Critical Power for Vanishing of Isolated Nodes in Wireless Networks with Log-Normal Shadowing

In this paper, we study the connectivity of multihop wireless networks with the log-normal shadowing model by investigating the critical transmission power required by each node for asymptotic vanishing of the isolated notes in the network. The vanishing of isolated nodes is not only a prerequisite but also a good indication of network connectivity. All previous known work on network connectivity under such a shadowing model were obtained only based on simulation studies or ignoring the important boundary effect to avoid the challenging technical analysis, and thus cannot be applied to any practical wireless networks. It is extremely challenging to take the complicated boundary effect into consideration under such a realistic model because the transmission area of each node is an irregular region other than a circular area. We assume the wireless nodes are represented by a Poisson point process with density n over a unit-area disk. With the complicated boundary effect taken into consideration, we first derive an explicit formula for the expected number of isolated nodes in the network. This explicit formula allows people to control the expected number of isolated nodes by tuning the node density or even the transmission power, and thus the desired level of network connectivity can be expected. Then we give an upper bound and a lower bound of the critical transmission power for asymptotic vanishing of the isolated nodes. Our upper and lower bounds for the critical transmission power used by each node are almost tight.