We study the connectivity in vehicular ad-hoc networks, where the motion of vehicles is constrained on a lattice-shaped road network. First, we theoretically investigate the connectivity under the Poisson-positioning assumption, where vehicles are positioned according to a Poisson process on each road at any arbitrary instants. We find that the Poisson-positioning assumption allows the existence of the finite critical-vehicle density; that is, if (and only if) the density of vehicles is greater than the finite critical density, then there exists a large (theoretically infinite) cluster of vehicles and an arbitrary pair of vehicles in the set is connected in single or multiple hops. We obtain an analytical expression for the critical density as a function of the transmission range of each vehicle and the distance between intersections. Next, we consider the connectivity under more realistic movement patterns of vehicles where the Poisson-positioning assumption does not hold. We numerically find that, even in non-Poisson-positioning cases, there exists the critical vehicle density. The critical density in non-Poisson-positioning cases is, however, larger than the one under the Poisson-positioning assumption. We also gain some insight on the efficiency of roadside-relay-station deployment to provide better connectivity between vehicles.
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