basics: branching (or how transit is like a river)

A short draft chapter from the book, overlapping the content of this recent post but with an extended BART example that I hope readers will enjoy and have comments on.

In 2011, cartographer Daniel Huffman thought it would be interesting to draw river systems as though they were subways.  Figure 1 shows part of his sketch of the Lower Mississippi.[i]

BRCH 01 Mississippi

Figure 1.  Lower Mississippi River System, Drawn as a Transit Map (Daniel Huffman)

It’s a fun idea, but it also points to an important insight.  If you travel upriver by boat, you expect the river to get smaller and smaller.  Every time you reach a branching point, the volume of water in the two rivers in front of you is the same as the volume in the river behind you.   If you keep going, you’ll eventually reach a river that’s too small for your boat.  Transit is like that too, because branching always divides frequency.

This is one of those too-obvious points that’s easy to forget in the heat of a transit debate.  For example, in 2003, the Bay Area Rapid Transit (BART) system opened a new extension to San Francisco International Airport (SFO) and also to Millbrae, an important connection point with the Caltrain commuter rail system.  The basic extension, southward from San Francisco, looks like the top image in Figure 2.

It’s a triangle, with tracks from San Francisco to both SFO and Millbrae, and also direct tracks between Millbrae and SFO.

If you’re unconsciously thinking like a motorist, looking at this as though it were a highway map, it looks fine.  All the points on the map are directly connected all the others.  But transit can’t run all these connections at the same high frequency, because of the effect of branching. The actual pattern of service will have to be one of images beneath it.  In these images, line-width represents frequency.


Figure 2.  BART San Francisco Airport (SFO) Terminus

For example, suppose you want service every 10 minutes to both SFO and Millbrae, but you can only afford 10 minute frequency on the line through San Bruno and on to San Francisco. You have to run one of the two ‘sequential’ options in the upper left of Figure 2, either run all service to Millbrae via SFO, or all service to SFO via Millbrae.  Either SFO passengers or Millbrae passengers are going to hate you.

Alternatively, we could branch the service at San Bruno, sending half of it to SFO and half of it to Millbrae.  But the branching will cut our frequency.  If we can only afford 10 minute frequency through San Bruno, then we’ll end up with 20 minute frequency at SFO and at Millbrae.

Finally, we can run everything every 10 minutes by forcing a connection.  One side of the loop would have a shuttle train, while the other would have through service.

There is one other option, though it’s not available for BART.   You could split the train in half, and send the front half on one branch and the rear half on the other.  This is very tricky; it requires a driver in position ready to take half of the train when it arrives.  It’s also hard to separate a train without at least a minute or two of delay, at least for the rear half of the divided train.

To sum up, we should suspicious whenever we see a branch drawn as though one line can effortlessly divide into two equal lines.  Often, such a branch will be called an extension, a very slightly misleading word because it suggests that an existing, known quantity of service is being extended.  In fact, a branch always means one of three things.  Either

  • points beyond the branching point have less frequent service or
  • one of the branches operates as a shuttle, requiring a connection, or
  • in a few rare cases, the train itself comes apart, with some cars proceeding along one branch and some along the other.

Geometrically, it has to mean one of those things, and it may not be the one you prefer.  So before you decide whether the service is useful to you, or whether you support a proposed transit project whose map looks like this, you may want to ask which of those it is.


[i] A collection of these spanning most of the US’s major river systems is at


22 Responses to basics: branching (or how transit is like a river)

  1. BBnet3000 February 24, 2011 at 5:54 pm #

    I would note that this particular branching isnt much of a sacrifice, as neither Millbrae nor SFO have huge ridership, and while Millbrae connects to Caltrain, nobody who rides Caltrain is going to complain about BART headways any time soon.
    The branching/frequency issue applies to BART very significantly though. The Transbay Tube is at capacity, and certain trains, particularly the Pittsburg/Bay Point trains, get very crowded at rush hour. The Bay Point line itself is running at ~1/4 of frequency capacity during rush hour, but cant run at any higher frequency because it is limited by the 2 track tunnel.
    Crowding at rush hour is becoming more of a problem for BART, and when theres train delays, it gets ugly fast. The trains absolutely must run maximum headways non-stop to keep the platform itself from crush-load. Maybe a month or two ago BART had some sort of delay, and they actually closed Embarcadero station (and possibly other Market St stations as well) because of the platform crowding.
    Fortunately, BART’s new and very expensive extensions wont generate enough ridership to exacerbate the crowding problem.

