In a number of large cities with both radial and circumferential urban rail service, there is a curious observation: there is express service on the radial lines, but not the circumferential ones. These cities include New York, Paris, and Berlin, and to some extent London and Seoul. Understanding why this is the case is useful in general: it highlights guidelines for urban public transport design that have implications even outside the distinction between radial and circumferential service. In brief, circumferential lines are used for shorter trips than radial lines, and in large cities connect many different spokes so that an express trip would either skip important stations or not save much time.
Berlin has three S-Bahn trunk lines: the Ringbahn, the east-west Stadtbahn, and the North-South Tunnel. The first two have four tracks. The last is a two-track tunnel, but has recently been supplemented with a parallel four-track North-South Main Line tunnel, used by regional and intercity trains.
The Stadtbahn has a straightforward local-express arrangement: the S-Bahn uses the local tracks at very high frequency, whereas the express tracks host less frequent regional trains making about half as many stops as well as a few intercity trains only making two stops. The north-south system likewise features very frequent local trains on the S-Bahn, and a combination of somewhat less frequent regional trains making a few stops on the main line and many intercity trains making fewer stops. In contrast, the Ringbahn has no systemic express service: the S-Bahn includes trains running on the entire Ring frequently as well as trains running along segments of it stopping at every station on the way, but the only express services are regional trains that only serve small slivers on their way somewhere else and only come once or twice an hour.
This arrangement is mirrored in other cities. In Paris, the entire Metro network except Line 14 is very local, with the shortest interstations and lowest average speeds among major world metro systems. For faster service, there is Line 14 as well as the RER system, tying the suburbs together with the city. Those lines are exclusively radial. The busiest single RER line, the RER A, was from the start designed as an express line parallel to Line 1, the Metro’s busiest, and the second busiest, the RER B, is to a large extent an express version of the Metro’s second busiest line, Line 4. However, there is no RER version of the next busiest local lines, the ring formed by Lines 2 and 6. For non-Metro circumferential service, the region went down the speed/cost tradeoff and built tramways, which have been a total success and have high ridership even though they’re slow.
In New York, the subway was built with four-track main lines from the start to enable express service. Five four-track lines run north-south in Manhattan, providing local and express service. Outside the Manhattan core, they branch and recombine into a number of three- and four-track lines in Brooklyn, Queens, and the Bronx. Not every radial line in New York has express service, but most do. In contrast, the circumferential Crosstown Line, carrying the G train, is entirely local.
In Seoul, most lines have no express service. However, Lines 1, 3, and 4 interline with longer-range commuter rail services, and Lines 1 and 4 have express trains on the commuter rail segments. They are all radial; the circumferential Line 2 has no express trains.
Finally, in London, the Underground has few express segments (all radial), but in addition to the Underground the city has or will soon have express commuter lines, including Thameslink and Crossrail. There are no plans for express service parallel to the Overground.
Is Tokyo really an exception?
Tokyo has express trains on many lines. On the JR East network, there are lines with four or six tracks all the way to Central Tokyo, with local and express service. The private railroads usually have local and express services on their own lines, which feed into the local Tokyo subway. But not all express services go through the primary city center: the Ikebukuro-Shibuya corridor has the four-track JR Yamanote Line, with both local services (called the Yamanote Line too, running as a ring to Tokyo Station) and express services (called the Saikyo or Shonan-Shinjuku Line, continuing north and south of the city); Tokyo Metro’s Fukutoshin Line, serving the same corridor, has a timed passing segment for express trains as well.
However, in three ways, the area around Ikebukuro, Shinjuku, and Shibuya behaves as a secondary city center rather than a circumferential corridor. The job density around all three stations is very high, for one. They have extensive retail as well, as the private railroads that terminated there before they interlined with the subway developed the areas to encourage more people to use their trains. This situation is also true of some secondary clusters elsewhere in Tokyo, like Tobu’s Asakusa terminal, but Asakusa is in a historically working-class area, whereas the Yamanote area was historically and still is wealthier, making it easier for it to attract corporate jobs.
Second, from the perspective of the transportation network, they are central enough that railroads that have the option to serve them do so, even at the expense of service to Central Tokyo. When the Fukutoshin Line opened, Tokyu shifted one of its two mainlines, the Toyoko Line, to connect to it and serve this secondary center, where it previously interlined with the Hibiya Line to Central Tokyo; Tokyu serves Central Tokyo via its other line, the Den-en-Toshi Line, which connects to the Hanzomon Line of the subway. JR East, too, prioritizes serving Shinjuku from the northern and southern suburbs: the Shonan-Shinjuku Line is a reverse-branch of core commuter rail lines both north and south, as direct fast service from the suburbs to Shibuya, Shinjuku, and Ikebukuro is important enough to JR East that it will sacrifice some reliability and capacity to Tokyo Station for it.
Third, as we will discuss below, the Yamanote Line has a special feature missing from circumferential corridors in Berlin and Paris: it has distinguished stations. A foreigner looking at satellite photos of land use and at a map of the region’s rail network without the stations labeled would have an easy time deciding where an express train on the line should stop: Ikebukuro, Shinjuku, and Shibuya eclipse other stations along the line, like Yoyogi and Takadanobaba. Moreover, since these three centers were established to some extent before the subway was built, the subway lines were routed to serve them; there are 11 subway lines coming from the east as well as the east-west Chuo Line, and of these, all but the Tozai and Chiyoda Lines intersect it at one of the three main stations.
