Precise Asymptotic Distribution of the Number of Isolated Nodes in Wireless Networks with Lognormal Shadowing

Abstract

In this paper, we study the connectivity of multihop wireless networks under the log-normal shadowing model by investigating the precise distribution of the number of isolated nodes. Under such a realistic shadowing model, all previous known works on the distribution of the number of isolated nodes were obtained only based on simulation studies or by 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. Assume that the wireless nodes are represented by a Poisson point process with densitynover a unit-area disk, and that the transmission power is properly chosen so that the expected node degree of the network equals ln n + ξ ( n ), where ξ ( n ) approaches to a constant ξ as n → ∞ . Under such a shadowing model with the boundary effect taken into consideration, we proved that the total number of isolated nodes is asymptotically Poisson with mean e$ {- ξ }. The Brun’s sieve is utilized to derive the precise asymptotic distribution. Our results can be used as design guidelines for any practical multihop wireless network where both the shadowing and boundary effects must be taken into consideration.