Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with Bernoulli nodes

Nodes in wireless ad hoc networks may become inactive or unavailable due to, for example, internal breakdown or being in the sleeping state. The inactive nodes cannot take part in routing/relaying and thus may effect the connectivity. A wireless ad hoc network containing inactive nodes is then said to be connected if each inactive node is adjacent to at least one active node and all active nodes form a connected network. This paper is the first installment of our probabilistic study of the connectivity of wireless ad hoc networks containing inactive nodes. We assume that the wireless ad hoc network consists of n nodes, which are distributed independently and uniformly in a unit-area disk and are active (or available) independently with probability p for some constant 0 < p /spl les/ 1. We show that if all nodes have a maximum transmission radius r/sub n/ = /spl radic/(ln n+c//spl pi/pn) for some constant c, then the total number of isolated nodes is asymptotically Poisson with mean e/sup -c/ and the total number of isolated active nodes is also asymptotically Poisson with mean pe/sup -c/.