We study wireless ad hoc networks with a large number of nodes, following the line of investigation initiated in [1] and continued in [2].We first focus on a network of n immobile nodes, each with a destination node chosen in random. We develop a scheme under which, in the absence of fading, the network can provide each node with a traffic rate λ1(n) = K1(n log n) − 1 2 . This result was first shown in [1] under a similar setting, however the proof presented here is shorter and uses only basic probability tools. We then proceed to show that, under a general model of fading, each node can send data to its destination with a rate λ2(n) = K2n − 1 2 (log n) − 3 2 . Next, we extend our formulation to study the effects of node mobility. We first develop a simple scheme under which each of the n mobile nodes can send data to a randomly chosen destination node with a rate λ3(n) = K3n − 1 2 (log n) − 3 2 , and with a fixed upper bound on the packet delay dmax that does not depend on n. We subsequently develop a scheme under which each of the nodes can send data to its destination with a rate λ4(n) = K4n d−1 2 (log n) − 5 2 , provided that nodes are willing to tolerate packet delays smaller than dmax(n) < K5n d , where 0 < d < 1. With both schemes, a general model of fading is assumed. In addition, nodes require no global topology or routing information, and only need to coordinate locally.The above results hold for an appropriate choice of values for the constants Ki, and with probability approaching 1 as the number of nodes n approaches infinity.
We present capacity results for three classes of wireless ad hoc networks, using a general framework that allows their unified treatment. The results hold with probability going to 1 as the number of nodes in the network approaches infinity, and under a general model for channel fading.We first study asymmetric networks that consist of n source nodes and around n d destination nodes, communicating over a wireless channel. Each source node creates data traffic that is directed to a destination node chosen at random. When 1 2 < d < 1, an aggregate throughput that increases with n as n 1 2 is achievable. If, however, 0 < d < 1 2 , bottlenecks are formed and the aggregate throughput can not increase faster than n d .We also consider cluster networks, that consist of n client nodes and around n d cluster heads, communicating over a wireless channel. Each of the clients wants to communicate with one of the cluster heads, but the particular choice of cluster head is not important. In this setting, the maximum aggregate throughput is on the order of n d , and it can be achieved with no transmissions taking place between client nodes.We conclude with the study of hybrid networks. These consist of n wireless nodes and around n d access points. The access points are equipped with wireless transceivers, but are also connected with each other through an independent network of infinite capacity. Their only task is to support the operation of the wireless nodes. When 1 2 < d < 1, an aggregate throughput on the order of n d is achievable, through the use of the infrastructure. If, however, 0 < d < 1 2 , using the infrastructure offers no significant gain, and the wireless nodes can achieve an aggregate throughput on the order of n 1 2 by using the wireless medium only.
Abstract-We investigate the spatial distribution of wireless nodes that can transport a given volume of traffic in a sensor network, while requiring the minimum number of wireless nodes. The traffic is created at a spatially distributed set of sources, and must arrive at a spatially distributed set of sinks. Under a general assumption on the physical and medium access control (MAC) layers, the optimal distribution of nodes induces a traffic flow identical to the electrostatic field that would exist if the sources and sinks of traffic were substituted with an appropriate distribution of electric charge.This analogy between Electrostatics and wireless sensor networks can be extended in a number of different ways. For example, Thomson's theorem on the distribution of electric charge on conductors gives the optimal distribution of traffic sources and sinks (that minimizes the number of nodes needed) when we have a limited degree of freedom on their initial placement. Electrostatics problems with Neumann boundary conditions and topologies with different types of dielectric materials can also be interpreted in the context of wireless sensor networks.The analogy also has important limitations. For example, if we move to a three dimensional topology, adapting our general assumption on the physical and MAC layers accordingly, or we stay in the two dimensional plane but use an alternative assumption, that is more suited to Ultra WideBand communication, the optimal traffic distribution is not in general irrotational, and so can not be interpreted as an electrostatic field. Finally, the analogy can not be extended to include networks that support more than one type of traffic.
Abstract-A spatially distributed set of sources is creating data that must be delivered to a spatially distributed set of sinks. A network of wireless nodes is responsible for sensing the data at the sources, transporting them over a wireless channel, and delivering them to the sinks. The problem is to find the optimal placement of nodes, so that a minimum number of them is needed.The critical assumption is made that the network is massively dense, i.e., there are so many sources, sinks, and wireless nodes, that it does not make sense to discuss in terms of microscopic parameters, such as their individual placements, but rather in terms of macroscopic parameters, such as their spatial densities.Assuming a particular interference-limited, capacity-achieving physical layer, and specifying that nodes only need to transport the data (and not to sense them at the sources, or deliver them at the sinks once their location is reached), the optimal node placement induces a traffic flow that is identical to the electrostatic field created if the sources and sinks are replaced by a corresponding distribution of positive and negative charges. Assuming a general model for the physical layer, and specifying that nodes must not only transport the data, but also sense them at the sources and deliver them at the sinks, the optimal placement of nodes is given by a scalar nonlinear partial differential equation found by calculus of variations techniques.The proposed formulation and derived equations can help in the design of large wireless sensor networks that are deployed in the most efficient manner, not only avoiding the formation of bottlenecks, but also striking the optimal balance between reducing congestion and having the data packets follow short routes.
We analyze the performance of an interferencelimited, decode-and-forward, cooperative relaying system that comprises a source, a destination, and N relays, placed arbitrarily on the plane and suffering from interference by a set of interferers placed according to a spatial Poisson process. In each transmission attempt, first the transmitter sends a packet; subsequently, a single one of the relays that received the packet correctly, if such a relay exists, retransmits it. We consider both selection combining and maximal ratio combining at the destination, Rayleigh fading, and interferer mobility.We derive expressions for the probability that a single transmission attempt is successful, as well as for the distribution of the transmission attempts until a packet is transmitted successfully. Results provide design guidelines applicable to a wide range of systems. Overall, the temporal and spatial characteristics of the interference play a significant role in shaping the system performance. Maximal ratio combining is only helpful when relays are close to the destination; in harsh environments, having many relays is especially helpful, and relay placement is critical; the performance improves when interferer mobility increases; and a tradeoff exists between energy efficiency and throughput.
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