How Many Tracks Do Train Stations Need?

A brief discussion on Reddit about my post criticizing Penn Station expansion plans led me to write a very long comment, which I’d like to hoist to a full post explaining how big an urban train station needs to be to serve regional and intercity rail traffic. The main principles are,

  • Good operations can substitute for station size, and it’s always cheaper to get the system to be more reliable than to build more tracks in city center.
  • Through-running reduces the required station footprint, and this is one of the reasons it is popular for urban commuter rail systems.
  • The simpler and more local the system is, the fewer tracks are needed: an urban commuter rail system running on captive tracks with no sharing tracks with other traffic and with limited branching an get away with smaller stations than an intercity rail station featuring trains from hundreds of kilometers away in any direction.

The formula for minimum headways

On subways, where usually the rush hour crunches are the worst, trains in large cities run extremely frequently, brushing up against the physical limitation of the tracks. The limit is dictated by the brick wall rule, which states that the signal system must at any point assume that the train ahead can turn into a brick wall and stop moving and the current train must be able to brake in time before it reaches it. Cars, for that matter, follow the same rule, but their emergency braking rate is much faster, so on a freeway they can follow two seconds apart. A metro train in theory could do the same with headways of 15 seconds, but in practice there are stations on the tracks and dealing with them requires a different formula.

With metro-style stations, without extra tracks, the governing formula is,

\mbox{headway } = \mbox{stopping time } + \mbox{dwell time } + \mbox{platform clearing time }

Platform clearing time is how long it takes the train to clear its own length; the idea of the formula is that per the brick wall rule, the train we’re on needs to begin braking to enter the next station only after the train ahead of ours has cleared the station.

But all of this is in theory. In practice, there are uncertainties. The uncertainties are almost never in the stopping or platform clearing time, and even the dwell time is controllable. Rather, the schedule itself is uncertain: our train can be a minute late, which for our purpose as passengers may be unimportant, but for the scheduler and dispatcher on a congested line means that all the trains behind ours have to also be delayed by a minute.

What this means that more space is required between train slots to make schedules recoverable. Moreover, the more complex the line’s operations are, the more space is needed. On a metro train running on captive tracks, if all trains are delayed by a minute, it’s really not a big deal even to the control tower; all the trains substitute for one another, so the recovery can be done at the terminal. On a mainline train running on a national network in which our segment can host trains to Budapest, Vienna, Prague, Leipzig, Munich, Zurich, Stuttgart, Frankfurt, and Paris, trains cannot substitute for one another – and, moreover, a train can be easily delayed 15 minutes and need a later slot. Empty-looking space in the track timetable is unavoidable – if the schedule can’t survive contact with the passengers, it’s not a schedule but crayon.

How to improve operations

In one word: reliability.

In two words: more reliability.

Because the main limit to rail frequency on congested track comes from the variation in the schedule, the best way to increase capacity is to reduce the variation in the schedule. This, in turn, has two aspects: reducing the likelihood of a delay, and reducing the ability of a delay to propagate.

Reducing delays

The central insight about delays is that they may occur anywhere on the line, roughly in proportion to either trip time or ridership. This means that on a branched mainline railway network, delays almost never originate at the city center train station or its approaches, not because that part of the system is uniquely reliable, but because the train might spend five minutes there out of a one-hour trip. The upshot is that to make a congested central segment more reliable, it is necessary to invest in reliability on the entire network, most of which consists of branch segments that by themselves do not have capacity crunches.

The biggest required investments for this are electrification and level boarding. Both have many benefits other than schedule reliability, and are underrated in Europe and even more underrated in the United States.

Electrification is the subject of a TransitMatters report from last year. As far as reliability is concerned, the LIRR and Metro-North’s diesel locomotives average about 20 times the mechanical failure rate of electric multiple units (source, PDF-pp. 36 and 151). It is bad enough that Germany is keeping some outer regional rail branches in the exurbs of Berlin and Munich unwired; that New York has not fully electrified is unconscionable.

Level boarding is comparable in its importance. It not only reduces dwell time, but also reduces variability in dwell time. With about a meter of vertical gap between platform and train floor, Mansfield has four-minute rush hour dwell times; this is the busiest suburban Boston commuter rail station at rush hour, but it’s still just about 2,000 weekday boardings, whereas RER and S-Bahn stations with 10 time the traffic hold to a 30-second standard. This also interacts positively with accessibility: it permits passengers in wheelchairs to board unaided, which both improves accessibility and ensures that a wheelchair user doesn’t delay the entire train by a minute. It is fortunate that the LIRR and (with one peripheral exception) Metro-North are entirely high-platform, and unfortunate that New Jersey Transit is not.

Reducing delay propagation

Even with reliable mechanical and civil engineering, delays are inevitable. The real innovations in Switzerland giving it Europe’s most reliable and highest-use railway network are not about preventing delays from happening (it is fully electrified but a laggard on level boarding). They’re about ensuring delays do not propagate across the network. This is especially notable as the network relies on timed connections and overtakes, both of which require schedule discipline. Achieving such discipline requires the following operations and capital treatments:

  • Uniform timetable padding of about 7%, applied throughout the line roughly on a one minute in 15 basis.
  • Clear, non-discriminatory rules about train priority, including a rule that a train that’s more than 30 minutes loses all priority and may not delay other trains at junctions or on shared tracks.
  • A rigid clockface schedule or Takt, where the problem sections (overtakes, meets, etc.) are predictable and can receive investment. With the Takt system, even urban commuter lines can be left partly single-track, as long as the timetable is such that trains in opposite directions meet away from the bottleneck.
  • Data-oriented planning that focuses on tracing the sources of major delays and feeding the information to capital planning so that problem sections can, again, receive capital investment.
  • Especial concern for railway junctions, which are to be grade-separated or consistently scheduled around. In sensitive cases where traffic is heavy and grade separation is too expensive, Switzerland builds pocket tracks at-grade, so that a late train can wait for a slot without delaying cross-traffic.

So, how big do train stations need to be?

A multi-station urban commuter rail trunk can get away with metro-style operations, with a single station track per approach track. However, the limiting factor to capacity will be station dwell times. In cases with an unusually busy city center station, or on a highly-interlinked regional or intercity network, this may force compromises on capacity.

In contrast, with good operations, a train station with through-running should never need more than two station tracks per approach track. Moreover, the two station tracks that each approach track splits into should serve the same platform, so that if there is an unplanned rescheduling of the train, passengers should be able to use the usual platform at least. Berlin Hauptbahnhof’s deep tracks are organized this way, and so is the under-construction Stuttgart 21.

Why two? First, because it is the maximum number that can serve the same platform; if they serve different platforms, it may require lengthening dwell times during unscheduled diversions to deal with passenger confusion. And second, because every additional platform track permits, in theory, an increase in the dwell time equal to the minimum headway. The minimum headway in practice is going to be about 120 seconds; at rush hour Paris pushes 32 trains per hour on the shared RER B and D trunk, which is not quite mainline but is extensively branched, but the reliability is legendarily poor. With a two-minute headway, the two-platform track system permits a straightforward 2.5-minute dwell time, which is more than any regional railway needs; the Zurich S-Bahn has 60-second dwells at Hauptbahnhof, and the Paris RER’s single-level trains keep to about 60 seconds at rush hour in city center as well.

All of this is more complicated at a terminal. In theory the required number of tracks is the minimum turn time divided by the headway, but in practice the turn time has a variance. Tokyo has been able to push station footprint to a minimum, with two tracks at Tokyo Station on the Chuo Line (with 28 peak trains per hour) and, before the through-line opened, four tracks on the Tokaido Main Line (with 24). But elsewhere the results are less optimistic; Paris is limited to 16-18 trains per hour at the four-track RER E terminal at Saint-Lazare.

