Aspects of Wireless Network Modeling based on Poisson Point Processes

In these notes we discuss stochastic models for a simplified wireless network that consists of a collection of spatially distributed stations equipped with emitters and/or receivers for transmission over a common communication channel. The modeling approach is based on using Poisson point processes for the spatial locations as well as for other signaling characteristics of the network nodes. Throughout we work from the premises that the transmission of signals is syncronized and slotted in time so that in a fixed time slot each sender attempts to emit the equivalent of one symbol. The signal power is affected by random fading and attenuation proportional to the traveled distance. Using simple Poisson models and superpositioning the effect of all stations in one slot it is possible to obtain some insights into the balance between node density and node interference. For traffic sessions over a fixed or random number of slots and under assumptions of Rayleigh fading, we propose a modeling scenario based on the Lévy gamma subordinator process and its relation to complex Gaussian waweforms. These models can be extended to sessions which are Poisson in both space and time, and have short-tailed or heavy-tailed random session duration times. It is possible also to include lognormal fading. This is a mechanism supposed to act on the slow time scale of sessions, which is in contrast to Rayleigh fading that generate random variation on the fast scale of slots. For the traffic session models it is possible to perform a scaling approximation to analyze the fluctuations that build up in the interference field.