Interstations and trip length
The optimal stop spacing depends on how long passenger trips are on the line: keeping all else equal, it is proportional to the square root of the average unlinked trip. The best formula is somewhat more delicate: widening the stop spacing encourages people to take longer trips as they become faster with fewer intermediate stops and discourages people from taking shorter ones as they become slower with longer walk distances to the station. However, to a first-order approximation, the square root rule remains valid.
The relevance is that not all lines have the same average trip length. Longer lines have longer trips than short lines. Moreover, circular lines have shorter average trips than straight lines of the same length, because people have no reason to ride the entire way. The Ringbahn is a 37-kilometer line on which trains take an hour to complete the circuit. But nobody has a reason to ride more than half the circle – they can just as well ride the shorter way in the other direction. Nor do passengers really have a reason to ride over exactly half the circle, because they can often take the Stadtbahn, North-South Tunnel, or U-Bahn and be at their destinations faster.
Circumferential lines are frequently used to connect to radial lines if the radial-radial connection in city center is inconvenient – maybe it’s missing entirely, maybe it’s congested, maybe it involves too much walking between platforms, maybe happens to be on the far side of city center. In all such cases, people are more likely to use the circumferential line for shorter trips than for longer ones: the more acute the angle, the more direct and thus more valuable the circle is for travel.
The relevance of this discussion to express service is that there’s more demand for express service in situations with longer optimum stop spacing. For example, the optimum stop spacing for the subway in New York based on current travel patterns is the same as that proposed for Second Avenue Subway, to within measurement error of parameters like walking speed; on the other trunk lines, the local trains have denser stop spacing and the express trains have wider stop spacing. On a line with very short optimum spacing, there is not much of a case for express service at all.
Distinguished stops versus isotropy
The formula for optimal stop spacing depends on the isotropy of travel demand. If origins and destinations are distributed uniformly along the line, then the optimal stop spacing is minimized: passengers are equally likely to live and work right on top of a station, which eliminates walk time, as they are to live and work exactly in the middle between two stations, which maximizes walk time. If the densities of origins and destinations are spiky around distinguished nodes, then the optimal stop spacing widens, because planners can place stations at key locations to minimize the number of passengers who have to walk longer. If origins are assumed to be perfectly isotropic but destinations are assumed to be perfectly clustered at such distinguished locations as city center, the optimum stop spacing is larger than if both are perfectly isotropic by a factor of .
Circumferential lines in large cities do not have isotropic demand. However, they have a great many distinguished stops, one at every intersection with a radial rail service. Out of 27 Ringbahn stops, 21 have a connection to the U-Bahn, a tramway, or a radial S-Bahn line. Express service would be pointless – the money would be better spent increasing local frequency, as ridership on short-hop trips like the Ringbahn’s is especially sensitive to wait time.
On the M2/M6 ring in Paris, there are 49 stops, of which 21 have connections to other Metro lines or the RER, one more doesn’t but really should (Rome, with a missed connection to an M14 extension), and one may connect to a future extension of M10. Express service is not completely pointless parallel to M2/M6, but still not too valuable. Even farther out, where the Paris region is building the M15 ring of Grand Paris Express, there are 35 stops in 69 kilometers of the main ring, practically all connecting to a radial line or located at a dense suburban city center.
The situation in New York is dicier, because the G train does have a distinguished stop location between Long Island City and Downtown Brooklyn, namely the connection to the L train at Bedford Avenue. However, the average trip length remains very short – the G misses so many transfers at both ends that end-to-end riders mostly stay on the radials and go through Manhattan, so the main use case is taking it a few stops to the connection to the L or to the Long Island City end.
A large urban rail network should be predominantly radial, with circumferential lines in dense areas providing additional connectivity between inner neighborhoods and decongesting the central transfer points. However, that the radial and circumferential lines are depicted together on the same metro or regional rail map does not mean that people use them in the same way. City center lies ideally on all radials but not on the circumferentials, so the tidal wave of morning commuters going from far away to the center is relevant only to the radials.
This difference between radials and circumferentials is not just about service planning, but also about infrastructure planning. Passengers make longer trips on radial lines, and disproportionately travel to one of not many distinguished central locations; this encourages longer stop spacing, which may include express service in the largest cities. On circumferential lines, they make shorter trips to one of many different connection points; this encourages shorter stop spacing and no express service, but rather higher local frequency whenever possible.
Different countries build rapid transit in radically different ways, and yet big cities in a number of different countries have converged on the same pattern: express service on the strongest radial corridors, local-only service on circumferential ones no matter how busy they are. There is a reason. Transportation planners in poorer cities that are just starting to build their rapid transit networks as well in mature cities that are adding to their existing service should take heed and design infrastructure accordingly.
The American rail activist term regional rail refers to any mainline rail service short of intercity, which lumps two distinct service patterns. In some German cities, these patterns are called S-Bahn and RegionalBahn, with S-Bahn referring to urban rail running on mainline tracks and RegionalBahn to longer-range service in the 50-100 km range and sometimes even beyond. It’s useful to distinguish the two whenever a city wishes to invest in its regional rail network, because the key infrastructure for the two patterns is different.
As with many this-or-that posts of mine, the distinction is not always clear in practice. For one, in smaller cities, systems that are labeled S-Bahns often work more like RegionalBahn, for example in Hanover. Moreover, some systems have hybrid features, like the Zurich S-Bahn – and what I’ve advocated in American contexts is a hybrid as well. That said, it’s worth understanding the two different ends of this spectrum to figure out what the priority for rail service should be in each given city.
