Criticality of large delay tolerant networks via directed continuum percolation in space-time

We study delay tolerant networking (DTN) and in particular, its capacity to store, carry and forward messages to their final destination(s). We approach this broad question in the framework of percolation theory. To this end, we assume an elementary mobility model, where nodes arrive to an infinite plane according to a Poisson point process, move a certain distance ℓ, and then depart. In this setting, we characterize the mean density of nodes required to support DTN style networking. Under the given assumptions, we show that DTN communication is feasible when the mean node degree ν is greater than 4 · η_c(γ), where parameter γ= ℓ/d is the ratio of the distance ℓ to the transmission range d, and η_c(γ) is the critical reduced number density of tilted cylinders in a directed continuum percolation model. By means of Monte Carlo simulations, we give numerical values for η_c(γ). The asymptotic behavior of η_c(γ) when γ tends to ∞ is also derived from a fluid flow analysis.