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.