Coverage and Meta Distribution Analysis in Ultra-Dense Cellular Networks With Directional Antennas

We develop an analytical framework to derive the meta distribution and moments of the conditional success probability (CSP), which is defined as success probability for a given realization of the transmitters, in large-scale co-channel uplink and downlink non-orthogonal multiple access (NOMA) networks with one NOMA cluster per cell. The moments of CSP translate to various network performance metrics such as the standard success or signal-to-interference ratio (SIR) coverage probability (which is the 1-st moment), the mean local delay (which is the −1st moment in a static network setting), and the meta distribution (which is the complementary cumulative distribution function of the success or SIR coverage probability and can be approximated by using the 1st and 2nd moments). For the uplink NOMA network, to make the framework tractable, we propose two point process models for the spatial locations of the inter-cell interferers by utilizing the base station (BS)/user pair correlation function. We validate the proposed models by comparing the second moment measure of each model with that of the actual point process for the inter-cluster (or inter-cell) interferers obtained via simulations. For downlink NOMA, we derive closed-form solutions for the moments of the CSP, success (or coverage) probability, mean local delay, and meta distribution for the users. As an application of the developed analytical framework, we use the closed-form expressions to optimize the power allocations for downlink NOMA users in order to maximize the success probability of a given NOMA user with and without latency constraints. Closed-form optimal solutions for the transmit powers are obtained for two-user NOMA scenario. We note that maximizing the success probability with latency constraints can significantly impact the optimal power solutions for low SIR thresholds and favor orthogonal multiple access.

The performance analysis of heterogeneous cellular networks (HetNets), that relied mostly on the homogeneous Poisson point process (PPP) for the spatial distribution of the users and base stations (BSs), has seen a major transition with the emergence of the Poisson cluster process (PCP)-based models. With the combination of PPP and PCP, it is possible to construct a general HetNet model which can capture formation of hotspots and spatial coupling between the users and the BSs. While the downlink coverage analysis of this model in terms of the distribution of the received downlink signal-to-interference ratio ( SIR ) is well understood by now, more fine grained analysis in terms of the meta distribution of SIR is an open problem. In this letter, we solve this problem by deriving the meta distribution of the downlink SIR assuming that the typical user connects to the BS providing the maximum received power.

Heterogeneous cellular networks (HCNs) constitute a necessary step in the evolution of cellular networks. In this paper, we apply the signal-to-interference ratio (SIR) meta distribution framework for a refined SIR performance analysis of HCNs, focusing on K-tier heterogeneous cellular networks based on the homogeneous independent Poisson point process (HIP) model, with range expansion bias (offloading bias) in each tier. Expressions for the b-th moment of the conditional success probability for both the entire network and each tier are derived, based on which the exact meta distributions and the beta approximations are evaluated and compared. Key performance metrics including the mean success probability, the variance of the conditional success probability, the mean local delay and the asymptotic SIR gains of each tier are obtained. The results show that the biases are detrimental to the overall mean success probability of the whole network and that the b-th moment curve (versus the SIR threshold) of the conditional success probability of each tier can be excellently approximated by the horizontal shifted versions of the first moment curve of the single-tier PPP network. We also provide lower bounds for the region of the active probabilities of the base stations to keep the mean local delay of each tier finite.

The Poisson bipolar model considers user-base station pairs distributed at random on a flat domain, similar to matchsticks scattered onto a table. Though this is a simple and tractable setting in which to study dense networks, it doesn’t properly characterise the stochastic geometry of user-base station interactions in some dense deployment scenarios, which may involve short and long range links, with some paired very nearby optimally, and others sub-optimally due to local crowding. Since the users will pair one-to-one with base stations, we can consider using the popular bipartite Euclidean matching (BEM) from spatial combinatorics, and study the corresponding (meta) distribution of the signal-to-interference-ratio (SIR). This provides detailed information about the proportion of links in the network meeting a target reliability constraint. We can then observe via comparison the impact of taking into account the variable/correlated short-range distances between the transmitter-receiver pairs on the communication statistics. We illustrate and quantify how the widely-accepted bipolar model fails to capture the network-wide reliability of communication in a typical ultra-dense setting based on a binomial point process. We also show how assuming a Gamma distribution for link distances may be a simple improvement on the bipolar model. Overall, BEMs provide good grounds for understanding more sophisticated pairing features in ultra-dense networks.

This article performs a unified analysis of three coordinated multipoint (CoMP) transmission strategies in the downlink of mmWave cellular networks, including the fixed-number base station (BS) cooperation (FNC), the fixed-region BS cooperation (FRC), and the interference-aware BS cooperation (IAC). We first develop a comprehensive framework for CoMP operation in cellular networks, and investigate the network performance under a Poisson point process (PPP) model together with mmWave spectrum. To show what fraction of users in the network achieve target reliability for a given signal to interference-plus-noise ratio (SINR)/signal-to-interference ratio (SIR), we derive the SINR/SIR meta distributions, and further obtain the coverage probability as well as mean local delay for the three cooperation strategies. A pivotal intermediate step to compute the performance metrics is the derivation of joint distributions of distances between a typical user and cooperative BSs. Our analysis demonstrates that parameters of blockage have a significant influence on the network performance for the three CoMP schemes. We find that the FRC scheme makes more users achieve the given link reliability for the scenario with a low density of BSs, while the IAC scheme provides better performance for the network with a high density of BSs. Moreover, the optimal CoMP scheme can be approximately selected by considering the nearest distance from the serving BS to user and the radius of the approximate line-of-sight (LoS) region in the cellular networks.

