We consider the capacity problem for wireless networks. Networks are modeled as random unit-disk graphs, and the capacity problem is formulated as one of finding the maximum value of a multicommodity flow. In this paper, we develop a proof technique based on which we are able to obtain a tight characterization of the solution to the linear program associated with the multiflow problem, to within constants independent of network size. We also use this proof method to analyze network capacity for a variety of transmitter receiver architectures, for which we obtain some conclusive results. These results contain as a special case (and strengthen) those of Gupta and Kumar for random networks, for which a new derivation is provided using only elementary counting and discrete probability tools.
As mentioned before, @cite_19 sparked significant interest in these problems. Follow up results from the same group were reported in @cite_10 @cite_29 . Some information theoretic bounds for large-area networks were obtained in @cite_30 . When nodes are allowed to move, assuming transmission delays proportional to the mixing time of the network, the total network throughput is @math , and therefore the network can carry a non-vanishing rate per node @cite_2 . Using a linear programming formulation, non-asymptotic versions of the results in @cite_19 are given in @cite_22 ; an extended version of that work can be found in @cite_15 . An alternative method for deriving transport capacity was presented in @cite_1 . The capacity of large Gaussian relay networks was found in @cite_3 . Preliminary versions of our work based on network flows have appeared in @cite_9 @cite_32 ; and network flow techniques have been proposed to study network capacity problems (cf., e.g., @cite_34 , [Ch. 14.10] cover-thomas:it-book ), and network coding problems @cite_0 . From the network coding literature, of particular relevance to this work is the work on multiple unicast sessions @cite_26 .
@cite_30 @cite_26 @cite_22 @cite_29 @cite_9 @cite_1 @cite_32 @cite_3 @cite_0 @cite_19 @cite_2 @cite_15 @cite_34 @cite_10
2110225498 2165510626 2131992719 2108190032 2138203492 2138928022 2137775453 59733848 1747529848 2105831729 2136812100
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. We define and study capacity regions for wireless ad hoc networks with an arbitrary number of nodes and topology. These regions describe the set of achievable rate combinations between all source-destination pairs in the network under various transmission strategies, such as variable-rate transmission, single-hop or multihop routing, power control, and successive interference cancellation (SIC). Multihop cellular networks and networks with energy constraints are studied as special cases. With slight modifications, the developed formulation can handle node mobility and time-varying flat-fading channels. Numerical results indicate that multihop routing, the ability for concurrent transmissions, and SIC significantly increase the capacity of ad hoc and multihop cellular networks. On the other hand, gains from power control are significant only when variable-rate transmission is not used. Also, time-varying flat-fading and node mobility actually improve the capacity. Finally, multihop routing greatly improves the performance of energy-constraint networks. We consider the problem of determining rates of growth for the maximum stable throughput achievable in dense wireless networks. We formulate this problem as one of finding maximum flows on random unit-disk graphs. Equipped with the max-flow min-cut theorem as our basic analysis tool, we obtain rates of growth under three models of communication: (a) omnidirectional transmissions; (b) "simple" directional transmissions, in which sending nodes generate a single beam aimed at a particular receiver; and (c) "complex" directional transmissions, in which sending nodes generate multiple beams aimed at multiple receivers. Our main finding is that an increase of Θlog2n in maximum stable throughput is all that can be achieved by allowing arbitrarily complex signal processing (in the form of generation of directed beams) at the transmitters and receivers. We conclude therefore that neither directional antennas, nor the ability to communicate simultaneously with multiple nodes, can be expected in practice to effectively circumvent the constriction on capacity in dense networks that results from the geometric layout of nodes in space. We address the problem of how throughput in a wireless network scales as the number of users grows. Following the model of Gupta and Kumar, we consider n identical nodes placed in a fixed area. Pairs of transmitters and receivers wish to communicate but are subject to interference from other nodes. Throughput is measured in bit-meters per second. We provide a very elementary deterministic approach that gives achievability results in terms of three key properties of the node locations. As a special case, we obtain spl Omega ( spl radic n) throughput for a general class of network configurations in a fixed area. Results for random node locations in a fixed area can also be derived as special cases of the general result