In 2018, Eric’s and my Brooklyn bus redesign proposal included a new route to run between Red Hook and Lower Manhattan using the Brooklyn-Battery Tunnel. This was not our idea; a junior planner we talked to suggested this. Our plan was not adopted, but in the formal process New York City Transit and consultancy Sam Schwartz engaged in, at community meetings riders proposed the same idea, and junior staff seemed to like it but it was still not adopted. Now, a coalition of neighborhood groups and city-wide transit advocacy groups is directly calling for such a bus to be included in the Brooklyn bus redesign, including ETA. My goal in this post is to look at some alignment possibilities, more carefully than we did in the 2018 proposal. On the Manhattan side, it is not too hard to hit Lower Manhattan jobs and subway transfers on a short bus loop, but on the Brooklyn side, the Red Hook street network and its connection to the tunnel force serious compromises.

Current conditions

There are express buses in the tunnel from points much farther out to Lower Manhattan, but they don’t make stops along the way. Red Hook is instead served exclusively within Brooklyn, in three directions: one north along Van Brunt to Downtown Brooklyn, one east along Lorraine to the Smith/9th Street subway station and Park Slope, and one also along Lorraine to Smith/9th but then going north to Downtown Brooklyn. The first two are together the B61 route, in an awkward C-shaped through-route; the third is the B57, which through-runs past Downtown Brooklyn to points northeast along Flushing Avenue.

The neighborhood has roughly three major destinations to serve. Visible in the center-bottom of the map are the Red Hook Houses, with a total of 6,000 residents. At the very bottom of the map is Ikea, the main destination for people coming into the neighborhood from elsewhere. Then on the left there is Van Brunt, the local commercial drag.

Per OnTheMap, the entire neighborhood has 6,700 jobs and 5,000 employed residents as of 2019; it is not at all a bedroom community. Ikea is not even one of the main job centers – the biggest are elsewhere, such as the nearby Amazon warehouse. The neighborhood’s residents work about 40% in Manhattan, 40% in Brooklyn, and 20% elsewhere, while the workers are half from Brooklyn with no other origin having much concentration (the second biggest county origin, Queens, is 14%). Only 300 people both live and work in Red Hook, so a transit system connecting the neighborhood to the rest of the city, for both origins and destinations, is vital.

Why the tunnel?

Red Hook’s current bus connections are only with the rest of Brooklyn. This materially slows down travel for the 40% of residents who work in Manhattan and roughly 10% who work in places one accesses via Manhattan, such as the Bronx or Long Island City. The on-street bus connections are slow, and the neighborhood is not well-located relative to the Brooklyn subway network. The B57 only kind of hits Smith/9th southbound, since Smith is one-way northbound and the southbound trip is one block west on Court. Smith/9th itself is not accessible, and is the highest subway station in the system above the local street level as it was built with high clearance below for shipping through the Gowanus Canal.

Let’s look at how fast it is to get to 42nd Street. Via the B57 or B61, it’s about 10 minutes by bus from Ikea to Smith/9th; the B61 runs every 12 minutes and the B57 every 15 or 20, for maximum inconvenience. Then from Smith/9th to Bryant Park, it’s 27 minutes on the F. A bus in the tunnel would get to Fulton Street in 25 minutes and then it’s 12 minutes on the A. In theory, it’s the same trip time from Ikea, and around three minutes faster in relative terms from the Red Hook Houses depending on the route. In practice, being able to connect in Manhattan means having a much wider variety of destinations than just what’s on the F, which doesn’t even get to Lower Manhattan. The benefits for Red Hook-to-rest-of-city commuters would be noticeable.

The Manhattan street section would have variable traffic. On the other hand, the tunnel is less congested than its approaches, and congestion pricing stands to reduce traffic exactly there, as on other roads into the Manhattan core. With no bus stops in the tunnel, the average speed would be reasonable even with a short loop through Lower Manhattan. Diverting ridership from slower buses to Downtown Brooklyn would save revenue-hours, which could then be spent on higher frequency on all remaining routes.

