A Tractable Approach to Coverage and Rate in Cellular Networks

Fractional Frequency Reuse (FFR) is a promising to improve the spectrum e ciency in the LongTerm Evolution (LTE) cellular network. In the literature, various research works have been conducted to evaluate the performance of FFR. However, the presented analytical approach only dealt with the special cases in which the users are divided into 2 groups and only two power levels are utilised. In this paper, we consider a general case of FFR in which the users are classified intogroups and each group is assigned a serving power level. The mathematical model of the general FFR is presented and analysed through a stochastic geometry approach. The derived analytical results in terms of average coverage probability can covered all the related well-known results in the literature.
 Keywords: 
 Fractional Frequency Reuse, LongTerm Evolution, coverage probability, stochastic geometry
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Accurate modeling of network interference, deep understanding of its impact on the achievable performance, and development of efficient techniques to mitigate or exploit it are three important and fundamental research assets in current and next–generation cellular networks. In this context, Andrews, Baccelli, and Ganti [1] have recently introduced a new analytical approach to estimate coverage and rate of cellular networks subject to other–cell interference. In this paper, we move from the approach developed in [1], and propose an alternative analytical\nderivation to compute the rate of cellular networks. More specifically, by using stochastic geometry and Poisson point processes theory, we derive a simple and easy–to–compute expression of the rate, which can be used for arbitrary network and channel parameters, e.g., path–loss exponent, receiver noise, density of Base Stations (BSs), etc. Compared to [1], our framework has two main distinguishable features: i) the rate can be computed via a single numerical integral, rather than via a three–fold numerical integral; and ii) the formula is applicable to arbitrary fading distributions on the intended link, rather than being useful for Rayleigh fading only. Our analytical derivation is substantiated through extensive Monte Carlo simulations.

This Letter proposes a novel tractable solution for the average rate in cellular networks. Specifically, the authors derive a closed-form solution based on bipolar point processes to simplify the computational complexity, which may be conveniently applied to the analysis of actual base station (BS) deployment. Proved by simulation, the results of closed-form solution are consistent with the numerical results. Furthermore, they analyse the BS activation effect under noisy Rayleigh fading channel conditions in 2D and 3D cases, respectively. Simulation shows that turning off the inactive BS can be used as an effective interference mitigation solution, and the interference of the system is mainly limited by the density of active users.

A K-tier heterogeneous downlink millimeter wave (mmWave) cellular network with user-centric small cell deployments is studied in this paper. In particular, we consider a heterogeneous network model with user equipments (UEs) being deployed according to a Poisson Cluster Process (PCP), i.e., Thomas cluster process, where the UEs are clustered around the base stations (BSs) and the distances between UEs and the BS are modeled as Gaussian distributed. In addition, distinguishing features of mmWave communications including directional beamforming and a sophisticated path loss model incorporating both line-of-sight (LOS) and non-line-of-sight (NLOS) transmissions, are taken into account. In this paper, the complementary cumulative distribution function (CCDF) and probability density function (PDF) of the path loss are provided. Also, using tools from stochastic geometry, we derive a general expression of the signal-to-interference-plus-noise ratio (SINR) coverage probability. Our results demonstrate that coverage probability can be improved by decreasing the size of UE clusters around BSs, and interference has noticeable influence on the coverage performance of our model.

In this paper, we provide an analytical framework to analyze heterogeneous downlink millimeter-wave (mm-wave) cellular networks consisting of $K$ 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 $\mu$ -wave frequency bands is addressed.

In this paper, we provide an analytical framework to analyze heterogeneous downlink mmWave cellular networks consisting of $K$ tiers of randomly located base stations (BSs) where each tier operates in a mmWave 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 mmWave communications such as directional beamforming and having different path loss laws for line-of-sight (LOS) and non-line-of-sight (NLOS) links are incorporated into the coverage analysis by assuming averaged biased-received power association and Nakagami-m fading. By using the noise-limited assumption for mmWave networks, a simpler expression requiring the computation of only one numerical integral for coverage probability is obtained. Finally, effect of beamforming alignment errors on the coverage probability analysis is investigated to get insight on the performance in practical scenarios.

