Spectral radii of asymptotic mappings and the convergence speed of the standard fixed point algorithm

Important problems in wireless networks can often be solved by computing fixed points of standard or contractive interference mappings, and the conventional fixed point algorithm is widely used for this purpose. Knowing that the mapping used in the algorithm is not only standard but also contractive (or only contractive) is valuable information because we obtain a guarantee of geometric convergence rate, and the rate is related to a property of the mapping called modulus of contraction. To date, contractive mappings and their moduli of contraction have been identified with case-by-case approaches that can be difficult to generalize. To address this limitation of existing approaches, we show in this study that the spectral radii of asymptotic mappings can be used to identify an important subclass of contractive mappings and also to estimate their moduli of contraction. In addition, if the fixed point algorithm is applied to compute fixed points of positive concave mappings, we show that the spectral radii of asymptotic mappings provide us with simple lower bounds for the estimation error of the iterates. An immediate application of this result proves that a known algorithm for load estimation in wireless networks becomes slower with increasing traffic.Comment: Paper accepted for presentation at ICASSP 201

Distributed Power Control in Multiuser MIMO Networks with Optimal Linear Precoding

Contractive interference functions introduced by Feyzmahdavian et al. is the newest approach in the analysis and design of distributed power control laws. This approach can be extended to several cases of distributed power control. One of the distributed power control scenarios wherein the contractive interference functions have not been employed is the power control in MIMO systems. In this paper, this scenario will be analyzed. In addition, the optimal linear precoder is employed in each user to achieve maximum point-to-point information rate. In our approach, we use the same amount of signaling as the previous methods did. However, we show that the uniqueness of Nash equilibria is more probable in our approach, suggesting that our proposed method improves the convergence performance of distributed power control in MIMO systems. We also show that the proposed power control algorithm can be implemented asynchronously, which gives a noticeable flexibility to our algorithm given the practical communication limitations.Comment: 6 pages, 3 figures, Presented in 7th International Symposium on Telecommunications (IST 2014

On the stability of positive nonlinear systems: Cooperative and concave system dynamics with applications to distributed networks

The most general class of systems we consider in this thesis is associated to sub-homogeneous vector fields, which includes as a special case concave vector fields. Conditions on the existence and uniqueness of an equilibrium point in the interior of the positive orthant are given and an estimate of the domain of attraction is made. We consider systems with irredubile, or reducible Jacobian matrix if the system is distribute

On Power and Load Coupling in Cellular Networks for Energy Optimization

We consider the problem of minimization of sum transmission energy in cellular networks where coupling occurs between cells due to mutual interference. The coupling relation is characterized by the signal-to-interference-and-noise-ratio (SINR) coupling model. Both cell load and transmission power, where cell load measures the average level of resource usage in the cell, interact via the coupling model. The coupling is implicitly characterized with load and power as the variables of interest using two equivalent equations, namely, non-linear load coupling equation (NLCE) and non-linear power coupling equation (NPCE), respectively. By analyzing the NLCE and NPCE, we prove that operating at full load is optimal in minimizing sum energy, and provide an iterative power adjustment algorithm to obtain the corresponding optimal power solution with guaranteed convergence, where in each iteration a standard bisection search is employed. To obtain the algorithmic result, we use the properties of the so-called standard interference function; the proof is non-standard because the NPCE cannot even be expressed as a closed-form expression with power as the implicit variable of interest. We present numerical results illustrating the theoretical findings for a real-life and large-scale cellular network, showing the advantage of our solution compared to the conventional solution of deploying uniform power for base stations.Comment: Accepted for publication in IEEE Transactions on Wireless Communication