At Paris’s levels of efficiency, which are well below global best practices, an unexpanded Penn Station without through-running would still need two permanent tracks for Amtrak, leaving 19 tracks for commuter traffic. With the Gateway tunnel built, there would be four two-track approaches, two from each direction. The approaches that share tracks with Amtrak (North River Tunnels, southern pair of East River Tunnels) would get four tracks each, enough to terminate around 18 trains per hour at rush hour, and the approaches that don’t would get five, enough for maybe 20 or 22. The worst bottleneck in the system, the New Jersey approach, would be improved from today’s 21 trains per hour to 38-40.

A Penn Station with through-running does not have the 38-40 trains per hour limit. Rather, the approach tracks would become the primary bottleneck, and it would take an expansion to eight approach tracks on each side for the station itself to be at all a limit.

124 comments

  1. df1982

    I can see the utility of a mainline central station a la Penn Station Amtrak having six platforms for two through tracks. An island platform with two tracks on each side for through services, and an island platform in the middle in case you need to turn the odd train at the central station, e.g. for long-distance that have no need of travelling through, or just for late trains. Berlin Hbf manages this with Gesundbrunnen, which is about a five-minute trip away.

    At terminals, how much do the station throats affect capacity? It surprises me, for instance, that more terminals don’t have grade separated junctions at the throats, so that they can effectively function like two adjacent branches for turnback purposes. Instead everything gets thrown at a mishmash of switches and interlockings.

    • Alon Levy

      At terminals, how much do the station throats affect capacity?

      Significantly. Good practices on this include,

      – Separating stations into multiple stations as you mention whenever possible; this is what I propose for Penn Station and South Station.
      – Increasing travel speed through the throat – American 10 mph throats force trains to spend more time in the most congested part of the system, whereas Stuttgart 21 is designed around 80-90 km/h.
      – Reliable timetabling to reduce throat conflict.

      Even then, the relationship between turnback capacity and the number of tracks at a terminal looks somewhat degressive to me – the RER turns 10 peak tph at CDG T2 on two tracks and yet only turns 16 on four tracks at Saint-Lazare.

    • xh

      “At terminals, how much do the station throats affect capacity”

      In China, we have always been told that the capacity of a railway station is determined by the following three kinds of capacities:

      1. capacities of platform tracks (i.e. how efficient trains dwell at platform tracks )
      2. capacities of connecting mainline tracks
      3. capacities of station throats

      Thus, whichever is the smallest would play a decisive role. In early 1900s, locomotive-hauled trains used to have very long platform dwell time. That’s probably why so many gigantic terminals, like the Grand Central Terminal, were built at that time: to increase overall capacities of platform tracks, which are limiting factors.

      Today, however, modern mainline EMUs could turn back in less that fifteen minutes. Overall platform track capacities are no longer limiting factors. Adding more tracks to a station may not necessarily lead to higer capacity, and sometimes it will lead to lower capacity, as station throats become longer.

      Lessons learned from early Chinese HSR constructions are good examples of showing this: Shanghai Hongqiao Station was designed when EMUs’ performances were significantly underestimated. The “high-speed” yard of Shanghai Hongqiao Station serves two high-speed lines (Beijing-Shanghai HSR and Shanghai-Hangzhou HSR) or four mainline tracks with 19 platform tracks. Even if Shanghai-Hangzhou HSR is equipped with modern CTCS level 3 (equivalent of ETCS level 2) signalling, with one-kilometer-long station throats at present, trains still cannot run reliably under a less-than-three-minute headway, according to a very recent research: https://kns.cnki.net/kcms/detail/detail.aspx?dbcode=CJFD&dbname=CJFDLAST2021&filename=TDYS202110001&uniplatform=NZKPT

      • Matthew Hutton

        The Americans (and to a lesser extent the Chinese) could make their stations more efficient by allowing passengers onto the platform before the train arrives.

        • adirondacker12800

          It’s unwise to let a few thousand people loiter around on a platform designed in 1905 for 142 parlor car and sleeping car passengers.

          • Matthew Hutton

            I mean you do in Europe and Japan with the partial exception of terminus stations. Perhaps platform edge doors should be added?

          • adirondacker12800

            You can wait on the platform at most stations in the U.S. too. Many suburban stations, there is no other place to wait.
            If you let people mill around in the waiting lounges, concourses and the retail in the main station, during the five minutes the track is open for passengers, the train is the platform edge screen. Darwin Award winners manage to defeat that very occasionally.

          • Krist van Besien

            Than you reduce the number of tracks so you can wide the platforms…

          • adirondacker12800

            Big city station with 5 islands and ten tracks…which 40 percent of the standing room only passengers on rush hour trains do you tell to work from home for a few years while the platforms are widened? 10 island, 20 track station like Penn Station in New York, which 20 percent?

          • Matthew Hutton

            The Odakyu Electric Railway at Shinjuku in Japan handles 500k vs 350k passengers a day with only ten platforms, I’m sure something could be done while the platforms were widened or platform barriers were installed.

          • adirondacker12800

            It is a lot cheaper to announce the express to someplace is on track whatever and the horde that has been milling around gets on the train. A few moments later the local is announced and that horde gets on the train.
            It’s going to take a year or two for demand to overwhelm Gateway when it opens. They could double deck 7 through 14, aligned so it could connect with the 5th and 6th tunnels under the Hudson and East rivers.

          • Eric2

            Lol adirondacker12800, you are a classic incurious American. The RER Chatelet Les Halles station handles more passengers per year than Penn Station, and it has just 4 platforms and 7 tracks. Other systems worldwide have similar passenger to track/platform ratios. But you don’t know about this, or else have chosen to pretend not to know, so you just make ignorant sarcastic comments about how 40% of the passengers will have to go home if a few platforms are eliminated.

          • adirondacker12800

            The RER has platforms the size of aircraft carrier decks.

          • Alon Levy

            Their limiting factor isn’t width, it’s vertical egress. You’d expect those 17 m platforms would support banks of six escalators at a time, but no, it’s two escalators per access point, and mild queues do form at rush hour.

            Penn Station is likewise limiting by egress, which LIRR work in the 1990s and then Moynihan phase 1 (not the useless phase 2) alleviated.

          • adirondacker12800

            17 meters from the far side of the track to the center of the island? The platform, the concrete bits where they let passengers circulate, is extra wide between track 18 and 19. The other ones aren’t. I’ve never taken out a tape measure and measured, I concentrate on not being trampled.
            Railfans just love to video the mighty Amtrak trains arriving and departing. The next platform over is in their lovingly shot videos. It’s not 17 meters. They also love to video the stampede when a track is announced. I don’t know which Penn Station you have been in but it’s not the one I’ve been using for 50 years.

          • Alon Levy

            17 meters of platform width at Les Halles. Obviously Penn is narrower – the platforms other than that wider one are around 5, which isn’t too bad by subway standards.

          • Eric2

            Chatelet Les Halles is about 80m wide, including all tracks and all platforms. Penn Station is about 200m wide. If you are short of platform space at Penn, you just pave over another couple tracks and you have all the space you need.

          • adirondacker12800

            Why is it so offensive to not know four days in advance the express is going to be on Track 13?
            Just because yokels from the hinterlands find two levels of concourse and 21 tracks confusing doesn’t mean we should spend billions of dollar to make it look more like their Toonerville Trolley that meets every train six times a day.

