The time domain inter-cell interference coordination techniques specified in LTE Rel. 10 standard improves the throughput of picocell-edge users by protecting them from macrocell interference. On the other hand, it also degrades the aggregate capacity in macrocell because the macro base station (MBS) does not transmit data during certain subframes known as almost blank subframes. The MBS data transmission using reduced power subframes was standardized in LTE Rel. 11, which can improve the capacity in macrocell while not causing high interference to the nearby picocells. In order to get maximum benefit from the reduced power subframes, setting the key system parameters, such as the amount of power reduction, carries critical importance. Using stochastic geometry, this paper lays down a theoretical foundation for the performance evaluation of heterogeneous networks with reduced power subframes and range expansion bias. The analytic expressions for average capacity and 5th percentile throughput are derived as a function of transmit powers, node densities, and interference coordination parameters in a heterogeneous network scenario, and are validated through Monte Carlo simulations. Joint optimization of range expansion bias, power reduction factor, scheduling thresholds, and duty cycle of reduced power subframes are performed to study the trade-offs between aggregate capacity of a cell and fairness among the users. To validate our analysis, we also compare the stochastic geometry based theoretical results with the real MBS deployment (in the city of London) and the hexagonal-grid model. Our analysis shows that with optimum parameter settings, the LTE Rel. 11 with reduced power subframes can provide substantially better performance than the LTE Rel. 10 with almost blank subframes, in terms of both aggregate capacity and fairness.
With PPPs, using simplifying assumptions, such as Rayleigh fading channel model, and a path-loss exponent of four, we can obtain closed form expressions for aggregate interference and outage probability. Therefore, use of PPP models for performance evaluation of HetNets is appealing due to their simplicity and tractability @cite_15 . Furthermore, the PPP based models provide reasonably close performance results when compared with the real BS deployments. In particular, results in @cite_17 show that, when compared with real BS deployments, PPP and hexagonal grid based models for BS locations provide a lower bound and an upper bound, respectively, on the outage probabilities of UEs. Also, the PPP based models are expected to provide a better fit for analyzing denser HetNet deployments due to higher degree of randomness in small-cell deployments @cite_20 . In this paper, due to their simplicity and reasonable accuracy, we will use PPP based models to characterize and understand the behavior of HetNets in terms of various design parameters.
@cite_15 @cite_20 @cite_17
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The growing complexity of heterogeneous cellular networks (HetNets) has necessitated a variety of user and base station (BS) configurations to be considered for realistic performance evaluation and system design. This is directly reflected in the HetNet simulation models proposed by standardization bodies, such as the 3rd Generation Partnership Project (3GPP). Complementary to these simulation models, stochastic geometry-based approach, modeling the locations of the users, and the @math tiers of BSs as independent and homogeneous Poisson point processes (PPPs), has gained prominence in the past few years. Despite its success in revealing useful insights, this PPP-based @math -tier HetNet model is not rich enough to capture spatial coupling between user and BS locations that exists in real-world HetNet deployments and is included in 3GPP simulation models. In this paper, we demonstrate that modeling a fraction of users and arbitrary number of BS tiers alternatively with a Poisson cluster process (PCP) captures the aforementioned coupling, thus bridging the gap between the 3GPP simulation models and the PPP-based analytic model for HetNets. We further show that the downlink coverage probability of a typical user under maximum signal-to-interference-ratio ( @math ) association can be expressed in terms of the sum-product functionals over PPP, PCP, and its associated offspring point process, which are all characterized as a part of our analysis. We also show that the proposed model converges to the PPP-based HetNet model as the cluster size of the PCPs tends to infinity. Finally, we specialize our analysis based on general PCPs for Thomas and Matern cluster processes. Special instances of the proposed model closely resemble the different configurations for BS and user locations considered in 3GPP simulations. Due to its tractability, a multitier model of mutually independent Poisson point processes (PPPs) for heterogeneous cellular networks (HCNs) has recently been attracting much attention. However, in reality, the locations of the BSs, within each tier and across tiers, are not fully independent. Accordingly, in this paper, we propose two HCN models with inter-tier dependence (Case 1) and intra-tier dependence (Case 2), respectively. In Case 1, the macro-base station (MBS) and the pico-base station (PBS) deployments follow a Poisson point process (PPP) and a Poisson hole process (PHP), respectively. Under this setup and conditioning on a fixed serving distance (distance between a user and its nearest serving BS), we derive bounds on the outage probabilities of both macro and pico users. We also use a fitted Poisson cluster process to approximate the PHP, which