Throughput-Delay Trade-off for Hierarchical Cooperation in Ad Hoc Wireless Networks

Hierarchical cooperation has recently been shown to achieve better throughput scaling than classical multihop schemes under certain assumptions on the channel model in static wireless networks. However, the end-to-end delay of this scheme turns out to be significantly larger than those of multihop schemes. A modification of the scheme is proposed here that achieves a throughput-delay trade-off $D(n)=(\log n)^2 T(n)$ for T(n) between $\Theta(\sqrt{n}/\log n)$ and $\Theta(n/\log n)$, where D(n) and T(n) are respectively the average delay per bit and the aggregate throughput in a network of n nodes. This trade-off complements the previous results of El Gamal et al., which show that the throughput-delay trade-off for multihop schemes is given by D(n)=T(n) where T(n) lies between $\Theta(1)$ and $\Theta(\sqrt{n})$. Meanwhile, the present paper considers the network multiple-access problem, which may be of interest in its own right.Comment: 9 pages, 6 figures, to appear in IEEE Transactions on Information Theory, submitted Dec 200

Product Multicommodity Flow in Wireless Networks

We provide a tight approximate characterization of the $n$-dimensional product multicommodity flow (PMF) region for a wireless network of $n$ nodes. Separate characterizations in terms of the spectral properties of appropriate network graphs are obtained in both an information theoretic sense and for a combinatorial interference model (e.g., Protocol model). These provide an inner approximation to the $n^2$ dimensional capacity region. These results answer the following questions which arise naturally from previous work: (a) What is the significance of $1/\sqrt{n}$ in the scaling laws for the Protocol interference model obtained by Gupta and Kumar (2000)? (b) Can we obtain a tight approximation to the "maximum supportable flow" for node distributions more general than the geometric random distribution, traffic models other than randomly chosen source-destination pairs, and under very general assumptions on the channel fading model? We first establish that the random source-destination model is essentially a one-dimensional approximation to the capacity region, and a special case of product multi-commodity flow. Building on previous results, for a combinatorial interference model given by a network and a conflict graph, we relate the product multicommodity flow to the spectral properties of the underlying graphs resulting in computational upper and lower bounds. For the more interesting random fading model with additive white Gaussian noise (AWGN), we show that the scaling laws for PMF can again be tightly characterized by the spectral properties of appropriately defined graphs. As an implication, we obtain computationally efficient upper and lower bounds on the PMF for any wireless network with a guaranteed approximation factor.Comment: Revised version of "Capacity-Delay Scaling in Arbitrary Wireless Networks" submitted to the IEEE Transactions on Information Theory. Part of this work appeared in the Allerton Conference on Communication, Control, and Computing, Monticello, IL, 2005, and the Internation Symposium on Information Theory (ISIT), 200

Linear Capacity Scaling in Wireless Networks: Beyond Physical Limits?

We investigate the role of cooperation in wireless networks subject to a spatial degrees of freedom limitation. To address the worst case scenario, we consider a free-space line-of-sight type environment with no scattering and no fading. We identify three qualitatively different operating regimes that are determined by how the area of the network A, normalized with respect to the wavelength lambda, compares to the number of users n. In networks with sqrt{A}/lambda < sqrt{n}, the limitation in spatial degrees of freedom does not allow to achieve a capacity scaling better than sqrt{n} and this performance can be readily achieved by multi-hopping. This result has been recently shown by Franceschetti et al. However, for networks with sqrt{A}/lambda > sqrt{n}, the number of available degrees of freedom is min(n, sqrt{A}/lambda), larger that what can be achieved by multi-hopping. We show that the optimal capacity scaling in this regime is achieved by hierarchical cooperation. In particular, in networks with sqrt{A}/lambda> n, hierarchical cooperation can achieve linear scaling.Comment: 10 pages, 5 figures, in Proc. of IEEE Information Theory and Applications Workshop, Feb. 201