          • Krist van Besien

            In Zürich you know a year in advance what platform is going to be. Several years in fact. SBB is currently working on the timetable for 2035…

          • Matthew Hutton

            Adiron, if people which platform the train is going to be on 10 minutes ahead they can be waiting on the platform ready to board the train before it arrives which means Amtrak trains could have a 1-2 minute dwell like the Shinkansen rather than a 15 minute dwell.

          • adirondacker12800

            Manhattan is quite the destination, when the train comes in lots and lots and lots of people get off the train.

          • Sassy

            If the platforms were known well in advance, you wouldn’t have to concentrate on not getting trampled. Maybe even get that tape measure out.

            Penn Station doesn’t eve have all that many people getting on and off each train. There’s at least one 1 platform 2 track station that has comparable ridership. The chaos at Penn Station is an indictment of the way things currently work.

          • adirondacker12800

            I don’t care what platform it’s on when it arrives.

          • Sassy

            You admit to having to concentrate on not getting trampled, and describe passengers as a stampede. Most people would consider those descriptions as a complaint. With a better layout, you would not have to concentrate on not getting trampled by a stampede of passengers.

            Whether or not you care about what platforms trains come in on, you do seem to care about the downstream effects of people knowing or not knowing the platform assignments well in advance.

          • adirondacker12800

            I don’t understand why I need to know where the train is going to be next week. At 5:07 I care where the 5:11 is. If at 5:10 I’m still on a subway train I won’t make it and don’t care where the 5:12 is because that doesn’t go to my station, the 5:14, 5:15, 5:17, 5:19, 5:20, 5:22 or the other 5:22, and I’m disappointed the 5:24 is a local not an express but it still gets me to where I’m going faster than the 5:44 express. YMMV depending which of the ten branches and subbranches hiding in them and which station you want. Stare at the schedules hard enough there can be peculiar things where the express that leaves later gets to the distant stations earlier than the local. If the local gets that far at all because this ain’t a blue line and a green line that cross over the orange line someplace.

          • Sassy

            If you, and everyone else, know where the trains are going to be well in advance, you wouldn’t have to concentrate on not getting trampled by a stampede of passengers.

            Also, it’s actually pretty convenient to just show up to the same platform every time and have everything just work. You should try it some day.

    • xh

      Penn Station partially remedies the problem casued by long station throats, by:

      1. Adding intermediate stop signals within stations throats, at the cost of complexity. This special design also makes it impossible to bring PTC protection within station limits. Without PTC, trains are only allowed move under 15 mph, and there’s no positive prevention of train collisions.

      2. Using pneumatic switch machines, rather than electric ones. Pneumatic switch machines are able to throw switches in less than a second. However, due to their speeds of action, they’re very unreliable, requiring intensive inspection or maintenance.

      • Max Wyss

        I don’t think that are good ideas.

        It would be better to slim up the station throat to allow 30 mph (or better). That would be sufficiently fast to prevent short blocks.

        Also, pneumatic switch machines are, as you state, unreliable. The two seconds an electric switch machine needs more are really neglectable. And there are drives which require way less maintenance, but are considerably reliable.

        • adirondacker12800

          They went and built skyscrapers on either side of Penn Station. What you see is what you are going to get unless you want to tear down skyscrapers and historic landmarks.

  2. Max Wyss

    Zürich is mentioned. I am looking at the Bahnhof Museumstrasse, underground, used by S-Bahn etc. only. It has the 4 track-2 island platforms design mentioned several times in the article. One side has a double track, which takes two lines, joining the previous stop (Hardbrücke). The other side goes into a tunnel to Stadelhofen (more about that later).

    Necessary dwell time is about 60 seconds, because it is the main passenger exchange place, and the low-entrance bi-levels are not the most efficient when it comes to passenger exchange. However, this station is also a “Zeitabgleichstation”, meaning that it sets depature times, but for arrivals, it has padding. This leads to stops up to 3 minutes during the day, and in many cases, a semi-legal connection from the following train is possible. Keep in mind that a train is missed, when the doors are closed and locked.

    On the other end of the tunnel is Stadelhofen, which has 3 tracks, but has the whole traffic of the line (for a track plan, look at OpenRailwayMaps). Track 1 is the home track, and it serves trains towards Hauptbahnhof. Tracks 2 and 3 have an island platform, and are fed from Hauptbahnhof. The tracks split towards Tiefenbrunnen (a single-track tunnel), and Stettbach (a double track tunnel). The separation is at grade. In order to improve reliability, there are a few services, which use track 2 towards Hauptbahnhof, allowing parallel entrance from the two tunnels. This is possible when there is no train towards Tiefenbrunnen. That it works (and it does so quite reliably), the system depends on being on time.

  3. Bob Campbell

    Alan, I want to make a few comments on train loading and some of the variables that sometimes can make a difference in possible headways. Philadelphia observations were mostly from the late 20th century. .Main stairs at just one far end of the station. So passengers crowd the front or back of the train to walk less distance. And the volume may yield a slower stop as opposed to when passengers can be spread out. . Passenger culture: In New York people rush to get on an overcrowded train even if there is very likely another train behind it (but unseen). In Philadelphia people tend more to assume that there really is a train coming along, it so the crowd stuffing on the first train is reduced and so is dwell time. Also tough is a concern. Standing passengers prefer not to touch each other vs NYC and Boston. . Complicated branching has random delays and late train may need to sit for a slot to open up if behind in schedule. . Very long train need to be cleaned, and there may not be a service crew when headways are long. Sometime the major stop downtown has a service crew (to reduce manpower). . Spacing of signals may affect headways. Gone are the days when SEPTA’s subway surface lines had human eyeballing when in the underground stations. (That is they acted lie streetcars).

    .Boston’s Red Line had doors opening on both sides of the train at Park Street (Under) to speed movement at the heaviest station. I think that maybe Chicago has a similar strategy at underground rapid stations. .Boston which had longer distances on its rapid transit stops New York and Philadelphia used to have many short bus routes to plug the gaps, which was expensive. Chicago began a program of thinning out the lower use rapid stops (mostly on the el) beginning after WW2. Skip stop is a strategy to increase speed but not overall ridership. This applies to rapid transit and commuter rail. Zoned stopping strategies are also used by commuter rail where the cluster of stops shares a similar fare. Bob Campbell (retired from the MTA).

  4. Max Wyss

    An interesting station to look at would be Bern Hauptbahnhof (also see OpenRailwayMap for the track plan). The station has 14 tracks, where one has no platform, and serves to connect the yards on either side of the station. Access from the East is a 4-track, all banalisé (meaning that each track can be operated in both directions), splitting a few kilometers to the East. Access from the West is two double tracks, however, one line can be reached from tracks 1 to 9, the other one from tracks 10 to 13 (further west, the lines run parallel, and there are interchanges.

    This looks like a very high number of tracks (theoretically, 8 would be enough). However, there is way more traffic towards the East than the West. At a main node time, 5 higher level trains (IC/IR/RE) reverse directions, and that fills up the tracks.

    Ah yeah, in the underground, there is the RBS station, 4 tracks stub end which has 20 tph reversing.

    • Sascha Claus

      Transfer nodes for integrated timetables seem to be manifestly not</em the subject of this article. 🙂 That would be: one track per line and direction, and another track if an additional, faster train in this direction is desired.

  5. adirondacker12800

    that New York has not fully electrified is unconscionable.

    Is it worth it for the occasional train that toddles into Port Jervis? Or Greenport? Or Hackettstown? Or the dozen or so summer weekends to Montauk when there is more than a car or two of demand?

