Spatial fluid limits for stochastic mobile networks

We consider Markov models of large-scale networks where nodes are characterized by their local behavior and by a mobility model over a two-dimensional lattice. By assuming random walk, we prove convergence to a system of partial differential equations (PDEs) whose size depends neither on the lattice size nor on the population of nodes. This provides a macroscopic view of the model which approximates discrete stochastic movements with continuous deterministic diffusions. This result may be used in practice to justify a space coarsening induced by the numerical PDE solver. We illustrate this by modeling a network of mobile nodes with on/off behavior performing file transfers with connectivity to 802.11 access points, empirically validated against discrete-event simulation. We show high quality of the PDE approximation in the case of moderate lattice granularity and population sizes.