Sometimes, Bus Stop Consolidation is Inappropriate

For the most part, the optimal average spacing between bus stops is 400-500 meters. North American transit agencies have standardized on a bus stop every 200-250 meters, so stop consolidation is usually a very good idea. But this is based on a model with specific inputs regarding travel behavior. In some circumstances, travel behavior is different, leading to different inputs, and then the model’s output will suggest a different optimum. In contrast with my and Eric’s proposal for harsh stop consolidation in Brooklyn, I would not recommend stop consolidation on the crosstown buses in Manhattan, and am skeptical of the utility of stop consolidation in Paris. In Vancouver I would recommend stop consolidation, but not on every route, not do we recommend equally sweeping changes on every single Brooklyn route.

The model for the optimal stop spacing

If demand along a line is isotropic, and the benefits of running buses more frequently due to higher in-vehicle speed are negligible, then the following formula holds:

\mbox{Optimum spacing} = \sqrt{2\cdot\frac{\mbox{walk speed}}{\mbox{walk penalty}}\cdot\mbox{stop penalty}\cdot\mbox{average trip distance}}

The most important complicating assumption is that if demand is not isotropic, but instead every trip begins or ends at a distinguished location where there is certainly a stop, such as a subway connection, then the formula changes to,

\mbox{Optimum spacing} = \sqrt{4\cdot\frac{\mbox{walk speed}}{\mbox{walk penalty}}\cdot\mbox{stop penalty}\cdot\mbox{average trip distance}}

The choice of which factor to use, 2 or 4, is not exogenous to the bus network. If the network encourages transferring, then connection points will become more prominent, making the higher factor more appropriate. Whether the network encourages interchanges depends on separate policies such as fare integration but also on the shape of the network, including bus frequency. Higher average bus speed permits higher frequency, which makes transferring easier. The model does not take the granularity of transfer ease into account, which would require a factor somewhere between 2 and 4 (and, really, additional factors for the impact of higher bus speed on frequency).

After the choice of factor, the most contentious variable is the walk speed and penalty. Models vary on both, and often they vary in directions that reinforce each other rather than canceling out (for example, certain disabilities reduce both walk speed and willingness to walk a minute longer to save a minute on a bus). In Carlos Daganzo’s textbook, the walk speed net of penalty is 1 m/s. For an able-bodied adult, the walk speed can exceed 1.5 m/s; penalties in models range from 1.75 (MTA) to 2 (a Dutch study) to 2.25 (MBTA). The lower end is probably more appropriate, since the penalty includes a wait penalty, and stop consolidation reduces waits even as it lengthens walk time.

Update 10/31: alert reader Colin Parker notes on social media that you can shoehorn the impact of walk time into the model relatively easily. The formula remains the same with one modification: the average trip distance is replaced with

\mbox{average trip distance} + \frac{\mbox{average distance between buses}\cdot\mbox{wait penalty}}{2}.

The factor of 2 in the formula comes from computing average wait time; for worst-case wait time, remove the 2 (but then the wait penalty would need to be adjusted, since the wait penalty is partly an uncertainty penalty).

The average distance between buses is proportional to the number of service-hours, or fleet size: it obeys the formula

\mbox{revenue service-hours per hour} = \frac{2\cdot\mbox{route-length}}{\mbox{average distance between buses}}.

The factor of 2 in the formula comes from the fact that route-length is measured one-way whereas revenue hours are for a roundtrip.

If we incorporate wait time into the model this way, then the walk and wait penalties used should be higher, since we’re taking them into account; the Dutch study’s factor of 2 is more reasonable. The conclusions below are not really changed – the optima barely increase, and are unchanged even in the cases where stop consolidation is not recommended.

The situation in New York

The average unlinked New York City Transit bus trip is 3.35 km: compare passenger-miles and passenger trips as of 2016. In theory this number is endogenous to the transit network – longer interstations encourage passengers to take the bus more for long trips than for short trips – but in practice the SBS routes, denoted as bus rapid transit in the link, actually have slightly shorter average trip length than the rest. For all intents and purposes, this figure can be regarded as exogenous to stop spacing.

