Variational analysis of functionals of Poisson processes

Let F(II) be a functional of a (generally nonhomogeneous) Poisson process I I with intensity measure mu. Considering the expectation EmuF(II) as a func tional of mu from the cone M of positive finite measures, we derive closed form expressions for its Frechet derivatives of an orders that generalize t he perturbation analysis formulae for Poisson processes. Variational method s developed for the space mm allow us to obtain first and second order suff icient conditions for various types of constrained optimization problems fo r EmuF. We study in detail optimization over the class of measures with a f ixed total mass a and develop a technique that often allows us to obtain th e asymptotic behavior of the optimal intensity measure in the high intensit y setting when a grows to infinity. As a particular application we consider the problem of optimal placement of stations in the Poisson model of a two -layer telecommunication network.