a, Stacked horizontal bar plots showing the start and end of the operation time (left and right ends of the stacked bars) for each vehicle in the minimum fleet assignment for various days of the week. The day in each panel represents results for days in the second week of each month. Vehicle active times are represented by a very thin coloured bar. The vehicles are sorted by the start of operation time, and stacking them horizontally creates each plot for each day. In all days (except for the outlier day of the second Saturday in March 2011), the patterns show high efficiency, with the majority of vehicles starting early in the day and operating until the end of the day. b, Stacked horizontal bar plots, this time representing the total operation time of the vehicle (the length of the bar). The bars are sorted based on the vehicle’s total operation time, the lowest bar corresponding to the vehicle with the longest operation time. A distinct pattern emerges on most of the weekends and on some days during the week. A substantial percentage of vehicles on most weekends operate for a short time to serve a small subset of trips, which we refer to as special-demand trips. We believe that the existence of these trips requires additional vehicles because of the way their pick-up and drop-off times and locations are distributed spatiotemporally. c, The q−q plot (where the quantile is the fraction of points below the given value) showing the percentage of trips served (vertical axis), using the vehicle-shareability minimum fleet optimization, with the percentage of vehicles represented on the horizontal axis. Vehicles are sorted on the basis of their total operation time, that is, the vehicles with longer operation times appear to the left of those with shorter operation times on the horizontal axis of these plots. Each panel corresponds to a day of the week and the curves in each panel represent all such days in the entire year (for example, all Mondays). On most weekends and consistent with the patterns observed in b, a large percentage of vehicles (between 5% and 10%) serve only a very small percentage of trips (<1%). This can be observed from the cusps appearing near the top of some of the curves.