Category: Urban Transit

Our Brooklyn Bus Redesign

Eric Goldwyn and I spent about six months working on a Brooklyn bus redesign. I mentioned some aspects of it before here, on social media, and in blog comments, but not the overall shape. Eric and I gave a pair of presentations about our plan, one two days ago at the MTA in front of senior MTA planners and NYC DOT people and one today at TransitCenter in front of activists and mid-level MTA planners. We have a still-unreleased writeup explaining everything we’re doing with references to both public reports from various cities and peer-reviewed literature. Here I’m going to condense the 8,000-word writeup into a blog post length, going over the main points, including of course the proposed map.

The map, in brief

The depicted version is 1.1. You can see a lower-resolution version 1.0 on Streetsblog, albeit with a different color code (the map we made for the presentation, reproduced on Streetsblog, uses red for the highest-frequency routes and blue for the lowest-frequency ones whereas the Google Earth version linked above is the opposite). It has 353 route-km, down from about 550 today, not including Grand and Metropolitan Avenues, which are Queens bus routes, shown on the map for completeness’s sake, without stopping pattern.

Some tails are cut due to low ridership or duplication of rail:

  • The B25 on Fulton goes.
  • The B37 on Third Avenue is consolidated into the B63 on Fifth.
  • The B45 and B65 are merged into one compromise route.
  • The B15 is cut east of the Long-Term Parking JFK AirTrain station (where service is free); ideally it would be cut east of City Line with passengers taking the subway to the AirTrain (as was the case in version 1.0), but I do not expect Port Authority to integrate AirTrain fares with the subway.
  • The B41 is cut north of Parkside Avenue, at the transfer to the B/Q.
  • Instead of two routes in Bed-Stuy between Nostrand (i.e. B44) and Malcolm X (i.e. B46), today’s B15 and B43, there’s just one route.
  • The B57 segment on Court and Smith Streets in South Brooklyn goes, as the subway serves the area in several directions.
  • The B39 over the Williamsburg Bridge goes.
  • The B32 and B62, providing north-south service through Williamsburg up to Long Island City, are merged into one compromise route.
  • The East New York bus network is circuitous (buses go to Gateway Center the long way around) and is straightened here.
  • In version 1.0, the B26 on Halsey was cut west of Franklin with a forced transfer to the subway, but the short distance to Downtown Brooklyn argues in favor of continuing to at least Flatbush.

Overall, this is a cut from 54 routes (including the separately-managed MTA Bus routes B100 and B103) to 37. The smaller network is far more frequent. The minimum frequency is,

  • Every 6 minutes between 6 am and 10 pm every day.
  • Every 10 minutes between 5 and 6 am and between 10 pm and midnight.
  • Every 30 minutes between midnight and 5 am; every 20 minutes with timed transfers to the subway is aspirational, but the subway doesn’t run reliably on a timetable overnight for such a system to be viable. The 30-minute night network could potentially involve mini-pulses in Downtown Brooklyn and smaller hubs (like East New York and Bay Ridge).

Routes depicted in red on the Google Maps link, or in blue on the map in the Streetsblog link, have exactly the minimum frequency. Routes depicted in green have higher frequency at the peak; routes depicted in blue on Google Maps or red on Streetsblog have higher frequency peak and off-peak. Higher frequency than the minimum is depicted as “Utica [2/4]” (buses on Utica run every 2 minutes peak, 4 off-peak) or “Avenue U [5/6]” (buses on Avenue U run every 5 minutes peak, 6 off-peak). Peak means 7-9 am and 5-7 pm on weekdays, in both directions; the morning peak is a little earlier and the afternoon peak a little later than the subway peak, but as buses are still mostly subway feeders, an earlier morning peak and a later afternoon peak are justifiable.

Speedup treatments

Pruning the network is not the only or even most important part of bus reform. Buses have to be sped up to be useful for people except as last-resort transit. In interviews about unrelated topics, people have volunteered to me that they do not take trips they used to take due to the degradation in bus speed and reliability. New York City Transit bus ridership peaked in 2002; the fare hike in 2003 led to a small dip in ridership that the mid-2000s oil crisis didn’t quite erase, and then in the recession and subsequent recovery bus ridership crashed. In Manhattan it’s 30% below the 2007 level; in Brooklyn it’s 20% below the 2007 level, with buses extending the subway or letting people connect to a better line (like the B41 and B35) particularly hit.

The current average speed in Brooklyn is about 11 km/h. Excluding limited-stop buses, it’s 10.8. We’re proposing to increase it to 15, even though the redesign is pruning buses in faster areas more than in slower ones. This is using four speedup treatments.