is shown to provide a good approximation of the interference and outage statistics. In Case 2, the MBSs and the PBSs follow a PPP and an independent Matern cluster process, respectively. Explicit expressions of the interference and the outage probability are derived first for fixed serving distance and second with random distance, and we derive the outage performance, the per-user capacity, and the area spectral efficiency (ASE) for both cases. It turns out that the proposed Case 2 model is a more appropriate and accurate model for a HCN with hotspot regions than the multitier independent PPP model since the latter underestimates some key performance metrics, such as the per-user capacity and the ASE, by a factor of 1.5 to 2. Overall, the two models proposed provide good tradeoffs between the accuracy, tractability, and practicability. It is common practice to model the base station (BS) locations in a cellular system by a grid, such as a hexagonal or square lattice. This model is usually analytically intractable as well as quite idealized. Therefore, system designers resort to complex simulations to evaluate network performance. In this paper, we introduce a new model for the base station locations based on a homogeneous Poisson point process (PPP), whereby the mobiles communicate with their nearest base stations. We obtain the distribution of the signal-to-interference-noise ratio (SINR), compute the average ergodic rate, and analytically verify the trade-off between coverage and rate with frequency reuse. We compare our results with actual BS locations as well as the grid model. In addition to being tractable, we also observe that the performance predicted by the PPP model lower bounds the actual performance, and is about as predictive as the grid model which provides upper bounds. Although the Poisson point process (PPP) has been widely used to model base station (BS) locations in cellular networks, it is an idealized model that neglects the spatial correlation among BSs. This paper proposes the use of the determinantal point process (DPP) to take into account these correlations, in particular the repulsiveness among macro BS locations. DPPs are demonstrated to be analytically tractable by leveraging several unique computational properties. Specifically, we show that the empty space function, the nearest neighbor function, the mean interference, and the signal-to-interference ratio (SIR) distribution have explicit analytical representations and can be numerically evaluated for cellular networks with DPP-configured BSs. In addition, the modeling accuracy of DPPs is investigated by fitting three DPP models to real BS location data sets from two major U.S. cities. Using hypothesis testing for various performance metrics of interest, we show that these fitted DPPs are significantly more accurate than popular choices such as the PPP and the perturbed hexagonal grid model. In cellular network models, the base stations are usually assumed to form a lattice or a Poisson point process (PPP). In reality, however, they are deployed neither fully regularly nor completely randomly. Accordingly, in this paper, we consider the very general class of motion-invariant models and analyze the behavior of the outage probability (the probability that the signal-to-interference-plus-noise-ratio (SINR) is smaller than a threshold) as the threshold goes to zero. We show that, remarkably, the slope of the outage probability (in dB) as a function of the threshold (also in dB) is the same for essentially all motion-invariant point processes. The slope merely depends on the fading statistics. Using this result, we introduce the notion of the asymptotic deployment gain (ADG), which characterizes the horizontal gap between the success probabilities of the PPP and another point process in the high-reliability regime (where the success probability is near 1). To demonstrate the usefulness of the ADG for the characterization of the SINR distribution, we investigate the outage probabilities and the ADGs for different point processes and fading statistics by simulations. The spatial correlations in transmitter node locations introduced by common multiple access protocols make the analysis of interference, outage, and other related metrics in a wireless network extremely difficult. Most works therefore assume that nodes are distributed either as a Poisson point process (PPP) or a grid, and utilize the independence properties of the PPP (or the regular structure of the grid) to analyze interference, outage and other metrics. But, the independence of node locations makes the PPP a dubious model for nontrivial MACs which intentionally introduce correlations, e.g., spatial separation, while the grid is too idealized to model real networks. In this paper, we introduce a new technique based on the factorial moment expansion of functionals of point processes to analyze functions of interference, in particular outage probability. We provide a Taylor-series type expansion of functions of interference, wherein increasing the number of terms in the series provides a better approximation at the cost of increased complexity of computation. Various examples illustrate how this new approach can be used to find outage probability in both Poisson and non-Poisson wireless networks. The spatial structure of transmitters in wireless networks plays a key role in evaluating the mutual interference and hence the performance. Although the Poisson point process (PPP) has been widely used to model the spatial configuration of wireless networks, it is not suitable for networks with repulsion. The Ginibre point process (GPP) is one of the main