# Suspended Railways

Suspended railways are not a common mode of transportation. In Europe, the best-known example is the Wuppertal Suspension Railway, opened in 1901. Two examples exist in Japan, which is more willing to experiment with nonstandard rail technology. With essentially just these three examples in normal urban rail usage, it is hard to make generalizations. But I believe that the technology is underrated, and more cities should be considering using it in lieu of more conventional elevated or underground trains.

The reason why suspended trains are better than conventional ones is simple: centrifugal force. Train cars are not perfectly rigid – they have a suspension system, which tolerates some angle between the bogies and the carbody. Under the influence of centrifugal force, the body leans a few degrees to the outside of each curve:

If the train is moving away from you, and is turning left, then the outside of the curve is to your right; this is where the body leans in the image on the right. This is because centrifugal force pushes everything to the right, including in particular the carbody. This increases the centrifugal force felt by the passengers – the opposite of what a tilt system does. A train is said to have soft suspension if this degree of lean is large, and rigid suspension if it is small. The depicted image is rotated 3 degrees, which turns 1 m/s^2 acceleration in the plane of the tracks into 1.5 m/s^2 felt by the passengers; this is the FRA’s current limit, and is close to the maximum value of emergency deceleration. There are no trains with perfectly rigid suspension, but the most recent Shinkansen trains have active suspension, which provides the equivalent of 1-2 degrees of tilt.

On a straddling train, this works in reverse. A straddling train moving away from you turning left will also suspend to the right:

It’s almost identical, except that now the floor of the train leans toward the inside of the curve, rather than to the outside. So the suspension system reduces the lateral acceleration felt by the passengers, rather than increasing it. By softening the suspension system, it’s possible to provide an arbitrarily large degree of tilt, limited only by the maximum track safety value of lateral acceleration, which is not the limiting factor in urban rail.

This is especially useful in urban rail. Longer-distance railroads can superelevate the tracks, especially high-speed tracks, where trains have to be reliable enough for other reasons that they never have to stop in the middle of a superelevated curve. Some urban rail lines have superelevation as well, but not all do. Urban rail lines with high crowding levels routinely stop the trains in the middle of the track to maintain sufficient spacing to the train ahead; this is familiar to my New York readers as “we are being delayed because of train traffic ahead of us,” but the same routinely happens in Paris on the RER. This makes high superelevation dicey: a stopped train leans to the inside of the curve, which is especially uncomfortable for passengers. High superelevation on urban rail is also limited by the twist, i.e. the rate at which the superelevation increases per linear meter (in contrast, on intercity rail, the limiting factor is jerk, expressed in superelevation per second).

Another reason why reducing curve radius is especially useful in urban rail is right-of-way constraints. It’s harder to build a curve of radius 200 meters in a dense city (permitting 60 km/h with light superelevation) than a curve of radius 3 km outside built-up areas (permitting 250 km/h with TGV superelevation and cant deficiency). Urban rail systems make compromises about right-of-way geometry, and even postwar systems have sharp curves by mainline rail standards; in 1969, the Journal of the London Underground Railway Society listed various European limits, including Stockholm at 200 meters. The oldest lines go well below that – Paris has a single 40-meter curve, and New York has several. Anything that permits urban rail to thread between buildings (if above ground), building foundations (if underground), and other lines without sacrificing speed is good; avoiding curves that impose 30 km/h speed limits is important for rapid transit in the long run.

Suspended railways are monorails, so they run elevated. This is not inherent to the technology. Monorails and other unconventional rail technologies can go underground. The reason they don’t is that a major selling point for monorails is that their sleek structures are less visually obtrusive when elevated. But underground they can still use the same technology – if anything, the difficulty of doing emergency evacuation on an elevated suspended monorail is mitigated on an underground line, where passengers can hop to the floor of the tunnel and walk.

I’d normally say something about construction costs. Unfortunately, the technology I am plugging has three lines in regular urban operation, opened in 1901, 1970, and 1988. The 1988 line, the Chiba Monorail, seems to have cost somewhat more per km than other contemporary elevated lines in Japan, but I don’t want to generalize from a single line. Underground there should not be a cost difference. And ultimately, cost may well be lower, since, at the same design speed, suspended monorails can round tighter curves than both conventional railroads and straddle monorails.

Despite its rarity, the technology holds promise in the most constrained urban environments. When they built their next new metro lines, disconnected from the older network, cities like New York, London, Paris, and Tokyo should consider using suspended railroads instead of conventional subways.

# Fare Integration

I said something on my Patreon page about fare integration between buses and trains, in the context of an article I wrote for the DC Policy Center about improving bus service, and got pushback of the most annoying kind, that is, the kind that requires me to revise my assumptions and think more carefully about the subject. The controversy is over whether fare integration is the correct policy. I still think it is, but there’s a serious drawback, which the positive features have to counterbalance.

First, some background: fare integration means that all modes of public transit charge the same fare within the same zone, or between the same pair of stations. Moreover, it means transfers are free, even between modes. Fare integration between city buses and urban rail seems nearly universal; big exceptions include Washington (the original case study) and London, and to a lesser extent Chicago. Fare integration between urban rail and regional rail is ubiquitous in Europe – London doesn’t quite have it, but it’s actually closer than fare integration between buses and the Underground – but does not exist in North America. In Singapore there is fare integration. In Tokyo, there are about twelve different rail operators, with discounted-but-not-free transfers between two (Tokyo Metro and Toei) and full-fare transfers between any other pair.

The reason North American commuter rail has no fare integration with other forms of transit is pure tradition: railroaders think of themselves as special, standing apart from mere urban transit. We can dispense with the idea that it is a seriously thought-out fare system. However, lack of integration between buses and trains in general does have some thought behind it. In London, the stated reason is that the Underground is at capacity, so its fares are jacked up to avoid overcrowding, while the buses remain cheap. In Washington, it’s that Metro is a better product than the buses, so it should cost more, in the same way first-class seats cost more than second-class seats on trains. Cap’n Transit made a similar point about this in the context of express buses.

There are really three different questions about fare integration: demand, supply, and network effects. The first one, as noted by Patreon supporters, favors disintegrated fares. The other two favor fare integration, for different reasons.

Demand just means charging more for a product that has higher demand. This is about revenue maximization, assuming fixed service provision: people will pay more for the higher speed of rapid transit, so it’s better to charge each mode of transportation the maximum it can bear before people stop taking trips altogether, or choose to drive instead. It’s related to yield management, which maximizes revenue by using a fare bucket system, using time of booking as a form of price discrimination; SNCF uses it on the TGV, and in its writeups for American high-speed rail from 2009, it said it boosted revenue by 4%. In either case, you extract from each passenger the maximum they can pay by making features like “don’t get stuck in traffic” cost extra.

Supply means giving riders incentives to ride the mode of transportation that’s cheaper to provide. In other words, here we don’t assume fixed (or relatively fixed) service provision, but variable service provision and relatively fixed ridership. Trains nearly universally have lower marginal operating costs than buses per passenger-km; in Washington the buses cost 40% more per vehicle-km, and perhaps 2.5 times as much per unit of capacity (Washington Metro cars are long). Using the fare system to incentivize passengers to take the train rather than the bus allows the transit agency to shift resources away from expensive buses, or perhaps to redeploy these resources to serve more areas. If anything, the bus should cost more. There are shades of this line in incentives some transit agencies give for passengers to switch from older fare media to smart cards: the smart card is more convenient and thus in higher demand, but it also involves lower transaction costs, and thus the agency incentivizes its use by charging less.

The network effect means avoiding segmenting the market in any way, to let passengers use all available options. The fastest way to get between two points may be a bus in some cases and a train in others, or a combined trip. This fastest way is often also the most direct, which both minimizes provision cost to the agency and maximizes passenger utility. This point argues in favor of free transfers especially, more so than fare integration. Tokyo fares are integrated in the sense that the different railroads charge approximately the same for the same distance; but transfers are not free, and monthly passes are station to station, with no flexibility for passengers who live between two parallel (usually competing) lines.

The dominant reason to offer integrated fares is network effects, more so than supply. Evidently, I am not aware of transit agencies that charge more for buses than for trains, only in the other direction. That fare integration allows transit agencies to reduce operating costs mitigates the loss of revenue coming from ending price discrimination; it is not the primary reason to integrate fares.

The issue at hand is partly frequency, and partly granularity. A typical transit corridor, supporting a reasonably frequent bus or a medium-size subway station, doesn’t really have the travel demand for multiple competing lines, even if it’s a parallel bus and a rail line. Fare disintegration ends up reducing the frequency on each option, sometimes beyond the point where it starts hurting ridership.