It is common practice to model the base station (BS) locations in a cellular system by a grid, such as a hexagonal or square lattice. This model is usually analytically intractable as well as quite idealized. Therefore, system designers resort to complex simulations to evaluate network performance. In this paper, we introduce a new model for the base station locations based on a homogeneous Poisson point process (PPP), whereby the mobiles communicate with their nearest base stations. We obtain the distribution of the signal-to-interference-noise ratio (SINR), compute the average ergodic rate, and analytically verify the trade-off between coverage and rate with frequency reuse. We compare our results with actual BS locations as well as the grid model. In addition to being tractable, we also observe that the performance predicted by the PPP model lower bounds the actual performance, and is about as predictive as the grid model which provides upper bounds.

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, we consider a cellular network in which the user equipments (UEs) connect to multiple base stations (BSs) for augmenting their signal to interference plus noise ratio (SINR) coverage. In literature, the improvement in the SINR performance with multi-connectivity, or macrodiversity has been well-investigated for several network scenarios, e.g., millimeter wave (mm-wave) networks, networks with random blockages, etc. However, the impact of such multi-connectivity schemes on the network throughput has relatively sparse literature. We bridge this gap by jointly characterizing the effect of multi-connectivity on the SINR coverage and the throughput coverage of the UEs in the network, leveraging tools of stochastic geometry. Such a characterization for generic n connections to the typical user is necessary for 5G and beyond cellular networks, precisely since the degree of multi-connectivity should necessarily be dynamic and self-organizing in nature. Our results show that although the SINR coverage of the UEs always improves with more number of connections, the per-UE throughput in the network may increase or decrease depending on the density and data-rate requirement of the UEs. Interestingly, if the data-rate requirement of the UEs increases, our results show that the UEs should connect to a fewer number of BSs to maximize the per-UE throughput.

Device-to-Device (D2D) communication has currently been emerging as a promising technology to increase capacity and to extend coverage area in cellular communication network. D2D communication allows direct communication between two or more devices such as mobile user equipments without any base station help as a relay. However, enabling D2D features in cellular communication network will reveal more complex interference problems, because D2D communication could share the same frequency resources as its underlain cellular communication network. This paper analyzes the interference problems in such D2D communications underlying cellular communication network for downlink transmission. This paper explores the use of power control methods to reduce the effect of interference. The decision whether to increase or to decrease the power level on base station (evolved Node B/eNB in Fourth Generation/4G Cellular Networks) or on the transmitter of D2D pair (Transmitter of D2D User Equipment/TUE) is based on the estimated current Signal to Interference plus Noise Ratio (SINR). First method of power control (PC-1) uses a fixed value to control the power level of the transmitter. Another one (PC-2) uses moving average of interference power values. The simulation was carried out to evaluate those two power control methods and its results in term of Cumulative Distribution Function (CDF) of SINR are compared to the system without power control method. The simulation results show that both power control methods contribute the improvement of performances; for one cellular equipment (CUE) and 100 pairs of D2D it achieved the improvement of SINR distribution at 5% with PC-1 and at 4% with PC-2 compared to the system without powr control, meanwhile for 1 D2D pair and 100 CUEs the CDF of SINR at 0 dB achieves 40%, 3%, and 0% for the systems without power control, with PC-1, and PC-2 methods, accordingly.

In this paper we consider a multi-user spatial multiplexing (SM) cellular network, where N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> streams are transmitted to N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> users in the cell. Specifically, we obtain the coverage and rate expressions for a system employing zero-forcing (ZF) receiver. Compared to single stream transmission (SST), it is interesting to see that SM degrades the rate for a notable percentage of users. For the case of two and four receiver antennas, the increase in mean rate of SM is modest compared to single stream transmission (SST) while SST provides a gain over SM for cell edge users.

