Localization in non-homogeneous one-dimensional wireless ad-hoc networks

In this paper, we study the hop-count properties of one-dimensional wireless ad-hoc networks, where the nodes are placed independently and identically according to a Poisson distribution with an arbitrary density function. We derive exact equations to calculate the probability mass function of the number of hops needed for a node located at an arbitrary location in the network to receive a message from the source (located at one end of the linear network). Based on the derived formulas, we then propose localization methods. Through simulations, we show that our best proposed localization method not only has a competitive performance for a range-free method, but also outperforms range-based methods with a local distance measurement error of 10% or more. An important feature of our methods is that they are applicable to arbitrary densities. This is unlike the existing methods that are limited only to the case of uniform node densities. Moreover, the hop-count equations derived in this work can be used in analyzing other aspects of broadcasting protocols such as location verification, quality of service, and delay.