    Metro-North are entirely high-platform, and unfortunate that New Jersey Transit is not.

    New Jersey is filled with people who think it’s still 1955. They then elect Republicans who also think it’s 1955. The few people who don’t think it’s 1955 go and get the station listed on the National Register of Historic Places which makes it very difficult to tear down the 125 year old station building so that level boarding could be put in. Even if they don’t have to tear down the 125 year old station building the people who think it’s 1955 object because it would ruin their fantasy that it’s 1955 and the people who want to put the station building on the National Register object because it ruins their fantasy that it’s 1895. Electrifying would ruin their view of the Christmas tree-like utility poles near the tracks. And most of the time it doesn’t matter because they elected a Republican governor who thinks it’s 1955, is apparently unaware of arithmetic and there is no money.

    …..Buses last 12 years on average. NJTransit didn’t do much of anything because the Republican governor was apparently unaware of arithmetic. He leaves office. When 18 year old buses start breaking down a lot it’s the Democratic governor’s fault. You are expecting people to actually consider Planet Earth and not live in their fantasies.

    • Matthew Hutton

      In Europe places which get more than half a dozen trains a day are getting electrified.

    • Transportation Justice CNY

      Port Jervis trains don’t share tracks with anything that runs into NYC so I guess I don’t care about them. (It sucks for people on the ex-Erie lines though.) Those other termini you named all run trains through Jamaica or Kearny, so their failures propagate onto electrified lines.

      • adirondacker12800

        They will in 2052 when the “Erie Loop” gets built because Gateway will be opening real soon.

    • Oreg

      This is the second sarcastic post that made me laugh out loud today. Thanks, adirondacker. 😀

  6. Phake Nick

    Remind me of Meitetsu Nagoya station, using 2 tracks and 3 platforms at city center to serve ~10 regional rail lines connecting into Nagoya from all around the city.

    • Borners

      Finally somebody other than me talks Meitetsu. 27 tph with a majority of services being various forms of express. Munich manages 30 but they’re all locals as far as I know. Meitetsu’s an interesting case convergent evolution with S-bahn practice, the tunnel actually predates Munich, use of the Spanish solution and a clockface timetable (off peak at least).

      And there’s basically zilch on it in English because we can’t have nice things. Though Meitetsu’s network as a whole is a bit of a warning since they got greedy in the 1960-1980’s, didn’t get Aichi to build them a through-running tunnel between Kamayama on the mainline to their two northern lines (Komaki and Seto). Then post-privatisation JR Central gutted their profits by building up their share of Nagoya station and running the paralleling Tokaido mainline competently. Very fiendish of them.

      Not mobilsing that legacy network properly is the no.1 reason why Nagoya is a meh transit city. Still Meitetsu’s problems are concrete not organisation, so they are a useful example for Western platform space whiners.

      • adirondacker12800

        It’s 150, 160 meters from track 1 somewhere in the general vicinity of 31st Street, at Penn Station New York and track 21 which is supposedly under 33rd. There are ten islands to serve them. North American trains are almost as wide as Shinkansen. How much concrete to people have to use and would it be wise to have 4,000 of them loitering around on it. And I don’t why it’s such great offense to wait in a concourse until five minutes or so before departure.

        • Sassy

          The North American trains that stop at Penn Station tend to be focused on providing lots of seats. Trains with more focus on standing capacity carry more people. And waiting on the platform for them works fine all around the world.

          Also, you can wait for the Shinkansen on the platform. The 4 track 2 island platform Shinkansen complex at Nagoya Station is about 36ish meters wide and serves up to 40 trains per hour over its 2 approach tracks. Dwell times are on the order of 1 minute. You could fit about 4 of those in that 150 meters, for a total of 16 tracks, 8 island platforms, and 8 approach tracks.

          • adirondacker12800

            You do understand that people are using Penn Station and closing it down for a few years, after they’ve torn down Madison Square Garden, is very unlikely to happen?

          • Sassy

            Considering relatively few passengers are served by each of those 10 island platforms at Penn Station, you wouldn’t even have to close it down. Platforms on average 80% the width should obviously be capable of handling well under 80% the passenger traffic, at least if improvements are made to how they are used.

          • adirondacker12800

            You have a source for an anti gravity machine that can hover Madison Square Garden for a few years while they do this?

          • Krist van Besien

            You do not need anti gravity. Just good engineers. The underground parts of Zurich HB have been built under an operating 19th century station…

          • adirondacker12800

            How far under the new tracks did the foundations for the old tracks go?
            The arena and the skyscraper office buildings have columns that go through platform level to the bedrock below. Most of the forest of columns visible at platform level are holding up the floor over them but some of the hold up arena, office buildings, two four track subway lines and the streets over the subway. There are unobstructed views across the arena, moving columns holding up the outside of the arena would be the trickiest part of shuffling stuff around. They have to be quite sturdy to span multiple tracks and platforms.

      • Andrew in Ezo

        FWIW back in 2019 Meitetsu announced plans to expand the station, doubling its size to a four track two island platform configuration (akin to say Tokyu’s Toyoko Line station at Shibuya), as part of a 400m long station building redevelopment project. Platform barrier doors as well as reduced track curvature were specified, allowing speeding up of the currently slow train approach speeds. However it seems the announcement and any official documentation has been scrubbed from the corporate website, presumably a victim of the economic uncertainty of the past two years. Anyway, the current layout and operations will continue for the foreseeable future for all to see and confound tourists.

        • Borners

          I am surprised it took them so long. But the mayor of Nagoya a populist austerion and for reasons I mentioned above Meitetsu had a very difficult post-1990 period. Its a multi-decade screwup, the combo of the lack of 1 seat rides on the mainline to the CBD at Sakai and the main station frequency bottleneck degraded the network which already dying at its extremes (Nishio and Hiromi lines). The real tell is that if you look at their property offerings, most of them are near subway stations.

          I’m still baffled they haven’t given in and begged to through-run with the Sakuradori line. I mean its not as though that line is at capacity in terms of frequency (less than 20 tph).

          • Andrew in Ezo

            It’s telling that most of the official documentation that MLIT puts out on run-through/interlining and other such improvements in urban networks is focused on the Kanto region- cartopias like the Tokai region with its positively N.Americanesque 3 space carports/single family household norm don’t seem to get much attention, perhaps understandably. And yeah, the Sakuradori Line seemed like a good candidate for interlining, though perhaps the hard turn left and dead end at Nakamurakuyakusho limits the appeal to a private railway like Meitetsu, as you ideally want an interconnection at both ends (such as with the Tsurumai Line in same city).

          • Borners

            Tokai is a cartopia compared to Tokyo maybe. But basic street layouts are pretty normal. Even the giant city centre roads aren’t too different compared to Osaka. Sprawl is typical Japanese walkable higgledy-piggledy, not American curve-mania. Where there is good rail service there is plenty of normal Japanese TOD. This meme about Nagoya being impossibly auto-centric is an excuse to cover consistent incompetence by rail and public transit operators. And if other opposition stronghold like Hokkaido or Osaka are able to get projects then Nagoya should be to.

            I think you exaggerate the problem of Nakamurakuyakusho. It still would be way better than the Namboku line in Tokyo. Though there is a case for extension to Nakamurakoen.

            And the limits of the attraction to Meitetsu of through-running are mostly having their terminal department store determine their rail choices. Which has worked terribly and is only going to get worse.