This paper provides an analytical framework with foundations in stochastic geometry to characterize the spatio-temporal interference correlation as well as the joint coverage probability at two spatial locations in a cellular network. In particular, modeling the locations of cellular base stations (BSs) as a Poisson point process, we study interference correlation at two spatial locations $\ell _{1}$ and $\ell _{2}$ separated by a distance $v$ , when the user follows the closest BS association policy at both spatial locations and moves from $\ell _{1}$ to $\ell _{2}$ . With this user displacement, two scenarios can occur: i) the user is handed off to a new serving BS at $\ell _{2}$ , or ii) no handoff occurs and the user is served by the same BS at both locations. After providing intermediate results, such as probability of handoff and distance distributions of the serving BS at the two user locations, we use them to derive exact expressions for spatio-temporal interference correlation coefficient and joint coverage probability for any distance separation $v$ . We also study two different handoff strategies: i) handoff skipping , and ii) conventional handoffs , and derive the expressions of joint coverage probability for both strategies. The exact analysis is not straightforward and involves a careful treatment of the neighborhood of the two spatial locations and the resulting handoff scenarios. To provide analytical insights, we also provide easy-to-use expressions for two special cases: i) static user ( $v =0$ ) and ii) highly mobile user ( $v \rightarrow \infty )$ . As expected, our analysis shows that the interference correlation and joint coverage probability decrease with increasing $v$ , with $v \rightarrow \infty $ corresponding to a completely uncorrelated scenario. Further design insights are also provided by studying the effect of few network/channel parameters, such as BS density and path loss on the interference correlation.

In cellular networks, each mobile station adjusts its power level under control of its base station, i.e., through uplink transmit power control, which is essential to reach desired signal-to-interference-plus-noise ratio (SINR) at the base station and to limit inter-cell interference. The optimal levels of transmit power in a network depend on path loss, shadowing, and multipath fading, as well as the network configuration. However, since path loss is distance dependent and the cell association distances are correlated due to the cell association policies, the performance analysis of the uplink transmit power control is very complicated. Consequently, the impact of a specific power control algorithm on network performance is hard to quantify. In this paper, we analyze three uplink transmit power control schemes. We assume the standard power-law path loss and composite Rayleigh-lognormal fading. Using stochastic geometry tools, we derive the cumulative distribution function and the probability density function of the uplink transmit power and the resulting network coverage probability. It is shown that the coverage is highly dependent on the severity of shadowing, the power control scheme, and its parameters, but invariant of the density of deployment of base stations when the shadowing is mild and power control is fractional. At low SINRs, compensation of both path loss and shadowing improves the coverage. However, at high SINRs, compensating for path loss only improves coverage. Increase in the severity of shadowing significantly reduces the coverage.

Multi-cell cooperation is a promising approach for mitigating inter-cell interference in dense cellular networks. Quantifying the performance of multi-cell cooperation is challenging as it integrates physical-layer techniques and network topologies. For tractability, existing work typically relies on the over-simplified Wyner-type models. In this paper, we propose a new stochastic- geometry model for a cellular network with multi-cell cooperation, which accounts for practical factors including the irregular locations of base stations (BSs) and the resultant path-losses. In particular, the proposed network-topology model has three key features: i) the cells are modeled using a Poisson random tessellation generated by Poisson distributed BSs, ii) multi-antenna BSs are clustered using a hexagonal lattice and BSs in the same cluster mitigate mutual interference by spatial interference avoidance, iii) BSs near cluster edges access a different sub- channel from that by other BSs, shielding cluster-edge mobiles from strong interference. Using this model and assuming sparse scattering, we analyze the shapes of the outage probabilities of mobiles served by cluster-interior BSs as the average number K of BSs per cluster increases. The outage probability of a mobile near a cluster center is shown to be proportional to e -c(2-√ν)2K where ν is the fraction of BSs lying in the interior of clusters and c is a constant. Moreover, the outage probability of a typical mobile is proved to scale proportionally with e -c′(1-√ν)2K where c′ is a constant. © 2011 IEEE.

Cellular operators are continuously densifying their networks to cope with the ever-increasing capacity demand. Furthermore, an extreme densification phase for cellular networks is foreseen to fulfill the ambitious fifth generation (5G) performance requirements. Network densification improves spectrum utilization and network capacity by shrinking base stations' (BSs) footprints and reusing the same spectrum more frequently over the spatial domain. However, network densification also increases the handover (HO) rate, which may diminish the capacity gains for mobile users due to HO delays. In highly dense 5G cellular networks, HO delays may neutralize or even negate the gains offered by network densification. In this paper, we present an analytical paradigm, based on stochastic geometry, to quantify the effect of HO delay on the average user rate in cellular networks. To this end, we propose a flexible handover scheme to reduce HO delay in case of highly dense cellular networks. This scheme allows skipping the HO procedure with some BSs along users' trajectories. The performance evaluation and testing of this scheme for only single HO skipping shows considerable gains in many practical scenarios.