Today, New York prefers to treat off-board fare collection as a special product available only on select buses (i.e. SBS). This should be changed to citywide prepayment, with all-door boarding. German-speaking cities do it; so does San Francisco. Data from San Francisco and from the TRB (PDF-p. 20) suggests a gain of about 2.5-3 seconds per passenger boarding, counting both boarding and alighting time. At Brooklyn’s bus ridership level, this suggests a saving of around 400-450 revenue-hours, or about 4% of total service-hours. This is not a big change, but it helps stabilize the schedule by slowing down the mechanism by which buses bunch.

How to get passengers to pay if not on-board remains an open question; there are several approaches. The Zurich model involves placing a ticket-vending machine (TVM) at every bus stop. While New York severely pays for TVMs on SBS (the RPA says $75,000 per stop), an ATM costs $3,000, so installing the required infrastructure need not cost a lot. But more commonly, passengers can board freely if they have transfers or unlimited monthlies and pay the driver (potentially after the bus has begun moving) otherwise.

Of note, the bus drivers are particularly interested in prepayment. Eric and I explained the issue in a CityLab article a few months back: the drivers are worried about being assaulted by riders who don’t want to pay.

Stop consolidation

About 60% of the time saving in our plan relative to current practices comes from stop consolidation. I discussed the issue here, and our forthcoming report has references to many studies in the literature optimizing stop spacing for minimum door-to-door travel time. With each deleted stop saving 20-30 seconds (say 25 seconds on average), our proposed stop consolidation, from an average of 220 meters to 490 excluding long tails (i.e. the B15’s long nonstop segment toward JFK) saves around a minute per km, cutting travel time from 5.5 minutes per km to 4.5.

Conceptually, stop spacing should be longer when trips are longer, or when relative density is less uniform. New York City Transit bus trips are short, as many are subway extenders, but relative density is extremely spiky, as a large number of people get off at a few dominant stops at the subway connection points. If the on/off density on a route is uniform, then lengthening the stop spacing means passengers have to walk longer at both ends; but if passengers are guaranteed a connection at one end (because of transfer points with the subway or other buses) then they only have to walk longer at the other end. Based on this principle, Utica and Nostrand get particularly long stop spacing. Conversely, routes with extremely short trips, like the Mermaid route inherited from the B74, have shorter stop spacing.

To improve network legibility, we have tried as far as possible to have buses stop on consistent streets. For example, south of Fulton Street (where it’s awkwardly between Nostrand and Franklin), Bedford Avenue gets a stop on every intersecting bus, including east-west routes but also the diagonal B41.

Every bus stop should have shelter. In Central Florida, North Florida, and London, this costs $10,000 per stop, give or take. Our 707-stop plan (700 in version 1.0) would cost $14 million at this cost. Even at Santa Ana’s higher cost of $35,000, it’s $50 million. NIMBYs who oppose stop consolidation argue that having many stops is necessary for people with disabilities, but people with disabilities would benefit from benches and shelter, without needing to stand for 15 minutes waiting for bunched buses.

Bus lanes

Every bus in an area with congestion should get dedicated lanes. SBS implementations so far, imperfect as they are, have saved around 30 seconds per km in traffic. Physically-separated median lanes should do better; the MTA and NYCDOT have so far avoided them on the theory that local and limited bus routes should coexist on the same route and limiteds should pass locals, but in reality, a single stopping pattern is better, and then there are no drawbacks to physical separation.

On wide streets, this is not a problem. On narrow ones, it is. The real headache is Nostrand, about 25 meters wide building to building, enough for just four lanes. The correct thing to do is a moving lane and a bus lane in each direction, with merchants told to park on side streets. If parking is unavoidable, then a contraflow bus lane, with parking on one side, is also feasible, but less safe for pedestrians (Boulevard Saint-Michel has this configuration and has to remind pedestrians crossing the street to look left).

Two-way buses are essential whenever streets are widely separated, as on avenues, in Brooklyn as well as Manhattan. Nostrand is just more important than Rogers and New York Avenue, where northbound B44s go today; today’s configuration forces east-west buses to make too many stops (the B35 limited makes 4 stops in a kilometer).

Signal priority

Buses should get priority at intersections and not just on the street. The studies we’ve seen find a 4-7% gain, bus only on individual bus routes, not gridded networks. In our proposed trip times we are not assuming any speedup from signal priority, just better timekeeping as more delayed buses get priority to stabilize the schedule. This is a counter-bunching mechanism more than a straight speedup.

A process, not an immutable product

Jarrett Walker’s bus network redesigns tend to come as complete products, changed rapidly from radial low-frequency networks. What we’re proposing is a longer process. Nova Xarxa began implementation in 2012 and is wrapping up now, installing a few routes at a time by cannibalizing parallel routes. The map we’re showing is what we estimate would be a good fit for 2022-3. Beyond that, more subway stops are going to be wheelchair-accessible, making it easier to prune more subway-parallel buses (like the B63).