          • Phake Nick

            I think, one thing that separate Tokyo from Nagoya is that Tokyo is large enough that they can support building a large subway network to replace streetcars as main traffic mode inside the city. In Nagoya, it seems more traffic of what used to be served by streetcars are still served by bus nowadays. And buses obviously can’t win against cars. So you have higher degree of motorization.

          • Eric2

            The Nagoya metro area is 10 million people! Way bigger than cities like Madrid and Barcelona which have extensive subway networks.

          • Andrew in Ezo

            @Eric2
            The Nagoya Subway is a municipal system (6 lines/93km route length) so serves the the city proper (pop. 2.3 million). You have to include the other rail lines which extend throughout the metro area, such as the aforementioned Meitetsu (total 156km for the big 3 lines-Main Line, Tokoname L., and Inuyama L.), Kintetsu *24km Nagoya to Kuwana*), and JR Central (Tokaido Main Line *70km Okazaki to Gifu* and Chuo Line *24km Nagoya to Kozoji*) which like most Japanese urban railways have both suburban and metro characteristics/functions. In that sense, and even while ignoring the other numerous smaller lines, metropolitan Nagoya has a decent if not outstanding network.

          • Borners

            @PhakeNick Nagoya can’t afford the density of Tokyo orbitals. They also suffer from some of the weakenesses of the Japanese systems of poor intermodal fare intergration outside commuter passes since buses are so central. That said its not like they use their orbital railways as they should. The Aichi Loop line is distinctly meh because JR Tokai sucks at sharing and the Linimo is deeply terrible.

            On a global level is fine, but considering the Nagoya is a megacity with a density that’s high by Western standards. And I’m willing to bet part of the reason the service sector centre is weak is because they haven’t developed it properly.
            The problem isn’t so much track length, but line design, the subway lines should follow Tokyo best practice and connect up existing legacy track because the CBD is so far from the train hub at Nagoya station.

            The one proper through-running line the Tsurumai is rubbish in terms of ridership partly because they through-run once hour which considering the Toyota line has nowhere else to go is mindboggling. The Inuyama connection doesn’t work so well because the timetable on the Nagoya mainline are so sensitive. And it skirts round the CBD to boot. It should have been connected to Meitetsu northern orphan lines. And the Sakuradori should have been connected to what is noq the Aonami line. All water under the bridge now. They should still build an Aonami line/JR Kansai line-Meitestu Komaki line connector.

      • Frederick

        All the big private rails in Japan make use of the Spanish solution in their stub-end terminals. But Meitetsu Nagoya is more special because it is a through-station.

  7. Joseph E

    > Cars … follow the same rule, but their emergency braking rate is much faster, so on a freeway they can follow two seconds apart.

    Not quite. At motorway speeds of 100 km/h (60 mph) a car takes 73 m (240 feet) to come to a stop, under idea conditions (https://www.theaa.com/breakdown-cover/advice/stopping-distances).

    At 100 kmh (60 mph) a car travels 28 meters (90 feet) per second, so you would need a trailing distance of 3 seconds in perfect conditions, and usually 4 or 5 seconds considering realist levels of attention by the driver, pavement conditions and brake function, to prevent collision if the car in front immediated decelerated to 0 velocity.

    Car drivers on a crowded motorway are are twice as close together as they would be based on perfectly safe operation. This usually seems to work, because a vehicle on a motorway is unlikely to stop suddenly by hitting a large stationary object, but usually decelerates over a few seconds. But when a bad crash does happen it is common for many more vehicles to get involved due to insufficent time to react.

      • adirondacker12800

        If it’s real busy, which is when people are following at risky distances, if you swerve there is a different car there.

          • Frederick

            They do, but they shouldn’t. Lower speeds allow smoother traffic. Less accidents and phantom traffic jams.

          • Tiercelet

            This is America, Frederick. It’s our divine right to fling our two-ton death machines at the fastest rate possible into the tiniest of available spaces. And the “real” speed limit is always at least 15 miles faster than the posted one, it’s right there in the Constitution.

            Oh sure, you’ll tell me that different behavior would lead to more fuel efficiency, better safety, less traffic, and just notably better outcomes for everyone, but that sounds like socialism, especially if it means I can’t mash my gas pedal and feel powerful.

          • adirondacker12800

            14. In most states. the fines and other penalties get much higher once you pass that. People have noticed that people are very very good at it because there are an usually high number of speeding tickets written for 14.

    • Eric2

      This sounds familiar – we demand extreme safety precautions from nuclear power plants, but tolerate a high level of damage from non-nuclear power (fossil fuel emissions, solar panel chemicals, dam collapses, etc) making nuclear relatively expensive compared to those. Similarly we demand extreme safety precautions from trains, but tolerate massive numbers of road deaths, making trains relatively more expensive.

    • Alex Mazur

      I never quite understood why trains are like that. Both trains and cars are heavy; they cannot come to a complete stop immediately (even if there’s a malfunction, or they hit something else). Why exactly do we assume that they turn into stationary objects anytime?

      • xh

        When the concept of moving block signalling was conceived in ~1970s, the industry did have considered braking models in which the leading train may still move at maximum deceleration when emergency happens. This is known as the “soft wall” model, compared with the “hard (brick) wall” model mentioned by Mr. Levy.

        Theroetically, “soft wall” model does guarantee some additional capacity. However, after thorough investigation, the industry concluded that there’re simply no “maximum decelerations”. If the leading train hits something that’s really solid, its deceleration can be arbitrarily large, or at least order of magnitudes larger than the emergency brake rate of the following train.

        Further more, for transit-style operation, “soft wall” model won’t provide any capacity gain. “Soft wall” model allows higer capacity since it takes the brake distance of the leading train into consideration. As the leading train stops, its brake distance goes to zero. This is when “soft wall” model degenerates into “hard wall” model. Transit trains stop at stations with only two tracks. “Soft wall” model won’t reduce headways at these stations, a major bottleneck as’ve mentioned by Mr. Levy in this post.

        • William

          There is a hard wall at switches: you need to be able to prove that a train can stop before a switch before releasing it.

          There has been some work on virtual coupling (relative braking distance) on open track though, but it hasn’t been used in revenue yet.

        • Krist van Besien

          In the Brussels North South railway the trains drive on sight. So at Brussel Centraal you will witness how the next train will already appear at one end of the platform when the end of the previous one hasn’t even cleared the platform yet. That is how they manage to overcome the unfortunate decision in the 50ies to only provide 3 platforms/6 tracks at Centraal. Moving block but with human processors…

      • xh

        The railroad industry is well-known for its safety regulations and/or records, while the automotive industry has a bad reputation in terms of safety. For someone not familiar with this issue:

        In railroad world, especially railroad signalling and control world, there’re vital (safety-critical) functions and non-vital functions. Vital functions must comply to the well-known fail-safe principle, that is, should any malfunction occur, the system should fall back to states that are known to be safe. Actually, the fail-safe principle itself originated from the railroad industry.

        Take track circuits as an example. Track circuits are mechanisms that detect the presence of trains along the track, by connecting some batteries to the running rails. If a train is present, its wheels will short out the running rails, closing the track circuit. A relay in the circuit, known as the “track relay”, will, in turn, de-energize. Now considers the following possible senarios in which track circuits may fail:

        1. Track circuits running out of batteries.
        2. Broken electric connections.
        3. Broken rails.

        Should any of the above occur, the “track relay” will remain de-energized. This is a safe indication for the signalling system, which will safely handle the situation as if there’s a train along the track.

        • Stephen Bauman

          “the fail-safe principle itself originated from the railroad industry.”

