The project includes not just double track but also an upgrade to 160 km/h.

Denmark is also investing heavily in electrification – see here for a list of projects, without costs. Costs for some of these projects are provided by Railway Gazette. The Fredericia-Aalborg line is 249 km and 4.7 billion DKK, the Roskilde-Kalundborg line is 56 km and 1.2 billion DKK, and the Esbjerg-Lunderskov line is 57 km and 1.19 billion DKK; all three lines are double-track. The longer line is $2.6 million per km, the shorter two are $2.9 million. This is much cheaper than in the core Anglosphere but more expensive than projects for which I have data in France, Israel, and New Zealand.

It’s cheap, but do people ride it?

Absolutely. Low construction costs can occur for projects that nobody has any reason to build, they’re so low-ridership, while some high-cost projects remain cost-effective if they have extremely high ridership, like Second Avenue Subway Phase 1.

In the case of the Nordic capitals, the recent extensions are well-patronized. The ridership prognosis for the City Circle Line is 289,000 per weekday, which means its cost is $11,800 per rider. The link above for the Stockholm T-bana extension projects 170,000 riders per day, which I believe means weekday rather than literal day; in that case, the projected cost per rider is $14,500. Løren’s ridership is 8,000 per day, which one former resident says is just boardings without alightings, which means total ridership is actually 16,000, making the cost of the line just shy of $9,000 per rider. And Helsinki’s West Metro is projected to get 100,000 daily riders, which means its cost is about $15,500 per rider.

Moreover, Stockholm’s overall use of public transportation is very healthy. The first 6 pages of this PDF comprise a report on modal split in Stockholm, out of all trips, not just work trips. In 2015, 32% of all trips in Stockholm County were by public transport, 38% were by car, 9% were by bike, and 16% were on foot. There had been a notable shift from cars to the other modes since 2004.

Converting this statistic to work trip mode share, the most stable metric and the one reported for the US, Canada, UK, and France, requires some additional work. However, where both statistics are available, they do provide some insight: in Hamburg in 2008, the overall car mode shares for all trips and for just work trips were similar (48% for work trips vs. 42% for all trips in the city, 65% vs. 63% in the suburbs); work trips alone exhibit much higher transit mode share (33% vs. 18% in the city, 16% vs. 8% in the suburbs), at the expense of non-motorized trips, which are disproportionately for short errands. It is very likely that the work trip public transport mode share in Stockholm County is comparable to Ile-de-France’s 43%, in a metro area one fifth the size.

Transit ridership in the other Nordic capitals is weaker, though still impressive for their size. Copenhagen lags in transit but has a strong bike network. Oslo had 118 million metro riders in 2017 (source, PDF-p. 31 – per same link you can also see the operating costs per car-km work out to just short of PPP$4, compared with a typical first-world range of $4-7), plus some additional commuter rail ridership (65 million nationwide, not just around Oslo). Helsinki had 63 million annual metro passengers in 2015, before the extension opened, and somewhat fewer additional commuter rail passengers, for a total ridership of perhaps 120 million. Both of the smaller cities have about the same metro area rail ridership per capita as New York, which is about fifteen times their size.

What does this mean?

Scandinavia has a reputation for efficient government at home as well as abroad. Right-wing pundits are far more likely to look for aspects of its governance that play to their desire for privatization, such as Sweden’s school voucher system or the contracting out of urban rail, than to assert that Scandinavia is a socialist failure. Unlike autocracies that have cultivated such reputation, the Nordic countries fully deserve this praise when it comes to building infrastructure cost-effectively. Sweden appears to consistently build rail for half the per-unit cost of Germany.

And yet, I don’t see that much praise for Nordic infrastructure. There are people in the English-speaking world making grandiose claims about how democratic countries need to be more like China and about how authoritarianism is just more efficient. I don’t know of any making that claim about how Nordic social democracy is more efficient, with its depoliticized state apparatus, multiparty elections, high levels of transparency, bureaucratic legalism, and near-universal collective bargaining.

Across all levels of public transportation investment, from high-speed rail down to routine track upgrades, we see inexpensive, efficient projects in the Nordic countries. They achieve high levels of rail usage without megacities in which only masochists drive, and keep expanding their networks in order to complete the green transition. Public transit managers in not just the laggard that is the US but also Germany and other relatively solid countries should make sure to study how things work in Scandinavia and how they can import Nordic success.

Circles

Rail services can be lines or circles. The vast majority are lines, but circles exist, and in cities that have them they play an important niche. Owing to an overreaction, they are simultaneously overused and underused in different parts of the world. However, that some places overuse circles does not mean that circles are bad, nor does it mean that specific operational problems in certain cities are universal.

In particular, what I think of as the ideal urban rapid transit network should feature circles once the network reaches a certain scale, as in the following diagram that I use as my Patreon avatar:

Circles and circumferentials

Circles are transit lines that run in a loop without having a definitive start or end. Circumferentials are lines that go around city center, connecting different branches without passing through the most congested part of the city. In the ideal diagram above, the purple line is both a circle and a circumferential. However, lines can be one without being the other, and in fact examples of lines that are only one of the two outnumber examples of lines that are both.

For example, here is the Paris Metro:

Paris has a circle consisting of Metro Lines 2 and 6, which are operationally lines; people wishing to travel on the arcs through the meeting points at Nation and Etoile must transfer. Farther out, there is an incomplete circle consisting of Tramway Line 3, where the forced transfer between 3a and 3b is Porte de Vincennes. Even farther out there is an under-construction line not depicted on the map, Line 15 of Grand Paris Express, which has a pinch point at its southeast end rather than continuous circular service. All three systems are great example of circumferential lines with very high ridership that are not operationally circles.

Another rich source of circumferential lines that are not circles is cities near bodies of water. In those cities, a circumferential line is likely to be a semicircle rather than a circle. This is responsible for the current state of the Singapore Circle Line, although in the future it will be closed to form a full circle. The G train in New York is a single-sided circumferential line to the east of Manhattan, not linking with anything to the west of Manhattan because of the combination of wide rivers and the political boundaries between New York and New Jersey.