Gradual implementation means starting from the easier parts of the network. East New York’s current network is so circuitous that straightening it should not be too controversial. Our proposed redesign there is also better at connecting to the 2, 3, 4, and 5 trains and not just the L, which should prove valuable during the L shutdown. In Southern Brooklyn, we are proposing more service, but this could be paired with stop consolidation. Central Brooklyn and Bed-Stuy require the most street redesigns and the most robust frequency network-wide (as they are already transfer-based grids, and nobody transfers at 12-15 minute off-peak frequency) and could be done later; the B25 itself should probably not be eliminated until Broadway Junction is made accessible on the A and C lines.

We are not even wedded to the map as a proposal for 2022. Some variations are always possible, as already seen in the differences between versions 1.0 and 1.1. The biggest addition we can think of is adding a second north-south route through Bed-Stuy: the existing one would be moved from Marcus Garvey to Throop (hitting the subway better), while the B17 could be extended up Troy and Lewis.

Overall, Brooklyn has 10,800 service-hours today. Our redesign uses just 10,000, with a 1% gain in efficiency from location relative to bus depots on top of that. There is room for service increases, or restoration of marginal routes required for political reasons, or slowdowns imposed by political unwillingness to install bus lanes.


In a modern developed country, it’s rare to find win-win situations. The US is blessed with these in transit (i.e. it’s so inefficient at construction it might as well be third-world), but not in urban bus networks. Stop consolidation is a net benefit to the average user of the route, but a few people would still see longer trips, e.g. those living at the exact midpoint between two widely-spaced stops. Route consolidation (as in Ocean Hill) is the same thing.

There are sociopolitical groups that would win out: labor would see higher ridership, reducing the pressure to cut jobs; regular commuters (who generally have low transfer penalties) would see faster trips; people with disabilities that make it difficult for them to stand (as is true of some people with chronic pain) would be able to sit at bus stops and wouldn’t need to sit for long. In contrast, small business owners would sometimes lose the ability to park in front of their stores, and occasional users who usually drive would see longer perceived trips because of stiff transfer penalties.

This is equally true on the level of neighborhoods. Southern Brooklyn generally gains, and Borough Park in general gains an extra north-south route (though this is canceled out by high transfer and access penalty among Haredis: in Israel they just won’t walk longer to better service). East New York sees much more direct routes. Flatbush and East Flatbush don’t see much change in network structure but do gain off-peak frequency. Red Hook gains a direct connection to Manhattan. But then Bed-Stuy loses north-south routes, South Brooklyn’s buses are completely gutted, and Williamsburg loses north-south routes.

A political system based on citywide (or nationwide) ideological groups could find the will to build the network we’re proposing or something like it. Could a system based on local representation, treating retirees and small business owners as a vanguard class, deliver the same? We will see in the next year or two.

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.

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.

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.

Guidelines for Driverless Buses

As I’ve said a few months ago in The American Prospect, driverless bus technology does not yet appear ready for mass deployment. However, research into this technology continues. Of particular note is Google’s work at Waymo, which a source within the Bay Area’s artificial intelligence community tells me is more advanced and more serious than the flops at Uber and Tesla; Waymo’s current technology is pretty good on a well-understood closed route, but requires laborious mapping work to extend to new routes, making it especially interesting for fixed-route buses rather than cars. But ultimately, automated vehicles will almost certainly eventually be mature and safe, so it is useful to plan around them. For this, I propose the following dos and don’ts for cities and transit agencies.

Install dedicated, physically-separated bus lanes

A bus with 40 people should get 40 times the priority of a car with one person, so this guideline should be adopted today already. However, it’s especially important with AVs, because it reduces the friction between AV buses and regular cars, which is where the accident in Las Vegas reference in my TAP article happened. The CityMobil2 paradigm involves AVs in increasingly shared traffic, starting from fully enclosed circuits (like the first line in Helsinki, at the zoo) and building up gradually toward full lane sharing. Dedicated lanes are a lower level of sharing than mixed traffic, and physical separation reduces the ability of cars to cut ahead of the bus.

If there is a mixture of AV and manual buses, both should be allowed in the dedicated lanes. This is because bus drivers can be trained to know how to deal with AVs. Part of the problem with AVs in mixed traffic is that human drivers are used to getting certain cues from other human drivers, and then when facing robot drivers they don’t have these cues and misread the car’s intentions. But professional drivers can be trained better. Professional bus drivers are also familiar with their own bus system and will therefore know when the AV is going to turn, make stops, and so on.

Use Kassel curbs to provide wheelchair accessibility

Buses are at a disadvantage compared with trams in wheelchair accessibility. Buses sway too much to have the precise alignment that permits narrow enough gaps for barrier-free access on trains. However, as a solution, some German cities have reconstructed the edges of the bus lane next to the bus stop platform, in order to ease the wheels into a position supporting step-free access on low-floor buses. Potentially, AVs could make this easier by driving more precisely or by having platform extenders similar to those of some regional trains (such as those of Zurich) bridging the remainder of the horizontal gap.

