# Bicycle schemes need big cities

In larger cities such as Lyon or even Dublin, bikeshare schemes are quite successful. In smaller ones like Limerick they struggle. I am convinced the problem is critical mass. As a scheme gets bigger, it provides disproportionately more possible useful journeys (as long as there is the population density to support it).

I want to model this. Let’s start by imagining cities that are big enough to sustain a square grid of bike stations, and let’s count the number of possible A-B journeys it provides (of different distances).

A 5 by 5 grid contains 25 stations, and 600 possible A-B journeys (25 by 24). I believe the ideal distance between stations is taken to be about 300m: if the stations are thus located, the distances of the 600 possible trajectories are distributed as follows:

5×5 distances Freq. Percent Cumulative
(metres) Percent
300 80 13.33 13.33
424 64 10.67 24.00
600 60 10.00 34.00
671 96 16.00 50.00
849 36 6.00 56.00
900 40 6.67 62.67
949 64 10.67 73.33
1082 48 8.00 81.33
1200 20 3.33 84.67
1237 32 5.33 90.00
1273 16 2.67 92.67
1342 24 4.00 96.67
1500 16 2.67 99.33
1697 4 0.67 100.00
Total 600 100.00

### 7 by 7 grid

If we (almost) double the size of the scheme to a 7 by 7 grid (49 stations), the number of possible journeys (almost) quadruples to 2,352:

7×7 distances Freq. Percent Cumulative
(metres) Percent
300 168 7.14 7.14
424 144 6.12 13.27
600 140 5.95 19.22
671 240 10.20 29.42
849 100 4.25 33.67
900 112 4.76 38.44
949 192 8.16 46.60
1082 160 6.80 53.40
1200 84 3.57 56.97
1237 144 6.12 63.10
1273 64 2.72 65.82
1342 120 5.10 70.92
1500 152 6.46 77.38
1530 96 4.08 81.46
1616 80 3.40 84.86
1697 36 1.53 86.39
1749 64 2.72 89.12
1800 28 1.19 90.31
1825 48 2.04 92.35
1897 40 1.70 94.05
1921 48 2.04 96.09
2012 32 1.36 97.45
2121 16 0.68 98.13
2163 24 1.02 99.15
2343 16 0.68 99.83
2546 4 0.17 100.00
Total 2,352 100.0

A 10 by 10 scheme offers 9,900 journeys, with a mean length of 1.572 km and a max of 3.818 km.

Clearly there is a really big return to scale. Even in the 10 by 10 grid, the longest journey is less than 4km, which is not long enough to put people off. Correspondingly, in the small 5 by 5 grid, 24% of journeys are less than 500m, which is possibly too short to bother checking out a bike.

Since there is no point putting a station in an area with low population density, city size (and possibly shape, topography) puts a hard limit on the size of the scheme. This is the hard reality facing bike schemes in Limerick.

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