Deriving the critical density (which is equivalent to deriving the critical radius or power) to achieve coverage and/or connectivity for random deployments is a fundamental problem in the area of wireless networks. The probabilistic conditions normally derived, however, have limited appeal among practitioners because they areoften asymptotic, i.e., they only make high probability guarantees in the limit of large system sizes. Such conditions are not very useful in practice since deployment regions are always finite. Another major limitation of most existing work on coverage and connectivity is their focus on thick deployment regions (such as a square or a disk). There is no existing work (including traditional percolation theory) that derives critical densities for thin strips (or annuli).   In this paper, we address both of these shortcomings by introducing new techniques for deriving reliable density estimates for finite regions (including thin strips). We apply our techniques to solve the open problem of deriving reliable density estimates for achieving  barrier coverage  and connectivity in thin strips, where sensors are deployed as a barrier to detect moving objects and phenomena. We use simulations to show that our estimates are accurate even for small deployment regions. Our techniques bridge the gap between theory and practice in the area of coverage and connectivity, since the results can now be readily used in real-life deployments.
['Paul Balister', 'Béla Bollobás', 'Amites Sarkar', 'Santosh Kumar']
Reliable density estimates for coverage and connectivity in thin strips of finite length