Driverless trains in Vancouver and even on Paris Metro Line 14 have roll-on wheelchair access: passengers in wheelchairs can board the train unassisted. In contrast, older manually-driven trains tend to tolerate large horizontal and vertical gaps blocking passengers in wheelchairs, to the point that New York has to have some special boarding zones for wheelchairs even at accessible stations. If the combination of precision driving and Kassel curbs succeeds in creating the same accessibility on a bus as on SkyTrain in Vancouver, then the bus driver’s biggest role outside of actually driving the bus is no longer necessary, facilitating full automation.

Don’t outsource planning to tech firms

Transit networks work best when they work in tandem. This means full fare and schedule integration within and across different modes, and coordinated planning. Expertise in maintaining such networks lies within the transit agencies themselves as well as with various independent consultancies that specialize in transportation.

In contrast, tech firms have little expertise in this direction. They prefer competition to cooperation, so that there would be separate fleets within each city by company – and moreover, each company would have an incentive to arrange schedules so that buses would arrive just ahead of the other companies to poach passengers, so there wouldn’t be even headways. The culture of tech involves brazen indifference to domain expertise and a preference for reinventing the wheel, hence Uber and Chariot’s slow realization that no, really, fixed-route buses are the most efficient way of carrying passengers on the street in dense cities. Thus, outsourcing planning is likely to lead to both ruinous competition and retarded adoption of best practices. To prevent this, cities should ban private operations competing with their public bus networks and instead run their own AVs.

Most of the world’s richest cities have deep pools of tech workers, especially the single richest, San Francisco. It would be best for Muni, RATP, NYCT, and other rich-city agencies to hire tech talent using the same methods of the private sector, and train them in transit network planning so that they can assist in providing software services to the transit system in-house.

Resist the siren song of attendants

Las Vegas’s trial run involved an attendant on each bus performing customer service and helping passengers in wheelchairs. A bus that has an attendant is no more a driverless bus than a subway with computer-controlled driving and an operator opening and closing doors is a driverless train. The attendant’s work is similar to that of a bus driver. If the hope of some private operators is that relabeling the driver as an attendant will allow them to de-skill the work and hire low-pay, non-union employees, then it’s based on a misunderstanding of labor relations: transit employees are a prime target for unionization no matter whether they are called drivers.

Ultimately, the difficulty of driving a bus is not much greater than that of dealing with annoying customers, being on guard in case passengers act aggressively or antisocially, and operating wheelchair lifts. Bus drivers get back pain at high rates since they’re at the wheel of a large vehicle designed for passenger comfort for many hours a day, but this may still be a problem on AVs, and all other concerns of bus drivers (such as the risk of assault by customers) remain true for attendants. Either get everything right to the point of not needing any employee on the bus, or keep manual driving with just some computer assistance.

Resist the siren song of small vehicles

All AV bus experiments I know of (which I know for a fact is not all AV buses that are trialing) involve van-size vehicles. The idea is that, since about 75% of the cost of running a bus today is the driver’s wage, there’s no real point in running smaller vehicles at greater frequency if there’s a driver, but once the driver is removed, it’s easy enough to run small vehicles to match passenger demand and reduce fuel consumption.

However, vans have two problems. First, they only work on thin routes. Thick routes have demand for articulated buses running at high frequency, and then vans both add congestion to the bus lane and increase fuel consumption (when the vehicles are full, bigger is always more fuel-efficient). And second, they lead to safety problems, as passengers may be afraid of riding a bus alone with 3-4 other passengers but not with 20 or more (Martha Lauren rides full London buses fearlessly but would make sure to sit near the driver on nearly-empty Baltimore buses).

Medium-size buses, in the range of 20-30 seats, could be more useful on thin routes. However, passenger safety problems are likely to remain if only a handful of people ride each vehicle.

Get your maintenance costs under control

If you remove the driver, the dominant factor in bus operating costs becomes maintenance. Assuming maintenance workers make the same average annual wage and get the same benefits as transit workers in general, the wages of maintenance workers are about 15% of the total operating costs of buses in Chicago and 20% in New York.

The importance of fuel economy grows as well, but fuel today is a much smaller proportion of costs. Around 3% in Chicago and 2% in New York. European fuel costs are much higher than Americans, but so are European bus fuel economy rates: in tests, Boris buses got 4.1 km per liter of diesel, which is maybe twice as good as the US average and three times as good as the New York average.

This suggests that with the driver gone, maybe 75% of the remaining variable operating cost is maintenance. Chicago does better than New York here, since it replaces 1/12 of its fleet every year, so every year 1/12 of the fleet undergoes mid-life refurbishment and work is consistent from year to year, whereas in New York the replacement schedule is haphazard and there is more variation in work needs and thus more idle time. The most important future need for AV procurement is not electric traction or small size, but low lifecycle costs.