In the opposite direction – circles that are not circumferentials – there are circular lines that don’t neatly orbit city center. The Yamanote Line in Tokyo is one such example: its eastern end is at city center, so it combines the functions of a north-south radial line with those of a north-south circumferential line connecting secondary centers west of Central Tokyo. London’s Circle Line is no longer operationally a circle but was one for generations, and yet it was never a circumferential – it combined the central legs of two east-west radial mainlines, the Metropolitan and District lines.

We can collect this distinction into a table:

Circle, not circumferential Circumferential, not a circle Circumferential circle
Yamanote Line
Osaka Loop Line
Seoul Metro Line 2
London Circle line (until 2009)
Madrid Metro Line 12		 Paris M2/6, T1, T2, T3, future M15
Copenhagen F train
New York G train, proposed Triboro
London Overground services
Chicago proposed Circle Line
Singapore Circle line (today)		 Moscow Circle Line, Central Circle
Berlin S41/S42
Beijing Subway Line 2, Line 10
Shanghai Metro Line 4
Madrid Metro Line 6

Operational concerns: the steam era

In the 19th century, it was very common to build circular lines in London. In the steam era, reversing a train’s direction was difficult, so railways preferred to build circles. This was the impetus for joining the Metropolitan and District lines to form the Circle line. Mainline regional rail services often ran in loops as well: these were as a rule never or almost never complete circles, but instead involved trains leaving one London terminus and then looping around to another terminus.

Another city with a legacy inherited from steam-era train operations is Chicago. The Loop was built to easily reverse the direction of trains heading into city center. At the outer ends they would need to reverse direction the traditional way, but there was no shortage of land for yards there, unlike in the Chicago CBD since named after the Loop.

As soon as multiple-unit control was invented in the 1890s, this advantage of circles evaporated. Subsequently rapid transit lines mostly stopped running as circles unless they were circumferential. London’s Central line, originally pitched as two long east-west lines forming a circle, became a single east-west line, on which trains would reverse direction.

Operational concerns: the modern era

Today, it is routine to reverse the direction of a rapid transit train. The vast majority of rapid transit routes run as lines rather than circles.

If anything, there have been complaints that circles are harder to run service on than lines. However, I believe these concerns are all specific to London, which changed its Circle line from a continuous loop to a spiral in 2009. I have heard concerns about the operations of the Ringbahn here, but as far as I can tell the people who express them are doing so in analogy with what happened in London, and are not basing them on the situation on the ground here. Moreover, there are no plans to make the Yamanote Line run as anything other than the continuous loop it is today.

The situation in London is that the Circle line has always shared tracks with both the Metropolitan and District lines. There has always been extensive branching, in which a delay on one train propagates to the entire network formed by these two mainlines. To this day, Transport for London does not expect the lines in the subsurface network to have the same capacity as the isolated deep tube lines: with moving block signaling it expects 32 trains per hour, compared with 36 on isolated lines.

What’s more, the junctions in London are generally flat. Trains running in opposite directions can conflict at such junctions, which makes the schedules more fragile. Until 2009, London ran the Circle line trains every 7 minutes, which was bound to create conflicts with other lines.

The importance of this London-specific background is that the argument against circles is that they make schedules more fragile. If there is no point on the line where trains are regularly taken out of service, then it is hard to recover from timetable slips, and delays compound throughout the day. However, this is relevant mainly in the context of an extensively-branching system like London’s. Berlin has some of that branching as well, but much less so; one of the sources of reverse-branching on the S-Bahn is a line that should get its own cross-city route anyway, and another is a Cold War relic swerving around West Berlin (S8/85).

The benefits of complete circles

The complete circle of the Yamanote Line or the Ringbahn can be compared with incomplete circles, such as the Oedo Line or the various circumferentials in Paris. From passengers’ perspective, it’s better to have a complete circle, because then they can undertake more trips.

Circumferential lines broadly have two purposes:

  1. They offer service on strong corridors that are orthogonal to the direction of city center, such as the various boulevards hosting the M2/6 ring as well as the Boulevards des Maréchaux hosting T3.
  2. They offer connections between two radial lines that may not connect in city center, or may connect so far from the route of the circumferential that transferring via the circumferential is faster.

Both purposes are enhanced when the route is continuous. In the case of Paris, a north-south trip east of Nation is difficult to undertake, as it requires a transfer at Porte de Vincennes. Passengers connecting from just south, on M8 or even on M7, may not save as much time traveling to lines just north, such as M9 or M3, and might end up transferring at the more central stations of Republique or Opera, adding to congestion there.

In contrast, in Berlin the continuous nature of the Ring makes trips across the main transfer points more feasible. Just today I traveled from my new apartment to a gaming event on the Ringbahn across Ostkreuz. At Ostkreuz the trains dwelled longer than the usual, perhaps 2 minutes rather than the usual 30 seconds, which I imagine is a way to keep the schedule. That delay was, all things considered, minor. Had I had to transfer to a new train, I would have almost certainly taken a different combination of trains altogether; the extra waiting time adds up.

Why are circles so uncommon?

The operational concerns of London aside, it’s still uncommon to see complete circles on rapid transit networks. They are the ideal for cities that grow beyond the scale of three or four radial trunks, but there are only a handful of examples. Why is that?

The answer is always some sort of special local concern. If city center is offset to one side of the built-up area, such as in a coastal city, then circumferential lines will be semicircles and not full circles. If there is some dominant transfer point that requires a pinch, then cities prefer to build a pinch into the system, as is the case for Porte de Vincennes on T3 or for some of the lines cobbled together to form the London Overground.

This is similar to the question of missed connections. Public transportation networks must work hard to ensure that whenever two lines meet, they will have a transfer. Nonetheless, missed connections exist in virtually all large rapid transit networks. Some of those are a matter of pure incompetence, but in many, rail networks that developed over generations may end up having one subway line that happens to intersect another far from any station on the older line, and there is little that can be done.

Likewise, it is useful to ensure that circumferential lines be complete circles whenever the city is symmetric enough to warrant circles. Paris, like other big cities with strong transit networks, is good but not perfect, and it is important to call it on the mistakes it makes, in this case building M15 to have a jughandle rather than running as a complete circle.

