ABSTRACT
CONTENTS 9.1 Introduction ............................................................................................... 176 9.2 Review of Prior Work ............................................................................... 177 9.3 Device Location Model ............................................................................ 178 9.4 Radio Propagation Model ....................................................................... 179 9.5 Connectivity between Network Components ...................................... 181
9.5.1 Mathematical Preliminaries ........................................................ 181 9.5.2 Distribution of the Number of Neighbors of a
Given Node/AP/GW .................................................................. 181 9.5.3 Distribution of the Number of Nodes Supported
by a Single AP ............................................................................... 184 9.5.4 Probability Distribution of Distance between a GW and an
AP with a Direct Connection to the GW ................................... 186 9.6 Measures of Coverage in the Overall Network ................................... 188
9.6.1 The Fraction of Nodes That Cannot Be Reached by a GW in Two Hops ....................................................................... 188 9.6.1.1 Probability That a Node Cannot Connect to a
Specific GW a Distance D Away in Two Hops .......... 188 9.6.1.2 Probability That an Arbitrary Node Cannot
Connect to Any GW in Two Hops .............................. 190 9.6.2 Design of the AP-to-GW Link to Avoid Overload ................... 191
9.6.2.1 Distribution of the Traffic Load on an AP Serving as a Root of a Tree Containing Other APs .................. 191
9.6.2.2 Overload Condition on an AP ..................................... 192 9.7 Summary .................................................................................................... 192 References ........................................................................................................... 193
We study a large wireless network involving three hierarchical layers. At the lowest layer we have subscriber stations, called “nodes’’ in the sequel. Each node contains a transceiver with certain common wireless capabilities. Nodes are typically installed at customer premises, and their locations cannot be predicted at the planning stage. The second hierarchical layer contains access points (APs) owned and installed by the service provider wherever he sees fit subject to such constraints as space availability, the parameters of the service provided (e.g., peak data rate), cost considerations, and the expected local penetration of the service. Like regular nodes, APs have no connection to the wired infrastructure and are all identical. To reach the wired infrastructure, e.g., the Internet, the APs rely on high data rate links to gateways (GWs), which form the third hierarchical layer. The GWs are connected to a fiber-optic infrastructure that, for all practical purposes, can be seen as having infinite capacity. Each GW is equipped with high antenna tower, requires significant real estate, and investment. Since GWs are co-located with fibers, and for economic reasons are expected to be relatively scarce, their locations cannot be positioned on the basis of wireless coverage considerations only. To account for this reality, we assume that the GWs, as well as the APs and nodes, are also placed uniformly and randomly over the service area. This location model should not be seen as a worst case scenario, but it is common in the literature and facilitates analysis. The density of APs is expected to be significantly lower than the density of nodes, and the density of GWs even lower.