Compromises on the route

The routing within the neighborhood for any bus route using the tunnel cannot be perfect; the neighborhood is not laid out for it. This is seen in how awkward the buses through Red Hook are today, as mentioned above; none of them even goes through the Red Hook Houses, which are the dominant origin. All of the following constraints require creating a single compromise bus route:

• The ridership potential is not there for more than one route. Whatever option is chosen, whether it’s a shuttle as I’m implying in this post or an extension of an existing route that goes deeper into Manhattan (or Brooklyn), that’s the only thing that can run. Even with one route, there may need to be compromises on frequency (by which I mean a bus every 8-10 minutes instead of 6, not 12).
• Van Brunt, Ikea plus the other waterfront jobs, and the Red Hook Houses are not at all collinear.
• The only place to get to the tunnel from Red Hook is the ramp from West 9th or Huntington, and West 9th is one-way west and may need to be converted to two-way. In particular, Van Brunt is too far, and the interface with the tunnel needs to be to and from the Red Hook Houses directly.

In effect, what all of the above implies is that a bus to Manhattan on Van Brunt is not likely to work. Here is one version of what could:

The circles along the path denote control points on Google Maps, and not stops. The western waterfront may have to just not be served; people could walk from Van Brunt across Coffey Park and it would be faster than taking the bus the long way around, down Van Brunt and then along Beard and up Columbia.

At the Manhattan end, the route would either loop just far enough north to hit the Fulton Street subway complex, or through-run. Fulton is necessary because the Wall Street stations are inaccessible, and is generally useful for the connection to World Trade Center. Beyond that, one option is to through-run to the M9, which hits more Manhattan destinations. That said, Manhattan bus speeds are so low that nearly all riders would switch to the subway; M9 frequencies are also low, every 15 minutes off-peak, and when there’s not much traffic this is almost unusably low for Red Hook Houses-Wall Street trips.

Two different issues in New York – the bus redesign process and the Interborough Express – are making me think about optimal stop spacing again. I blogged about it in general about buses a few days ago, but crosstown routes present their own special issues, and this is noticeable on rail more than on buses. Circumferential rail routes, in particular, can justify wider stop spacing than radial routes in certain circumstances. This can explain why, over the iterations of Triboro RX leading to the current IBX proposal, the stop spacing has widened: the Third Regional Plan-era effort in the 2000s had a stop every half mile in Brooklyn and Queens, but more recent efforts proposed fewer stops, and the current one if anything has too few and misses a transfer.

Density and isotropy

The tradeoff in stop spacing on both buses and trains is that more stops reduce the amount of walking to the station but increase the in-vehicle trip time for people going through the stop without getting on or off.

Density by itself does not affect this tradeoff. A uniform increase in density along a line equally increases the costs and benefits of changing the stop spacing. However, relative density matters: stop spacing should be tighter in areas with higher density and wider in areas with lower density, both relative to other areas along the same line. This is because higher relative density means passengers are disproportionately likely to have their origin or destination in this area, and disproportionately less likely to be traveling through it, both of which argue in favor of tighter stop spacing, and lower relative density means the opposite.

This then leads to the issue of isotropy. On an isotropic network, relative density is by definition always the same; spikes in relative density make travel less isotropic. As my previous post explains with bus stop spacing formulas, also valid on rail with different parameters, less isotropic density should mean not just that there should be more stops in some places and fewer in others, but also that there should be fewer stops overall. In the simplest case of non-isotropy, assume everyone is traveling to the same distinguished node, which on a rail line can be thought of as city center (let’s say there’s just one central transfer point) and on a bus can be thought of as the connection to the subway. Then, all passengers can be guaranteed to be going to a place with a station, and therefore the cost of widening the stop spacing is halved, since only the origin walk time is increased, not the destination walk time.

Isotropy and circumferential routes

Successful circumferential routes live off of their ability to connect to the rest of the network. Over time, those connection points may grow to become large destinations in their own right – this is the story of how Ikebukuro, Shinjuku, and Shibuya, all at the intersection of the Yamanote Line with radial rail links (JR, private, or subway), became large business districts. But the connections have to come first. If passengers can’t conveniently transfer, then the route has to live off of origin-and-destination traffic just on the line, and then, because it is circumferential and by definition doesn’t go to city center, traffic will be low. This principle is why the G train in New York is so weak: it may connect the two largest non-Manhattan job centers in the region, but that’s still neither Manhattan nor service to the entire city, and with poor transfers, it has to live off of the small number of people living in Williamsburg and Bedford-Stuyvesant working in Long Island City or Downtown Brooklyn.