by verifying the growth rate of three parameters. For example, as a simple corollary of our result we obtain a stronger (almost sure) version of the spl radic n spl radic (logn) throughput for random node locations in a fixed area obtained by Gupta and Kumar. Results for some other interesting non-independent and identically distributed (i.i.d.) node distributions are also provided. The capacity of a particular large Gaussian relay network is determined in the limit as the number of relays tends to infinity. Upper bounds are derived from cut-set arguments, and lower bounds follow from an argument involving uncoded transmission. It is shown that in cases of interest, upper and lower bounds coincide in the limit as the number of relays tends to infinity. Hence, this paper provides a new example where a simple cut-set upper bound is achievable, and one more example where uncoded transmission achieves optimal performance. The findings are illustrated by geometric interpretations. The techniques developed in this paper are then applied to a sensor network situation. This is a network joint source-channel coding problem, and it is well known that the source-channel separation theorem does not extend to this case. The present paper extends this insight by providing an example where separating source from channel coding does not only lead to suboptimal performance-it leads to an exponential penalty in performance scaling behavior (as a function of the number of nodes). Finally, the techniques developed in this paper are extended to include certain models of ad hoc wireless networks, where a capacity scaling law can be established: When all nodes act purely as relays for a single source-destination pair, capacity grows with the logarithm of the number of nodes. We take a new look at the issue of network capacity. It is shown that network coding is an essential ingredient in achieving the capacity of a network. Building on recent work by (see Proc. 2001 IEEE Int. Symp. Information Theory, p.102), who examined the network capacity of multicast networks, we extend the network coding framework to arbitrary networks and robust networking. For networks which are restricted to using linear network codes, we find necessary and sufficient conditions for the feasibility of any given set of connections over a given network. We also consider the problem of network recovery for nonergodic link failures. For the multicast setup we prove that there exist coding strategies that provide maximally robust networks and that do not require adaptation of the network interior to the failure pattern in question. The results are derived for both delay-free networks and networks with delays. 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. We introduce a methodology for studying wireless ad hoc networks in a multihop traffic environment. Our approach is to use theoretical upper bounds on network performance for evaluating the effects of various design choices: we focus on power control, the queuing discipline, the choice of routing and media access protocols, and their interactions. Using this framework, we then concentrate on the problem of medium access for wireless multihop networks. We first study CSMA CA, and find that its performance strongly depends on the choice of the accompanying routing protocol. We then introduce two protocols that outperform CSMA CA, both in terms of energy efficiency and achievable throughput. The progressive back off algorithm (PBOA) performs medium access jointly with power control. The progressive rump up algorithm (PRUA) sacrifices energy efficiency in favor of higher throughput. Both protocols slot time, and are integrated with queuing disciplines that are more relaxed than the first in first out (FIFO) rule. They are totally distributed and the overhead they require does not increase with the size and node density of the network. We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems. We study communication networks of arbitrary size and topology and communicating over a general vector discrete memoryless channel (DMC). We propose an information-theoretic constructive scheme for obtaining an achievable rate region in such networks. Many well-known capacity-defining achievable rate regions can be derived as special cases of the proposed scheme. A few such examples are the physically degraded and reversely degraded relay channels, the Gaussian multiple-access channel, and the Gaussian broadcast channel. The proposed scheme also leads to inner bounds for the multicast and allcast capacities. Applying the proposed scheme to a specific wireless network of n nodes located in a region of unit area, we show that a transport capacity of spl Theta (n) bit-meters per second (bit-meters s) is feasible in a certain family of networks, as compared to the best possible transport capacity of spl Theta ( spl radic n) bit-meters s in (2000), where the receiver capabilities were limited. Even though the improvement is shown for a specific class of networks, a clear implication is that designing and employing more sophisticated multiuser coding schemes can provide sizable gains in at least some large wireless networks.