examples of determinantal point processes that can be used to model random phenomena where repulsion is observed. Considering the accuracy, tractability and practicability tradeoffs, we introduce and promote the @math -GPP, an intermediate class between the PPP and the GPP, as a model for wireless networks when the nodes exhibit repulsion. To show that the model leads to analytically tractable results in several cases of interest, we derive the mean and variance of the interference using two different approaches: the Palm measure approach and the reduced second moment approach, and then provide approximations of the interference distribution by three known probability density functions. Besides, to show that the model is relevant for cellular systems, we derive the coverage probability of the typical user and also find that the fitted @math -GPP can closely model the deployment of actual base stations in terms of the coverage probability and other statistics. Pushing data traffic from cellular to WiFi is an example of inter radio access technology (RAT) offloading. While this clearly alleviates congestion on the over-loaded cellular network, the ultimate potential of such offloading and its effect on overall system performance is not well understood. To address this, we develop a general and tractable model that consists of M different RATs, each deploying up to K different tiers of access points (APs), where each tier differs in transmit power, path loss exponent, deployment density and bandwidth. Each class of APs is modeled as an independent Poisson point process (PPP), with mobile user locations modeled as another independent PPP, all channels further consisting of i.i.d. Rayleigh fading. The distribution of rate over the entire network is then derived for a weighted association strategy, where such weights can be tuned to optimize a particular objective. We show that the optimum fraction of traffic offloaded to maximize SINR coverage is not in general the same as the one that maximizes rate coverage, defined as the fraction of users achieving a given rate. The spatial structure of base stations (BSs) in cellular networks plays a key role in evaluating the downlink performance. In this paper, different spatial stochastic models (the Poisson point process (PPP), the Poisson hard-core process (PHCP), the Strauss process (SP), and the perturbed triangular lattice) are used to model the structure by fitting them to the locations of BSs in real cellular networks obtained from a public database. We provide two general approaches for fitting. One is fitting by the method of maximum pseudolikelihood. As for the fitted models, it is not sufficient to distinguish them conclusively by some classical statistics. We propose the coverage probability as the criterion for the goodness-of-fit. In terms of coverage, the SP provides a better fit than the PPP and the PHCP. The other approach is fitting by the method of minimum contrast that minimizes the average squared error of the coverage probability. This way, fitted models are obtained whose coverage performance matches that of the given data set very accurately. Furthermore, we introduce a novel metric, the deployment gain, and we demonstrate how it can be used to estimate the coverage performance and average rate achieved by a data set. 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. Powering the massive number of small computing devices in the Internet of Things (IoT) is a major challenge because replacing their batteries or powering them with wires is very expensive and impractical. A viable option to enable a perpetual network operation is through the use of far-field Wireless Power Transfer (WPT). We study a large network architecture that uses a combination of WPT and backscatter communication. The network consists of power beacons (PBs) and passive backscatter nodes (BNs), and their locations are modeled as points of independent Poisson point processes (PPPs). The PBs transmit a sinusoidal continuous wave (CW) and the BNs reflect back a portion of this signal while harvesting the remaining part. A BN harvests energy from multiple nearby PBs and modulates its information bits on the composite CW through backscatter modulation. The analysis poses real challenges due to the double fading channel, and its dependence on the PPPs of both BNs and PBs. With the help of stochastic geometry, we derive the coverage probability and the capacity of the network in tractable and easily computable expressions. These expressions depend on the density of both PB and BN, both forward and backward path loss exponents, transmit power of the PB, backscattering efficiency, and number of PBs in a harvesting region. We observe that the coverage probability decreases with an increase in the density of the BNs, while the capacity of the network improves. We compare the performance of this network with a regular powered network in which the BNs have a reliable power source and show that for certain parameters the coverage of the former network approaches that of the regular powered network. We study the local connectivity, i.e., the node isolation probability of two coexisting wireless ad hoc networks (a primary network vs. a secondary network), where two users can communicate if the signal-to-interference ratio (SIR) at the receiver is larger than a threshold. Assuming the primary users are distributed as a Poisson point process (PPP) and the secondary users are distributed as a Matern cluster process (MCP), we investigate the impact of network parameters on the node isolation probability. Upper and lower bounds for the node isolation probability of the heterogeneous