Circumferential Lines and Express Service

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.

The situation

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 \sqrt{2}.

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.


  1. SB

    Both Osaka Loop line and Osaka Higashi (Outer Loop) Line have rapid trains which bypass some of the stations.
    Both have double tracks with occasional bypass tracks.
    The Osaka Loop even has some express trains connecting Kansai Airport/Wakayama to Kyoto.

    • yuuka

      The Osaka Loop Line rapid service skips only a handful of stations, hard for me to consider that a meaningful express service.

      As for the Haruka, well, using the loop is the only way it can serve both Osaka and Kyoto. Not like that really matters for this discussion since it’s a premium fare limited express anyway.

  2. Herbert

    “[Those in charge of Paris public transit] built tramways, which have been a total success and have high ridership even though they’re slow.”

    How could such a thing every possibly have happened? Has it ever happened elsewhere?

    • Alon Levy


      In Paris the most successful trams are circumferential, i.e. T1, T2, and T3. The other trams aren’t so strong. Radial trams elsewhere aren’t that strong, either – the radial trams in Munich get around half the ridership per route-km of the Paris system. The point is that because people take circumferential transit over very short distances, it’s less important to build fast express lines, which by the same token makes tramways more valuable.

      I wrote something similar, more specific to New York, in 2016:

      • Herbert

        But if a city only has relatively short distances to begin with, why not cover it all with trams?

        What’s the point of a one minute subway ride if you have to schlep down to a poorly lit tunnel for three minutes and then schlep up again?

        • Alon Levy

          Yeah, in a city the size of Karlsruhe, Basel, or Strasbourg, urban transit can be done exclusively with trams; tunnels may be necessary but only for longer-range S-Bahns if the main station is at an awkward spot as in Karlsruhe. But once your urban area is much more than a million people, you need more – Prague and Budapest both have more riders on the metros than on the trams, and the Budapest tramway network seems fairly circumferential here.

          • Herbert

            Most cities in Germany at least are not above one million. There’s one that’s clearly above it, two that are above it and one that hovers around it.

            Yet for some crazy reason Nuremberg decided to build a subway, Hamburg decided to shut down its trams and Cologne calls its Stadtbahn U-Bahn…

          • Alon Levy

            A bunch of contiguous urban areas are well above 1 million – Berlin, Hamburg, Munich, Stuttgart, Frankfurt, Cologne (separately), Düsseldorf, the Ruhr proper. In Frankfurt there were at some point plans to annex suburbs like Hanau and Offenbach, which would push the city proper well into the millions. I get that there are a bunch of polycentric regions that don’t have cores this big, like Rhine-Neckar or (at much larger scale) Rhine-Ruhr, but in those areas public transit has to involve extensive regional rail just to link all the different cores, which isn’t the same as either U-Bahn or tramways.

          • Herbert

            An U-Bahn from Frankfurt to the burbs would go through a lot of pretty empty space… You can do that with a tram, but with a subway, it becomes cost prohibitive…

          • Alon Levy

            Yeah, so you do it at-grade outside the built-up area, same way around half the route-lengths of London Underground and New York City Subway aren’t actually underground, but in rail rights-of-way or (in NY only) elevated over streets. Essentially that’s the high-intensity S-Bahn strategy used here or in Hamburg or Munich or Paris – at-grade in the suburbs, in tunnel in city center.

          • Herbert

            At grade subways – especially if they don’t even stop – have a lot of negative effects on the surrounding area and few positive ones.

            The same can’t be said for trams…

          • Herbert

            Well, a public transit line is supposed to attract people, not repel them…

            So would you replace Tram4 in Nuremberg with a subway through Friedrich Ebert Platz and build the planned extension to Erlangen and on to Herzogenaurach as an at grade subway (What the current plan is can be seen under NIMBYs are particularly up in arms about the planned crossing of the river Regnitz)

          • Herbert

            At any rate, one shouldn’t count areas not served by a subway when counting the population base.

            Berlin subway doesn’t cross the city boundary. Munich subway serves only one place outside of Munich proper (Garching). Nuremberg subway could be renamed Nuremberg-Fürth subway due to its handful of stations there…

  3. Eric

    Also, express lines are built only when there’s demand for 4 tracks. Circumferential routes don’t have that demand, except maybe in Tokyo.

    • Herbert

      Many cities with old legacy rail lines have or had at some point a circumferential freight bypass. Even Nuremberg has a Ringbahn built for freight

        • Herbert

          Those can be used for passengers and some have been converted to pax use along parts of them

          • Eric

            I don’t think any of them have been converted to 4-track local-and-express service. Even in cities that could do it easily, like Berlin. That is the point, there is not demand for such 4-track service.

