A Stochastic Geometry Approach to the Interference and Outage Characterization of Large Wireless Networks

This thesis characterizes the relationship between the geometry of a wireless network and its performance. The geometry of the network is largely influenced by the wireless node locations and the large-scale path loss and in part by fading and the transmit power levels. The performance of a wireless link is governed by interference and signal power which in turn depend on the node locations. Interference characterization is important in understanding the performance of a wireless system. But unfortunately the distribution of the interference is known only for few spatial distributions of node locations. More specifically, interference was characterized when the nodes were either distributed as a Poisson point process or as a lattice process. But in reality, the wireless nodes may neither be so random nor so regular but somewhere in between. In this thesis the location of nodes is modeled as a general stationary and isotropic point process, and we use stochastic geometry to analyze the interference and outage. We prove that the interference distribution depends critically on the path-loss model under consideration. When the path-loss is unbounded at the origin, i.e., ℓ(x) = --x --alpha, interference has a heavy-tailed distribution with parameter 2/alpha. When the path-loss is bounded, i.e., ℓ( x) = (1 + --x--alpha)--1 the interference distribution depends on the fading statistics. We prove that the interference has an exponential distribution when the fading is exponential and has a heavy-tailed distribution when the fading is heavy-tailed. This proves that Gaussian modeling of interference is not appropriate. We also provide the temporal and spatial correlation of interference when ALOHA is used for MAC scheduling. Wireless nodes may cluster because of physical constraints or because of MAC scheduling. For example, soldiers (with radios) cluster on the battle field or sensor networks are clustered for energy reduction. But it is not clear if clustering of transmitters is beneficial compared to randomizing the transmissions from the perspective of link outages. We derive the outage probability of a Poisson clustered network by obtaining its conditional probability generating functional. It is difficult for the base station in a cellular network to connect to mobile stations on the cell boundary because of the distance and the inter-cell interference. It has therefore been proposed for the base station to communicate with the mobile at the cell use using multiple hops. We use stochastic geometry to analyze the outage probability in a two-hop cellular system. We provide the asymptotic gain of a two-hop system over direct transmission, for three different relay selection schemes. The major emphasis of the thesis is the inclusion of the spatial statistics of the node locations in the performance analysis of a wireless network. To that end we concentrate on effect of the spatial distribution of nodes on the interference distribution. We provide results on the PDF, correlation and the tail behavior of the interference. One of the main contributions of the thesis is the methodology and the tools of analysis that we develop. The thesis concentrates on developing spatial analysis techniques that have wide applicability rather than concentrating on very specific details of a communication system.