We propose analytical models for the interference power distribution in a cellular system employing MIMO beamforming in rich and limited scattering environments, which capture non line-of-sight signal propagation in the microwave and mm-wave bands, respectively. Two candidate models are considered: the inverse Gaussian and the inverse Weibull, both are two-parameter heavy tail distributions. We further propose a mixture of these two distributions as a model with three parameters. To estimate the parameters of these distributions, three approaches are used: moment matching, individual distribution maximum likelihood estimation (MLE), and mixture distribution MLE with a designed expectation maximization algorithm. We then introduce simple fitted functions for the mixture model parameters as polynomials of the channel path loss exponent and shadowing variance. To measure the goodness of these models, the information-theoretic metric relative entropy is used to capture the distance from the model distribution to a reference one. The interference models are tested against data obtained by simulating a cellular network based on stochastic geometry. The results show that the three-parameter mixture model offers remarkably good fit to simulated interference power. The mixture model is further used to analyze the capacity of a cellular network employing joint transmit and receive beamforming and confirms a good fit with simulation.
To evaluate an interference model, we apply it to a cellular network and analyze the system performance. For cellular network analysis, stochastic geometry is shown to capture the main performance trends with analytical tractability . Stochastic geometry is used to develop a tractable model for downlink heterogeneous cellular networks @cite_26 , which is applied to analyze coordinated multipoint beamforming @cite_19 . These stochastic geometry based networks are useful in verifying that the performance of a cellular network matches experimental trends, and can also be used to verify the accuracy of an interference model.
@cite_19 @cite_26
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Cellular systems are becoming more heterogeneous with the introduction of low power nodes including femtocells, relays, and distributed antennas. Unfortunately, the resulting interference environment is also becoming more complicated, making evaluation of different communication strategies challenging in both analysis and simulation. Leveraging recent applications of stochastic geometry to analyze cellular systems, this paper proposes to analyze downlink performance in a fixed-size cell, which is inscribed within a weighted Voronoi cell in a Poisson field of interferers. A nearest out-of-cell interferer, out-of-cell interferers outside a guard region, and cross-tier interferers are included in the interference calculations. Bounding the interference power as a function of distance from the cell center, the total interference is characterized through its Laplace transform. An equivalent marked process is proposed for the out-of-cell interference under additional assumptions. To facilitate simplified calculations, the interference distribution is approximated using the Gamma distribution with second order moment matching. The Gamma approximation simplifies calculation of the success probability and average rate, incorporates small-scale and large-scale fading, and works with co-tier and cross-tier interference. Simulations show that the proposed model provides a flexible way to characterize outage probability and rate as a function of the distance to the cell edge. 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. In this paper, we introduce an analytical framework to compute the average rate of downlink heterogeneous cellular networks. The framework leverages recent application of stochastic geometry to other-cell interference modeling and analysis. The heterogeneous cellular network is modeled as the superposition of many tiers of Base Stations (BSs) having different transmit power, density, path-loss exponent, fading parameters and distribution, and unequal biasing for flexible tier association. A long-term averaged maximum biased-received-power tier association is considered. The positions of the BSs in each tier are modeled as points of an independent Poisson Point Process (PPP). Under these assumptions, we introduce a new analytical methodology to evaluate the average rate, which avoids the computation of the Coverage Probability (Pcov) and needs only the Moment Generating Function (MGF) of the aggregate interference at the probe mobile terminal. The distinguishable characteristic of our analytical methodology consists in providing a tractable and numerically efficient framework that is applicable to general fading distributions, including composite fading channels with small- and mid-scale fluctuations. In addition, our method can efficiently handle correlated Log-Normal shadowing with little increase of the computational complexity. The proposed MGF-based approach needs the computation of either a single or a two-fold numerical integral, thus reducing the complexity of Pcov-based frameworks, which require, for general fading distributions, the computation of a four-fold integral. In this paper, we provide an analytical framework to analyze heterogeneous downlink millimeter-wave (mm-wave) cellular networks consisting of @math tiers of randomly located base stations (BSs), where each tier operates in an mm-wave frequency band. Signal-to-interference-plus-noise ratio (SINR) coverage probability is derived for the entire network using tools from stochastic geometry. The distinguishing features of mm-wave communications, such as directional beamforming, and having different path loss laws for line-of-sight and non-line-of-sight links are incorporated into the coverage analysis by assuming averaged biased-received power association and Nakagami fading. By using the noise-limited assumption for mm-wave networks, a simpler expression requiring the