A Study of k-Coverage and Measures of Connectivity in 3D Wireless Sensor Networks

In a wireless sensor network (WSN), connectivity enables the sensors to communicate with each other, while sensing coverage reflects the quality of surveillance. Although the majority of studies on coverage and connectivity in WSNs consider 2D space, 3D settings represent more accurately the network design for real-world applications. As an example, underwater sensor networks require design in 3D rather than 2D space. In this paper, we focus on the connectivity and k-coverage issues in 3D WSNs, where each point is covered by at least k sensors (the maximum value of k is called the coverage degree). Precisely, we propose the Reuleaux tetrahedron model to characterize k-coverage of a 3D field and investigate the corresponding minimum sensor spatial density. We prove that a 3D field is guaranteed to be k-covered if any Reuleaux tetrahedron region of the field contains at least k sensors. We also compute the connectivity of 3D k-covered WSNs. Based on the concepts of conditional connectivity and forbidden faulty sensor set, which cannot include all the neighbors of a sensor, we prove that 3D k-covered WSNs can sustain a large number of sensor failures. Precisely, we prove that 3D k-covered WSNs have connectivity higher than their coverage degree k. Then, we relax some widely used assumptions in coverage and connectivity in WSNs, such as sensor homogeneity and unit sensing and communication model, so as to promote the practicality of our results in real-world scenarios. Also, we propose a placement strategy of sensors to achieve full k-coverage of a 3D field. This strategy can be used in the design of energy-efficient scheduling protocols for 3D k-covered WSNs to extend the network lifetime.