S-Bahn as urban rail
The key feature of the S-Bahn (or the Paris RER) is that it has a trunk that acts like a conventional urban rapid transit line. There are 6-14 stations on the trunks in the examples to keep in mind, often spaced toward the high end for rapid transit so as to provide express service through city center, and all trains make all stops, running every 3-5 minutes all day. Even if the individual branches run on a clockface schedule, people do not use the trunk as a scheduled railroad but rather show up and go continuously.
Moreover, the network layout is usually complementary with existing urban rail. The Munich S-Bahn was built simultaneously with the U-Bahn, and there is only one missed connection between them, The Berlin S-Bahn and U-Bahn were built separately as patchworks, but they too have one true missed connection and one possible miss that depends on which side of the station one considers the crossing point to be on. The RER has more missed connections with the Metro, especially on the RER B, but the RER A’s station choice was designed to maximize connections to the most important lines while maintaining the desired express stop spacing.
Urban rail lines rarely terminate at city center, and the same is true for S-Bahn lines. In cities whose rail stations are terminals, such as Paris, Munich, Frankfurt, and Stuttgart, there are dedicated tunnels for through-service; London is building such a tunnel in Crossrail, and built one for Thameslink, which has the characteristics of a hybrid. In Japan, too, the first priority for through-running is the most local S-Bahn-like lines – when there were only six tracks between Tokyo and Ueno, the Yamanote and Keihin-Tohoku Lines ran through, as did the Shinkansen, whereas the longer-range regional lines terminated at the two ends until the recent through-line opened.
The difference between an S-Bahn and a subway is merely that the subway is self-contained, whereas the S-Bahn connects to suburban branches. In Tokyo even this distinction is blurred, as most subway lines connect to commuter rail lines at their ends, often branching out.
RegionalBahn as intercity rail
Many regional lines descend from intercity lines that retooled to serve local traffic. Nearly every trunk line entering London from the north was built as a long-range intercity line, most commuter rail mainlines in New York are inner segments of lines that go to other cities or used to (even the LIRR was originally built to go to Boston, with a ferry connection), and so on.
In Germany, it’s quite common for such lines to maintain an intercity characteristic. The metropolitan layout of Germany is different from that of the English-speaking world or France. Single-core metro regions are rather small, except for Berlin. Instead, there are networks of independent metropolitan cores, of which the largest, the Rhine-Ruhr, forms an urban complex almost as large as the built-up areas of Paris and London. Even nominally single-core metro regions often have significant independent centers with long separate histories. I blogged about the Rhine-Neckar six months ago as one such example; Frankfurt is another, as the city is ringed by old cities including Darmstadt and Mainz.
But this is not a purely German situation. Caltrain connects what used to be two independent urban areas in San Francisco and San Jose, and many outer ends of Northeastern American commuter lines are sizable cities, such as New Haven, Trenton, Providence, and Worcester.
The intercity characteristic of such lines means that there is less need to make them into useful urban rail; going express within the city is more justifiable if people are traveling from 100 km away, and through-running is a lower priority. Frequency can be lower as well, since the impact of frequency is less if the in-vehicle travel time is longer; an hourly or half-hourly takt can work.
S-Bahn and RegionalBahn combinations
The S-Bahn and RegionalBahn concepts are distinct in history and service plan, but they do not have to be distinct in branding. In Paris, the distinction between Transilien and the RER is about whether there is through-running, and thus some lines that are RegionalBahn-like are branded as RER, for example the entire RER C. Moreover, with future extension plans, the RER brand will eventually take over increasingly long-distance regional service, for example going east to Meaux. Building additional tunnels to relieve the worst bottlenecks in the city’s transport network could open the door to connecting every Transilien line to the RER.
Zurich maintains separate brands for the S-Bahn and longer-distance regional trains, but as in Paris, the distinction is largely about whether trains terminate on the surface or run through either of the tunnels underneath Hauptbahnhof. Individual S-Bahn branches run every half hour, making extensive use of interlining to provide high frequency to urban stations like Oerlikon, and many of these branches go quite far out of the city. It’s not the same as the RER A and B or most of the Berlin S-Bahn, with their 10- and 15-minute branch frequencies and focus on the city and innermost suburbs.
But perhaps the best example of a regional rail network that really takes on lines of both types is that of Tokyo. In branding, the JR East network is considered a single Kanto-area commuter rail network, without distinctions between shorter- and longer-range lines. And yet, the rapid transit services running on the Yamanote, Keihin-Tohoku, and Chuo-Sobu Lines are not the same as the highly-branched network of faster, longer-range lines like Chuo Rapid, Yokosuka, Sobu Rapid, and so on.
The upshot is that cities do not need to neatly separate their commuter rail networks into two separate brands as Berlin does. The distinction is not one of branding for passengers, but one of planning: should a specific piece of infrastructure be S-Bahn or RegionalBahn?
Highest and best use for infrastructure
Ordinarily, the two sides of the spectrum – an S-Bahn stopping every kilometer within the city, and a RegionalBahn connecting Berlin with Magdeburg or New York with New Haven – are so different that there’s no real tradeoff between them, just as there is no tradeoff between building subways and light rail in a city and building intercity rail. However, they have one key characteristic leading to conflict: they run on mainline track. This means that transportation planners have to decide whether to use existing mainline tracks for S-Bahn or RegionalBahn service.