In Washington it’s especially bad, because of reverse-branching. The street network makes it hard for the same bus to serve multiple downtown destinations (or offer transfers to other buses for downtown service). Normally, riders would be able to just take a bus to the subway station and get to their destination, but Washington plans buses and trains separately, so two of the trunk routes, running on 14th an 16th Streets, reverse-branch. The hit to frequency (16-18 minutes per destination off-peak) is so great that even without fare integration it’s worthwhile to prune the branches. But such situations are not unique to Washington, and can occur anywhere.

The required ingredients are a city center that is large enough, or oriented around a long axis, with a street network that isn’t a strict grid and isn’t oriented around the axis of city center. New York is such a city: if it didn’t have fare integration, buses would need to reverse-branch from the north to serve the East Side and West Side, and from anywhere to serve Midtown and Lower Manhattan.

The granularity issue is that there isn’t actually a large menu of options for riders with different abilities to pay. This is especially a problem in American suburbs, with nothing between commuter rail (expensive, infrequent off- and reverse-peak, assumes car ownership) and the bus (in the suburbs, a last-ditch option for people below the poverty line). I wrote about this for Streetsblog in the context of Long Island; there’s also a supply angle – different classes of riders travel in opposite directions, so it’s more efficient to put them on one vehicle going back and forth – but this is fundamentally a problem of excessive market segmentation.

This also explains how Tokyo manages without fare integration between different rail operators. Its commuter rail lines are not the typical transit corridor. With more than a million riders per day (not weekday) on many lines, there is enough demand for very high frequency even with disintegrated fares. A passenger between two competing lines can only get a monthly pass on one, but it’s fine because the one line is frequent and the trains run on time.

The rest of the world is not Tokyo. Branches in Outer London and the Paris suburbs aren’t terribly frequent, and only hit one of the city centers, necessitating free transfers to distribute passengers throughout the city. They also need to collect all possible traffic, without breaking demand between different modes. If RER fares were higher than Metro fares, some areas would need to have a Metro line (or bus line) paralleling the RER, just to collect low-income riders, and the frequency on either line would be weaker.

The demand issue is still real. Fare integration is a service, and it costs money, in terms of lost revenue. But it’s a service with real value for passengers, independently of the fact that it also reduces operating costs. The 99.5% of the world that does not live in Tokyo needs this for flexible, frequent transit choices.

# When Buses are a Poor Guide to Corridor Demand, Redux

Generally, the best guide to where a city should build rail lines is where the busiest buses are. However, there are exceptions. I have written two posts about this giving examples of exceptions, and am going to give a third exception; I also intend to write a separate post soon giving a fourth exception.

The first post, from four years ago, deals with cases where the bus alignment has to stay on a major street, but some major destinations are just away from the street; a subway can deviate to serve those destinations. Examples include Old Jaffa in Tel Aviv near the north-south spine of bus lines 1 and 25, and Century City near the Wilshire corridor. Here, buses are a good guide to corridor demand, but the rail line should serve microdestinations just outside the corridor.

The second post, from last year, is more properly about corridors. It describes street networks that are hostile to surface transit, by featuring narrow, meandering streets. The main example is Boston, especially the Green Line Extension, in a rail right-of-way in a city infamous for its labyrinthine streets. Another example is the Evergreen extension in Vancouver, serving Coquitlam; the bus the extension replaced, the 97-B, meandered through Coquitlam since the streets were so poorly configured, while the extension uses a short tunnel and runs parallel to a railroad.

In this post I’d like to expand on a point I made, obliquely, in the Voice of San Diego. In San Diego, there’s an under-construction light rail extension, in a rail right-of-way, into an area with not-great bus ridership. Consult the following map:

Preexisting light rail (“Trolley”) is in black, the extension (of the Blue Line) in blue, the parallel north-south arterial in purple, and two buses in green and red. The bus ridership on Ingraham is very low: the bus route running on it, 9, has 1,500 riders per weekday (source). The top bus in San Diego, the 7 (going north of downtown, then east), has 11,000. So on the surface, this suggests there isn’t much demand for north-south transit in that area of the city, called Pacific Beach.

But that’s wrong, because in an auto-oriented city like any US city except New York, the major streets are determined by car access. The relentless grids of so many North American cities – Chicago, Los Angeles, Toronto, Vancouver – are not just where the buses go, but also where the cars go. Even in Manhattan, if you have the misfortune to find yourself going east-west in a car, you will probably use one of the major two-way streets, like 14th or 42nd, which are less clogged than the one-way streets in between. Non-gridded street networks for the most part obey this rule too – the commercial streets tend to be the wider ones used by car through-traffic.

Freeways throw a wrench into this system. They offer a convenient route for cars, but are abominable for commerce. Locations 5 minutes by car from the freeway are good; locations right along the freeway are not, unlike ones right along an arterial road. The main car route from Pacific Beach to the CBD is taking an east-west arterial to the I-5, not going south on Ingraham. This means that the demand for north-south traffic actually shows as strong commerce on east-west streets, hosting bus routes 27 and 30, and not on Ingraham. The 27 has weak ridership, and the 30 has strong ridership but not right along the I-5. But in a sense it doesn’t really matter, because, like the car- and bus-hostile narrow streets of old city centers, the freeway-centric road network in that part of San Diego suppresses bus ridership relative to future rail ridership.

In the presence of rail, the strong routes are the ones orthogonal to the rail line. Here, the 27 and 30 already preexist; there is a planned Trolley stop at the intersection with the 27, and presumably the 30 will be rerouted to serve that intersection rather than to duplicate the trains along the freeway. (I tried talking to the transit agency about this, but didn’t get any useful answers.) So the decent east-west bus ridership in Pacific Beach is actually an argument in favor of a north-south rail extension.

Like every exception to a general rule, this is not a common scenario. So where else are there cases where this special case holds? The necessary elements are,

1. The city must be auto-oriented enough that car access is crucial to nearly all commercial drags. In Paris, it doesn’t matter how you reach the Peripherique by car, because car ownership is so low.
2. The city should not have a strong mainline rail network, which leads to a hierarchical transit network (buses feeding train stations), in which both buses and cars use the same major streets to reach train stations. This means that Sydney and Melbourne are out, as are German cities short of Berlin and Munich’s transit mode shares.
3. The city must have a strong network of urban freeways, disrupting the street network to the point of siphoning traffic away from the surface streets that would otherwise be the main routes.

As it happens, all three elements are present in Tel Aviv. North-south travel within the region uses Ayalon Freeway, inconveniently east of the traditional city center; the city has been building a CBD closer to the freeway, but it’s still not quite there. This suggests that traffic is suppressed on the north-south arterials to the west – Ibn Gabirol (hosting the planned second line of the subway) and Dizengoff (possibly hosting the third) – is suppressed, and those streets require subways. This is in part why, before the Red Line began construction, I argued in favor of putting a north-south subway under Ibn Gabirol, and not under freeway-adjacent Namir Road, where the Red Line goes.

In the future, this pattern suggests that Tel Aviv should make sure to build north-south subways under Ibn Gabirol and Dizengoff, and extend them north. The significance of the northern direction is that the effect I’m describing in this post only works when car ownership is high; Israel is poor enough that car ownership is not universal, and in the poorer southern suburbs it is low enough that the buses do give a good guide to corridor demand, whereas in the northern suburbs everyone owns a car. There is likely to be suppressed transit demand in Herzliya, Ramat HaSharon, and northeastern Tel Aviv (including Ramat HaHayal, an edge city with many tech jobs). Thus ridership on a subway line going elevated over Sokolov in Ramat HaSharon and Herzliya, or on Raoul Wallenberg to Ramat HaHayal, is likely to be higher than present-day bus ridership suggests.

An American example is Washington’s suburbs. The Metro extensions are planned with little regard for bus ridership. While the Silver Line is bad for multiple reasons – high construction costs, service to too far exurbs, too much branching on an overloaded trunk – the extension to Tysons Corner is its one good aspect. There is no point in discussing bus ridership at an edge city like Tysons – conventional buses wouldn’t be following the same route that the cars follow, and freeway express buses almost universally have trivial ridership.

Finally, Vancouver. While Vancouver itself is gridded, its suburbs are much less so. In the suburbs served by the Trans-Canada Highway, especially Surrey, it’s likely that car traffic mostly follows roads feeding the highway. People drive to their jobs in Downtown, Central Broadway, Metrotown, or any of Surrey’s internal centers; there aren’t a lot of park-and-rides at SkyTrain stations, which instead emphasize transit-oriented development, and in Surrey there are actually more park-and-ride spaces at the freeways, with express bus access, than at the one SkyTrain stop with parking, Scott Road. This suggests that there is suppressed bus ridership in Surrey and Langley parallel to the Trans-Canada, along Fraser Highway. Extending SkyTrain in that direction is on a distant priority list for the region, and this theory suggests that it should be moved up, to be just behind the Broadway subway to UBC.