But the same principle also means that non-transfer stops lose value. This doesn’t mean there shouldn’t be any of them, but it does mean that agencies can afford to be pickier about where to place them. They’re unlikely to be destinations, only origins, and even as origins their value is discounted since some passengers use the circumferential line as the second leg of a three-legged trip, between two radial lines.

The impact on IBX

I used to criticize the decision to build fewer stops on IBX. For example, here, when it was still an RPA proposal in what would later become the Fourth Regional Plan, I outlined several criticisms of the then-Triboro route. I think some of them stand, especially the section on the plan to have the route go into the Bronx and provide local commuter rail service to Coop City. However, on the matter of stop spacing, I must withdraw the criticism.

That said, a station at every connection with a radial rail line remains nonnegotiable. IBX errs in only stopping in East New York at Atlantic Avenue, connecting to the L and the LIRR, with no direct connection to Broadway Junction for the A/C and J. The distance between these two locations is only 350 meters, and it may be awkward to have two stops in short succession, but the meaning of high relative density is exactly that it’s okay to have more closely spaced stops. Alternatively, there could be one stop at a compromise location, with in-system connections at both ends, but then the walk times would be higher, which is less desirable.

There are arguments over bus stop spacing in my Discord channel. As the Queens bus redesign process is being finalized, there’s a last round of community input, and as one may expect, community board members amplify the complaints of people who reject any stop consolidation on “they’re taking my stop, I’ll have to walk longer” grounds. I wrote about this in 2018, as Eric and I were releasing our proposed Brooklyn bus redesign, which included fairly aggressive consolidation, to an average interstation of almost 500 meters, up from the current value of about 260. I’d like to revisit this issue in this post, first because of its renewed relevance, and second because there’s a complication that I did not incorporate into my formula before, coming from the fact that the city comprises discrete blocks rather than perfectly isotropic distribution of residents along an avenue.

The formula for bus stop spacing

The tradeoff is that stop consolidation means people have to walk longer to the bus stop but then the bus is faster. In practice, this means the bus is also more frequent by a proportionate amount – the resources required to operate a bus depend on time rather than distance, chiefly the driver’s wage, but also maintenance and fuel, since stops incur acceleration and idling cycles that stress the engines and consume more fuel.

The time penalty of each stop can be modeled as the total of the amount of time the bus needs to pull into the stop, the minimum amount of time it takes to open and close the doors, and the time it takes to pull out. Passenger boardings are not included, because those are assumed to be redistributed to other stops if a stop is deleted. In New York and Vancouver, the difference in schedules between local and limited stop buses in the 2010s was consistent with a penalty of about 25 seconds per stop.

The optimum stop spacing can be expressed with the following formula:

To explain in more detail:

• d is a dimensionless factor indicating how far one must walk, based on the stop spacing; the more isotropic passenger travel is, the lower d is, to a minimum of 2. The specific meaning of d is that if the stop spacing is n, then the average walk is n/d. For example, if there is perfect isotropy, then passengers’ distance from the nearest bus stop is uniformly distributed between 0 and n/2, so the average is n/4, and this needs to be repeated at the destination end, summing to n/2.
• Walk speed and walk penalty take into account that passengers prefer spending time on a moving bus to walking to the bus. In the literature that I’ve seen, the penalty is 2. Usually the literature assumes the walk speed is around 5 km/h, or 1.4 m/s; able-bodied adults without luggage walk faster, especially in New York, but the speed for disabled people is lower, around 1 m/s for the most common cases.
• Stop penalty, as mentioned above, can be taken to be 25 s.
• Average trip length is unlinked; for New York City Transit in 2019, counting NYCT local buses including SBS but not express buses, the average was 3,421 meters.
• Average bus spacing is the headway between buses on the route measured in units of distance, not speed; it’s expressed this way since the resources available can be expressed in how many buses can circulate at a given time, and then the frequency is the product of this figure with speed. In Brooklyn in the 2010s, this average was 1,830 m; our proposed network, pruning weaker routes, cut it to 1,180. The Queens figure as of 2017 appears similar to the Brooklyn figure, maybe 1,860 m. Summing the average trip length and average bus spacing indicates that passengers treat wait time as a worst-case scenario, or, equivalently, that they treat it as an average case but with a wait penalty of 2, which is consistent with estimates in the papers I’ve read.