The stop penalty, judging by the difference between local and limited routes, is different for different routes. The range among the routes I have checked looks like 20-40 seconds. However, Eric tells me that in practice the B41, which on paper has a fairly large stop penalty, has little difference in trip times between the local and limited versions. The local-SBS schedule difference is consistent with a stop penalty of about 25 seconds, at least on the B44 and B46.

As a sanity check, in Vancouver the scheduled stop penalty on 4th Avenue is 22 seconds – the 84 makes 19 fewer stops than the 4 between Burrard and UBC and is 7 minutes faster – and the buses generally run on schedule. The actual penalty is a little higher, since the 4 has a lot of pro forma stops on the University Endowment Lands that almost never get any riders (and thus the bus doesn’t stop there). This is consistent with 25 seconds at a stop that the bus actually makes, or even a little more.

Plugging the numbers into the formula yields

\mbox{Optimum spacing} = \sqrt{4\cdot\frac{1.5}{1.75}\cdot 25\cdot 3350} = 536 \mbox{ meters}

if we assume everyone connects to the subway (or otherwise takes the bus to a distinguished stop), or

\mbox{Optimum spacing} = \sqrt{2\cdot\frac{1.5}{1.75}\cdot 25\cdot 3350} = 379 \mbox{ meters}

if we assume perfectly isotropic travel demand. In reality, a large share of bus riders are connecting to the subway, which can be seen in fare revenue, just $1.16 per unlinked bus trip compared with $1.91 per subway trip (linked or unlinked, only one swipe is needed). In Brooklyn, it appears that passengers not connecting to the subway disproportionately go to specific distinguished destinations, such as the hospitals, universities, and shopping centers, or Downtown Brooklyn, making the higher figure more justified. Thus, our proposed stop spacing, excluding the long nonstop segments across the Brooklyn-Battery Tunnel and between borough line and JFK, is 490 meters.

Update 10/31: if we incorporate wait time, then we need to figure out the average distance between buses. This, in turn, depends on network shape. Brooklyn today has 550 km of bus route in each direction, which we propose to cut to 350. With around 600 service hours per hour – more at the peak, less off-peak – we get an average distance between buses of 1,830 meters today or 1,180 under our proposal. Using our proposed network, and a wait and walk penalty of 2, we get

\mbox{Optimum spacing} = \sqrt{4\cdot\frac{1.5}{2}\cdot 25\cdot (3350 + \frac{2\cdot 1180}{2}} = 583 \mbox{ meters}

or

\mbox{Optimum spacing} = \sqrt{2\cdot\frac{1.5}{2}\cdot 25\cdot (3350 + \frac{2\cdot 1180}{2})} = 412 \mbox{ meters}.

Short bus routes imply short stop spacing

Our analysis recommending 490 meter interstations in Brooklyn depends on the average features of New York’s bus network. The same analysis ports to most of the city. But in Manhattan, the situation is different in a key way: the crosstown buses are so short that the average trip length cannot possibly match city average.

Manhattan is not much wider than 3 km. Between First and West End Avenues the distance is 2.8 km. The likely average trip length is more than half the maximum, since the typical use case for the crosstown buses is travel between the Upper East Side and Upper West Side, but the dominant destinations are not at the ends of the line, but close to the middle. With Second Avenue Subway offering an attractive two-seat ride, there is less reason to take the crosstown buses to connect to the 1/2/3 (and indeed, the opening of the new line led to prominent drops in ridership on the M66, M72, M79, M86, and M96); the best subway connection point is now at Lexington Avenue, followed by Central Park West. On a long route, the location of the dominant stop is not too relevant, but on a short one, the average trip length is bounded by the distance between the dominant stop and the end of the line.

