A scalable framework is developed to allocate radio resources across a large number of densely deployed small cells with given traffic statistics on a slow timescale. Joint user association and spectrum allocation is first formulated as a convex optimization problem by dividing the spectrum among all possible transmission patterns of active access points (APs). To improve scalability with the number of APs, the problem is reformulated using local patterns of interfering APs. To maintain global consistency among local patterns, inter-cluster interaction is characterized as hyper-edges in a hyper-graph with nodes corresponding to neighborhoods of APs. A scalable solution is obtained by iteratively solving a convex optimization problem for bandwidth allocation with reduced complexity and constructing a global spectrum allocation using hyper-graph coloring. Numerical results demonstrate the proposed solution for a network with 100 APs and several hundred user equipments. For a given quality of service (QoS), the proposed scheme can increase the network capacity several fold compared to assigning each user to the strongest AP with full-spectrum reuse.
The stochastic geometry framework way proposed to evaluate and optimize the expected system performance over random topologies and channel conditions @cite_7 @cite_16 . The approach does not apply to the optimization of resource allocation to all possible interference patterns as considered in this paper.
@cite_16 @cite_7
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Stochastic geometry proves to be a powerful tool for modeling dense wireless networks adopting random MAC protocols such as ALOHA and CSMA. The main strength of this methodology lies in its ability to account for the randomness in the nodes' location jointly with an accurate description at the physical layer, based on the SINR, that allows to consider also random fading on each link. Existing models of CSMA networks adopting the stochastic geometry approach suffer from two important weaknesses: 1) they permit to evaluate only spatial averages of the main performance measures, thus hiding possibly huge discrepancies in the performance achieved by individual nodes; 2) they are analytically tractable only when nodes are distributed over the area according to simple spatial processes (e.g., the Poisson point process). In this paper we show how the stochastic geometry approach can be extended to overcome the above limitations, allowing to obtain node throughput distributions as well as to analyze a significant class of topologies in which nodes are not independently placed. This volume bears on wireless network modeling and performance analysis. The aim is to show how stochastic geometry can be used in a more or less systematic way to analyze the phenomena that arise in this context. It first focuses on medium access control mechanisms used in ad hoc networks and in cellular networks. It then discusses the use of stochastic geometry for the quantitative analysis of routing algorithms in mobile ad hoc networks. The appendix also contains a concise summary of wireless communication principles and of the network architectures considered in the two volumes. Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance. We develop a flexible and accurate framework for device-to-device (D2D) communication in the context of a conventional cellular network, which allows for time-frequency resources to be either shared or orthogonally partitioned between the two networks. Using stochastic geometry, we provide accurate expressions for SINR distributions and average rates, under an assumption of interference randomization via time and or frequency hopping, for both dedicated and shared spectrum approaches. We obtain analytical results in closed or semi-closed form in high SNR regime, that allow us to easily explore the impact of key parameters (e.g., the load and hopping probabilities) on the network performance. In particular, unlike other models, the expressions we obtain are tractable, i.e., they can be efficiently optimized without extensive simulation. Using these, we optimize the hopping probabilities for the D2D links, i.e., how often they should request a time or frequency slot. This can be viewed as an optimized lower bound to other more sophisticated scheduling schemes. We also investigate the optimal resource partitions between D2D and cellular networks when the dedicated spectrum approach is used. This paper proposes a new approach for communication networks planning based on stochastic geometry. We first summarize the state of the art in this domain, together with its economic implications, before sketching the main expectations of the proposed method. The main probabilistic tools are point processes and stochastic geometry. We show how several performance evaluation and optimization problems within this framework can actually be posed and solved by computing the mathematical expectation of certain functionals of point processes. We mainly analyze models based on Poisson point processes, for which analytical formulae can often be obtained, although more complex models can also be analyzed, for instance via simulation. This paper presents a stochastic geometry model for the performance analysis and the planning of dense IEEE 802.11 networks. This model allows one to propose heuristic formulas for various properties of such networks like the probability for users to be covered, the probability for access points to be granted access to the channel or the average long term throughput provided to end-users. The main merit of this model is to take the effect of interferences and that of CSMA into account within this dense network context. This analytic model, which is based on Matern point processes, is partly validated against simulation. It is then used to assess various properties of such networks. We show for instance how the long term throughput obtained by end-users behaves when the access point density increases. We also briefly show how to use this model for the planning of managed networks and for the economic modeling of unplanned networks. We study constrained stochastic programs where the decision vector at each time slot cannot be chosen freely but is tied to the realization of an underlying random state vector. The goal is to minimize a general objective function subject to linear constraints. A