Gibbsian Method for the Self-Optimization of Cellular Networks

In this work, we propose and analyze a class of distributed algorithms performing the joint optimization of radio resources in heterogeneous cellular networks made of a juxtaposition of macro and small cells. Within this context, it is essential to use algorithms able to simultaneously solve the problems of channel selection, user association and power control. In such networks, the unpredictability of the cell and user patterns also requires distributed optimization schemes. The proposed method is inspired from statistical physics and based on the Gibbs sampler. It does not require the concavity/convexity, monotonicity or duality properties common to classical optimization problems. Besides, it supports discrete optimization which is especially useful to practical systems. We show that it can be implemented in a fully distributed way and nevertheless achieves system-wide optimality. We use simulation to compare this solution to today's default operational methods in terms of both throughput and energy consumption. Finally, we address concrete issues for the implementation of this solution and analyze the overhead traffic required within the framework of 3GPP and femtocell standards.Comment: 25 pages, 9 figures, to appear in EURASIP Journal on Wireless Communications and Networking 201

A Gibbsian model for message routeing in highly dense multihop networks

We investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin. Given the (random) transmitter locations, we weight the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured in terms of the number of pairs of hops using the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of an optimization of the individual trajectories. In the limit of high spatial density of users, we describe the totality of all the message trajectories in terms of empirical measures. Employing large deviations arguments, we derive a characteristic variational formula for the limiting free energy and analyse the minimizer(s) of the formula, which describe the most likely shapes of the trajectory flow. The empirical measures of the message trajectories well describe the interference, but not the congestion; the latter requires introducing an additional empirical measure. Our results remain valid under replacing the two penalization terms by more general functionals of these two empirical measures.Comment: 40 page

Routeing properties in a Gibbsian model for highly dense multihop networks

We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signal-to-interference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the high-density limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equally-sized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a game-theoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalize

Routeing properties in a Gibbsian model for highly dense multihop networks

We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution, which favours trajectories with low interference, measured in terms of signal-to-interference ratio. This model was introduced in our earlier paper [KT18], where we expressed, in the limit of a high density of users, the typical distribution of the family of trajectories in terms of a law of large numbers. In the present work, we derive its qualitative properties. We analytically identify the emerging typical scenarios in three extreme regimes. We analyse the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory approaches a straight line quickly, in regime (1) with equal hop lengths. Interestingly, in regime (1), the typical length of a hop diverges logarithmically in the distance of the transmitter to the base station. We further analyse (3) local and global repulsive effects of a densely populated subarea on the trajectories.Comment: 36 pages, 5 figure

A Gibbsian model for message routing in highly dense multi-hop networks

We investigate a probabilistic model for routing in relay-augmented multihop ad-hoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure

Limit theory for geometric statistics of point processes having fast decay of correlations

Let $P$ be a simple,stationary point process having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $P_n:= P \cap W_n$ be its restriction to windows $W_n:= [-{1 \over 2}n^{1/d},{1 \over 2}n^{1/d}]^d \subset \mathbb{R}^d$. We consider the statistic $H_n^\xi:= \sum_{x \in P_n}\xi(x,P_n)$ where $\xi(x,P_n)$ denotes a score function representing the interaction of $x$ with respect to $P_n$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and CLT for $H_n^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_n^{\xi} := \sum_{x \in P_n} \xi(x,P_n) \delta_{n^{-1/d}x}$, as $W_n \uparrow \mathbb{R}^d$. This gives the limit theory for non-linear geometric statistics (such as clique counts, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes having fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear statistics. It also gives the limit theory for geometric U-statistics of $\alpha$-permanental point processes and the zero set of Gaussian entire functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and Takahashi (2003), which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the CLT for $\mu_n^\xi$ via an extension of the cumulant method.Comment: 62 pages. Fundamental changes to the terminology including the title. The earlier 'clustering' condition is now introduced as a notion of mixing and its connection to Brillinger mixing is remarked. Newer results for superposition of independent point processes have been adde

Probabilistic Image Models and their Massively Parallel Architectures : A Seamless Simulation- and VLSI Design-Framework Approach

Algorithmic robustness in real-world scenarios and real-time processing capabilities are the two essential and at the same time contradictory requirements modern image-processing systems have to fulfill to go significantly beyond state-of-the-art systems. Without suitable image processing and analysis systems at hand, which comply with the before mentioned contradictory requirements, solutions and devices for the application scenarios of the next generation will not become reality. This issue would eventually lead to a serious restraint of innovation for various branches of industry. This thesis presents a coherent approach to the above mentioned problem. The thesis at first describes a massively parallel architecture template and secondly a seamless simulation- and semiconductor-technology-independent design framework for a class of probabilistic image models, which are formulated on a regular Markovian processing grid. The architecture template is composed of different building blocks, which are rigorously derived from Markov Random Field theory with respect to the constraints of \it massively parallel processing \rm and \it technology independence\rm. This systematic derivation procedure leads to many benefits: it decouples the architecture characteristics from constraints of one specific semiconductor technology; it guarantees that the derived massively parallel architecture is in conformity with theory; and it finally guarantees that the derived architecture will be suitable for VLSI implementations. The simulation-framework addresses the unique hardware-relevant simulation needs of MRF based processing architectures. Furthermore the framework ensures a qualified representation for simulation of the image models and their massively parallel architectures by means of their specific simulation modules. This allows for systematic studies with respect to the combination of numerical, architectural, timing and massively parallel processing constraints to disclose novel insights into MRF models and their hardware architectures. The design-framework rests upon a graph theoretical approach, which offers unique capabilities to fulfill the VLSI demands of massively parallel MRF architectures: the semiconductor technology independence guarantees a technology uncommitted architecture for several design steps without restricting the design space too early; the design entry by means of behavioral descriptions allows for a functional representation without determining the architecture at the outset; and the topology-synthesis simplifies and separates the data- and control-path synthesis. Detailed results discussed in the particular chapters together with several additional results collected in the appendix will further substantiate the claims made in this thesis