To understand the spatial deployment of base stations (BSs) is the first step to analyze the performance of cellular networks and further design efficient networking protocols. Poisson point process (PPP), which has been widely adopted to characterize the deployment of BSs and established the reputation to give tractable results in the stochastic geometry analyses, usually assumes a static BS deployment density in homogeneous PPP (HPPP) models or delicately designed location-dependent density functions in in-homogeneous PPP models. However, the simultaneous existence of attractiveness and repulsiveness among BSs practically deployed in a large-scale area defies such an assumption, and the @math -stable distribution, one kind of heavy-tailed distributions, has recently demonstrated superior accuracy to statistically model the varying BS density in different areas. In this paper, we start with these new findings and investigate the intrinsic feature (i.e., the spatial self-similarity) embedded in the BSs. Afterwards, we refer to a generalized PPP setup with @math -stable distributed density and theoretically derive the related coverage probability. In particular, we give an upper bound of the derived coverage probability for high signal-to-interference-plus-noise ratio thresholds and show the monotonically decreasing property of this bound with respect to the variance of BS density. Besides, we prove that our model could reduce to the single-tier HPPP for some special cases and demonstrate the superior accuracy of the @math -stable model to approach the real environment.
As discussed later in Section , the PPP model has provided many useful performance trends. However, the concern about total independence between nodes (e.g., BSs) has never stopped. Hence, in order to reduce the modeling gap between the single-tier HPPP model and the practical BS deployment, some researchers have adopted two-tier or multiple tier HPPP models @cite_70 @cite_91 @cite_54 @cite_6 @cite_63 @cite_67 @cite_81 @cite_69 . Though these models may lead to some tractable results, but it lacks the reasonable explanation to divide the gradually deployed and increasingly denser heterogeneous cells @cite_34 into different tiers. On the contrary, the @math -stable model contributes to understanding the spatial self-similarity and bridging the gap between cellular networks and other social behavior-based complex networks.
@cite_67 @cite_69 @cite_91 @cite_70 @cite_54 @cite_6 @cite_81 @cite_63 @cite_34
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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. 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. 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 heterogeneous cellular networks spatial characteristics of base stations (BSs) influence the system performance intensively. Existing models like two-dimensional hexagonal grid model or homogeneous spatial poisson point process (SPPP) are based on the assumption that BSs are ideal or uniformly distributed, but the aggregation behavior of users in hot spots has an important effect on the location of low power nodes (LPNs), so these models fail to characterize the distribution of BSs in the current mobile cellular networks. In this paper, firstly existing spatial models are analyzed. Then, based on real data from a mobile operator in one large city of China, a set of spatial models is proposed in three typical regions: dense urban, urban and suburban. For dense urban area, “Two Tiers Poisson Cluster Superimposed Process” is proposed to model the spatial characteristics of real-world BSs. Specifically, for urban and suburban area, conventional SPPP model still can be used. Finally, the fundamental relationship between user behavior and BS distribution is illustrated and summarized. Numerous results show that SPPP is only appropriate in the urban and suburban regions where users are not gathered together obviously. Principal parameters of these models are provided as reference for the theoretical analysis and computer simulation, which describe the complex spatial configuration more reasonably and reflect the current mobile cellular network performance more precisely.