Update: by the same token, it’s important to keep a lid on vehicle procurement costs. New York spends $500,000 on a standard-length bus and $750,000 on an articulated bus; the Boris buses, which are bilevel and similar in capacity to an artic, cost about $500,000, which is locally considered high, and conventional artic or bilevel buses in London cost $300,000-350,000. American cities replace buses every 12 years, compared with every 15 years in Canada, and the depreciation in New York is around 6% of total bus operating costs. Cutting bus procurement costs to London levels would only save New York a small percent of its cost, but in an AV future the saving would represent around 12% of variable costs.

Plan for higher frequency

AVs represent an opportunity to reduce marginal operating costs. This means transit agencies should plan accordingly:

  • Lower marginal costs encourage running buses more intensively, running almost as much service off-peak and on weekends as at rush hour.
  • Very high frequency encourages passengers to transfer more, so the value of one-seat rides decreases.
  • Higher frequency always increases capacity, but its value to passengers in terms of reduced wait times is higher when the starting frequency is low, which means agencies should plan on running more service on less frequent routes and only add service on routes that already run every 5 minutes or less if the buses are overcrowded.

The Role of Local Expertise in Construction Costs

When I first looked at construction costs, I looked exclusively at developed countries. Eventually I realized that the difference in average costs between rich and poor countries is small. But then I noticed a different pattern in the third world: some places, like India, Bangladesh, Nigeria, and Indonesia, spend much more than China does. Why is that? While I’ve had a bunch of different explanations over the years, I believe today that the difference concerns local expertise versus reliance on first-world consultants.

The facts, as far as I can tell, are as follows:

  • Construction costs in China are about $250 million per km, a little more than the average for Continental Europe.
  • Construction costs in post-communist Europe are all over, but are the same range as in Western Europe. Bulgaria is pretty cheap; in this post I bring up a line that costs around $200 million/km in today’s money but other extensions built this decade are cheaper, including one outer one at $50 million/km. In contrast, Warsaw’s Line 2 is quite expensive.
  • Latin American construction costs have the same range as Europe, but it seems more compressed – I can’t find either $50 million/km lines or $500 million/km ones.
  • Africa and the parts of Asia that used to be colonies have high construction costs: India and Egypt are expensive, and here I give two expensive examples from Bangladesh and Indonesia. The Lagos Metro is spending subway money on an el in the middle of a wide road and is reminiscent of American costs.
  • When the first world had comparable income levels to those of the third world today, in the early 20th century, its construction costs were far lower, around $30-50 million per underground km. First-world cost growth in the last 100 years has mostly tracked income growth – it’s been somewhat faster in New York and somewhat slower in Paris, but on average it’s been similar.

For a while, I had to contend with the possibility that Chinese autocracy is just better at infrastructure than Indian (or Bangladeshi, or Indonesian, or Nigerian) democracy. The nepotism and corruption in India are globally infamous, and it’s still well-governed compared with Indonesia and Nigeria, which have personality-based politics. But then, in the developed world, authoritarian states aren’t more efficient at construction (Singapore’s construction costs are high); moreover, post-communist democracies like Bulgaria and Romania manage low construction costs.

What I instead think the issue is is where the state’s infrastructure planning comes from. China learned from the USSR and subsequently added a lot of domestic content (such as the use of cut-and-cover in some situations) fitting its particular needs; as a result, its construction costs are reasonable. The post-communist world learned from the USSR in general. There’s a wide range, with Romania near one end and Poland near the other, but the range is comparable to that of Western Europe today. Overall it seems that Eastern Europe can competently execute methods geared to the middle-income world (as the second world was in the Cold War) as well as, thanks to assistance from the EU, the high-income world.

Latin America, too, uses domestically-developed methods. The entire region is infamous in the economic development literature for having begun an inward economic turn in the Great Depression, cutting itself off from global markets and generally stagnating. Government functions are likewise done domestically or maybe outsourced to domestic contractors (and if international ones are involved, it’s in construction, not planning). Evidently, Latin America developed bus rapid transit, a mode of transportation optimally designed for countries with low incomes (so paying armies of bus drivers is cheaper than building rail tracks) and relatively strong currencies (so importing buses from richer countries isn’t ruinously expensive).

The situation in the ex-colonies is completely different. Even relatively protectionist ones outsource much of their planning to the developed world or increasingly to China, out of a combination of cultural cringe and shortage of domestic capital. The metro lines I have data for in India, Bangladesh, and Indonesia all involve Japanese technology and planning, with no attempt to adapt the technology to local conditions. So insistent is Japan on following its domestic recipe exactly that India’s high-speed rail construction is using standard gauge rather than broad gauge and Shinaknsen-size trains rather than larger Indian trains (which are 3.7 meters wide and can fit people 6-abreast). Elsewhere, China contributes capital and planning as part of the Belt and Road Initiative, and then its methods are geared toward middle income and not low income.

