Mean field analysis of stochastic networks with reservation

Christine Fricker (1) , Hanene Mohamed (2)

1 DYOGENE - Dynamics of Geometric Networks
2 MODAL'X - Modélisation aléatoire de Paris X

The problem of reservation in a large distributed system is analysed via a new mathematical model. The target application is car-sharing systems. This model is precisely motivated by the large station-based car-sharing system in France, called Autolib'. This system can be described as a closed stochastic network where the nodes are the stations and the customers are the cars. The user can reserve the car and the parking space. In the paper, we study the evolution of the system when the reservation of parking spaces and cars is effective for all users. The asymptotic behaviour of the underlying stochastic network is given when the number $N$ of stations and the fleet size $M$ increase at the same rate. The analysis involves a Markov process on a state space with dimension of order $N^2$. It is quite remarkable that the state process describing the evolution of the stations, whose dimension is of order $N$, converges in distribution, although not Markov, to an non-homogeneous Markov process. We prove this mean-field convergence. We also prove, using combinatorial arguments, that the mean-field limit has a unique equilibrium measure when the time between reserving and picking up the car is sufficiently small. This result extends the case where only the parking space can be reserved.

Mots clés: Stochastic Networks; Reservation; Mean-Field; Non-Linear Markov Process; Finite Capacity Queues; Product-Form Distribution

Domaines: Mathématiques [math]

Christine Fricker, Hanene Mohamed. Mean field analysis of stochastic networks with reservation. 2023. ⟨hal-03539104v2⟩