arXiv subjects

Francois Baccelli

Publications and source records attributed to Francois Baccelli.

52 records · Page 3Linked to original sources

arXiv2013

Coverage by Pairwise Base Station Cooperation under Adaptive Geometric Policies

We study a cooperation model where the positions of base stations follow a Poisson point process distribution and where Voronoi cells define the planar areas associated with them. For the service of each user, either one or two base stations are involved. If two, these cooperate by exchange of user data and reduced channel information (channel phase, second neighbour interference) with conferencing over some backhaul link. The total user transmission power is split between them and a common message is encoded, which is coherently transmitted by the stations. The decision for a user to choose service with or without cooperation is directed by a family of geometric policies. The suggested policies further control the shape of coverage contours in favor of cell-edge areas. Analytic expressions based on stochastic geometry are derived for the coverage probability in the network. Their numerical evaluation shows benefits from cooperation, which are enhanced when Dirty Paper Coding is applied to eliminate the second neighbour interference.

cs.IT↗

arXiv2013

On Association Cells in Random Heterogeneous Networks

Characterizing user to access point (AP) association strategies in heterogeneous cellular networks (HetNets) is critical for their performance analysis, as it directly influences the load across the network. In this letter, we introduce and analyze a class of association strategies, which we term stationary association, and the resulting association cells. For random HetNets, where APs are distributed according to a stationary point process, the area of the resulting association cells are shown to be the marks of the corresponding point process. Addressing the need of quantifying the load experienced by a typical user, a "Feller-paradox" like relationship is established between the area of the association cell containing origin and that of a typical association cell. For the specific case of Poisson point process and max power/SINR association, the mean association area of each tier is derived and shown to increase with channel gain variance and decrease in the path loss exponents of the corresponding tier.

cs.IT↗

arXiv2013

A Stochastic Geometry Framework for Analyzing Pairwise-Cooperative Cellular Networks

Cooperation in cellular networks has been recently suggested as a promising scheme to improve system performance, especially for cell-edge users. In this work, we use stochastic geometry to analyze cooperation models where the positions of Base Stations (BSs) follow a Poisson point process distribution and where Voronoi cells define the planar areas associated with them. For the service of each user, either one or two BSs are involved. If two, these cooperate by exchange of user data and channel related information with conferencing over some backhaul link. Our framework generally allows variable levels of channel information at the transmitters. In this paper we investigate the case of limited channel state information for cooperation (channel phase, second neighbour interference), but not the fully adaptive case which would require considerable feedback. The total per-user transmission power is further split between the two transmitters and a common message is encoded. The decision for a user to choose service with or without cooperation is directed by a family of geometric policies depending on its relative position to its two closest base stations. An exact expression of the network coverage probability is derived. Numerical evaluation allows one to analyze significant coverage benefits compared to the non-cooperative case. As a conclusion, cooperation schemes can improve system performance without exploitation of extra network resources.

cs.IT↗

arXiv2012

Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks

Cellular networks are in a major transition from a carefully planned set of large tower-mounted base-stations (BSs) to an irregular deployment of heterogeneous infrastructure elements that often additionally includes micro, pico, and femtocells, as well as distributed antennas. In this paper, we develop a tractable, flexible, and accurate model for a downlink heterogeneous cellular network (HCN) consisting of K tiers of randomly located BSs, where each tier may differ in terms of average transmit power, supported data rate and BS density. Assuming a mobile user connects to the strongest candidate BS, the resulting Signal-to-Interference-plus-Noise-Ratio (SINR) is greater than 1 when in coverage, Rayleigh fading, we derive an expression for the probability of coverage (equivalently outage) over the entire network under both open and closed access, which assumes a strikingly simple closed-form in the high SINR regime and is accurate down to -4 dB even under weaker assumptions. For external validation, we compare against an actual LTE network (for tier 1) with the other K-1 tiers being modeled as independent Poisson Point Processes. In this case as well, our model is accurate to within 1-2 dB. We also derive the average rate achieved by a randomly located mobile and the average load on each tier of BSs. One interesting observation for interference-limited open access networks is that at a given SINR, adding more tiers and/or BSs neither increases nor decreases the probability of coverage or outage when all the tiers have the same target-SINR.

