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 isin (2,3], cooperative communication is order optimal, and for large path-loss exponents alpha > 3, multihop 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 isin (2,3], we show that our scheme is order optimal for all node placements, and achieves exactly the same throughput scaling as in Ozgur. This shows that the regularity of the node placement does not affect the scaling of the achievable rates for alpha isin (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 ldquointerpolaterdquo between multihop 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.
@cite_7: We derive upper bounds on the transport capacity of wireless networks. The bounds obtained are solely dependent on the geographic locations and power constraints of the nodes. As a result of this derivation, we are able to conclude the optimality, in the sense of scaling of transport capacity with the number of nodes, of a multihop communication strategy for a class of network topologies.
@cite_9: An achievable bit rate per source-destination pair in a wireless network of n randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same 1 radicn transmission rate of arbitrarily located nodes. This contrasts with previous results suggesting that a 1 radicnlogn reduced rate is the price to pay for the randomness due to the location of the nodes. The network operation strategy to achieve the result corresponds to the transition region between order and disorder of an underlying percolation model. If nodes are allowed to transmit over large distances, then paths of connected nodes that cross the entire network area can be easily found, but these generate excessive interference. If nodes transmit over short distances, then such crossing paths do not exist. Percolation theory ensures that crossing paths form in the transition region between these two extreme scenarios. Nodes along these paths are used as a backbone, relaying data for other nodes, and can transport the total amount of information generated by all the sources. A lower bound on the achievable bit rate is then obtained by performing pairwise coding and decoding at each hop along the paths, and using a time division multiple access scheme
@cite_1: When n identical randomly located nodes, each capable of transmitting at W bits per second and using a fixed range, form a wireless network, the throughput spl lambda (n) obtainable by each node for a randomly chosen destination is spl Theta (W spl radic (nlogn)) 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 spl Theta (W spl radic An) bit-meters per second. Thus even under optimal circumstances, the throughput is only spl Theta (W spl radic n) 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.
@cite_3: Wireless networks with a minimum inter-node separation distance are studied where the signal attenuation grows in magnitude as 1 ρδ with distance ρ. Two performance measures of wireless networks are analyzed. The transport capacity is the supremum of the total distance-rate products that can be supported by the network. The energy cost of information transport is the infimum of the ratio of the transmission energies used by all the nodes to the number of bit-meters of information thereby transported.If the phases of the attenuations between node pairs are uniformly and independently distributed, it is shown that the expected transport capacity is upper-bounded by a multiple of the total of the transmission powers of all the nodes, whenever δ > 2 for two-dimensional networks or δ > 5 4 for one-dimensional networks, even if all the nodes have full knowledge of all the phases, i.e., full channel state information. If all nodes have an individual power constraint, the expected transport capacity grows at most linearly in the number of nodes due to the linear growth of the total power. This establishes the best case order of expected transport capacity for these ranges of path-loss exponents since linear scaling is also feasible.If the phases of the attenuations are arbitrary, it is shown that the transport capacity is upper-bounded by a multiple of the total transmission power whenever δ > 5 2 for two-dimensional networks or δ > 3 2 for one-dimensional networks, even if all the nodes have full channel state information. This shows that there is indeed a positive energy cost which is no less than the reciprocal of the above multiplicative constant. It narrows the transition regime where the behavior is still open, since it is known that when δ < 3 2 for two-dimensional networks, or δ < 1 for one-dimensional networks, the transport capacity cannot generally be bounded by any multiple of the total transmit power.
@cite_5: We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuation regime where either the medium is absorptive, a situation which generally prevails, or the path loss exponent is greater than 3. We study the transport capacity, defined as the supremum over the set of feasible rate vectors of the distance weighted sum of rates. We consider two assumption sets. Under the first assumption set, which essentially requires only a mild time average type of bound on the fading process, we show that the transport capacity can grow no faster than O(n), where n denotes the number of nodes, even when the channel state information (CSI) is available noncausally at both the transmitters and the receivers. This assumption includes common models of stationary ergodic channels; constant, frequency-selective channels; flat, rapidly varying channels; and flat slowly varying channels. In the second assumption set, which essentially features an independence, time average of expectation, and nonzeroness condition on the fading process, we constructively show how to achieve transport capacity of spl Omega (n) even when the CSI is unknown to both the transmitters and the receivers, provided that every node has an appropriately nearby node. This assumption set includes common models of independent and identically distributed (i.i.d.) channels; constant, flat channels; and constant, frequency-selective channels. The transport capacity is achieved by nodes communicating only with neighbors, and using only point-to-point coding. The thrust of these results is that the multihop strategy, toward which much protocol development activity is currently targeted, is appropriate for fading environments. The low attenuation regime is open.