computation of only one numerical integral for coverage probability is obtained. Also, the effect of beamforming alignment errors on the coverage probability analysis is investigated to get insight on the performance in practical scenarios. Downlink rate coverage probability is derived as well to get more insights on the performance of the network. Moreover, the effect of deploying low-power smaller cells and the impact of biasing factor on energy efficiency is analyzed. Finally, a hybrid cellular network operating in both mm-wave and @math -wave frequency bands is addressed. The analysis of the success probability or signal-to-interference ratio (SIR) distribution for coherent joint transmission (JT) based on stochastic geometry is an open issue. In this letter, we study coherent JT in downlink heterogeneous cellular networks and provide an upper bound of the success probability for the general user and the worst-case user (cell-corner user), and an approximation for the general user. Simulation results show that the derived upper bound and approximation are quite accurate and thus provide an analytical approach to quantify the SIR performance of coherent JT. 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. In this paper, a comprehensive study of the downlink performance in a heterogeneous cellular network (or HetNet) is conducted via stochastic geometry. A general HetNet model is considered consisting of an arbitrary number of open-access and closed-access tiers of base stations (BSs) arranged according to independent homogeneous Poisson point processes. The BSs within each tier have a constant transmission power, random fading factors with an arbitrary distribution and arbitrary path-loss exponent of the power-law path-loss model. For such a system, analytical characterizations for the coverage probability are derived for the max-SINR connectivity and nearest-BS connectivity models. Using stochastic ordering, interesting properties and simplifications for the HetNet downlink performance are derived by relating these two connectivity models to the maximum instantaneous received power (MIRP) connectivity model and the maximum biased received power (MBRP) connectivity models, providing good insights about HetNets and their downlink performance in these complex networks. Furthermore, the results also demonstrate the effectiveness and analytical tractability of the stochastic geometric approach to study the HetNet performance. 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. The equivalent-in-distribution (EiD)-based approach to the analysis of single-input-single-output (SISO) cellular networks for transmission over Rayleigh fading channels has recently been introduced [1]. Its rationale relies upon formulating the aggregate other-cell interference in terms of an infinite summation of independent and conditionally distributed Gaussian random variables (RVs). This approach leads to exact integral expressions of the error probability for arbitrary bi-dimensional modulations. In this paper, the EiD-based approach is generalized to the performance analysis of multiple-input-multiple-output (MIMO) cellular networks for transmission over Rayleigh fading channels. The proposed mathematical formulation allows us to study a large number of MIMO arrangements, including receive-diversity, spatial-multiplexing, orthogonal space-time block coding, zero-forcing reception and zero-forcing precoding. Depending on the MIMO setup, either exact or approximate integral expressions of the error probability are provided. In the presence of other-cell interference and noise, the error probability is formulated in terms of a two-fold integral. In interference-limited cellular networks, the mathematical framework simplifies to a single integral expression. As a byproduct, the proposed approach enables us to study SISO cellular networks for transmission over Nakagami- @math fading channels. The mathematical analysis is substantiated with the aid of extensive Monte Carlo simulations. In this letter, we propose a new coordinated multipoint technique based on mutual information (MI) accumulation using rateless codes. Using a stochastic geometry model for the cellular downlink, we quantify the performance enhancements in coverage probability and rate due to MI accumulation. By simulation and analysis, we show that MI accumulation using rateless codes leads to remarkable improvements in coverage and rate for general users and specific cell edge users. 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. There have been a bulk of analytic results about the performance of cellular networks where base stations are regularly located on a hexagonal or square lattice. This regular model cannot reflect the reality, and tends to overestimate the network performance. Moreover, tractable analysis can be performed only for a fixed location user (e.g., cell center or edge user). In this paper, we use the stochastic geometry approach, where base stations can be modeled as a homogeneous Poisson point process. We also consider the user density, and derive the user outage probability that an arbitrary user is under outage owing to low signal-to-interference-plus-noise ratio or high congestion by multiple users. Using the result, we calculate the density of success transmissions in the downlink cellular network. An interesting observation is that the success transmission density increases with the base station density, but the increasing rate diminishes. This means that the number of base stations installed should be more than @math -times to increase the network capacity by a factor of @math . Our results will provide a framework for performance analysis of the wireless infrastructure with a high density of access points, which will significantly reduce the burden of network-level simulations. This paper characterizes the performance of coordinated beamforming with dynamic clustering. A downlink model based on stochastic geometry is put forth to analyze the performance of such a base