Poisson-Voronoi Spanning Trees with Applications to the Optimization of Communication Networks
Résumé
We define a family of random trees in the plane. Their nodes of level $k, k=0\thru m$ are the points of a homogeneous Poisson point process $\Pi_k$, whereas their arcs connect nodes of level $k$ and $k+1$, according to the least distance principle: if $V$ denotes the Voronoi cell w.r.t. $\Pi_k+1$ with nucleus $x$, where $x$ is a point of $\Pi_k+1$, then there is an arc connecting $x$ to all the points of $\Pi_k$ which belong to $V$. This creates a family of stationary random trees rooted in the points of $\Pi_m$. These random trees are useful to model the spatial organization of several types of hierarchical communication networks. In relation with 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 the average cost in function of the intensity parameters of the Poisson processes. The formulas which are derived for the average value of the cost function are then exploited for parametric optimization purposes.