We provide a comprehensive overview of mathematical models and analytical techniques for millimeter wave (mmWave) cellular systems. The two fundamental physical differences from conventional sub-6-GHz cellular systems are: 1) vulnerability to blocking and 2) the need for significant directionality at the transmitter and/or receiver, which is achieved through the use of large antenna arrays of small individual elements. We overview and compare models for both of these factors, and present a baseline analytical approach based on stochastic geometry that allows the computation of the statistical distributions of the downlink signal-to-interference-plus-noise ratio (SINR) and also the per link data rate, which depends on the SINR as well as the average load. There are many implications of the models and analysis: 1) mmWave systems are significantly more noise-limited than at sub-6 GHz for most parameter configurations; 2) initial access is much more difficult in mmWave; 3) self-backhauling is more viable than in sub-6-GHz systems, which makes ultra-dense deployments more viable, but this leads to increasingly interference-limited behavior; and 4) in sharp contrast to sub-6-GHz systems cellular operators can mutually benefit by sharing their spectrum licenses despite the uncontrolled interference that results from doing so. We conclude by outlining several important extensions of the baseline model, many of which are promising avenues for future research.

This paper proposes a stochastic geometry framework to analyze the signal-to-noise-and-interference ratio (SINR) and rate performance in a large-scale uplink massive multiple-input and multiple-output (MIMO) network. Based on the model, expressions are derived for spatial average SINR distributions over user and base station distributions with maximum ratio combining (MRC) and zero-forcing (ZF) receivers. We show that, using massive MIMO, the uplink SINR in certain urban marcocell scenarios is limited by interference. In the interference-limited regime, the results reveal that for MRC receivers, a superlinear (polynomial) scaling law between the number of base station antennas and scheduled users per cell preserves the uplink signal-to-interference ratio (SIR) distribution, while a linear scaling applies to ZF receivers. ZF receivers are shown to outperform MRC receivers in the SIR coverage, and the performance gap is quantified in terms of the difference in the number of antennas to achieve the same SIR distribution. Numerical results verify the analysis. It is found that the optimal compensation fraction in fractional power control to optimize rate is generally different for MRC and ZF receivers. Besides, simulations show that the scaling results derived from the proposed framework apply to the networks, where base stations are distributed according to a hexagonal lattice.

Massive multiple-input multiple-output (MIMO) is a potential physical layer technology for 5G cellular networks. This paper leverages stochastic geometry to derive the uplink signal-to- interference (SIR) distribution in massive MIMO networks. Based on the derived expressions, a scaling law between the number of base station antennas and scheduled users per cell is provided to preserve the uplink SIR distribution, where the impacts of correlation in small-scale fading and power control in the form of fractional path loss compensation are taken account. Numerical results verify the analysis, and show that fractional power control with a compensation fraction of 0.5 is nearly optimal for the average achievable rate in certain cases.

Wireless communication at millimeter wave (mmWave) frequencies is attractive for cellular, local area, and ad hoc networks due to the potential for channels with large bandwidths. As a byproduct of directional beamforming and propagation differences, some studies have claimed that mmWave networks will be noise rather than interference limited. This paper presents a derivation of the instantaneous interference-to-noise ratio (INR) distribution of a mmWave ad hoc network. Random network model of transmitters represented by a Poisson point process with a narrowband channel model is used to derive an approximation of the INR distribution. The analysis shows that the shape of the INR distribution is determined largely by the line-of-sight interferers, which depends on the overall network density and building blockage. A main conclusion drawn is that even with highly directional beamforming, interference can only sometimes be neglected in an ad hoc network. With a reasonable choice of system parameters, the interference is nearly always stronger than the noise power in dense networks.