  2. anonymouse February 24, 2011 at 7:53 pm #

    The branching becomes a huge pain when BART doesn’t have enough money to run a shuttle, and you end up either delaying people going to SF from Millbrae to make the detour to the airport (on weekends and evenings), or forcing a transfer for travelling from Millbrae to the airport, meaning that you have to make three transfers (Caltrain to BART to BART to BART to AirTrain), and going only one stop on each leg too, where before you could just transfer from Caltrain to a bus that took you directly to your terminal.
    I think more than anything else this is a perfect illustration of the sheer insidiousness of referring to any building of transportation infrastructure as an “improvement” (see also Central Subway and Oakland Airport Connector). Whenever you hear the word “improvement”, you really have to ask just what is being improved, and for whom.

  3. ant6n February 24, 2011 at 9:22 pm #

    While at it I’d like to throw in this map. It shows the train “groups” of the Berlin S-Bahn, which is a system fairly similar to BART. Every line represents a fixed interval service of 20 minutes, so where 5 lines overlap, there are 15 tph. This is an internal map.
    You can see that the line thickness corresponds directly to how much service is provided (at least in terms of frequency). And branching will basically half the frequency along every branch.
    But this actually makes a lot of sense – the required frequency goes down in relation with the distance from the center; at the same time the amount of branching goes up with distance from the center. I think it’s pretty neat how these ideas fit together. The farther you live from the center the less frequent trains you get, because there’s less ridership on the fringes. But at the same time the coverage is actually pretty good.
    And the service even at the end is quite tolerable since these are 20min fixed interval schedules.

  4. Winston February 24, 2011 at 10:55 pm #

    One option that would help reduce crowding on BART’s Pittsburg/Bay Point line would be to create an additional service that travels between Concord and Hayward with a timed transfer to the under-utilized Dublin trains. Of course, there aren’t that many people In Concord who want to go south of Hayward or vice versa.

  5. Anon256 February 25, 2011 at 1:04 am #

    Given that BART’s schedule adherence is good enough that timed transfers work elsewhere in the system, I really don’t see why the shuttle options are so bad. Isn’t Jarett always telling us that “connections” are nothing to fear? Though since San Bruno has only two tracks, timed transfers could never be as convenient as they are at e.g. MacArthur.

  6. hkp February 25, 2011 at 5:54 am #

    An editorial comment:
    In describing figure 2 you say: “either run all service to Millbrae via SFO, or all service via SFO to Millbrae.”
    There is no difference there. I think you mean: “either run all service to Millbrae via SFO, or all service to SFO via Millbrae.”
    Don’t know whether that just turns up in the post or whether it’s in the draft, but it probably shouldn’t make it into the final version.

  7. Tom West February 25, 2011 at 6:30 am #

    Following up on ant6n’s post, Network Rail (rail infrastuture owner in Great Britiain) does somethign similar. See, for example, , starting on page 23 of the PDF.

  8. Alexis February 25, 2011 at 8:33 am #

    As a former user of this set of lines/transfers, I disagree with bbnet3000. The key is not BART headways but time investment and transfer complications, especially for the time-sensitive nature of travel to the airport and transfers to Caltrain. Less BART service means fewer transfer options and more missed transfers.
    BART’s decision to run a strange mix of the branching, via, and shuttle services seems to me to be an indication that they are conflicted about the best decision for this setup as well, and may not have initially thought through all the compromises. Especially since the ridership on this segment was projected to be far higher than it turned out to be.

  9. ant6n February 25, 2011 at 8:34 am #

    does Network Rail actually make the schedules, and the operators buy ‘slots’ in the schedule?

