The Limits of Clockface Scheduling

This is morally the last post in my series on improving the MBTA: see here, here, and here for the three previous posts. However, it’s a more general principle concerning interlined regional rail services.

Good practice for running transit service that isn’t at show-up-and-go frequency – say, anything that comes every 10 minutes or more, certainly anything that comes every 15 minutes or more – is to have regular clockface intervals. This is memorable for passengers, and works as a baseline with which to work on providing extra connections. In addition, if there is interlining, then it makes it easy to schedule trains to come at a uniform frequency on the share segment. If service is uniform throughout the day, then this is very easy. The problems start when it is not.

Normally, if extra peak service is required, then rigid clockface systems, such as those found in the German-speaking world, will usually interpolate in the middle of the period. In other words, if a station gets inbound trains at :00 and :30 every hour, then in the peak it will also get them at :15 and :45. This is what’s done in Stuttgart on two of the S-Bahn lines, interpreting peak very liberally, and less rigidly on the TER in Nice. Many systems instead use similar peak and midday service, dropping service only in the evening, such as the Berlin S-Bahn, and BART.

To see where problems could occur, let us look at Berlin again. There are three services on the Stadtbahn: the S5, the S7, and the S75. At the peak, all three run every ten minutes, with westbound trains departing Ostkreuz at :02, :00, and :05 respectively. Off-peak, the S75 drops to 20-minute frequencies, introducing 8-minute gaps into a schedule whose average headway is 4 minutes.

For a cleaner, contrived example, let’s say we interline two services, each with a 15/30 frequency; a factor-of-2 difference in frequency is more or less the norm on commuter lines in Tokyo and Paris, which do not have rigid clockface schedules – more local lines have a slightly smaller gradient than more long-distance lines. There is an inherent tradeoff between uniform frequency at the peak and uniform frequency off-peak. It’d be much easier to do if both services were bound to have the same frequency but the frequency varied continuously, as it does on most subways; however, what works on a dedicated line when passengers show up and go fails when passengers consult schedules and when timed connections or overtakes are involved.

More concretely, if Line 1 leaves a station at :00 and Line 2 leaves at :08, providing uniform peak frequency of 7-8 minutes, then off-peak we will have a 22-minute gap when we reduce to half-hourly frequency on each line; and if Line 2 instead leaves nearly at :15 to provide uniform off-peak frequency, then there will be a gap of nearly 15 minutes at the peak. The sum of the largest peak and off-peak gaps is necessarily 30 minutes, whereas the ideal would be for the sum to be 22.5 minutes.

Extra constraints can force one choice of gaps. For example, the Providence and Stoughton Lines are (or should be) constrained by the need to fit faster intercity trains on the line, at least in the future; for details of those constraints, see my posts on MBTA-HSR compatibility. In short, if we choose the symmetry axis to be :00, then Providence Line trains are compelled to leave South Station at :02 to meet up with trains to Woonsocket, and high-speed trains leave South Station at :10-11 and begin to overtake Providence trains at Readville at :15. Stoughton trains should then leave immediately after the high-speed trains so that they can leave the line toward Stoughton just before they’d get overtaken (at Sharon, if they continued), which means at :12-13. Thus we obtain about a 10-minute gap at the peak and a 20-minute gap off-peak, which is an acceptable compromise.

In contrast, one thing that clockface scheduling does not limit is short-turns. Indeed the Berlin S-Bahn often short-turns every other train, without trouble. Moreover, it is not difficult to drop to half the peak frequency with short-turns. If a train that leaves at :00 runs all the way to the end and a train that leaves at :08 short-turns (both repeating every 15 minutes), then it is not difficult to change things so that in the off-peak, a train that leaves at :00 or :30 runs to the end and a train that leaves at :15 and :45 short-turns. People beyond the short-turn point would still only need to memorize at what minute between :00 and :29 the trains serve their stations, and it would be regular all day; people before the short-turn point would again only need to memorize one number, from :00 to :14. In the case of the MBTA, this means that the Fairmount Line (which should get a train that turns at Readville for every train that continues toward the Franklin Line) can get a perfectly regular timetable.


  1. ant6n

    The length of trains can be varied as well. Keeping with the Berlin Example, a full train consists of 4 pairs of cars, or 8 cars (about 160m long). Then there exist 3/4, 1/2 and 1/4 trains, with 6, 4 and 2 cars, respectively. 1/4 trains are basically non-existent now, but other lengths still exist and allow to keep a fixed schedule with variable costs and capacity – although naturally the cost savings of running a shorter train may not be that big.