Hierarchical Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks

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. 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 n^{2/3} of Aeron and Saligrama, 2006. 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 n^{2-alpha/2} for 2<alpha<3 and sqrt{n} for alpha>3. The best known earlier result (Xie and Kumar 2006) identified the scaling law for alpha > 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 MIMO communication. The key ingredient is a hierarchical and digital architecture for nodal exchange of information for realizing the cooperation.Comment: 56 pages, 16 figures, submitted to IEEE Transactions on Information Theor

Information Theoretic Operating Regimes of Large Wireless Networks

In analyzing the point-to-point wireless channel, insights about two qualitatively different operating regimes--bandwidth- and power-limited--have proven indispensable in the design of good communication schemes. In this paper, we propose a new scaling law formulation for wireless networks that allows us to develop a theory that is analogous to the point-to-point case. We identify fundamental operating regimes of wireless networks and derive architectural guidelines for the design of optimal schemes. Our analysis shows that in a given wireless network with arbitrary size, area, power, bandwidth, etc., there are three parameters of importance: the short-distance SNR, the long-distance SNR, and the power path loss exponent of the environment. Depending on these parameters we identify four qualitatively different regimes. One of these regimes is especially interesting since it is fundamentally a consequence of the heterogeneous nature of links in a network and does not occur in the point-to-point case; the network capacity is {\em both} power and bandwidth limited. This regime has thus far remained hidden due to the limitations of the existing formulation. Existing schemes, either multihop transmission or hierarchical cooperation, fail to achieve capacity in this regime; we propose a new hybrid scheme that achieves capacity.Comment: 12 pages, 5 figures, to appear in IEEE Transactions on Information Theor

Diversity-Multiplexing Tradeoff for the MIMO Static Half-Duplex Relay

In this work, we investigate the diversity-multiplexing tradeoff (DMT) of the multiple-antenna (MIMO) static half-duplex relay channel. A general expression is derived for the DMT upper bound, which can be achieved by a compress-and-forward protocol at the relay, under certain assumptions. The DMT expression is given as the solution of a minimization problem in general, and an explicit expression is found when the relay channel is symmetric in terms of number of antennas, i.e. the source and the destination have n antennas each, and the relay has m antennas. It is observed that the static half-duplex DMT matches the full-duplex DMT when the relay has a single antenna, and is strictly below the full-duplex DMT when the relay has multiple antennas. Besides, the derivation of the upper bound involves a new asymptotic study of spherical integrals (that is, integrals with respect to the Haar measure on the unitary group U(n)), which is a topic of mathematical interest in itself.Comment: 19 pages, 2 figures, submitted to the IEEE Transactions on Information Theor

Set inversion for χ-algorithms with application to guaranteed robot localization

International audienceCharacterizing the set of all parameter vectors such that their image by a vector function belongs to a given set is a set-inversion problem. The algorithm SIVIA (Set Inversion Via Interval Analysis) makes it possible to perform this task in an approximate but guaranteed way. In the examples treated so far, the function to be inverted was given either explicitly or by a sequential algorithm. In this paper, this approach is extended to the case of branching algorithms involving if statements. As an illustration, the static localization of a robot from bounded-error range measurements is considered. The notion of remoteness, introduced for an archetypal but realistic sonar model, allows this problem to be cast into the set-inversion framework

Analytical Model for the Magnetic Field Distribution in a Flux Modulation Superconducting Machine

International audienceThis paper presents a theoretical analysis of an axial field machine using High Temperature Superconductors (HTS) wires and bulks. The air-gap magnetic field obtained with the HTS coil and modulated by the HTS bulks is predicted by two 2D axisymmetric models. Analytical models are based on the solution of Laplace's equation by the separation of variable method. The torque is obtained by a quick numerical integration of the Laplace force that acts on the armature winding. The proposed model is compared with 3D finite element simulations and good agreement is obtained. This model can be used with an optimization design procedure with a large reduction of the computational time