Using different language, I talked about this dilemma in Boston’s context in 2012. The situation of Boston is instructive even in other cities, even outside the United States, purely because its commuter rail service is so bad that it can almost be viewed as blank slate service on existing infrastructure. On each of the different lines in Boston, it’s worth asking what the highest and best use for the line is. This really boils down to two questions:
- Would the line fill a service need for intra-urban travel?
- Does the line connect to important outlying destinations for which high speed would be especially beneficial?
In Boston, the answer to question 1 is for the most part no. Thirty to forty years ago the answer would have been yes for a number of lines, but since then the state has built subway lines in the same rights-of-way, ignorant of the development of the S-Bahn concept across the Pond. The biggest exceptions are the Fairmount Line through Dorchester and the inner Fitchburg Line through suburbs of Cambridge toward Brandeis.
On the Fairmount Line the answer to question 2 is negative as well, as the line terminates within Boston, which helps explain why the state is trying to invest in making it a useful S-Bahn with more stops, just without electrification, high frequency, fare integration, or through-service north of Downtown Boston. But on the Fitchburg Line the answer to question 2 is positive, as there is quite a lot of demand from suburbs farther northwest and a decent anchor in Fitchburg itself.
The opposite situation to that of Fairmount is that of the Providence Line. Downtown Providence is the largest job center served by the MBTA outside Boston; the city ranks third in New England in number of jobs, behind Boston and Cambridge and ahead of Worcester and Hartford. Fast service between Providence and Boston is obligatory. However, Providence benefits from lying on the Northeast Corridor, which can provide such service if the regional trains are somewhat slower; this is the main justification for adding a handful of infill stops on the Providence Line.
In New York, the situation is the most complicated, befitting the city’s large size and constrained location. On most lines, the answers to both questions is yes: there is an urban rail service need, either because there is no subway service (as in New Jersey) or because there is subway service and it’s overcrowded (as on the 4/5 trains paralleling the Metro-North trunk and on the Queens Boulevard trains paralleling the LIRR trunk); but at the same time, there are key stations located quite far from the dense city, which can be either suburban centers 40 km out or, in the case of New Haven, an independent city more than 100 km out.
Normally, in a situation like New York’s, the solution should be to interline the local lines and keep the express lines at surface terminals; London is implementing this approach line by line with the Crossrail concept. Unfortunately, New York’s surface terminals are all outside Manhattan, with the exception of Grand Central. Penn Station has the infrastructure for through-running because already in the 1880s and 90s, the ferry transfers out of New Jersey and Brooklyn were onerous, so the Pennsylvania Railroad invested in building a Manhattan station fed by east-west tunnels.
I call for complete through-running in New York, sometimes with the exception of East Side Access, because of the island geography, which makes terminating at the equivalent of Gare du Nord or Gare de Lyon too inconvenient. In other cities, I might come to different conclusions – for example, I don’t think through-running intercity trains in Chicago is a priority. But in New York, this is the only way to guarantee good regional rail service; anything else would involve short- and long-range trains getting in each other’s way at Penn Station.
Does the absolute size of a country matter for public transport planning? Usually it does not – construction costs do not seem to be sensitive to absolute size, and the basics of rail planning do not either. That Europe’s most intensely used mainline rail networks are those of Switzerland and the Netherlands, two geographically small countries, is not really about the inherent benefits of small size, but about the fact that most countries in Europe are small, so we should expect the very best as well as the very worst to be small.
But now Germany is copying Swiss and Dutch ideas of nationally integrated rail planning, in a way that showcases where size does matter. For decades Switzerland has had a national clockface schedule in which all trains are coordinated for maximum convenience of interchange between trains at key stations. For example, at Zurich, trains regularly arrive just before :00 and :30 every hour and leave just after, so passengers can connect with minimum wait. Germany is planning to implement the same scheme by 2030 but on a much bigger scale, dubbed Deutschlandtakt. This plan is for the most part good, but has some serious problems that come from overlearning from small countries rather than from similar-size France.
In accordance with best industry practices, there is integration of infrastructure and timetable planning. I encourage readers to go to the Ministry of Transport (BMVI) and look at some line maps – there are links to line maps by region as well as a national map for intercity trains. The intercity train map is especially instructive when it comes to scale-variance: it features multihour trips that would be a lot shorter if Germany made a serious attempt to build high-speed rail like France.
Before I go on and give details, I want to make a caveat: Germany is not the United States. BMVI makes a lot of errors in planning and Deutsche Bahn is plagued by delays; these are still basically professional organizations, unlike the American amateur hour of federal and state transportation departments, Amtrak, and sundry officials who are not even aware Germany has regional trains. As in London and Paris, the decisions here are defensible, just often incorrect.
Run as fast as necessary
Switzerland has no high-speed rail. It plans rail infrastructure using the maxim, run trains as fast as necessary, not as fast as possible. Zurich, Basel, and Bern are around 100 km from one another by rail, so the federal government invested in speeding up the trains so as to serve each city pair in just less than an hour. At the time of this writing, Zurich-Bern is 56 minutes one-way and the other two pairs are 53 each. Trains run twice an hour, leaving each of these three cities a little after :00 and :30 and and arriving a little before, enabling passengers to connect to onward trains nationwide.
