TITLE:
Precise Asymptotic Distribution of the Number of Isolated Nodes in Wireless Networks with Lognormal Shadowing
AUTHORS:
Lixin Wang, Alberto Argumedo, William Washington
KEYWORDS:
Connectivity, Asymptotic Distribution, Random Geometric Graph, Isolated Nodes, log-Normal Shadowing
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.15,
August
8,
2014
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 lnn + 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.