High-Speed Rail in India

India’s economic development lags China’s by about 15 years, so it shouldn’t be surprising that it’s beginning to construct a high-speed rail network. The first line, connecting Mumbai and Ahmedabad via Surat, began construction at the end of last year, with completion targeted within four years; the two states served, Maharashtra and Gujarat, are more or less India’s two richest large states, and are also both deeply right-wing, with nearly every constituency backing Modi. There are some severe problems with the system, stemming from its use of turnkey Japanese technology. But more broadly, India’s geography is just difficult for high-speed rail, especially by comparison with other high-population density countries at similar level of development, like Pakistan and Indonesia.

Japanese technology

The Mumbai-Ahmedabad corridor is to use imported Shinkansen technology, with Japanese financing. India has a vast railway network using broad gauge, with extensive regional rail (the Mumbai Suburban Railway has 2.6 billion riders per year) as well as legacy intercity rail.

However, to maintain Shinkansen compatibility, India has chosen to use standard gauge. This is based on a misunderstanding of why HSR uses standard gauge. Spain uses near-Indian gauge on its legacy network but standard gauge on HSR to maintain compatibility with the French TGV network, and Japan has narrow gauge on the legacy network and standard gauge on Shinkansen because narrow-gauge trains can’t run as fast. Neither of these justifications applies to India, and evidently, in another country where they don’t apply, Russia, HSR is to use broad gauge. With standard gauge, India will not be able to run HSR through to the legacy network, connecting to cities beyond the initial line, such as Delhi, nor will it be able to stage future construction to build lines in phases, the way France did, with through-service to lower-speed territory.

Even worse, alone in the world, India is using the Shinkansen’s loading gauge on HSR: trains are 3.35 meters wide, enough for 5-abreast seating. Indian Railways has a loading gauge allowing 3.66 meter trains, enough for 6-abreast seating with the same compromises on comfort familiar to every airline economy passenger. I don’t know what the standards for track centers are to be on India’s HSR: Indian Railways’ manual says 5.3 meters, which is wide enough for everything, but Shinkansen standards specify 4.3 meters, which is tight enough that a future widening of the track and loading gauges may pose difficulties for passing at high speed (at low speed it’s easy, India’s legacy track centers are 4.265 meters, and standard-gauge America’s are 3.7 meters on the slower parts of the Northeast Corridor).

During construction, the decision to use the wrong-size trains is fixable. Even after service opens, if the track centers are not too narrow, it’s possible to add a third rail to permit a transition to broad gauge. If the track centers are as narrow as the Shinkansen then might still be possible, if the third rails are on the outside (it would widen the track centers by the difference between the gauges, or 23.3 cm), but then the platforms would need to be shaved for wider trains. In the medium and long runs, such gauge widening is critical as India builds out its network.

But today, so complete is India’s reliance on Japanese technology that the training for drivers will be conducted in Japan, in Japanese; train drivers will be required to speak Japanese, as the Shinkansen trainers will not all speak English. It goes without saying that without a large body of Japanese speakers, India will be forced to pay first-world or near-first-world wages, forgoing its advantage in having low labor costs.

Construction costs

The projected construction cost of the 508-kilometer line is 1.1 trillion (“lakh crore”) rupees, which is $15 billion in exchange rate terms and about $55 billion in PPP terms. Per Wikipedia, the route includes only one tunnel, a 21-km approach to Mumbai with suburban and underwater tunneling (even if the gauges were compatible, using existing tracks like TGVs is impossible due to the use of every approach track by overcrowded Suburban Railway trains). The rest of the route is predominantly elevated, but the decision to runs the trains elevated rather than at-grade is only responsible for about 10% of its cost.

Despite the complexity of such a tunnel, there is no excuse for the high construction cost. In exchange rate terms it’s reasonable. Japan’s domestic Shin-Aomori extension of the Shinkansen cost about $55 million per kilometer, including a 26 km tunnel consisting of a third of the route and additional tunnels totaling a majority of the route. More recently, Japan’s new bout of Shinkansen construction costs about $30 billion for 389 km, but tunneling is extensive, with the Hokkaido route planned to be 76% in tunnel.

With India’s complete reliance on Japanese technology, paying the same as Japan in exchange rate terms is not surprising. It’s a disaster for India, which has to pay in depreciated rupees instead of leveraging its low-cost labor, but as far as Japan is concerned, it’s a perfect copy of the domestic Shinkansen system. Similar high costs can be observed for some Asian metro projects using Japanese financing, namely Dhaka (the world’s highest-cost elevated metro, even worse than in the US) and Jakarta.

In contrast, where India improves its rail network by tapping into Indian Railways’ own expertise, costs are low. Nearly half of India’s rail network is electrified, and to save money on expensive fuel, the country is rapidly electrifying the system, targeting 100% electrification. A plan to electrify 13,675 route-km in the next four years is to cost 12,134 crore rupees, about $123,000/km in exchange rate terms or $450,000/km in PPP terms. In the developed world, $1-1.5 million/km for electrification is reasonable, and the unreasonably expensive UK, US, and Canada go up to $5-10 million/km. Left to its own devices, Indian Railways can build things cheaply.

Network structure

India’s geography for high-speed rail is not easy. Mumbai, Surat, and Ahmedabad are the only three cities in the top 20 that lie on a straight line at easy HSR range. Delhi-Mumbai, Delhi-Kolkata, and Mumbai-Chennai are all just outside the best range for HSR (and Kolkata-Chennai is well outside it), having to rely on intermediate cities like Ahmedabad, Hyderabad, and Kanpur for ridership. Within Uttar Pradesh, Kanpur and Lucknow are both large cities, but the line connecting them is almost perpendicular to that connecting Delhi and Kolkata, so that only one can be served on the main line. In the South, there is a similar situation with Mumbai-Chennai, via either Bangalore or Hyderabad (and there, both routes should be built as Bangalore and Hyderabad are both near-megacities). Mumbai itself requires extensive tunneling in all directions: north toward Gujarat and Delhi, south toward Pune, and possibly also northeast toward the interior cities of Maharashtra.

I drew a possible map for a nationwide network. The total length is 17,700 double-track-km. It’s about the same length as most American proposals, and less than half as much as what China aims to build by 2025, but India has four times the population of the US and far higher population density, and its density is also several times that of China. For a better comparison, consider Pakistan: it is slightly less dense than India and has about 15% India’s population, and yet two spines totaling about 1,800 km, Karachi-Lahore and Lahore-Islamabad-Peshawar, would connect nearly every major city. Lying on the Indus, much of Pakistan has a linear population distribution, facilitating rail connections.

With a difficult urban geography for HSR, India has to take especial care to reduce construction costs. This means, in turn, that it needs to rely on indigenous expertise and standards whenever possible. When imported technology is unavoidable, it needs to provide its own financing (with an annual budget of 29 trillion rupees, it can afford to do so) and force Japanese, Korean, and European vendors to compete. A Chinese-style tech transfer (read: theft) is not possible – the vendors got burned once and won’t agree to the same again – but domestic driver training, with the foreign role restricted to the rolling stock (built to Indian standards) and engineering, is essential and unlikely to bother the global industry.

Is Missing Middle Really Missing?

There’s en emerging concept within North American urbanism and planning called missing middle. This refers to housing density that’s higher than suburban single-family housing but lower than urban mid- and high-rise buildings. The context is that in some cities with rapid housing construction, especially Toronto, the zoning code is either single-family or high-density, with nothing in between. The idea of allowing more missing middle housing has become a mainstay of New Urbanism as well as most North American YIMBY movements, underpinning demands such as the abolition of single-family zoning in California and Seattle.

Unfortunately, it’s an overrated concept. It applies to Toronto, but not Vancouver or the most expensive American cities, which are replete with missing middle density. The most in-demand neighborhoods have far too many people who want to move in to make do with this density level. Moreover, missing middle density in its New Urbanist form is not even really transit-oriented: low-rise construction spread over a large area is unlikely to lead middle-class workers to take transit when cars are available. The density required to encourage transit ridership and reduce housing costs is much higher, including mid- and high-rise residences.

What’s missing middle density?

A website created by Opticos Design, an architecture firm specializing in this kind of housing, has a helpful graphical definition:

Many of the missing middle housing forms are part of the vernacular architecture of American cities. In New England, this is the triple-decker, a three-story building with an apartment per floor. In Chicago, this is the fourplex, a two-story building with two apartments per floor. In Los Angeles this is the dingbat, with two or three inhabited floors on top of ground floor parking. In Baltimore and Philadelphia (and in London) this is the rowhouse. This history makes it easier to accept such buildings as both part of the local culture and as affordable to the lower middle class.

The triple-deckers in the parts of Providence and Cambridge I am most familiar with have a floor area ratio of about 1-1.5: they have 2.5 to 3 floors (counting sloped roofs as half a floor) and build on one third to one half the lot. A quick look at some Philadelphia rowhouses suggests they, too, have a floor area ratio in that range. Somerville has a population density just short of 7,000 people per km^2, with little non-residential land and some mid-rise and single-family areas canceling out to missing middle density. Kew Gardens Hills has about 12,000 people per km^2, and has a mixture of missing middle and mid-rise housing.

In Continental Europe, the vernacular architecture is instead mid-rise. In Scandinavia and Central Europe the euroblock has 4-7 floors and a floor area ratio of 2.5-4; Urban Kchoze shows many examples with photos, mostly from Prague, and Old Urbanist finds a euroblock in Berlin with a floor area ratio of 4.3. Central Stockholm’s residential buildings are almost entirely euroblocks, and residential density is 17,000/km^2 in Södermalm, 21,000/km^2 in Vasastan, and 28,000/km^2 in Östermalm. Parisian density is even higher – the floor area ratio of the traditional buildings looks like 4-5, with about 30,000-40,000 people per km^2.

Is missing middle really missing?

In Europe the answer is obviously no: lower-density cities like London are largely missing middle in their inner areas, and higher-density ones like Paris have missing middle density in their outer areas. But even in North America, where the term is popular, the expensive cities where people call for abolishing single-family zoning have missing middle housing. In addition to the above-listed vernacular examples, New York has brownstones all over Brooklyn (the term Brownstone Brooklyn refers to the gentrified inner neighborhoods, but this density is also seen in outer neighborhoods like Bay Ridge and Sheepshead Bay).

Vancouver is an especially instructive example. English Canada’s big cities are fast-growing, and a zoning regime that’s historically been friendlier to developers than to local NIMBYs has encouraged high-rise growth. Moreover, the high-rises are built in the modern boxy style (earning the ire of people who hate modern architecture) and tend to target middle-class and high-skill immigrant buyers (earning the ire of people who blame high housing costs on new construction). In contrast, vast swaths of Toronto and Vancouver are zoned for single-family housing.

