arXiv subjects

François Baccelli

Publications and source records attributed to François Baccelli.

At least 55 records · Page 3Linked to original sources

arXiv2018

Doeblin Trees

This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a bridge graph, generated by paths started in a fixed state at any time. The bridge graph is made into a unimodular network by marking it and selecting a root in a specified fashion. The unimodularity of this network is leveraged to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e., recurrence both forwards and backwards in time, is introduced and shown to be a key property in uniquely distinguishing paths in the Doeblin graph, and also a decisive property for Markov chains indexed by $\mathbb{Z}$. Properties related to simulating the bridge graph are also studied.

math.PR↗

arXiv2018

The stochastic geometry of unconstrained one-bit data compression

A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of $\mathbb{R}^n$ or a stationary Poisson point process in $\mathbb{R}^n$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and in source coding.

math.PR↗

arXiv2018

Reach of Repulsion for Determinantal Point Processes in High Dimensions

Goldman [7] proved that the distribution of a stationary determinantal point process (DPP) $Φ$ can be coupled with its reduced Palm version $Φ^{0,!}$ such that there exists a point process $η$ where $Φ= Φ^{0,!} \cup η$ in distribution and $Φ^{0,!} \cap η= \emptyset$. The points of $η$ characterize the repulsive nature of a typical point of $Φ$. In this paper, the first moment measure of $η$ is used to study the repulsive behavior of DPPs in high dimensions. It is shown that many families of DPPs have the property that the total number of points in $η$ converges in probability to zero as the space dimension $n$ goes to infinity. It is also proved that for some DPPs there exists an $R^*$ such that the decay of the first moment measure of $η$ is slowest in a small annulus around the sphere of radius $\sqrt{n}R^*$. This $R^*$ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high dimensional Boolean models is given.

math.PR↗

arXiv2018

Poisson Cox Point Processes for Vehicular Networks

This paper analyzes statistical properties of the Poisson line Cox point process useful in the modeling of vehicular networks. The point process is created by a two-stage construction: a Poisson line process to model road infrastructure and independent Poisson point processes, conditionally on the Poisson lines, to model vehicles on the roads. We derive basic properties of the point process, including the general quadratic position of the points, the nearest distance distribution, the Laplace functional, the densities of facets of the Cox-Voronoi tessellation, and the asymptotic behavior of the typical Voronoi cell under vehicular densification. These properties are closely linked to features that are important in vehicular networks.

cs.IT↗

arXiv2018

Renewal Population Dynamics and their Eternal Family Trees

Based on a simple object, an i.i.d. sequence of positive integer-valued random variables, $\{a_n\}_{n\in \mathbb{Z}}$, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time $n$ and dies at time $n+a_n$. This dynamics is that of a D/GI/$\infty$ queue, with arrivals at integer times and service times given by $\{a_n\}_{n\in \mathbb{Z}}$. Second, the directed random graph $T^f$ on $\mathbb{Z}$ generated by the random map $f(n)=n+a_n$. Only assuming $\mathbb{E}[a_0]<\infty$ and $\mathbb{P}[a_0=1]>0$, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regenerative epochs. We identify a unimodular structure in this dynamics. More precisely, $T^f$ is a unimodular directed tree, in which $f(n)$ is the parent of $n$. This tree has a unique bi-infinite path. Moreover, $T^f$ splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regenerative epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regenerative integers form stationary and mixing point processes on $\mathbb{Z}$.

math.PR↗

arXiv2017

An Analytical Framework for Coverage in Cellular Networks Leveraging Vehicles

This paper analyzes an emerging architecture of cellular network utilizing both planar base stations uniformly distributed in Euclidean plane and base stations located on roads. An example of this architecture is that where, in addition to conventional planar cellular base stations and users, vehicles also play the role of both base stations and users. A Poisson line process is used to model the road network and, conditionally on the lines, linear Poisson point processes are used to model the vehicles on the roads. The conventional planar base stations and users are modeled by independent planar Poisson point processes. The joint stationarity of the elements in this model allows one to use Palm calculus to investigate statistical properties of such a network. Specifically, this paper discusses two different Palm distributions, with respect to the user point processes depending on its type: planar or vehicular. We derive the distance to the nearest base station, the association of the typical users, and the coverage probability of the typical user in terms of integral formulas. Furthermore, this paper provides a comprehensive characterization of the performance of all possible cellular transmissions in this setting, namely vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I), infrastructure-to-vehicle (I2V), and infrastructure-to-infrastructure (I2I) communications.

cs.IT↗

arXiv2017

SINR and Throughput of Dense Cellular Networks with Stretched Exponential Path Loss