What Berlin is Building is Not What It Needs to Build

Berlin has a deceptively simple S-Bahn network. There’s the Ringbahn circling city center. There’s the elevated east-west Stadtbahn, which has two tracks dedicated to S-Bahn service and two for everything else, including longer-range regional trains and intercity trains. And there’s the two-track North-South Tunnel, which only carries S-Bahn traffic; longer-range traffic uses the four-track north-south mainline through Berlin Hauptbahnhof, whereas the North-South Tunnel intersects the Stadtbahn one station east of Hauptbahnhof, at Friedrichstrasse.

The main S-Bahn capacity needs in Berlin are east-west; meanwhile, the North-South mainline is underfull, with Wikipedia listing around 7 trains per hour. And yet, Berlin’s big S-Bahn capital project is a new tunnel, dubbed S21, adding yet another north-south track pair through Hauptbahnhof. Fortunately, the project is salvageable, but only if the city and the federal government act quickly, within a few years, to change yet-unbuilt phases to run in the right direction.

Berlin urban rail traffic map

Here is a map of traffic demand on every interstation on the combined Berlin U- and S-Bahn network (source, p. 6):

The numbers are in thousands of passengers per weekday in both directions combined.

The U-Bahn is in blue. It’s a weird-looking network because two lines (U7, running northwest-southeast in the west, and U9, running north-south also in the west) were built in the Cold War to serve West Berlin’s center around Kurfürstendamm, whereas the S-Bahn and the older U-Bahn lines serve the historic center. Since reunification, Germany has made an effort to move the Berlin central business district back to the historic center, and S21 is to reinforce that, serving the western end of Mitte.

Unfortunately, as we see in the green lines, that’s not where the pressing S-Bahn capacity needs are. First, the Stadtbahn is busier than the North-South Tunnel. Second, the busiest branches heading into the city come from the east, with substantially more traffic than from the north and south.

And then there’s the Görlitz Railway. It is the line heading to the southeast, without its own trunk line through the city – it reverse-branches to the two directions of the Ringbahn. Moreover, going north there’s additional reverse-branching, to the Stadtbahn (S9) and around the Ring to the northern branches (S8, S85), with each service running only every 20 minutes. Total traffic across these services is quite high, 107,000 weekday passengers, compared with 144,000 on the Prussian Eastern Railway (S5, S7, S75; S5 is the mainline), 128,000 between the two branches feeding the North-South Tunnel from the south, and 133,000 between the two branches feeding the North-South Tunnel from the north. The brief segment where S9 runs alongside the Ring has 184,000 weekday passengers, the city’s busiest.

S21: what Berlin is actually building

Berlin Hauptbahnhof is a new station. It only opened in 2006, when the North-South Intercity Line opened. The new four-track line has ample capacity for additional S-Bahn traffic, but nonetheless it hosts no S-Bahn trains in regular service. Instead, there are plans for two additional S-Bahn tracks, mostly in tunnel, parallel to the line, with service to Hauptbahnhof:

The map does not show the phasing. The segment from the Ringbahn in the north down to Hauptbahnhof is just about complete, with opening expected soon. The segment from Hauptbahnhof to Potsdamer Platz, which contrary to the map is to be nonstop, is in early stages of construction, and Wikipedia says it is expected to open in 2023.

Farther south of Potsdamer Platz is still not under construction, and frankly should not be built as is. The only real addition this would give to the network is the stop at Gleisdreieck, where the line intersects the east-west U1; the North-South Tunnel intersects U1 without a connection, the only place in the city where there is a missed U-Bahn/S-Bahn connection unless one counts the marginal U9/Stadtbahn miss in which the next station, Zoologischer Garten, is a transfer.

However, the North-South Main Line’s tunnel portal lies just south of Gleisdreieck, and thus it should be feasible if nontrivial to add platforms there for two of the tracks. Farther south, at Yorckstrasse, it is well outside the portal and adding platforms should be fairly easy.

Görlitz Railway S-Bahn: what Berlin should be building

A radial rail network with three lines should aim to have them meet at a triangle in city center. Berlin has two S-Bahn radial lines, and S21 is to add a third. Instead of running parallel to the North-South Tunnel, it should provide a third trunk line. North of Potsdamer Platz the route is already baked in, but farther south, the Görlitz Railway route is a perfect legacy line to link to. It is quite busy, and the likely locations of the intermediate stops between existing infrastructure and Potsdamer Platz are busy U-Bahn stations in their own right.

I was delighted to see this already discussed on the technical transit blog Zukunft Mobilität. It has a long list of potential Berlin rail extensions, some in accordance with current long-term plans, some not. It specifically criticizes S21 for duplicating existing infrastructure, and proposes an extension to the southeast, mentioning that there were plans to that effect in the 1930s. There are two variants, one through Hermannplatz and one through the old route of the Görlitz Railway.

A higher-zoom 11 MB image is available here.

The dashed lines denote under-construction lines, including S21 to Potsdamer Platz, the 4.5-kilometer Siemens Railway to the northwest, and the U5-U55 connection. Dotted lines denote lines I am proposing: either variant connecting S21 toward the southeast, paired with the Siemens Railway as well as two new-build lines through the area of Tegel Airport, which is slated for redevelopment after the Berlin-Brandenburg Airport finally opens. Two branches are depicted toward Tegel, one toward airport grounds to be redeveloped, and one going farther north taking over S25; there are already discussions of a rapid transit line to Tegel, branching off of U6, but this option does not force the outer parts of U6 to contend with reduced frequency.

The two branches should of course not both be built. The main advantage of the southern option is that it hits Hermannplatz, one of the busiest stations in the system: the above diagram of rail ridership shows a large change in U8 demand north and south of the station, and a factsheet from 2010 asserts that it is the second busiest U-Bahn station, closely behind Alexanderplatz. In effect, it functions as an express link from Neukölln to city center. U8 isn’t especially crowded – nothing in Berlin is – but it’s busiest than the North-South Tunnel; this link is at least as justified as the S21 tunnel to the south. This would require about 7 km of tunnel. While S-Bahn tunnels cost more than U-Bahn tunnels, this is deliberately an express line, so keeping costs down to the per-km level of the U5-U55 connection (€525 million for 2.2 km) is reasonable, making it a €1.8 billion project or thereabout.

