Christine Fricker
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Christine Fricker has a total of 24 co-authors. Here are the top co-authors by number of shared publications:
- Philippe Robert
Université Côte d’Azur, Cognition Behaviour Technology Lab (CoBTeK), Nice France
11 shared publications - Fabrice Guillemin
Orange Labs, Lannion, France
9 shared publications - Nelson Antunes
Center for Computational and Stochastic Mathematics, University of Lisbon, Lisbon, Portugal
5 shared publications - Hanene Mohamed
MODAL’X, UMR CNRS 9023, UPL, Université Paris Nanterre, Nanterre, France
3 shared publications - James Roberts
INRIA France
3 shared publications - Yousra Chabchoub
INRIA, Le Chesnay
3 shared publications
Latest Publications
- Open Access
Stability and renormalization of Jackson networks with non-idling mobile servers
A tandem of two queues sharing a pool of servers, where users take the time to switch to thesecond queue, is used to model a typical pathway through an emergency department (ED), wherepatients undergo two consultations separated by diagnostic tests. In this article, explicit conditionsfor ergodicity and transience are given and proven via Foster’s criterion, using a linear Lyapunovfunction. This result is extended to a Jackson network, with the key difference that the nodes sharea pool of servers, with a non-idling service policy. Further, the delay times for customers to movefrom one node to another must be taken into account. This covers some of the main features ofmodels for emergency departments, namely priorities (triage) between patients. In the case of thetandem queue, by scaling the arrival rate and the number of servers by N, we obtain a renormalizedprocess converging to the solution of an ordinary differential equation (ODE) subject to boundaryconditions. When the system is ergodic, we discuss the solution of this ODE as t → ∞.
Open AccessStability and renormalization of Jackson networks with non-idling mobile servers
A tandem of two queues sharing a pool of servers, where users take the time to switch to thesecond queue, is used to model a typical pathway through an emergency department (ED), wherepatients undergo two consultations separated by diagnostic tests. In this article, explicit conditionsfor ergodicity and transience are given and proven via Foster’s criterion, using a linear Lyapunovfunction. This result is extended to a Jackson network, with the key difference that the nodes sharea pool of servers, with a non-idling service policy. Further, the delay times for customers to movefrom one node to another must be taken into account. This covers some of the main features ofmodels for emergency departments, namely priorities (triage) between patients. In the case of thetandem queue, by scaling the arrival rate and the number of servers by N, we obtain a renormalizedprocess converging to the solution of an ordinary differential equation (ODE) subject to boundaryconditions. When the system is ergodic, we discuss the solution of this ODE as t → ∞.
- 18 October 2025
- Published by Springer Nature in Queueing Systems
- Vol. 109 (4) , 1-35
Stochastic averaging and mean-field for a large system with fast varying environment with applications to free-floating car-sharing
- 18 October 2025
- Published by Springer Nature in Queueing Systems
- Vol. 109 (4) , 1-35
Stochastic averaging and mean-field for a large system with fast varying environment with applications to free-floating car-sharing
- Open Access
Thermodynamical limits for models of car-sharing systems: the Autolib example
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 $K$.Open AccessThermodynamical limits for models of car-sharing systems: the Autolib example
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 $K$. - 6 January 2025
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 62 (2) , 832-858
Mean-field analysis of stochastic networks with reservation
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.- 6 January 2025
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 62 (2) , 832-858
Mean-field analysis of stochastic networks with reservation
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.- 8 April 2024
- Published by Springer Nature in La Matematica
- Vol. 3 (2) , 651-676
An Incentive Algorithm for a Closed Stochastic Network: Data and Mean-Field Analysis
- 8 April 2024
- Published by Springer Nature in La Matematica
- Vol. 3 (2) , 651-676
An Incentive Algorithm for a Closed Stochastic Network: Data and Mean-Field Analysis
- Open Access
Stationary analysis of the shortest queue problem
A simple analytical solution is proposed for the stationary loss system of two parallel queues with finite capacity $K$, in which new customers join the shortest queue, or one of the two with equal probability if their lengths are equal. The arrival process is Poisson, service times at each queue have exponential distributions with the same parameter, and both queues have equal capacity. Using standard generating function arguments, a simple expression for the blocking probability is derived, which as far as we know is original. Using coupling arguments and explicit formulas, comparisons with related loss systems are then provided. Bounds are similarly obtained for the average total number of customers, with the stationary distribution explicitly determined on $\{K, \dots, 2K \}$, and elsewhere upper bounded. Furthermore, from the balance equations, all stationary probabilities are obtained as explicit combinations of their values at states $(0,k)$ for $0 \le k \le K$. These expressions extend to the infinite capacity and asymmetric cases, i.e., when the queues have different service rates. For the initial symmetric finite capacity model, the stationary probabilities of states $(0,k)$ can be obtained recursively from the blocking probability. In the other cases, they are implicitly determined through a functional equation that characterizes their generating function. The whole approach shows that the stationary distribution of the infinite capacity symmetric process is the limit of the corresponding finite capacity distributions. For the infinite capacity symmetric model, we provide an elementary proof of a result by Cohen which gives the solution of the functional equation in terms of an infinite product with explicit zeroes and poles.Open AccessStationary analysis of the shortest queue problem
