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f = (g) if g = O( f ); f = (g) if f = O(g) as well as g = O( f ). Thus, all O( ) results are upper bounds, all ( ) results me lower bounds, and all ( ) results are sharp order estimates for the transport capacity. For n nodes located in an area of A square meters, it is shown in Section 19.2 that the transport capacity is of order O[ (An)] under a noninformation theoretic protocol model. If A itself grows like n, i.e. A = (n), then the scaling law is O[ (An)] = O(n), which coincides with the information-theoretic scaling law here. In fact, A must grow at least this rate since nodes are separated by a minimum distance min > 0, i.e. A = (n), and so the O(n) result here is slightly stronger than the O[ (An)] result in the previous section. (r3) If either > 0 or > 2 in any linear network, then CT where c2 ( , , min ) Ptotal 2 (19.53)
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(r4) For any linear network, if either > 0 or > 2, then the transport capacity is upper-bounded as follows: c2 ( , , min )Pind n 2 where c2 ( , , min ) is as in Equation (19.44). CT
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19.3.5 Multihop and feasible lower bounds under high attenuation The O(n) upper bound on transport capacity is tight for regular planar networks in media with > 0 or > 3, and it is achieved by multihop. The multihop strategy is de ned as the following. Let denote the set of all paths from source s to destination d , where by such a path we mean a sequence (s = j0 , j1 , . . . , . . . , jz = d ) with jq = jr for q = r . The total traf c rate R to be provided to the source destination pair (s , d ) is split over the paths in in such a way that if traf c rate 0 is to be carried over path , then = R . On each path , packets are relayed from node to next node. On each P such hop, each packet is fully decoded, treating all interference as noise. Thus, only pointto-point coding is used, and no network coding or multiuser estimation is employed. Such a strategy is of great interest and it is currently the object of much protocol development activity. The following result implies that, when > 0 or > 3, the sharp order of the transport capacity for a regular planar network is (n), and that it can be attained by multihop. (r5) In a regular planar network with either > 0 or > 1, and individual power constraint Pind CT S e 2 Pind c3 ( , )Pind + 2 n
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and S(x) denotes the Shannon function S(x) := log(1 + x) /2. This order of distance weighted sum of rates is achievable by multihop. Multihop is order-optimal in a random scenario over a regular planar network in media with > 0 or > 3, providing some theoretical justi cation for its use in situations where traf c is diffused over the network. Consider a regular planar network with either > 0 or > 1, and individual power constraint Pind . The n source destination pairs are randomly chosen as follows: every source s is chosen as the node nearest to a randomly (uniformly i.i.d.) chosen point in the domain, and similarly for every destination d . Then lim Prob [ R = c/ (n log n) is feasible for n every {1, 2, . . , n}] = 1 for some c > 0. Consequently, a distance weighted sum of . rates CT = [n/ (log n)] is supported with probability approaching one as . This is within a factor l/ log n of the transport capacity (n) possible when > 3. (r6) A vector of rates (R1 , R2 , . . . , Rm ) can be supported by multihop in a planar network in media with > 0 or > 1, if the traf c can be load balanced such that no node is overloaded and no hop is too long. This is a fairly straightforward result saying nothing about order optimality, and is provided only in support of the above theme that multihop is an appropriate architecture for balanceable scenarios. (r7) A set of rates (R1 , R2 , . . . Rm ) for a planar network can be supported by multihop if no hop is longer than a distance , and for every 1 i n, the traf c to be relayed by node i
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