@cite_13: How much information can be carried over a wireless network with a multiplicity of nodes, and how should the nodes cooperate to transfer information? To study these questions, we formulate a model of wireless networks that particularly takes into account the distances between nodes, and the resulting attenuation of radio signals, and study a performance measure that weights information by the distance over which it is transported. Consider a network with the following features. I) n nodes located on a plane, with minimum separation distance spl rho sub min >0. II) A simplistic model of signal attenuation e sup - spl gamma spl rho spl rho sup spl delta over a distance spl rho , where spl gamma spl ges 0 is the absorption constant (usually positive, unless over a vacuum), and spl delta >0 is the path loss exponent. III) All receptions subject to additive Gaussian noise of variance spl sigma sup 2 . The performance measure we mainly, but not exclusively, study is the transport capacity C sub T :=sup spl Sigma on sub spl lscr =1 sup m R sub spl lscr spl middot spl rho sub spl lscr , where the supremum is taken over m, and vectors (R sub 1 ,R sub 2 ,...,R sub m ) of feasible rates for m source-destination pairs, and spl rho sub spl lscr is the distance between the spl lscr th source and its destination. It is the supremum distance-weighted sum of rates that the wireless network can deliver. We show that there is a dichotomy between the cases of relatively high and relatively low attenuation. When spl gamma >0 or spl delta >3, the relatively high attenuation case, the transport capacity is bounded by a constant multiple of the sum of the transmit powers of the nodes in the network. However, when spl gamma =0 and spl delta <3 2, the low-attenuation case, we show that there exist networks that can provide unbounded transport capacity for fixed total power, yielding zero energy priced communication. Examples show that nodes can profitably cooperate over large distances using coherence and multiuser estimation when the attenuation is low. These results are established by developing a coding scheme and an achievable rate for Gaussian multiple-relay channels, a result that may be of interest in its own right.
@cite_12: We derive an information-theoretic upper bound on the rate per communication pair in a large ad hoc wireless network. We show that under minimal conditions on the attenuation due to the environment and for networks with a constant density of users, this rate tends to zero as the number of users gets large.
@cite_11: n source and destination pairs randomly located in an area want to communicate with each other. Signals transmitted from one user to another at distance r apart are subject to a power loss of r-alpha as well as a random phase. We identify the scaling laws of the information-theoretic capacity of the network when nodes can relay information for each other. In the case of dense networks, where the area is fixed and the density of nodes increasing, we show that the total capacity of the network scales linearly with n. This improves on the best known achievability result of n2 3 of Aeron and Saligrama. In the case of extended networks, where the density of nodes is fixed and the area increasing linearly with n, we show that this capacity scales as n2-alpha 2 for 2lesalpha 4. Thus, much better scaling than multihop can be achieved in dense networks, as well as in extended networks with low attenuation. The performance gain is achieved by intelligent node cooperation and distributed multiple-input multiple-output (MIMO) communication. The key ingredient is a hierarchical and digital architecture for nodal exchange of information for realizing the cooperation.
Since @cite_1 , the problem has received a considerable amount of attention. One stream of work @cite_13 @cite_7 @cite_12 @cite_5 @cite_3 @cite_9 @cite_11 has progressively broadened the conditions on the channel model and the communication model, under which multi-hop communication is order optimal. Specifically, with a power loss of @math for signals sent over distance @math , it has been established that under signal attenuation @math and random node placement, the best achievable per-node rate @math for random source-destination pairing scales essentially like @math and that this scaling is achievable with multi-hop communication.