Properties of 1-D infrastructure-based wireless multi-hop networks

Many real wireless multi-hop networks are deployed with some infrastructure support, where the results on ad-hoc networks cannot be readily extended to understand the properties of those networks. In this paper, we study those networks in 1-D. Specifically, we consider two types of nodes in the networks: ordinary nodes and powerful nodes, where ordinary nodes are i.i.d and Poissonly distributed in a unit interval and powerful nodes are arbitrarily distributed within the same unit interval. These powerful nodes are inter-connected via some backbone infrastructure. The network is said to be connected, i.e. any two nodes can communicate with each other, if each ordinary node is connected to at least one of the powerful nodes. We call this type of connectivity type-II connectivity. Exact and simplified asymptotic formulas for type-II connectivity probability and the average hop count between two arbitrary nodes are obtained. Further we prove that equi-distant powerful nodes deployment delivers the optimum performance which maximizes the type-II connectivity probability. These results are important for the design and deployment of 1-D infrastructure-based networks and provide useful insights into the analysis of higher dimensional networks.