          • Sascha Claus

            If you get the freight trains out of the way. In some places, this already happened after WWII, in some places not completely and some are in the midst of a resurgence of railfreight.

    • yuuka

      The Yamanote “express” *did* start out as a freight line, for what it’s worth.

      After the Musashino Line opened in 1973, freight was banned from the Yamanote area, so it sat unused until the Saikyo Line opened in the 80s and the Shonan-Shinjuku Line in 2001. As built it had a connection up to Ikebukuro, but apparently JNR felt it pretty necessary to provide a one seat ride to Shinjuku. As for Shibuya, that opened in 1996, hence the awkward location of the Saikyo/Shonan-Shinjuku Line platforms (compared to the Yamanote, Tokyu, etc) which JR East is fixing for the Olympics.

      • yuuka

        Hmm, I remember being told it was banned, but it appears that there are still a few daily freight runs.

        There was an incident where a train carrying jet fuel to the US air bases collided with another train and blew up, but that was in the late 60s. It was a major impetus for building the outer-circumferential Musashino Line though.

    • Alon Levy

      The Ringbahn evidently does have four tracks, but yeah, the two express tracks are mostly for Mushroom Concept intercity trains.

      • Herbert

        There is some passenger service (including S-Bahn) on the Güteraußenring and some even advocate an “outer ring” either as S-Bahn or RE…

        • Alon Levy

          Yeah, I feel weird about that. The population density isn’t there for frequent service on the entire Outer Ring (as opposed to a short branch like S75), so it would have to be less frequent RegionalBahn, and somehow there would need to be timed connections to radial lines at the transfer points, which seems hard to schedule. And on the western part of the Outer Ring, in Brandenburg, it doesn’t really pass through the existing suburbs, just the airport, Blankenfelde, Golm, and Hohen Neuerdorf.

          • Herbert

            On that subject, what’s your opinion re Heidekrautbahn and Dresdner Bahn (with Wowereit being among the NIMBYs in the latter case)

          • Alon Levy

            Heidekrautbahn: I don’t know. I imagine the idea is to add S-Bahn service there? Or is it just to beef up RB on that line? I can see something happening there depending on core capacity, but at least on Google Earth the area looks way less developed than the areas on S1 and S2.

            Dresdner Bahn: it’s a key project cutting 20 minutes on trains between Berlin and Dresden, right? It seems really useful. If there were a Berlin-Halle HSR line under construction right now I might ask if it’s possible to just branch off to Dresden from there, around Teltow, and avoid construction in the built-up area of Berlin, but there isn’t such a line and those 20 minutes are valuable on a connection that’s around 2 hours. There’s also the airport connection, I guess, but I’m not sure how I feel about splitting city-BER frequency across so many lines – S-Bahn, RB to the Stadtbahn, RB to the Ring, RB to the North-South Main Line…

            EDIT: I should crayon Berlin mainlines more completely than I did in April, but right now my crayon is to feed all the intercity service through two tracks of the North-South Main Line and turn over the express tracks of the Stadtbahn to RB designed to have decent trunk frequency to help take pressure off the S-Bahn tracks. But the Dresdner Bahn works are pretty much independent of this and seem generally useful.

          • Herbert

            The Heidekrautbahn was one of many many rail lines in the Berlin area affected by partition. It used to end in Wilhelmsruh and have a stop at… Märkisches Viertel…

            Current plans call for reconstruction of that former route and through running to Gesundbrunnen

          • Herbert

            There are certainly people who wish to have the trains running all the way to Hauptbahnhof, but the official plan is only to go to Gesundbrunnen and have people change trains there…

          • Alon Levy

            Why is that? The North-South Main Line is ridiculously underfull, there are like 7 trains per hour in each direction and 4 tracks, and even in the D-Takt plan it only goes up to 14.5 north of Hbf, with a couple trains terminating at Hbf from the south.

          • Herbert

            I do not know. None of the sources I consulted tell… Maybe you can ask Mr. Alexander Kaczmarek…

          • Sascha Claus

            Don’t say too loud
            that the outer ring passes through BER, lest someone believes it! 😉 The outer ring only skirts the old terminal (SXF), which is to be closed when BER opens.
            The new terminal sits above a branch of the outer ring, which will slow down Berlin-Dresden trains should the try to serve it.