In the most isotropic case, with d = 2, plugging in the numbers gives,

However, isotropy is more complex than this. For one, if we’re guaranteed that all passengers are connecting to one distinguished stop, say a subway connection point, then consolidating stops will still make them walk longer at the other end, but it will not make them walk any longer at the guaranteed end, since that stop is retained. In that case, we need to set d = 4 (because the average distance to a bus stop if the interstation is n is n/4 and at the other end we’re guaranteed there’s no walk), and the same formula gives,

The Queens bus redesign recognizes this to an extent by setting up what it calls rush routes, designed to get passengers from outlying areas in Eastern Queens to the subway connection points of Flushing and Jamaica; those are supposed to have longer interstations, but in practice this difference has shrunk in more recent revisions.

That said, even then, there’s a complication.

City blocks and isotropy

The models above assume that passengers’ origins are equally distributed along a line. For example, here is Main Street through Kew Gardens Hills, the stretch I am most likely to use a New York bus on:

I always take the bus to connect to Flushing or Jamaica, but within Kew Gardens Hills, the assumption of isotropy means that passengers are equally likely to be getting on the bus at any point along Main Street.

And this assumption does not really work in any city with blocks. In practice, neighborhood residents travel to Main Street via the side streets, which are called avenues, roads, or drives, and numbered awkwardly as seen in the picture above (72nd Avenue, then 72nd Road, then 72nd Drive, then 73rd Avenue, then 75th Avenue…). The density along each of those side streets is fairly consistent, so passengers are equally likely to be originating from any of these streets, for the most part. But they are always going to originate from a side street, and not from a point between them.

The local bus along Main, the Q20, stops every three blocks for the most part, with some interstations of only two blocks. Let’s analyze what happens if the system consolidates from a stop every three blocks, which is 240 meters, to a stop every six, which is 480. Here, we assume isotropy among the side streets, but not continuous isotropy – in other words, we assume passengers all come from a street but are equally likely to be coming from any street.

With that in mind, take a six-block stretch, starting and ending with a stop that isn’t deleted. Let’s call this stretch 0th Street through 6th Street, to avoid having to deal with the weird block numbering in Kew Gardens Hills; we need to investigate the impact of deleting a stop on 3rd Street. With that in mind: passengers originating on 0th and 1st keep going to 0th Street and suffer no additional walk, passengers originating on 5th keep going to 6th and also suffer no additional walk, passengers originating on 2nd and 4th have to walk two blocks instead of one, and passengers originating on 3rd have to walk three blocks instead of zero. The average extra walk is 5/6 of a block. This is actually more than one quarter of the increase in the stop spacing; if there is a distinguished destination at the other end (and there is), then instead of d = 4, we need to use d = 3/(5/6) = 3.6. This shrinks the optimum a bit, but still to 576 m, which is about seven blocks.

The trick here is that if the stop spacing is an even number of blocks, then we can assume continuous isotropy – passengers are equally likely to be in the best circumstance (living on a street with a stop) and in the worst (living on a street midway between stops). If it’s an odd number of blocks, we get a very small bonus from the fact that passengers are not going to live on a street midway between stops, because there isn’t one. The average walk distance, in blocks, with stops every 1, 2, 3, … blocks, is 0, 0.5, 2/3, 1, 1.2, 1.5, 12/7, 2, … Thus, ever so slightly, planners should perhaps favor a stop every five or seven blocks and not every six, in marginal cases. To be clear, the stop spacing on each stretch should be uniform, so if there are 12 blocks between two distinguished destinations, there should be one intermediate stop at the exact midpoint, but, perhaps, if there are 30 blocks with no real internal structure of more or less important streets, a stop should be placed every five and not six blocks, especially if destinations are not too concentrated.

The ongoing designs for the Interborough Express are making me think about bus redesign again. Before the Transit Costs Project, Eric and I worked on a proposal for a bus redesign in Brooklyn, which sadly was not adopted. The redesign was based on the reality of 2017 – the ridership patterns, the bus speeds, the extent of the system, etc. Since then the subway map has not changed, but IBX stands to change the map, and with it, the buses should change as well.

With our program having produced both the bus redesign proposal and soon a comprehensive proposal for how to change the city’s built-up layout to take advantage of the new line, I should probably say something about how the buses should change. I say I and not we, because so far we don’t have a project under our program for this; for now, this is just a blog post, though one informed by past work on the subject.