station (BS) coordination strategy. Analytical expressions for the complementary cumulative distribution function (CCDF) of the instantaneous signal-to-interference ratio (SIR) are derived in terms of relevant system parameters, chiefly the number of BSs forming the coordination clusters, the number of antennas per BS, and the pathloss exponent. Utilizing this CCDF, with pilot overheads further incorporated into the analysis, we formulate the optimization of the BS coordination clusters for a given fading coherence. Our results indicate that: 1) coordinated beamforming is most beneficial to users that are in the outer part of their cells yet in the inner part of their coordination cluster and that 2) the optimal cluster cardinality for the typical user is small and it scales with the fading coherence. Simulation results verify the exactness of the SIR distributions derived for stochastic geometries, which are further compared with the corresponding distributions for deterministic grid networks. In this paper, a new mathematical framework to the analysis of millimeter wave cellular networks is introduced. Its peculiarity lies in considering realistic path-loss and blockage models, which are derived from recently reported experimental data. The path-loss model accounts for different distributions of line-of-sight and non-line-of-sight propagation conditions and the blockage model includes an outage state that provides a better representation of the outage possibilities of millimeter wave communications. By modeling the locations of the base stations as points of a Poisson point process and by relying on a noise-limited approximation for typical millimeter wave network deployments, simple and exact integral as well as approximated and closed-form formulas for computing the coverage probability and the average rate are obtained. With the aid of Monte Carlo simulations, the noise-limited approximation is shown to be sufficiently accurate for typical network densities. The noise-limited approximation, however, may not be sufficiently accurate for ultra-dense network deployments and for sub-gigahertz transmission bandwidths. For these case studies, the analytical approach is generalized to take the other-cell interference into account at the cost of increasing its computational complexity. The proposed mathematical framework is applicable to cell association criteria based on the smallest path-loss and on the highest received power. It accounts for beamforming alignment errors and for multi-tier cellular network deployments. Numerical results confirm that sufficiently dense millimeter wave cellular networks are capable of outperforming micro wave cellular networks, in terms of coverage probability and average rate. Cellular networks are usually modeled by placing the base stations on a grid, with mobile users either randomly scattered or placed deterministically. These models have been used extensively but suffer from being both highly idealized and not very tractable, so complex system-level simulations are used to evaluate coverage outage probability and rate. More tractable models have long been desirable. We develop new general models for the multi-cell signal-to-interference-plus-noise ratio (SINR) using stochastic geometry. Under very general assumptions, the resulting expressions for the downlink SINR CCDF (equivalent to the coverage probability) involve quickly computable integrals, and in some practical special cases can be simplified to common integrals (e.g., the Q-function) or even to simple closed-form expressions. We also derive the mean rate, and then the coverage gain (and mean rate loss) from static frequency reuse. We compare our coverage predictions to the grid model and an actual base station deployment, and observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks. 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. 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 We consider a heterogeneous cellular network (HetNet) where a macrocell tier with a large antenna array base station (BS) is overlaid with a dense tier of small cells (SCs). We investigate the potential benefits of incorporating a massive MIMO BS in a TDD-based HetNet and we provide analytical expressions for the coverage probability and the area spectral efficiency using stochastic geometry. The duplexing mode in which SCs should operate during uplink macrocell transmissions is optimized. Furthermore, we consider a reverse TDD scheme, in which the massive MIMO BS can estimate the SC interference covariance matrix. Our results suggest that significant throughput improvement can be achieved by exploiting interference nulling and implicit coordination across the tiers due to flexible and asymmetric TDD operation. This paper analyzes the carrier-to-interference ratio of the so-called shotgun cellular system (SCS). In the SCS, base-stations are placed randomly according to a two-dimensional Poisson point process. Such a system can model a dense cellular or wireless data network deployment, where the base station locations end up being close to random due to constraints other than optimal coverage. The SCS is a simple cellular system where we can introduce several variations and design scenarios such as shadow fading, power control features, and multiple channel reuse groups, and assess their impact on the performance. We first derive an analytical expression for the characteristic function of the inverse of the carrier-to-interference ratio. Using this result, we show that the carrier-to-interference ratio is independent of the base station density and further, we derive a semi-analytical expression for the tail-probability. These results enable a complete characterization of the cellular performance of the SCS. Next, we incorporate shadow fading into the SCS and demonstrate that it merely scales the base station density by a constant. Hence, the cellular performance of the SCS is independent of shadow fading. These results are further used to analyze dense cellular scenarios.