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Mean-field analysis of stochastic networks with reservation

Part of: Markov processes Special processes

Published online by Cambridge University Press:  06 January 2025

Christine Fricker  and
Hanene Mohamed
Christine Fricker*
Affiliation:
DI-ENS, CNRS, INRIA de Paris
Hanene Mohamed*
Affiliation:
MODAL’X, UMR CNRS 9023, UPL, Université Paris Nanterre
*
*Postal address: 48 rue Barrault, 75013 Paris, France. Email: christine.fricker@inria.fr
**Postal address: 200 avenue de la République, 92000 Nanterre, France. Email: hanene.mohamed@parisnanterre.fr
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Abstract

The problem of reservation in a large distributed system is analyzed via a new mathematical model. The target application is car-sharing systems. This model is 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 a car and a parking space. 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 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 a 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.

Keywords

Stochastic networksreservationmean fieldnon-linear Markov processfinite-capacity queuesproduct-form distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 27
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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