Incentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity

Bike-sharing systems are becoming important for urban transportation. In these systems, users arrive at a station, pick up a bike, use it for a while, and then return it to another station of their choice. Each station has a finite capacity: it cannot host more bikes than its capacity. We propose a stochastic model of an homogeneous bike-sharing system and study the effect of the randomness of user choices on the number of problematic stations, i.e., stations that, at a given time, have no bikes available or no available spots for bikes to be returned to. We quantify the influence of the station capacities, and we compute the fleet size that is optimal in terms of minimizing the proportion of problematic stations. Even in a homogeneous city, the system exhibits a poor performance: the minimal proportion of problematic stations is of the order of the inverse of the capacity. We show that simple incentives, such as suggesting users to return to the least loaded station among two stations, improve the situation by an exponential factor. We also compute the rate at which bikes have to be redistributed by trucks for a given quality of service. This rate is of the order of the inverse of the station capacity. For all cases considered, the fleet size that corresponds to the best performance is half of the total number of spots plus a few more, the value of the few more can be computed in closed-form as a function of the system parameters. It corresponds to the average number of bikes in circulation.