There is little benefit in speeding up Switzerland’s domestic trains further. If SBB increases the average speed to 140 km/h, comparable to the fastest legacy lines in Sweden and Britain, it will be able to reduce trip times to about 42 minutes. Direct passengers would benefit from faster trips, but interchange passengers would simply trade 10 minutes on a moving train for 10 minutes waiting for a connection. Moreover, drivers would trade 10 minutes working on a moving train for 10 minutes of turnaround, and the equipment itself would simply idle 10 minutes longer as well, and thus there would not be any savings in operating costs. A speedup can only fit into the national takt schedule if trains connect each city pair in just less than half an hour, but that would require average speeds near the high end of European high-speed rail, which are only achieved with hundreds of kilometers of nonstop 300 km/h running.
Instead of investing in high-speed rail like France, Switzerland incrementally invests in various interregional and intercity rail connections in order to improve the national takt. To oversimplify a complex situation, if a city pair is connected in 1:10, Switzerland will invest in reducing it to 55 minutes, in order to allow trains to fit into the hourly takt. This may involve high average speeds, depending on the length of the link. Bern is farther from Zurich and Basel than Zurich and Basel are from each other, so in 1996-2004, SBB built a 200 km/h line between Bern and Olten; it has more than 200 trains per day of various speed classes, so in 2007 it became the first railroad in the world to be equipped with ETCS Level 2 signaling.
With this systemwide thinking, Switzerland has built Europe’s strongest rail network by passenger traffic density, punctuality, and mode share. It is this approach that Germany seeks to imitate. Thus, the Deutschlandtakt sets up control cities served by trains on a clockface schedule every 30 minutes or every hour. For example, Erfurt is to have four trains per hour, two arriving just before :30 and leaving just after and two arriving just before :00 and leaving just after; passengers can transfer in all directions, going north toward Berlin via either Leipzig or Halle, south toward Munich, or west toward Frankfurt.
Flight-level zero airlines
Richard Mlynarik likes to mock the idea of high-speed rail as conceived in California as a flight-level zero airline. The mockery is about a bunch of features that imitate airlines even when they are inappropriate for trains. The TGV network has many flight-level zero airline features: tickets are sold using an opaque yield management system; trains mostly run nonstop between cities, so for example Paris-Marseille trains do not stop at Lyon and Paris-Lyon trains do not continue to Marseille; frequency is haphazard; transfers to regional trains are sporadic, and occasionally (as at Nice) TGVs are timed to just miss regional connections.
And yet, with all of these bad features, SNCF has higher long-distance ridership than DB, because at the end of the day the TGVs connect most major French cities to Paris at an average speed in the 200-250 km/h range, whereas the fastest German intercity trains average about 170 and most are in the 120-150 range. The ICE network in Germany is not conceived as complete lines between pairs of cities, but rather as a series of bypasses around bottlenecks or slow sections, some with a maximum speed of 250 and some with a maximum speed of 300. For example, between Berlin and Munich, only the segments between Ingolstadt and Nuremberg and between Halle and north of Bamberg are on new 300 km/h lines, and the rest are on upgraded legacy track.
Even though the maximum speed on some connections in Germany is the same as in France, there are long slow segments on urban approaches, even in cities with ample space for bypass tracks, like Berlin. The LGV Sud-Est diverges from the classical line 9 kilometers outside Paris and permits 270 km/h 20 kilometers out; on its way between Paris and Lyon, the TGV spends practically the entire way running at 270-300 km/h. No high-speed lines get this close to Berlin or Munich, even though in both cities, the built-up urban area gives way to farms within 15-20 kilometers of the train station.
The importance of absolute size
Switzerland and the Netherlands make do with very little high-speed rail. Large-scale speedups are of limited use in both countries, Switzerland because of the difficulty of getting Zurich-Basel trip times below half an hour and the Netherlands because all of its major cities are within regional rail distance of one another.
But Germany is much bigger. Today, ICE trains go between Berlin and Munich, a distance of about 600 kilometers, in just less than four hours. The Deutschlandtakt plan calls for a few minutes’ speedup to 3:49. At TGV speed, trains would run about an hour faster, which would fit well with timed transfers at both ends. Erfurt is somewhat to the north of the midpoint, but could still keep a timed transfer between trains to Munich, Frankfurt, and Berlin if everything were sped up.
Elsewhere, DB is currently investing in improving the line between Stuttgart and Munich. Trains today run on curvy track, taking about 2:13 to do 250 km. There are plans to build 250 km/h high-speed rail for part of the way, targeting a trip time of 1:30; the Deutschlandtakt map is somewhat less ambitious, calling for 1:36, with much of the speedup coming from Stuttgart21 making the intercity approach to Stuttgart much easier. But with a straight line distance of 200 km, even passing via Ulm and Augsburg, trains could do this trip in less than an hour at TGV speeds, which would fit well into a national takt as well. No timed transfers are planned at Augsburg or Ulm. The Baden-Württemberg map even shows regional trains (in blue) at Ulm timed to just miss the intercity trains to Munich. Likewise, the Bavaria map shows regional trains at Augsburg timed to just miss the intercity trains to Stuttgart.
The same principle applies elsewhere in Germany. The Deutschlandtakt tightly fits trains between Munich and Frankfurt, doing the trip in 2:43 via Stuttgart or 2:46 via Nuremberg. But getting Munich-Stuttgart to just under an hour, together with Stuttgart21 and a planned bypass of the congested Frankfurt-Mannheim mainline, would get Munich-Frankfurt to around two hours flat. Via Nuremberg, a new line to Frankfurt could connect Munich and Frankfurt in about an hour and a half at TGV speed; even allowing for some loose scheduling and extra stops like Würzburg, it can be done in 1:46 instead of 2:46, which fits into the same integrated plan at the two ends.
