arXiv subjects

Christine Fricker

Publications and source records attributed to Christine Fricker.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotics of Parking Search in Hyperfractal Networks

We study the asymptotic behaviour of the distance to the first available parking slot in a recursive Manhattan street network endowed with a hyperfractal intensity structure, where slot-release events occur according to Poisson processes along the streets. We establish, by analysing the associated self-similar harmonic sums via Mellin-transform asymptotics, a power-law decay of the expected distance as the total intensity grows, with exponent equal to the inverse of the hyperfractal dimension. In particular, the scaling exponent depends only on the large-scale geometry of the network. We further prove that this exponent is robust under random multiplicative modulations of the street intensities: mild stochastic heterogeneity affects only the multiplicative constant. Similar scaling behaviour holds for the variance, the number of turns before parking, and for a jump-over variant of the search strategy.

math.PR↗

arXiv2025

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 $K$.

math.PR↗

arXiv2022

Mean field analysis of stochastic networks with reservation

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.

math.PR↗

arXiv2017

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.

math.PR↗

arXiv2017

Allocation Schemes of Ressources with Downgrading

We consider a server with large capacity delivering video files encoded in various resolutions. We assume that the system is under saturation in the sense that the total demand exceeds the server capacity $C$. In such a case, requests may be rejected. For the policies considered in this paper, instead of rejecting a video request, it is downgraded. When the occupancy of the server is above some value $C_0{<}C$, the server delivers the video at a minimal bit rate. The quantity $C_0$ is the bit rate adaptation threshold. For these policies, request blocking is thus replaced by bit rate adaptation. Under the assumptions of Poisson request arrivals and exponential service times, we show that, by rescaling the system, a process associated to the occupancy of the server converges to some limiting process whose invariant distribution is computed by a Wiener-Hopf technique. This allows us to derive an asymptotic expression of the key performance measure of such a policy, the equilibrium probability that a request is transmitted without downgrading. Numerical applications of these results are presented.

cs.NI↗

arXiv2016

Analysis of an offloading scheme for data centers in the framework of fog computing

In the context of fog computing, we consider a simple case when data centers are installed at the edge of the network and assume that if a request arrives at an overloaded data center, then it is forwarded to a neighboring data center with some probability. Data centers are assumed to have a large number of servers and that traffic at some of them is causing saturation. In this case the other data centers may help to cope with this saturation regime by accepting some of the rejected requests. Our aim is to qualitatively estimate the gain achieved via cooperation between neighboring data centers. After proving some convergence results, related to the scaling limits of loss systems, for the process describing the number of free servers at both data centers, we show that the performance of the system can be expressed in terms of the invariant distribution of a random walk in the quarter plane. By using and developing existing results in the technical literature, explicit formulas for the blocking rates of such a system are derived.

cs.NI↗

arXiv2015

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.

math.PR↗

arXiv2015

A Stochastic Model for Car-Sharing Systems

Vehicle-sharing systems are becoming important for urban transportation. In these systems, users arrive at a station, pick up a vehicle, use it for a while and then return it to another station of their choice. Depending on the type of system, there might be a possibility to book vehicles before picking-up and/or a parking space at the chosen arrival station. Each station has a finite capacity and cannot host more vehicles and reserved parking spaces than its capacity. We propose a stochastic model for an homogeneous car-sharing system with possibility to reserve a parking space at the arrival station when picking-up a car. We compute the performance of the system and the optimal fleet size according to a specific metric. It differs from a similar model for bike-sharing systems because of reservation that induces complexity, especially when traffic increases.

math.PR↗

arXiv2014

Impatience in mobile networks and its application to data pricing

We consider in this paper an import Quality of Experience (QoE) indicator in mobile networks that is reneging of users due to impatience. We specifically consider a cell under heavy load conditions and compute the reneging probability by using a fluid limit analysis. By solving the fixed point equation, we obtain a new QoE perturbation metric quantifying the impact of reneging on the performance of the system. This metric is then used to devise a new pricing scheme accounting of reneging. We specifically propose several flavors of this pricing around the idea of having a flat rate for accessing the network and an elastic price related to the level of QoE perturbation induced by communications.

cs.NI↗

arXiv2014

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.

nlin.AO↗

arXiv2014

Incentives and Redistribution in Homogeneous Bike-Sharing Systems with Stations of Finite Capacity

Bike-sharing systems are becoming important for urban transportation. In such systems, users arrive at a station, take a bike and use it for a while, then return it to another station of their choice. Each station has a finite capacity: it cannot host more bikes than its capacity. We propose a stochastic model of an homogeneous bike-sharing system and study the effect of users random choices on the number of problematic stations, i.e., stations that, at a given time, have no bikes available or no available spots for bikes to be returned to. We quantify the influence of the station capacities, and we compute the fleet size that is optimal in terms of minimizing the proportion of problematic stations. Even in a homogeneous city, the system exhibits a poor performance: the minimal proportion of problematic stations is of the order of (but not lower than) the inverse of the capacity. We show that simple incentives, such as suggesting users to return to the least loaded station among two stations, improve the situation by an exponential factor. We also compute the rate at which bikes have to be redistributed by trucks to insure a given quality of service. This rate is of the order of the inverse of the station capacity. For all cases considered, the fleet size that corresponds to the best performance is half of the total number of spots plus a few more, the value of the few more can be computed in closed-form as a function of the system parameters. It corresponds to the average number of bikes in circulation.

