Spent a bit thinking about the special case we started with*--walking ALONG the transit line. Basically comes down to two cases:
1) Get off at Station X, walk to destination.
For Case 1
--If destination is less than halfway between Station X and Station X+1, walk back.
--If destination is MORE than halfway between Station X & Station X+1, walk to station X+1
Case 2 is much more interesting, and has to do with the walk ratio between the two.
Getting off at station X+1 means you have to walk to the destination, and then walk back to Station X+1.
Now, for the transit option (Case 2) to be useful, it must be quicker than the transit option (all else equal).
The time to reach a destination using transit is*:
t=d/T + 2x/W
where t = time, d= distance, T = transit travel speed, x = distance of destination from Station X+1, and W is walk speed.
t=d/T + 2x/W
where t = time, d= distance, T = transit travel speed, x = distance of destination from Station X+1, and W is walk speed.
It's 2x, because leaving Station X+1, you still have to walk back to x+1.
Case 1 is always just d/W.
So an area is more accessible to Case 2 than Case 1 when d/T + 2x/W<d/W
If transit is 2x as fast as walking, so that T=2W... (2W, 3W, 4W,6W) we get:
d/2W + 2x/W<d/W
d/3W + 2x/W<d/W
d/4W + 2x/W<d/W
d/6W + 2x/W<d/W
d/2W + 2x/W<d/W
d/3W + 2x/W<d/W
d/4W + 2x/W<d/W
d/6W + 2x/W<d/W
Now I'll assume that the distance between Station X and Station X+1 is 12, and W =1 (making the math simply)
12/2*1 + 2x/1<12/1 --- > 6+ 2x < 12 --- > x <3
12/3*1 + 2x/1<12/1 --- > 4+ 2x < 12 --- > x <4
12/4*1 + 2x/1<12/1 --- > 3+ 2x < 12 --- > x <4.5
12/6*1 + 2x/1<12/1 --- > 2+ 2x < 12 --- > x <5
Effectively, there is a little 'bubble' around Station X+1 that it is faster to use transit to access, and that bubble gets bigger in proportion to the ratio between the two. However, that bubble does not grow in proportion to the increase in speed.
12/3*1 + 2x/1<12/1 --- > 4+ 2x < 12 --- > x <4
12/4*1 + 2x/1<12/1 --- > 3+ 2x < 12 --- > x <4.5
12/6*1 + 2x/1<12/1 --- > 2+ 2x < 12 --- > x <5
Effectively, there is a little 'bubble' around Station X+1 that it is faster to use transit to access, and that bubble gets bigger in proportion to the ratio between the two. However, that bubble does not grow in proportion to the increase in speed.
BUBBLE | SPEED
4 3*W
5 6*W
Rare is the urban transit that is 6 times the speed of walking. My intuition tells me that most urban rail* is about 3 times the speed of walking, on city street. For any location less than half way, it is always faster to walk. For any location less than 2/3, it is almost certainly faster to walk.
To some extent, this explains the value of streetcars as economic development tools. As 'slow' transit (2.x walking speed), the 'bubble' at the end is very small, so it's very rational to get off and walk the full distance between stops, so that all locations along the corridor benefit from the pass-by pedestrian traffic. If economic development is the sole aim of streetcars, going slower (so long as they are faster than walking) doesn't actually hurt. But getting people to ride streetcars in the first place requires them to provide transportation benefit to a degree where waiting for the streetcar and riding the streetcar is faster than walking.
To some extent, this explains the value of streetcars as economic development tools. As 'slow' transit (2.x walking speed), the 'bubble' at the end is very small, so it's very rational to get off and walk the full distance between stops, so that all locations along the corridor benefit from the pass-by pedestrian traffic. If economic development is the sole aim of streetcars, going slower (so long as they are faster than walking) doesn't actually hurt. But getting people to ride streetcars in the first place requires them to provide transportation benefit to a degree where waiting for the streetcar and riding the streetcar is faster than walking.
For longer distances (of light rail magnitude), let me double the distance. I'll assume that the distance between Station X and Station X+1 is 24**
24/2*1 + 2x/1<24/1 --- > 6+ 2x < 24 --- > x < 9
24/3*1 + 2x/1<24/1 --- > 4+ 2x < 24 --- > x < 10
24/4*1 + 2x/1<24/1 --- > 3+ 2x < 24 --- > x < 10.5
24/6*1 + 2x/1<24/1 --- > 2+ 2x < 24 --- > x < 11
24/3*1 + 2x/1<24/1 --- > 4+ 2x < 24 --- > x < 10
24/4*1 + 2x/1<24/1 --- > 3+ 2x < 24 --- > x < 10.5
24/6*1 + 2x/1<24/1 --- > 2+ 2x < 24 --- > x < 11
Bigger distances, bigger transit accessible 'bubble' at Station X+1, (all else equal). Makes less sense to make the full walk. This suggests more frequent stops, more economic development, as more pedestrians walk along the transit corridor. The historic form of retail during the transit-centric streetcar age (thin, deep stores) confirms this. Access to street frontage is what matters. Tempting to make really long stop spacing, but there is a distance decay on how far people are willing to walk. Pretty sure their is a way to compare this to walk-trip distance decay (1/x^2) to determine optimum stop spacing for accessibility.
I am not sure how walking ALONG transit compares to TOD access in terms of overall accessibility provided. We habitually calculate 'area accessible to transit' using PI*r^2. When we actually consider how people travel in an urban environment (along street frontages), travel along*** a transit corridor is where the action is. More unique urban land is exposed to a pedestrian (extracted from their car) walking along a transit corridor, than to a pedestrian walking to a destination in proximity to transit station (as there is no return trip along the corridor). Note I do not say the pedestrian has more accessibility (as per our discussions of accessibility isochrones around stations), as each station imposes a time-cost that reduces the average travel speed on transit, and thus the area accessible from the transit network.
I am not sure how walking ALONG transit compares to TOD access in terms of overall accessibility provided. We habitually calculate 'area accessible to transit' using PI*r^2. When we actually consider how people travel in an urban environment (along street frontages), travel along*** a transit corridor is where the action is. More unique urban land is exposed to a pedestrian (extracted from their car) walking along a transit corridor, than to a pedestrian walking to a destination in proximity to transit station (as there is no return trip along the corridor). Note I do not say the pedestrian has more accessibility (as per our discussions of accessibility isochrones around stations), as each station imposes a time-cost that reduces the average travel speed on transit, and thus the area accessible from the transit network.
-Matt.
**This could probably be turned into a graphic like the attached.
***This suggests that a retail arcade between two stations (even if only along 1 side of the tracks) would do rather well, if placed between two closely spaced stations.
***This suggests that a retail arcade between two stations (even if only along 1 side of the tracks) would do rather well, if placed between two closely spaced stations.