Routing policy-dependent hop-count distribution in wireless ad hoc networks

We derive asymptotic expressions for the probability mass function (PMF) of the number of hops (a.k.a hop-count distribution) required to reach a designated distance D in random, connected and uniformly distributed one- and two-dimensional ad hoc networks. The elegance of the result is owned to the application of Renewal Theory, a generalization of Poisson processes. The method is general, requiring only that the distribution of the hop-length under the desired policy be known, such that application to two-dimensional networks is straightforward, amounting to the problem of computing the projection of the average hop-length onto a straight. Consequently, the formulae are remarkably simple and accurate compared to current knowledge, allowing for the effect of hopping policies (e.g. random, closest- and furthest-neighbor) to be easily taken into account; and yielding counter-intuitive insight on this important parameter of ad hoc networks. For instance, the analysis reveals that the hop-count distribution is nothing but a scaled version of the node-number1 distribution, with the scale parameter inversely proportional the average hop-length times the network density. In the linear case, where the node-count distribution is Poisson, this result both elucidates the equivalence and quantifies the impact of node-density and hopping policy on the likely number of hops to reach any destination.