Poisson-Voronoi Spanning Trees with Applications to the Optimization of Communication Networks

Abstract

We define a family of random trees in the plane. Their nodes of level k, k ϭ 0, . . . , m are the points of a homogeneous Poisson point process ⌸ k , whereas their arcs connect nodes of level k and k ϩ 1, according to the least distance principle: If V denotes the Voronoi cell w.r.t. ⌸ kϩ1 with nucleus x, where x is a point of ⌸ kϩ1 , then there is an arc connecting x to all the points of ⌸ k that belong to V. This creates a family of stationary random trees rooted in the points of ⌸ m . These random trees are useful to model the spatial organization of several types of hierarchical communication networks. In relation to these communication networks, it is natural to associate various cost functions with such random trees. Using point process techniques, like the exchange formula between two Palm measures, and integral geometry techniques, we show how to compute these average costs as functions of the intensity parameters of the Poisson processes. The formulas derived for the average value of these cost functions can then be exploited for parametric optimization purposes. Several applications to classical and mobile cellular communication networks are presented.

Minimizing Energy Consumption in Large-Scale Sensor Networks Through Distributed Data Compression and Hierarchical Aggregation

“…Our objective is to study how to best arrange such hierarchies so as to minimize the overall energy costs. Due to the complexity of and spatial character of problem, we will use the methodology proposed in [18]. The idea is to use crude stochastic geometric models to capture the salient features of the system.…”

“…Similarly one might think it reasonable to assume each sensor sends its traffic to the closest compressor or, if it is closer, directly to a sink. As discussed in [18] such a hierarchical organization is induced by the Voronoi tessellation generated by the locations of the sinks and compressors. A Voronoi tessellation with respect to a set of points Π can be defined as follows.…”

A Scalable Model for Energy Load Balancing in Large-scale Sensor Networks

“…Finally we are inspired by studies modelling network structure via hierarchies of Voronoi tessellations and exploit such geometric structures for various purposes. Notably modelling of telecommunication networks using stochastic geometry has been proposed in [6] for analyzing the cost of a network with a hierarchy associated with proximity, where we borrow their framework and notations in part.…”

“…By multiplying it by g w (O, y j , z 0 ) we obtain the energy burden density experienced at the origin when the aggregated traffic N + yj is forwarded to a sink at z 0 originating from a AGN 1 at y j . Using Neveu exchange formula [6], one can rewrite (1) as…”

“…When the strips are originated from y in the region K w (x)∩ B w/2 (x), the energy density contribution changes with y. If we consider y ∈ K w (x) ∩ B w 2 (x) satisfying |x − y| is constant, h w (x, y, O) is also constant since it depends only on |x − y|, which facilitates the evaluation of the surface integral in (6). Also we denote the sector xBCA by C w (x) and approximate the region of integration…”