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.