On the Probability Distribution of the Minimal Number of Hops 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 a given intensity. 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 focus on the probability distribution of the minimum number of hops between a source and a destination node known to be at distance D from the source. When the distribution of source-to-destination distances is known, the distribution of the minimal number of hops can also be found. Many variations of this same problem have been studied in the literature. However, as far as we know, no exact closed-form analytic results for fading environments have been presented before.