And yet, Vancouver has considerable missing middle housing, too. The population density in Mount Pleasant, Fairview, Kitsilano, and West Point Grey is similar to that of Somerville and Eastern Queens. Buildings there are in modern style, but the housing typologies are not modernist towers in a park, but rather mostly buildings with 2-4 floors with the medium lot coverage typical of missing middle. I lived in an eight-unit, three-story building. Across from me there was a high-rise, but it was atypical; for the most part, that part of Vancouver is low-rise.

Shaughnessy offends people in its extravagance and wealth. In one Twitter conversation, an interlocutor who blamed absent landlords and foreigners (read: Chinese people) for Vancouver’s high housing costs still agreed with me that Shaughnessy, a white Canadian-born single-family area, shares the blame with its low-density zoning and very high residential space per person. Legalizing accessory dwelling units (“granny flats”) and townhouses in such a neighborhood faces local political headwind from the neighbors (who are still nowhere near as empowered to block rezoning as they would south of the border), but not from citywide social movements.

And yet, the density in the inner Westside neighborhoods near Broadway and Fourth Avenue is insufficient, too. It’s of course much higher than in Shaughnessy – I never really missed not owning a car living in Kitsilano – but the price signal screams “build more housing in Kits and Point Grey.”

Is missing middle transit-oriented?

Not really. In Providence the answer is absolutely not: car ownership is expected of every person who can afford it. The nearby supermarket, East Side Market, has an enormous parking lot; I’d walk, but it was obvious to me that my mode choice was not the intended use case. Even some Brown grad students owned cars (though most didn’t); at Columbia, car ownership among people below tenure-track faculty rank approaches zero. Once they own cars, people use them to take trips they wouldn’t otherwise have made, reorienting their travel patterns accordingly.

In Cambridge, car use is lower, but still substantial. The same is true of Vancouver (where outside Downtown and the West End the entire region’s density is at most missing middle, even if the typology is towers in a park and not uniformly low-rise). In Kew Gardens Hills, people seem to mostly drive as well.

This is not a universal feature of the urban middle class. In Stockholm, my postdoc advisor as far as I can tell does not own a car, and commutes to work by bike. Both there and in Basel, biking and using transit are normal and expected even among people who earn tenured academic salaries. At 7,000 people per km^2, people can forgo driving if they really want to, but most people will not do so. Only at the higher mid-rise density will they do so.

Sioux City: Straightening Buses and Getting Route-Length Right

A few days ago, Sandy Johnston linked to a diagram of the single bus route in South Sioux City, Nebraska, a suburb of Sioux City, Iowa. While South Sioux City has a traditional main street in Dakota Avenue, the bus does not follow it; it meanders, hitting destinations on and off Dakota. Many destinations are on US Route 77, an arterial bypass around the built-up area, with recent auto-oriented retail and office uses, including a Wal-Mart (in small-town America often the biggest bus trip generator). The discussion around what to do with this region’s bus network made me realize a crucial concept in planning infrequent transit: getting route-length right. To start with, here is a map of the bus, numbered Route 9 within the Sioux City area:

Here is a PDF map of the entire network. It has 10 routes, using 12 buses running hourly, with a timed meet at the center of Sioux City (just off the above map) at :30 every hour. Most routes run as loops, with highly separated inbound and outbound legs. Route 9 above runs one-way southbound on Dakota Avenue in the northern and southern legs but then meanders to run southbound on Route 77; the Dakota Avenue leg in between the two major east-west runs is one-way northbound.

I asked, why need it be so complicated? The major destinations are all on Dakota or Route 77. It should be easy to run two distinct routes, one on each, right? Without the east-west meanders, there would be the same total service-hours, right?

But no. The route runs hourly. The scale of the map is small: from the bridge over the Missouri in the north to I-129 in the south it’s 4.1 km. There is so little traffic that in the evening rush Google Maps said it would be just 10 minutes by car from Downtown Sioux City to the southern edge of Dakota Avenue near I-129. The roundtrip time would be 25-30 minutes, so the bus would sit idle half the time due to the hourly pulse.

Getting route-length right

When designing regional rail schedules, as well as my take on night buses in Boston (since reduced to a single meandering route), I’ve taken great care to deal with roundtrip route length not always being an integer multiple of the headway. A train that comes every half hour had better have a roundtrip length that’s just less than an integer or half-integer number of hours, counting turnaround times, to minimize the time the train sits at the terminal rather than driving in revenue service. The same is true of buses, except that scheduling is less precise.

In Boston, the plan at the time was for hourly buses, and has since changed to half-hourly, but the principle remains. The roundtrip length of each leg of the night bus network, should it expand beyond one (double-ended) route, should be an integer or half-integer number of hours. In practice this means a one-way trip time of about 25-26 minutes, allowing for a little recovery time and for delays for passengers getting on or off; overnight there is no traffic and little ridership, so 25 minutes of driving time correspond to just less than 30 minutes of actual time.

Thus, on each corridor, the bus should extend about 25 minutes of one-way nighttime driving time from the connection point, and the choice of which routes to serve and where to end each route should be based on this schedule. Of course on some shorter routes 12 minutes (for a half-hour roundtrip) and on some long routes 38 minutes (for a 90-minute roundtrip) are feasible with half-hourly frequencies, but in Boston’s case the strong night bus routes in practice would all be 25.

Length and frequency

In the case of Sioux City, hourly buses meeting at the center should have a one-way trip time of 25 minutes. However, the city is so lightly populated that there is little traffic, and the average traffic speed is so high that 25 minutes puts one well outside the built-up area. The driving time from city center to the edge of the built-up area, around I-129, Lakefront Shopping Center, and the various Wal-Marts ringing the city, is around 10 minutes.

Moreover, a car travel time of 10 minutes corresponds to not much longer on a bus. Frequent commenter Zmapper notes that in small American cities, taking the driving time in traffic and multiplying by 1.2, or 1.3 with recovery time, is enough. A one-way driving time of 11-12 minutes involves a roundtrip bus time of half an hour.

With such a small urban extent, then, the bus frequency should be bumped to a bus every half hour, leveraging the fact that few important destinations lie more than 11-12 minutes outside city center. The question is then how to restructure the network to allow for doubling frequency without doubling operating expenses.

The importance of straight routes

Some of Sioux City’s bus routes go beyond the 12-minute limits, such as route 6 to the airport. But most stay within that limit, they’re just incredibly circuitous. Look at the map of route 9 again. It jumps between two main corridors, has multiple loops, and enters the parking lots of the Wal-Mart and other destinations on US 77.

The reason for the meanders is understandable. US 77 is a divided highway without sidewalks or crosswalks, and none of the destinations thereon fronts the road itself. From the wrong side of the road to Wal-Mart it’s 330 meters, and a few other retail locations are more than 100 meters off. Many agencies wince at making passengers walk this long.

However, understandable does not mean justifiable. Traversing even 330 meters takes only about 4 minutes, and even with a hefty walking penalty it’s much less than the inconvenience caused by hourly headways. The other routes in the Sioux City area have the same problem: not a single one runs straight between city center and its outer destination.

With straighter routes, the savings in service-hours would permit running every half hour. A single bus could run every half hour if the one-way car travel time were at most 11-12 minutes; up to 23 minutes, two buses would provide half-hourly service. With 12 buses, there is room to replace route 9 with two routes, one on Dakota and one on US 77 (possibly entering the Wal-Mart, since the route is so short it may be able to get closer to Wal-Mart while still staying under 12 minutes). The Lakeport Commons and Southern Hills Mall area could get buses at the entrance, as it is the logical end of the line (route 1, to Southern Hills).

Some pruning would still be required. Some low-density areas far from the main corridors would have to be stranded. Some circumferential lines would be pruned as well, such as route 10 (to the Commons) on US 75 and route 2 (on Pierce Jackson) to Wal-Mart. Circumferential lines at such a low frequency are not useful unless the transfers to the spokes are timed, which is impossible without breaking the city center interchange since the lines take different amounts of time to get between city center with the plausible connection point. Ultimately, replacing the hourly routes with half-hourly routes would guarantee better service to everyone who’d still get any service, which is nearly everyone.

It’s not just Sioux City

I focus on Sioux City because it’s a good toy model, at such scale that I could redesign the buses in maybe two weeks of part-time work. But it’s not the only place where I’ve seen needlessly circuitous routes wreck what should be a decent bus network for the city’s size and density. In 2014 Sandy wrote about the bus network in New Haven, which has okay trunks (I only needed to hitchhike because of a bus delay once – the other four or five times I took the bus it was fine) but splits into indescribably complex branches near its outer ends.

More recently, I looked at the network in Ann Arbor, partly out of prurient interest, partly out of having gone to two math conferences there and had to commute from the hotel to the university on the city’s most frequent bus, route 4. Zoomed out, the Ann Arbor map looks almost reasonable (though not quite – look at routes 5 and 6), but the downtown inset shows how route 4 reverse-branches. Ann Arbor is a car-oriented city; at my last math conference, in Basel, a professor complained that despite the city’s leftist politics, people at the math department were puzzled when the professor biked to campus. The buses are designed to hit every destination someone who’s too poor to own a car might go to, with speed, frequency, and reliability not the main concerns.

The underlying structure of bus networks in small American cities – radial buses converging on city center, often with a timed transfer – is solid. The problem is that the buses run every hour when cities should make an effort to run them every half hour, and the routes themselves are circuitous. In very small cities like Sioux City, increasing the base frequency is especially urgent, since their built-up extent is so compact a direct bus would reach the limit of the serviceable area in 10-12 minutes, perfect for a half-hourly schedule, and not the 25 minutes more typical of hourly schedules. Sometimes, scaling down requires maintaining higher frequency than the bare minimum, to avoid wasting drivers’ time with low-value meanders.

Cities and Cultural Cringe

Following up on my last post’s promise to tackle both cultural theory of risk and cultural cringe, here is my take on the latter issue.

It is normal for people to have some degree of national pride and fervor. Cultural cringe refers to the opposite trend: when, in some circumstances, people in certain countries feel national shame and develop an inferiority complex. The term cultural cringe itself was coined by A. A. Phillips in 1950, describing Australia’s inferiority complex toward Britain in literary fields: Australians thought their literature was too provincial and perhaps too incomprehensible to the British readers, and as a result many authors were uncomfortable making the local references celebrated in the literary canon of Britain, France, Russia, the US, etc. This notion has been generalized elsewhere. Amos Oz says he felt uncomfortable writing books in such a peripheral country as Israel until he read Sherwood Anderson’s Winesburg, Ohio, showing how literature of and by the provinces can thrive.

From its origin in Australian literature, the idea of the cultural cringe has expanded to other fields, including the law, social relations, technology, and business. It seems endemic in former colonies, especially ones that are not rich. One writer in Nigeria argues how best practices thinking is cultural cringe by giving an example of a recent legal importation that turns out to already exist in traditional Yoruba law. In Australia itself, political scientist L. J. Hume pushed back against the notion that there is cultural cringe, arguing it is true of literature but not economics of other fields. But in mass culture, the vast majority of countries, both developed and developing, consider American film and television superior to their own and have domestic industries that focus on arthouse films or low-budget flicks.