The northern option works differently. It doesn’t hit anything as interesting as Hermannplatz on the way, but it does serve Alt-Treptow, one of the bigger rapid transit deserts inside the Ring. The infill station would also break what is the second or third longest interstation on the Ring. Closer in, it has better coverage in the center – Checkpoint Charlie offers another CBD station in addition to Potsdamer Platz. The cost is more of an open question here. From Görlitzer Bahnhof to Potsdamer Platz it’s about 4 km; east of Görlitzer Bahnhof it’s plausible that the line could reuse the Görlitz Railway’s right-of-way and run elevated, or at worst underground with cut-and-cover. However, the per km cost of the tunnel would be higher, since proportionately more of it is in city center, and it has the same number of stations over shorter length; my vague guess is somewhat less than €1.5 billion.

The Berlin S-Bahn would become a system with three radial lines, meeting at Hauptbahnhof, Friedrichstrasse, and Potsdamer Platz. All reverse-branching would cease: the various branches on the Görlitz Railway, including the existing ones as well as an under-construction one to the airport-to-be, would feed into the S-Bahn trunk, rather than to the Ring or the Stadtbahn. The removal of S25 from the North-South Tunnel would create space for the S8 and S85 services in Pankow to use the North-South Tunnel instead of diverting to the Ring and Görlitz Railway. Potentially, the North-South Tunnel could also be realigned to serve Gleisdreieck, as depicted on the map. Finally, with S9 removed from the Stadtbahn, there would be room to beef up service on S3 and/or end the current practice in which S75 trains from the east stop at Ostbahnhof rather than running through.

Germany isn’t perfect

Writing about North America, I talk a lot about how it can Germanize its regional rail network. But it’s important to understand that while far better than North America, Germany is not perfect. It makes mistakes of many kinds: some involving high construction costs, some involving schedule slips, some involving unnecessary prestige projects. These can mostly be prefaced by “by Continental standards,” though the Berlin-Brandenburg Airport disaster is bad even by the standards of the Anglosphere and its billion-dollars-per-kilometer subways.

The Berlin S-Bahn is a case in point. It has a pretty hefty peak-to-base ratio by German standards – the Ring lines (S41 and S42) run every 5 minutes peak and every 10 off-peak, and a number of other lines have a peak-to-base ratio of 2 as well. It also has a peculiarity in that S75 trains only run east of Ostbahnhof; I can’t tell if there’s a problem with track capacity or demand mismatch, but if it’s the former then it’s strange since peak S-Bahn traffic on the Stadtbahn is only 18 trains per hour (Munich achieves 30 through its central tunnel, with much higher crowding levels), and if it’s the latter then it’s again strange – why not run through to Westkreuz like S5?

S21 is another of these little mistakes. It’s a prestige project on the heels of the construction of Hauptbahnhof, rather than a solution to a transportation need. There are six north-south tracks through Berlin between the S-Bahn and the mainline and they’re not anywhere near capacity; the mind boggles at why anyone would add seventh and eighth tracks before adding fifth and sixth east-west tracks.

Fortunately, the mistake is fixable. Germany’s dragging infrastructure timeline means that there’s often room for modifications to make things more useful. The airport is a lost cause, but S21 is not. From Potsdamer Platz south there’s a good option that adds S-Bahn service exactly where it is needed and simplifies citywide schedules by making it feasible to eliminate reverse-branching. In lieu of building more autobahns, Berlin should commit to building the southeastern extension via Alt-Treptow or Neukölln.

Stop Spacing and Route Spacing

Six months ago I blogged a model for optimal stop spacing on an urban transit route. These models exist in the published literature, but they assume that the speed benefit of stop consolidation reduces operating costs, which requires introducing new variables for the value of time. My model assumes the higher speed of stop consolidation is plugged into higher frequency, which means only five variables are needed, and only two of them vary substantially between different cities and their networks. The formula is a square root.

In this post, I’m going to extend this formula to optimizing route spacing on a grid.

I’m using mode-neutral language like “vehicle,” but this is really just about buses, because to a good approximation, urban rail networks are never grids. I’m sorry, Mexico City, I know your Metro network does its best to pretend you have an isotropic city, but your three core radial lines are just far busier than the tangential ones.

Optimal stop spacing: a recap

My previous post uses words rather than symbolic language, since there are only five relevant parameters. Here I’m going to use symbols for the variables to make the calculation even somewhat tractable. All units I’m using are base SI units, so speed is expressed in meters per second rather than kilometers per hour, but the dimensional analysis works out so that it’s not necessary to pick units in advance.

  • s: stop spacing
  • v: walk speed
  • p: stop penalty
  • d: average distance traveled
  • w: walk/wait penalty, expressed as a ratio of perceived walk or wait time to in-vehicle time
  • λ: average distance between successive vehicles, or in other words headway in units of distance, not time

The variables v and p are fairly consistent from place to place. The variable w is as well, but may well differ by circumstance, e.g. people with luggage may have a higher walk penalty and a lower wait penalty, and people who are more familiar with the system usually have lower w. The parameter λ is a function of how much service runs on the line, as we will see when we expand to cover route spacing.

A key assumption in this model is that d does not change based on the network. This is a simplification: if s is too low then it will drag down d with it, as people who are discouraged by the slow in-vehicle speed avoid long trips or choose other modes of travel, whereas if s is too high then it will drag d up, as people who have to walk too long to the stop may just walk all the way to their destination if it’s nearby. In Carlos Daganzo’s textbook this situation is resolved by replacing an empirically determined d with the size of the city, assuming travel is isotropic, but the effect is essentially the same as just setting d to be half the length of a square city.

The formula for perceived travel time is

\frac{sw}{2v} + \frac{dp}{s} + \frac{\lambda wp}{2s}

if travel along the line is isotropic, or

\frac{sw}{4v} + \frac{dp}{s} + \frac{\lambda wp}{2s}

if one end of the travel (e.g. the residential end) is isotropic and the other is at a fixed node (e.g. a subway transfer). In either case, in-vehicle time excluding stops is omitted, as it is constant.

