Compared to the standard success (coverage) probability, the meta distribution of the signal-to-interference ratio (SIR) provides much more fine-grained information about the network performance. We consider general heterogeneous cellular networks (HCNs) with base station tiers modeled by arbitrary stationary and ergodic non-Poisson point processes. The exact analysis of non-Poisson network models is notoriously difficult, even in terms of the standard success probability, let alone the meta distribution. Hence we propose a simple approach to approximate the SIR meta distribution for non-Poisson networks based on the ASAPPP ("approximate SIR analysis based on the Poisson point process") method. We prove that the asymptotic horizontal gap @math between its standard success probability and that for the Poisson point process exactly characterizes the gap between the @math th moment of the conditional success probability, as the SIR threshold goes to @math . The gap @math allows two simple approximations of the meta distribution for general HCNs: 1) the per-tier approximation by applying the shift @math to each tier and 2) the effective gain approximation by directly shifting the meta distribution for the homogeneous independent Poisson network. Given the generality of the model considered and the fine-grained nature of the meta distribution, these approximations work surprisingly well.
The works in @cite_7 @cite_11 @cite_13 obtained analytically tractable results for the HIP model. For HCNs with non-Poisson deployments, it is often the case that it is hard to perform an exact mathematical analysis of key performance metrics such as the SIR distribution (sometimes called the coverage probability). Even if an exact expression of the SIR distribution exists, it is available in a complex form that does not help gain insights about the performance of the network for different network parameters @cite_22 @cite_21 @cite_23 @cite_18 @cite_10 .
@cite_18 @cite_22 @cite_7 @cite_10 @cite_21 @cite_23 @cite_13 @cite_11
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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. We consider spatial stochastic models of downlink heterogeneous cellular networks (HCNs) with multiple tiers, where the base stations (BSs) of each tier have a particular spatial density, transmission power and path-loss exponent. Prior works on such spatial models of HCNs assume, due to its tractability, that the BSs are deployed according to homogeneous Poisson point processes. This means that the BSs are located independently of each other and their spatial correlation is ignored. In the current paper, we propose two spatial models for the analysis of downlink HCNs, in which the BSs are deployed according to @a-Ginibre point processes. The @a-Ginibre point processes constitute a class of determinantal point processes and account for the repulsion between the BSs. Besides, the degree of repulsion is adjustable according to the value of @[email protected]?(0,1]. In one proposed model, the BSs of different tiers are deployed according to mutually independent @a-Ginibre processes, where the @a can take different values for the different tiers. In the other model, all the BSs are deployed according to an @a-Ginibre point process and they are classified into multiple tiers by mutually independent marks. For these proposed models, we derive computable representations for the coverage probability of a typical user-the probability that the downlink signal-to-interference-plus-noise ratio for the typical user achieves a target threshold. We exhibit the results of some numerical experiments and compare the proposed models and the Poisson based model. 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 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. Future mobile networks are visualized as networks that consist of more than one type of base station to cope with rising user demands. Such networks are referred to as heterogeneous networks. There have been various attempts at modeling and optimization of such networks using spatial point processes, some of which are alluded to (later) in this paper. We model a heterogeneous network consisting of two types of base stations by using a particular Poisson cluster process model. The main contributions are two-fold. First, a complete description of the interference in heterogeneous networks is derived in the form of its Laplace functional. Second, using an asymptotic convergence result which was shown in our previous work, we derive the expressions for the mean and variance of the distribution to which the interference converges. The utility of this framework is discussed for both the contributions. Motivated by the ongoing discussion on coordinated multipoint in wireless cellular standard bodies, this paper considers the problem of base station cooperation in the downlink of heterogeneous cellular networks. The focus of this paper is the joint transmission scenario, where an ideal backhaul network allows a set of randomly located base stations, possibly belonging to different network tiers, to jointly transmit data, to mitigate intercell interference and hence improve coverage and spectral efficiency. Using tools from stochastic geometry, an integral expression for the network coverage probability is derived in the scenario where the typical user located at an arbitrary location, i.e., the general user, receives data from a pool of base stations that are selected based on their average received power levels. An expression for the coverage probability is also derived for the typical user located at the point equidistant from three base stations, which we refer to as the worst case user. In the special case where cooperation is limited to two base stations, numerical evaluations illustrate absolute gains in coverage probability of up to 17 for the general user and 24 for the worst case user compared with the noncooperative case. It is also shown that no diversity gain is achieved using noncoherent joint transmission, whereas full diversity gain can be achieved at the receiver if the transmitting base stations have channel state information. The Signal to Interference Plus Noise Ratio (SINR) on a wireless link is an important basis for consideration of outage, capacity, and throughput in a cellular network. It is therefore important to understand the SINR distribution within such networks, and in particular heterogeneous cellular networks, since these are expected to dominate future network deployments . Until recently the distribution of SINR in heterogeneous networks was studied almost exclusively via simulation, for selected scenarios representing pre-defined arrangements of users and the elements of the heterogeneous network such as macro-cells, femto-cells, etc. However, the dynamic nature of heterogeneous networks makes it difficult to design a few representative simulation scenarios from which general inferences can be drawn that apply to all deployments. In this paper, we examine the downlink of a heterogeneous cellular network made up of multiple tiers of transmitters (e.g., macro-, micro-, pico-, and femto-cells) and provide a general theoretical analysis of the distribution of the SINR at an arbitrarily-located user. Using physically realistic stochastic models for the locations of the base stations (BSs) in the tiers, we can compute the general SINR distribution in closed form. We illustrate a use of this approach for a three-tier network by calculating the probability of the user being able to camp on a macro-cell or an open-access (OA) femto-cell in the presence of Closed Subscriber Group (CSG) femto-cells. We show that this probability depends only on the relative densities and transmit powers of the macro- and femto-cells, the fraction of femto-cells operating in OA vs. Closed Subscriber Group (CSG) mode, and on the parameters of the wireless channel model. For an operator considering a femto overlay on a macro network, the parameters of the femto deployment can be selected from a set of universal curves.