Pricing in vehicle sharing systems: optimization in queuing networks with product forms

One-way vehicle sharing systems (VSS) such as Vélib’ Paris are flourishing. The usefulness of VSS for users is highly impacted by the availability of vehicles and parking spots. Most existing systems are ruled by the trips of users. We study the potential interest of influencing the users to improve the performance of the system. We assume that each user is associated with a pair origin–destination (O–D) of stations, and only interacts with the system if his O–D trip is available. We consider leverage that can influence the rate of user requests for each pair O–D, such as a price that will be prohibitive for a prescribed proportion of users. We focus on optimizing the number of trips taken in the system. To provide exact formulas and analytical insights, transportation times are assumed to be null, stations to have infinite capacities and the demand to be stationary over time. In other words, VSS are modelled as closed queuing networks with infinite buffer capacity and Markovian demands. We propose a heuristic based on computing a Maximum Circulation on the demand graph together with a convex integer program solved optimally by a greedy algorithm. For M stations and N vehicles, the performance ratio of this heuristic is proved to be exactly N/(N+M-1). We discuss our understanding on the possibility of extending this result to more realistic models in the perspectives. The complexity of computing optimum policies remains open. Insights on this issue are provided in the Appendix. The Appendix also contains an example showing that VSS can have poor performances without regulation.