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In this paper, we study the critical transmitting range (CTR) for connectivity in mobile ad hoc networks. We prove that r_{\cal M}=c\sqrt{{\ln n\over \pi n}} for some constant c\ge 1, where r_{\cal M} is the CTR in the presence of {\cal M}{\hbox{-}}\rm like node mobility and n is the number of network nodes. Our result holds for an arbitrary mobility model {\cal M} such that: 1) {\cal M} is obstacle free and 2) nodes are allowed to move only within a certain bounded area. We also investigate in detail the case of random waypoint mobility, which is the most common mobility model used in the simulation of ad hoc networks. Denoting with r_p^w the CTR with random waypoint mobility when the pause time is set to p and node velocity is set to v, we prove that r_p^w= {p+ {0.521405\over v}\over p}\sqrt{{\ln n\over \pi n}} if p>0 and that r_0^w \gg \sqrt{{\ln n\over n}}. The results of our simulations also suggest that if n is large enough (n\ge 50), r_0^w is well approximated by {r\over 4}\ln n, where r is the critical range in case of uniformly distributed nodes. The results presented in this paper provide a better understanding of the behavior of a fundamental network parameter in the presence of mobility and can be used to improve the accuracy of mobile ad hoc network simulations.
Critical transmitting range, connectivity, random waypoint model, mobility modeling, ad hoc networks.

P. Santi, "The Critical Transmitting Range for Connectivity in Mobile Ad Hoc Networks," in IEEE Transactions on Mobile Computing, vol. 4, no. , pp. 310-317, 2005.
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