If we take the average trip length to be 1.6 km and plug it into the formula, we get

\mbox{Optimum spacing} = \sqrt{4\cdot\frac{1.5}{1.75}\cdot 25\cdot 1600} = 370 \mbox{ meters}

or

\mbox{Optimum spacing} = \sqrt{2\cdot\frac{1.5}{1.75}\cdot 25\cdot 1600} = 262 \mbox{ meters.}

A crosstown bus stopping at First, Second, Third, Lex, Madison, Fifth, Central Park West, Columbus, Amsterdam, Broadway, and West End makes 10 stops in 2.8 km, for an average of 280 meters. There isn’t much room for stop consolidation. If the bus continues to Riverside, lengthening the trip to 3 km at the latitude of 96th Street, then it’s possible to drop West End. If the buses running up Third and down Lex are converted to two-way running, presumably on Lex for the subway connections, then Third could be dropped, but this would still leave the interstation at 330 meters, much tighter than anything we’re proposing in Brooklyn.

The only other places where avenues are too closely spaced are poor locations for stop removal. Amsterdam and Broadway are very close, but Amsterdam carries a northbound bus, and if the Columbus/Amsterdam one-way pair is turned into two two-way avenues, then Amsterdam is a better location for the bus than Columbus because it provides better service to the Far West Side. Fifth and Madison are very close as well, but the buses using them, the M1 through M4, are so busy (a total of 32 buses per hour at the peak) that if the two avenues are converted to two-way running then both should host frequent bus trunks. It’s not possible to skip either.

Within Brooklyn, there is one location in which the same issue of short bus routes applies: Coney Island. The B74 and B36 act as short-hop connectors from Coney Island the neighborhood to Coney Island the subway station. The routes we propose replacing them have 7 stops each from the subway connection west, over distances of 2.5 and 2.7 km respectively, for interstations of 360 and 390 meters.

Vancouver supplies two more examples of routes similar to the B74 and B36: the 5 and 6 buses, both connecting the West End with Downtown. The 6 is only 2 km between its western end and the Yaletown SkyTrain station, and the 5 is 2.3 km from the end to the Burrard station and 2.8 km to city center at Granville Street. The average trip length on these buses is necessarily short, which means that stop consolidation is not beneficial, unlike on the main grid routes outside Downtown.

Update 10/31: incorporating wait time into this calculation leads to the same general conclusion. The short routes in question – the Manhattan crosstowns, the B36 and B74, and the 5 and 6 in Vancouver – have high frequency, or in other words short distance between buses. For example, the M96 runs every 4 minutes peak, 6 off-peak, and takes 22-24 minutes one-way, for a total of 6 circulating buses per direction peak (which is 500 meters), or 4 off-peak (which is 750 meters). This yields

\mbox{Optimum spacing} = \sqrt{2\cdot\frac{1.5}{2}\cdot 25\cdot (1600 + \frac{2\cdot 500}{2})} = 281 \mbox{ meters.}

A network that discourages transferring should have more stops as well

In Paris the average interstation on buses in the city looks like 300 meters; this is not based on a citywide average but on looking at the few buses for which Wikipedia has data plus a few trunks on the map, which range from 250 to 370 meters between stations.

The short stop spacing in Paris is justified. First of all, the average bus trip in Paris is short: 2.33 km as of 2009 (source, PDF-p. 24). Parisian Metro coverage is so complete that the buses are not useful for long trips – Metro station access time is short enough that the trains overtake the buses on total trip time very quickly.

Second, there is little reason to transfer between buses here, or to transfer between buses and the Metro. The completeness of Metro coverage is such that buses are just not competitive unless they offer a one-seat ride where the Metro doesn’t. Another advantage of buses is that they are wheelchair-accessible, whereas the Metro is the single least accessible major urban rail network in the world, with nothing accessible to wheelchair users except Line 14 and the RER A and B. It goes without saying that people in wheelchairs are not transferring between the bus and the Metro (and even if they could, they’d have hefty transfer penalties). The New York City Subway has poor accessibility, but nearly all of the major stations are accessible, including the main bus transfer points, such as Brooklyn College and the Utica Avenue stop on the 3/4.

