F. Baccelli et S. Zuyev, Poisson-Voronoi spanning trees with applications to the optimization of communication networks, OPERAT RES, 47(4), 1999, pp. 619-631
We define a family of random trees in the plane. Their nodes of level k, k
= 0,..., m are the points of a homogeneous Poisson point process Pi(k), whe
reas 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 nucle
us x, where x is a point of Pi(k+1), then there is an are connecting x to a
ll the points of Pi(k) that belong to V. This creates a family of stationar
y random trees rooted in the points of Pi(m). These random trees are useful
to model the spatial organization of several types of hierarchical communi
cation networks. In relation to these communication networks, it is natural
to associate various cost functions with such random trees. Using point pr
ocess 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 formul
as derived for the average value of these cost functions can then be exploi
ted for parametric optimization purposes. Several applications to classical
and mobile cellular communication networks are presented.