Cultural cringe in legal, political, or technological fields remains endemic in many other developed countries. In one recent example, Emmanuel Macron said France is inherently resistant to change and (by implication) ungovernable, comparing it negatively with Denmark. In business, 1980s-era America was replete with books telling managers how to think like a Japanese or German, which trend ended when the Japanese lost decade and the economic crisis of German unification made these countries less fashionable.

Lying in the intersection of business, politics, and technology, urbanism and transportation are amenable to analysis using this concept. As in the Nigerian example, the third world tends to have too much cultural cringe and too much faith in the merits of importing first-world methods. Conversely, the United States (and to a large extent Canada) today is resistant to outside ideas and does not know how to be a periphery.

Urban layout: there’s a world outside Europe

During the SB 827 debate in California, supporters reassured restive city residents that the density the bill promoted – up to 7 floors right next to transit lines and up to 5 a little farther away – was gentle. “Paris density,” they said. Everyone likes Paris as a tourist. Everyone recognizes Paris as good urbanism.

There is very little cultural cringe in the United States – on the contrary, Americans are solipsistic in every field. However, one of very few exceptions is that the American middle class vacations in Europe and is familiar with how walkable European cities are. (It’s even referenced on Mad Men when a minor character goes on walks in their car-oriented New York suburb.) Paris is the largest and richest city Americans of a certain wealth and education level can be expected to be familiar with and like, but by the same token the YIMBYs could mention Barcelona, Amsterdam, and Rome.

But it’s useful to think of what was not mentioned. Certainly not Hong Kong or Dubai, which seem to be mentioned almost exclusively negatively in Western discourse. Not Tokyo, which Westerners are much less likely to visit to the point that the Western blogs talking about Japanese urbanism (like Urban Kchoze) are notable for it. Nothing in the middle-income world, including some old cities (like Mexico City and Istanbul) that have building height, street width, and stylistic variation that first-world urbanists would approve of (and do if they’ve been there).

In this situation, the invocation of famous European cities feels less like a dialogue and more like an attempt to induce cringe defensively, to make people feel less attached to their cities’ American auto-oriented character. In effect, it’s an attack on “it will change the character of our neighborhood,” a line that’s much less common in countries that are used to thinking of themselves as inferior to whatever they consider the metropolitan core (such as the first world writ large in Israel, or the former colonial master in ex-colonies).

Transportation: a little cringe is good, but not too much

In the developing world, there is extensive cringe. Without using that term, I suggested it as a reason behind high construction costs in the third world, which are similar to the costs of the first world today and several times as high as those of the first world from back when its income levels were comparable to those of subway-building third-world countries, in the early 1900s. In Latin America and China, development is more inward-looking, and China in particular learned to build subways from the USSR in the 1950s, not a rich country. In former colonies, there seems to be a greater willingness to import methods from either the former colonizer or from countries that aggressively invest in third-world infrastructure, like Japan and China; the result is very high construction costs for projects for which I have data in India and other countries of that development level.

In some cases, like India’s high-speed rail program, the country imports technology wholesale, and Japan (or China) may insist on an exact copy of its methods. As it is, Japan refuses to call Taiwan High-Speed Rail a Shinkansen system even though it runs Shinkansen rolling stock: construction methods were European, so Japan only calls THSR a high-speed rail system using Shinkansen-based technology.

However, decisions like India’s standard-gauge metro lines happen even in indigenous systems. Delhi Metro uses standard gauge not for some turnkey technological import, but purely because it feels more modern whereas Indian mainline trains feel dinghy and dangerous. Evidently, Delhi Metro electrification is 25 kV, which is standard on mainline trains but unheard of on first-world metros; modifying subways for high-voltage electrification requires expensive concrete pouring, since high-voltage catenary requires more generous clearances to avoid arcing, whereas modifying rail gauge is routine since the European vendors are used to selling to broad-gauge Finland and Spain and the Japanese ones are used to their country’s multitude of gauges.

And if India errs on the side of too much shiny adoption of foreign technology, the US errs on the side of adopting too little. Americans do not think their country is inferior. American authors do not think they need to experience another country or speak another language before they write. There was a time when the American business community felt outcompeted, but today it feels like it’s at the top of the world, Silicon Valley having long left Japanese corporations in the dust; I stopped seeing complaints that American cars were inferior to German and Japanese ones not long after Obama’s auto industry bailout.

The American policy sphere seems especially constrained. There is some cultural cringe toward London, leading thinktanks like the Regional Plan Association and TransitCenter to overlearn from London’s peculiarities (like the Oyster fare cap and contactless credit card payment), but not much toward Continental Europe and practically none toward Japan. Instead, the attitude toward non-English-speaking countries is one of dismissal. When Richard Mlynarik pointed out to a Caltrain official that Japanese trains turned much faster at terminals than Caltrain thought possible, the official replied, “Asians don’t value life the way we do.”

If India fails to understand where its own methods could be superior despite being a peripheral country, the United States fails to understand that it’s a peripheral country in the first place. Transportation innovation rarely happens in North America. It happens in Western Europe and Japan, and to some extent in developing countries that have less cultural cringe than former colonies, such as Brazil and Colombia and their invention of BRT or Colombia, Bolivia, and Mexico’s use of aerial gondolas in mountainous suburban areas.

Urban development: you are not New York

I’ve been reading Aaron Renn’s blog, the Urbanophile, since maybe 2008. At the time he was still in Indianapolis, in (I believe) management consulting, writing about how his city was trying to become culturally and economically bigger than it was, and sometimes but not always succeeding. A recurrent theme in his writings has been that Midwestern American cities are desperate for development. They keep saying they need more creative people, more venture capital, or whatever else is in vogue. (In contrast, he says, Rhode Island, where he lived later, doesn’t even understand how peripheral it is.)

However, the way the Midwestern cities he focuses on try to attract this elusive development is through cheap copying. An old post of his I can no longer find contrasts world-class Indianapolis with world class in Indianapolis. The former involves investing in some city institution to make it world-class, or more realistically notable enough that boosters can call it world-class with a straight face. The latter involves inviting a starchitect or another person with international cachet (such as Richard Florida) to build something in Indianapolis that’s notable and is exactly as notable as what this person might build in any other city of that size, with no particular connection to the city itself.

In the transportation field, many American cities build mixed-traffic downtown streetcars and beam with pride if they get 4,000 riders per weekday. Often this mentality overrides any attempt to provide services to city residents: thus, the streetcar in Detroit is not integrated with the city’s bus network, and in fact a bus runs on the same street, on different lanes from the streetcar. This isn’t about some mythical preference for rail over bus: these cities build whatever they hear is in vogue and will get them noticed by New York media, whether it’s peak-only commuter rail, a downtown streetcar, a limited bus that calls itself BRT, or now a bus network redesign around untimed 15-minute frequencies.

Cringe vs. dialogue

It’s important to distinguish dialogue with a foreign culture and cultural cringe toward it. One difference is that cringe implies infatuation; however, infatuation can also develop among immigrants who are steeped in the metropole’s culture after having lived there even while maintaining ties to the old country. A bigger difference is the extent of two-way dialogue. Israelis use the expression “unbroken country” to refer to the mythical average first-world country in which you can get things done without having to tell government bureaucrats that you served in the military with their bosses; however, few have lived abroad long enough to know the details of what makes these countries tick better.

With limited knowledge of the core, the periphery can worship at the feet of the few people who do know, which leads to political bias. This is where moral panics of no-go zones come from: there is an Israeli television show purporting to portray how things are in Europe, but any connection between Belleville (or other racially diverse Paris neighborhoods) and what they depict is completely incidental. In that case, the bias is right-wing. In the opposite direction, left-wing bias can occur when American liberals and socialists are enamored by European health care and education systems and elide a thousand details that distinguish them from American renditions of single-payer health care or free college tuition.

But the biased reaction is only common in places that care little about how to govern. “Well, actually Tower Hamlets is a no-go zone” is not a blueprint for reducing nonwhite immigration to the United States or Israel. Instead, in the policy sphere a more common reaction is a shrug. Dialogue is threatening: the people capable of it are typically not the top pundits on this issue. Instead, it’s more common to aggressively dismiss knowledge that’s hard to access, even among people who at the same time invoke the cringe. In Israel it takes the form of self-denigrating lines like “this is Israel, not Finland.” Cultural cringe leads to lower expectations this way.

When Phillips criticized Australian authors who deracinated their writing to appeal to British taste, he was implicitly saying that Australians couldn’t root their literature in British experience. Oz, similarly, felt constrained about writing when he was young because living in Israel, he could not root his books in Paris, Milan, and other flashy cities whose books he devoured. The economic (or legal, or technological) analogue of this observation is that the reason there is cultural cringe is that people in peripheral areas (which in transportation include the United States) are too unfamiliar with the core and cannot dialogue with it the way people in different parts of the core can.

Urbanism is not literature. One doesn’t need extraordinary sensitivity and a lifetime (short as it may be) in a culture to produce very good insights about transportation, housing, or municipal governance. It’s possible to break out of the cringe by acquiring detailed knowledge of how the core operates. In the case of the third world and subway construction, it means learning enough about current and historical construction methods to be able to propose ways to build infrastructure at low costs commensurate with these cities’ low wages; in the case of the United States, it means learning enough about what makes European, Japanese, Latin American, etc. urbanism tick that it can be adopted domestically.

Urbanism is not literature in a far more important sense: there really are better and worse traditions there. It’s not enough to have pride in what you have when what you have is a third-world city where the poor don’t have running water, or for that matter an American city that would shut down instantly were gas prices to rise to levels necessary to stop global warming. Learning from the core is crucial. It’s just equally important to do so through dialogue and not through the ignorant self-denigration that is cultural cringe.

Cities and Cultural Theory

I ran a Patreon poll about sociological theories as applied to urbanism, offering two options: cultural theory of risk, and cultural cringe. The poll was tied, so I feel compelled to do one post on each (when cultural theory was ahead I was outlining two separate posts on it, one about transit and one about housing).

Psychologists and sociologists have long known that people’s perceptions of risk can vary widely from actual risks (e.g. people are more afraid of flying than of driving even though planes are safer), and, moreover, different people have different evaluations of risk. Early theories analyzed differences in risk perception along lines of class, race, or gender, but subsequently a group of social scientists, many (though not all) libertarians, argued for an ideology-based cultural identity. In 1982, the anthropologist Mary Douglas and the political scientist Aaron Wildavksy published Risk and Culture, arguing for three different identities (later expanded to four). Douglas used her past insights from analyzing premodern societies’ social taboos to analyzing risk perception within industrialized societies, especially the rise of the environmental movement during a time of falling pollution levels.