The minimum travel time occurs at

s = \sqrt{2\cdot \frac{v}{w}\cdot p\cdot(d + \frac{\lambda w}{2})}

if travel is isotropic and

s = \sqrt{4\cdot \frac{v}{w}\cdot p\cdot(d + \frac{\lambda w}{2})}

if there is a distinguished node at one end of the trip.

Observe that there is negative interaction between stop consolidation and other aspects of bus modernization. First, higher frequency, as expressed in concentrating service on strong routes, reduces the value of λ and therefore slightly reduces the optimal stop spacing. Second, the model assumes the same penalty w for walking and waiting, but sometimes these two activities have distinct penalties, and then the walk penalty is responsible for the occurrence of w in the denominator in the formula whereas the wait penalty supplies the appearance of w in the numerator. Improving bus stop facilities reduces the wait penalty, pushing the optimal s farther down, even though at the same time it’s cheaper to improve bus stops if there are fewer of them.

The empirically determined values of the five variables in the formula are as follows:

  • v is 1.45 m/s in Forde-Daniel, 1.3-1.4 m/s in Bohannon, and 1.38 in TRB Part 4, PDF-p. 16; I take v = 4/3
  • p is 25 seconds based on examining the differences in schedules between local and limited buses in New York and Vancouver
  • d is 3,360 meters per unlinked trip per the NTD
  • w is around 2 for waiting in Fan-Guthrie-Levinson, 2 in general for buses in Teulings-Ossokina-de Groot, PDF-p. 25, 1.75 in the New York MTA’s internal model, 2.25 in the MBTA’s (as mentioned in one of Reinhard Clever’s papers), and a range of 2-3 in Lago-Mayworm-McEnroe; I take w = 2
  • λ is single-lane network length (that is, twice the route-length, modulo one-way loops) divided by fleet size in actual use, which is 1,830 meters in Brooklyn today and 1,160 based on what Eric Goldwyn and I recommend

This leads to optimal stop spacing equal to

s = \sqrt{2\cdot \frac{4/3}{2}\cdot 25\cdot(3360 + \frac{1160\cdot 2}{2})} = 388 \mbox{ meters}

if travel is isotropic and

s = \sqrt{4\cdot \frac{4/3}{2}\cdot 25\cdot(3360 + \frac{1160\cdot 2}{2})} = 549 \mbox{ meters}

if there is a distinguished node. The numbers are slightly lower than in my older post since I’m using a slightly lower walk speed, 1.33 m/s rather than 1.5.

Optimal route spacing: stops at intersection points

Studying route spacing has to incorporate stop spacing for a simple reason: there should be a stop at every intersection between routes, and therefore the route spacing should be an integer multiple of the stop spacing. There are three modifications required to the above formula, of which the first is easy, the second requires defining more parameters but is mathematically still easy, and the third is very hard:

  1. Passengers need to walk not just along the route to their stop but also from their origin to the route, which increases walk time
  2. The value of λ may change, since fewer routes imply more vehicles per route and thus denser vehicle spacing, and in particular wait time depends not just on how many stops are on the way but also on the speed net of stops
  3. Increasing the route and stop spacing in tandem reduces the number of stops involved in waiting for the bus (this is λ again) twice, that is quadratically

The first modification means that instead of traveling an average distance of s/4 to the stop at each end, assuming isotropy, people have to travel a distance of s/4 along the route and also s/4 to the route itself. In the travel time formula, we replace sw/2v with just sw/v with isotropic travel.

To deal with the second modification, we define the following variables, in addition to the ones from the section above on stop spacing:

  • f: fleet size in independent vehicles in actual revenue operation (buses or trains, not train cars)
  • a: area of the network to be covered by the grid, e.g. a city, metro area, or borough
  • u: speed assuming there are no stops along the route

If the area is a, then we can approximate it as a square of side \sqrt{a}, which has \sqrt{a}/s north-south and \sqrt{a}/s east-west routes, each of length \sqrt{a}, and thus the total two-way network length is 2a/s. Since the value of λ is the one-way length divided by fleet size, we write

\lambda = \frac{4a}{sf}

Moreover, people wait an additional λw/2u; in the previous section this wait existed as well but was ignored in the formula as it did not depend on s, but here it does, and thus we need to add this wait factor.

We deal with the third modification by replacing λ with 4a/sf in the formula for wait time. If people travel isotropically and do not transfer, the travel time formula is now

\frac{sw}{v} + \frac{dp}{s} + \frac{d}{u} + \frac{2aw}{sfu} + \frac{2awp}{fs^{2}}

The summand d/u is constant but is included for completeness here, in analogy with the no-longer-constant summand 2aw/sfu.

But it’s the last summand that gives the most problems: it turns the optimization problem from extracting a square root to solving a cubic. This is technically possible, but the formula is opaque and does not really help showcase how the parameters affect the final outcome. We need to solve for s:

\frac{w}{v}s^{3} - (dp + \frac{2aw}{fu})s - \frac{4apw}{f} = 0

We can plug in the above values of w, v, d, and p, as well as the following values of the new variables, and use any cubic solver:

  • f = 612 buses in Brooklyn, excluding vehicles in turnaround, non-revenue service, etc. (it’s actually slightly lower today, around 600, but our network is a bit more efficient with depot moves)
  • a = 180,000,000 m^2 for Brooklyn
  • u = 5.3 m/s net of stops, assuming our other proposals, such as bus lanes, are implemented

The cubic formula turns into

1.5s^{3} - 305976s - 58823529 = 0

for which the positive solution is s = 528 meters.

We can complicate this formula in two ways.

First, we can let go of the assumption of isotropy. If there is a distinguished node at one end, then walk time is halved, as in the formula for stop spacing on a given route. The overall travel time is equal to

\frac{sw}{2v} + \frac{dp}{s} + \frac{d}{u} + \frac{2aw}{sfu} + \frac{2awp}{fs^{2}}

and this is optimized when

\frac{w}{2v}s^{3} - (dp + \frac{2aw}{fu})s - \frac{4apw}{f} = 0.

