Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

In this paper, we compute the expected number of vehicles with at least one two-hop path to a fixed roadside unit (RSU) in a multi-hop, one-dimensional vehicular ad hoc network (VANET) where other cars can act as relays. The pairwise channels experience Rayleigh fading in the random connection model, and so exist, with a probability given by a function of the mutual distance between the cars, or between the cars and the RSU. We derive exact expressions for the expected number of cars with a two-hop connection to the RSU when the car density $\rho$ tends to zero and infinity, and determine its behaviour using an infinite oscillating power series in $\rho$, which is accurate for all regimes of traffic density. We also corroborate those findings with a realistic scenario, using snapshots of actual traffic data. Finally, a normal approximation is discussed for the probability mass function of the number of cars with a two-hop connection to the RSU.