Urbanism and public transit are intimately connected with environmentalism. A large fraction of transit advocacy is environmental in nature, and both early NIMBYs and present-day YIMBYs come from green progressivism. Even when the arguments are not explicitly ecological, the parallels are unavoidable: Jane Jacobs’ critique of urban renewal has strong similarities with Rachel Carson’s critique of DDT. Legally, the mechanisms that exist to protect both endangered species and neighborhoods are often the same (e.g. the American environmental impact report process). Thus, understanding a sociological theory developed originally to analyze environmentalism should have straightforward applications to cities and urban transportation.

Grid-group typology

Cultural theory begins with the distinction between markets and hierarchies. These are two distinct ways of organizing society, leading to different institutions and different social views. Douglas and Wildavsky’s innovation is to distinguish two different axes of separation between markets and hierarchies, which they call group and grid, leading to a 2*2 chart:

Group measures group solidarity among members of the system; grid measures the restrictions placed on the individual’s ability to exit the system. While individualism and hierarchy are politically stronger than the other two cultural identities, group and grid are fairly independent on the level of personal politics and there are numerous examples of egalitarianism and fatalism.

I strongly recommend reading the original book, but this review does it and the theory’s subsequent developments justice.

Individualism arises in institutions that are atomized and like it. The free market is the best example, but professions with mostly independent workers (like academia and the law, especially historically) also fit. Individualists view nature as resilient, returning to a stable equilibrium no matter what happens, and thus business control of the environment is to be celebrated as development; I had this aspect of cultural theory in mind when I wrote one of my early posts critiquing the idea that cities have a single equilibrium. Rejecting systemic or environmental risks, individualists focus on risks that disrupt the market’s operation, like war or recession.

Hierarchy arises in institutions where everyone has a predetermined role to play. Examples include the military, premodern feudalism, and modern bureaucracies. Hierarchists view nature as perverse or tolerant, capable of adapting to change to an extent but not beyond circumscribed limits, and therefore employ what their society considers expert opinion (e.g. scripture, bureaucratic process, big science, etc.) to figure these limits. Hierarchists focus on risks that indicate social deviance, like crime.

Douglas and Wildavsky call the above two tendencies the center, distinguished from what they call the border, whose growth they ascribe to the erosion of trust in institution in the 1960s and 70s (coming in the US from the Vietnam War and Watergate, in France from the reaction to the social protests of 1968, etc.).

Egalitarianism was the border tendency studied in Risk and Culture, which polemically called it sectarianism. It occurs in groups that rely on intensive solidarity among members but cannot enforce their collective will on the individual, and thus require other mechanisms to encourage people not to leave. These include internal equality, to stave off discontentment, and the precautionary principle, to prevent change from inducing disaffected members to exit. Thus they view nature as fragile, prone to collapse at any moment if the system endures any change in direction, and focus on low-probability, high-impact risks (such as environmental collapse), which enhance the group’s internal solidarity against outside enemies.

One of the key oppositions Douglas and Wildavsky point out is between the Hutterites and the Amish. Both denominations are high-group, socializing almost exclusively among their own kind, adhering to strict religious principles. But despite their common Anabaptist origin, they differ in one crucial aspect: the Hutterites have communal ownership of property, the Amish don’t. This makes the Hutterites high-grid, since members who leave start from zero, whereas Amish who leave get to keep their land. The Amish openly adhere to the precautionary principle, which they famously interpret extremely conservatively; the Hutterites have formal rules for group size and adopt modern farming technology easily.

Fatalism is the last tendency, so politically weak that it was ignored in the original book and only discussed in subsequent refinements of the theory. It arises in institutions whose lower-ranked members (whether by market poverty or low rank in the hierarchy) are disaffected, unable to leave and yet not sharing any of the group’s purported values. They tend to view nature as capricious, moving without clear direction, and do not have any particular risk focus, but tend to be especially concerned about things they do not understand (such as unfamiliar or complex technology). Transgressive fiction like The Wire tends to depict fatalist institutions; geekier readers may also recognize H. P. Lovecraft’s mythos as fatalist, portraying a universe so far beyond human understanding that any who begins to figure any of it out goes insane or slowly becomes a monster.

Some political movements have obvious cultural identities. Libertarianism is individualist. The New Left is egalitarian. The far right is hierarchist: Cas Mudde calls it pathological normalcy, and its issue focus (crime, immigration as genetic pollution, terrorism) is hierarchical, even as it rejects traditional hierarchical institutions. However, the broader left vs. right distinction does not neatly map to any of the four cultural biases. About the only generalization that can be made is that activists are usually not fatalists.

Cultural theory and transportation

Transportation planning is an inherently hierarchical industry. The technologies involved are old and continuously tweaked within well-understood parameters. With so much accumulated knowledge, work experience matters, requiring companies in the industry to adopt a hierarchical setup. Moreover, the transportation network itself is complex and interconnected, with changes in one region cascading to others. Changes to the bus network, the train schedule, etc. are possible but only if the people implementing them know what they’re doing, creating a picture of the network much like the hierarchical view of nature as tolerant up to a limit.

The individualist ethos of tech companies – move fast and break things – works for fast-growing industries. Individualism is by far the fastest of the four biases in reacting to sudden changes. The tech industry’s denigration of public transit as an old hat has to be understood as individualists reacting poorly to an industry that has to be run by a business culture they find alien.

Readers who have been following me closely may ask, well, what about me? I’m an individualist. I evidently talk more to startups than to transportation consulting megacorps. One reader notes that I’ve called for people in positions of authority to be fired for incompetence so many times that a post like this one may read as hesitant purely because I only call for removing the governor of Massachusetts and the secretary of transportation and not also for firing planners.

The answer is that while there is extensive accumulated knowledge about good public transit in Western Europe, Japan, and South Korea, there is very little in the area I’m most involved in, North America. This is especially true when it comes to regional rail: the existing mainline rail in the US should be treated as more or less tabula rasa. Adopting best practices requires extensive expert knowledge, but the methods in which they should be implemented have little to do with the internal bureaucracy of hierarchical organizations, since the railroads that would ordinarily be in charge (like the LIRR or the MBTA) are the problem and not the solution.

But if the actual process of running a transportation network is hierarchical, the politics are completely different. As with left-right politics, the politics of public transit don’t neatly fit into any of the four tendencies. Center-right hierarchists tend to support extensions of the status quo, which means more urban transit in New York, London, Paris, and other large cities, as well as high-speed rail on strong corridors (High Speed 2 in the UK is bipartisan), but more roads everywhere else. Individualists on the right tend to be anti-rail, partly because it looks so hierarchical, partly because of peculiarities like Koch funding of American libertarianism (which has been exported to Israel, at least).

Egalitarian environmentalists tend to be pro-transit, but their discomfort with hierarchy sometimes shows up as mistrust of big infrastructure projects. The radical environmentalist Chris Clarke, opposed early attempts to fast-track California High-Speed Rail and called Robert Cruickshank of California HSR Blog a shill for developer interests. Jane Jacobs herself ended up arguing late in her life that mass transit was at the wrong scale and instead cities should encourage community jitney services.

The process itself has issues of trust that activate egalitarians and fatalists, the latter often reflexively opposing reforms since they assume things must always get worse. It leads to tension between community outreach, which helps defuse this opposition, and speed of implementation.

Cultural theory and housing development

Whereas transportation politics isn’t neatly slotted into the grid-group paradigm, the politics of urban development is: YIMBY is an individualist movement, with near-universal support from people who identify with that cultural bias. The other three tendencies are split. The market urbanist proposition of abolition or near-abolition of zoning doesn’t appeal to hierarchists (who want to be able to control where housing goes) or egalitarians (who worry about the consequences of empowering market actors); but there are egalitarian left-YIMBYs and hierarchical city leaders who favor transit-oriented development.

In fact, when analyzing NIMBYism, it’s useful to slot it not by class or political opinion, but by cultural identity. There is much less difference between working-class and middle-class NIMBYs than leftists posit, and in some cases anti-gentrification politics and racist opposition to fair housing blend together (as in South Tel Aviv, where the local far right has argued black refugees are part of a gentrification ploy).

The key is that egalitarianism really consists of two distinct concepts, both necessary to maintain high group solidarity without grid: internal equality, and strong boundedness (which refers to sharp distinctions between insiders and outsiders). The cultural geographer Stentor Danielson argued once that surveys consistently show people approve of internal equality but not of strong boundedness, which is why egalitarian communities are so rare even though many people agree with most of their tenets.

Thus, when NIMBYs argue that more development would bring outsiders or change the character of the neighborhood, this is as compatible with egalitarianism as with hierarchy. Gentrification is just the name for when these outsiders are not begging for scraps. The real difference is in where this is taken. Egalitarian NIMBYism emphasizes irrevocable change, high-impact risks (e.g. that a new development would induce runaway gentrification), and trust. Hierarchical NIMBYism instead talks about behavioral norms, usually referring to middle-class moral panics about crime, but occasionally flipping to black American fears that white people would call the police more often.

The fatalists, too, have their own criticism of redevelopment – namely, that it represents another sudden change involving forces they have no control over. “Nobody asked us” has to be understood as a fatalist and not egalitarian cry, even though egalitarians often try to organize fatalists.

It’s not really possible to promise any of the other groups what it really wants: protection from change for egalitarians, a more concrete relationship between development and their actual lives for fatalists, or ethnic or other kinds of homogeneity for hierarchists. Nonetheless, alliances are possible with some egalitarians and hierarchists. SF YIMBY has to be viewed as an attempt at an individualist-egalitarian alliance for more housing, ceding ground on rent control to curry favor with ideological socialists (and its East Bay offshoot is run by actual socialists). In the other direction, Theresa May’s making noises about releasing more land for housing to get young people on the “housing ladder,” invoking a hierarchical sense of normality regarding when it’s appropriate to buy a house.

Overbuilding for Future Capacity

I ran a Patreon poll with three options for posts about design compromises: overbuilding for future capacity needs, building around compromises with unfixably bad operations, and where to build when it’s impossible to get transit-oriented development right. Overbuilding won with 16 votes to bad operations’ 10 and development’s 13.

It’s generally best to build infrastructure based exactly on expected use. Too little and it gets clogged, too much and the cost of construction is wasted. This means that when it comes to rail construction, especially mainline rail, infrastructure should be sized for the schedule the railroad intends to run in the coming years. The Swiss principle that the schedule comes first was just adopted in Germany; based on this principle, infrastructure construction is geared around making timed transfers and overtakes and shortening schedules to be an integer (or half-integer) multiple of the headway minus turnaround time for maximum equipment utilization.

And yet, things aren’t always this neat. This post’s topic is the issue of diachronic optimization. If I design the perfect rail network for services that come every 30 minutes, I will probably end up with a massive upgrade bill if ridership increases to the point of requiring a train every 20 minutes instead. (I chose these two illustrative numbers specifically because 30 is not a multiple of 20.) In some cases, it’s defensible to just build for higher capacity – full double-tracking even if current ridership only warrants a single track with passing sidings, train stations with more tracks in case more lines are built to connect to them, and so on. It’s a common enough situation that it’s worth discussing when what is technically overbuilding is desirable.