# Future Los Angeles Metro Investments

I just put up an article on Urbanize complaining about Los Angeles’s uniquely high operating costs on the subway and light rail. In the article, I offered a few explanations, but also said that none of them seems satisfying: high wages (wages are as high in Chicago), low frequency (frequency is as low in Atlanta), low train operator efficiency (the gap with London is too small), few lines with two different technologies (Atlanta has just two lines and Miami one).

Long-time readers may be used to my sneering at American transit operations for being primitive compared with European ones, but here, the best American system (Chicago) outperforms the four Western European systems for which I have data, and one more (Philadelphia) is within those four European systems’ range. Per car-km, Chicago spends $5 in operating costs, London/Paris/Berlin$6, Philadelphia and Madrid $7, New York$9-10, and Los Angeles $12. So Los Angeles is special. Lisa Schweitzer suggests my discounting the frequency and system size explanations is in error, and when I brought up Atlanta on social media, she noted that Atlanta’s labor costs are lower than Los Angeles’s. Assuming this is correct (Southern California uniquely combines high nominal wages with a tiny subway network), Los Angeles should expect subway operating costs to come down as it builds its urban rail network. Some lines, like the Regional Connector, the Wilshire subway, and the Crenshaw light rail line, are already under construction. But as the system grows (especially the subway system, which is technologically incompatible with the light rail lines, even the fully grade-separated Green Line), average operating costs will fall, which suggests that marginal operating costs are low. If Los Angeles has not figured this into its calculation, this means that the finances of future subway lines are better than projected. I drew this map of what rapid transit Los Angeles should build. The map isn’t new, but I want to use it to explain how I think cities should be building subways. 1. Every line is rapid transit, even lines built out of light rail lines today, like the Blue Line and the Expo Line. Unprotected grade crossings and street running, even in dedicated tracks, limit capacity and reliability elsewhere down the line, even though they do not reduce speed on other segments of the line. The Orange Line is replaced by a subway, not light rail. 2. Branching is rare. Only three subway lines branch. Two tunnel through Sepulveda Pass (where Let’s Go LA suggests four branches on each side of the tunnel), with each line branching into two in the north, in the Valley, where demand on each corridor is lower. The third is on Vermont, with a branch west to Torrance. 3. Many lines run elevated, in less dense areas with very wide streets. South Vermont is this south of Gage. This also includes the four north-south lines in the San Fernando Valley heading from the Sepulveda tunnel. 4. There are three distinct regional rail lines, all electrified, with two through-running; the branch to the airport is elevated. One branches, the others don’t. Local and express trains could happen, but the acceleration and reliability boosts from electrification are so great that speeds in the 70-80 km/h range are possible even with all the infill stops. The line to LAX could also host some intercity trains, provided it has four tracks. The dark blue line, labeled the I-5 line, should have four tracks at the very least on the shared segment, and likely longer, for planned high-speed rail; some of the work is already being done, but there is still going to be track sharing with freight trains. 5. The system is really a hybrid of a typical radial rail system and a grid, like the Mexico City Metro. There are fifteen lines, including commuter rail; eight, including the commuter lines, serve the CBD. Some (the Pink, Orange, and Atlantic Lines, and the southern half of the Green Line) are fully circumferential, the others (Harbor/Azure, Red, Crenshaw/Brown) serve secondary CBDs and try to avoid being too much like bad combinations of radial and circumferential transit. The reason for this structure is that Los Angeles has very strong secondary centers, including Century City, Burbank, El Segundo, Santa Monica, and Koreatown. 6. Much of the system assumes reasonable upzoning, for example the northern extension of the West Santa Ana/Lime Line to La Crescenta and Sun Valley. This includes replacing single-family zoning with multifamily zoning everywhere, and building up CBDs at major connection points such as Vermont/Wilshire and El Segundo. 7. There is a lot of service in LA County, but not much in the other counties except lines to the CBD. It’s possible to build up a fuller system in Orange County, extending the Purple Line east and also adding some grid routes, assuming extensive residential upzoning everywhere and commercial upzoning in Santa Ana, Anaheim, and the beach cities. 8. At LA construction costs (about$400-500 million per km underground), the entire map should be doable for maybe $90 billion; at reasonable costs, make it$40 billion. LA is spending comparable amounts of money on transportation out of the recent ballot measures, it just spends a lot of it on operational waste, on BRT (the current plans for Vermont are BRT, even though the corridor is busy enough to deserve a subway), or on roads.

# Anti-Infill on Surface Transit

I wrote about infill stops on commuter rail two weeks ago, and said I cannot think of any example of anti-infill on that mode. But looking at Muni Metro reminded me that there is need for anti-infill on surface transit. This is called stop consolidation normally, and I only use the term anti-infill to contrast with the strategy of adding more stops on commuter trains.

The root of the problem is that in North America, transit agencies have standardized on 200-250 meters as the typical spacing between bus stops. In Europe, Australasia, and East Asia, the standard is instead 400-500 meters. Even without off-board fare collection, the difference in speed is noticeable. In Vancouver, the difference between the local 4 and the express 84 is substantial: on the shared segment between Burrard and Tolmie, a distance of 4.8 km, the 84 makes 5 stops and takes 10 minutes, the 4 makes 18 stops and takes 16 minutes. A bus with the normal first-world stop spacing would make 10-12 stops and take, linearly, 12-13 minutes. 23 km/h versus 18 km/h.

With off-board fare collection, the impact of stop spacing on speed grows. The reason is that a bus’s stop penalty consists of the time taken to stop and open its doors, plus the time it takes each passenger to board. The former time is independent of the fare collection method but depends on stop spacing. The latter time is the exact opposite: if the stop spacing widens, then there are more passengers per bus stop, and unless the change in stop spacing triggers changes in ridership, overall passenger boarding and alighting time remains the same. Another way to think about it is that judging by Vancouver data, there appears to be a 30-second stop penalty, independent of ridership. Off-board fare collection increases bus speed, so the 30-second stop penalty becomes more important relative to overall travel time; the same is true of other treatments that increase bus speed, such as dedicated lanes and signal priority.

In New York, there aren’t a lot of places with local and limited-stop buses side by side in which the limited-stop bus has on-board fare collection. One such example is the M4, meandering from Washington Heights down the 5th/Madison one-way-pair, over 15.3 km. At rush hour, the local takes 1:45, the limited-stop takes 1:30: 9 vs. 10 km/h. But the limited-stop bus runs local for 6 km, and over the other 9.3 km it skips 26 local stops if I’ve counted right. The B41 has a limited-stop version over 8.3 km (the rest is local), skipping about 17 stops; the time difference is 10 minutes.

One possible explanation for why the stop penalty in New York seems a little higher than in Vancouver is that the M4 and B41 routes are busier than the 4/84 in Vancouver, so every stop has at least one passenger, whereas the 4 in Vancouver often skips a few stops if there are no passengers waiting. Conversely, the higher passenger traffic on buses in New York comes from higher density and more traffic in general, which slows down the buses independently of stopping distance.

On subways, there’s reason to have more densely-spaced stops in denser areas, chief of which is the CBD. On surface transit, it’s less relevant. The reason is that absolute density doesn’t matter for stop spacing, except when expected ridership at once station is so high it would stress the egress points. What really matters is relative density. Putting more stops in an area means slowing down everyone riding through it in order to offer shorter station access times to people within it. On surface transit, relative density gradients aren’t likely to lead to variations in stop spacing, for the following reasons:

1. Historically, surface transit stop spacing was always shorter than rapid transit stop spacing because of its lower top speed and the faster braking capabilities of horses vs. steam trains; often people could get off at any street corner they chose. So it induced linear development, of roughly constant density along the corridor, rather than clusters of high density near stations.
2. If there is considerable variation in density along a surface transit line, then either density is medium with a few pockets of high density, which would probably make the line a good candidate for a subway, or density is low with a few pockets of higher density, and the bus would probably skip a lot of the low-density stops anyway.

Most importantly, the 400-meter standard is almost Pareto-faster than the 200-meter standard. In the worst case, it adds about 4 minutes of combined walking time at both the start and the end of the trip, for an able-bodied, healthy person not carrying obscene amounts of luggage. The breakeven time on 4 minutes is 8 skipped stops, so 3.2 km compared with the 200-meter standard. Bus trips tend to be longer than this, except in a few edge cases. In New York the average unlinked bus trip is 3.4 km (compare boardings and passenger-km on the NTD), but many trips involve a transfer to another bus or the subway, probably half judging by fare revenue, and transfer stations would never be deleted. If the destination is a subway station, guaranteed to have a stop, then the breakeven distance is 1.6 km.