With little interchange and a mostly isotropic city density, the correct formula for the optimal bus stop spacing within Paris is

\mbox{Optimum spacing} = \sqrt{2\cdot\frac{1.5}{1.75}\cdot 25\cdot 2330} = 316 \mbox{ meters,}

which is close to the midpoint of the range of interstations I have found looking at various routes.

Conclusion

The half-kilometer (or quarter-to-a-third-of-a-mile) rule for bus stop spacing is an empirical guideline. It is meant to describe average behavior in the average city. It is scale-invariant – the density of the city does not matter, only relative density does, and the size of the city only matters insofar as it may affect the average trip length. However, while scale itself does not lead to major changes from the guideline, special circumstances might.

If the geography of the city is such that bus trips are very short, then it’s correct to have closer stop spacing. This is the case for east-west travel in Manhattan. It is also common on buses that offer short-hop connections to the subway from a neighborhood just outside walking range, such as the B36 and B74 in Coney Island and the 5 and 6 in Vancouver’s West End.

Note that even in New York, with its 3.3 km average trip length, stop consolidation is still beneficial and necessary on most routes. North American transit agencies should not use this article as an excuse not to remove extraneous stops. But nor should they stick to a rigid stop spacing come what may; on some routes, encouraging very short trips (often 1.5 km or even less), closely spaced stations are correct, since passengers wouldn’t be riding for long enough for the gains from stop consolidation to accumulate.

14 comments

  1. Fbfree

    A few other considerations that may be added to this model.

    Your model for stop spacing considers either an average or maximum stop penalty, but doesn’t consider the effect of stop penalty variations and its effect on bunching. This is a problem for stop spacing or times of day that approach 0.5 to 1 customers per stop per bus, where a large variation in the number of stops called at may occur. Generally, stop consolidation on long routes also fixes this problem.

    I would also add a note about wayfinding to stops, but you already covered that nicely in your Brooklyn redesign post.

    • Fbfree

      One equates the total time lost by customers on the bus (stop penalty * # of stops * 1/2 (average loading)) to the total time lost walking (distance between stops / 4 / walking speed x 1 or 2 ends), where (# of stops = average distance travelled / distance between stops).

    • Alon Levy

      It’s the optimum bus stop spacing from the perspective of minimizing the total in-vehicle and walking time (with walking time incurring a penalty).

      Specifically, set up the following variables:
      x is the stop spacing
      v is walk speed
      w is walk penalty
      p is the bus’s stop penalty
      d is the average trip distance

      Total in-vehicle travel time is a fixed number independent of the stop spacing plus dp/x. The walk distance from the average location to the nearest stop is x/4: the worst-case scenario is in the middle between two stops, which is x/2, and the average is half that. If both origins and destinations are isotropic then total walk distance is x/2 and then walk time with penalty is xw/2v; if the destination definitely has a stop then you only have to walk at the origin end so walk time with penalty is xw/4v.

      Now set

      f(x) = \frac{dp}{x} + \frac{xw}{2v}

      and differentiate. The minimum value will occur at

      x = \sqrt{2dpv/w}.

      If you instead assume the destination is guaranteed to have a stop then all appearances of 2 in the formulas change to 4.

  2. Adam

    Elon musk says he has the digging down to a cost of $10 mil per mile. But caveat he’s using a much smaller tunnel diameter so the cost needs to be converted to $/cubic meter excavated rather than $/distance in order to make a proper comparison .

    It’s interesting regarding the size of tunnels, tunnels are big because trains are big, trains are big because they carry a lot of people but trains are also big because they have a couple meters of drain train under the passenger floor (so we have high platforms for level boarding)

    I’m thinking Elon is thinking that if you get rid of all the stuff underneath the floor of a train, you can greatly reduce weight and equally importantly height, meaning you’ve decreased acceleration and deceleration penalities meaning you can have more tph, and if you’ve made them smaller you can decrease the size of the tunnel and every tunnel size decrease is worth it’s weight in gold, and if you can send a lot more smaller lighter trains, even if they don’t carry as many people per train you can probably still move a lot of people per hour and if trains are coming more frequently users perceive much less time penalty for their use, creating a virtuous cycle.

