Stochastic averaging and mean-field for a large system with fast varying environment with applications to free-floating car-sharing

This work is motivated by free-floating car-sharing systems. In these systems, free-floating cars share public parking space with a much larger number of private cars. We propose an adapted model, which also includes free-floating car reservations. The service area is divided into N zones. The capacity of each zone is the number of parking spaces of the public space, also of order N. In the model, $$M_N \sim sN$$ M N ∼ s N type 1 particles, move between N sites whose dynamics also depend on a random environment. The environment is made up of numerous other particles which enter and leave each site independently. The environment and the type 1 particles interact due to the finite capacity CN of each site. The main feature of the model is that, at each site, the environment evolves on a faster timescale than the type 1 particles. It yields that, in the limit, a site behaves as a M/M/CN/CN loss queue, disturbed by a small number of type 1 particles. A phase transition is obtained between an underloaded regime where the type 1 particles can enter a site with probability 1 and an overloaded regime where a type 1 particle cannot enter a site with some positive probability depending on the parameters of the environment. We prove an averaging principle in a large-scale system. In the overloaded regime, when the system becomes large, the limiting stationary number of empty slots and the limiting stationary number of type 1 particles are independent, with geometric distributions whose parameters have explicit expressions. It is used to show that the operator can increase the size of the car-sharing fleet without reducing the number of available public parking spaces, even if they are scarce. As a result, a dimensioning problem concerning the optimal fleet size is solved: the more shared cars, the better the system.