Plugging the usual values of the parameters, we get

0.75s^{3} - 305976s - 58823529 = 0,

for which the positive solution is s = 719 meters. The ratio between the results with isotropy and a distinguished node is 1.36, close to the square root of 2 that we get in the formula for stop spacing on a predetermined route; the reason is that in the cubic formula the linear term is much larger than the constant term near the root, so the effect of changing the cubic term is much closer to the square root than to the cube root.

The second complication is introducing transfers. Transfers do not change the walk time – the walking time between platforms or curbside waiting areas is small and constant – but introduce additional wait time, which means we need to double both terms that include waits. But if we have transfers we need to restore the assumption of isotropic travel, since for the most part the distinguished nodes for Brooklyn buses involve subway transfers.

In that case, the travel time formula is

\frac{sw}{v} + \frac{dp}{s} + \frac{d}{u} + \frac{4aw}{sfu} + \frac{4awp}{fs^{2}}

which is minimized at the positive root of the cubic

\frac{w}{v}s^{3} - (dp + \frac{4aw}{fu})s - \frac{8apw}{f} = 0.

We need to figure out the value of d, which is difficult in this case – the New York bus network discourages bus-to-bus transfers through low frequency and poor bus stop amenities. That the formulas I’m using do not allow for how the shape of the network influences d is a real drawback here. But if we let d be the usual 3,360 meters that it is for unlinked trips, and plug the usual values of the other parameters, we get,

1.5s^{3} - 527951s - 117647059 = 0

to which the solution is s = 683 meters.

Optimal route spacing: the general case

The above section makes a critical assumption about route spacing and stop spacing: they must be equal, making every stop a transfer. However, this assumption is not strictly necessary. Indeed, if we assume isotropy, and let the route spacing be 860 meters, then it’s better for passengers to double the density of stops to one every 430 meters just from looking at the formula for stop spacing.

In this section, we look at the optimal formulas assuming route spacing is twice or thrice the stop spacing. Then in the next section we will compare everything together.

We keep all the variable names from before, and set s to be the stop spacing, not the route spacing. Instead, we will find formulas for route spacing equal to 2s and 3s and compare their optima with that for the special case in which stop and route spacing are equal.

We need to modify the formula in the previous section in two ways. First, walk time is, in the isotropic case, half the stop spacing plus half the route spacing. And second, the dependence of λ on the shape of the network comes from route spacing rather than stop spacing. If route spacing is 2s, the formula for travel time is

\frac{3sw}{2v} + \frac{dp}{s} + \frac{d}{u} + \frac{aw}{sfu} + \frac{awp}{fs^{2}}

and its minimum is at the positive solution to

\frac{3w}{2v}s^{3} - (dp + \frac{aw}{fu})s - \frac{2apw}{f} = 0.

We retain the New York- and Brooklyn-oriented variables from the above sections and obtain

2.25s^{3} - 194989s - 29411765 = 0.

The solution is s = 352 meters, i.e. routes are to be spaced 704 meters apart, with one intermediate station on each route between each pair of successive crossing routes.

If we have three interstation segments between two successive routes, then we need to solve the cubic

\frac{2w}{v}s^{3} - (dp + \frac{2aw}{3fu})s - \frac{4apw}{3f} = 0

or

3s^{3} - 157992s - 19607843 = 0

to which the solution is s = 276 meters.

In the above section we also looked at two potential complications: introducing transfers, and introducing non-isotropy. Non-isotropy, expressed as an isotropic origin and a distinguished destination, halves the cubic term; transfers double the wait times and thus double the constant term and the larger of the two summands adding up to the linear term.

If the route spacing is exactly twice the stop spacing, then the non-isotropic formula is

\frac{3w}{4v}s^{3} - (dp + \frac{aw}{fu})s - \frac{2apw}{f} = 0

or, using the same parameters as always,

1.125s^{3} - 194989s - 29411765 = 0.

The solution is s = 420 meters, with routes spaced 840 meters apart.

The isotropic cubic with transfers is

\frac{3w}{2v}s^{3} - (dp + \frac{2aw}{fu})s - \frac{4apw}{f} = 0

and with the usual parameters, again sticking with d = 3,360 even though in practice it is likely to be higher, this is

2.25s^{3} - 305976s - 58823529 = 0

and then the root is s = 442 meters, with routes spaced 884 meters apart.

We conclude this section with the same formulas assuming the route spacing is not 2s but 3s. The non-isotropic, one-seat ride formula is

\frac{w}{v}s^{3} - (dp + \frac{2aw}{3fu})s - \frac{4apw}{3f} = 0

or with the usual parameters

1.5s^{3} - 157992s - 19607843 = 0,

of which the positive root is s = 374 meters, with routes spaced 1,123 meters apart,

The transfer-based isotropic formula is,

\frac{2w}{v}s^{3} - (dp + \frac{4aw}{3fu})s - \frac{8apw}{3f} = 0

or

3s^{3} - 231984s - 39215686 = 0.

The positive root is s = 340 meters, with routes spaced 1,021 meters apart.

What’s the best route spacing?

We have optimums based on assumptions about the interaction between stop and route spacing, but so far we have not compared these assumptions with each other. In this section, we do. For each scenario – isotropic, transfer-free travel; a distinguished node along transfer-free travel; and isotropic travel with a transfer – we look at the optimal values of route spacing equal to one, two, or three times the stop spacing.

In the table below, the walk and wait times are without penalty; but the penalty is applied to them when summed with in-vehicle time.

Scenario Component Route spacing = s Route spacing = 2s Route spacing = 3s
Isotropy; 1-seat ride Optimal s 528 352 276
Walk time 396 396 414
Wait time 262.954 216.997 198.394
In-vehicle time 793.053 872.599 938.31
Total time 2110.962 2098.593 2163.097
Distinguished node; 1-seat ride Optimal s 719 420 374
Walk time 269.625 236.25 280.5
Wait time 182.811 173.812 133.965
In-vehicle time 750.791 833.962 858.561
Total time 1655.663 1654.086 1687.49
Isotropy; 2-seat ride Optimal s 683 442 340
Walk time 512.25 497.25 510
Wait time 388.05 326.378 302.432
In-vehicle time 756.949 824.008 881.021
Total time 2557.549 2471.263 2505.885

 

The table implies that in all scenarios it’s optimal to have two interstations between parallel routes, though if there’s a distinguished node the difference with having just one interstation between parallel routes is very small. The three-interstation option is never optimal, but is also never far from the optimum, only half a minute to a minute worse.