cs.IT↗

arXiv2012

Gibbsian Method for the Self-Optimization of Cellular Networks

In this work, we propose and analyze a class of distributed algorithms performing the joint optimization of radio resources in heterogeneous cellular networks made of a juxtaposition of macro and small cells. Within this context, it is essential to use algorithms able to simultaneously solve the problems of channel selection, user association and power control. In such networks, the unpredictability of the cell and user patterns also requires distributed optimization schemes. The proposed method is inspired from statistical physics and based on the Gibbs sampler. It does not require the concavity/convexity, monotonicity or duality properties common to classical optimization problems. Besides, it supports discrete optimization which is especially useful to practical systems. We show that it can be implemented in a fully distributed way and nevertheless achieves system-wide optimality. We use simulation to compare this solution to today's default operational methods in terms of both throughput and energy consumption. Finally, we address concrete issues for the implementation of this solution and analyze the overhead traffic required within the framework of 3GPP and femtocell standards.

math.OC↗

arXiv2012

Generating Functionals of Random Packing Point Processes: From Hard-Core to Carrier Sensing

In this paper we study the generating functionals of several random packing processes: the classical Matérn hard-core model; its extensions, the $k$-Matérn models and the $\infty$-Matérn model, which is an example of random sequential packing process. We first give a sufficient condition for the $\infty$-Matérn model to be well-defined (unlike the other two, the latter may not be well-defined on unbounded spaces). Then the generating functional of the resulting point process is given for each of the three models as the solution of a differential equation. Series representations and bounds on the generating functional of the packing models are also derived. Last but not least, we obtain moment measures and Palm distributions of the considered packing models departing from their generating functionals.

math.PR↗

arXiv2012

Stochastic Geometry based Medium Access Games in Mobile Ad hoc Networks

This paper studies the performance of Mobile Ad hoc Networks (MANETs) when the nodes, that form a Poisson point process, selfishly choose their Medium Access Probability (MAP). We consider goodput and delay as the performance metric that each node is interested in optimizing taking into account the transmission energy costs. We introduce a pricing scheme based on the transmission energy requirements and compute the symmetric Nash equilibria of the game in closed form. It is shown that by appropriately pricing the nodes, the selfish behavior of the nodes can be used to achieve the social optimum at equilibrium. The Price of Anarchy is then analyzed for these games. For the game with delay based utility, we bound the price of anarchy and study the effect of the price factor. For the game with goodput based utility, it is shown that price of anarchy is infinite at the price factor that achieves the global optima.

cs.GT↗

arXiv2011

Poisson Hail on a Hot Ground

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACS) of the Euclidean space. These RACS arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACS can be served simultaneously and service is in the First In First Out order: only the hailstones in contact with the ground melt at speed 1, whereas the other ones are queued; a tagged RACS waits until all RACS arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

math.PR↗

arXiv2011

A Tractable Approach to Coverage and Rate in Cellular Networks

Cellular networks are usually modeled by placing the base stations on a grid, with mobile users either randomly scattered or placed deterministically. These models have been used extensively but suffer from being both highly idealized and not very tractable, so complex system-level simulations are used to evaluate coverage/outage probability and rate. More tractable models have long been desirable. We develop new general models for the multi-cell signal-to-interference-plus-noise ratio (SINR) using stochastic geometry. Under very general assumptions, the resulting expressions for the downlink SINR CCDF (equivalent to the coverage probability) involve quickly computable integrals, and in some practical special cases can be simplified to common integrals (e.g., the Q-function) or even to simple closed-form expressions. We also derive the mean rate, and then the coverage gain (and mean rate loss) from static frequency reuse. We compare our coverage predictions to the grid model and an actual base station deployment, and observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks.

cs.IT↗

arXiv2011

Interference Networks with Point-to-Point Codes

The paper establishes the capacity region of the Gaussian interference channel with many transmitter-receiver pairs constrained to use point-to-point codes. The capacity region is shown to be strictly larger in general than the achievable rate regions when treating interference as noise, using successive interference cancellation decoding, and using joint decoding. The gains in coverage and achievable rate using the optimal decoder are analyzed in terms of ensemble averages using stochastic geometry. In a spatial network where the nodes are distributed according to a Poisson point process and the channel path loss exponent is $β> 2$, it is shown that the density of users that can be supported by treating interference as noise can scale no faster than $B^{2/β}$ as the bandwidth $B$ grows, while the density of users can scale linearly with $B$ under optimal decoding.

