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It is suspected that that an upper bound of order (W n) b-m/s may actually hold. In the special case where the ratio Pmax /Pmin between the maximum and minimum powers that transmitters can employ is bounded above by , then an upper bound is in fact CT = (W 8n/ )/ ( Pmin /Pmax ) 1

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Both bounds suggest that transport capacity improves when is larger, i.e. when the signal power decays more rapidly with distance. 19.2.2 Random networks In this case, n nodes are randomly located, i.e. independently and uniformly distributed, either on the surface S 2 of a three-dimensional sphere of area 1 m2 , or in a disk of area 1 m2 in the plane. The purpose in studying S 2 is to separate edge effects from other phenomena. Each node has a randomly chosen destination to which it wishes to send (n) b/s. The destination for each node is independently chosen as the node nearest to a randomly located point, i.e. uniformly and independently distributed. (Thus destinations are on the order of 1 m away on average.) All transmissions employ the same nominal range or power (homogeneous nodes). As for arbitrary networks, both a protocol model and a physical model are considered. 19.2.2.1 The protocol model All nodes employ a common range r for all their transmissions. When node X i transmits to a node X j over the mth subchannel, this transmission is successfully received by X j if |X i X j | r and for every other node X k simultaneously transmitting over the same subchannel |X k X j | (1 + )r .

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19.2.2.2 The physical model All nodes choose a common power level P for all their transmissions. Let {X k ; k T } be the subset of nodes simultaneously transmitting at some time instant over a certain subchannel. A transmission from a node X i , i T , is successfully received by a node X j if S/I , where S = P/ X i X j | and I =N+

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19.2.2.3 The throughput capacity of random networks The throughput is de ned in the usual manner as the time average of the number of bits per second that can be transmitted by every node to its destination. 19.2.2.4 Feasible throughput A throughput of (n) b/s for each node is feasible if there is a spatial and temporal scheme for scheduling transmissions, such that by operating the network in a multihop fashion and buffering at intermediate nodes when awaiting transmission, every node can send (n) b/s on average to its chosen destination node. That is, there is a T < such that in every time interval [(i 1)T, iT] every node can send T (n) b to its corresponding destination node. 19.2.2.5 The throughput capacity of random wireless networks We say that the throughput capacity of the class of random networks is of order b/s if there are deterministic constants c > 0 and c < + such that

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lim inf Prob [ (n) = c f (n) is feasible] < l It will be shown in the next section that, in the case of both the surface of the sphere and a planar disk, the order of the throughput capacity is (n) = [W/ (n log n)] b/s for the protocol model. For the upper bound for some c ,

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Speci cally, there are deterministic constants c and c not depending on n, W , such or that (n) = c W/ (1 + )2 (n log n) b/s is feasible, and (n) = c W/ 2 (n log n) b/s is infeasible, both with probability approaching 1 as n . It will be also shown that, for the physical model, a throughput of (n) = cW/ (n log n) b/s is feasible, while (n) = c W/ n bits per second is not, for appropriate c, c , both with probability approaching 1 as n . Speci cally, there are deterministic constants c and

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