          The disciplines of reliability, digital circuit design, etc. have progressed far beyond the railroads “fail-safe” principles and hardware like vital relays and track circuits, etc. that were used to implement them.

          The railroad industry and their regulators have ignored newer principles and hardware. They expect the world to conform to their principles because they still believe it’s the beginning of the 20th century and the Pennsylvania Railroad is the largest employer in the US.

          Electronic product prices have been in free fall for half a century with the introduction of integrated circuits. The hardware prices for the railroad industry’s niche market have been increasing exponentially during the same period.

          It turns out the railroad industry’s “fail-safe” principles are not that fail safe. A non-conductive layer that’s on train wheels or on tracks will defeat the track circuit’s “fail-safe” voltage-present means safe mantra. There’s a duality principle in logic circuit design that derives from DeMorgan’s Principle and Shannon’s extension to it. These show the “fail-safe’s” fallacy.

          There is also no way to show that the design, whether implemented by relays, vital relays, computer programs, programmable logic controllers, etc., will perform in a safe manner under all conditions. This is a consequence of of Goedel’s Theorem. One homework exercise from my digital circuit course from 50+ years ago, was to design a machine that would duplicate the behavior for any machine for an arbitrary but finite number of input sequences but give a contradictory output for the next one.

          The electronics industry has developed techniques to build reliable and inexpensive systems despite these limitations. It’s about time the railroad industry was brought into the technology of the second half of the 20th century.

          • Alon Levy

            The hardware prices for the railroad industry’s niche market have been increasing exponentially during the same period.

            What do you mean? Real rolling stock costs are not increasing over time, this isn’t construction.

          • xh

            “It turns out the railroad industry’s “fail-safe” principles are not that fail safe. A non-conductive layer that’s on train wheels or on tracks will defeat the track circuit’s “fail-safe” voltage-present means safe mantra.”

            Loss-of-shunt has long been a solved problem in the industry. The cure is to use the “sequential” logic. For example: if a track circuit block goes from “occupied” to “clear” while the following block goes from “clear” to “occupied”, then logically set the first to “clear”; however, if an “occupied” block clears to nowhere, then logically set it to “occupied”, which is an indication of “loss-of-shunt”.

            Such logic is known as “三点检查法” (checking a sequence of three blocks) in China Railways, which uses more track circuits than any other rail operators else in the world. Every single HSR in China detect trains by track circuits, and all are protected by this vital logic. No accidents have been caused by loss-of-shunt problems.

          • Stephen Bauman

            “What do you mean? Real rolling stock costs are not increasing over time, this isn’t construction.”

            I should have been specific. I was referring to signal systems. Real electronics costs are decreasing (in free fall) over time – especially if capabilities are included.

          • xh

            “The electronics industry has developed techniques to build reliable and inexpensive systems despite these limitations.”

            What matters on railroads is not just reliability, but distinct indicators known collectively as RAMS: Reliability, availability, maintainability and safety. Reliability itself does not imply safety. The industry’s requirements are roughly:

            Reliability: > 100,000 – 1,0000,000 hrs of MTBF (mean time between failure)
            Availability: >99.99% – 99.999%
            Maintainability: Less than 0.5 – 1 hr
            Safety: SIL 4, < 10^{-9} hazardous incidents per hour

            No "inexpensive" systems are able to deliver the required RAMS's. Vendors must make significant adaptions to their systems (i.e. to adopt a 2oo2 redundant structure) and go through strict certification processes, making the production costs of the hardware almost irrelevant.

          • Stephen Bauman

            “The cure is to use the “sequential” logic. For example: if a track circuit block goes from “occupied” to “clear” while the following block goes from “clear” to “occupied”, then logically set the first to “clear”; however, if an “occupied” block clears to nowhere, then logically set it to “occupied”, which is an indication of “loss-of-shunt”.

            Thank you for that explanation. This is an example of using logic to overcome a hardware fault.

            A more modern approach would be to use multiple transducers to detect the presence of a train within a block. It’s not that difficult to imagine transducers that can detect the presence of a multi-ton rail car directly above it. The positive outputs for all transducers (including the track circuit) would be logically or’ed to determine the presence of a train within the block.

            Here’s a fail-unsafe scenario, based on the sequential logic narrative.

            I will need a failure to clear event, as well as failure to make (occupy) events. Suppose a train is in a station. Somehow the train’s wheels get covered with the non-conducting coating. The train also opens the battery connection to the station’s track circuit.

            This means the train will remain in the station, so far as the signal system is concerned. The station block has no track circuit power – so it is occupied. The following N blocks don’t see a passing train because the train’s wheels are covered with the non-conducting coating. How many blocks back does the logic look to determine loss-of-shunt? I can always postulate one more block to make it fail-unsafe.

            You may argue that this scenario is pathological and out of spec for the sequential logic cure. You would be right. That’s the point of extending Goedel’s Theorem to logical circuits. There’s no way to determine whether a logic circuit is complete. It’s just a matter of time as to when one of these pathological events occurs.

          • xh

            “Here’s a fail-unsafe scenario … The station block has no track circuit power – so it is occupied.”

            It’s not a fail-unsafe scenario. The station block has no track circuit power, so the physical signal in front of it is red. All following trains have to stop before the block. Nothing is actually unsafe here.

            And even if it is, statistics from China revealed that the incidence of loss-of-shunt incidents is of the order of magnitude of 10^{-6} – 10^{-7} per block per hour, while the incidence of false-track-occupancy incidents is of the order of magnitude of 10^{-5} – 10^{-6} per block per hour. As the causes of these two type of incidents do not overlap, they’re probabilistically independent. Thus, the incidence of missed loss-of-shunt detection is of the order of magnitude of 10^{-11} – 10^{-13} per block per hour, well below the bar set by SIL 4 requirement.

          • Stephen Bauman

            “…Reliability: > 100,000 – 1,0000,000 hrs of MTBF (mean time between failure)
            Availability: >99.99% – 99.999%
            Maintainability: Less than 0.5 – 1 hr
            Safety: SIL 4, < 10^{-9} hazardous incidents per hour

            No "inexpensive" systems are able to deliver the required…"

            I assume the MTBF refer to individual components and not the entire system. There are 8760 hours in a year, so a MTBF of 100,000 hours is a MTBF of 11.4 years. That's a bit excessive for a land based system. It's not for a system that's located in space or under water. A system that fails due to a single fault, would need individual components with that MTBF.

            There is a way to use inexpensive components to achieve any required MTBF. Here's a link to an 1956 article by Moore and Shannon, entitled "Reliable Circuits Using Less Reliable Relays."

            https://cctbio.ece.umn.edu/wiki/images/3/30/Moore_Shannon_Reliable_Circuits_Using_Less_Reliable_Relays.pdf

            Their methodology is to achieve reliability through redundancy so that multiple failures would be required for a component failure. Let’s consider the failure rate and cost vital relays vs. signal relays.

            The relay MTBF parameter is expressed as number of operations not time as the base. Let’s assume a train passing at Penn Sta requires about 10 operations and there are 24 tph for 24/7. That amounts to 2 million operations per year. An 11.4 year MTBF (100K hrs) would require 23 million failure free operations for a relay.

            A $2.50 signal relay can be expected to operate in excess of 10 million operations between failures. This corresponds to a failure rate of 10^(-7). Placing 4 such relays with the coils in parallel and contacts in a series-parallel configuration would reduce the failure rate 10^(-14) per operation. Given that 2 million operations per year are expected, the MTBF for this $10 4-relay combination is [(10^14)/(2*10^6)] or 50 million years (4.4 x 10^11 hours). These failure rates are considerably better than those required of the vital relays at a small fraction of their cost.

