Séminaire MODAL'X : Alessia Rigonat (Inria et Modal'X, Université Paris Nanterre)

Résumé : 
This talk deals with the mean-field limit of M_N ∼ sN type 1 particles, moving
between N sites whose dynamics also depend on a random environment. In the
model, the environment consists of many other particles which enter and leave
each site independently. The environment and the type 1 particles interact due
to 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. The aim
of the study is to prove this averaging principle in a large-scale system. In
the overloaded regime, when the system becomes large, the limiting stationary
numbers of empty slots and of type 1 particles are independent, with geometric
distributions whose parameters have explicit expressions.
The model is motivated by free-floating car sharing systems. It gives a new
approach which takes into account the interactions between free-floating cars
and private cars sharing the public space. The service area is divided into N
zones. The capacity of each zone is the number of parking spaces of the public
space, also with order N . We proved 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, the dimensioning problem is also
solved: the more shared cars, the better the system.