Connectivity and Minimum Total Transmit Energy per Packet between any Pair of Nodes in a Bounded Wireless Ad-Hoc Network Subject to Fading

We investigate an ad hoc wireless network where node locations are distributed according to a homogeneous Poisson process. All nodes are equipped with identical wireless transceivers and require a certain minimal received power for proper reception. Our link model depends on the length of the link and on random log-normal fading. Each node may function as a source and destination of data packets, and may also serve as a repeater. We consider a source and destination node a given distance apart and consider only paths that take at most two hops (i.e., use at most one repeater). We provide exact analytic results for (a) the probability that no such path exists given the peak transmit power of a node, and (b) the distribution of the total transmit energy required per data packet. We provide figures showing the benefit of allowing two hops over just one.