The value of a tightly integrated schedule is at its highest on regional rail networks, on which trains run hourly or half-hourly and have one-way trip times of half an hour to two hours. On metro networks the value is much lower, partly because passengers can make untimed transfers if trains come every five minutes, and partly because when the trains come every five minutes and a one-way trip takes 40 minutes, there are so many trains circulating at once that the run-as-fast-as-necessary principle makes the difference between 17 and 18 trainsets rather than that between two and three. In a large country in which trains run hourly or half-hourly and take several hours to connect major cities, timed transfers remain valuable, but running as fast as necessary is less useful than in Switzerland.
The way forward for Germany
Germany needs to synthesize the two different rail paradigms of its neighbors – the integrated timetables of Switzerland and the Netherlands, and the high-speed rail network of France.
High investment levels in rail transport are of particular importance in Germany. For too long, planning in Germany has assumed the country would be demographically stagnant, even declining. There is less justification for investment in infrastructure in a country with the population growth rate of Italy or of last decade’s Germany than in one with the population growth rate of France, let alone one with that of Australia or Canada. However, the combination of refugee resettlement and a very strong economy attracting European and non-European work migration is changing this calculation. Even as the Ruhr and the former East Germany depopulate, we see strong population growth in the rich cities of the south and southwest as well as in Berlin.
The increased concentration of German population in the big cities also tilts the best planning in favor of the metropolitan-centric paradigm of France. Fast trains between Berlin, Frankfurt, and Munich gain value if these three cities grow in population whereas the smaller towns between them that the trains would bypass do not.
The Deutschlandtakt’s fundamental idea of a national integrated timed transfer schedule is good. However, a country the size and complexity of Germany needs to go beyond imitating what works in Switzerland and the Netherlands, and innovate in adapting best practices for its particular situation. People keep flying domestically since the trains take too long, or they take buses if the trains are too expensive and not much faster. Domestic flights are not a real factor in the Netherlands, and barely at all in Switzerland; in Germany they are, so trains must compete with them as well as with flexible but slow cars.
The fact that Germany already has a functional passenger rail network argues in favor of more aggressive investment in high-speed rail. The United States should probably do more than just copy Switzerland, but with nonexistent intercity rail outside the Northeast Corridor and planners who barely know that Switzerland has trains, it should imitate rather than innovating. Germany has professional planners who know exactly how Germany falls short of its neighbors, and will be leaving too many benefits on the table if it decides that an average speed of about 150 km/h is good enough.
Germany can and should demand more: BMVI should enact a program with a budget in the tens of billions of euros to develop high-speed rail averaging 200-250 km/h connecting all of its major cities, and redo the Deutschlandtakt plans in support of such a network. Wedding French success in high-speed rail and Swiss and Dutch success in systemwide rail integration requires some innovative planning, but Germany is capable of it and should lead in infrastructure construction.
I’ve written a lot about the importance of radial network design for urban metros, for examples here, here, here, here, and here. In short, an urban rail network should look something like the following diagram:
That is, every two radial routes should intersect exactly once, with a transfer. In this post I am going to zoom in on a specific feature of importance: the location of the intersection points. In most cities, the intersection points should be as close as possible to the center, first in order to serve the most intensely developed location by all lines, and second in order to avoid backtracking.
The situation in Berlin
Here is the map of the central parts of Berlin’s U- and S-Bahn network, with my apartment in green and three places I frequently go to in red:
(Larger image can be found here.)
The Ring is severed this month due to construction: trains do not run between Ostkreuz, at its intersection with the Stadtbahn, and Frankfurter Allee, one stop to the north at the intersection with U5. As a result, going to the locations of the two northern red dots requires detours, namely walking longer from Warschauer Strasse to the central dot, and making a complex trip via U7, U8, and U2 to the northern dot.
But even when the Ring is operational, the Ring-to-U2 trip to the northern dot in Prenzlauer Berg is circuitous, and as a result I have not made it as often as I’d have liked; the restaurants in Prenzlauer Berg are much better than in Neukölln, but I can’t go there as often now. The real problem is not just that the Ring is interrupted due to construction, but that the U7-U2 connection is at the wrong place for the city’s current geography: it is too far west.
As with all of my criticism of Berlin’s U-Bahn network layout, there is a method to the madness: most of the route of U7 was built during the Cold War, and if you assumed that Berlin would be divided forever, the alignment would make sense. Today, it does not: U7 comes very close to U2 in Kreuzberg but then turns southwest to connect with the North-South Tunnel, which at the time was part of the Western S-Bahn network, running nonstop in the center underneath Mitte, then part of the East.
On hindsight, a better radial design for U7 would have made it a northwest-southeast line through the center. West of the U6 connection at Mehringdamm it would have connected to the North-South Tunnel at Anhalter Bahnhof and to U2 at Mendelssohn Park, and then continued west toward the Zoo. That area between U1/U2 and Tiergarten Park is densely developed, with its northern part containing the Cold War-era Kulturforum, and in the Cold War the commercial center of West Berlin was the Zoo, well to the east of the route of U7.
