Connectivity and Transmit Energy Considerations Between any Pair of Nodes in a Bounded Wireless Ad-hoc Network Subject to Fading

We investigate an ad hoc network where node locations are distributed according to a homogeneous Poisson process with intensity $lambda_{text{n}}$. We assume that all the nodes are equipped with an identical wireless transceiver capable of operating satisfactorily up to a certain maximal link loss. Our link model depends on the length of the link and on random lognormal fading. Each node functions as a source and destination of data packets, and may also serve as a repeater to transport packets over multi-hop routes as determined by the network router. We study two important properties of the network. The first is connectivity, viz., given the peak transmit power of a node, the probability that a node cannot communicate with a random destination node at distance $D$, when at most $t$ hops are allowed. We provide exact analytical results for $t=1$ and $t=2$, and an iterative lower bound for $t > 2$. We calculate the average number of hops of the minimum hop- count route between a source and destination at distance $D$ apart. The second property relates to power consumption -- an important parameter when the nodes are battery operated. We derive the cumulative distribution function of the total transmit energy required per data packet when the distance between the source and the destination node is $D$, and only one or at most two hops are allowed. We show graphically the benefit of allowing two hops over just one.