Distance-based attenuation is a critical aspect of wireless communications. As opposed to the ubiquitous power-law path loss model, this paper proposes a stretched exponential path loss model that is suitable for short-range communication. In this model, the signal power attenuates over a distance $r$ as $e^{-αr^β}$, where $α,β$ are tunable parameters. Using experimental propagation measurements, we show that the proposed model is accurate for short to moderate distances in the range $r \in (5,300)$ meters and so is a suitable model for dense and ultradense networks. We integrate this path loss model into a downlink cellular network with base stations modeled by a Poisson point process, and derive expressions for the coverage probability, potential throughput, and area spectral efficiency. Although the most general result for coverage probability has a double integral, several special cases are given where the coverage probability has a compact or even closed form. We then show that the potential throughput is maximized for a particular BS density and then collapses to zero for high densities, assuming a fixed SINR threshold. We next prove that the area spectral efficiency, which assumes an adaptive SINR threshold, is non-decreasing with the BS density and converges to a constant for high densities.

cs.IT↗

arXiv2017

Eternal Family Trees and Dynamics on Unimodular Random Graphs

This paper is centered on covariant dynamics on unimodular random graphs and random networks, namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues the stable manifold of the dynamics. The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three classes of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils). An infinite connected component of the graph of a vertex-shift on a random network forms an infinite tree with one selected end which is referred to as an Eternal Family Tree. Such trees can be seen as stochastic extensions of branching processes. Unimodular Eternal Family Trees can be seen as extensions of critical branching processes. The class of offspring-invariant Eternal Family Trees, which is introduced in the paper, allows one to analyze dynamics on networks which are not necessarily unimodular. These can be seen as extensions of non-necessarily critical branching processes. Several construction techniques of Eternal Family Trees are proposed, like the joining of trees or moving the root to a far descendant. Eternal Galton-Watson Trees and Eternal Multi-type Galton-Watson Trees are also introduced as special cases of Eternal Family Trees satisfying additional independence properties. These examples allow one to show that the results on Eternal Family Trees unify and extend to the dependent case several well known theorems of the literature on branching processes.

math.PR↗

arXiv2016

On Shared Rate Time Series for Mobile Users in Poisson Networks

This paper focuses on modeling and analysis of the temporal performance variations experienced by a mobile user in a wireless network and its impact on system level design. We consider a simple stochastic geometry model: the infrastructure nodes are Poisson distributed while the user's motion is the simplest possible i.e., constant velocity on a straight line. We first characterize variations in the SNR process, and associated downlink Shannon rate, resulting from variations in the infrastructure geometry seen by the mobile. Specifically, by making a connection between stochastic geometry and queueing theory the level crossings of the SNR process are shown to form an alternating renewal process whose distribution can be completely characterized. For large or small SNR levels, and associated rare events, we further derive simple distributional (exponential) models. We then characterize the second major contributor to variation, associated with changes in the number of other users sharing infrastructure. Combining these two effects, we study what are the dominant factors (infrastructure geometry or sharing number) given mobile experiences a very high or low shared rate. These results are then used to evaluate and optimize the system-level Quality of Service (QoS) and system-level capacity to support mobile users sharing wireless infrastructure, including mobile devices streaming video which proactively buffer content to prevent rebuffering and mobiles which are downloading large files. Finally, we use simulation to assess the fidelity of this model and its robustness to factors which are presently taken into account.

cs.IT↗

arXiv2016

Shape Theorems for Poisson Hail on a Bivariate Ground

We consider the extension of the Euclidean stochastic geometry Poisson Hail model to the case where the service speed is zero in some subset of the Euclidean space and infinity in the complement. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process and on the geometrical properties of the zero speed set.

math.PR↗

arXiv2016

Point-Shift Foliation of a Point Process

A point-shift $F$ maps each point of a point process $Φ$ to some point of $Φ$. For all translation invariant point-shifts $F$, the $F$-foliation of $Φ$ is a partition of the support of $Φ$ which is the discrete analogue of the stable manifold of $F$ on $Φ$. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot$, the orbits of which are the $F$-foils of $Φ$, and which are measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.

math.PR↗

arXiv2016

Point-Map-Probabilities of a Point Process and Mecke's Invariant Measure Equation

A compatible point-shift $F$ maps, in a translation invariant way, each point of a stationary point process $Φ$ to some point of $Φ$. It is fully determined by its associated point-map, $f$, which gives the image of the origin by $F$. It was proved by J. Mecke that if $F$ is bijective, then the Palm probability of $Φ$ is left invariant by the translation of $-f$. The initial question motivating this paper is the following generalization of this invariance result: in the non-bijective case, what probability measures on the set of counting measures are left invariant by the translation of $-f$? The point-map probabilities of $Φ$ are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map probability exists, is uniquely defined, and if it satisfies certain continuity properties, it then provides a solution to this invariant measure problem. Point-map probabilities are objects of independent interest. They are shown to be a strict generalization of Palm probabilities: when $F$ is bijective, the point-map probability of $Φ$ boils down to the Palm probability of $Φ$. When it is not bijective, there exist cases where the point-map probability of $Φ$ is singular with respect to its Palm probability. A tightness based criterion for the existence of the point-map probabilities of a stationary point process is given. An interpretation of the point-map probability as the conditional law of the point process given that the origin has $F$-pre-images of all orders is also provided. The results are illustrated by a few examples.