nlin.AO↗

arXiv2012

A versatile and accurate approximation for LRU cache performance

In a 2002 paper, Che and co-authors proposed a simple approach for estimating the hit rates of a cache operating the least recently used (LRU) replacement policy. The approximation proves remarkably accurate and is applicable to quite general distributions of object popularity. This paper provides a mathematical explanation for the success of the approximation, notably in configurations where the intuitive arguments of Che, et al clearly do not apply. The approximation is particularly useful in evaluating the performance of current proposals for an information centric network where other approaches fail due to the very large populations of cacheable objects to be taken into account and to their complex popularity law, resulting from the mix of different content types and the filtering effect induced by the lower layers in a cache hierarchy.

cs.NI↗

arXiv2012

Impact of traffic mix on caching performance in a content-centric network

For a realistic traffic mix, we evaluate the hit rates attained in a two-layer cache hierarchy designed to reduce Internet bandwidth requirements. The model identifies four main types of content, web, file sharing, user generated content and video on demand, distinguished in terms of their traffic shares, their population and object sizes and their popularity distributions. Results demonstrate that caching VoD in access routers offers a highly favorable bandwidth memory tradeoff but that the other types of content would likely be more efficiently handled in very large capacity storage devices in the core. Evaluations are based on a simple approximation for LRU cache performance that proves highly accurate in relevant configurations.

cs.NI↗

arXiv2010

Traffic Capacity of Large WDM Passive Optical Networks

As passive optical networks (PON) are increasingly deployed to provide high speed Internet access, it is important to understand their fundamental traffic capacity limits. The paper discusses performance models applicable to wavelength division multiplexing (WDM) EPONs and GPONs under the assumption that users access the fibre via optical network units equipped with tunable transmitters. The considered stochastic models are based on multiserver polling systems for which explicit analytical results are not known. A large system asymptotic, mean-field approximation, is used to derive closed form solutions of these complex systems. Convergence of the mean field dynamics is proved in the case of a simple network configuration. Simulation results show that, for a realistic sized PON, the mean field approximation is accurate.

cs.NI↗

arXiv2010

Upstream traffic capacity of a WDM EPON under online GATE-driven scheduling

Passive optical networks are increasingly used for access to the Internet and it is important to understand the performance of future long-reach, multi-channel variants. In this paper we discuss requirements on the dynamic bandwidth allocation (DBA) algorithm used to manage the upstream resource in a WDM EPON and propose a simple novel DBA algorithm that is considerably more efficient than classical approaches. We demonstrate that the algorithm emulates a multi-server polling system and derive capacity formulas that are valid for general traffic processes. We evaluate delay performance by simulation demonstrating the superiority of the proposed scheduler. The proposed scheduler offers considerable flexibility and is particularly efficient in long-reach access networks where propagation times are high.

cs.NI↗

arXiv2009

On the Statistical Characterization of Flows in Internet Traffic with Application to Sampling

A new method of estimating some statistical characteristics of TCP flows in the Internet is developed in this paper. For this purpose, a new set of random variables (referred to as observables) is defined. When dealing with sampled traffic, these observables can easily be computed from sampled data. By adopting a convenient mouse/elephant dichotomy also dependent on traffic, it is shown how these variables give a reliable statistical representation of the number of packets transmitted by large flows during successive time intervals with an appropriate duration. A mathematical framework is developed to estimate the accuracy of the method. As an application, it is shown how one can estimate the number of large TCP flows when only sampled traffic is available. The algorithm proposed is tested against experimental data collected from different types of IP networks.

cs.NI↗

arXiv2009

An identification problem in an urn and ball model with heavy tailed distributions

We consider in this paper an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are i.i.d. random variables with a heavy tailed probability distribution, for instance a Pareto or a Weibull distribution. We draw a small fraction $p\ll 1$ of the total number of balls. The basic problem addressed in this paper is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and Chen-Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. We finally establish explicit bounds between the two distributions when each ball is drawn with fixed probability $p$.

cs.NI↗

arXiv2009

Adaptive algorithms for identifying large flows in IP traffic

We propose in this paper an on-line algorithm based on Bloom filters for identifying large flows in IP traffic (a.k.a. elephants). Because of the large number of small flows, hash tables of these algorithms have to be regularly refreshed. Recognizing that the periodic erasure scheme usually used in the technical literature turns out to be quite inefficient when using real traffic traces over a long period of time, we introduce a simple adaptive scheme that closely follows the variations of traffic. When tested against real traffic traces, the proposed on-line algorithm performs well in the sense that the detection ratio of long flows by the algorithm over a long time period is quite high. Beyond the identification of elephants, this same class of algorithms is applied to the closely related problem of detection of anomalies in IP traffic, e.g., SYN flood due for instance to attacks. An algorithm for detecting SYN and volume flood anomalies in Internet traffic is designed. Experiments show that an anomaly is detected in less than one minute and the targeted destinations are identified at the same time.

cs.NI↗