Avoiding three-seat rides
If the interchange points between lines are all within city center, then the optimal route between any two points is at worst a two-seat ride. This is important: transfers are pretty onerous, so transit planners should minimize them when it is reasonably practical. Two-seat rides are unavoidable, but three-seat rides aren’t.
The two-seat ride rule should be followed to the spirit, not the letter. If there are two existing lines with a somewhat awkward transfer, and a third line is built that makes a three-seat ride better than connecting between those two lines, then the third line is not by itself a problem, and it should be built if its projected ridership is sufficient. The problem is that the transfer was at the wrong location, or maybe at the right location but with too long a walk between the platforms.
Berlin’s awkward U-Bahn network is such that people say that the travel time between any two points within the Ring is about 30 minutes, no matter what. When I tried pushing back, citing a few 20-minute trips, my interlocutors noted that with walking time to the station, the inevitable wait times, and transfers, my 20-minute trips were exceptional, and most were about 30 or slightly longer.
The value of an untimed transfer rises with frequency. Berlin runs the U-Bahn every 5 minutes during the daytime on weekdays and the S-Bahn mostly every 5 minutes (or slightly better) as well; wait times are shorter in a city like Paris, where much of the Metro runs every 3 minutes off-peak, and only drops to 5 or 6 minutes late in the evening, when Berlin runs trains every 10 minutes. However, Parisian train frequencies are only supportable in huge cities like Paris, London, and Tokyo, all of which have very complex transfers, as the cities are so intensely built that the only good locations for train platforms require long walks between lines.
New York of course has the worst of all worlds: a highly non-radial subway network with dozens of missed connections, disappointing off-peak frequencies, and long transfer corridors in Midtown. In New York, three-seat rides are ubiquitous, which may contribute to weak off-peak ridership. Who wants to take three separate subway lines, each coming every 10 minutes, to go 10 kilometers between some residential Brooklyn neighborhood and a social event in Queens?
New York is engaging in the process of redesigning its urban bus network borough by borough. The first borough is the Bronx, with an in-house redesign; Queens is ongoing, to be followed by Brooklyn, both outsourced to firms that have already done business with the MTA. The Bronx redesign draft is just out, and it has a lot of good and a great deal of bad.
What does the redesign include?
Like my and Eric Goldwyn’s proposal for Brooklyn, the Bronx redesign is not just a redrawing of lines on a map, but also operational treatments to speed up the buses. New York City Transit recognizes that the buses are slow, and is proposing a program for installing bus lanes on the major streets in the Bronx (p. 13). Plans for all-door boarding are already in motion, to be rolled out after the OMNY tap card is fully operational; this is incompetent, as all-door boarding can be implemented with paper tickets, but at this stage this is a delay of just a few years, probably about 4 years from now.
But the core of the document is the network redesign, explained route by route. The map is available on p. 14; I’d embed it, but due to file format issues I cannot render it as a large .png file, so you will have to look yourselves.
The shape of the network in the core of the Bronx – that is, the South Bronx – seems reasonable. I have just one major complaint: the Bx3 and Bx13 keep running on University Avenue and Ogden Avenue respectively and do not interline, but rather divert west along Washington Bridge to Washington Heights. For all of the strong communal ties between University Heights and Washington Heights, this service can be handled with a high-frequency transfer at the foot of the bridge, which has other east-west buses interlining on it. The subway transfer offered at the Washington Heights end is low-quality, consisting of just the 1 train at the GWB bus station; a University-Ogden route could instead offer people in University Heights a transfer to faster subway lines at Yankee Stadium.
Outside the South Bronx, things are murkier. This is not a damn by faint praise: this is an acknowledgement that, while the core of the Bronx has a straightforward redesign since the arterials form a grid, the margins of the Bronx are more complicated. Overall the redesign seems fairly conservative – Riverdale, Wakefield, and Clasons Point seem unchanged, and only the eastern margin, from Coop City down to Throgs Neck, sees big changes.
The issue of speed
Unfortunately, the biggest speed improvement for buses, stop consolidation, is barely pursued. Here is the draft’s take on stop consolidation:
The spacing of bus stops along a route is an important factor in providing faster and more reliable bus service. Every bus stop is a trade-off between convenience of access to the bus and the speed and reliability of service. New York City buses spend 27 percent of their time crawling or stopped with their doors open and have the shortest average stop distance (805 feet/245 m) of any major city. London, which has the second closest stop spacing of peer cities, has an average distance between stops of 1,000 ft/300 m.
Bus stop spacing for local Bronx routes averages approximately 882 feet/269 meters. This is slightly higher than the New York City average, but still very close together. Close stop spacing directly contributes to slow buses and longer travel times for customers. When a bus stops more frequently along a route, exiting, stopping, and re-entering the flow of traffic, it loses speed, increases the chance of being stopped at a red traffic signal, and adversely affects customers’ travel time. By removing closely-spaced and under-utilized stops throughout the Bronx, we will reduce dwell time by allowing buses to keep moving with the flow of traffic and get customers where they need to go faster.
Based on what I have modeled as well as what I’ve seen in the literature, the optimal bus stop spacing for the Bronx, as in Brooklyn, is around 400-500 meters. However, the route-by-route descriptions reveal very little stop consolidation. For example, on the Bx1 locals, 3 out of 93 stops are to be removed, and on the Bx2, 4 out of 99 stops are to be removed.
