We are interested in giving a mathematical formula of a cost in adhoc network model. In our model, the cost is formulated as an application of first-passage percolation and the motion of devices is random, and an asymptotic density of devices is formulated by hydrodynamic limit. Under some technical assumptions, we give asymptotics of a cost in adhoc network model. In order to formulate this model, we extend the results of first-passage percolation given by Howard Newman  to that in inhomogeneous environments.
"On scaling limit of a cost in adhoc network model." Kodai Math. J. 42 (2) 247 - 273, June 2019. https://doi.org/10.2996/kmj/1562032830