    (most other German S-Bahn systems use trains that are 67m long, and are used in consists of 1 (short), 2 (long), 3 (extra-long) trains, giving a maximum train length of 210ish m)

    • Alon Levy

      Yes, that too is possible. Richard tells me that this is pretty much all they do in Switzerland, where the extra train frequency in the peak is very small – much less than the factor of 2 difference in Paris and Tokyo.

      For the record, current MBTA practice is to keep train lengths constant. Off-off-peak, when there isn’t enough demand to fill 6 cars, they only open 2 cars and have the rest deadhead, with bars making it hard for passengers to cross from the open cars to them.

        • Alon Levy

          Probably very little – when the cars are pulled by a locomotive, especially one that was adapted from freight, there is very little variance in fuel consumption based on train size.

          • dejv

            That depends on stop frequency and full throttle/idle consumption ratio. Every ton that has to be pulled from standstill to top running speed makes difference in acceleration time and subsequently time share of high-consumption. Put another way: the extra energy that goes waste during the braking has to be given to the train during acceleration.

            (You can google that full throttle-coasting-brake is the most energy efficient way of driving train with short stop spacing)

          • Alon Levy

            The Fairmount Line DMU study does not specify the operating assumptions it uses in computing fuel efficiency. It sources the numbers to the MBTA’s reporting to the NTD in 2004, so presumably that’s the average for MBTA trains of each length, in which case the stop spacing is fairly wide.

          • Richard Mlynarik

            (You can google that full throttle-coasting-brake is the most energy efficient way of driving train with short stop spacing)

            No you can’t! Just think about it. (OK, maybe you can. In which case there is something wrong on the internet.) Lower speeds are more energy efficient in a world with non-linear resistive losses.

            Pedal-to-the-metal is the most time-efficient (shortest stop to stop interval) way of driving, however, as ought to be obvious as one considers any variation from that strategy.

      • ant6n

        One can also create a clockface system that allows going from 2 to 3 trains within the base-takt while minimizing variance. For example, assuming a 20 minute takt, one could run a train at :0 and :12, giving a 8/12 minute schedule with a 10 minute average. If one adds a possibly short-turning train at :6, one has a 6/6/8 schedule. If you then make the train after the long wait a long train, you could have pretty even load balancing, and fairly even waiting times.

        (The math geek inside me tells me the smallest ratio between the long and the short wait is sqrt(2), i.e. there’s a 2/sqrt(s) schedule, and a 1/1/sqrt(2) schedule – although in practice one might want to have less variance for the long wait).

        • Alon Levy

          Yes, that’s also a possibility.

          The main drawback, though, is that like other things that break the rigid takt, it’s more useful at high frequency than at low frequency. 8/12 and 6/6/8 works better than 24/36 and 18/18/24.

          The smallest ratio between the long and short wait if the average frequency ratio is 1:2 is 1:2, i.e. there’s a 1/2/1/2 schedule and a 4/2 schedule.

        • Richard Mlynarik

          One can also create a clockface system that allows going from 2 to 3 trains within the base-takt while minimizing variance. …


          But I think these are somewhat “artificial” situations.

          I say “artificial” because my local system does exactly that: changing from 15 minute takt on five routes (four of them interlined in the major CBD through San Francisco, and two making a timed transfer in the secondary CBD of Oakland) to three routes at 20 minute takt (of which two are interlined in San Francisco, and two make Oakland timed transfers.)

          (I simplify a little, because one of the peak routes is actually on 7.5 minute headway. And on Saturdays before 19:00 (but not Sundays!) all five routes operate at 20 minute headway.)

          But it is “artificial” because the 20 minute headway service to very distant and very low traffic stations at all times is entirely a political decision. Rather than simply cutting service to the suburbs from 15 minute to 30 minute headways — which is more than can be justified in some cases — every line goes to 20 minute headways. Train lengths are reduced, but even here the politics means the trains are too long, since they are sized for central urban loads but then continue nearly empty for long distances into exurban areas.

          This makes for some off service gaps during the transition, as Alon mentions, but few notice them.

          The real downside, and the reason that this 15-to-20 transition is artificial, is because there are no takt-coordinated transfers with any other services. (It’s hard to understand, but that’s how things operate in the SF Bay Area and the US in general!) If there were co-ordinated transfers with regular headway buses and trams and other rail systems around the network, then the changeover becomes unmanageable: traffic towards the CBD goes from 15 to 20 headway an hour and a half before traffic from the CBD does, and the transition times differ at every station on the line.