The correct way for countries in the per capita income range of Nigeria, India, and Bangladesh to build subways is to open up their main roads, which are often very wide, and put in four tracks in a cut-and-cover scheme similar to that of early-20th century New York. If they can elevate the tracks instead, they should use the same methods used to build Lines 2 and 6 in Paris in the early 20th century, which use concrete columns and are quiet enough that, unlike in New York, people can carry a conversation under the viaduct while a train passes. If the line needs to deviate from roads, then the city should buy property and carve up a new street (as New York did with Seventh Avenue South and Sixth Avenue in the Village) or else learn to implement late Victorian and Edwardian London’s techniques of deep boring.

However, actually implementing Belle Epoque construction methods requires particular knowledge that international consultants don’t have. Most of these consultants’ income comes from the first world, where wages are so high that the optimal construction methods involve extensive automation, using machinery rather than battalions of navvies with shovels. The technical support required for a tunnel boring machine is relatively easy in a rich country with a deep pool of qualified engineers and mechanics and a nightmare in a poor one where all such expertise has to be imported or trained from scratch. Thus, the consultants are likely to recommend the first-world methods they are familiar with, and if they do try to adapt to low wages, they may make mistakes since they have to reinvent ideas or read historical sources (which they are typically not trained to do – they’re consultants, not historians).

The result is that even though open economies tend to grow faster overall, economies with a history of closure tend to do better on this specific topic, where international consultants are not very useful for the needs of the developing world. India in particular needs to get better at indigenizing its construction and avoid mindlessly copying the first world out of cultural cringe, because even though it is almost a middle-income country by now, its wages remain a fraction of those of North America, Western Europe, and Japan, and its future growth trajectory is very different, requiring extensive adaptations. Both the overall extent of planning and the specific construction methods must be tailored to local conditions, and so far India seems bad at both (hence the undersized, expensive high-speed trains).

The Formula for Frequent Transit Networks

As I’m working on refining a concrete map for Brooklyn buses, I’m implementing the following formula:

Daily service hours * average speed per hour = daily frequencies * network length

In this post I’m going to go over what this formula really means and where it is relevant.

Operating costs

The left-hand side represents costs. The operating costs of buses are proportional to time, not distance. A few independent American industry sources state that about 75-80% of the cost of bus service is the driver’s wage; these include Jarrett Walker as well as a look at the payrolls in Chicago. The remaining costs are fuel, which in a congested city tracks time more than distance (because if buses run slow it’s because of stop-and-go traffic and idling at stops or red lights), and maintenance, which tracks a combination of time and distance because acceleration and braking cycles stress the engine.

This means that the number of service hours is fixed as part of the budget. My understanding is that the number in Brooklyn is 10,000 per weekday. I have seen five different sources about bus speeds and service provision in New York (or Brooklyn) and each disagrees with the others; the range of hours is between 9,500 and 12,500 depending on source, and the range of average speeds is between 9.7 km/h (imputed from the NTD and TransitCenter’s API) and 11 km/h (taken from schedules). The speed and hours figures are not inversely correlated, so some sources believe there are more service-km than others.

On a rail network, the same formula applies but the left-hand side should directly include service-kilometers, since rail operating costs (such as maintenance and energy) are much more distance- than time-dependent; only the driver’s wage is time-dependent, and the driver’s wage is a small share of the variable costs of rail operations.

Creating more service

Note that on a bus network, the implication of the formula is that higher speed is equivalent to more service-hours. My current belief, based on the higher numbers taken from schedules, is that 14 km/h is a realistic average speed for a reformed bus network: it’s somewhat lower than the average scheduled speed of the B44 SBS and somewhat higher than that of the B46 SBS, and overall the network should have somewhat denser stop spacing than SBS but also higher-quality bus lanes canceling out with it. The problem is that it’s not clear that SBS actually averages 14 km/h; my other sources for these two routes are in the 12-13 km/h range, and I don’t yet know what is correct. This is on top of the fact that faster transit attracts more paying riders.

Another way to create more service is to reduce deadheading and turnaround times. This is difficult. Bus depots are not sited based on optimal service. They are land-intensive and polluting and end up in the geographic and socioeconomic fringes of the city. The largest bus depot in New York (named after TWU founder Mike Quill) is in Hudson Yards, but predates the redevelopment of the area. In Brooklyn the largest depots appear to be East New York (more or less the poorest neighborhood in the city) and Jackie Gleason (sandwiched between a subway railyard and a cemetery). Figuring out how to route the buses in a way that lets them begin or end near a depot so as to reduce deadheading is not an easy task, but can squeeze more revenue-hours out of an operating cost formula that is really about total hours including turnaround time and non-revenue moves.

