When identical randomly located nodes, each capable of transmitting at bits per second and using a fixed range, form a wireless network, the throughput () obtainable by each node for a randomly chosen destination is 2 log bits per second under a noninterference protocol. If the nodes are optimally placed in a disk of unit area, traffic patterns are optimally assigned, and each transmission's range is optimally chosen, the bit-distance product that can be transported by the network per second is 2() bit-meters per second. Thus even under optimal circumstances, the throughput is only 2 bits per second for each node for a destination nonvanishingly far away. Similar results also hold under an alternate physical model where a required signal-to-interference ratio is specified for successful receptions. Fundamentally, it is the need for every node all over the domain to share whatever portion of the channel it is utilizing with nodes in its local neighborhood that is the reason for the constriction in capacity. Splitting the channel into several subchannels does not change any of the results. Some implications may be worth considering by designers. Since the throughput furnished to each user diminishes to zero as the number of users is increased, perhaps networks connecting smaller numbers of users, or featuring connections mostly with nearby neighbors, may be more likely to be find acceptance. Index Terms-Ad hoc networks, capacity, multihop radio networks, throughput, wireless networks. I. INTRODUCTION W IRELESS networks consist of a number of nodes which communicate with each other over a wireless channel. Some wireless networks have a wired backbone with only the last hop being wireless. Examples are cellular voice and data networks and mobile IP. In others, all links are wireless.

Abstract-Coding strategies that exploit node cooperation are developed for relay networks. Two basic schemes are studied: the relays decode-and-forward the source message to the destination, or they compress-and-forward their channel outputs to the destination. The decode-and-forward scheme is a variant of multihopping, but in addition to having the relays successively decode the message, the transmitters cooperate and each receiver uses several or all of its past channel output blocks to decode. For the compressand-forward scheme, the relays take advantage of the statistical dependence between their channel outputs and the destination's channel output. The strategies are applied to wireless channels, and it is shown that decode-and-forward achieves the ergodic capacity with phase fading if phase information is available only locally, and if the relays are near the source node. The ergodic capacity coincides with the rate of a distributed antenna array with full cooperation even though the transmitting antennas are not colocated. The capacity results generalize broadly, including to multiantenna transmission with Rayleigh fading, single-bounce fading, certain quasi-static fading problems, cases where partial channel knowledge is available at the transmitters, and cases where local user cooperation is permitted. The results further extend to multisource and multidestination networks such as multiaccess and broadcast relay channels.

In wireless data networks each transmitter's power needs to be high enough to reach the intended receivers, while generating minimum interference on other receivers sharing the same channel. In particular, if the nodes in the network are assumed to cooperate in routing each others' packets, as is the case in ad hoc wireless networks, each node should transmit with just enough power to guarantee connectivity in the network. Towards this end, we derive the critical power a node in the network needs to transmit in order to ensure that the network is connected with probability one as the number of nodes in the network goes to infinity. It is shown that if n nodes are placed in a disc of unit area in !R 2 and each node transmits at a power level so as to cover an area of lrT2 = (log n + c(n))/n, then the resulting network is asymptotically connected with probability one if and only if c(n) -+ +00. W. M. McEneaney et al. (eds.), Stochastic Analysis, Control, Optimization and Applications

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In recent work, Ozgur, Leveque, and Tse (2007) obtained a complete scaling characterization of throughput scaling for random extended wireless networks (i.e., $n$ nodes are placed uniformly at random in a square region of area $n$). They showed that for small path-loss exponents $\alpha\in(2,3]$ cooperative communication is order optimal, and for large path-loss exponents $\alpha > 3$ multi-hop communication is order optimal. However, their results (both the communication scheme and the proof technique) are strongly dependent on the regularity induced with high probability by the random node placement. In this paper, we consider the problem of characterizing the throughput scaling in extended wireless networks with arbitrary node placement. As a main result, we propose a more general novel cooperative communication scheme that works for arbitrarily placed nodes. For small path-loss exponents $\alpha \in (2,3]$, we show that our scheme is order optimal for all node placements, and achieves exactly the same throughput scaling as in Ozgur et al. This shows that the regularity of the node placement does not affect the scaling of the achievable rates for $\alpha\in (2,3]$. The situation is, however, markedly different for large path-loss exponents $\alpha >3$. We show that in this regime the scaling of the achievable per-node rates depends crucially on the regularity of the node placement. We then present a family of schemes that smoothly "interpolate" between multi-hop and cooperative communication, depending upon the level of regularity in the node placement. We establish order optimality of these schemes under adversarial node placement for $\alpha > 3$.Comment: 38 pages, 6 figures, to appear in IEEE Transactions on Information Theor