Bus stop spacing with heterogeneous trip lengths and elastic demand

This paper develops models of a bus route in which (i) stop spacing can vary; (ii) trip lengths are heterogeneous; (iii) demand is elastic; and (iv) passengers delay the bus. Since wider spacings make sufficiently long trips faster, and sufficiently short trips slower, they induce long trips and repel short trips. We explore two continuum-approximation models: one with fixed headways and another in which headways depend on the spacing. The pattern of induced/repelled trips means the ridership-maximizing spacing is shorter than the one that maximizes passenger-km traveled. The same pattern also makes the average trip length endogenous to spacing. In the model with endogenous headways, when spacing is very narrow, a rise in spacing can reduce the expected wait time by more than it increases the expected walk time. We draw several lessons for practice and use a discrete simulation to confirm results from the continuous approximation models.