          • Matthew Hutton

            Yes but that requires a public sector employee to care about value for money which doesn’t happen much.

          • Stephen Bauman

            ““Here’s a fail-unsafe scenario … The station block has no track circuit power – so it is occupied.”

            It’s not a fail-unsafe scenario. The station block has no track circuit power, so the physical signal in front of it is red. All following trains have to stop before the block. Nothing is actually unsafe here.”

            I should have been more expansive in my explanation of the unsafe condition. I was referring to trains after the station, not those entering it.

            Let’s assume block 0 is in the station and the train that has left the station with the coated wheels has traveled N blocks. Also, there’s a merging point at block n, 0 < n n. Let’s assume that the leader, with the non-conductive coated wheels, encounters a hard or soft wall on block N. The signal system will not prevent the follower that merged onto the track at block n for hitting the stopped leader at block N.

          • adirondacker12800

            The non conductive coating that magically appeared gets crushed off after it’s moved a bit?

          • Frederick

            Yes. All those non-conductive stuffs, be it oil or rust, will normally get removed by friction between tracks, wheels, and the braking mechanisms that clamp down the wheels.

            If there is significant risk of failure of track circuit, the solution will be to augment the track circuit with other means of of detection, e.g. treadles, axle counters, and CCTV.

          • Matthew Hutton

            > They do, all the time, what are you talking about?

            Admittedly this goes back a fair while – but I’ve heard lots of stories from retired salesmen that when selling to the public sector they stuck a zero on the end of the cost and no-one ever challenged it.

            More recently I’ve heard several tales from public sector employees about how much their mobile phone contracts cost – and it’s far more than a domestic one would. They also provided internal advice saying you should buy a mobile phone on contract rather than outright and getting a SIM only contract which is completely wrong from a value-for-money perspective.

            With mobile phones you don’t need to go to Japan to work out how much that should cost – you know from your own personal life so there’s no excuse.

          • Alon Levy

            The gap between “civil servants pay an extra 30 quid a month for a phone contract” and “civil servants don’t care about cost control” is vast.

          • adirondacker12800

            That’s strange. Large employers whether it’s public or private negotiate contracts with vendors. Whether it’s for rental cars or the fuel to put in them or cell phones or office supplies or… and that’s who the employees use. One of the perks of working for a large employer can be that the vendors extend the corporate rate to individual employees. And for things like computers/smart phones they are the corporation’s, you will run this software, you won’t run that software, they can access anything on it or wipe it clean.

          • Matthew Hutton

            I think the problem is that government enters that same big business process except they don’t bother to negotiate the price.

          • adirondacker12800

            Governments, down to tiny little villages where it may just be one of the things the village clerk does, has a purchasing department that shops for the best rate. Sometimes that isn’t the lowest price.

          • Matthew Hutton

            The best price for a mobile phone contract is to be on a SIM only MVNO on the best network with a contract with a suitable number of minutes, texts and data for the job being done. They weren’t doing that, which is why they were paying about twice as much as retail.

            And it’s not just about phones, it’s about buying custom software when commodity software would work fine with a process change, it’s about paying a good price for stationary, and shopping around for cheaper utilities etc etc etc.

          • Alon Levy

            Stationery costs are really not what you want to optimize for, is the point. For example, they’re commoditized in ways that purchasing a railway tunnel isn’t. They’re also such a peripheral cost you should not be worrying about them.

          • Matthew Hutton

            But if they are overspending on their mobile phone contract by £30/month/employee over a 200 person department that’s the total cost of employment for a full time employee. Certainly something worth spending a couple of days investigating and making the case for change and getting it approved.

            And if stuff like that doesn’t get dealt with quickly who is to believe staff costs and staff time use are being handled well? That’s where the real money is.

      • xh

        Another good example is the safe braking model underlying the “hard wall” rule, first saw usage in transit automatic train control systems (https://trid.trb.org/view/86883 for reference), then in any modern train control systems like CBTC, ETCS/CTCS or PTC. The model calculates the safe emergency braking distance from the following the worst of theworst-case senarios:

        1. An emergency occurs. Upon its detection, the on-board signalling subsystem sends the emergency brake command to the train’s control subsystem.
        2. The model assumes the train to accelerate at the maximum possible accleration (determined by the train’s maximum output and the grade of the track), until the train’s control system cuts out the traction.
        3. The train’s control subsystem also calls the braking subsystem to action. But until full emergency braking force is established, the model assumes the train to continue to roll along the track, and subject to further acceleration if it’s downgrade.
        4. Now the model assumes the train to brake at the minimum guaranteed emergency brake rate (GEBR) to stop. The GEBR is determined by both the train’s performance and wheel-rail adhesion.

        The automotive world don’t have something even comparable to this. Their “braking models” are pretty naive, by estimating the time to collision.

      • xh

        Since 1980s, computer-based systems have been extensively (and in some cases, exclusively) used on railroads to carry out designed vital functions, but only in a vital way.

        Hardware side, to prevent accidents caused by random hardware error, American vendors usually utilize “reaction” fail-safe, by additional “watchdog modules” independent of the main processors that check and validate the output of the processors. Once an error is detected, the “watchdog modules” will safely cut the power off, via some vital relays. European vendors, on the other hand, tend to use “combination” fail-safe. Their products usually have processors in a two-out-of-two (2oo2) or three-out-of-two (3oo2) redundant structure, which cross-validate each other. Thus, random error occured on a single processor no longer poses a safety threat.

        Software side, implementation of vital functions must be deterministic: No pointer dereference, no dynamic memory allocation, no overflow, no probabilistic modeling, no machine learning. Usually each function that maps the vital input states (e.g. position and integrity of trains) to vital output states (movement authority) is implemented via multiple programs, for cross-validation. To make the programs completely bug-free, formal methods are extensively used. The idea is to take programs as mathematical statements written in special languages, then use provers to formally prove them. Paris Metro Line 14, a fully automated metro line opened in 1998, made extensive use of formal methods, which turns out to be a great success.

        Today, these systems are subjected to strict regulations. Most European railway operators require SIL 4 certification, meaning that the incidence of hazard incidents in these systems must be lower than 10^{-9} per hour, or one per a billion hours.

        • Gok

          Should pointed out that the tech stack used for formal methods with the Line 14 software (B-Method) were also used for Ariane 5, where it didn’t work so well.

    • Henry Miller

      Don’t forget reaction time. Half of that two seconds is consumed by noticing a problem, figuring out what to do and then moving your foot to the brake.

      Brick walls don’t just appear. If you want to know brick wall time look at stop light yellow cycles which are more like 5 seconds.

      I suspect a derailed train is going to stop a lot faster than the following one so it makes sense to have a longer following distance. I don’t have numbers on this, but I assume someone does.

    • Matthew Hutton

      I am pretty sure those standard stopping distances are assuming 4 drum brakes and no ABS or stability control. So a modern car with 4 disk brakes, ABS, traction control and modern legal tyres will be able to stop quicker than that in dry or damp conditions (but not very wet or icy obviously)

      • adirondacker12800

        Apparently the almost self driving wundercars are deeply confused by snow. It can be great fun to watch the dog or the cat figure out their first snowfall. They figure it out. Teaching sand to do stuff animals do is going to be a lot more difficult than the computer boyz think it will be.

        • Henry Miller

          Humans are terrible drivers, even when we are not distracted. Driving a car better than a human is not a very high bar. Sure it is hard, but you can be better than humans while still killing a lot of people.

