Critical density for coverage and connectivity in two-dimensional fixed-orientation directional sensor networks using continuum percolation

Given an initially uncovered field, and as more and more directional sensors (sensors with sector shape sensing area) are continuously added to the sensor network, the size of partial covered areas increases. At some point, the situation abruptly changes from small fragmented covered areas to a single large covered area. This abrupt change is called the sensing-coverage phase transition (SCPT). Likewise, given an originally disconnected sensor network, as more and more sensors are added, the number of connected components changes such that the sensor network suddenly becomes connected. This sudden change is called the network connectivity phase transition (NCPT). Such phase transitions occur in a certain density which is called critical density and finding it is a central topic of Percolation Theory. In this paper, we introduce fixed-orientation directional sensor networks (FIODSNs) and analytically compute critical density of nodes for both SCPT and NCPT in FIODSNs, for all field-of-view angles between 0 and π using continuum percolation. In FIODSNs which are the most common type of directional sensor networks, sensor nodes are deployed based on Poisson Point Process, and orientation of them is distributed between 0 and 2π, independently and uniformly. Due to percolation theory, the critical density is the infimum density for densities above it SCPT and NCPT almost surely occur. Therefore, the results could be used to prepare coverage and connectivity in FIODSNs. Moreover, to solve the SCPT and NCPT problems together, we propose a model for percolation in directional sensor networks which could be used in other related researches.