We analyze the so-called mean-field equations obtained for models motivated by a large station-based car-sharing system (in France called Autolib’). Users reserve a parking space when they take a car. In a first model, capacity constraints are ignored and the reservation of parking spaces is effective for all users. The model is carried out in thermodynamical limit, that is when the number n of stations and the number of cars M_n tend to infinity, with U = lim n→∞ M_n/n. In this case, the limit distribution of cars and reservations at a typical station is, at time t, given by a non-linear partial differential equation (PDE) obtained from forward Kolmogorov’s equations of a two-dimensional time-inhomogeneous Markov process. We analytically demonstrate that this PDE has a unique solution, which, as t→∞, converges exponentially fast to an equilibrium point representing the stationary joint distribution of two queues in tandem (reservations + cars). Two related models with capacity constraints are also briefly presented: the simplest, without reservation (the Vélib’ system), leads to a one-dimensional problem; the second corresponds to our first model with a finite total capacity K at each station.

This is a joint work with Christine Fricker (Inria Paris).