Generating Functionals of Random Packing Point Processes: From Hard-Core to Carrier Sensing

Abstract

In this paper we study the generating functionals of several random packing processes: the classical Matérn hard-core model; its extensions, the $k$-Matérn models and the $\infty$-Matérn model, which is an example of random sequential packing process. We first give a sufficient condition for the $\infty$-Matérn model to be well-defined (unlike the other two, the latter may not be well-defined on unbounded spaces). Then the generating functional of the resulting point process is given for each of the three models as the solution of a differential equation. Series representations and bounds on the generating functional of the packing models are also derived. Last but not least, we obtain moment measures and Palm distributions of the considered packing models departing from their generating functionals.