Expected growth rates

A fast-growing area can expect future rail traffic to rise, which implies that building for future capacity today is good. However, there are two important caveats. The first is that higher growth usually also means higher uncertainty: maybe our two-track commuter line designed around a peak of 8 trains per hour in each direction will need 32 trains per hour, or maybe it will stay at 8 for generations on end – we usually can’t guarantee it will rise steadily to 16.

The second caveat, applicable to fast-growing developing countries, is that high growth raises the cost of capital. Early British railroads were built to higher standard than American ones, and the explanation I’ve seen in the rail history literature is that the US had a much higher cost of capital (since growth rates were high and land was free). Thus mainlines in cities (like the Harlem) ran in the middle of the street in the US but on elevated structures in Britain.

But with that in mind, construction costs have a secular increase. Moreover, in constrained urban areas, the dominant cost of above-ground infrastructure cost is finding land for multiple tracks of railroad (or lanes of highway), and those are definitely trending up. The English working class spent 4-5% of its income on rent around 1800 (source, PDF-p. 12); today, spending one third of income on rent is more typical, implying housing costs have grown faster than incomes, let alone the general price index.

The upshot is that cities that can realistically expect large increases in population should overbuild more, and optimize the network around a specific level of traffic less. Switzerland and Germany, both of which are mature, low-population growth economies, can realistically predict traffic many decades hence. India, not so much.

Incremental costs

The expected growth rate helps determine the future benefits of overbuilding now, including reduced overall costs from fronting construction when costs are expected to grow. Against these benefits, we must evaluate the costs of building more than necessary. These are highly idiosyncratic, and depend on precise locations of needed meets and overtakes, potential connection points, and the range of likely train frequencies.

On the Providence Line, the infrastructure today is good for an intercity train at current Amtrak speed every 15 minutes and a regional train making every stop every 15 minutes. There is one overtake segment at Attleboro, around three quarters of the way from Boston to Providence, and the line is otherwise double-track with only one flat junction, with the Stoughton branch. If intercity trains are sped up to the maximum speed permitted by right-of-way geometry, an additional overtake segment is required about a quarter of the way through, around Readville and Route 128. If the trains come every 10 minutes, in theory a mid-line overtake in Sharon is required, but in practice three overtakes would be so fragile that instead most of the line would need to be four-tracked (probably the entire segment from Sharon to Attleboro at least). This raises the incremental costs of providing infrastructure for 10-minute service – and conversely, all of this is in lightly developed areas, so it can be deferred without excessive future increase in costs.

An even starker example of high incremental costs is in London. Crossrail 2 consists of three pieces: the central tunnel between Clapham Junction and Euston-St. Pancras, the northern tunnel meandering east to the Lea Valley Lines and then back west to connect to the East Coast Main Line, and the southern tunnel providing two extra tracks alongside the four-track South West Main Line. The SWML is held to be at capacity, but it’s not actually at the capacity of an RER or S-Bahn system (as I understand it, it runs 32 trains per hour at the peak); the two extra tracks come from an expectation of future growth. However, the extreme cost of an urban tunnel with multiple new stations, even in relatively suburban South London, is such that the tunnel has to be deferred in favor of above-ground treatments until it becomes absolutely necessary.

In contrast, an example of low incremental costs is putting four tracks in a cut-and-cover subway tunnel. In absolute terms it’s more expensive than adding passing tracks in suburban Massachusetts, but the effect on capacity is much bigger (it’s an entire track pair, supporting a train every 2 minutes), and moreover, rebuilding a two-track tunnel to have four tracks in the future is expensive. Philadelphia most likely made the right choice to build the Broad Street Line four-track even though its ridership is far below the capacity of two – in the 1920s it seemed like ridership would keep growing. In developing countries building elevated or cut-and-cover metros, the same logic applies.

Sundry specifics

The two main aspects of every infrastructure decision are costs and benefits. But we can discern some patterns in when overbuilding is useful:

  1. Closing a pinch point in a network, such as a single- or double-track pinch point or a flat junction, is usually worth it.
  2. Cut-and-cover or elevated metro lines in cities that are as large as prewar New York (which had 7 million people plus maybe 2 million in the suburbs) or can expect to grow to that size class should have four tracks.
  3. On a piece of infrastructure that is likely to be profitable, like high-speed rail, deferring capacity increases until after operations start can be prudent, since the need to start up the profitable system quickly increase the cost of capital.
  4. Realistic future projections are imperative. Your mature first-world city is not going to triple its travel demand in the foreseeable future.
  5. Higher uncertainty raises the effective cost of capital, but it also makes precise planning to a specific schedule more difficult, which means that overbuilding to allow for more service options becomes reasonable.
  6. The electronics before concrete principle extends to overbuilding: it’s better to complete a system (such as ETCS signaling or electrification) even if some branches don’t merit it yet just because of the benefits of having a single streamlined class of service, and because of the relatively low cost of electronics.

Usually cities and countries should not try to build infrastructure ahead of demand – there are other public and private priorities competing for the same pool of money. But there are some exceptions, and I believe these principles can help agencies decide. As a matter of practice, I don’t think there are a lot of places in the developed world where I’d prescribe overbuilding, but in the developing world it’s more common due to higher future growth rates.

The Dynamics of Bus Bunching

I’ve been wanting to write a paper about how to use dynamical systems to analyze failure modes for transportation networks. So far I haven’t been able to analyze this more carefully, but there’s one relatively simple example, namely bunching along a single bus line. This intersects to some extent with what I did in math academia, although the mathematical tools I’m using are fairly primitive within dynamics, going back to the early 20th century and not to the advanced machinery that dynamicists have developed in the last forty years (like the Mandelbrot set). As a caution, despite the math jargon and the math paper structure, it’s a blog post, and not something I’d even be comfortable uploading to the arXiv.

The upshot of the mathematical model in this post is that several already-understood reforms can seriously reduce bus bunching: speeding up boarding through prepayment and all-door boarding, using bigger buses with many doors on the busiest routes, implementing signal priority and enforcing bus lanes better, and improving dispatching to tell bus drivers to maintain even headways leaving each terminus. Section 1 provides mathematical background, and people who know some dynamics can skip it; it’s meant to be accessible to a general audience (if you’ve heard of derivatives, you should be fine). Section 2 constructs the model for bus schedule variations, section 3 explains how the model predicts bunching, and section 4 goes into how the above interventions can improve the situation. The mathematics I’m using is not terribly advanced, but it may benefit from careful reading, especially around the formulas.

1. Background on dynamics and chaos

Before I left academia, when people asked me to explain my research, I’d use the following example. In dynamics, we study what happens when we take a function and iterate it many times. We are specifically interested in chaotic behavior, which arises when two very close numbers can end up widely separated after sufficient iteration. There is no chaos if we only look at linear functions, so the simplest example is quadratic:

f(x) = x^{2}

The simplest behavior of any number when we apply a function many times is if nothing changes. A point where this happens is called a fixed point. Two numbers are fixed points for the function x^2: 0 and 1. But in practice, it’s useful to view infinity as a number, so that instead of being far away from each other, the numbers 1,000,000, 1,000,000,000, and -1,000,000 should be viewed as all very close to infinity. Under the squaring function, infinity is a fixed point as well.

The key to understanding the dynamics of a function is to look at the behavior of the function near a fixed point. Near the point 0, if we take the square of a number, it gets much smaller. For example, 0.1^2 = 0.01. This means that if a number x is close to zero, then as we iterate the function x^2 we will get closer and closer to 0, very quickly. This behavior is called attracting. Near 0, small changes in initial conditions don’t matter much: the numbers 0.1 and 0.11 are close, and if we keep squaring them, we will approach 0 either way. Infinity is attracting as well once you get used to thinking of very large (or very large with a negative sign) numbers as close to infinity: 1,000 is pretty close to infinity and 1,000^2 = 1,000,000, even larger, i.e. closer.

However, near the point 1, we get the opposite behavior: 1.1^2 = 1.21, which is about twice as far away from 1 as 1.1, and similarly 0.9^2 = 0.81, again about twice as far away from 1 as 0.9. We then say that 1 is a repelling point. Near a repelling point, we have chaotic behavior, because two points that start out close, like 0.9 and 1.1 or even 0.99 and 1.01, end up widely separated after iteration (0.99 eventually approaches 0, 1.01 eventually approaches infinity).

If you’ve learned calculus, your reaction to the line about how 1.21 is about twice as far away from 1 is “the derivative is 2!”. In general, the way to figure out whether a fixed point is attracting or repelling is to take the derivative f‘(x) at the point (technically it’s called the multiplier). If the absolute value of the derivative is less than 1, the point is attracting; if it’s more than 1, the point is repelling; if it’s exactly equal to 1, we say the point is indifferent and then the behavior near the fixed point depends on further details that I don’t want to get into. As a note of caution, taking the derivative anywhere except at a fixed point won’t tell you anything about the function – for example, the derivative of x^2 (which is 2x) is 4 when x = 2 but 2 is not repelling, it’s a non-fixed point that ends up going off to infinity.

(You may wonder what it exactly means to take the derivative at infinity. The answer is that if f is a polynomial, then the multiplier at infinity is always 0. If f is not a polynomial, there is a definition on Wikipedia.)

Attracting points are in every sense nicer to deal with than repelling points. Unfortunately, chaos is everywhere: most points of every nonlinear function are repelling. More precisely: in addition to fixed points, there are periodic points (i.e. fixed points of iterates). The periodic points of x^2 are a little hard to unpack if you’re not used to complex numbers: they’re solutions to equations like x^4 = x, x^8 = x, etc., and these are all complex numbers on the unit circle. We can compute multipliers there too (take the derivative of the iterate for which they’re fixed) and classify them as attracting or repelling. One of the foundational theorems of dynamics is that all but finitely many periodic points are repelling – and the number of non-repelling points is at most 2d-2 where d is the degree of the function (and if the function is a polynomial of degree d, then there are at most d-1, not counting infinity). My main contribution to math is to extend this result to a certain number-theoretic application.

2. Modeling buses using dynamics

The key insight of why buses bunch is that maintaining the exact schedule is a repelling fixed point, so small variations from the schedule (due to traffic, slow passengers, or random noise in passenger numbers) will compound over time, just as the variation of 0.9 or 1.1 from 1 compounds over time as you apply the squaring function.

More precisely, let’s say buses on a certain street run every 10 minutes. Eventually we will call the scheduled headway h, but to make this concrete, let’s say h = 10 minutes. Every few hundred meters, the buses stop to pick up and drop off passengers. Before San Francisco instituted prepayment, each additional passenger took on average 3.9 seconds to board and another 3.9 to disembark (link, PDF-p. 14); the TRB claims the average is 3 seconds to board (link, PDF-p. 20). We will call the extra boarding time per passenger b, and right now set b = 3 seconds = 0.05 minutes.

