Variational analysis of functionals of Poisson processes (Q2757649)

If \(\Pi\) is a (generally nonhomogeneous) Poisson point process with the intensity measure \(\mu\), the corresponding expected value of a functional \(F(\Pi)\), i.e., a function defined on realizations of the process, is denoted by \({\mathbf E}_\mu F(\Pi)\). The latter depends on \(\mu\) and can be considered as a functional defined on a cone \(\mathbb M\) of (nonnegative) measures. The authors develop a variational calculus for such functionals directly on its parameter space \(\mathbb M\). Closed form expressions for its Fréchet derivatives of all orders that generalize the perturbation analysis formulae for Poisson processes are derived. It is shown that in the case of a finite intensity measure, the expectation \({\mathbf E}_\mu F(\Pi)\) is analytic on \(\mathbb M\) for most ``reasonable'' functionals \(F\). The obtained formulae for derivatives are used to solve variational problems of minimizing \({\mathbf E}_\mu F(\Pi)\) over a class of admissible intensity measures \(\mu\). In order to formulate useful first and second order necessary optimality conditions for the constrained optimization problem on \(\mathbb M\), abstract optimization results for general Banach spaces are obtained. These general results are used to derive specific conditions for a minimum in the class of nonnegative measures with a fixed total mass. Asymptotic behavior of solutions for a fixed total mass problem when this mass increases to infinity is studied. A method that often makes possible an explicit calculation of the asymptotically optimal measure in this high intensity framework is developed. A practical example illustrating the developed technique is given and the asymptotically optimal density of telecommunication servers that minimizes the total connection cost given the density of subscribers is found. This method differs fundamentally from optimization technique for stochastic systems whose distribution depends on a finite number of parameters.