This also suggests that different routes may have different stop spacing. Very short routes should have shorter stop spacing, for example the 5 and 6 buses in Vancouver. Those routes compete with walking anyway. This may create a spurious relationship with density: the 5 and 6 buses serve the very dense West End, but the real reason to keep stop spacing on them short is that they are short routes, about 2 km each. Of course, West End density over a longer stretch would justify a subway, so in a way there’s a reason short optimal stop spacing correlates with high bus stop density.

The situation on subways is murkier. The stop penalty is slightly higher, maybe 45 seconds away from CBD stations with long dwell times. But the range of stop distances is such that more people lose out from having fewer stops. Paris has a Metro stop every 600 meters, give or take. Some of the busiest systems in countries that were never communist, such as Tokyo, Mexico City, and London, average 1.2 km; in former communist bloc countries, including Russia and China, the average is higher, 1.7 km in Moscow. The difference between 600 meters and 1.2 km is, in the worst case, another 1.2 km of walking, about 12 minutes; breakeven is 16 deleted stops, or 20 km, on the long side for subway commutes.

One mitigating factor is that subway-oriented development clusters more, so the worst case is less likely to be realized, especially since stops are usually closer together in the CBD. But on the other hand, at 1.2 km between stations it’s easy for transfers to be awkward or for lines to cross without a transfer. London and Tokyo both have many locations where this happens, if not so many as New York; Mexico City doesn’t (it’s the biggest subway network in which every pair of intersecting lines has a transfer), but it has a less dense network in its center. Paris only has three such intersections, two of them involving the express Metro Line 14. Even when transfers do exist, they may be awkward in ways they wouldn’t have been if stop spacing had been closer (then again, Paris is notorious for long transfers at Chatelet and Montparnasse).

In all discussions of subway stop spacing, New York is sui generis since the lines have four tracks. On paper its subway lines stop every 600-700 meters when not crossing water, but many trains run express and stop every 2 km or even more. Average speed is almost the same as in Tokyo and London, which have very little express service, and it used to be on a par until recent subway slowdowns. This distinction, between longer stop spacing and shorter stop spacing with express runs, also ports to buses. Buses outside the US and Canada stop every 400-500 meters and have no need for limited-stop runs – they really split the difference between local and limited buses in North America.

On a subway, the main advantage of the international system over the New York system is obvious: only two tracks are required rather than four, reducing construction costs. On a bus line, the advantages are really the same, provided the city gives the buses enough space. A physically separated bus lane cannot easily accommodate buses of different speeds. In New York, this is the excuse I’ve heard in comments for why the bus lanes are only painted, not physically separated as in Paris. Mixing buses of different speeds also makes it hard to give buses signal priority: it is easy for buses to conflict, since the same intersection might see two buses spaced a minute apart.

Buses also benefit from having a single speed class because of the importance of frequency. In Vancouver, the off-peak weekday frequency on 4th Avenue is an 84 rapid bus every 12 minutes, a 44 rapid bus every 20 minutes, and a local 4 every 15 minutes. The 84 keeps going on 4th Avenue whereas the 4 and 44 divert to Downtown, but the 4 and 44 could still be consolidated into a bus coming every 10 minutes. If there were enough savings to boost the 84 to 10 minutes the three routes could vaguely be scheduled to come every 5 minutes on the common section, but without dedicated lanes it’s probably impossible to run a scheduled service at that frequency (pure headway management and branching don’t mix).

The example of 4th Avenue gets back to my original impetus for this post, Muni Metro. Only diesel buses can really run in regular surface mode mixing different speed classes. Trolleys can’t. Vancouver runs trolleys on the local routes and diesels on the limited routes. At UBC, it has different bus loops for diesels and trolleys, so people leaving campus have to choose which type of bus to take – they can’t stand at one stop and take whatever comes first.

On rail, this is of course completely impossible. As a result, American subway-surface trolleys – the Boston Green Line, SEPTA’s Subway-Surface Lines, and Muni Metro – all run at glacial speed on the surface, even when they have dedicated lanes as in Boston. In Boston there has been some effort toward stop consolidation on the Green Line’s busiest branch, the B, serving Boston University. This is bundled with accessibility – it costs money to make a trolley stop wheelchair-accessible and it’s cheaper to have fewer stops. Muni Metro instead makes one stop every 3-5 accessible (on paper), but keeps stopping at all the other stops. It would be better to just prune the surface stops down to one every 400-500 meters, which should be accessible.

If you view rail as inherently better than bus, which I do, then it fits into the general framework: anti-infill on surface transit has the highest impact on the routes with the best service quality. Higher speed makes the speed gain of stop consolidation more important relative to travel time; trolleywire makes it impossible to compensate for the low speed of routes with 200-meter interstations by running limited-stop service. Even on local buses, there is never a reason for such short stop spacing, and it’s important for North American cities to adopt best industry practice on this issue. But it’s the most important on the highest-end routes, where the gains are especially large.

# Infrastructure for Mature Cities

A post by Aaron Renn just made me remember something I said in the Straphangers Campaign forum ten years ago. I complained that New York was building too little subway infrastructure – where were Second Avenue Subway, Utica, Nostrand, various outer extensions in Queens and the Bronx that we crayonistas liked? Shanghai, I told people in the forum, was building a lot of subway lines at once, so why couldn’t New York? The answer is not about construction costs. Ten years ago, China’s construction costs relative to local incomes were about the same as those of New York; even today, the difference is small. Rather, it is that China is a fast-growing economy that’s spending a lot of its resources on managing this growth, whereas the US is a mature economy without infrastructure problems as urgent as those of developing countries.

Aaron posits that American cities are too conservative, in the sense of being timid rather than in the sense of being on the political right. He gives examples of forward-looking infrastructure projects that New York engaged in from the early 19th century to the middle of the 20th century: the Manhattan grid, the Erie Canal, the Croton Aqueduct, the subway, the Robert Moses-era highways and parks. Today, nothing of the sort happens. Aaron of course recognizes that “New, rapidly growing cities need lots of new infrastructure and plans. Mature cities need less new infrastructure.” The difference is that for me, this is where this line of questioning ends. New York is a mature city, and doesn’t need grand plans; it needs to invest in infrastructure based on the assumption that it will never again grow quickly.

If not grand plans like building the Manhattan grid far beyond the city’s then-built up area, then what should a mature city do? Aaron talks about dreaming big, and there is something to that, but it would take a profoundly different approach from what New York did when its population grew by 50% every decade. I stress that, as with my last post critiquing another blog post, I agree with a substantial part of what Aaron says and imagine that Aaron will treat many of the solutions I posit here as positive examples of thinking big.

Rationalization of Government

Mature societies have accumulated a great deal of kludge at all levels, coming from social structures and government programs that served the needs of previous generations, often with political compromises that are hard to understand today. Welfare programs are usually a kludge of different social security programs (for the disabled, for retirees, for various classes of unemployed people, sometimes even for students), housing benefits, reduced tax rates for staple goods like food, child credit, and in the US food stamps. A good deal of the impetus for basic income is specifically about consolidating the kludge into a single cash benefit with a consistent effective marginal tax rate.

In transportation, bus networks have often evolved incrementally, with each change making sense in local context. When a new housing development opened, the nearest bus would be extended to serve it. In Israel, which grew late enough to grow around buses and not rail, this was also true of dedicated industrial zones. In cities that used to have streetcar networks, some buses just follow the old streetcar routes; the Washington bus system even today distinguishes between former streetcars (which have numbers) and routes that were never streetcars (which use letters). Jarrett Walker‘s bus network redesigns are partly about reorganizing such systems around modern needs, based on modern understanding of the principles behind transit ridership.

Governance often needs to be rationalized as well. In the early 20th century, it was important to connect outlying neighborhoods to city center, and connections between lines were less important. This led to excessively radial surface transit (rapid transit is always radial), but also to rail lines that don’t always connect to one another well. Sometimes due to historical contingency the lines are run by separate agencies and have uncoordinated schedules and different fare systems charging extra for transfers. Occasionally even the same agency charges for bus-rail transfers, often because of a history of separate private operators before the public takeover. In the US and Canada, the special status of commuter rail, with different unions, fares, schedules, and management is of particular concern, because several cities could use commuter rail to supplement the rest of the transit network.