To understand why bunching occurs, let’s say that our bus falls behind schedule by a minute. It’s now 11 minutes behind the bus ahead (which we’ll assume is on schedule), not 10 minutes. On average, there will be 10% more passengers to pick up at each stop (passengers arrive at stops at a uniform rate). Let’s say the bus gets 60 boardings per hour (which is the Brooklyn-wide average). Typically we expect the bus to get 10 boardings in the next 10 minutes, but because there are 10% more passengers per stop, there will instead be 11 boardings. The one extra boarding will slow the bus down by 3 more seconds. The bus will then be 1:03 minutes behind. It’s a small difference, but over time it compounds.

There will also be more alightings as the bus gets more crowded, with a lag time equal to average passenger trip length. But in practice, to avoid introducing exponential factors, complicating the analysis, it’s best to just think of boardings plus alightings as a single metric, which if there are 60 boardings per hour equals 120 per hour or 2 per minute, and take note that a 1-minute delay only starts accumulating half a lag time in the future (e.g. 10 minutes if the average unlinked passenger trip is 20 minutes, as in New York). We call the number of boardings plus alightings per hour r, and in our example case r = 2.

If we choose our unit of time to be the minute, then the formula for the average delay a minute after our bus was x minutes behind the bus ahead is,

f(x) = x + \frac{x - h}{h}\cdot r\cdot b

In the example we worked out above, it took 10 minutes to accumulate an additional 6-second delay (3 from boarding, 3 from alighting). Using the numbers h = 10, x = 11, b = 0.05, r = 2, verify that the formula spits out 0.01, or in other words an extra 0.6-second delay per minute. If x = h, that is if the headway between our bus and the bus ahead is exactly as timetabled, then there is no additional delay, making the correct headway a fixed point, but a repelling one. The multiplier is equal to,

1 + \frac{r\cdot b}{h}

Note that choosing units is important. The reason is that the mathematics I’m using assumes there are discrete steps: you apply the squaring function (or any other nonlinear function) once at a time. In reality, time is continuous. So to model it using discrete dynamics, it matters which unit of time we pick; this is the equivalent of choosing between x^2, x^4, x^16, or any other iterate. Fixed points will stay attracting or repelling (or indifferent) no matter what, but the exact value of the multiplier will change.

With this in mind, when our quantum of time is a minute, the multiplier with our usual values of h, b, and r is equal to 1.01. Every minute, a delay multiplies by a factor of 1.01. Within an hour, this factor grows to 1.82. This doesn’t seem too bad – it means a 1-minute delay turns into a 1:49-minute delay within an hour.

3. How bunching occurs

In section 2 we showed that if h is the scheduled headway, b is the average boarding time per passenger, r is the average number of boardings and alightings per unit of scheduled service time, and x is the current distance (in units of time) between our bus and the bus ahead of us, then within a minute we expect the distance to grow to

f(x) = x + \frac{x - h}{h}\cdot r\cdot b

The multiplier is 1 + rb/h, which doesn’t seem too bad. However, there are complications. For one, the initial delay may be not 1 minute but longer. In Eric Goldwyn’s interviews with drivers, they cited traffic as the top reason why they believed bunching occurs, and barely mentioned passenger boardings. In math we cannot conflate popular perception with reality, but that the drivers complain about traffic suggests that there is widespread variation in the extent of the initial delays coming from missing a light, drivers blocking the bus lane, etc. If the initial delay is 2 minutes, then all delay numbers are naturally doubled over 1 minute.

But more importantly, our delayed bus will never bunch with the bus ahead. It will bunch with the bus behind. And the effect of the model of cascading delays on the bus behind us is exactly double what I described above. The reason is that if our bus is a minute behind – for example, 11 minutes behind the bus ahead when it should be 10 minutes behind – then the bus behind us, if it starts out on schedule, is now a minute ahead, only 9 minutes behind us when it should be 10 minutes. This means that within 10 minutes, we fall 6 seconds behind (and are thus 11:06 behind the bus ahead of us), but by the same token the bus behind us advances 6 seconds ahead (and is thus 8:48 behind us). In practice, the quantity relevant to bunching is the distance between two successive buses, and behind us, the multiplier is not 1.01 but 1.02. Within an hour a 1-minute delay reduces the gap between our bus and the bus behind us by 3:17, and a 2-minute delay reduces it by 6:34, almost two thirds of the way to catching up with us. If the initial delay is 1 minute, the bus behind us will actually catch up with us within log_{1.02} 10 = 116:17 minutes. If the initial delay is 2 minutes, it will catch up within log_{1.02} 5 = 81:16 minutes.

And third, the multiplier grows as r grows and h falls – that is, it’s higher when the frequency is higher and when there are more riders per service hour. Keeping r at 2 (again, this is 60 boardings and 60 alightings per hour) but lowering h to 5 raises the multiplier to 1.02 ahead of us and 1.04 behind us. A multiplier of 1.04 with a headway of 5 minutes means the bus behind us will catch us within log_{1.04} 5 = 41:02 minutes with just a 1-minute delay.

The real limiting factor to the capacity of city buses is not minimum stopping distance, unlike with trains. It’s that as the headway h decreases, bunching becomes so routine that adding more buses does not actually add capacity. If a bus runs every 2.5 minutes, keeping b = 0.05 and r = 2 gives us a multiplier of 1 + 0.05*2/2.5 = 1.04 ahead of us and then 1.08 behind us; the bus behind us will catch our bus within 12 minutes.

4. How to reduce bunching

The formula for the multiplier of the dynamical system formed by bus performance is 1 + rb/h where r is the rate at which passengers board and alight, b is boarding time per passenger, and h is scheduled headway. However, since as our bus gets further and further behind, the bus behind us gets further ahead relative to schedule and certainly relative to us, the multiplier relevant to bunching is 1 + 2rb/h. The bus behind us will catch ours in

log_{1 + 2rb/h} h/d

minutes, where d is the initial delay (so we start the calculation from x = h + d). On very frequent buses, this will happen very quickly: 2.5-minute headways with New Yorkish assumptions on passenger traffic density and conservative assumptions on boarding speed yield a catchup time of just 12 minutes. So how do we prevent this?

4.1. Reduce boarding time per passenger

Off-board fare collection allows passengers to board the bus more quickly, without paying the driver. This has the effect of greatly reducing b, from 3 seconds to about 1.2 per the TRB. Prepayment also allows all-door boarding, effectively halving the average boarding time per passenger at stops without large volumes of disembarking passengers.

But in addition to prepayment, there are other ways of reducing b. Low-floor buses allow passengers to get on and off more easily; the reason San Francisco’s numbers are higher than the TRB’s is that San Francisco assumes a mostly high-floor bus fleet, whereas on the low-floor fleets boarding is much faster (in fact, faster on low-floor buses without prepayment than on high-floor buses with).

Adding more doors is desirable as well. The typical 12-meter bus has two doors, but some cities have purchased three-door buses, such as Nice and Florence. The typical 18-meter accordion bus has three doors, but in Florence I have seen four-door accordion buses; in contrast, the older accordion buses in New York only had two doors, slowing down boarding and alighting at busy stations. Per TRB data, three-door buses reduce b from 3 seconds to 0.9, or 0.015 minutes. One third the multiplier means roughly three times the time it takes to bunch.

4.2. Use bigger buses

The multiplier depends only on the rate at which passengers wish to board our route. Adding more bus service will reduce r (by spreading boardings across more service-hours) and h (by adding more frequency) at the same rate, but make it take less time for bunching to occur. Just running less service means passengers take longer to get on each bus, but also means that the passenger load per stop is less sensitive to fixed delays occurring upstream.

Of course, running less service is cruel to passengers and can discourage ridership due to a negative frequency-ridership spiral or (on the busiest routes) inadequate capacity. But running bigger buses to compensate can provide the necessary capacity while also helping reduce b through faster access and egress. As noted in section 4.1, accordion buses should have four doors, to minimize loading time.

4.3. Reduce random variability

None of this discussion would matter if we were guaranteed that buses would run exactly on schedule. Of course, they don’t and we cannot get such guarantee. However, we can look for treatments that would make initial delays less common. All-door boarding is one such treatment, in addition to its effect on average boarding time per passenger, because one of the factors that can cause delays is an unexpected wave of passengers all getting on and having to queue one at a time, for example if they come off class or transfer from a full train. Schools with synchronized class times can overload transit networks for the first few minutes after classes end. And in Shanghai, I had to wait 20 minute to buy a metro ticket coming off a full intercity train since two of the three ticket machines were broken and the train was full of visitors who didn’t already have the Shanghai Public Transportation Card.

But as Eric’s interviews with drivers suggest, the biggest single source of variability is traffic and not unusual passenger loads. Bus lanes reduce the impact of traffic, but may not reduce variability, since cars may block them unexpectedly. This suggests that better enforcement of bus lanes could improve schedule regularity and reduce bunching further downstream.

Another source of variability is traffic lights. Traffic lights are discrete: they’re red or green, and a bus that misses the light will be delayed by a full phase, which in New York means about 45 seconds (and in Bangkok means 3 or even 5 minutes). Giving buses signal priority even in its weakest form entails lengthening the green phase in the direction the bus travels in for a few seconds to let a bus through and avoid making it wait a full red phase. This would keep a lid on the maximum variability that a single intersection can produce. Note also that it’s very easy for a bus to be delayed at two successive intersections, for example if traffic is such that it’s a hair slower than usual, forcing it to miss two lights in rapid succession. In this case, 1.5-minute initial delays are routine, setting up bunching later.

4.4. Dispatch buses to maintain even headways at terminals

The most brutal way to eliminate bunching is to have dispatchers tell bus drivers to sit still for a few minutes if the bus either behind or ahead of them is too far behind. The subway in New York would do that to the trains to maintain something that to a manager at control center looked like even headways (“wait assessment”), which multiple independent sources have told me is responsible for falling subway speeds and increased delays. This brutal approach is unlikely to provide better service to riders.

However, telling bus drivers to sit still to maintain even headways has no such downside when it is done at the terminus of the bus route. At most, agencies would have to pay bus drivers some overtime, which is probably swamped by the positive effect reducing bunching has on ridership (or for that matter the fact that reducing bunching permits the transit agency to provide the same effective frequency with fewer service-hours).

4.5. More empirical research is needed

This section is a lot less quantitative than sections 1, 2, and 3, owing to the fact that we are stepping away from strict modeling. While quantifying the effect of low floors, more doors, bigger buses, and prepayment is easy within the confines of the model, quantifying the initial shock to ridership discussed in subsection 4.3 is more difficult. There is a range of plausible shocks, and the serious questions to ask are along the lines of “what is the 90% confidence interval of the travel time on each segment?”.

The literature review I’ve done for signal priority in particular is not comprehensive, but suggestive that there is no research yet in that direction. Figuring out exactly how common initial delays are and which treatments can reduce them by how much must be the next step.