With so little stop consolidation, NYCT plans to retain the distinction between local and limited buses, which reduces frequency to either service pattern. The Bx1 and Bx2 run mostly along the same alignment on Grand Concourse, with some branching at the ends. In the midday off-peak, the Bx1 runs limited every 10 minutes, with some 12-minute gaps, and the Bx2 runs local every 9-10 minutes; this isn’t very frequent given how short the typical NYCT bus trip is, and were NYCT to eliminate the local/limited distinction, the two routes could be consolidated to a single bus running every 4-5 minutes all day.
How much frequency is there, anyway?
The draft document says that consolidating routes will allow higher frequency. Unfortunately, it makes it difficult to figure out what higher frequency means. There is a table on p. 17 listing which routes get higher frequency, but no indication of what the frequency is – the reader is expected to look at it route by route. As a service to frustrated New Yorkers, here is a single table with all listed frequencies, weekday midday. All figures are in minutes.
|Route||Headway today||Proposed headway|
|Bx38 (28 variant)||17||discontinued|
|Bx42 (40 variant)||20||cut to a shuttle, 15|
A few cases of improving frequency on a trunk are notable, namely on the Bx28/38 and Bx40/42 pairs, but other problem spots remain, led by the Bx1/2 and the local and limited variants on some routes.
The principle of interchange
A transfer-based bus network can mean one of two things. The first, the one usually sold to the public during route redesigns, is a grid of strong routes. This is Nova Xarxa in Barcelona, as well as the core of this draft. Eric’s and my proposal for Brooklyn consists entirely of such a grid, as Brooklyn simply does not have low-density tails like the Bronx, its southern margin having high population density all the way to the boardwalk.
But then there is the second meaning, deployed on networks where trunk routes split into branches. In this formulation, instead of through-service from the branches to the trunk, the branches should be reduced to shuttles with forced transfers to the trunk. Jarrett Walker’s redesign in Dublin, currently frozen due to political opposition (update: Jarrett explains that no, it’s not really frozen, it’s in revision after public comments), has this characteristic. Here’s a schematic:
The second meaning of the principle of interchange is dicey. In some cases, it is unavoidable – on trains, in particular, it is possible to design timed cross-platform transfers, and sometimes it’s just not worth it to deal with complex junctions or run diesels under the catenary. On buses, there is some room for this principle, but less than on trains, as a bus is a bus, with no division into different train lengths or diesels vs. electrics. Fundamentally, if it’s feasible to time the transfers at the junctions, then it’s equally possible to dispatch branches of a single route to arrive regularly.
New York’s bus network is already replete with the first kind of interchange, and then the question is where to add more of it on the margins. But the Bronx draft includes some of the second, justified on the grounds of breaking long routes to improve reliability. Thus, for example, there is a proposed 125th Street crosstown route called the M125, which breaks apart the Bx15 and M100. Well, the Bx15 is a 10.7 km route, and the M100 is an 11.7 km route. The Bx15 limited takes 1:15-1:30 end to end, and the M100 takes about 1:30; besides the fact that NYCT should be pushing speedup treatments to cut both figures well below an hour, if routes of this length are unreliable, the agency has some fundamental problems that network redesign won’t fix.
In the East Bronx, the same principle of interchange involves isolating a few low-frequency coverage routes, like the Bx24 and Bx29, and then making passengers from them transfer to the rest of the network. The problem is that transferring is less convenient on less frequent buses than on more frequent ones. The principle of interchange only works at very high frequency – every 8 minutes is not the maximum frequency for this but the minimum, and every 4-6 minutes is better. It would be better to cobble together routes to Country Club and other low-density neighborhoods that can act as tails for other trunk lines or at least run to a transfer point every 6-8 minutes.
Is any of this salvageable?
The answer is yes. The South Bronx grid is largely good. The disentanglement of the Bx36 and Bx40 is particularly commendable: today the two routes zigzag and cross each other twice, whereas under any redesign, they should turn into two parallel lines, one on Tremont and one on 180th and Burnside.
But outside the core grid, the draft is showing deep problems. My semi-informed understanding is that there has been political pressure not to cut too many stops; moreover, there is no guarantee that the plans for bus lanes on the major corridors will come to fruition, and I don’t think the redesign’s service hours budget takes this into account. Without the extra speed provided by stop consolidation or bus lanes, there is not much room to increase frequency to levels that make transfers attractive.
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.
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.
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.
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.
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 where a and b are empirical parameters depending on walking speed and other relevant variables.
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.
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.
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.
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.
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|
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
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:
- 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.
- 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.
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
if travel along the line is isotropic, or
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
if travel is isotropic and
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
if travel is isotropic and
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:
- Passengers need to walk not just along the route to their stop but also from their origin to the route, which increases walk time
- 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
- 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 , which has north-south and east-west routes, each of length , 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
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
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:
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
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
and this is optimized when
Plugging the usual values of the parameters, we get
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
which is minimized at the positive root of the cubic
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,
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
and its minimum is at the positive solution to
We retain the New York- and Brooklyn-oriented variables from the above sections and obtain
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
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
or, using the same parameters as always,
The solution is s = 420 meters, with routes spaced 840 meters apart.
The isotropic cubic with transfers is
and with the usual parameters, again sticking with d = 3,360 even though in practice it is likely to be higher, this is
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
or with the usual parameters
of which the positive root is s = 374 meters, with routes spaced 1,123 meters apart,
The transfer-based isotropic formula is,
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|
|Distinguished node; 1-seat ride||Optimal s||719||420||374|
|Isotropy; 2-seat ride||Optimal s||683||442||340|
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.