math.PR↗

arXiv2015

Best Signal Quality in Cellular Networks: Asymptotic Properties and Applications to Mobility Management in Small Cell Networks

The quickly increasing data traffic and the user demand for a full coverage of mobile services anywhere and anytime are leading mobile networking into a future of small cell networks. However, due to the high-density and randomness of small cell networks, there are several technical challenges. In this paper, we investigate two critical issues: \emph{best signal quality} and \emph{mobility management}. Under the assumptions that base stations are uniformly distributed in a ring shaped region and that shadowings are lognormal, independent and identically distributed, we prove that when the number of sites in the ring tends to infinity, then (i) the maximum signal strength received at the center of the ring tends in distribution to a Gumbel distribution when properly renormalized, and (ii) it is asymptotically independent of the interference. Using these properties, we derive the distribution of the best signal quality. Furthermore, an optimized random cell scanning scheme is proposed, based on the evaluation of the optimal number of sites to be scanned for maximizing the user data throughput.

cs.IT↗

arXiv2014

Statistical Modeling and Probabilistic Analysis of Cellular Networks with Determinantal Point Processes

Although the Poisson point process (PPP) has been widely used to model base station (BS) locations in cellular networks, it is an idealized model that neglects the spatial correlation among BSs. The present paper proposes the use of determinantal point process (DPP) to take into account these correlations; in particular the repulsiveness among macro base station locations. DPPs are demonstrated to be analytically tractable by leveraging several unique computational properties. Specifically, we show that the empty space function, the nearest neighbor function, the mean interference and the signal-to-interference ratio (SIR) distribution have explicit analytical representations and can be numerically evaluated for cellular networks with DPP configured BSs. In addition, the modeling accuracy of DPPs is investigated by fitting three DPP models to real BS location data sets from two major U.S. cities. Using hypothesis testing for various performance metrics of interest, we show that these fitted DPPs are significantly more accurate than popular choices such as the PPP and the perturbed hexagonal grid model.

cs.IT↗

arXiv2014

On Scaling Limits of Power Law Shot-noise Fields

This article studies the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, it is shown that the shot-noise field can be scaled suitably to have a $α$-stable limit, intensity of the underlying point process goes to infinity. It is also shown that the finite dimensional distributions of the limiting random field have i.i.d. stable random components. We hence propose to call this limte the $α$- stable white noise field. Analogous results are also obtained for the extremal shot-noise field which converges to a Fréchet white noise field. Finally, these results are applied to the analysis of wireless networks.

math.PR↗

arXiv2014

The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Consider a family of Boolean models, indexed by integers $n \ge 1$, where the $n$-th model features a Poisson point process in ${\mathbb{R}}^n$ of intensity $e^{n ρ_n}$ with $ρ_n \to ρ$ as $n \to \infty$, and balls of independent and identically distributed radii distributed like $\bar X_n \sqrt{n}$, with $\bar X_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: $τ_d$ the degree threshold; $τ_p$ the percolation threshold; and $τ_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $ρ< τ_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for $τ_d< ρ< τ_p$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for $τ_p< ρ< τ_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for $τ_d< ρ< τ_v$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $ρ> τ_v$, the whole space covered. The analysis of this asymptotic regime is motivated by related problems in information theory, and may be of interest in other applications of stochastic geometry.

math.PR↗

arXiv2013

On the Generating Functionals of a Class of Random Packing Point Processes

Consider a symmetrical conflict relationship between the points of a point process. The Matérn type constructions provide a generic way of selecting a subset of this point process which is conflict-free. The simplest one consists in keeping only conflict-free points. There is however a wide class of Matérn type processes based on more elaborate selection rules and providing larger sets of selected points. The general idea being that if a point is discarded because of a given conflict, there is no need to discard other points with which it is also in conflict. The ultimate selection rule within this class is the so called Random Sequential Adsorption, where the cardinality of the sequence of conflicts allowing one to decide whether a given point is selected is not bounded. The present paper provides a sufficient condition on the span of the conflict relationship under which all the above point processes are well defined when the initial point process is Poisson. It then establishes, still in the Poisson case, a set of differential equations satisfied by the probability generating functionals of these Matérn type point processes. Integral equations are also given for the Palm distributions.

math.PR↗

arXiv2013

Queuing Networks with Varying Topology -- A Mean-Field Approach

We consider the queuing networks, which are made from servers, exchanging their positions. The customers, using the network, try to reach their destinations, which is complicated by the movements of the servers, taking their customers with them, while they wait for the service. We develop the general theory of such networks, and we establish the convergence of the symmetrized version of the network to the Non-Linear Markov Process.

math.PR↗