But please interpret the table with caution, especially the two-seat ride section. The total time for a 3.36-kilometer trip without applying the walk or wait penalty is about 28 minutes regardless of whether the route to stop spacing ratio is 1, 2, or 3. This is still faster than walking, but not by much, and riders may well be so discouraged as to walk the entire way. If the trip is much shorter than 3.36 kilometers or the rider’s particular disutility of walking is much lower than 2 then transit will not be competitive with walking. In turn, a network set up with the stop spacing implied by the above formulas will only get transfer trips if they’re much longer, which should raise the optimal interstation somewhat. If d = 6,000 then in the transfer scenario the optimum if stop and route spacing are equal is 711 meters and that if route spacing is twice as high as stop spacing is 470 meters, and the latter option is noticeable faster.

How does our bus redesign compare with the theory?

We drew our redesigned map with full knowledge of how to optimize stop spacing on a single route, but we didn’t look at route spacing optimization. Of course, the assumption of regular route spacing is less realistic than that of regular stop spacing, as some areas have higher demand, or more distinguished arterials. But we can still discuss the average route spacing in our plan, by comparing our proposed route-length with Brooklyn’s land area.

With a 356-kilometer network in a borough of 180 km^2, effective route spacing is 1,010 meters. This is a little longer than I expected; in Southern Brooklyn the north-south and east-west routes we propose are spaced around 800-850 meters apart, and in Bed-Stuy the east-west routes tighten to 600 meters as they’re all radial toward Downtown Brooklyn and quite busy. The reason the answer is 1,010 meters is that there are margins of the borough with no service (like Floyd Bennett Field) or grid interruptions due to parks (such as Prospect Park) or already-good subway service (South Brooklyn).

The stop spacing we use is 480 meters, excluding nonstop freeway segments in the Brooklyn-Battery Tunnel and toward JFK. In the Southern Brooklyn grid, we’re pretty close to a regular spacing of two interstations between parallel routes. In the Bed-Stuy grid, the north-south routes have a stop per crossing route since the east-west routes are so densely placed, and the east-west routes have one, two, or three interstations between crossing routes, but the average is two.

To the extent the optimization formulas tell us anything, it’s that we should consider adding a few more routes. Target additions include another north-south Bed-Stuy route, an east-west route in South Brooklyn restoring the discontinued B71, and a north-south route through Southern Brooklyn on 16th Avenue. Altogether this would add around 20 km to our network. Beyond that, additional routes would duplicate subway routes, which my analysis above excludes even when they form a coherent grid with the buses.

Rules of thumb for your city

If your city has streets that form a coherent grid, then you can design a bus grid without too many constraints. By constraints I mean street networks that interrupt the grid so often so as to force you to use particular streets at particular spacing, for example the Bronx or Queens. Constraints in a way make planning easier, by reducing the search space; I contend Brooklyn is the hardest of the four main boroughs to redesign precisely because it has the fewest constraints in its grids and yet its grid is just interrupted enough that it cannot be treated as tabula rasa.

In general, you probably want buses spaced around 800 meters to a kilometer apart. While the value of d will differ between cities, the optimum route spacing isn’t that sensitive to it. If d rises to as high as 10,000, the optimal s in the scenario with transfers is 753 meters if route spacing equals stop spacing and 511 meters if it equals twice stop spacing, compared with 683 and 442 meters respectively with d = 3,360; the one-interstation-per-parallel-route scenario becomes better than the two-interstation scenario, but the difference is half a minute, compared with a minute and a half in favor of two interstations with d = 3,360.

In practice I don’t know of any city whose grid is so unconstrained and so isotropic that you can seriously debate 700, 800, 900, 1,000, etc. meters between routes. At that resolution you’re always constrained by arterial spacing, which in American cities tends to be 800 because it’s half a mile and in Canada is irregular (de facto close to a mile) due to constant grid interruptions on intermediate would-be arterials in both Toronto and Vancouver. In this range of arterial spacing, you want exactly two interstations between parallel routes; if you want more or fewer then you should have a very good reason, such as a major destination such as a hospital located at an awkward offset.

Something that does matter very much is fleet size relative to the area served – the quantity a/f. If you aren’t running much service, then you need wider route spacing just to avoid reducing frequency to unusable levels. If instead of f = 612 we use f = 200, then the optimum with one interstation per parallel routes with the transfer scenario is s = 1087, with two it’s s = 676, with three it’s s = 508, and with four it’s s = 414, and among these three is best and even four is a few seconds faster than two. In that case route spacing of about a kilometer and a half, which may be a mile in American arterials, is fully justified.

Conversely, if buses are faster, that is if u is higher, then the optimal interstations fall in all cases. This is because the impact of u comes from its effect on wait times, so faster buses mean that it’s less important to reduce λ.

The effects of a/f and u relate again to the negative interactions between various components of bus reform. Running more service means it’s justifiable to have more closely-spaced routes, since pruning routes to increase frequency from 10 to 5 minutes is much less valuable than pruning them to increase frequency from 30 to 15 minutes. Likewise, running faster service means wait times fall, again reducing the need to prune routes.

If you’re tasked with designing bus routes, then make sure to use correct values for a, f, u, and d for your city, as they are likely to be very different from those of New York. The formulas are more intricate when optimizing route spacing and it’s useful to play with them until you get comfortable with them on an intuitive level, but ultimately they do give reasonable answers for how to design a bus network.

How Ambitious is Mayor de Blasio’s Bus Plan?

