Continuum Equilibria and Global Optimization for Routing in Dense Static Ad Hoc Networks

We consider massively dense ad hoc networks and study their continuum limits as the node density increases and as the graph providing the available routes becomes a continuous area with location and congestion dependent costs. We study both the global optimal solution as well as the non-cooperative routing problem among a large population of users where each user seeks a path from its origin to its destination so as to minimize its individual cost. Finally, we seek for a (continuum version of the) Wardrop equilibrium. We first show how to derive meaningful cost models as a function of the scaling properties of the capacity of the network and of the density of nodes. We present various solution methodologies for the problem: (1) the viscosity solution of the Hamilton-Jacobi-Bellman equation, for the global optimization problem, (2) a method based on Green's Theorem for the least cost problem of an individual, and (3) a solution of the Wardrop equilibrium problem using a transformation into an equivalent global optimization problem

Magnetworks: how mobility impacts the design of Mobile Networks

In this paper we study the optimal placement and optimal number of active relay nodes through the traffic density in mobile sensor ad-hoc networks. We consider a setting in which a set of mobile sensor sources is creating data and a set of mobile sensor destinations receiving that data. We make the assumption that the network is massively dense, i.e., there are so many sources, destinations, and relay nodes, that it is best to describe the network in terms of macroscopic parameters, such as their spatial density, rather than in terms of microscopic parameters, such as their individual placements. We focus on a particular physical layer model that is characterized by the following assumptions: i) the nodes must only transport the data from the sources to the destinations, and do not need to sense the data at the sources, or deliver them at the destinations once the data arrive at their physical locations, and ii) the nodes have limited bandwidth available to them, but they use it optimally to locally achieve the network capacity. In this setting, the optimal distribution of nodes induces a traffic density that resembles the electric displacement that will be created if we substitute the sources and destinations with positive and negative charges respectively. The analogy between the two settings is very tight and have a direct interpretation in wireless sensor networks

DESIGN AND IMPLEMENTATION OF INFORMATION PATHS IN DENSE WIRELESS SENSOR NETWORKS

In large-scale sensor networks with monitoring applications, sensor nodes are responsible to send periodic reports to the destination which is located far away from the area to be monitored. We model this area (referred to as the distributed source) with a positive load density function which determines the total rate of traffic generated inside any closed contour within the area. With tight limitations in energy consumption of wireless sensors and the many-to-one nature of communications in wireless sensor networks, the traditional definition of connectivity in graph theory does not seem to be sufficient to satisfy the requirements of sensor networks. In this work, a new notion of connectivity (called implementability) is defined which represents the ability of sensor nodes to relay traffic along a given direction field, referred to as information flow vector field $\vec{D}$. The magnitude of information flow is proportional to the traffic flux (per unit length) passing through any point in the network, and its direction is toward the flow of traffic. The flow field may be obtained from engineering knowledge or as a solution to an optimization problem. In either case, information flow flux lines represent a set of abstract paths (not constrained by the actual location of sensor nodes) which can be used for data transmission to the destination. In this work, we present conditions to be placed on $\vec{D}$ such that the resulting optimal vector field generates a desirable set of paths. In a sensor network with a given irrotational flow field $\vec{D}(x,y)$, we show that a density of $n(x,y)=O(|\vec{D}(x,y)|^2)$ sensor nodes is not sufficient to implement the flow field as $|\vec{D}|$ scales linearly to infinity. On the other hand, by increasing the density of wireless nodes to $n(x,y)=O(|\vec{D}(x,y)|^2 \log |\vec{D}(x,y)|)$, the flow field becomes implementable. Implementability requires more nodes than simple connectivity. However, results on connectivity are based on the implicit assumption of exhaustively searching all possible routes which contradicts the tight limitation of energy in sensor networks. We propose a joint MAC and routing protocol to forward traffic along the flow field. The proposed tier-based scheme can be further exploited to build lightweight protocol stacks which meet the specific requirements of dense sensor networks. We also investigate buffer scalability of sensor nodes routing along flux lines of a given irrotational vector field, and show that nodes distributed according to the sufficient bound provided above can relay traffic from the source to the destination with sensor nodes having limited buffer space. This is particularly interesting for dense wireless sensor networks where nodes are assumed to have very limited resources

