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Bike-sharing systems are becoming important for urban transportation. In suchsystems, users arrive at a station, take a bike and use it for a while, thenreturn it to another station of their choice. Each station has a finitecapacity: it cannot host more bikes than its capacity. We propose a stochasticmodel of an homogeneous bike-sharing system and study the effect of usersrandom choices on the number of problematic stations, i.e., stations that, at agiven time, have no bikes available or no available spots for bikes to bereturned to. We quantify the influence of the station capacities, and wecompute the fleet size that is optimal in terms of minimizing the proportion ofproblematic stations. Even in a homogeneous city, the system exhibits a poorperformance: the minimal proportion of problematic stations is of the order of(but not lower than) the inverse of the capacity. We show that simpleincentives, such as suggesting users to return to the least loaded stationamong two stations, improve the situation by an exponential factor. We alsocompute the rate at which bikes have to be redistributed by trucks to insure agiven quality of service. This rate is of the order of the inverse of thestation capacity. For all cases considered, the fleet size that corresponds tothe best performance is half of the total number of spots plus a few more, thevalue of the few more can be computed in closed-form as a function of thesystem parameters. It corresponds to the average number of bikes incirculation.

Incentives and Redistribution in Homogeneous Bike-Sharing Systems with Stations of Finite Capacity