Service provision

The right-hand side of the equation describes how much service is provided. The network length is just the combined length of all routes. Daily frequency is measured in the average number of trips per day, which is not an easily understandable metric, so it’s better to convert it to actual frequencies:

Frequency Daily trips
15 minutes 6 am-9 pm, 30 minutes otherwise 5-1 am 70
15 minutes 24/7 96
5 minutes 7-9 am, 5-7 pm, 10 minutes otherwise 6 am-10 pm, 30 minutes 10 pm-12 am 124
5 minutes 7-9 am, 5-7 pm, 7.5 minutes otherwise 6 am-10 pm, 15 minutes 10 pm-12 am, 30 minutes overnight 164
6 minutes 6 am-10 pm, 10 minutes otherwise 5-12 am, 30 minutes overnight 188
5 minutes 6 am-10 pm, 10 minutes otherwise 5-12 am, 20 minutes overnight 228
3 minutes 7-9 am, 5-7 pm, 5 minutes otherwise 6 am-10 pm, 10 minutes otherwise 5-12 am, 20 minutes overnight 260

Daily trips are given per direction; for trips in both directions, multiply by 2. There are internal tradeoffs to each number of daily trips between peak and off-peak frequency and between midday frequency and span. But for the most part the tradeoff is between the average number of daily trips per route and the total route-length. This is the quantitative version of Jarrett’s frequency-coverage tradeoff. In reality it’s somewhat more complicated – for example, average speeds are lower at the peak than off-peak and lower in the CBD than outside the CBD, so in practice adding more crosstown routes with high off-peak frequency costs less than providing the same number of revenue-km on peaky CBD-bound buses.

It’s also important to understand that this calculation only really works for frequent transit, defined to be such that the ratio of the turnaround time to the frequency and length of each route is small. On low-frequency routes, or routes that are so short that their total length is a small multiple of the headway, the analysis must be discrete rather than continuous, aiming to get the one-way trip time plus turnaround time (including schedule padding) to be an even multiple of the headway, to avoid wasting time. On regional rail, which often has trains coming every half hour on outer tails and which is much more precisely scheduled than a street bus ever could be, it’s better to instead get the length of every route from the pulse point to the outer end to be an integer or half-integer multiple of the clockface headway minus the turnaround time.

Where is New York?

All of my numbers for New York so far should be viewed as true up to a fudge factor of 10-15% in each direction, as  my source datasets disagree. But right now, Brooklyn has about 10,500 revenue-hours per weekday (slightly more on a school day, slightly fewer on a non-school day) and an average speed of about 10.5 km/h, for a total of 110,000 revenue-km. Its bus network is 550 km long, counting local and limited versions of the same bus route as a single route but counting two bus routes that interline (such as the B67 and B69) separately; interlining is uncommon in Brooklyn, and removing it only shortens the network by a few km. This means that the average bus gets 200 runs per day, or 100 per direction.

Based on the above table, 100 runs per direction implies a frequency somewhat worse than every 5 minutes peak and every 10 off-peak. This indeed appears to be the case – nearly half of Brooklyn’s network by length has off-peak weekday frequency between 10 and 15 minutes, and the median is 12. At the peak, the median frequency, again by route-length, is 7 minutes. 7 minutes peak, 12 off-peak with some extra evening and night service works out to just less than 100 runs a day in each direction.

This exercise demonstrates the need to both shrink the network via rationalization to reduce the number of route-km and increase speed to raise the left-hand side of the equation. SBS treatments increased the speed on the B44 and B46 by 30-40% relative to the locals (not the limiteds), but just keeping the network as is would onl permit 130-140 buses per weekday per direction, which is more frequency but not a lot of frequency. The 7.5-minute standard that appears to be used in Toronto and Vancouver requires more; Barcelona’s range of 3-8 minutes implies an average of 5-6 and requires even more.

Where could New York be?

It’s definitely possible to get the number of daily frequencies on the average Brooklyn bus route to more than 200 in each direction. In Manhattan this appears true as well (the big question is whether the avenues can get two-way service), and in the Bronx 250 is easy. But even 200 in Brooklyn (which implies perhaps 350 km of network) requires some nontrivial choices about which routes get buses and which don’t, cutting some buses that are too close to other routes or to the subway. I’m not committing to anything yet because the margin calls happen entirely within the 10-15% fudge factor in my datasets.

The main reason I post this now is that I believe the formula is of general interest. In any city that wants to rationalize its transit system (bus or rail), the formula is a useful construction for the tradeoffs involved in transit provision. You can look at the formula and understand why some systems choose to branch: at the same average frequency the busiest parts of the network would get more service. You can also understand why some systems choose not to branch: at some ranges of frequency, the outer ends would get so little frequency that it would discourage ridership.

What is high frequency?

I’m using 5-6 minutes as a placeholder value beyond which there’s no point in raising frequency if there’s no capacity crunch. This isn’t quite true – on a 15-minute bus trip, going from 6 minutes between buses to 3 is a 14% cut in worst-case trip time including wait – but at this point higher frequency is at best a second-order factor. It’s not like now, when going from 15 minutes to 6 would reduce the worst-case trip time on the same bus trip by 30%.