  10. JGrant February 25, 2011 at 12:17 pm #

    “points beyond the branching point have less frequent service”
    This is of course true, however, when extending a system via a branch, it is important to consider whether the branch will add to or subtract from the frequency along the trunk line. In Vancouver, when the Millennium Line was completed, the frequency along the existing Expo line almost doubled where the lines combined (trunk). The 5-6 minute headway on each branch converged on the trunk line to create 2.5-3 minute headways. This will also be the case along the Millennium Line trunk when the evergreen line is completed.
    The point I am making is that you have two options when branching, the first option divides the frequency on the trunk by two (as Jarrett has explained) and the second option multiplies the frequency on the trunk by two.

  11. Ron February 25, 2011 at 3:16 pm #

    If this pic stull represents the inerior of a Bart car, then remove some seats! Instead of 2 by 2 – there should only be one seat on east side. That would allow many more standees.

  12. david vartanoff February 26, 2011 at 10:38 am #

    The SFO extension is a complete disaster. The tracks should have terminated at a trimodal station nearer the airport where the Airtrain circulator could have taken riders to the various terminals. The Millbrae garage is the Trojan Horse for BART’s dream of killing and replacing Caltrain and was necessary for the ridership projections which have never been achieved.
    As to the service patterns, BART has tried several, but all have generated very low ridership. Mounting losses triggered a lawsuit as BART tried to enforce a contract with the San Mateo County Transit system to pay for empty trains. The ill designed routing has saddled the public with recurring losses as well as rider inconvenience.
    BART’s original agit prop claimed 90 sec headways–never achieved on a regular basis. Crowding on the P’burg BP line illustrates why extensions of BART are a mistake. The sprawlburb riders are very rush hour centric. When there was an outage of the Bay Bridge, they overfilled the trains with no room for the regular west of the hills urban riders.

  13. Ted King February 26, 2011 at 3:13 pm #

    Re : BART’s New Cars
    ” BART Directors Consider Design Concepts for New Rail Cars” (StreetsBlog article, 8 May 2009)
    “New Train Car Project” ( article, last upd. 11 Feb. 2011)
    I think that some of the ideas being put forward are flame bait (e.g. the leaning pads and mostly transverse seating). I remember the long runs from 24th+Mission to Pleasant Hill back in the 1980’s as good times to take a nap or read a book. Jostling (from starting and stopping) plus transverse seating would have a fair number of riders in each other’s laps. [The SFMuni version is jerkiness due to the often abrupt starts and stops.]
    Less controversial, a modest narrowing of the seats (say from 22″ down to 20″) would make the aisles wider and save money on seat costs. I also like the idea of a third set of doors because I’ve seen the crowds that are waiting to board get bunched up at the current doors when boarding in downtown S.F. As one who is trying to get off of the train I’ve felt like a cowboy facing a stampede of longhorns (well chilled longnecks would be more fun).

  14. Mike Fogel February 26, 2011 at 11:37 pm #

    I had to deal with this f-up of bart for three years. I used to live right by civic center station and commute to the south bay. It was actually easier for me to go right past civic center, and head all the way to the 4th and King Caltrain station than deal with the problem of trying to transfer to/from Bart and Caltrain.
    @BBnet3000 The bart headways aren’t really the main issue. It’s that there is no coordination between the bart and caltrain schedules.
    I have to wonder if this extension were built today if this mistake would make it through. Seems to me that back in 2003 there wasn’t as much of a transit community as we enjoy today… I’d like to think that by 2011 the game has changed – experts in transit (and other fields) have a stronger voice via blogs/twitter/FB/etc. It’s not hard to point out a number of projects that are going through today that the majority of the community would label as ‘not optimal’ (OAK connector, bart to SJ). But I don’t think those rival the Milbrae/SFO split in sheer stupidity.