In New York, this points toward the following agenda:

• Modernization of commuter rail, with full fare integration with the subway and buses, proof-of-payment fare collection to reduce operating costs, high off-peak frequency on the local lines, and through-running where there is infrastructure for it (i.e. Penn Station).
• Some bus service reorganization. New York already has extensive frequent buses, but some of its network is still questionable, for example some branches of the Third/Lexington and Madison/Fifth one-way pairs in Harlem.
• Subway reorganization. The subway branches too much, and at several places it could have higher capacity if it reduced the extent of reverse-branching; see discussion here and in comments here. Some elevated lines could also see their stops change to support better transfers, including the J/M/Z at Broadway and Manhattan to transfer to the G, and maybe even the 7 at 108th Street to enable a transfer to a straightened Q23 bus.
• Fare integration with PATH, and demolition of the false walls between the PATH and the F/M trains on Sixth Avenue, to enable cross-platform transfers.

Serve, Don’t Shape

There are two models for building new infrastructure: serve, and shape. Serve means focusing on present-day economic and demographic patterns. Shape means expecting the project to change these patterns, the “build it and they will come” approach. When New York built the 7 train to Flushing, Flushing already existed as a town center but much of the area between Long Island City and Flushing was open farmland. I’ve argued before that third-world cities should use the shape model. In contrast, mature cities, including the entire developed world except a few American Sunbelt cities and analogs in Canada and Australia, should use the serve model.

The serve model flies in the face of the belief that public transit can induce profound changes in urban layout. In reality, some local transit-oriented development is possible, but the main center of New York will remain Midtown; so far Hudson Yards seems like a flop. In the suburbs, more extensive redevelopment is possible, with apartment buildings and mixed uses near train stations. But these suburbs, built after WW2, are less mature than the city proper. In fast-growing cities in North America outside the traditional manufacturing belt the shape model still has validity – Vancouver, still a relatively new city region in the 1980s, got to shape itself using SkyTrain. But in New York, there is no chance.

This also has some ethnic implications. Jarrett likes to plan routes without much regard for social circumstances, except perhaps to give more bus service to a lower-income area with lower car ownership. But in reality, it is possible to see ethnic ties in origin-and-destination transit trips. This is why there are internal Chinatown buses connecting Chinatown, Flushing, and Sunset Park, and a bus connecting two different ultra-Orthodox neighborhoods in Brooklyn. In Washington, there is origin and destination data, and there are noticeable ties between black neighborhoods, such as Anacostia and Columbia Heights.

In a mature city with stable ethnic boundaries (Harlem has been black for ninety years), it is possible to plan infrastructure around ethnic travel patterns. This means that as New York disentangles subway lines to reduce branching, it should try choosing one-seat rides that facilitate known social ties, such as between Harlem and Bedford-Stuyvesant. While New York’s ethnic groups are generally integrated, this has special significance in areas with a mixture of linguistic or religious groups with very little intermarriage, such as Israel, which has two large unassimilated minorities (Arabs, and ultra-Orthodox Jews); Israeli transportation planning should whenever possible take into account special ultra-Orthodox travel needs (e.g. large families) and intra-ethnic connections such as between Bnei Brak and Jerusalem or between Jaffa and Nazareth.

Integrated Planning

A few years ago, I wrote a post I can no longer find talking about building the minimum rail infrastructure required for a given service plan. In comments, Keep Houston Houston replied that no, this makes it really difficult to add future capacity if demand grows. For example, a single-track line with meets optimized for half-hourly service requires total redesign if demand grows to justify 20-minute frequency. In a growing city, this means infrastructure should be planned for future-proofing, with double track everywhere, no reliance on timed overtakes, and so on. In a mature city, this isn’t a problem – growth is usually predictable.

It is relatively easy to integrate infrastructure planning and scheduling based on today’s travel patterns, and impossible to integrate them based on the future travel patterns of a fast-growing city such as Lagos or Nairobi. But in a slow-growing city like New York, future integration isn’t much harder than present-day integration. Alone among North American cities, New York has high transit mode share, making such integration even easier – transit usage could double with Herculean effort, but there is no chance that a real transit revival would quadruple it or more, unlike in cities that are relatively clean slates like Los Angeles.

Since the mature city does not need too much new infrastructure, it is useful to build infrastructure to primarily use existing infrastructure more efficiently. One example of this is S-Bahn tunnels connecting two stub-end lines; these are also useful in growing cities (Berlin built the Stadtbahn in the 1880s), but in mature cities their relative usefulness is higher, because they use preexisting infrastructure. This is not restricted to commuter rail: there is a perennial plan in New York to build a short tunnel between PATH at World Trade Center and the 6 train at City Hall and run through-service, using the fact that PATH’s loading gauge is similar to that of the numbered subway lines.

In New York, this suggests the following transit priorities:

• Open commuter rail lines and stations based on the quality of transfers to the subway and the key bus routes. For example, Penn Station Access for Metro-North should include a stop at Pelham Parkway for easy transfer to the Bx12 bus, and a stop at Astoria for easy transfer to the subway.
• Investigate whether a PATH-6 connection is feasible; it would require no new stations, but there would be construction difficulties since the existing World Trade Center PATH station platforms are in a loop.
• Change subway construction priorities to emphasize lines that reduce rather than add branching. In particular, Nostrand may be a higher priority than Utica, and both may be higher priorities than phases 3 and 4 of Second Avenue Subway. A subway line under Northern Boulevard in Queens may not be feasible without an entirely new Manhattan trunk line.
• Build commuter rail tunnels for through-running. The Gateway project should include a connection to Grand Central rather than Penn Station South, and should already bake in a choice of which commuter lines on each side match to which commuter lines on the other side. Plan for commuter rail lines through Lower Manhattan, connecting the LIRR in Brooklyn with New Jersey Transit’s Erie Lines, and, accordingly, do not connect any of the lines planned for this system to Penn Station (such as with the circuitous Secaucus Loop in the Gateway project).

Conclusion

New York still needs infrastructure investment, like every other city. Such investment requires thinking outside the box, and may look radical if it forces different agencies to cooperate or even amalgamate. But in reality the amount of construction required is not extensive. More deeply, New York will not look radically different in the future from how it looks today. Technological fantasies of driverless flying cars aside, New York’s future growth is necessarily slow and predictable, and cities in that situation need to invest in infrastructure accordingly.

In my post about third-world transit, I posited an epistemological principle that if the presence of a certain trait makes a certain solution more useful, then the absence of the trait should make the solution less useful. The shape vs. serve argument comes from this principle. The same is true of the emphasis on consolidating the kludge into a coherent whole and then building strategically to support this consolidation. A fast-growing city has no time to consolidate, and who’s to say that today’s consolidation won’t be a kludge in thirty years? A mature city has time, and has little to worry about rapid change obsoleting present-day methods.

But at the same time, the same epistemology means that these changes are less critical in a mature city. In the third world, everything is terrible; in the first world, most things are fine. New York’s transportation problems are painful for commuters, but ultimately, they will not paralyze the city. It will do well even if it doesn’t build a single kilometer of subway in the future. Nothing is indispensable; this means that, in the face of high costs, often the correct alternative may be No Build. This illustrates the importance of improving cost-effectiveness (equally important in the third world, but there the problem is the opposite – too many things are indispensable and there isn’t enough money for all of them).

I emphasize that this does not mean transportation is unimportant. That New York will not be destroyed if it stops building new infrastructure does not mean that new infrastructure is of no use for the city. The city needs to be able to facilitate future economic and demographic growth and solve lingering social problems, and better infrastructure, done right, can play a role in that. New York will most likely look similar in 2067 to how it looks in 2017, but it can still use better infrastructure to be a better and more developed city by then.

# Branching and Transfer Breaking

This is the winning option in a poll I conducted among my Patreon backers. Thanks to everyone who participated. Another option, about commuter rail infill stops, came a close second, and I will likely tackle it later this month.

The New York City Subway is unusually branched. I’ve written about the general concept here, and specifically criticized reverse-branching on the subway here. In this post, I want to talk about a more specific feature of complex branching arrangements: they have station locations that make it hard to disentangle the branches without breaking transfers.

The left image is a common way junctions are set up. In this image, it’s possible to travel from any leg to any leg; an example of this is BART, with its three-way junction in Oakland between the East Oakland Line carrying trains to Fremont and Pleasanton, the line to the north carrying trains to Berkeley, and the line to the west carrying trains to San Francisco. In many other cases, the branching is simpler, with a clear trunk and two branches, and it’s often not possible for trains to travel between the two branches without backing up; this is like the depicted image with one of the connections missing.

New York has one current example like the left image: the A/C/F/G junction in Downtown Brooklyn has a northern leg (A/C/F), an eastern leg (A/C/G), and a southern leg (F/G). All legs have four tracks and not every track pair connects to every other track pair, but each leg connects to both other legs. It has one former example: the junction between Sixth Avenue Line and 53rd Street Line, with the B/D going south-to-west (then north), the E going east-to-west (then south), and the F going south-to-east. The E/F shared tracks to the east, but neither service shared tracks with the B/D to the south or west.

