Section: New Results

Stochastic Networks: Large Bike Sharing Systems

Participants : Christine Fricker, Hanene Mohamed, Danielle Tibi.

This is a collaboration with Nicolas Gast (EPFL) starting in December 2010. Bike sharing systems were launched by numerous cities as a part of urban transportation, for example Velib in 2007 (20 000 bikes, 1 500 stations). One of the major issues is the availability of the resources: bikes or free slots. These systems become a hot topic in Operation Research but studies on these stochastic networks are very few. To our knowledge, no theoretical study of such bike sharing systems exist taking into account the limited capacity of the stations.

We modeled this system in a symmetric case. Mean field limit gives the dynamic of a large system and the limiting stationary behavior of a single station as the system gets large. Analytical results are obtained and convergence proved in the standard model via Lyapounov functions. It allows to find the best ratio of bikes par station and to measure the improvement of incentive mechanisms, as choosing among two stations for example. Redistribution by trucks is also investigated. See [26] .

Further results have been obtained for some heterogeneous systems. By mean field techniques, analytical results are obtained with Hanene Mohamed for systems with clusters (see [17] ).

In a work in progress with Danielle Tibi , a more direct method is used when the network has a product form invariant measure by central and local limit theorem. It is a way to prove in this case the equivalence of ensembles, known in physic statistics. It applies to the simplest non homogeneous model. It gives a way to generalize the cluster case.