          In contrast, with a simple cut-back from 15 to 30 minute headway, connecting services throughout a coordinated Verkhrsverbund can be cut back together while preserving the same timed connections, and, most importantly, preserving the same human memory schedules. (My stop is “:07 past the hour, every 30 minutes, with bonus extra service at peak” is easy to remember that “:07 past the hour every 15 minutes, except after 19:47 on weekdays, and :12 past the hour every 20 minutes evenings after 20:12 and on weekends”.)

          • ant6n

            Well, the example I gave is one where two trains stay on a completely fixed schedule within every 20 minute period (so you have a train every 12/8 minutes), and a third one can be added in (a train every 6/6/8 minutes).

            From the structure it’s very similar to a half hour schedule (a train every 30 minutes), where a train gets added during rush (a train every 15 minutes). It’s just that this doubles frequency, which may be undesirable at times, whereas the example I gave only adds 1/2 more trains, making it more manageable.

            The point was to show that in this scenario you can still have two of three trains operate on a complete clockface schedule at all times. The example of BART that you give, of falling from 15 minute headways to 20 minute headways does not do that.

            On a side note: In Berlin in general you actually have a maximum headway of 20 minutes. Some of the suburban termini use 10 minute headways, or 10 minute headways during weekdays, and 20 minute headways otherwise. But there are some termini that have a fixed 20 minute schedule at all times, during the rush hour and sundays. From a user perspective that’s a good thing – there’s always service on an easy to remember schedule.

          • ComradeFrana

            Little off-topic, but that timed transfer (between Richmond – Fremont and Pittsburg/Bay Point – Daly City lines) is pretty “interesting”.

            In the southbound direction the Fremont and Daly City bound trains (except peak trains from Concord) make a timed transfer at MacArthur after which the Fremont train has to wait 2 minutes, because MacArthur-12st Oakland Center segment has only three tracks (two northbound, one southbound).

            On the other hand Richmond and Pittsburg/Bay Point bound trains run parallel all the way from 12st Oakland Center to MacArthur having 3 timed transfers in a row.

            Obviously the diagonal transfers (Fremont – Daly City and Richmond – Pittsburg/Bay Point) are not timed and not even cross-platform. (not counting the “timed” but not cross-platform transfer from Fremont to Richmond train to Richmond to Daly City/Millbrae train which I doubt is humanely possible).

  2. R. W. Rynerson

    We use clock headways most of the day on Denver’s Light Rail lines for several reasons:

    1. We have many wide headway feeder bus routes in our low density areas, and I learned years ago in another city that if a main line schedule is written “North American” style — i.e., every scheduler for him/herself — with many variances in clock times to make it seem to work perfectly — that it will either disrupt the feeder routes that need memory pattern headways or in most cases, the other schedulers will throw out any concern for main line connections and write what pleases them on each route. A subset of this is that Ray Perkins, long-gone GM of long-gone Rose City Transit Co. explained to me that it was more efficient to run wide, clock headways with good connections than to throw service onto lines in the hope that random connections would improve. His firm was interested in making a profit, but in a sprawl city, resources are stretched thin and I often borrow from things he taught me.

    2. We learned from the 16th Street Mall Shuttle that with a cooperative effort with city traffic engineers, that we could write schedules that mesh with the traffic signals, offering the sensation of transit pre-emption without the disruptive effects. The trade-off is that our Central trunk line must operate on multiples of three minutes (90-second traffic signals using the new national standards for slow walking pedestrians). That does not have to result in clock scheduling, but it works best to do so. In peak hours, we run 18 trains in 20 slots an hour, converging from three lines and diverging onto two lines and return, so if a route shifted times, it would require the whole schedule to shift.

    3. Although employers offer wildly flexible work schedules in this era, most of our customers prefer to keep conventional clock hours. In the CBD and at college campuses, customers show up in waves that correspond with clock patterns and we try to accommodate them.

    I have a couple of YouTube videos posted that show portions of this. One shows our Night Meets, a three-way connection on 30-minute headways after 10 p.m., favoring the outbound travel directions (SE to SW, Central to SE/SW, Union Station to SE/SW). This gets most customers through their transfers without waiting long. The other video shows customers transfering in the AM peak from the 3-minute headway inbound trains to the 6-minute headway inbound Limited stop bus, which gets them to the corner of Downtown Denver that does not have a rail line. The clock headway on the rail line leads logically to the clock headway on the bus.

