Probability distribution of multi-hop-distance in one-dimensional sensor networks

In spatially random sensor networks, estimating the Euclidean distance covered by a packet in a given number of hops carries a high importance for various functions such as localization and distance estimations. The inaccuracies in such estimations motivate this study on the distribution of the Euclidean distance covered by a packet in spatially random sensor networks in a given number of hops. Although a closed-form expression of distance distribution cannot be obtained, highly accurate approximations are derived for this distribution in one-dimensional spatially random sensor networks. Using statistical measures and numerical examples, it is also shown that the presented distribution approximation yields very high accuracy even for small number of hops. Furthermore, the distribution of hop distance that covers a given Euclidean distance is also approximated.