The problem with this arrangement is that it makes the schedules more fragile. A delay on one branch can cascade. Toronto at one point ran its subway line this, with an eastern and western leg under Bloor Street (continuing to Danforth to the east), and a southern leg under University (looping back north under Yonge); it subsequently ended branching by extending the University leg to the north along the Spadina Expressway right-of-way and operating two independent lines.

The rub is that such an extension usually breaks transfers. Look at the right image: running the lines without branching means no transfers, since there is no station located at the crossing. Toronto dodged this problem because of how the original branching was laid out – in fact, there are two adjacent transfer stations. But usually, it is not hard to convert a branching like the left image into two lines with a simple transfer in the middle.

The 53rd/Sixth situation in New York is a good example of the problem. New York realized it needed more capacity going east, toward Queens, since there were only three track pairs – 53rd Street, plus two more disconnected from the system depicted. For this, it built a tunnel under 63rd Street, and connected it to Sixth Avenue, routing the F through it and creating a new service for Sixth-East 53rd trains, then called the V and now called the M. The junction now looks like an incomplete version of the right image, missing the two upper arcs. The F continues north under Sixth, and only diverts east under 63rd, and has no transfer with the E, which runs east-west under 53rd. The next transfer between the two services to the south is at West 4th Street; the next transfer to the east is at Roosevelt Avenue/74th Street, well into Queens, since the alignment of 63rd Street Tunnel into Queens prevents it from intersecting the E closer in, at Queens Plaza in Long Island City.

The highly-branched nature of the subway in New York makes sure that it is possible to travel between legs even when there’s no transfer, provided one is okay transferring between lines with not-great frequency. The first station south of the junction on Sixth, 47th-50th Streets-Rockefeller Center, lets passengers transfer wrong-way, between southbound and northbound trains. I have used this before to transfer from the B/D to the F on my way between Columbia and Queens, which are not well-connected to each other. Going from east to south is already easy on the M; going from east to north is possible via the M and F, but is unusual, since ultimately both legs lead into the same line in Queens.

However, it is hard to disentangle this to reduce branching. If one believes that reducing branching is useful for reliability and capacity, then one must believe it is necessary for New York to figure out how to split branching in the least painful ways. Partial data from the London Underground is suggestive (see international benchmarking, PDF-p. 15) – the non-branching Victoria and Piccadilly lines are more reliable than the complexly-branching Northern line. Moreover, the intensive service in Moscow, topping at 39 trains per hour without any automation, only works since none of the lines branches. This compels New York and other cities with highly branched systems to disentangle lines.

In the Bay Area, the situation is relatively easy, in the sense of requiring relatively little capital construction. There is no real need for a one-seat ride between East Oakland and Berkeley. The reason there are any trains running on that leg is that Downtown Oakland is on the leg to Berkeley and not on the leg to San Francisco. This was bad planning, and was noted as bad planning even in the 1960s.

What is required is a short bypass tunnel. There are two options. First, a tunnel from the east, replacing the Lake Merritt station with a station a few blocks to the north, effectively moving the junction one station north, so that 12th Street-Oakland City Center can be on the western leg toward San Francisco. Second, a tunnel from the west, between West Oakland and Downtown Oakland. This would not move any station, and put 12th Street on the eastern leg toward East Oakland; Downtown Oakland has a second station, at 19th Street, which would stay on the northern leg, for Berkeley-Downtown Oakland service. Either option would break East Oakland-Berkeley transfers, but make the remaining system more robust.

In New York, disentangling reverse-branches is considerably more difficult. On the numbered lines, it isn’t too difficult to shuffle the 2, 3, 4, and 5 so that the only track sharing is between the 2 and 3, and between the 4 and 5. On the lettered lines, first of all one key connection has to be severed: 11th Street Connection, letting the R go between 60th Street Tunnel toward Manhattan and the Queens Boulevard Line. All trains via 60th Street would go to Astoria; in comments, Alexander Rapp suggests flipping the connection at Queensboro Plaza, letting trains from 60th Street (such as the R) go to Flushing and the 7 go to Astoria, matching the busier line in Queens (Flushing) with the more popular route into Manhattan (60th Street). Queensboro Plaza and Queens Plaza have no transfer, and one would need to be constructed, but even with moving walkways, transferring would involve several minutes of walking between platforms.

Then, the Queens Boulevard Line would be left with local and express services, feeding 53rd and 63rd Street Tunnels. Trains on 63rd have to go to Sixth Avenue. This requires all 53rd Street trains to serve 8th Avenue – the east-west line shown in the images. No more M, just a more frequent E train, with implications for how the A/C run (probably both express between 145th Street and Chambers, where the E terminates). This breaks the transfer, and there is no possible way to create a new one. Transfers between the E and trains on 63rd would only be at Roosevelt and West 4th, and trips from East 53rd to Sixth would require a wrong-way transfer on the western leg using the B/D.

It’s possible to keep the limited reverse-branching and have Queens Boulevard trains of either type, local and express feed either 53rd or 63rd Street Tunnel. Local-local transfers would then be available immediately east of Queens Plaza. The problem is that this still introduces schedule dependence, on what is most likely the most crowded line in the city now that Second Avenue Subway has taken pressure off of 4/5/6 on Lexington. Conversely, without reverse-branching, both choices of how to match lines have drawbacks: sending locals to 63rd and expresses to 53rd means there is no connection between local stops in Queens and Long Island City, whereas doing the opposite makes the connections better but matches the busier Queens trunk (the express tracks) with the less desirable Manhattan connection (63rd).

That said, despite the drawbacks, something like this disentanglement is requires. New York needs more capacity, and shuffling trains like this effectively creates another half a tunnel’s worth of capacity between Queens and Manhattan and allows higher frequency on Second Avenue Subway, useful given the high population density in the part of the Upper East Side that it serves.

For other cities, let this be your lesson: do not build infrastructure that looks like the left image, unless you know how you can convert it to two intersecting lines with a transfer, the way Toronto did. Branching may look like a nifty way to provide one-seat rides between more pairs of origins and destinations, but it will reduce your capacity, and in the distant future force you into difficult choices in which anything you do, including the no action alternative, will screw someone over. What looked like good planning when the IND built subways under Sixth and 53rd in the 1930s turns out to be bad planning today with what we know of how subways operate around the world.

# The Lagos Metro

Lagos is the second-largest city in the world without rail rapid transit (the largest is Karachi). Sources disagree on its population, but it looks like 21 million in the built-up area, consisting of most of Lagos State plus a few suburbs just outside it, such as Ota and Ijoko – in total, 1,000 km^2, someone more including the suburbs. All of the problems that rapid transit intends to fix – traffic jams, pollution, long commutes, overcrowding, unpopular jitneys – are present. I don’t want to repeat the case I made in this post in favor of aggressive investment in subway systems in Lagos and other big third-world cities. Instead, I want to talk about concrete features.

Here is the map – it’s slightly different from the version I circulated in previous posts. Note the following features:

1. Four of the lines – East-West (blue), Main Line (red), Ikorodu (brown), North-South (purple) – are four-track. This is because Lagos is so big that its outer margins require very wide stop spacing, which requires four tracks. The first three also run in wide enough rights-of-way for significant stretches, making four-track elevated structures easier to construct. Express stops are denoted with squares, local ones with circles. If two four-track lines intersect, the stop is denoted with a square, and is express for both lines. All four-track lines have two-track tails – the farthest-out square is the last station with four tracks, where locals turn, while expresses keep going to the end. Only one four-track line has an end without a tail: the Main Line terminates all trains at Leventis.

2. On the four-track stretches, the average interstation is a little less than 4 km. Average speed can approach 60 km/h, given that the rights-of-way are often quite straight. On the local tails of these lines the average looks like 2 km, or around 50 km/h. On the two-track radial lines, the average interstation is a little less than 1.5 km, or around 40 km/h; the difference with the local tails of the four-track lines is that the local tails are in suburban areas.

3. Every pair of radials intersects, usually in Lagos Island but sometimes elsewhere. Four circumferentials – Apapa (dark red), University (ochre), Ishaga (dark green), Idimu (magenta) – intersect all the radials. Whenever two lines intersect, there’s a station. Doing this requires a lot of lines to run in the same alignment in the center. This is not track sharing, but multi-track tunnels. The only eight-track tunnel, where the Main and Ikorodu Lines run together between Eko Bridge and Leventis, has a wide road to go under; the only ten-track el, where the East-West, Ota, and Ikorodu Lines parallel around the National Theatre, has an entire expressway to go over.