You have to give Bill de Blasio credit: when someone else forces his hand, he will immediately claim that he was on the more popular-seeming side all along. After other people brought up the idea of a bus turnaround, starting with shadow agencies like TransitCenter and continuing with his frontrunning successor Corey Johnson, the mayor released an action plan called Better Buses. The plan has a bold goal: to speed up buses to 16 km/h using stop consolidation and aggressive enforcement of bus priority. And yet, elements of the plan leave a bad taste in my mouth.

Bus speeds

The Better Buses plan asserts that the current average bus speed in New York is 8 miles per hour, and with the proposed treatments it will rise to 10. Unfortunately, the bus speed in New York is lower. The average according to the NTD is 7.05 miles per hour, or 11.35 km/h. This includes the Select Bus Service routes, whose average speed is actually a hair less than the New York City Transit average, since most of them are in more congested parts of the city. The source the report uses for the bus speed is an online feed that isn’t reliable; when I asked one of the bus planners while working on the Brooklyn route redesign, I was told the best source to use was the printed schedules, and those agree with the slower figures.

In Brooklyn, the average bus speed based on the schedules is around 11 km/h. But the starting point for the speed treatment Eric Goldwyn and I recommended is actually somewhat lower, around 10.8 km/h, for two reasons: first, the busiest routes already have faster limited-stop overlays, and second, the redesign process itself reduces the average speed by pruning higher-speed lightly-used routes such as the B39 over the Williamsburg Bridge.

The second reason is not a general fact of bus redesigns. In Barcelona, Nova Xarxa increased bus speeds by removing radial routes from the congested historic center of the city. However, in Brooklyn, the redesign marginally slows down the buses. While it does remove some service from the congested Downtown Brooklyn area, most of the pruning in is outlying areas, like the industrial nooks and crannies of Greenpoint and Williamsburg. Without having drawn maps, I would guess the effect in Queens should be marginal in either direction, for essentially the same set of reasons as in Brooklyn, but in the Bronx it should slow down the buses by pruning coverage routes in auto-oriented margins like Country Club.

With all of the treatments Eric and I are proposing, the speed we are comfortable promising if our redesign is implemented as planned is 15 km/h and not 16 km/h.

How does the plan compare with the speaker’s?

City Council Speaker Johnson’s own plan for city control of NYCT proposes a bus turnaround as well. Let us summarize the differences between the two plans:

Aspect Johnson’s plan De Blasio’s plan
Route redesign Yes Yes
Bus shelters Yes Probably
Stop consolidation Not mentioned Yes
Bus lanes 48 km installed per year 16-24 km installed per year
Bus lanes vs. cars Parking removal if needed Not mentioned
Physically separated bus lanes Yes 3 km pilot
Median bus lanes Probably Maybe
Signal priority 1000 intersections equipped per year 300 intersections equipped per year

For the most part, the mayor’s plan is less ambitious. The question of bus lanes is the most concerning. What Eric and I think the Brooklyn bus network should look like is about 350 km. Even excluding routes that already have bus lanes (like Utica) or that have so little congestion they don’t need bus lanes (like the Coney Island east-west route), this is about 300 km. Citywide this should be on the order of 1,000 km. At the speaker’s pace this is already too slow, taking about 20 years, but at the mayor’s, it will take multiple generations.

The plan does bring up median lanes positively, which I appreciate: pp. 10-11 talk about center-running lanes in the context of the Bx6, which has boarding islands similar to those I have observed on Odengatan in Stockholm and Boulevard Montparnasse in Paris. Moreover, it suggests physically separated lanes, although the picture shown for the Bx6 involves a more obtrusive structure than the small raised curbs of Paris, Stockholm, and other European cities where I’ve seen such separation. Unfortunately, the list of tools on pp. 14-15 assumes bus lanes remain in or near the curb, talking about strategies for curb management.

The omission of Nostrand

The mayor’s plan has a long list of examples of bus lane installation. These include some delicate cases, like Church Avenue. However, the most difficult, Nostrand, is entirely omitted.

Nostrand Avenue carries the B44, the second busiest bus in the borough and fifth in the city. The street is only 24 meters wide and therefore runs one-way southbound north of Farragut Avenue, just north of the crossing with Flatbush Avenue and Brooklyn College. Northbound buses go on New York Avenue if they’re local or on Rogers if they’re SBS, each separated from Nostrand by about 250 meters. The argument for the split is that different demographics ride local and SBS buses, and they come from different sides of Nostrand. The subway is on Nostrand and so is the commerce. And yet, parking is more important to the city than a two-way bus lane on the street to permit riders to access the main throughfare of the area most efficiently.

Moreover, even the bus lanes that the plan does discuss leave a lot to be desired. The second most important street in Brooklyn to equip with high-quality physically separated bus lanes, after Nostrand, is Church, like Nostrand a 24-meter street where something has to give. The plan trumpets its commitment to transit priority, and yet on Church it includes a short segment with curb lanes partly shared with delivery trucks using curb management. Limiting merchant complaints is more important to the mayor than making sure people can ride buses that are reliably faster than a fast walk.

Can the city deliver?

Probably not.

The mayor has recurrently prioritized the needs of people who are used to complaining at public meetings, who are typically more settled in the city, with a house and a car. New York may have a majority of its households car-free, but to many of them car ownership remains aspirational and so does home ownership, to the point that the transit-oriented lifestyle remains a marker of either poverty or youth, to be replaced with the suburban auto-oriented lifestyle as one achieves middle-class status. Even as there is cultural change and this mentality is increasingly not true, the city’s political system keeps a process that guarantees that millions of daily transit users must listen to drivers who complain that they have to park a block away.

The plan has an ambitious number: 16 km/h. But when it comes to actually implementing it, it dithers. Its examples of bus lanes are half-measures. There’s no indication that the city is willing to overrule merchants who think they have a God-given right to the street that their transit-riding customers do not. Without this, bus lanes will remain an unenforced joke, and the vaunted speed improvements will be localized to too small a share of bus route-km to truly matter.

The most optimistic take on Better Buses is that the mayor is signaling that he’s a complete nonentity when it comes to bus improvement, rather than an active obstacle. But more likely, the signal is that the mayor has heard that there are political and technical efforts to improve bus service in the city and he wants to pretend to participate in them while doing nothing.