  3. Andrew in Ezo

    A note to add about Japan- though urban lines run on a show up and go frequency, certain complicated lines with various levels of train service and interline running do use a variation on the clockface schedule to help passengers. Called the “pattern diagram”, it is a regular repeating (every 60 min) schedule lineup of trains, typically used during the mid-day off-peak (11:00 to 17:00). To take Kita-Senju Station on the Tobu Isezaki Line as an example, the noon cycle begins with 3 trains (two long distance zone rapids for different destinations and a local) departing simultaneously, as the Isezaki Line here is on two levels and quad tracked. They are followed five minutes later by an express, eight minutes later by a zone semi-express, ten minutes later by a local, and so forth, with limited express trains interspersed between (4 trains/hour). Locals come every ten minutes as do expresses. Zone semi-expresses come every twenty minutes. The whole process is repeated in the following hour. Passengers don’t have to remember the actual times, but rather know if a they miss a local, they only have to wait for 10 minutes for the next local. Or if they see an express departing, they know five minutes later the local will arrive.

    • Alon Levy

      Yes, this is true. Something similar also happens on the Tokaido Main Line to Odawara. But those lines tend to break the clockface pattern completely in the peak. This is similar to the RER, which (at least on lines A and B) is rigidly clockface in the afternoon off-peak, but changes the spacing and service pattern organically at the peak, and in contrast to systems in German-speaking cities, the Kodama, and the TER in the Riviera, which keep the basic clockface pattern in the peak but add supplementary trains.

      And of course the local lines have a completely show up and go schedule, as they should. When the service pattern is “Trains travel in a circle and share tracks with nothing else and run every 5 minutes until midnight,” there’s not much need for a clockface schedule.

  4. Richard Mlynarik

    To com back to your subject of the limits of clockface scheduling, and in particular the examples of problems you posit (which are for a single route, with some interlining): I suggest that there is a happy near-coincidence between the threshold at which humans treat the service as “show and go”, with no attention to timetables, and that at which it becomes more seriously technically challenging to maintain strictly evenly-spaced service, and that pair of thresholds are 12-minute-ish .

    Certainly at individual route headways below 10 minutes it doesn’t make practical sense to time (meaning plan for, and plan schedule padding for and provide infrastructure padding for) anything but the most straightforward transfer (ie that across a single platform.)

    The real limit to clockface scheduling aren’t with interlined routes on one track, but with the need for number of parallel routes into large transfer junction stations serving diverging routes, together with the number of platforms at those stations, and the speed of pedestrian movement between them. (There’s also a big takt-y power spike for the traction supply to manage.) It doesn’t seem likely that any station in North America will serve dozens or even half dozens of takt-coordinated mainline rail routes any time soon, so this is a bit of an hypothetical downside in our context. (It’s a big deal for, say, Bern, with 14 departures scheduled within 7 minutes every half hour at :00 and :30, but not here.)

    In short: regular interval scheduling works pretty much perfectly until you get down to the the frequency of service that riders don’t need to or care to consult a schedule anyway.)

  5. Adirondacker12800

    It doesn’t seem likely that any station in North America will serve dozens or even half dozens of takt-coordinated mainline rail routes any time soon, so this is a bit of an hypothetical downside in our context.

    …. Change at Jamaica…. Once East Side Access opens they are going to abandon timed cross platform transfers and go with “the connecting train will be here ‘soon’ ” model. Everything will be directed to Penn Station or Grand Central. If you want Brooklyn, change to the shuttle. MTA is aiming for 2016, FTA says 2018.

  6. DGL

    Add this to the list of infuriatingly imperfect solutions.. Couldn’t you just have a transition train on one line? This minimizes the impact of strange spacing AND it does something else cool, which is that if, say, you decide you want to transition frequencies at a certain hour, one line transitions precisely at the hour and the other splits the difference across the hour with something resembling finicky 20 minute spacing (actually 22.5 which is exactly midway between 15 and 30). I am scratching my head at why this isn’t more of an automatic reaction, since it seems like there are basically zero drawbacks.

    :00 :10 :20 :30 :40 :50
    :30 :40 :50 :00 :10 :20
    :00 :10 :20 :30 :40 :50
    :15 :25 :35 :45 :55 :05
    :30 :40 :50 :00 :10 :20
    :45 :55 :05 :15 :25 :35
    :00 :10 :20 :30 :40 :50
    :15 :25 :35 :45 :55 :05
    :45 :55 :05 :15 :25 :35
    :07 :17 :27 :37 :47 :57
    :22 :32 :42 :52 :02 :12
    :37 :47 :57 :07 :17 :27
    :52 :02 :12 :22 :32 :42
    :07 :17 :27 :37 :47 :57

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