DISTRIBUTED FLOW OPTIMIZATION IN DENSE WIRELESS NETWORKS

Due to large number of variables, optimizing information flow in a dense wire- less network using discrete methods can be computationally prohibitive. Instead of treating the nodes as discrete entities, these networks can be modeled as continuum of nodes providing a medium for information transport. In this scenario multi-hop information routes transform into an information flow vector field that is defined over the geographical domain of the network. At each point of the network, the orientation of the vector field shows the direction of the flow of information, and its magnitude shows the density of information flow. Modeling the dense network in continuous domain enables us to study the large scale behavior of the network under existing routing policies; furthermore, it justifies incorporation of multivariate calculus techniques in order to find new routing policies that optimize a suitable cost function, which only depend on large scale properties of the network. Thus, finding an optimum routing method translates into finding an optimal information flow vector field that minimizes the cost function. In order to transform the optimal information flow vector field into a routing policy, connections between discrete space (small scale) and continuous space (large scale) variables should be made and the question that how the nodes should interact with each other in the microscopic scale in order that their large scale behavior become optimal should be answered. In the past works, a centralized method of calculating the optimal information flow over the entire geographical area that encompasses the network has been suggested; however, using a centralized method to optimize information flow in a dynamic network is undesirable. Furthermore, the value of information flow vector field is needed only at the locations of randomly scattered nodes in the network, thus calculation of the information flow vector field for the entire network region (as suggested in previous models) is an unnecessary overhead. This poses a gap between the continuous space and discrete space models of information flow in dense wireless networks. This gap is how to calculate and apply the optimum information flow derived in continuous domain to a network with finite number of nodes. As a first step to fill this gap, a specific quadratic cost function is considered. In previous works, it is proved that the the vector field that minimizes this cost function is irrotational, thus it is written as the gradient of a potential function. This potential function satisfies a Poisson Partial Differential Equation (PDE) which in conjunction with Neumann boundary condition has a unique solution up to a constant. In this thesis the PDE resulted by optimization in continuous domain is discretized at locations of the nodes. By assuming a random node distribution with uniform density, the symmetries present enable us to solve the PDE in a distributed fashion. This solution is based on Jacobi iterations and requires only neighboring nodes to share their local information with each other. The gradient of the resulting potential defines the routes that the traffic should be forwarded. Furthermore, based on a graph theory model, we generalize our distributed solution to a more general cost function, namely, the p-norm cost function. This model also enables us to enhance the convergence rate of the Jacobi iterations

Distributed Detection and Estimation in Wireless Sensor Networks

In this article we consider the problems of distributed detection and estimation in wireless sensor networks. In the first part, we provide a general framework aimed to show how an efficient design of a sensor network requires a joint organization of in-network processing and communication. Then, we recall the basic features of consensus algorithm, which is a basic tool to reach globally optimal decisions through a distributed approach. The main part of the paper starts addressing the distributed estimation problem. We show first an entirely decentralized approach, where observations and estimations are performed without the intervention of a fusion center. Then, we consider the case where the estimation is performed at a fusion center, showing how to allocate quantization bits and transmit powers in the links between the nodes and the fusion center, in order to accommodate the requirement on the maximum estimation variance, under a constraint on the global transmit power. We extend the approach to the detection problem. Also in this case, we consider the distributed approach, where every node can achieve a globally optimal decision, and the case where the decision is taken at a central node. In the latter case, we show how to allocate coding bits and transmit power in order to maximize the detection probability, under constraints on the false alarm rate and the global transmit power. Then, we generalize consensus algorithms illustrating a distributed procedure that converges to the projection of the observation vector onto a signal subspace. We then address the issue of energy consumption in sensor networks, thus showing how to optimize the network topology in order to minimize the energy necessary to achieve a global consensus. Finally, we address the problem of matching the topology of the network to the graph describing the statistical dependencies among the observed variables.Comment: 92 pages, 24 figures. To appear in E-Reference Signal Processing, R. Chellapa and S. Theodoridis, Eds., Elsevier, 201

