Asymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networks with Unreliable Nodes and Links

In randomly-deployed wireless ad hoc networks with reliable nodes and links, vanishment of isolated nodes asymptotically implies connectivity of networks. However, in a realistic system, nodes may become inactive, and links may become down. The inactive nodes and down links cannot take part in routing/relaying and thus may affect the connectivity. In this paper, we study the connectivity of a wireless ad hoc network that is composed of unreliable nodes and links by investigating the distribution of the number of isolated nodes in the network. We assume that the wireless ad hoc network consists of n nodes which are distributed independently and uniformly in a unit-area disk or square. Nodes are active independently with probability 0<p 1les1, and links are up independently with probability 0<p 2les1. A node is said to be isolated if it doesn't have an up link to an active node. We show that if all nodes have a maximum transmission radius r n=radiclnn+xi/pip 1 p 2 n for some constant xi, then the total number of isolated nodes is asymptotically Poisson with mean e -xi and the total number of isolated active nodes is also asymptotically Poisson with mean p 1 e -xi. In addition, the work can be extended for secure wireless networks which adopt m-composite key predistribution schemes in which a node is said to be isolated if it doesn't have a secure link. Let p denote the probability of the event that two neighbor nodes have a secure link. We show that if all nodes have a maximum transmission radius r n=radiclnn+xi/pipn for some constant xi, then the total number of isolated nodes is asymptotically Poisson with mean e -xi.

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