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Thermodynamical limits for models of car-sharing systems: the Autolib example

Guy Fayolle, Christine Fricker

TL;DR

This work analyzes the thermodynamic limit of a large, station-based car-sharing system with parking reservations. By formulating time-inhomogeneous Markov dynamics in mean-field terms, it derives a nonlinear forward Kolmogorov system that governs the joint evolution of reservations and cars, proving existence/uniqueness and exponential convergence to a product-form stationary distribution reflecting a two-queue tandem. A closed-form equilibrium is obtained, with explicit dependence on system load $U$ and the ratio $\rho=\lambda/\mu$, via a mass-conservation constraint that determines a parameter $\beta$. The paper also extends the analysis to two capacity-constrained variants, yielding single-queue and tandem-with-capacity models, each admitting a unique equilibrium and a tractable generating-function description. These results provide theoretical guarantees and explicit steady-state characterizations for Autolib-like CSS, offering insight into how reservations and capacity constraints shape long-run performance in large-scale, time-varying deployments.

Abstract

We analyze mean-field equations obtained for models motivated by a large station-based car-sharing system in France called Autolib. The main focus is on a version where users reserve a parking space when they take a car. In a first model, the reservation of parking spaces is effective for all users (see [4]) and capacity constraints are ignored. The model is carried out in thermodynamical limit, that is when the number $N$ of stations and the number of cars $M_N$ tend to infinity, with $U = \lim_{N\to\infty} M_N/N$. This limit is described by Kolmogorov equations of a two-dimensional time-inhomogeneous Markov process depicting the numbers of reservations and cars at a station. It satisfies a non-linear differential system. We prove analytically that this system has a unique solution, which converges, as $t\to\infty$, to an equilibrium point exponentially fast. Moreover, this equilibrium point corresponds to the stationary distribution of a two queue tandem (reservations, cars), which is here always ergodic. The intensity factor of each queue has an explicit form obtained from an intrinsic mass conservation relationship. Two related models with capacity constraints are briefly presented in the last section: the simplest one with no reservation leads to a one-dimensional problem; the second one corresponds to our first model with finite total capacity $K$.

Thermodynamical limits for models of car-sharing systems: the Autolib example

TL;DR

This work analyzes the thermodynamic limit of a large, station-based car-sharing system with parking reservations. By formulating time-inhomogeneous Markov dynamics in mean-field terms, it derives a nonlinear forward Kolmogorov system that governs the joint evolution of reservations and cars, proving existence/uniqueness and exponential convergence to a product-form stationary distribution reflecting a two-queue tandem. A closed-form equilibrium is obtained, with explicit dependence on system load and the ratio , via a mass-conservation constraint that determines a parameter . The paper also extends the analysis to two capacity-constrained variants, yielding single-queue and tandem-with-capacity models, each admitting a unique equilibrium and a tractable generating-function description. These results provide theoretical guarantees and explicit steady-state characterizations for Autolib-like CSS, offering insight into how reservations and capacity constraints shape long-run performance in large-scale, time-varying deployments.

Abstract

We analyze mean-field equations obtained for models motivated by a large station-based car-sharing system in France called Autolib. The main focus is on a version where users reserve a parking space when they take a car. In a first model, the reservation of parking spaces is effective for all users (see [4]) and capacity constraints are ignored. The model is carried out in thermodynamical limit, that is when the number of stations and the number of cars tend to infinity, with . This limit is described by Kolmogorov equations of a two-dimensional time-inhomogeneous Markov process depicting the numbers of reservations and cars at a station. It satisfies a non-linear differential system. We prove analytically that this system has a unique solution, which converges, as , to an equilibrium point exponentially fast. Moreover, this equilibrium point corresponds to the stationary distribution of a two queue tandem (reservations, cars), which is here always ergodic. The intensity factor of each queue has an explicit form obtained from an intrinsic mass conservation relationship. Two related models with capacity constraints are briefly presented in the last section: the simplest one with no reservation leads to a one-dimensional problem; the second one corresponds to our first model with finite total capacity .
Paper Structure (23 sections, 11 theorems, 96 equations, 1 figure)

This paper contains 23 sections, 11 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A particle model with reservation
  3. A queueing description of the model
  4. Main outcomes
  5. Thermodynamical limit in the unbounded capacity case
  6. Mean-field equations viewed as a tandem of two queues
  7. Results
  8. Proof of Theorem \ref{['thm:S1']}
  9. Subsidiary equations
  10. Equation for
  11. Equation for
  12. Eqution for
  13. Assumption H1
  14. A differential system for
  15. Speed of convergence of
  16. ...and 8 more sections

Key Result

Proposition 2.1

Assume $\vec{\alpha}^N(0)$ converges weakly, as $N\to\infty$, to some fixed distribution $\vec{\alpha}_0\in \Lambda$. Then the empirical measure $\vec{\alpha}^N(t)$ converges in distribution to a deterministic dynamical system denoted by $\vec{\alpha}(t)$, which satisfies the following infinite syst where $\vec{\alpha}(t)\in \Lambda$, $\vec{\alpha}(0)= \vec{\alpha}_0$ and $b(t) \stackrel{\hbox{\ti

Figures (1)

  • Figure 2.1: Dynamics of the tandem. The first queue is of $M(t)/M/\infty$ type and the second queue is a simple $./M/1/\infty$ queue.

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • ...and 14 more