On the Hop Count Statistics in Wireless Multihop Networks Subject to Fading

Consider a wireless multihop network where nodes are randomly distributed in a given area following a homogeneous Poisson process. The hop count statistics, viz. the probabilities related to the number of hops between two nodes, are important for performance analysis of the multihop networks. In this paper, we provide analytical results on the probability that two nodes separated by a known euclidean distance are k hops apart in networks subject to both shadowing and small-scale fading. Some interesting results are derived which have generic significance. For example, it is shown that the locations of nodes three or more hops away provide little information in determining the relationship of a node with other nodes in the network. This observation is useful for the design of distributed routing, localization, and network security algorithms. As an illustration of the application of our results, we derive the effective energy consumption per successfully transmitted packet in end-to-end packet transmissions. We show that there exists an optimum transmission range which minimizes the effective energy consumption. The results provide useful guidelines on the design of a randomly deployed network in a more realistic radio environment.