Random Environments: Cox Point Processes

Abstract

Modeling a system of telecommunication devices in space via a homogeneous PPP represents a situation where no information about the environment or any preferred behavior of the devices is available. To some degree this can be compensated by the use of a non-homogeneous PPP with general intensity measure μ, where areas can be equipped with higher or lower device density. Thereby we leave the mathematically nicer setting of spatial stationarity, but at least we keep the spatial independence. Nevertheless, the independence of devices is an assumption that is often violated in the real world since user behavior is usually correlated. One way to incorporate dependencies into the distribution of devices in space is to use Cox point processes , which are PPPs with a random intensity measure representing the environment. In the simplest case, the scalar intensity λ > 0 of a homogeneous PPP can now be seen as a random variable Λ taking values in [0, ∞) representing, for example temporal fluctuations in the intensity of devices in a city. The joint distribution remains stationary, but it is highly spatially correlated and in particular not ergodic, see Chap. 6.