Mean field analysis of stochastic networks with reservation

TL;DR

This paper develops a mean-field mathematical model to analyze reservation dynamics in large stochastic networks, such as car-sharing systems, revealing convergence properties and equilibrium behavior as the system scales.

Contribution

It introduces a novel mean-field analysis for reservation systems in large stochastic networks, extending previous models to include both cars and parking space reservations.

Findings

The system's state process converges to a non-homogeneous Markov process.

The mean-field limit has a unique equilibrium when reservation time is small.

The model extends prior work by incorporating reservations for both cars and parking spaces.

Abstract

The problem of reservation in a large distributed system is analyzed via a new mathematical model. A typical application is a station-based car-sharing system which 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 behavior of the underlying stochastic network is given when the number $N$ of stations and the fleet size 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.