Abstract: This paper presented an analytical method to evaluate the hop-distance relationship for wireless sensor network in a shadow fading envirorunent,where sensor nodes are randomly and independently deployed in a circular region by a Poisson process. Different from the UDG model, the shadow fading model has the non-uniformity property for connectivity. We derived an approximate recursive expression of the probability of the hop count by a given geographic dislance. The multi-hop dependence problem was also taken into consideration. Furthermore, we gave the expressions about the distribution of distance by a known hop count by the Bayer formula .The simulation results show that the analytical expressions can closely agree with the statistical data and the border effects cannot be ignored.

Key words: Internet of things, Wireless sensor networks, Shadow fading model, Hop-distance relationship