4. The southern ends of the Ota (yellow) and Ikeja (silver) Lines are in open water today. This is because the area is slated for land reclamation and intense development, called Eko Atlantic. The western margin of that area, around the station I call Eko Atlantic, is already reclaimed, but still undeveloped. That entire area (Victoria Island) is the favored quarter of Lagos, and is commercializing; farther east, in Lekki, there are grand plans for suburban redevelopment, and an immense amount of casual marketing in the media (“buy now and the land will triple its price in five years”).

The system I’m proposing differs from the current proposal. The current proposal only has five radials (with apparently just one crossing from the Mainland to Lagos Island), corresponding to my East-West Line, Main Line, Ota Line, and a hybrid of the Ikorodu Line and the two branches of the North-South Line; there is one circumferential, vaguely corresponding to my Idimu Line.

Some of the features of the current proposal are a good start. It’s not possible to build eighteen lines at once, and some prioritization is required. It’s even fine to start with shared tracks into Lagos Island and subsequently give lines their own way into Lagos Island. (On my map, everything paralleling the three existing road bridges is a bridge, the rest are tunnels).

However, the existing proposal suffers from several shortcomings, which will need to be fixed later:

• The stop spacing is too wide even in the center, intermediate between my local and express tracks.
• There is far too much branching. In any reasonable sequencing of the lines, the East-West Line will fill before the later ones (i.e. the ones going north-northeast) open, requiring new routes into Lagos Island.
• There isn’t enough service in Victoria Island, which is developing as a CBD, as is common for rich neighborhoods.

But the worst problem is that the under-construction route goes elevated along the Ring Road, skirting Lagos Island, with just two stops, Ebute Ero and Marina. This segment has little value and should be demolished once direct underground routes with more stops open. According to a paper that’s no longer online, one of the reasons third-world subway lines tend to underperform ridership expectations is that many of them use available rights-of-way and skirt the CBD, instead of building short tunnels to get to the center. This paper in turns is one of the references used by Bent Flyvbjerg in his paper arguing that in general urban rail has cost overruns and ridership shortfalls.

The big obstacle for constructing any subway system is cost. This is especially true in Lagos, where the construction cost of the current project (the Blue Line, corresponding to the inner part of the western half of my East-West Line) looks like $180 million per km, entirely elevated. I say “looks like” because while the cost is consistently$2-2.4 billion in exchange rate terms (around \$5 billion in PPP terms), it’s not completely clear whether this is the Blue Line or also the Red Line.

My proposal is largely elevated, but the Lagos Island segments have to be tunneled. So do some other segments, for examples the inner parts of the North-South and Ikeja Lines, and most circumferentials. The system, totaling about 900 km, of which 200 is four-tracked, looks like it could be built only about one-quarter underground, or slightly more. Moreover, much of the underground construction could be cut-and-cover, including several segments in Lagos Island, and most of the circumferentials; only the underwater crossings and a few off-grid Lagos Island connections have to be bored.

The main analytic point here, carrying over to other cities, is the importance of prior planning. My map has 18 lines. This is useful because figuring out where stations should go, which stations on four-track lines get to be express stops, and how to sequence the lines should be based on long-term considerations. As I hinted in a post written right after I finished the first version of this map, about subway lines that intersect without a transfer, building the best lines 1-3 requires having a good idea where lines 4-12 will go. So even if Nigeria runs out of money after the first two lines are built, figuring out the built-out network is not useless – it informs the current network, and makes adding future lines less painful.

This is true for reducing construction costs, and not just for a good network. One of the positive features of the Paris Metro is that nearly all of it was planned as a whole, and as a result, difficult stations like Chatelet, Etoile, and Nation were built with the intention to have multiple Metro lines serve them. This meant that the stations could be built once, rather than multiple times, once for each line. Two lines, Metro 8 and 9, even run alongside each other for 1.7 km under the Grands Boulevards, which are wide enough for a four-track subway.

The planners of every system, regardless of whether it’s a metro for a city that doesn’t have any urban rail or an extension in a city with fifteen lines, should always think ahead. What are the future plans? What are the future needs? What is the expected future growth? In Lagos, the answer to the last question is “fast demographic and economic growth,” and this means the city should plan on a large system – hence my almost gridlike system of radials going north and northwest and circumferentials crossing the Mainland going southwest to northeast. But really, every city needs to ask itself how it wants its rail network to look, and plan the highest-priority segments accordingly.

# Line Spacing, and Intersections Without Transfers (Hoisted from Comments)

This post is inspired by two separate things. The first is my work on a fantasy subway map for Lagos; here is the current live version. There are twelve radial lines, all serving the western half of Lagos Island, converging on nine transfer stations. Under the principle that whenever two lines intersect there should be a transfer station, this greatly constrains the paths the lines can take. Result: the path between two CBD stations, Eko Bridge and Leventis, carries ten tracks underneath it. This is under a wide street, and it might be possible with a double-deck four-track-wide tunnel and two more tracks deep-boring around it, but it’s not easy to construct.

The second inspiration is a post by Brian Stokle about subway line spacing. Brian looks not at spacing between successive stops on one line, but at spacing between parallel lines, averaging a few North American examples. The average is a little higher than half a kilometer, narrower than the typical stop spacing. On Twitter, Joshua Mello notes Boston’s spacing was narrower; in comments, I add examples from New York and Paris, which are a bit narrower than Brian’s examples but wider than Boston (New York is one block in Midtown, so 280 meters, and Paris is 300 between Metro 3 and Metro 8 and 9).

These two examples together illustrate the tradeoff in subway construction. Most subways have a stop every 1-1.5 km; newer systems are at the high end of this range, mostly because of the demographic weight of China. It’s normal for stop spacing to tighten in the core, but not to a large extent. In Tokyo, the average stop spacing is 1.2 km, and in Central Tokyo it’s perhaps 800 meters. In London, the Tube lines seem to tighten from an average of 1.2-1.5 km to 600-800 meters in Central London.

At the same time, subway line spacing is necessarily short. The reason is that modern CBDs are geographically small. Midtown is maybe 4 km^2, from 30th to 60th Streets and from between 2nd and 3rd Avenues to between 8th and 9th. The Paris CBD, from just west of Les Halles to just east of Etoile, is also about 4 km^2 (see job density on PDF-p. 6 here). The Tokyo CBD, defined around Otemachi, Nihonbashi, Hibiya, Shimbashi, and increasingly Roppongi, is maybe 5-6 km^2, in a metro area of 38 million people.

Subway networks in such CBDs are necessarily crowded. The CBD is where people want to go. A subway line can get away with skirting it – Paris M4 does, and is in a near-tie with M1 for highest ridership per km. But avoiding it entirely is a ridership killer, except specifically for circumferential lines concentrating off-CBD travel: in Paris this suppresses M10 ridership, and in New York, it suppresses ridership on the J/Z (even though they serve Lower Manhattan) and the L (even though it serves Union Square). This means that the CBD of a large city will have many subway lines converging on a relatively small area. New York has its five north-south lines through Midtown.

Ensuring that every pair of intersecting subway lines has a transfer in this environment is difficult. Line spacing is usually narrower than station spacing, requiring kludges like the block-long walkways in New York, such as between Times Square at 42nd/7th and Port Authority at 42nd/8th. Paris managed to have an almost perfect network – before M14 was built, it only had one missed connection, between M9 and M12 (built by a competing private company) – but only by having very short station spacing, unusual even by the standards of the early 1900s, ruling out significant suburban extensions of the kind that are routine in London and Tokyo.

The situation in smaller cities is actually easier. The CBD is very small, often smaller than a square kilometer, but there are fewer lines, so it’s easier to make sure lines intersect properly. It’s also much easier to get line spacing right outside the CBD, where there’s less intense demand, allowing line spacing compatible with stop spacing on any intersecting or circumferential line.

The fundamental issue here is really about planning for the future. It’s not hard to gets lines 1, 2, and 3 to intersect nicely, or even lines 1-6. But beyond that, a city will often find itself in a situation where the best street alignment for line 7 happens to be right between two stations on line 1, spaced too far apart for a transfer. This is what happened to Tokyo. In New York, the three constituent systems (IRT, BMT, and IND) were each internally planned cohesively, so when two lines within the same system intersect, there’s a transfer, and, with difficulty, the IRT/BMT intersections have transfers as well. But the IND connects poorly to the other two systems, sometimes deliberately, and the IND’s layout made future extensions and service changes break transfers. My proposal to reduce reverse-branching in New York runs into the problem of breaking the transfers designed by the IND around a specific service plan.