The actual values depend on trip length. An intercontinental flight every hour is frequent; a regional train every hour is infrequent; a city bus every hour might as well not exist. One fortunate consequence is that bus trips tend to be shorter in precisely the cities that can most afford to run intensive service: dense cities with large rail networks for the buses to feed. New York’s average NYCT bus trip (excluding express buses) is 3.5 km; Chicago’s is 4.1 km; Los Angeles’s is 6.7 km. Los Angeles can’t afford to run 6-minute service on its grid routes, but trips are long enough that 10-minute service may be good enough to start attracting riders who are not too poor to own a car.

I Saw a Stampede on the Metro

France won the World Cup. Once the final ended, people all over Paris went out to the streets to celebrate. At Nation I saw impromptu dancing, drivers waving tricolore flags, and car passengers climbing out of their cars to wave their own flags. But the real celebration was elsewhere, on Champs-Elysees in the central business district. This was well covered in the media; the Guardian cites an estimate of one million people going to Champs-Elysees to celebrate, and ESPN reports riots (which I didn’t witness but can easily believe happened given the general conduct I did see) and 110,000 police and gendarmerie officers.

The sidewalks were crowded and it was difficult to move; there were too few street closures, so pedestrians were confined to narrow zones for the most part. But the crowding was worst at the Metro stations, and RATP should learn from this example and do better next time there are large celebrations, perhaps next Bastille Day.

The problem is cascading closures. In London, where the Underground platforms are narrower and have fewer cross-passageways than the Metro platforms here, closures are routine at Bank because often the passageways get dangerously overcrowded. These closures cascade: once Bank is closed to limit crowding, passengers swarm the adjacent stations, such as Moorgate and London Bridge, which are not built to handle the typical Bank crowds, forcing TfL to close them as well.

France won the game around 7 in the evening Paris time. By 8, some stations on Champs-Elysees were closed, and as I sat on my severely delayed Metro Line 1 train, with passengers banging on the train’s walls and ceiling, I heard that they were closing more, ultimately going express from Palais-Royal to Argentine and skipping all the CBD stations, including Etoile. I got off at Argentine, as did practically the entire train. Not designed to handle the crowds of the entire CBD at once, Argentine’s platform was jammed. I spent maybe ten minutes trying to make my way from where I got off to the front end of the platform, where the only exits were, and failed, and at a few points the mass of passengers was such that I thought a stampede was likely. The only reason nobody fell onto the tracks was the platform edge doors, installed during the automation of Line 1.

Trains kept serving the station, dumping more and more people. The only mechanism preventing more passengers from getting on was that the crowding was so intolerable that some people started getting back onto the trains, including eventually me. I couldn’t even get off at the next stop, Porte Maillot – the platform was fine but the train was too crowded – so I got off in the suburbs, at Les Sablons, and walked back east.

Perhaps RATP did eventually close Argentine. But both RATP and the city made crucial mistakes that evening, which they should fix in the future.

First, they should have made the trains free to improve passenger circulation. Paying at the turnstiles takes time. This is especially bad in Paris, where there are separate gates for entry (which are turnstiles) and exit (which are one-way doors), unlike the two-way turnstiles of New York. Moreover, unlike New York, Paris has no large emergency doors that can be opened. All passengers were going in one direction – out – so RATP should have propped the exit doors open to let passengers out more smoothly.

Free transit for special events is routine in Paris. The trains are free around New Year’s, in order to encourage people to take the train rather than add to car traffic and pollution (and perhaps drunk driving). Bastille Day celebrations and any future victory at the World Cup or Euro Cup should be added to the list of free transit events, not to discourage people from driving but to prevent stampedes.

And second, the city should have closed the surrounding area to non-emergency car traffic. Champs-Elysees was closed, but there wasn’t much place to spill over; the side street I took once I tried leaving had a narrow sidewalk, and police cars were parked in a way to restrict people to a constrained exit path. There is no parallel street that can act as a spillover route, and between the Rond-Point and Etoile there is only one crossing street wider than about 25 meters, Avenue George V on the south side (whereas almost all rail alternatives to the Metro Line 1 are on the north side). With narrow side streets, it’s especially important to dedicate space to pedestrians and emergency vehicles and not to cars. This was as far as I can tell not done, making it hard for people to leave the most crowded areas. In contrast, Etoile itself, with twelve avenues radiating from its circle, was not so crowded, as people had escape routes.

World Cup victories are rare enough that cities understandably don’t design their entire layout based on them. But when they do happen, it’s critical to have a plan, and the same is true of other big celebrations, which often occur annually on national days. If passengers are overwhelming the subway, it’s critical to quickly do whatever the agency can to increase throughput at station passageways as well as on the tracks. And if pedestrians are overwhelming the streets above ground, it’s critical to give them more street space, including for entry and exit.