This paper studies the asymptotic properties of average area spectral efficiency (ASE) of a downlink cellular network in the limit of very dense base station (BS) and user densities. This asymptotic analysis relies on three assumptions: 1) interference is treated as noise; 2) the BS locations are drawn from a Poisson point process; and 3) the path loss function is bounded above satisfying mild regularity conditions. We consider three possible definitions of the average ASE, all of which give units of bits per second per unit bandwidth per unit area. When there is no constraint on the minimum operational signal-to-interference-plus-noise ratio (SINR) and instantaneous full channel state information (CSI) is available at the transmitter, the average ASE is proven to saturate to a constant, which we derive in a closed form. For the other two ASE definitions, wherein either a minimum SINR is enforced or CSI is not available, the average ASE is instead shown to collapse to zero at high BS density. We provide several familiar case studies for the class of considered path loss models, and demonstrate that our results cover most previous models and results on ultradense networks as special cases.
To obtain mathematically tractable results on wireless network performance without resorting to scaling laws, stochastic geometry @cite_33 @cite_42 @cite_6 @cite_17 has emerged as a key toolset, in particular for deriving the signal-to-interference-plus-noise ratio (SINR) distribution and related metrics, as achieved for ad hoc networks in @cite_39 @cite_18 . One of the key conclusions of early stochastic geometry results for cellular networks such as @cite_36 @cite_15 was that the SINR distribution is insensitive to the BS density, once the network is interference-limited. This indicated that the average ASE should increase linearly with the BS density: an encouraging result. Both @cite_36 @cite_15 , as well as most prior and subsequent literature including the aforementioned scaling laws results, use a standard power-law path loss function of the form @math , where @math is the distance, @math is a constant, and @math is the path loss exponent.
@cite_18 @cite_33 @cite_36 @cite_42 @cite_6 @cite_39 @cite_15 @cite_17
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Existing cellular network analyses, and even simulations, typically use the standard path loss model where received power decays like @math over a distance @math . This standard path loss model is quite idealized, and in most scenarios the path loss exponent @math is itself a function of @math , typically an increasing one. Enforcing a single path loss exponent can lead to orders of magnitude differences in average received and interference powers versus the true values. In this paper, we study multi-slope path loss models, where different distance ranges are subject to different path loss exponents. We focus on the dual-slope path loss function, which is a piece-wise power law and continuous and accurately approximates many practical scenarios. We derive the distributions of SIR, SNR, and finally SINR before finding the potential throughput scaling, which provides insight on the observed cell-splitting rate gain. The exact mathematical results show that the SIR monotonically decreases with network density, while the converse is true for SNR, and thus the network coverage probability in terms of SINR is maximized at some finite density. With ultra-densification (network density goes to infinity), there exists a phase transition in the near-field path loss exponent @math : if @math unbounded potential throughput can be achieved asymptotically; if $ 0 , ultra-densification leads in the extreme case to zero throughput. Distance-based attenuation is a critical aspect of wireless communications. As opposed to the ubiquitous power-law path loss model, this paper proposes a stretched exponential path loss model that is suitable for short-range communication. In this model, the signal power attenuates over a distance @math as @math , where @math and @math are tunable parameters. Using experimental propagation measurements, we show that the proposed model is accurate for short to moderate distances in the range @math meters and so is a suitable model for dense and ultradense networks. We integrate this path loss model into a downlink cellular network with base stations modeled by a Poisson point process, and derive expressions for the coverage probability, potential throughput, and area spectral efficiency. Although the most general result for coverage probability has a double integral, several special cases are given, where the coverage probability has a compact or even closed form. We then show that the potential throughput is maximized for a particular BS density and then collapses to zero for high densities, assuming a fixed signal-to-interference-plus-noise ratio (SINR) threshold. We next prove that the area spectral efficiency, which assumes an adaptive SINR threshold, is nondecreasing with the BS density and converges to a constant for high densities. 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, 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.