Energy Efficient, Cooperative Communication in Low-Power Wireless Networks

The increased interest in massive deployment of wireless sensors and network densification requires more innovation in low-latency communication across multi-hop networks. Moreover, the resource constrained nature of sensor nodes calls for more energy efficient transmission protocols, in order to increase the battery life of said devices. Therefore, it is important to investigate possible technologies that would aid in improving energy efficiency and decreasing latency in wireless sensor networks (WSN) while focusing on application specific requirements. To this end, and based on state of the art Glossy, a low-power WSN flooding protocol, this dissertation introduces two energy efficient, cooperative transmission schemes for low-power communication in WSNs, with the aim of achieving performance gains in energy efficiency, latency and power consumption. These approaches apply several cooperative transmission technologies such as physical layer network coding and transmit beamforming. Moreover, mathematical tools such as convex optimization and game theory are used in order to analytically construct the proposed schemes. Then, system level simulations are performed, where the proposed schemes are evaluated based on different criteria. First, in order to improve over all latency in the network as well as energy efficiency, MF-Glossy is proposed; a communication scheme that enables the simultaneous flooding of different packets from multiple sources to all nodes in the network. Using a communication-theoretic analysis, upper bounds on the performance of Glossy and MF-Glossy are determined. Further, simulation results show that MF-Glossy has the potential to achieve several-fold improvements in goodput and latency across a wide spectrum of network configurations at lower energy costs and comparable packet reception rates. Hardware implementation challenges are discussed as a step towards harnessing the potential of MF-Glossy in real networks, while focusing on key challenges and possible solutions. Second, under the assumption of available channel state information (CSI) at all nodes, centralized and distributed beamforming and power control algorithms are proposed and their performance is evaluated. They are compared in terms of energy efficiency to standard Glossy. Numerical simulations demonstrate that a centralized power control scheme can achieve several-fold improvements in energy efficiency over Glossy across a wide spectrum of network configurations at comparable packet reception rates. Furthermore, the more realistic scenario where CSI is not available at transmitting nodes is considered. To battle CSI unavailability, cooperation is introduced on two stages. First, cooperation between receiving and transmitting nodes is proposed for the process of CSI acquisition, where the receivers provide the transmitters with quantized (e.g. imperfect) CSI. Then, cooperation within transmitting nodes is proposed for the process of multi-cast transmit beamforming. In addition to an analytical formulation of the robust multi-cast beamforming problem with imperfect CSI, its performance is evaluated, in terms of energy efficiency, through numerical simulations. It is shown that the level of cooperation, represented by the number of limited feedback bits from receivers to transmitters, greatly impacts energy efficiency. To this end, the optimization problem of finding the optimal number of feedback bits B is formulated, as a programming problem, under QoS constraints of 5% maximum outage. Numerical simulations show that there exists an optimal number of feedback bits that maximizes energy efficiency. Finally, the effect of choosing cooperating transmitters on energy efficiency is studied, where it is shown that an optimum group of cooperating transmit nodes, also known as a transmit coalition, can be formed in order to maximize energy efficiency. The investigated techniques including optimum feedback bits and transmit coalition formation can achieve a 100% increase in energy efficiency when compared to state of the art Glossy under same operation requirements in very dense networks. In summary, the two main contributions in this dissertation provide insights on the possible performance gains that can be achieved when cooperative technologies are used in low-power wireless networks