Parallel and orthogonal buses

In general, when a new line opens, it reduces demand on parallel bus routes, which it outcompetes, and increases it on orthogonal ones, which feed it. However, what counts as parallel and what counts as orthogonal are not always obvious.

Case in point: when Second Avenue Subway opened at the end of 2016, ridership on the east-west buses between the Upper East and West Sides fell. The new line in theory runs north-south, but it undulates from the Upper East Side to Times Square, where passengers can connect to trains to the Upper West Side and points north; when I lived at 72nd and York and commuted to Columbia by bus and subway in 2009-10, I calculated that if Second Avenue Subway had been open already, a two-seat subway ride with a Times Square connection would have cut my one-way commute from 50 to 37 minutes.

This means that to understand how a new rail line will impact buses, it’s necessary to look beyond just the line itself, and think what it connects to.

For example, note on the above map that the increase in job access at the Flatbush Avenue station, intersecting the Nostrand Avenue Line, is relatively small, and doesn’t have a big north-south footprint along Nostrand. This is because the location already has subway service connecting to Manhattan, a much larger job center than anything IBX would connect to; the buses at the station, the B41 on Flatbush and B44 on Nostrand, already function as connectors to the subway at this point, and are unlikely to acquire more ridership as a result.

In contrast, the stations at Myrtle and Metropolitan are both seen to have a large increase in job access, and in particular a large increase in job access along those two avenues even somewhat away from the stations. On Myrtle, the current buses are the B54 and Q55; the B54 connects to the M train, but it’s one branch, and then the bus continues to Downtown Brooklyn, to which there’s no good subway connection from the future IBX station. The B54 is likely to lose ridership to Downtown Brooklyn but gain it to the new IBX station, and the Q55 is likely to gain in general, as they ferry passengers to a station where they can quickly and with one change go to any number of express lines. Metropolitan has a similar issue – the Q54 already connects to the M, but at least from points west, nobody has any reason to make that connection since it would just double back, whereas with IBX, the Q54 would efficiently connect people to Jackson Heights, and with an additional change to anywhere on the Queens Boulevard and Flushing Lines.

New nodes

Public transit lines serve two functions: to run along a corridor, and to connect nodes. New York usually thinks in terms of corridors, and indeed names nearly all subway lines after the streets they run on (such as a Manhattan avenue) rather than after where they go. But nodes are important as well. Some of that is reflected in the above analysis of the Flatbush-Nostrand Avenue station, currently Brooklyn College on the Nostrand Avenue Line: it really needs to be thought of as a node, and IBX will strengthen it, but not by enough to require running more B41 and B44 buses. In contrast, other nodes will be strengthened enough that bus service increases are warranted.

East New York/Broadway Junction is the biggest standout. East New York’s bus network today is not much of a grid – instead, buses connect outlying areas to the nearest subway station; the bus redesign we did for Brooklyn would make it more of a grid but still follow the logic of feeding the subway wherever it is closest. However, IBX makes Broadway Junction and the Atlantic Avenue station more interesting, which should leads to some changes, turning the new station into more of a node for buses. Buses avoiding this node should instead make sure to stop not just at the subway but also at a new IBX stop, such as Linden.

Jackson Heights is the other. It is a node to some extent today, served by the Q32, Q33, Q47, Q49, Q53, and Q70. But in that general area, the intersection of Woodhaven and Queens Boulevard is an even larger node, and in Queens writ large, the ends of the subway in Jamaica and Flushing are far and away the biggest ones. With IBX, more buses should run to Jackson Heights; for example, all Woodhaven buses, and not just the Q53, should continue along Queens Boulevard and Broadway to reach the station.

Substitutions

In Queens, the street network connecting Jackson Heights with the neighborhoods near the borough line with Brooklyn is not at all conducive for good transit. Buses are usually a good indicator of relative demand along a corridor, but sometimes they aren’t; the situation of IBX is generally one in which they are not, but this is especially bad in Queens. This means that the question of which buses would see demand fall as IBX substitutes for them is even harder than on Second Avenue Subway, the north-south line that efficiently substitutes for east-west buses.