Fare Payment Without the Stasi

Last year, I saw a tip by the Metropolitan Police: if you witness any crime on a London bus and wish to report it later, you should tell the police the number on your Oyster card and then they’ll already be able to use the number to track which bus you rode and then get the names and bank accounts of all other passengers on that bus. Londoners seem to accept this surveillance as a fact of life; closed-circuit TV cameras are everywhere, even in front of the house where Orwell lived and wrote. Across the Pond, transit agencies salivate over the ability to track passenger movements through smartcards and contactless credit cards, which is framed either as the need for data or as a nebulous anti-crime measure. Fortunately, free countries have some alternative models.

In Germany, the population is more concerned about privacy. Despite being targeted by a string of communist terrorist attacks in the 1970s and 80s, it maintained an open system, without any faregates at any train station (including subways); fare enforcement in German cities relies on proof of payment with roving inspectors. Ultimately, this indicates the first step in a transit fare payment system that ensures people pay their fares without turning the payment cards into tracking devices. While Germany resists contactless payment, there are ways to achieve its positive features even with the use of more modern technology than paper tickets.

The desired features

A transit fare payment system should have all of the following features:

  1. Integration: free transfers between different transit vehicles and different modes should be built into the system, including buses, urban rail, and regional rail.
  2. Scalability: the system should scale to large metro areas with variable fares, and not just to compact cities with flat fares, which are easier to implement. It should also permit peak surcharges if the transit agency wishes to implement them.
  3. No vendor lock: switching to a different equipment manufacturer should be easy, without locking to favored contractors.
  4. Security: it should be difficult to forge a ticket.
  5. Privacy: it should not be possible to use the tickets to track passengers in most circumstances.
  6. Hospitality: visitors and occasional riders should be able to use the system with ease, with flexible options for stored value (including easy top-up options) and daily, weekly, and monthly passes, and no excessive surcharges.

Smartcard and magnetic card systems are very easy to integrate across operators; all that it takes is political will, or else there may be integrated fare media without integrated fares themselves, as in the Bay Area (Clipper can store value but there are no free transfers between agencies). Scalability is easy on the level of software; the hardest part about it is that if there are faregates then every station must have entry and exit gates, and those may be hard to retrofit. Existing smartcard technologies vary in vendor lock, but the system the US and Britain are standardizing on, contactless credit cards, is open. The real problem is in protecting privacy, which is simply not a goal in tracking-obsessed Anglo-American agencies.

The need for hospitality

Hospitality may seem like a trivial concern, but it is important in places with many visitors, which large transit cities are. Moreover, universal design for hospitality, such as easily recognizable locations for topping up stored value, is also of use to regular riders who run out of money and need to top up. Making it easy to buy tickets without a local bank account is of use to both visitors and low-income locals without full-service bank accounts. In the US, 7% of households are unbanked and another 20% are underbanked; I have no statistics for other countries, but in Sweden banks will not even give debit cards to people with outstanding debts, which suggests to me that some low-income Swedes may not have active banking cards.

New York’s MetroCard has many faults, but it succeeds on hospitality better than any other farecard system I know of: it is easy to get the cards from machines, there is only a $1 surcharge per card, and season tickets are for 7 or 30 days from activation rather than a calendar week or month. At the other end of the hospitality scale, Navigo requires users to bring a passport photo and can only load weekly and monthly passes (both on the calendar); flexible 5-day passes cost more than a calendar weekly pass.

In fact, the main reason not to use paper tickets is that hospitality is difficult with monthly passes printed on paper. Before the Compass Card debacle, Vancouver had paper tickets with calendar monthly passes, each in a different color to make it easy for the driver to see if a passenger was flashing a current or expired pass. The tickets could be purchased at pharmacies and convenience stores but not at SkyTrain stations, which only sold single-ride tickets.

ID cards and privacy

The Anglosphere resists ID cards. The Blair cabinet’s attempt to introduce national ID cards was a flop, and the Britons I was reading at the time (such as the Yorkshire Ranter) were livid. And yet, ID cards provide security and privacy. Passports are extremely difficult to forge. Israel’s internal ID cards are quite difficult to forge as well; there are occasional concerns about voter fraud, but nothing like the routine use of fake drivers’ licenses to buy drinks so common in American college culture.

At the same time, in countries that are not ruled by people who think 1984 was an uplifting look at the future, ID cards protect privacy. The Yorkshire Ranter is talking about the evils of biometric databases, and Israeli civil liberties advocates have mounted the same attack against the government’s attempt at a database. But German passports, while biometric, store data exclusively on the passport, not in any centralized database. ID cards designed around proving that you paid your fare don’t even have to use biometrics; the security level is lower than with biometrics, but the failure mode is that the occasional forger can ride without paying $100 a month (which is much less than the cost of the forgery), not that a ring of terrorists can enter the country.

Navigo’s ID cards are not hospitable, but allowing passengers to ride with any valid state-issued ID would be. Visitors either came in from another country and therefore have passports, drove in and therefore have drivers’ licenses, or flew in domestically and therefore still have ID cards. Traveling between cities without ID is still possible here and in other free European countries, but everyone has national ID cards anyway; the ID problem is mainly in the US with its low passport penetration (and secondarily Canada and Australia), and the US has no intercity public transit network to speak of outside the Northeast Corridor.

What this means is that the best way to prevent duplication of transit passes is to require ID cards. Any ID card must be acceptable, including a passport (best option), a national ID card (second best), or an American driver’s license (worst).

Getting rid of the faregates

There are approximately three first-world Western cities that have any business having faregates on their urban rail networks: London, Paris, New York. Even there, I am skeptical that the faregates are truly necessary. The Metro’s crowd control during the World Cup victory celebration was not great. New York’s faregates sometimes cause backups to the point that passengers just push the emergency doors open to exit, and then rely on an informal honor system so that passengers don’t use the open emergency doors to sneak in without payment.

Evidently, the Munich S-Bahn funnels all traffic through a single two-track city center tunnel and has 840,000 weekday users, without faregates. Only one or two trunk lines are busier in Paris, the RER A with about a million, and possibly the RER B and D if one considers them part of the same trunk (they share a tunnel but no platforms); in London, only the Central, Victoria, and Jubilee lines are busier, none by very much; in New York, none of the two-track trunks is as busy. Only the overcrowded lines in Tokyo (and a handful in Osaka, Beijing, and Shanghai) are clearly so busy that barrier-free proof-of-payment fare enforcement is infeasible.

The main reason not to use faregates is that they are maintenance-intensive and interfere with free passenger flow. But they also require passengers to insert fare media, such as a paper ticket or a contactless card, at every station. With contactless cards the system goes well beyond exact numbers of users by station, which can be obtained with good accuracy even on barrier-free systems like Transilien using occasional counts, and can track individual users’ movements. This is especially bad on systems that do not have flat fares (because then passengers tag on and off) and on systems that involve transferring with buses or regional trains and not just the subway (because then passengers have to tag on and off at the transfer points too).

Best industry practice here is then barrier-free systems. To discourage fare evasion, the agency should set up regular inspections (on moving vehicles, with unarmed civilian inspectors), but at the same time incentivize season passes. Season passes are also good for individual privacy, since all the system registers is that the passenger loaded up a monthly pass at a certain point, but beyond that can’t track where the passenger goes. All cities that have faregates except for the largest few should get rid of them and institute POP, no matter the politics.

Tickets and ID cards

In theory, the ID card can literally be the ticket. The system can store in a central database that Alon Levy, passport number [redacted], loaded a monthly pass valid for all of Ile-de-France on 2018-08-16, and the inspector can verify it by swiping my machine-readable passport. But in practice, this requires making sure the ticket machine or validator can instantly communicate this to all roving fare inspectors.

An alternative approach is to combine paper tickets with ID cards. The paper ticket would just say “I am Alon Levy, passport number [redacted], and I have a pass valid for all of Ile-de-France until 2018-09-14,” digitally signed with the code of the machine where I validated the ticket. This machine could even be a home printer, via online purchase, or a QR code displayed on a phone. Designing such a system to be cryptographically secure is easy; the real problem is preventing duplication, which is where the ID card comes into play. Without an ID card, it’s still possible to prevent duplication, but only via a cumbersome system requiring the passenger to validate the ticket again on every vehicle (perhaps even every rail car) when getting on or off.

The same system could handle stored value. However, without printing a new ticket every time a passenger validates, which would be cumbersome, it would have to fall back on communication between the validator and the handheld readers used by the inspectors. But fortunately, such communication need not be instant. Since passengers prepay with stored value, the ticket itself, saying “I am Alon Levy, passport number [redacted], and I loaded 10 trips,” is already valid, and the only communication required is when passengers run out of money; moreover, single-use tickets have a validity period of 1-2 hours, so any validator-to-inspector communication lag time of less than the validity period will be enough to ensure not to validate expired tickets. The same system can also be used to have a daily cap as in Oyster, peak surcharges, and even generally-undesirable station-to-station rather than zonal fares.

It’s even possible to design a system without single-use tickets at all. Zurich comes close, in that a 24-hour pass costs twice as much as a single-use ticket (valid for just an hour), so passengers never have any reason to get a single-use ticket. In this system there would not be any stored value, just passes for a day or more, valid in prescribed zones, with printable tickets if regular riders in one zone occasionally travel elsewhere.

The upshot here is that advanced technology is only required for printing and reading QR codes. The machines do not need to be any more complicated than ATMs or Bitcoin ATMs (insert money, receive a Bitcoin slip of paper); I don’t know how much Bitcoin ATMs cost, but regular ATMs are typically $2,000-3,000, and the most expensive are $8,000, unlike the $75,000 ticket machines used at New York SBS stations. The moving parts are software and not hardware, and can use multi-vendor cryptographic protocols. In effect, the difficult part of verifying that there is no duplication or forgery is offloaded to the state ID system.

Bus Stop Spacing and Network Legibility

I had an interesting interview of the annoying kind, that is, one where my source says something that ends up challenging me to the point of requiring me to rethink how I conceive of transportation networks. On the surface, the interview reaffirmed my priors: my source, a mobility-limited New Yorker, prefers public transit to cars and is fine with walking 500 meters to a bus stop. But one thing my source said made me have to think a lot more carefully about transit network legibility. At hand was the question of where buses should stop. Ages ago, Jarrett suggested that all other things equal (which they never are), the best stop spacing pattern is as follows:

The bus stops on the north-side arterials are offset in order to slightly improve coverage. The reason Jarrett cites this doesn’t occur much in practice is that there would also be east-west arterials. But maybe there aren’t a lot of east-west arterials, or maybe the route spacing is such that there are big gaps between major intersections in which there’s choice about which streets to serve. What to do then? My source complained specifically about unintuitive decisions about which streets get a bus stop, forcing longer walks.