    TLDR, Elon’s gonna use rails for small trains built on modified auto chassis bodies within tunnels that are smaller and therefore much more cost effective to build.

    • Alon Levy

      Musk constantly makes things up. For example, the idea that cubic meters of excavation matter is incorrect. Tunnel boring costs scale with tunnel diameter and not cross-sectional area, because removing dirt is easy, it’s the concrete lining that’s hard. For another example, the limiting factor to braking distance on trains is not train mass but passenger safety. He’s a flim-flam artist who’s so bad at transportation he’s harassing short sellers on social media.

    • Michael James

      As of now, Musk hasn’t brought a single innovation to tunneling. He is using a second-hand TBM that was previously used to bore sewers in Oakland. It has only just finished the short bit of tunnel in Hawthorn LA and he announced last week that he would be giving free rides to people at 250km/hr before Xmas! It is raw tunnel so it is yet another wild bit of fantasy. I don’t see how he could even get permission to put living breathing people, whether volunteers or not, on any new e-sled system (of which I don’t believe he has any such thing developed; it’s maglev so unless he is buying something off the shelf from the Japanese or Germans, it is a major R&D program that would take many years). The whole thing is bizarre but is consistent with his recent spate of nutty behaviour.

  3. James Sinclair

    You continue to look at this from the perspective of a bus – hardware – rather than people – customers. From the bus perspective, the most efficient route has zero stops.

    But customers care about a lot of things. Not just time. They care a whole lot about safety. A large avenue can be a significant traffic safety risk which people dont want to cross. Crime, or the perception of crime, varies from block to block, by season, and time of day. Weather can’t be brushed away, especially in areas where sidewalk snow clearing isnt done well, forcing people to walk on the street or navigate mountains of snow.

    Admittedly, most of these issues dont apply to central Manhattan, but they certainly do where the buses are essential to get around.

    I dont believe it is appropriate to consolidate stops using a formula. You need to look at each one individually. Who is using the stop? Why are they using that stop versus the next one?

    • Alon Levy

      Everything you’re saying – safety, weather, sidewalk snow – is already incorporated into the walk and wait penalties. Of course there’s a tradeoff, and the model suggests that at the average trip distance in New York, the passenger-centric optimum is about 500 meters.

      • James Sinclair

        I dont think it is incorporated at all, because walking 200m west (across an 8 lane roadway, past a homeless shelter and 2 liquor stores) may be very different than walking 200 meters east (past a police station, no major roadway, and a playground).

        • Alon Levy

          Your assumption that the average New York bus rider wants to walk next to a police station is unwarranted. But anyway, the models for walk penalties take into account that people have bigger walk penalties in hilly terrain or across freeways, and this shows in tighter bus stop spacing in hilly areas. Freeways are mostly invisible to average stop spacing, though – I did make sure not to force people to cross freeways in the Brooklyn redesign, it just didn’t show in the average stop spacing because the freeways are surrounded by dead zones anyway.

          Arterials that function as major destinations would just get bus stops at the intersection. If pedestrians can’t cross them safely, that’s not something that can be fixed with bus stop spacing unless the bus stops on both sides of the road, which is excessive.

          • crazytrainmatt

            In NYC, police stations are some of the worst blocks for walking given that the cops generally park their personal vehicles on the sidewalk!

  4. Oreg

    Other than the cross-park sections I find cross-town buses in Manhattan excruciatingly slow to the point of being useless. They don’t seem much faster than walking. That’s not just because of the very short distance between stops but also the on-board fare collection and single-door boarding. Maybe if they would bring those into the 21st century the buses would start becoming relevant.

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