overlaid networks are presented. We also study the node isolation probability when the primary users are distributed as a MCP and the secondary users are distributed as a PPP. Numerical and simulation results indicate that the secondary users local connectivity has a performance degradation due to the heterogeneity of the secondary network, while the local connectivity is improved due to the heterogeneity of the primary network. The Poisson point process (PPP) has been widely employed to model wireless networks and analyze their performance. The PPP has the property that nodes are conditionally independent from each other. As such, it may not be a suitable model for many networks, where there exists repulsion among the nodes. In order to address this limitation, we adopt a Poisson hardcore process (PHCP), in which no two nodes can be closer than a repulsion radius from one another. We consider two-tier heterogeneous networks, where the spatial distributions of transmitters in the first-tier and the second-tier networks follow a PHCP and a PPP, respectively. To alleviate inter-tier interference, we consider a guard zone for the first-tier network and presume that the second-tier transmitters located in the zone are deactivated. Under this setup, the activated second-tier transmitters form a Poisson hard-core hole process. We first derive exact computable expressions of the coverage probability and introduce a method to efficiently evaluate the expressions. Then, we provide approximations of the coverage probability, which have lower computational complexities. In addition, as a special case, we investigate the coverage probability of single-tier networks by modeling the locations of transmitters as a PHCP. This paper surveys and unifies a number of recent contributions that have collectively developed a metric for decentralized wireless network analysis known as transmission capacity. Although it is notoriously difficult to derive general end-to-end capacity results for multi-terminal or adhoc networks, the transmission capacity (TC) framework allows for quantification of achievable single-hop rates by focusing on a simplified physical MAC-layer model. By using stochastic geometry to quantify the multi-user interference in the network, the relationship between the optimal spatial density and success probability of transmissions in the network can be determined, and expressed-often fairly simply-in terms of the key network parameters. The basic model and analytical tools are first discussed and applied to a simple network with path loss only and we present tight upper and lower bounds on transmission capacity (via lower and upper bounds on outage probability). We then introduce random channels (fading shadowing) and give TC and outage approximations for an arbitrary channel distribution, as well as exact results for the special cases of Rayleigh and Nakagami fading. We then apply these results to show how TC can be used to better understand scheduling, power control, and the deployment of multiple antennas in a decentralized network. The paper closes by discussing shortcomings in the model as well as future research directions. Using stochastic geometry, we develop a tractable uplink modeling paradigm for outage probability and spectral efficiency in both single and multi-tier cellular wireless networks. The analysis accounts for per user equipment (UE) power control as well as the maximum power limitations for UEs. More specifically, for interference mitigation and robust uplink communication, each UE is required to control its transmit power such that the average received signal power at its serving base station (BS) is equal to a certain threshold ρ o . Due to the limited transmit power, the UEs employ a truncated channel inversion power control policy with a cutoff threshold of ρ o . We show that there exists a transfer point in the uplink system performance that depends on the following tuple: BS intensity λ, maximum transmit power of UEs P u , and ρ o . That is, when P u is a tight operational constraint with respect to (w.r.t.) λ and ρ o , the uplink outage probability and spectral efficiency highly depend on the values of λ and ρ o . In this case, there exists an optimal cutoff threshold ρ o *, which depends on the system parameters, that minimizes the outage probability. On the other hand, when P u is not a binding operational constraint w.r.t. λ and ρ o , the uplink outage probability and spectral efficiency become independent of λ and ρ o . We obtain approximate yet accurate simple expressions for outage probability and spectral efficiency, which reduce to closed forms in some special cases. Cognitive radio networks using relays have been introduced to enhance secondary network performance by having secondary users (SUs) transmit in two hops. This paper develops a framework for a random cognitive relay network (RCRN) where the primary, secondary, and relay nodes are randomly distributed according to Poisson point processes (PPPs) to investigate the effect of the random nature of the relays on secondary outage performance. We first derive the optimal relay location in terms of the outage probability of the secondary network. In addition, we provide the outage probability of the RCRN based on the optimal relay selection strategy that compares the instantaneous channel state information (CSI) of all relays. We then propose an efficient relay selection strategy that reduces the selection complexity and the feedback burden by limiting candidate relays to those located close to the optimal relay location. This paper sheds new light on the design of an efficient RCRN with respect