In the case of the most important streets, it’s easy enough to declare that they should get stops. In Brooklyn, this means subway stations (whenever possible), intersecting bus routes, and important throughfares like Eastern Parkway or Flatbush. Right now the B44 Select Bus Service on Nostrand misses Eastern Parkway (and thus the connection to the 3 train) and the M15 SBS on First and Second Avenues misses 72nd Street (and thus the southernmost connection to Second Avenue Subway). However, there is a bigger question at hand, regarding network legibility.

Bus networks are large. Brooklyn’s current bus network is 550 km, and even my and Eric Goldwyn’s plan only shrinks it to about 340, still hefty enough that nobody can be expected to memorize it. Passengers will need to know where they can get on a stop. For the sake of network legibility, it’s useful to serve consistent locations whenever possible.

This is equally true of sufficiently large subway networks. Manhattan subway riders know that the north-south subway lines all have stops in the vicinity of 50th Street, even though the street itself isn’t especially important, unlike 42nd or 34th. In retrospect, it would have been better to have every line actually stop at 50th, and not at 49th or 51st, but the similarity is still better than if some line (say) stopped at 47th and 54th on its way between 42nd and 59th. A bad Manhattan example would be the stop spacing on the 6 on the Upper East Side, serving 68th and 77th Streets but not the better-known (and more important) 72nd and 79th.

There are similar examples of parallel subway lines, some stopping on consistent streets, and some not. There are some smaller North American examples, i.e. Toronto and Chicago, but by far the largest subway network in the world in a gridded city is that of Beijing. There, subway stops near city center are forced by transfer locations (Beijing currently has only one missed connection, though several more are planned), but in between transfers, they tend to stop on consistent streets when those streets are continuous on the grid.

But outside huge cities (or cities with especially strong grids like Chicago, Philadelphia, and Toronto), consistent streets are mostly a desirable feature for buses, not subways. Bus networks are larger and less radial, so legibility is more important there than on subways. Buses also have shorter stop spacing than subways, so people can’t just memorize the locations of some neighborhood centers with subway stops (“Nation,” “Porte de Vincennes,” etc.).

In the other direction, in cities without strong grids, streets are usually not very long, and the few streets that are long (e.g. Massachusetts Avenue in Boston) tend to be so important that every transit route intersecting them should have a stop. However, streets that are of moderate length, enough to intersect several bus lines, are common even in interrupted grids like Brooklyn’s or ungridded cities like Paris (but in London they’re rarer). Here is the Paris bus map: look at the one-way pair in the center on Rue Reaumur and Boulevard Saint-Denis (and look at how the northbound bus on Boulevard de Strasbourg doesn’t stop at Saint-Denis, missing a Metro transfer). There are a number of streets that could form consistent stops, helping make the Parisian bus network more legible than it currently is.

As with all other aspects of legibility, the main benefits accrue to occasional users and to regular riders who unfamiliar with one particular line or region. For these riders, knowing how to look for a bus stop (or subway station, in a handful of large cities) is paramount; it enables more spontaneous trips, without requiring constantly consulting maps. These occasional spontaneous trips, in turn, are likelier to happen outside the usual hours, making them especially profitable for the transit agency, since they reduce rather than raise the peak-to-base ratio. (Bus operating costs mostly scale with service-hours, but very peaky buses tend to require a lot of deadheading because they almost never begin or end their trip at a bus depot.)

The main takeaway from this is that bus network redesigns should aim to stop buses on parallel routes at consistent streets whenever possible, subject to other constraints including regular stop spacing, serving commercial nodes, and providing connections to the rail network. To the extent cities build multiple parallel subway lines, it’s useful to ensure they serve stations on consistent streets as well when there’s a coherent grid; this may prove useful if New York ever builds a subway under Utica and extends the Nostrand Avenue Line, both of which extensions were on the drawing board as recently as the 1970s.

Radial Metro Networks for Portions of Cities

I’ve harped about the necessity of radial metro networks, looking much like the following schematic:

However, in practice such pure radial networks are rare. Some networks have parallel lines (such as Paris and Beijing), nearly all have lines intersecting without a transfer at least once (the largest that doesn’t is Mexico City), some have chordal lines and not just radial or circular lines, and nearly all have lines that meet twice. Often these variations from pure radii are the result of poor planning or a street network that makes a pure radial system infeasible, but there are specific situations in which it’s reasonable for lines to meet multiple times (or sometimes even be parallel). These come from the need to built an optimal network not just for the whole city but also notable portions of it.

The unsegmented city

The diagram depicted above is a city with a single center and no obvious sub-areas with large internal travel demand. If the city is on a river, it’s not obvious from the subway map where the river passes, and it’s unlikely its non-CBD bank has a strong identity like that of the Left Bank of Paris, South London, or Brooklyn and Queens.

Among the largest metro networks in the world, the one most akin to the diagram above is Moscow. It has seven radial lines through city center (numbered 1-10, omitting the one-sided 4, the circular 5, and the yet-incomplete 8). They have some missed connections between them (3/6, 3/7, 6/9), and one pair of near-parallel lines (2/10, meeting only at Line 10’s southern terminus), but no parallel lines, and no case in which two lines cross twice. And Moscow’s development is indeed oriented toward connecting outlying areas with city center. Connections between areas outside the center are supposed to use the circular lines (5 and 14, with 11 under construction).

In a relatively monocentric city, this is fine. Even if this city is on the river, which Moscow is, it doesn’t matter too much if two neighborhoods are on the same side of the river when planning the network. Even in polycentric cities, this is fine if the sub-centers get connections via circular lines or the odd chordal line (as will eventually happen when Los Angeles builds a real subway network with such chords as Vermont and Sepulveda).

The segmented city

London and Paris are both segmented by their rivers, and their wrong sides (South London, Left Bank) both have strong regional identities, as does to some extent East London. New York, partitioned into boroughs by much wider waterways than the Thames and Seine, has even stronger sub-identities, especially in Brooklyn. I do not know of a single New Yorker whose commute to work or school involves crossing a bridge over a river on foot, nor of any case of anyone crossing a bridge in New York on foot (or bike) except for recreational purposes; in Paris I do so habitually when visiting the Latin Quarter, and at a conference in 2010 another attendee biked from Porte de Vincennes to Jussieu every day.

With a difficult water boundary, the wrong-side part of the city became a center in its own right. Downtown Brooklyn and the Latin Quarter should both be viewed as sub-centers that failed to become CBDs. The Latin Quarter, the oldest part of Paris outside the Ile de la Cite, declined in favor of the more commercial Right Bank as the city grew in the High and Late Middle Ages; Downtown Brooklyn declined in favor of more concentration in Manhattan and more dispersion to other centers (often in Queens) over the course of the 20th century.

Early 20th century New York and Paris were not polycentric cities. There was no everywhere-to-everywhere demand. There was demand specifically for travel within Brooklyn and within the Left Bank. To this day, the connections to the Latin Quarter from Right Bank neighborhoods not on Line 4 are not great, and from Nation specifically the alternatives are a three-seat ride and a long interchange at Chatelet. Ultimately, this situation occurs when you have a region with a strong identity and strong demand for internal travel larger than a neighborhood (which can be served by a few subway stops on a single line) but smaller than an entire city.

In this case, a radial subway network (which neither the New York City Subway nor the Paris Metro is) could justifiably have multiple crossings between two lines, ensuring that lines provide a coherent network for internal travel. South London is a partial example of this principle: not counting the Wimbledon branch of the District line, the South London Underground network is internally connected, and the best route between any two South London stations stays within South London. In particular, the Victoria and Northern lines cross twice, once at Stockwell and once at Euston, in a city that has a generally radial metro system.

Don’t go overboard

The need to serve internal travel within portions of a city is real, and it’s worthwhile to plan metro networks accordingly. But at the same time, it’s easy to go overboard and plan lines that serve only travel within such portions. Most of the examples I give of weak chordal lines – the G train in New York, Line 10 and the RER C in Paris, Line 6 in Shanghai – serve internal demand to the wrong side of a city divided by a river; only Shanghai’s Line 3 is an exception to this pattern, as a weak chordal line that doesn’t come from city segmentation.

In the cases of the G and M10, the problem is partly that the lines have compromises weakening them as radials. The G has too many missed connections to radial lines, including the J/Z and the entire Atlantic-Pacific complex; M10 terminates at Austerlitz instead of extending east to the library, which is the second busiest Left Bank Metro stop (after Montparnasse) and which has a particularly strong connection to the universities in the Latin Quarter.

But Line 6 is constrained because it doesn’t serve Lujiazui, just Century Avenue, and the RER C does serve the library but has exceptionally poor connections to the CBD and other Right Bank destinations. It’s important to ensure the network is coherent enough to serve internal demand to a large segment of the city but also to serve travel demand to the rest of the city well.

Good transfers

Serving the entire city hinges on good transfers. The most important destination remains city center, so lines that aren’t circumferential should still aim to serve the center in nearly all cases. Internal demand should be served with strategic transfers, which may involve two lines crossing multiple times, once in or near city center and once on the wrong side of the river.

The main drawback of multiple crossings is that they are less efficient than a pure radial network with a single city center crossing between each pair of lines, provided the only distinguished part of the city is the center. Once internal travel to a geographic or demographic segment is taken into account, there are good reasons to slightly reduce the efficiency of the CBD-bound network if it drastically raises the efficiency of the secondary center-bound network. While demographic trends may come and go (will Flushing still be an unassimilated Chinese neighborhood in 50 years?), geographic constraints do not, and place identities like “Left Bank” and “Brooklyn” remain stable.

Note the qualifiers: since the CBD remains more important than any secondary center, it’s only acceptable to reduce CBD-bound efficiency if the gain in secondary center-bound efficiency is disproportionate. This is why I propose making sure there are good transfers within the particular portion of the city, even at the cost of making the radial network less perfect: this would still avoid missed connections, a far worse problem than having too many transfer points.

So what?

In New York, London, and Paris, the best that can be done is small tweaks. However, there exist smaller or less developed cities that can reshape their transit networks, and since cities tend to form on rivers and bays, segmentation is common. Boston has at least two distinguished wrong-side segments: East Boston (including Chelsea and Revere) and Greater Cambridge. East Boston can naturally funnel transit through Maverick, but in Greater Cambridge there will soon be two separate subway spines, the Red and Green Lines, and it would be worthwhile to drag a rail connection between them. This is why I support investing in rail on the Grand Junction, turning it into a low-radius circular regional rail line together with the North-South Rail Link: it would efficiently connect the Green Line Extension with Kendall.

More examples of segmented cities include the Bay Area (where the wrong-side segment is the East Bay), Istanbul (where Europe and Asia have separate metro networks, connected only by Marmaray), Stockholm (where Södermalm and Söderort are separated by a wide channel from the rest of the city, and Kungsholmen is also somewhat distinguished), and Washington (where the wrong side is Virginia). In all of these there are various compromises on metro network planning coming from the city segmentation. Stockholm’s solution – making both the Red and Green Lines serve Slussen – is by far the best, and the Bay Area could almost do the same if BART were connected slightly differently around Downtown Oakland. But in all cases, there are compromises.