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.