cs.IT↗

arXiv2011

Series Expansion for Interference in Wireless Networks

The spatial correlations in transmitter node locations introduced by common multiple access protocols makes the analysis of interference, outage, and other related metrics in a wireless network extremely difficult. Most works therefore assume that nodes are distributed either as a Poisson point process (PPP) or a grid, and utilize the independence properties of the PPP (or the regular structure of the grid) to analyze interference, outage and other metrics. But,the independence of node locations makes the PPP a dubious model for nontrivial MACs which intentionally introduce correlations, e.g. spatial separation, while the grid is too idealized to model real networks. In this paper, we introduce a new technique based on the factorial moment expansion of functionals of point processes to analyze functions of interference, in particular outage probability. We provide a Taylor-series type expansion of functions of interference, wherein increasing the number of terms in the series provides a better approximation at the cost of increased complexity of computation. Various examples illustrate how this new approach can be used to find outage probability in both Poisson and non-Poisson wireless networks.

cs.IT↗

arXiv2010

Information-Theoretic Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements

This paper studies the Shannon regime for the random displacement of stationary point processes. Let each point of some initial stationary point process in $\R^n$ give rise to one daughter point, the location of which is obtained by adding a random vector to the coordinates of the mother point, with all displacement vectors independently and identically distributed for all points. The decoding problem is then the following one: the whole mother point process is known as well as the coordinates of some daughter point; the displacements are only known through their law; can one find the mother of this daughter point? The Shannon regime is that where the dimension $n$ tends to infinity and where the logarithm of the intensity of the point process is proportional to $n$. We show that this problem exhibits a sharp threshold: if the sum of the proportionality factor and of the differential entropy rate of the noise is positive, then the probability of finding the right mother point tends to 0 with $n$ for all point processes and decoding strategies. If this sum is negative, there exist mother point processes, for instance Poisson, and decoding strategies, for instance maximum likelihood, for which the probability of finding the right mother tends to 1 with $n$. We then use large deviations theory to show that in the latter case, if the entropy spectrum of the noise satisfies a large deviation principle, then the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is done for two classes of mother point processes: Poisson and Matérn. The practical interest to information theory comes from the explicit connection that we also establish between this problem and the estimation of error exponents in Shannon's additive noise channel with power constraints on the codewords.

cs.IT↗

arXiv2009

Optimal Paths on the Space-Time SINR Random Graph

We analyze a class of Signal-to-Interference-and-Noise-Ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on the SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. This paper studies optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both "positive" and "negative" results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.

math.PR↗

arXiv2007

The radial spanning tree of a Poisson point process

We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root. We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than $k$ generations away from the origin. This class of spanning trees has applications in many fields and, in particular, in communications.

math.PR↗

arXiv2007

Stationary flows and uniqueness of invariant measures

In this short paper, we consider a quadruple $(Ω, Å, θ, μ)$,where $Å$ is a $σ$-algebra of subsets of $Ω$, and $θ$ is a measurable bijection from $Ω$ into itself that preserves the measure $μ$. For each $B \in Å$, we consider the measure $μ_B$ obtained by taking cycles (excursions) of iterates of $θ$ from $B$. We then derive a relation for $μ_B$ that involves the forward and backward hitting times of $B$ by the trajectory $(θ^n ω, n \in \Z)$ at a point $ω\in Ω$. Although classical in appearance, its use in obtaining uniqueness of invariant measures of various stochastic models seems to be new. We apply the concept to countable Markov chains and Harris processes.

math.PR↗

arXiv2004

Moments and tails in monotone-separable stochastic networks

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework, which was first introduced to derive the stability region for stochastic networks with stationary and ergodic driving sequences, is revisited. It contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. Our purpose is the analysis of the tails of the stationary state variables in the particular case of i.i.d. driving sequences. For this, we establish general comparison relationships between networks of this class and the GI/GI/1/\infty queue. We first use this to show that two classical results of the asymptotic theory for GI/GI/1/\infty queues can be directly extended to this framework. The first one concerns the existence of moments for the stationary state variables. We establish that for all α\geq 1, the (α+1)-moment condition for service times is necessary and sufficient for the existence of the α-moment for the stationary maximal dater (typically the time to empty the network when stopping further arrivals) in any network of this class. The second one is a direct extension of Veraverbeke's tail asymptotic for the stationary waiting times in the GI/GI/1/\infty queue.

math.PR↗