How would a cellular network designed for maximal energy efficiency look like? To answer this fundamental question, tools from stochastic geometry are used in this paper to model future cellular networks and obtain a new lower bound on the average uplink spectral efficiency. This enables us to formulate a tractable uplink energy efficiency (EE) maximization problem and solve it analytically with respect to the density of base stations (BSs), the transmit power levels, the number of BS antennas and users per cell, and the pilot reuse factor. The closed-form expressions obtained from this general EE maximization framework provide valuable insights on the interplay between the optimization variables, hardware characteristics, and propagation environment. Small cells are proved to give high EE, but the EE improvement saturates quickly with the BS density. Interestingly, the maximal EE is achieved by also equipping the BSs with multiple antennas and operate in a "massive MIMO" fashion, where the array gain from coherent detection mitigates interference and the multiplexing of many users reduces the energy cost per user.
The EE analysis of multi-cellular networks is much more involved than in the single-cell case due to the complicated network topology and the arising inter-cell interference. The simplest approach is to rely on heavy Monte Carlo simulations. Attempts in this direction can be found in @cite_36 and @cite_37 . Unfortunately, Monte Carlo simulated results are often anecdotal since one cannot separate fundamental properties from behaviors induced by parameter selection. Alternatively, simplified network topologies can be considered, such as the Wyner model @cite_31 or the symmetric grid-based deployment @cite_42 . While attractive for its analytical simplicity, the Wyner model does not well capture the essential characteristics of real and practical networks @cite_7 . Similarly, the symmetric grid-based models cannot capture the irregular structure of small-cell network deployments.
@cite_37 @cite_7 @cite_36 @cite_42 @cite_31
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Cellular networks are usually modeled by placing the base stations on a grid, with mobile users either randomly scattered or placed deterministically. These models have been used extensively but suffer from being both highly idealized and not very tractable, so complex system-level simulations are used to evaluate coverage outage probability and rate. More tractable models have long been desirable. We develop new general models for the multi-cell signal-to-interference-plus-noise ratio (SINR) using stochastic geometry. Under very general assumptions, the resulting expressions for the downlink SINR CCDF (equivalent to the coverage probability) involve quickly computable integrals, and in some practical special cases can be simplified to common integrals (e.g., the Q-function) or even to simple closed-form expressions. We also derive the mean rate, and then the coverage gain (and mean rate loss) from static frequency reuse. We compare our coverage predictions to the grid model and an actual base station deployment, and observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks. Cellular networks are usually modeled by placing the base stations according to a regular geometry such as a grid, with the mobile users scattered around the network either as a Poisson point process (i.e. uniform distribution) or deterministically. These models have been used extensively for cellular design and analysis but suffer from being both highly idealized and not very tractable. Thus, complex simulations are used to evaluate key metrics such as coverage probability for a specified target rate (equivalently, the outage probability) or average sum rate. We develop general models for multi-cell signal-to-noise-plus-interference ratio (SINR) based on homogeneous Poisson point processes and derive the coverage probability, which is one minus the outage probability. Under very general assumptions, the resulting expressions for the SINR cumulative distribution function involve quickly computable integrals, and in some important special cases of practical interest these integrals can be simplified to common integrals (e.g., the Q-function) or even to exact and quite simple closed-form expressions. We compare our coverage predictions to the standard grid model and an actual base station deployment. We observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in urban cellular networks with highly variable coverage radii. In this paper, a new mathematical framework to the analysis of millimeter wave cellular networks is introduced. Its peculiarity lies in considering realistic path-loss and blockage models, which are derived from recently reported experimental data. The path-loss model accounts for different distributions of line-of-sight and non-line-of-sight propagation conditions and the blockage model includes an outage state that provides a better representation of the outage possibilities of millimeter wave communications. By modeling the locations of the base stations as points of a Poisson point process and by relying on a noise-limited approximation for typical millimeter wave network deployments, simple and exact integral as well as approximated and closed-form formulas for computing the coverage probability and the average rate are obtained. With the aid of Monte Carlo simulations, the noise-limited approximation is shown to be sufficiently accurate for typical network densities. The noise-limited approximation, however, may not be sufficiently accurate for ultra-dense network deployments and for sub-gigahertz transmission bandwidths. For these case studies, the analytical approach is generalized to take the other-cell interference into account at the cost of increasing its computational complexity. The proposed mathematical framework is applicable to cell association criteria based on the smallest path-loss and on the highest received power. It accounts for beamforming alignment errors and for multi-tier cellular network deployments. Numerical results confirm that sufficiently dense millimeter wave cellular networks are capable of outperforming micro wave cellular networks, in terms of coverage probability and average rate. The equivalent-in-distribution (EiD)-based approach to the analysis of single-input-single-output (SISO) cellular networks for transmission over Rayleigh fading channels has recently been introduced [1]. Its rationale relies upon formulating the aggregate other-cell interference in terms of an infinite summation of independent and conditionally distributed Gaussian random variables (RVs). This approach leads to exact integral expressions of the error probability for arbitrary bi-dimensional modulations. In this paper, the EiD-based approach is generalized to the performance analysis of multiple-input-multiple-output (MIMO) cellular networks for transmission over Rayleigh fading channels. The proposed mathematical formulation allows us to study a large number of MIMO arrangements, including receive-diversity, spatial-multiplexing, orthogonal space-time block coding, zero-forcing reception and zero-forcing precoding. Depending on the MIMO setup, either exact or approximate integral expressions of the error probability are provided. In the presence of other-cell interference and noise, the error probability is formulated in terms of a two-fold integral. In interference-limited cellular networks, the mathematical framework simplifies to a single integral expression. As a byproduct, the proposed approach enables us to study SISO cellular networks for transmission over Nakagami- @math fading channels. The mathematical analysis is substantiated with the aid of extensive Monte Carlo simulations.