Ad hoc networks consistently under perform compared to cellular networks. Protocols either create residual interference that leads to poor signal-to-interference-plus-noise ratios or try to coordinate transmissions leading to fewer transmission opportunities. Communication with millimeter-wave (mmWave) devices offers the potential for higher bandwidth communication channels and reduced interference in ad hoc networks. This paper uses a stochastic geometry approach to characterize the coverage probability of a mmWave ad hoc network with directional antennas and random building blockages. The coverage probability in the presence of noise and both line-of-sight and non-line-of-sight interference is analyzed and used to derive the transmission capacity. Performance of mmWave is then analyzed in terms of area spectral efficiency and rate coverage. The results show that mmWave networks support larger densities, higher area spectral efficiencies, and better rate coverage compared to traditional, lower-frequency ad hoc networks.

Millimeter wave (mmWave) holds promise as a carrier frequency for fifth generation cellular networks. Because mmWave signals are sensitive to blockage, prior models for cellular networks operated in the ultra high frequency (UHF) band do not apply to analyze mmWave cellular networks directly. Leveraging concepts from stochastic geometry, this paper proposes a general framework to evaluate the coverage and rate performance in mmWave cellular networks. Using a distance-dependent line-of-site (LOS) probability function, the locations of the LOS and non-LOS base stations are modeled as two independent non-homogeneous Poisson point processes, to which different path loss laws are applied. Based on the proposed framework, expressions for the signal-to-noise-and-interference ratio (SINR) and rate coverage probability are derived. The mmWave coverage and rate performance are examined as a function of the antenna geometry and base station density. The case of dense networks is further analyzed by applying a simplified system model, in which the LOS region of a user is approximated as a fixed LOS ball. The results show that dense mmWave networks can achieve comparable coverage and much higher data rates than conventional UHF cellular systems, despite the presence of blockages. The results suggest that the cell size to achieve the optimal SINR scales with the average size of the area that is LOS to a user.

In this paper, we study the influence of 3-dimensional (3D) base station (BS) antenna patterns on the of downlink random heterogeneous cellular networks. To assess the impact of downtilt and vertical beamwidth, we propose a mathematical framework based on stochastic geometry (to model BS locations) and random shape theory (to model building geometry). Using this framework, we study the impact of different antenna parameters at the pico cell on the coverage and area spectral efficiency of the network. Through simulations we find that narrowing the vertical beamwidth of the pico cell must be accompanied by downtilt to result in improved coverage and capacity. We also find that high building density leads to improved network performance, motivating the deployment of heterogeneous networks in dense urban areas.

Large-scale blockages such as buildings affect the performance of urban cellular networks, especially at higher frequencies. Unfortunately, such blockage effects are either neglected or characterized by oversimplified models in the analysis of cellular networks. Leveraging concepts from random shape theory, this paper proposes a mathematical framework to model random blockages and analyze their impact on cellular network performance. Random buildings are modeled as a process of rectangles with random sizes and orientations whose centers form a Poisson point process on the plane. The distribution of the number of blockages in a link is proven to be a Poisson random variable with parameter dependent on the length of the link. Our analysis shows that the probability that a link is not intersected by any blockages decays exponentially with the link length. A path loss model that incorporates the blockage effects is also proposed, which matches experimental trends observed in prior work. The model is applied to analyze the performance of cellular networks in urban areas with the presence of buildings, in terms of connectivity, coverage probability, and average rate. Our results show that the base station density should scale superlinearly with the blockage density to maintain the network connectivity. Our analyses also show that while buildings may block the desired signal, they may still have a positive impact on the SIR coverage probability and achievable rate since they can block significantly more interference.

Cellular networks are taking on a more heterogeneous character with the introduction of low power nodes. Besides the aggregate performance over the entire network, it is also of interest to evaluate the coverage probability conditioning on a specific user location in a cell. This letter proposes a method to compute the location-specific coverage probability inside the inscribing ball of a weighted Voronoi cell in heterogeneous networks. Simulations show that the proposed closed-form approximation accurately characterizes the coverage probability at locations of interest inside the inscribing ball. Moreover the expression can also be applied to evaluate the performance at locations outside the inscribing ball with minor loss.