In Brooklyn, I think the answer is relatively straightforward, in that the main crosstown routes, like the B35 on Church, exhibit substitutability. In Queens, it’s harder, and I don’t have concrete answers, only general thoughts that we can turn into a report if there turns out to be demand for it:

• If a bus has sections along the corridor but also away from it, like the Q18 or Q47, then it should be cut to just connect to the line, in these two case at Jackson Heights.
• If a bus runs directly between two nodes that could get faster service via a subway-IBX connection, and it doesn’t serve much along the way, then it’s likely to be analogous to the east-west buses across Central Park, and see reduced ridership demand.
• In general, the routes in Central Queens zigzag so much that IBX is likely to represent a massive improvement in trip times, making such buses less useful.

I have annoying commenters. They nitpick what I say and point out errors in my thinking – or if there are no errors, they take it beyond where I thought it could be taken and find new ways of looking at it. After I wrote about frequency relative to trip length last week, Colin Parker pointed out on the Fediverse that this can be simplified into thinking about frequency not in units of time (trains or buses per hour), but in units of distance (trains or buses per km of route). This post is dedicated to developing this idea on various kinds of transit service, including buses and trains.

The key unit throughout, as Colin points out, is the number of buses available per route, the assumption being that the average trip length is proportional to the average route length. However, this is not a perfect assumption, because then the introduction of network effects changes things – generally in the direction of shorter average trip length, as passengers are likelier to transfer, in turn forcing agencies to run more vehicles on a given route to remain useful. Conversely, timed transfers permit running fewer vehicles, or by the same token more routes with the same resources – but the network had better have a strong node to connect to after a series of vehicle changes, more like the Swiss rail network than like a small American city’s bus network.

Frequency and resources

On a bus network with even frequency across all routes, the following formula governs frequency, as I discussed six years ago:

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

When Eric and I proposed our Brooklyn bus redesign, we were working with a service-hour budget of about 10,800 per weekday; status quo as of 2017-8 was 11 km/h, 550 km, and thus 216 daily trips (108 per direction), averaging around a bus per 11 minutes during the daytime, while we were proposing speed up treatments and a redesign to change these figures to 15 km/h, 355 km, and thus 456 daily trips (228 per direction). The six-minute service ideal over 16 hours requires 188 trips per direction; the difference between 188 and 228 is due to higher frequencies on the busiest routes, which need the capacity.

To express this in units of length, we essentially eliminate time from the above dimensional analysis. Daily service hours is a dimensionless quantity: 10,800 hours per weekday means 450 buses circulating at a given time on average, in practice about 570 during the daytime but not many more than 100 buses circulating overnight. If there are 570 buses circulating at a given time, then a 550 km network will average a bus every 1.9 km and a 355 km one will average a bus every 1.25 km. With pre-corona New York bus trips averaging 3.4 km unlinked, a bus every 1.9 km means the maximum headway is a little higher than half the trip time, and a bus every 1.25 km means the maximum headway is a little higher than one third the trip time, independently of speed.

This calculation already illustrates one consequence of looking at frequency in units of distance and not time: your city probably needs to aggressively prune its bus network to limit the wait times relative to overall trip times.

Route length and trip length

On an isolated bus or train route, serving an idealized geography with a destination at its center and isotropic origins along the line, the average trip length is exactly one quarter of the route length. The frequency of service in units of distance should therefore be one eighth of the route length, requiring 16 vehicles to run service plus spares and turnarounds. This is around 18-20 vehicles in isolation, though bear in mind, the 10,800 service hours/day figure for Brooklyn buses above is only for revenue service, and thus already incorporates the margin for turnaround times and deadheads.

Colin points out that where he lives, in Atlanta, bus routes usually have around four vehicles circulating per route at a given time, rather than 16. With the above assumptions, this means that the average wait is twice the average trip time, which goes a long way to explaining why Atlanta’s bus service quality is so poor.

But then, different assumptions of how people travel can reduce the number 16:

1. If destinations are isotropic, then the average trip length rises from one quarter of the route length to 3/8 of the route length, and then the frequency should be 1.5/8 of the route length, which requires 11 vehicles in revenue service.
2. If origins are not isotropic, then the average trip length can rise or fall, depending on whether they are likelier to be farther out or closer in. A natural density gradient means origins are disproportionately closer-in, but then in a city with a natural density gradient and only four buses to spare per route, the route is likely to be cut well short of the end of the built-up area. If the end of the route is chosen to be a high-density anchor, then the origins relative to the route itself may be disproportionately farther out. In the limiting case, in which the average trip is half the route length, only eight buses are needed to circulate.