A simple analytical solution is proposed for the stationary loss system of two parallel queues with finite capacity $K$, in which new customers join the shortest queue, or one of the two with equal probability if their lengths are equal. The arrival process is Poisson, service times at each queue have exponential distributions with the same parameter, and both queues have equal capacity. Using standard generating function arguments, a simple expression for the blocking probability is derived, which as far as we know is original. Using coupling arguments and explicit formulas, comparisons with related loss systems are then provided. Bounds are similarly obtained for the average total number of customers, with the stationary distribution explicitly determined on $\{K, \dots, 2K \}$, and elsewhere upper bounded. Furthermore, from the balance equations, all stationary probabilities are obtained as explicit combinations of their values at states $(0,k)$ for $0 \le k \le K$. These expressions extend to the infinite capacity and asymmetric cases, i.e., when the queues have different service rates. For the initial symmetric finite capacity model, the stationary probabilities of states $(0,k)$ can be obtained recursively from the blocking probability. In the other cases, they are implicitly determined through a functional equation that characterizes their generating function. The whole approach shows that the stationary distribution of the infinite capacity symmetric process is the limit of the corresponding finite capacity distributions. For the infinite capacity symmetric model, we provide an elementary proof of a result by Cohen which gives the solution of the functional equation in terms of an infinite product with explicit zeroes and poles. - Top Cited PapersOpen Access
- 1 August 2016
- Published by Elsevier in EURO Journal on Transportation and Logistics
- Vol. 5 (3) , 261-291
Incentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity
- 1 August 2016
- Published by Elsevier in EURO Journal on Transportation and Logistics
- Vol. 5 (3) , 261-291
Top Cited PapersOpen AccessIncentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity
- 26 November 2015
- Published by Springer Nature in Queueing Systems
- Vol. 82 (1-2) , 173-197
Two-choice regulation in heterogeneous closed networks
- 26 November 2015
- Published by Springer Nature in Queueing Systems
- Vol. 82 (1-2) , 173-197
Two-choice regulation in heterogeneous closed networks
- Open Access
Equivalence of ensembles for large vehicle-sharing models
For a class of large closed Jackson networks submitted to capacity constraints, asymptotic independence of the nodes in normal traffic phase is proved at stationarity under mild assumptions, using a Local Limit Theorem. The limiting distributions of the queues are explicit. In the Statistical Mechanics terminology, the equivalence of ensembles - canonical and grand canonical - is proved for specific marginals. The framework includes the case of networks with two types of nodes: single server/finite capacity nodes and infinite servers/infinite capacity nodes, that can be taken as basic models for bike-sharing systems. The effect of local saturation is modeled by generalized blocking and rerouting procedures, under which the stationary state is proved to have product-form. The grand canonical approximation can then be used for adjusting the total number of bikes and the capacities of the stations to the expected demand.
Open AccessEquivalence of ensembles for large vehicle-sharing models
For a class of large closed Jackson networks submitted to capacity constraints, asymptotic independence of the nodes in normal traffic phase is proved at stationarity under mild assumptions, using a Local Limit Theorem. The limiting distributions of the queues are explicit. In the Statistical Mechanics terminology, the equivalence of ensembles - canonical and grand canonical - is proved for specific marginals. The framework includes the case of networks with two types of nodes: single server/finite capacity nodes and infinite servers/infinite capacity nodes, that can be taken as basic models for bike-sharing systems. The effect of local saturation is modeled by generalized blocking and rerouting procedures, under which the stationary state is proved to have product-form. The grand canonical approximation can then be used for adjusting the total number of bikes and the capacities of the stations to the expected demand.
- Open Access
Two-choice regulation in heterogeneous closed networks
A heterogeneous closed network with one-server queues with finite capacity and one infinite-server queue is studied. A target application is bike-sharing systems. Heterogeneity is taken into account through clusters whose queues have the same parameters. Incentives to the customer to go to the least loaded one-server queue among two chosen within a cluster are investigated. By mean-field arguments, the limiting queue length stationary distribution as the number of queues gets large is analytically tractable. Moreover, when all customers follow incentives, the probability that a queue is empty or full is approximated. Sizing the system to improve performance is reachable under this policy.
Open AccessTwo-choice regulation in heterogeneous closed networks
A heterogeneous closed network with one-server queues with finite capacity and one infinite-server queue is studied. A target application is bike-sharing systems. Heterogeneity is taken into account through clusters whose queues have the same parameters. Incentives to the customer to go to the least loaded one-server queue among two chosen within a cluster are investigated. By mean-field arguments, the limiting queue length stationary distribution as the number of queues gets large is analytically tractable. Moreover, when all customers follow incentives, the probability that a queue is empty or full is approximated. Sizing the system to improve performance is reachable under this policy.