The critical transmitting range for Connectivity in Mobile Packet Radio Networks

It is known that the critical transmitting range for connectivity in stationary packet radio networks is $r=c sqrt{frac{log n}{n}}$, for some constant $c!>!0$, under the assumption that $n$ nodes are uniformly distributed in $R=[0,1]^2$. In this note, we investigate how mobility affects this asymptotic result. We consider the well known random waypoint mobility model, whose asymptotic node spatial distribution has been recently derived. We prove that as long as the spatial distribution has a non-null uniform component, the mobile critical transmitting range differs from the stationary one at most by a constant factor. On the contrary, when the uniform component is null there is an asymptotic gap between the mobile and stationary case, i.e. $r!gg!c sqrt{frac{log n}{n}}$ for any constant $c!>!0$. Hence, the asymptotic behavior of the mobile critical transmitting range depends on the choice of the mobility parameters