To be clear, this is for a two-tailed route; a one-tailed route, connecting city center at one end to outlying areas at the other, needs half the bus service, but then a city needs twice as many such routes for its network.

The impact of transfers

Transfers can either reduce the required amount of service for it to be worth running or increase it, depending on type. The general rule is that untimed transfers occurring at many points along the line reduce the average unlinked trip and therefore force the city to run more service, while timed transfers occurring at a central node lengthen the effective trip relative to the wait time and therefore permit the city to run less service. In practice, this describes both how existing bus practices work in North America, and even why the Swiss rail network is so enamored with timed connections.

To the point of untimed transfers, their benefit is that there can be very many of them. On an idealized grid – let’s call it Toronto, or maybe Vancouver, or maybe Chicago – every grid corner is a transfer point between an east-west and a north-south route, and passengers can get from anywhere to anywhere. But then they have to wait multiple times; in transit usage statistics, this is seen in low average unlinked trip lengths. New York, as mentioned above, averages 3.4 km bus trips, with a network heavily based on bus-subway transfers; Chicago averages a not much higher 3.9 km. This can sort of work for New York with its okay if not great relative frequency, and I think also for Chicago; Vancouver proper (not so much its suburbs) and Toronto have especially strong all-day frequencies. But weaker transit networks can’t do this – the transfers can still exist but are too onerous. For example, Los Angeles has about the same total bus resources as Chicago but has to spread them across a much larger network, with longer average trip times to boot, and is not meaningfully competitive. The untimed grid, then, is a good feature for transit cities, which have the resources and demand to support the required frequencies.

Not for nothing, rapid transit networks love untimed transfers, and often actively prefer to spread them across multiple stations, to avoid overwhelming the transfer corridors. Subways are only built on routes that are strong enough to have many vehicles circulating, to the point that all but the shortest trips have low ratios of wait to in-vehicle times. They are also usually radial, aiming to get passengers to connect between any pair of stations with just one transfer; Berlin, Paris, and New York are among the main exceptions. These features make untimed transfers tolerable, in ways they aren’t on weaker systems; not for nothing, a city with enough resources for a 100 km bus network and nothing else does not mimic a 100 km subway network.

Timed transfers have the opposite effect as untimed transfers. By definition, a timed transfer means the wait is designed to be very short, ideally zero. At this point, the unlinked trip length ceases to be meaningful – the quantity that should be compared with frequency is the entire trip with all timed transfers included. In particular, lower frequencies may be justifiable, because passengers travel to much more than just the single bus or rail route.

This can be seen in small-city American bus networks, or some night bus networks, albeit not with good quality. It can be seen much more so on transfer-based rail networks like Switzerland’s. The idealized timed transfer network comprises many routes all converging on one node where they are timed to arrive and depart simultaneously, with very short transfers; this is called a knot in German transit planning and a pulse in American transit planning. American networks like this typically run a single bus circulating on each one-tailed route; the average wait works out to be four times longer than the average unlinked trip, and still twice as long as a transfer trip, which helps explain why ridership on such networks is a rounding error, and this system is only used for last-resort transportation in small cities where transit is little more than a soup kitchen or on night bus networks that are hardly more ridden. It would be better to redo such networks, pruning weaker routes to run more service on stronger ones, at least two per one-tailed route and ideally more.

But then the Swiss rail network is very effective, even though it’s based on a similar principle: there’s no way to fill more frequent trains than one every hour to many outlying towns, and even what are midsize cities by Swiss standards can’t support more than a train every half hour, so that many routes have a service offer of two to four vehicles circulating at a given time. However, on this network, the timed transfers are more complex than the idealized pulse – there are many knots with pulses, and they work to connect people to much bigger destinations than could be done with sporadic one-seat rides. A succession of timed connections can get one from a small town in eastern Switzerland to St. Gallen, then Zurich, then Basel, stretching the effective trip to hours, and making the hourly base frequency relatively tolerable. The key feature is that the timed transfers work because while individual links are weak enough to need them, there are some major nodes that they can connect to, often far away from the towns that make the most use of the knot system.