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Conference Papers ACM SIGMETRICS Performance Evaluation Review Year : 2020

## A mean field analysis of a stochastic model for reservation in car-sharing systems

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Christine Fricker
• Function : Author
• PersonId : 831323
Cedric Bourdais
• Function : Author
• PersonId : 1089965

#### Abstract

Over the past decade, vehicle-sharing systems have appeared as a new answer to mobility challenges, like reducing congestion or pollution for numerous cities. In this paper we analyze a simple homogeneous stochastic model for station-based car-sharing systems with one-way trips where users reserve the parking space when the car is picked up. In these systems, users arrive at a station, pick up a vehicle while reserve a parking space at a destination station, use it for a while and then return it at the reserved parking space. Each station has a finite capacity and cannot host more vehicles and reserved parking spaces than its capacity. If the user cannot pick up a car or reserve at destination, he leaves the system. For this model, the large scale behavior is investigated via mean field approach. We derive asymptotics of the empirical measure process when the number of cars and stations are large together, such that their ratio tends to a constant. This gives the limiting distribution of the state of a station as the solution of a differential equation, called the Fokker-Planck equation. Then the main result is that the equilibrium point, characterized using queuing theory, exists and is unique. The proof uses a monotonicity argument as for bike-sharing systems, but also needs implicit function theorem and combinatorial arguments. It allows to study the system performance in terms of large scale stationary proportion of empty and full stations, especially the influence of the fleet size. For the optimal fleet size, we give asymptotics for this quantity in light and heavy traffic. We prove that, in light traffic case, reservation has little impact, unlike the heavy traffic case.

#### Domains

Mathematics [math]

### Dates and versions

hal-03127674 , version 1 (02-02-2021)

### Identifiers

• HAL Id : hal-03127674 , version 1

### Cite

Christine Fricker, Hanene Mohamed, Cedric Bourdais. A mean field analysis of a stochastic model for reservation in car-sharing systems. MAMA - ACM SIGMETRICS 2020 Workshop on MAthematical performance Modeling and Analysis, Jun 2020, Boston / Virtual, United States. pp.18-20. ⟨hal-03127674⟩

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