to elements such as the effects of network parameters and the candidate relay set size on the RCRN outage probability. Cellular networks are in a major transition from a carefully planned set of large tower-mounted base-stations (BSs) to an irregular deployment of heterogeneous infrastructure elements that often additionally includes micro, pico, and femtocells, as well as distributed antennas. In this paper, we develop a tractable, flexible, and accurate model for a downlink heterogeneous cellular network (HCN) consisting of K tiers of randomly located BSs, where each tier may differ in terms of average transmit power, supported data rate and BS density. Assuming a mobile user connects to the strongest candidate BS, the resulting Signal-to-Interference-plus-Noise-Ratio (SINR) is greater than 1 when in coverage, Rayleigh fading, we derive an expression for the probability of coverage (equivalently outage) over the entire network under both open and closed access, which assumes a strikingly simple closed-form in the high SINR regime and is accurate down to -4 dB even under weaker assumptions. For external validation, we compare against an actual LTE network (for tier 1) with the other K-1 tiers being modeled as independent Poisson Point Processes. In this case as well, our model is accurate to within 1-2 dB. We also derive the average rate achieved by a randomly located mobile and the average load on each tier of BSs. One interesting observation for interference-limited open access networks is that at a given , adding more tiers and or BSs neither increases nor decreases the probability of coverage or outage when all the tiers have the same target-SINR. The intensity matching approach for tractable performance evaluation and optimization of cellular networks is introduced. It assumes that the base stations are modeled as the points of a Poisson point process (PPP) and leverages stochastic geometry for system-level analysis. Its rationale relies on observing that system-level performance is determined by the intensity measure of transformations of the underlaying spatial PPP. By approximating the original system model with a simplified one, whose performance is determined by a mathematically convenient intensity measure, tractable yet accurate integral expressions for computing area spectral efficiency and potential throughput are provided. The considered system model accounts for many practical aspects that, for tractability, are typically neglected, e.g., line-of-sight (LOS) and non-LOS propagation, antenna radiation patterns, traffic load, practical cell associations, and general fading channels. The proposed approach, more importantly, is conveniently formulated for unveiling the impact of several system parameters, e.g., the density of base stations and blockages. The effectiveness of this novel and general methodology is validated with the aid of empirical data for the locations of base stations and for the footprints of buildings in dense urban environments. In a two-tier heterogeneous network (HetNet) where femto access points (FAPs) with lower transmission power coexist with macro base stations (BSs) with higher transmission power, the FAPs may suffer significant performance degradation due to inter-tier interference. Introducing cognition into the FAPs through the spectrum sensing (or carrier sensing) capability helps them avoiding severe interference from the macro BSs and enhance their performance. In this paper, we use stochastic geometry to model and analyze performance of HetNets composed of macro BSs and cognitive FAPs in a multichannel environment. The proposed model explicitly accounts for the spatial distribution of the macro BSs, FAPs, and users in a Rayleigh fading environment. We quantify the performance gain in outage probability obtained by introducing cognition into the femto-tier, provide design guidelines, and show the existence of an optimal spectrum sensing threshold for the cognitive FAPs, which depends on the HetNet parameters. We also show that looking into the overall performance of the HetNets is quite misleading in the scenarios where the majority of users are served by the macro BSs. Therefore, the performance of femto-tier needs to be explicitly accounted for and optimized. Heterogeneous cellular networks (HetNets) are to be deployed for future wireless communication to meet the ever-increasing mobile traffic demand. However, the dense and random deployment of small cells and their uncoordinated operation raise important concerns about various costs issues, among which notably is energy efficiency. Base station (BS) cooperation is set to play a key role in managing interference in HetNets. In this paper, we consider BS cooperation in the downlink HetNets where BSs from different tiers within the respective cooperative clusters jointly transmit the same data to a typical user, and in particular focus on the optimization of the energy efficiency performance. First, based on a proposed clustering model, we derive the spectral efficiency using tools from stochastic geometry. Furthermore, we formulate a power minimization problem with a minimum spectral efficiency constraint and derive the optimal received signal strength (RSS) thresholds under certain approximation. Building upon these results, we could address the problem of how to design appropriate RSS thresholds, taking into account the tradeoff between spectral efficiency and energy efficiency. Simulations show that the proposed clustering model is more energy-saving than the geometric clustering model, and deploying a multitier HetNet is significantly more energy-saving compared to a macro-only network.