          • Herbert

            Unfortunately in a city otherwise obsessed with “Bauvorleistungen” for some reason the terminal of BER was not built with any provision for a U7 extension…

          • Sascha Claus

            Have you heard of these old plans (from non-official planners) to route Berlin – Dresden ICEs via Jüterbog und Riesa (Röderau)?
            By using existing Ausbaustrecken (ABS, upgraded legacy lines) for about 2/3 of the distance, you’d get almost the same bang for much less buck. (Maybe even some portion working from Berlin to Dresden and Chemnitz.)

          • Alon Levy

            I’ve heard people (I think Threestationsquare?) suggest this on Twitter when I complained about the current roundabout route out of Berlin.

            That said, is it really faster to Chemnitz than going via Leipzig? (I guess under any HSR program completing Berlin-Halle, that would be far and away the fastest way to get to Chemnitz…)

          • Sascha Claus

            I don’t know how much money would be needed and how fast each line would be afterward, but Riesa – Chemnitz is already electrified, so a (slow) service could start immediately.
            And I would expect more passengers between Hamburg and Berlin than between Berlin and Dresden and Prague, so splitting trains looks obvious.
            From where would you take an ICE for Leipzig – Chemnitz? Splitting the one Berlin – Munich? 😉

          • Herbert

            Dresden Prague is an incredibly busy freight line which is why they’re seriously considering an ore mountains base tunnel…

  4. yuuka

    In Singapore we’re building the 50-60km Cross Island Line. The plan is to get from Jurong to the airport in under 45 minutes.

    It’s supposed to be circumferential, but given geography it’s basically a point to point line since it cuts across the long axis of the diamond shaped island. They were considering running express services on it, but apparently that plan was canned due to high cost and not enough traffic to warrant it. Even so the resultant line still has quite a high interstation distance.

  5. adirondacker12800

    there is a curious observation: there is express service on the radial lines, but not the circumferential ones. …. If origins and destinations are distributed uniformly along the line…

    Nobody lives in SimCity. Geography and real estate developers are cruel.

    If the center has local and express service it does because it has demand for that. Sometimes with more than two tracks. With some exceptions like the Flushing Line. Which runs local in Manhattan and local and express in peak direction in Queens. If the center is defined by demand, when the line skirting the perimeter had enough demand for four tracks it wouldn’t be the perimeter. It would be the center or one of the centers.

    the circumferential Crosstown Line, carrying the G train, is entirely local.

    It’s two radial locals that happen to meet in Greenpoint. It’s setup to have higher frequency along what we now call the Culver Line and Queens Blvd. It ran local all the way out to Continental for decades. It doesn’t have to be. It could run express on Queens Blvd alllll the way out to Jamaica. And in peak direction to Coney Island. It’s nice that facilitates terminating at Court Square. Because instead of building something like 63rd in the 30s, they finagled the 60th Street connector and whatever the “Second System” was going to do is difficult to see. What was going to go on for Queens and Midtown is very hazy in anything I’ve see. Partly because Midtown was still predominantly residential in the 20s.

    namely the connection to the L train at Bedford Avenue

    Bedford-Nostrand Avenue station is four stops away on the G train from where you can connect to the L train at Lorimer Street. It’s quite a hike from the Metropolitan Avenue station on the G, where you can connect to the L, to the Bedford Ave stop on the L. Very rude of the NYC system to use the same names over and over. What’s up with all those stations named 14th Street? Or 23rd? That PATH uses too.

    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

    If they keep building condos in the outer boroughs Second Avenue will need four tracks. The travel patterns won’t be like today’s. Just like today’s aren’t like what they envisioned a century ago for the “Second System”. The plans I’ve seen for the Second System were for six tracks, on Second Avenue, four of them express. And lots and lots of service to Downtown Brooklyn and Wall Street. The…. Crosstowns.. were going to cross over themselves in the terminal in Jackson Heights. Squint at them, they had plans to six track Sixth Avenue or Eighth Avenue. Or both. The Riverdale express would have to go somewhere. Midtown went and happened. The Plan for Action had the subway “super” express using the LIRR ROW. It would be interesting to see how they planned on doing that and send LIRR trains to the East Side too. Squint even harder I think they were going to use the Montague Street tunnel for LIRR service to Chambers Street. Travel and work patterns have changed over the past half century. Those pesky pesky subway riders went and used the capacity. Or on the LIRR. They will probably change again as Midtown sprawls out even more and Wall Street creeps north of Chambers.

    Two tracks of Second Avenue subway means someday the Bronx gets half of a local. Four tracks means you can do something like run local on Third to Fordham Road, local to E180th or Co-Op City, on the excess capacity of Amtrak’s Hell Gate Line and express to Dyre Avenue. It’s unfortunate they allowed condos to be built on the ROW that once had.

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