Connectivity and Transmit-Energy Considerations Between Any Pair of Nodes in a 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. 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 log-normal fading. Each node functions as a source and a destination of data packets and may also serve as a repeater to transport the packets over multihop 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 the 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 a destination at a 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 graphically show the benefit of allowing two hops over just one.