  15. Jack Horner February 27, 2011 at 12:44 am #

    For another example of triangular service problems see,_Sydney
    $80 million spent building the 3rd side of the triangle between two lines in suburban Sydney, to allow a direct service between two regional centres (ie a downstream portion, traverse the third side of the triangle, then an upstream portion on the other line, bypassing the junction which had always been a transfer point).
    This is in addition to the services which have always run and continue to run from each of the these centres, through the bypassed junction station, to the main city further away.
    The new service is predictably poorly patronised (these are regional centres after all, neither of which is more than a tenth as big as the main city centre which is the focus of the rest of the network).
    Within 10 years the new service is canned. The Y link is a white elephant.
    I would have put the money into more frequent services on the existing lines, with some ‘weaving’ grade separation at the junction which was bypassed by the Y, to ensure cross platform interchange between the downstream and upstream trains on each line.

  16. Ted King February 27, 2011 at 2:57 am #

    I’ve read the Wikipedia article and agree that in some ways it was a waste of money. Are there any festivals in either regional center that would justify special service ? I’m thinking along the lines of sports (e.g. soccer, rugby, Highland Games), Aggie Fairs, etc.
    Granted it was a waste in terms of regular service but I see it as an investment in case of a disaster at either far end of the wye. I followed your link on up to CityRail and then down to their map [ “File:Sydney railway map.gif” (fits most screens)]. The Cumberland Line is a rough “7” (seven) on the west side of greater Sydney with the turn approximately in the middle of the image.

  17. jack horner February 27, 2011 at 1:47 pm #

    @ Ted King “are there any festivals etc to justify special service?’
    No. The Cumberland line (Parramatta to Liverpool direct bypassing the normal transfer point at Granville) is still noted in Sydney rail information because it has a token twice daily peak hour peak direction service.
    Saving distance (as opposed to avoiding a transfer) was never a significant point as the new triangle has sides no more than 600 metres long.

  18. teme February 27, 2011 at 2:09 pm #

    “You could split the train in half, and send the front half on one branch and the rear half on the other. This is very tricky; it requires a driver in position ready to take half of the train when it arrives. It’s also hard to separate a train without at least a minute or two of delay, at least for the rear half of the divided train.”
    I’ve ridden Hamburg’s U-Bahn to the airport that was IIRC split to two trains without any noticeable delay. Could be an automatic line, really don’t know and too lazy to google, or they just have two drivers. Some tram systems do that too. Plus if the capacity is frequency contstrained you get more trains.

  19. ant6n February 27, 2011 at 6:50 pm #

    Adding to what teme said:
    I also just came across that the S-Bahn Hamburg does indeed do a split. The S1 splits at a station, and the front half continues to the airport through a tunnel, the rest goes to the suburbs. I find this surprising because this is happening on a line that has service every 10 minutes – basically a subway, indeed.
    The station dwell time seems to be 2 minutes for the split, and 3 minutes for the join (at least for the part of the train connecting to the airport).
    More info in this (English/German) pdf/flyer:

  20. Kenny March 3, 2011 at 12:50 pm #

    I believe that BART has operated at various points with several of these structures. I think they normally use one of the sequential routings these days, but during rush hour they may use the branching route. It was originally labeled as having the branching route plus the Millbrae-SFO shuttle.

  21. Eugene Wong March 20, 2011 at 7:54 pm #

    I don’t recall frequency doubling along the Expo main trunk. I remember it staying the same.
    Also, for some time, the trains have been running at the highest frequency during rush hours. I know for a fact, that the remaining stations on the Expo line have decreased.
    What I don’t understand is why the trains aren’t broken up during the weekends, to give more frequent service. It’s so frustrating to stand out in the cold, waiting for a train to arrive in 8 minutes, and then have a 4 car train pull up, mostly empty.
    Maybe you could do something about this, when you work with Translink. I find it frustrating that we don’t get full frequency during the weekends.
    I favour splitting the trains up at Columbia Station. It’s annoying, but it makes life easier. People won’t necessarily have to switch cars.
    As it is, about every 3rd train will be a Millennium Line train, if I understand correctly. Maybe every 2nd and 3rd train could be split up, to maintain good frequency.

  22. Nathanael May 25, 2011 at 12:40 am #

    The Sydney case looks suspiciously to me like it was built for the purposes of freight and merely advertised for passengers. I could be wrong, but it has all the hallmarks. It would certainly help Canberra-Broken Hill freight traffic.