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On the Critical Total Power for k -Connectivity in Wireless Networks

This study investigates the critical total power required to maintain asymptotic k-connectivity in wireless networks. The research considers the heterogeneous case where each node can choose its own transmission power. The problem statement revolves around determining the minimal power required to maintain k-connectivity as node density approaches infinity.

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On the Critical Total Power for k -Connectivity in Wireless Networks

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  1. On the Critical Total Power for k-Connectivity in Wireless Networks Honghai Zhang, Jennifer C. Hou Department of Computer Science University of Illinois at Urbana-Champaign

  2. Wireless Networks • Characteristics of wireless networks • Nodes may be mobile • Energy is often supplied by batteries • Benefit of reducing transmission power • Extending network lifetime • Increasing network capacity • Mitigating MAC-level interference

  3. How to Reduce Transmission Power • Two ways of reducing transmission power • All nodes use a minimum common power • All nodes may choose different power • Which one is better? • Intuitively, the latter is better • How do we define “better”? • Total power consumption How much power can be saved by using power control compared with using common power ?

  4. Problem Statement • We investigate the critical total power required to maintain asymptotic k-connectivity on a unit square S=[0,1]2. • We consider the heterogeneous case in which each node can choose its own transmission power. • Formulation: • Wt,i : critical transmission power node i uses • Rt,i: corresponding transmission range of node i Wt,i = Rt,ia. • Total power: • Problem: Given the assumption that wireless nodes are distributed on S according to a Poisson point process with density l, how does the minimal power Wc scale with l as l infinity?

  5. Network Model • Nodes are randomly distributed as a Poisson point process with density n in a unit-area square • A good approximation for uniformly random distribution withnnodes 1 1

  6. Network Model (con’t) • Xi: location of nodei • Ri : transmission range of nodei • Link (j,i) exists if Rj  | Xi– Xj | • : transmission power of nodei • Total power: • k-connectivity: requires removing at least k nodes to disconnect the network • Critical total power Wc: minimum total power W for maintaining k-connectivity 1 Xj Rj Ri Xi 1

  7. Previous Studies • All nodes choose the common power • [Gupta & Kumar 98] studied the critical transmission range rn for 1-connectivity • [Wan & Yi 04] for k-connectivity • All nodes choose different power • [Blough 02] Critical total power for 1-connectivity • Our study: critical total power for k-connectivity

  8. Major Results • Main theory: • The critical total power for maintaining k-connectivity is with probability approaching 1 • Comparison with common power • The critical total power for k-connectivity with common power is • Allowing power control reduces the total power by a factor of

  9. Sketched Proof of the Main Theory • Main theory: the critical total power for maintaining k-connectivity is with high probability • We derive a lower bound based on the necessary condition that every node has to be able to reach at least k nearest nodes in order to maintain strong k-connectivity. • We derive an upper bound based on an assertion that the network is strongly k-connected if every node can reach k other nodes in each of its four quadrants.

  10. Proof of the Lower Bound Every node has to be able to reach at least k nearest nodes • Ri: the distance from node i to its kth nearest neighbor • is a lower bound for the critical power Ri r

  11. Proof of the Lower Bound (con’t) • Goal • We have got • Suffices to show Chebyshev Inequality

  12. Ri Proof of the Upper Bound • Each node has a coordinate system centered at itself • All x-axes and y-axes are in parallel, respectively • Lemma 1: The network has k-connectivity if every node can reach at least k nearest neighbors in each of the four quadrants of its own coordinate system • If a quadrant contains less than k nodes, it is sufficient to reach all of them.

  13. Proof of Lemma 1 (for k=1) • (xi,yi): Node i’s location • Proof using contradiction • Choose the pair of nodes A, B such that • No path from A to B, and • A, B have the minimum • Goal: find another pair of nodes X, Y such that • No path from X to Y, and

  14. Proof of Lemma 1 for k=1 • Assume B is in the first quadrant of A’s coordinate system • A must be able to reach the nearest neighbor C in the same quadrant • If C is not on x-axis or y-axis, then (C,B) is the pair needed for contradiction • No path from C to B • The case for C on x-axis or y-axis in in the paper! B C A x

  15. Proving • Assume B C C’’ 45o A C’ D B’ x Contradiction!

  16. Conclusion • The critical total power for k-connectivity is • Using power control can save the power by a factor of compared with using common power • This can make a difference between bounded value and infinite value

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