When lines are designed together, it’s easier to avoid this problem. Paris M8 and M9 share a route through the center, as they were built simultaneously as the street is wide enough for four tracks. In contrast, building a line under or next to an existing line is much more difficult; New York did it anyway, under Sixth Avenue, but this led to cost overruns that doomed the IND’s early plans for further expansion. It is also difficult to build a new station under an existing transfer station, as it usually requires underpinning; in Paris, this problem means that transfer stations tend not to have closely-aligned platforms, requiring long walks between lines. When I’m proposing running multiple lines in the same tunnel in Lagos, this is from the point of view of assuming coordinated planning, with sequencing that allows entire streets to be dug up at once.

However, in reality, even coordinated design has its limitations. Subway networks take multiple decades to build, and in the interim, the city changes. Planners can attempt to use zoning to shape city development in a way that facilitates further expansion, but some tendencies are too uncontrollable. For instance, high-income neighborhoods tend to commercialize; I mentioned Roppongi as a growing part of the Tokyo CBD earlier, which is an example of this trend. The hottest new part of New York commercial development, the Meatpacking District, is really not a subway hub. This means that even if a city plans out lines 1-12 to share tunnels appropriately, it may not be able to control where there will arise the most demand for line 13. Coordinated long-term planning makes things easier, but it will not solve the basic problem of optimal subway spacing and CBD size.

# Modeling Jitney-Bus Competition

I just published a piece at the Bay City Beacon about Lyft Shuttle, the company’s foray into fixed-route buses. In the piece, I mentioned briefly that adding a competing operator can reduce the average quality of service. The piece, nominally 500 words, really 700, was far too short for me to expound on the model, even though people on social media had asked me about it before. So here it is.

The first assumption is that the routes are identical. This is largely true for some of the more northern Lyft Shuttle routes, duplicating Muni’s 2 and 41 buses (the 41 duplicator skips some inner stops, though).

The second assumption is that the bus has a fixed schedule, but the jitney can schedule itself in response to the bus. This is true of the case studies I know of (and, to make it very clear, “case study” means “I’ve read an article” or “I’ve heard it discussed on social media”): Israel’s Sherut vans, and Hudson County’s private jitneys. Both use dynamic schedules, which in plain English means drivers radioing each other (in Israel) or employing lookouts (in New Jersey); the bus companies have fixed schedules that are slower to change.

The third assumption is that marginal riders take whichever route they see first. This is the case with regular jitneys. Not all riders are marginal: there might be an explicit ethnic dimension, e.g. dollar vans in New York are more popular with unassimilated immigrants, because they provide amenities like use of their own language rather than English. Lyft Shuttle is booked via app, so the situation there is murkier; however, most likely the use case will involve passengers buying a ticket a few minutes in advance, possibly checking against a bus app to see which will come first. Some people will stick with one system over the other (tech boosters seem to hate public services, people without other reasons to download the Lyft app will probably stick to the bus), but some will be flexible.

Let us work this out first assuming perfect scheduling and then introducing schedule irregularities due to bunching.

Now, let’s assume the public bus arrives every 6 minutes. If the transit agency wishes to double service, it will schedule additional service at the midpoint between each pair of successive trips, providing 3-minute frequency. But if additional service comes from a private operator, the incentive is to schedule to compete and not to cooperate. Say the public bus arrives at my station at :00 on a 6-minute takt. The private operator can schedule itself to arrive at :03 on a 6-minute takt and get half of my station’s traffic, or it can schedule itself at :05 and get 5/6 of its traffic. Let’s set the fudge factor at 1 minute for now: if the separation is smaller, for example if the jitney comes at :05:30 and the public bus at :06, then I see both at the same time and am indifferent as to which to ride.

By itself, this still means adding service. However, it’s likely that there is going to be some service diversion – that some public bus riders will switch to the private service (e.g. because it usually comes first). If one quarter of the bus riders switch, then the schedule is cut to a bus every 8 minutes. The jitney still aims to arrive 1 minute before the bus for maximum revenue, perhaps getting bigger vehicles if it needs the capacity, as the New Jersey jitneys did. So now my bus comes at :00, :08, etc., and the jitney comes at :07, :15, etc. Average wait time is 3.125 minutes, whereas before it was 3 minutes.

The formula in general if both vehicles come every x minutes and are separated by 1 minute is $\frac{(x-1)^2 + 1}{2x}$ and this equals 3 at $x = 4 \pm \sqrt{14} \approx 7.74$, corresponding to about 23.5% reduction in transit use; any higher reduction makes average wait times worse. If the original headway was y, then we have $x = 1 + y/2 \pm \sqrt{(1 + y/2)^2 - 2}$, or just under 2 extra minutes; the fixed separation, 1 minute, means this calculation is not scale-invariant.

Now, let’s introduce schedule irregularity into this system. As a toy model, let’s look at what happens if the bus can be up to 1 minute behind or ahead. If the route only has public buses, and one bus is off by 1 minute, then instead of two 6-minute gaps there’s a 5-minute gap and a 7-minute gap, for an average wait of 3:05. If the route has public buses and jitneys, scheduled 1 minute apart as above, then a 1-minute error is good for passengers if it evens out the schedules (it converts a 3.125-minute wait to a 2.5-minute wait), but bad for passengers if it makes the bus and the jitney arrive at exactly the same time (the wait rises to 4 minutes).

But really, bus schedules are an unstable equilibrium. (So are train schedules, but the instability is too slow to cause bunching). If a bus is a minute behind, then at the next bus stop it will have an above-average crowd size, since people had more time to show up and wait. Boarding time is a significant fraction of bus travel time, so the bus will go behind even further, until the bus behind it will catch up, at which point the two buses will leapfrog each other. It’s possible to reduce this effect by cutting boarding time, via low-floor buses and off-board fare collection with all-door boarding; Muni has implemented both, but this is still not enough to remove the instability. In practice, this imposes a minimum headway of about 3 minutes – below it, buses bunch so much that adding service doesn’t add any capacity. In theory it’s possible to go lower, but when ridership is high enough to justify 3-minute headways, it’s high enough for dwell times to make lower headways infeasible.

Jitneys do not have a 3-minute minimum headway; they’re more flexible about running express if they’re already full. This, in turn, means that schedules on jitney routes are more irregular than on buses, making wait times less predictable, and ultimately longer (since, at equal service, less even intervals translate to longer average waits). But more to the point, jitneys still aim to schedule themselves to come just before the bus does. This means that jitney arrival irregularity largely tracks bus arrival irregularity. So with this in mind, if the city bus gets delayed by a minute again, and the jitney adjusts, then we have the following service gaps, in minutes: 8, 1, 6, 1. Average wait time is now 3.1875 minutes.

So the question is, what effect does demand diversion from buses to jitneys have on bus irregularity? The answer is, not much. This depends on additional assumptions on dwell times and the initial delay. Delays compound exponentially, but the exponent (“Lyapunov exponent” in dynamics) is low; on a 6-minute bus route, a 1-minute delay means that future dwell times go up by 1/6, and at 2.5 seconds of variable dwell time per boarding or alighting (Muni average with all-door boarding, see PDF-p. 16 here) it takes 60*6/2.5 = 144 boardings and alightings to delay the schedule by another minute. Muni averages 127 boardings and alightings per hour, so it takes more than an hour for the 1 minute of delay to compound by another minute (strictly speaking, by $\sqrt[6]{e} \approx 1.18$ minutes). The busiest bus in North America, Vancouver’s 99, averages 320, so the compounding takes just 27 minutes (almost a full one-way trip); Vancouver buses with comparable ridership to the routes Lyft is duplicating average maybe half that.

The point of this exercise is that on the timescales relevant to a bus route, schedule irregularity is approximately linear. So demand diversion from buses to jitneys has a linear effect on schedule irregularity: 25% diversion means that instead of a 1-minute delay there’s only a 45-second delay. A 1-minute delay with 6-minute buses is still better than no delay with 8-minute buses an jitney spaced a minute apart, so the jitneys still remove value. At higher delays, this is no longer true (average wait time is quadratic in the delay); the point of equality is about 1.61 minutes of bus delay due to schedule instability (other delays, like traffic, affect all routes equally regardless of frequency). So jitneys can add value on long, busy routes (don’t forget – the lower the headway, the more significant the assumed 1-minute separation is). But the routes the Lyft Shuttle jitney runs are less busy, and more importantly are short, with less opportunity for delay compounding; delays do not compound past the terminus with decent dispatching.

The upshot is that jitneys do not universally remove value from transportation networks. But on short 6-minute routes, they do even under mild assumptions on ridership diversion. There’s more service in operation, but from the riders’ perspective, wait times have increased and regularity has degraded. The logic of competing private companies is not always the logic of better service for passengers. Sometimes, having a government monopoly inherently improves efficiency.