typical scenario where such programs appear is opportunistic scheduling over a network of time-varying channels, where the random state vector is the channel state observed, and the control vector is the transmission decision which depends on the current channel state. We consider a primal-dual type Frank-Wolfe algorithm that has a low complexity update during each slot and that learns to make efficient decisions without prior knowledge of the probability distribution of the random state vector. We establish convergence time guarantees for the case of both convex and non-convex objective functions. We also emphasize application of the algorithm to non-convex opportunistic scheduling and distributed non-convex stochastic optimization over a connected graph. We consider the problem of estimating the unreliability of a stochastic flow network, defined as the probability that the maximum flow value from a source node to a terminal node in a directed network with stochastic link capacities, is less than a specified demand level. The link capacities are assumed to be continuous random variables with a known joint distribution. We are interested in the situation where the unreliability is very small, in which case a crude Monte Carlo is not viable. We show how a Monte Carlo splitting algorithm can be adapted to handle this problem effectively. This paper presents a tutorial on stochastic geometry (SG)-based analysis for cellular networks. This tutorial is distinguished by its depth with respect to wireless communication details and its focus on cellular networks. This paper starts by modeling and analyzing the baseband interference in a baseline single-tier downlink cellular network with single antenna base stations and universal frequency reuse. Then, it characterizes signal-to-interference-plus-noise-ratio and its related performance metrics. In particular, a unified approach to conduct error probability, outage probability, and transmission rate analysis is presented. Although the main focus of this paper is on cellular networks, the presented unified approach applies for other types of wireless networks that impose interference protection around receivers. This paper then extends the unified approach to capture cellular network characteristics (e.g., frequency reuse, multiple antenna, power control, etc.). It also presents numerical examples associated with demonstrations and discussions. To this end, this paper highlights the state-of-the-art research and points out future research directions. For more than three decades, stochastic geometry has been used to model large-scale ad hoc wireless networks, and it has succeeded to develop tractable models to characterize and better understand the performance of these networks. Recently, stochastic geometry models have been shown to provide tractable yet accurate performance bounds for multi-tier and cognitive cellular wireless networks. Given the need for interference characterization in multi-tier cellular networks, stochastic geometry models provide high potential to simplify their modeling and provide insights into their design. Hence, a new research area dealing with the modeling and analysis of multi-tier and cognitive cellular wireless networks is increasingly attracting the attention of the research community. In this article, we present a comprehensive survey on the literature related to stochastic geometry models for single-tier as well as multi-tier and cognitive cellular wireless networks. A taxonomy based on the target network model, the point process used, and the performance evaluation technique is also presented. To conclude, we discuss the open research challenges and future research directions. Stochastic geometry models for wireless communication networks have recently attracted much attention. This is because the performance of such networks critically depends on the spatial configuration of wireless nodes and the irregularity of the node configuration in a real network can be captured by a spatial point process. However, most analysis of such stochastic geometry models for wireless networks assumes, owing to its tractability, that the wireless nodes are deployed according to homogeneous Poisson point processes. This means that the wireless nodes are located independently of each other and their spatial correlation is ignored. In this work we propose a stochastic geometry model of cellular networks such that the wireless base stations are deployed according to the Ginibre point process. The Ginibre point process is one of the determinantal point processes and accounts for the repulsion between the base stations. For the proposed model, we derive a computable representation for the coverage probability—the probability that the signal-to-interference-plus-noise ratio (SINR) for a mobile user achieves a target threshold. To capture its qualitative property, we further investigate the asymptotics of the coverage probability as the SINR threshold becomes large in a special case. We also present the results of some numerical experiments. This work presents joint iterative power allocation and interference suppression algorithms for spread spectrum networks with multiple relays and the amplify and forward cooperation strategy. A joint constrained optimization framework that considers the allocation of power levels across the relays subject to individual and global power constraints and the design of linear receivers for interference suppression is proposed. Constrained minimum mean-squared error (MMSE) expressions for the parameter vectors that determine the optimal power levels across the relays and the parameters of the linear receivers are derived. In order to solve the proposed optimization problems efficiently, stochastic gradient (SG) algorithms for adaptive joint iterative power allocation, and receiver and channel parameter estimation are developed. The results of simulations show that the proposed algorithms obtain significant gains in performance and capacity over existing cooperative and non-cooperative schemes. Volume I first provides a compact survey on classical stochastic geometry models, with a main focus on spatial shot-noise processes, coverage processes and random tessellations. It then focuses on signal to interference noise ratio (SINR) stochastic geometry, which is the basis for the modeling of wireless network protocols