Laurent Decreusefond

An analytical model for evaluating outage and handover probability of cellular wireless networks

We consider stochastic cellular networks where base stations locations form a homogeneous Poisson point process and each mobile is attached to the base station that provides the best mean signal power. The mobile is in outage if the signal-to-noise-plus-interference ratio falls below some threshold. The handover decision has to be made if the mobile is in outage during several time slots. The outage probability and the handover probabilities are evaluated taking into account the effect of path loss, shadowing, Rayleigh fast fading, frequency factor reuse and conventional beamforming. The main assumption is that the Rayleigh fast fading changes at each time slot while other network components remain static during the period of study.

Connectivity-Based Distributed Coverage Hole Detection in Wireless Sensor Networks

Coverage is considered as an important measure of quality of service provided by a wireless sensor network (WSN). Yet, coverage holes may appear in the target field due to random deployment, depletion of sensor power or sensor destruction. Discovering the boundaries of coverage holes is important for patching the sensor network. In this paper, we adopt two types of simplicial complexes called

Simplicial homology of random configurations

\check{\textrm{C}}

Homology-based distributed coverage hole detection in wireless sensor networks

ech complex and Rips complex to capture coverage holes and classify coverage holes to be triangular and non-triangular. A distributed algorithm with only connectivity information is proposed for non-triangular holes detection. Some hole boundary nodes are found first and some of them initiate the process to detect coverage holes. Simulation results show that the area percentage of triangular holes is always below 0.03\% when the ratio between communication radius and sensing radius of a sensor is two. It is also shown that our algorithm can discover most non-triangular coverage holes.

Accuracy of homology based approaches for coverage hole detection in wireless sensor networks

Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.

A note on the simulation of the Ginibre point process

Homology theory provides new and powerful solutions to address the coverage problems in wireless sensor networks (WSNs). They are based on algebraic objects, such as Čech complex and Rips complex. Čech complex gives accurate information about coverage quality, but requires a precise knowledge of the relative locations of nodes. This assumption is rather strong and hard to implement in practical deployments. Rips complex provides an approximation of Čech complex. It is easier to build and does not require any knowledge of nodes location. This simplicity is at the expense of accuracy. Rips complex cannot always detect all coverage holes. It is then necessary to evaluate its accuracy. This work proposes to use the proportion of the area of undiscovered coverage holes as performance criteria. Investigations show that it depends on the ratio between communication and sensing radii of a sensor. Closed-form expressions for lower and upper bounds of the accuracy are also derived. For those coverage holes that can be discovered by Rips complex, a homology-based distributed algorithm is proposed to detect them. Simulation results are consistent with the proposed analytical lower bound, with a maximum difference of 0.5%. Upper-bound performance depends on the ratio of communication and sensing radii. Simulations also show that the algorithm can localize about 99% coverage holes in about 99% cases.

A Case Study on Regularity in Cellular Network Deployment

Homology theory provides new and powerful solutions to address the coverage problems in wireless sensor networks (WSNs). They are based on algebraic objects, such as Cech complex and Rips complex. Cech complex gives accurate information about coverage quality but requires a precise knowledge of the relative locations of nodes. This assumption is rather strong and hard to implement in practical deployments. Rips complex provides an approximation of Cech complex. It is easier to build and does not require knowledge of nodes location. This simplicity is at the expense of accuracy. Rips complex can not always detect all coverage holes. It is then necessary to evaluate its accuracy. This work proposes to use the area of undiscovered coverage holes per unit of surface as performance criteria. Investigations show that it depends on the ratio of communication and sensing ranges of each sensor. Closed form expressions for lower and upper bounds of the accuracy are also derived. Simulation results are consistent with the proposed analytical lower bound, with a maximum difference of 0.4%. Upper bound performance depends on the ratio of communication and sensing ranges.

Reduction algorithm for simplicial complexes

The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

Moment formulae for general point processes

This letter aims to validate the β-Ginibre point process as a model for the distribution of base station locations in a cellular network. The β-Ginibre is a repulsive point process in which repulsion is controlled by the β parameter. When β tends to zero, the point process converges in law towards a Poisson point process. If β equals to one it becomes a Ginibre point process. Simulations on real data collected in Paris, France, show that base station locations can be fitted with a β-Ginibre point process. Moreover, we prove that their superposition tends to a Poisson point process as it can be seen from real data. Qualitative interpretations on deployment strategies are derived from the model fitting of the raw data.

Analyzing interference from static cellular cooperation using the Nearest Neighbour Model

In this paper, we aim at reducing power consumption in wireless sensor networks by turning off supernumerary sensors. Random simplicial complexes are tools from algebraic topology which provide an accurate and tractable representation of the topology of wireless sensor networks. Given a simplicial complex, we present an algorithm which reduces the number of its vertices, keeping its homology (i.e. connectivity, coverage) unchanged. We show that the algorithm reaches a Nash equilibrium, moreover we find both a lower and an upper bounds for the number of vertices removed, the complexity of the algorithm, and the maximal order of the resulting complex for the coverage problem. We also give some simulation results for classical cases, especially coverage complexes simulating wireless sensor networks.