          I’m still waiting on independent verification, but Tesla keeps claiming that despite highly publicized incidences they are a lot safer than human drivers. (my guess is independent verification would probably take out some particularly bad humans and fine Tesla isn’t better than normal humans, but they are improving all the time and so in a few years computer drivers will be significantly better than the best drivers in all conditions even while killing people)

          Driving a train is a lot easier than a car. The train is on the tracks, or it derailed and blocked the tracks. Either way you know where it is and where it is going well in advance. (trains have other controls that make them hard)

          • adirondacker12800

            By the time we are allowed to drive we understand the concept of snow too. May not have ever seen it in person but understand the concept. And can extend that concept to dust storms or sand. Tumbleweeds.
            Recently one of the car magazines took one of the advanced test cars out for a drive. In Brooklyn. It was making incorrect decisions or giving up. Passed that video around to a few friends and we all asked “What does it do at the toll plaza to where it’s lidar or radar doesn’t work well? Or a zipper lane? It can’t do snow, it can’t do complicated intersections or toll plazas, it’s advanced cruise control for limited access highways.
            Teaching sand how to navigate complicated three dimensional space is a lot harder than the computer boiz think it is.

          • Tom M

            Tesla doesn’t use lidar or radar, only cameras which may be part of their recent problems.

          • adirondacker12800

            It wasn’t Tesla they were testing and I don’t remember or care what it was. Tesla agrees that having 100 cars at a busy intersection allllll broadcasting lidar or radar or sonar or some combination isn’t going to work out. I don’t know or care, what kind or whiz bangery they were using. Brooklyn got it confused. Teaching sand how to do things animals do is going to be a lot harder than the computer boyz think it will be.

          • Tom M

            Pretty sure this is the now infamous test you are referring too (https://www.youtube.com/watch?v=2PMu7MD9GvI). It was a Tesla and it wasn’t a car magazine (it was CNN). Tesla agrees with who? Not the majority of the other autonomous vehicle efforts who use a combination of sensors and other information such as high-definition maps.

          • Henry Miller

            Computers are allowed to get better, and a lot of smart people are working on the problem. Computers are not ready to drive everywhere yet, but the time is coming. Do not let the issues of the past make you unwilling to accept the day when computers are better drivers than humans. I don’t know when that day is, but it is coming.

            Haven’t grown up in snow, humans are not good at it either, we go in the ditch all the time, people slide enough each other all the time, and other such things. Snow is hard at best and sometimes impossible. In the north we get enough of it that we need to accept it and get by as best we can, but the south is right to cancel all non-emergency actives when there is even a small dust of snow.

          • adirondacker12800

            Physics is a cruel mistress and the computer makers are bumping against more than one physical limit. The software is going to be much more complicated than computer boiz think it will be.

  8. adirondacker12800

    roughly in proportion to either trip time or ridership.
    Pesky pesky passengers attempting to use a passenger railroad. Just think of how fast Flushing Line trains could be bouncing back and forth between Flushing and Hudson Yards if they didn’t have to stop for passengers! And how reliably! Or Lexington Ave. locals. There are anecdotes about how yokels from the hinterlands who have moved to Bed-Stuy don’t understand that you have to let people off before you can get on and want to board the center cars.

    because the train might spend five minutes there out of a one-hour trip.
    If the train is engaging in the magical through-running why is it spending five minutes anywhere except at the suburban terminals?

  9. Isean Bhalla

    One thing you note is the dwell times of both the RER and the Zurich S-Bahn, both of which use bilevels. How does this calculus change with single levels and more doors per side? Could you have something like 45 second or even 30 second dwell times at central stations, and would that even be preferable?

    • Alon Levy

      With single-level trains and more doors, you can push dwell times down, yes – this is why Tokyo doesn’t use bilevels on its most crowded commuter trains, except on the lower-occupancy Green Cars.

      But I’m still squeamish about 30 s dwells in the most intense cases. The RER B runs single-deckers with four door pairs per car and peak dwells at Les Halles and Gare du Nord are still in the 45-60 s range; the interior of the car isn’t great for egress but that shouldn’t make the difference between 60 and 30 s.

  10. Paul Dyson

    Alon, May I have your permission to reproduce this in the next Steel Wheels, which will be 3Q22, July-August timeframe? Thank you, PD

    Paul Dyson, Commissioner, BGPAARailPAC President Emeritus Editor, Steel Wheels Magazine 818 371 9516

  11. Mark N.

    Alon, especially since you praised Stuttgart’s S21 general concept, do you have an opinion about the capacity concerns raised by many, who think that the reduction of 16 terminating tracks to 8 through tracks (served by 4 tunnels) won’t accommodate the desired increase in trains per hour and their ability to maintain the targeted takt?

  12. Oreg

    I was wondering about that, too. The point I heard was that a Takt requires longer dwell times of a few minutes and, therefore, needs more tracks to allow changing between all trains.

    • Alon Levy

      It depends. The Swiss intercity Takt requires this, because it’s based on timed connections between multiple nodes every hour or at best 30 minutes. However, before the Swiss Takt, there were commuter rail Takts, starting with the Munich S-Bahn, using metro-style operations through city center. The purpose of the commuter rail Takt is not to facilitate timed transfers between trains in the core, but rather to,

      1. Permit outer branches to remain partly single-track or partly shared with intercity trains, with timed meets.
      2. Facilitate timed connections with buses at key suburban nodes, so that the train has a normal dwell time but the buses stay a few minutes for the transfer.
      3. Facilitate sporadic same-direction or diagonal (wrong-direction) timed transfers on branches.

      I know that #1 occurs in Munich extensively; #3 occurs in Berlin internally to the U-Bahn (Mehringdamm) and between the U-Bahn and the S-Bahn (Wuhletal) every 10 minutes.

      • Oreg

        OK, but Deutschlandtakt follows the Swiss Takt model, right? Isn’t that a problem for Stuttgart 21 with only 8 tracks?

        • Alon Levy

          Yes, exactly, and the people who want Germany to be Switzerland (and not, say, a country with 300 km/h trains connecting metro areas larger than Zurich, Basel, Bern, and Geneva combined) have complained that S21 doesn’t fit the D-Takt plans. But then S21 does fit the sort of service Germany should be running as it keeps building more high-speed bypasses like Frankfurt-Mannheim and Stuttgart-almost-to-Munich.

          • Oreg

            Thank you, Richard. That article indeed makes the case that S21 will struggle to support the Deutschlandtakt.

            Alon, would you care to write about your Japanes-model-for-Germany concept sometimes? I don’t think I’ve seen that before. Given the current shared infrastructure and rolling-stock orders it would be very far off, right?

          • Mark N.

            Thanks, Richard (and Alon!). There has been talk of building a connector between the Feuerbach and Cannstatt tunnels, which would at least allow a better distribution from those directions. According to the project site, switches and such were put in place when building the tunnels to accommodate this expansion.

            That provides a somewhat better outlook for connections in three of the cardinal directions, but one (south) looks like it will maintain its current status of FUBAR for many years to come.Its truly kind of mind-blowing how neglected the planning for the Zurich-Stuttgart line has been from the start. And now, only three years from the projected opening, it looks like they’ve come to the conclusion that an additional 12 km tunnel (!) will be the best solution, although it means capping the line in a suburb for ten years (!) and forcing passengers to switch to the S-Bahn to get to the shiny new station. The (continued) insistence of a stop at the Stuttgart airport (a minor hub) is hard to fathom, but as with much of S21, it seems that planning has been driven by commercial and real-estate interests more than anything.

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