and architectures considered in Volume II. It also contains an appendix on mathematical tools used throughout Stochastic Geometry and Wireless Networks, Volumes I and II. In this paper, a novel non-orthogonal multiple access (NOMA)-enabled framework for cloud-radio access networks (CRANs) is proposed. In this framework, two users are scheduled in the same resources according to NOMA; however the performance of cell-edge users is enhanced by means of coordinating beamforming. Stochastic geometry is invoked for modeling the proposed framework, where the positions of BSs follow a cluster point process. In an effort to characterize the performance of the proposed framework, simple expressions in terms of outage probability are derived for both nearby users and the cell-edge users. It is analytically demonstrated that the average beamforming gain is a liner function with respect to the number of cooperating base stations. Numerical results verify the accuracy of analysis and reveal that the proposed framework is capable of greatly enhancing the performance of cell-edge users. We develop new stochastic optimization methods that are applicable to a wide range of structured regularizations. Basically our methods are combinations of basic stochastic optimization techniques and Alternating Direction Multiplier Method (ADMM). ADMM is a general framework for optimizing a composite function, and has a wide range of applications. We propose two types of online variants of ADMM, which correspond to online proximal gradient descent and regularized dual averaging respectively. The proposed algorithms are computationally efficient and easy to implement. Our methods yield O(1 √T) convergence of the expected risk. Moreover, the online proximal gradient descent type method yields O(log(T) T) convergence for a strongly convex loss. Numerical experiments show effectiveness of our methods in learning tasks with structured sparsity such as overlapped group lasso. This paper contains a quantitative evaluation of probabilistic traffic assignment models and proposes an alternate formulation. First, the concept of stochastic-user-equilibration (S-U-E) is formalized as an extension of Wardrop's user-equilibration criterion. Then, the stochastic-network-loading (S-N-L) problem (a special case of S-U-E for networks with constant link costs) is analyzed in detail and an expression for the probability of route choice which is based on two general postulates of user behavior is derived. The paper also discusses the weaknesses of existing S-N-L techniques with special attention paid to Dial's multipath method and compares them to the suggested approach. The proposed model seems reasonable and does not exhibit the inherent weaknesses of the logit model when applied to sets of routes which overlap heavily. The discussion is supported by several numerical examples on small contrived networks. The paper concludes with the discussion of two techniques that can be used to approximate the link flows resulting from the proposed model in large networks. The binary symmetric stochastic block model deals with a random graph of @math vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability @math within clusters and @math across clusters. In the asymptotic regime of @math and @math for fixed @math and @math , we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe14 . Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to @math . In this paper, we adopt stochastic geometry theory to analyze the optimal macro micro BS (base station) density for energy-efficient heterogeneous cellular networks with QoS constraints. We first derive the upper and lower bounds of the optimal BS density for homogeneous scenarios and, based on these, we analyze the optimal BS density for heterogeneous networks. The optimal macro micro BS density can be calculated numerically through our analysis, and the closed-form approximation is also derived. Our results reveal the best type of BSs to be deployed for capacity extension, or to be switched off for energy saving. Specifically, if the ratio between the micro BS cost and the macro BS cost is lower than a threshold, which is a function of path loss and their transmit power, the micro BSs are preferred, i.e., deploy more micro BSs for capacity extension or switch off certain macro BSs for energy saving. Otherwise, the optimal choice is the opposite. Our work provides guidance for energy efficient cellular network planning and dynamic operation control.1 This paper presents a novel stochastic optimisation approach to determining the feasible optimal solution of the economic dispatch (ED) problem considering various generator constraints. Many practical constraints of generators, such as ramp rate limits, prohibited operating zones and the valve point effect, are considered. These constraints make the ED problem a non-smooth non-convex minimisation problem with constraints. The proposed optimisation algorithm is called self-tuning hybrid differential evolution (self-tuning HDE). The self-tuning HDE utilises the concept of the 1 5 success rule of evolution strategies (ESs) in the original HDE to accelerate the search for the global optimum. Three test power systems, including 3-, 13- and 40-unit power systems, are applied to compare the performance of the proposed algorithm with genetic algorithms, the differential evolution algorithm and the HDE algorithm. Numerical results indicate that the entire performance of the proposed self-tuning HDE algorithm outperforms the other three algorithms. We present a methodology for determining the outage probability of wideband ad hoc networks with random wireless channels. Assuming that the nodes are Poisson distributed and subject to a required SINR constraint, we develop a simple framework that gives upper and lower bounds on the outage probability. These bounds are important in that they can be manipulated to obtain bounds on the transmission capacity, i.e., the maximum permissible spatial density of transmissions ensuring an acceptably low outage probability. In this paper, we derive the outage probability of wireless ad hoc networks under path loss and shadowing, which are the dominant large-scale effects in wideband ad hoc networks. The analytical framework is rooted in stochastic geometry, employing marked point processes, void probabilities, Palm measure, and Campbell’s Theorem.