Non-Poisson Point Processes

Motivation

In Chapters 5–7 we see how Poisson network models often give tractable results, and how random propagation effects such as fading and shadowing can render a network to appear more Poisson. Nevertheless, the Poisson model may still not be appropriate for certain network layouts. For example, although the Poisson process statistically does not exhibit either repulsion or clustering, it is possible for the clusters or voids of points to exist in network layouts.

Appropriate point processes

In Chapter 3 we see just a small selection of the possible spatial point processes. To choose appropriate point processes for network models, researchers have statistically analyzed networks located in various cities to demonstrate that other point processes may be more appropriate, although of course such analyses can be only as good as the data. But when developing stochastic geometry models of cellular networks, the number of suitable and tractable point processes quickly becomes limited. In terms of the SINR for a single user, there are two important and necessary features of a potential point process. One is the knowledge of the Palm distribution, as covered in Section 3.2.4, which immediately limits the choice of possible point processes. The other is being able to write down a calculable expression (at least, numerically) of the Laplace functional of the point process.

In short, although there is a rich range of possible point processes, the bulk of them unfortunately lack the necessary properties that make them tractable in our setting, which partly explains why the clear majority of network models are based on the Poisson point process. Consequently, we restrict ourselves to results for point processes from two general families that exhibit repulsion and clustering, respectively, determinantal processes and a specific shot noise Cox process.

Choice of the base station and propagation effects

In Section 5.1.3, we see the two different model assumptions for how a user at the typical location (the origin) chooses the serving base station under the Poisson model.