Coverage in One-Dimensional Wireless Networks With Infrastructure Nodes and Relay Extensions

Abstract

We consider a wireless network comprising two types of nodes, namely, sinks and relays. The sink nodes are connected to a wireline infrastructure, while the relay nodes are used to extend the region covered by providing multi-hop paths to the sink nodes. Restricting to the one-dimensional setting, our objective is to characterize the fraction of covered region as a function of sink and relay node densities. We first compare and contrast our infrastructure-based model with the traditional setting where every node is a sink, and hence a location is covered if it simply lies within the range of some node. Then, drawing an analogy between the connected components of the network and the busy periods of an M/D/∞ queue (and using renewal theoretic arguments) we derive a closed-form expression for the average vacancy (complement of coverage). We also compute an upper bound for vacancy by introducing the notion of left-coverage (i.e., coverage by a node on the left); a lower bound is derived by coupling our model with an independent-disk model, where the sinks' coverage regions are independent and identically distributed. Through an extensive theoretical and numerical study, we investigate the problem of minimizing network deployment cost subject to a constraint on the average vacancy. We also conduct simulations to understand the properties of a general notion of coverage, obtained by introducing hop-counts into the definition. Parameterized approximations for the hop-constrained cluster lengths (around a sink) are proposed, whose efficacy is evaluated numerically. In particular, there exists a range of parameter values where our cluster-length approximation is good. Finally, hop-constrained cost optimization is conducted to demonstrate the efficacy of the infrastructure-based design.