SIR analysis for non-uniform HetNets with joint decoupled association and interference management

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.

Edge caching is emerging as the most promising solution to reduce the content retrieval delay and relieve the huge burden on the backhaul links in the ultra-dense networks by proactive caching popular contents in the small base station (SBS). However, constraint cache resource of individual SBSs significantly throttles the performance of edge caching. In this paper, we propose a device-to-device (D2D) assisted cooperative edge caching (DCEC) policy for millimeter (mmWave) dense networks, which cooperatively utilizes the cache resource of users and SBSs in proximity. In the proposed DCEC policy, a content can be cached in either users' devices or SBSs according to the content popularity, and a user can retrieve the requested content from neighboring users via D2D links or the neighboring SBSs via cellular links to efficiently exploit the cache diversity. Unlike existing cooperative caching policies in the lower frequency bands that require complex interference management techniques to suppress interference, we take advantage of directional antenna in mmWave systems to ensure high transmission rate whereas mitigating interference footprint. Taking the practical directional antenna model and the network density into consideration, we derive closed-form expressions of the backhaul offloading performance and content retrieval delay based on the stochastic information of network topology. In addition, analytical results indicate that, with the increase of the network density, the content retrieval delay via D2D links increases significantly while that via cellular links increases slightly. Comprehensive simulations validate our theoretical analysis and demonstrate that the proposed policy can achieve higher performance in offloading the backhaul traffic and reducing the content retrieval delay compared with the state-of-the-art most popular caching policy.

Future mobile networks are converging toward heterogeneous multi-tier networks, where various classes of base stations (BS) are deployed based on user demand.So it is quite necessary to utilize the BSs resources rationally when BSs are sufficient.In this paper, we develop a more realistic model that fully considering the inter-tier dependence and the dependence between users and BSs, where the macro base stations (MBSs) are distributed according to a homogeneous Poisson point process (PPP) and the small base stations (SBSs) follows a Matern cluster process (MCP) whose parent points are located in the positions of the MBSs in order to offload the users from the over-loaded MBSs.We also assume the users are just randomly located in the circles centered at the MBSs.Under this model, we derive the association probability and the average ergodic rate by stochastic geometry.An interesting result that the density of MBS and the radius of the clusters jointly affect the association probabilities in a joint form is obtained.We also observe that using the clustered SBSs results in aggressive offloading compared with previous cellular networks.

The growing complexity of heterogeneous cellular networks (HetNets) has necessitated the need to consider variety of user and base station (BS) configurations for realistic performance evaluation and system design. This is directly reflected in the HetNet simulation models considered by standardization bodies, such as the third generation partnership project (3GPP). Complementary to these simulation models, stochastic geometry-based approach modeling the user and BS locations 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 model is not rich enough to capture all the spatial configurations that appear in real-world HetNet deployments (on which 3GPP simulation models are based). In this paper, we bridge the gap between the 3GPP simulation models and the popular PPP-based analytical model by developing a new unified HetNet model in which a fraction of users and some BS tiers are modeled as Poisson cluster processes (PCPs). This model captures both non-uniformity and coupling in the BS and user locations. For this setup, we derive exact expression for downlink coverage probability under maximum signal-to-interference ratio (SIR) cell association model. As intermediate results, we define and evaluate sum-product functionals for PPP and PCP. Special instances of the proposed model are shown to closely resemble different configurations considered in 3GPP HetNet models. Our results concretely demonstrate that the performance trends are highly sensitive to the assumptions made on the user and SBS configurations.

Device-to-device (D2D) communications is a promising approach for wireless communications in terms of energy and spectrum efficiency. However, such communications increase interferences in a cellular network. In this paper, we propose to lower these interferences by using millimeter wave directional antennas. To analyze the impact of these techniques, we introduce mathematical sectored antenna models that are deduced from mmWave antenna radiation patterns. Moreover, recent studies consider a constant transmit power for the devices. Nevertheless, modern communications use power control techniques to mitigate energy consumption and interferences. The main contribution of our work is the consideration of channel inversion, which is more realistic than the commonly used transmit model. Most works dealing with conventional D2D communications propose to use stochastic geometry to model a D2D-enabled network in order to evaluate the impact of interferences and noise on the various links. The objective of this paper is to analyze the SINR and the energy efficiency of outband D2D links for UEs equipped with directional mmWave antennas. To do so, we implement an energy efficiency calculation that considers both directional antennas and channel inversion. We propose to highlight the advantages and drawbacks of directional mmWave antennas in outband D2D for diverse antenna designs and different environments.

Millimeter Wave (mmWave) communication is considered as an enabling technology for the next generation of cellular networks because it offers much larger bandwidth and higher data rate than the current lower-frequency cellular systems to satisfy the exponential growth of mobile data demand. High gain directional antennas are needed to overcome high propagation losses in mmWave bands. However, the reliance on highly directional antennas will result in a more complicated initial cell search procedure since both base station and mobile device have to look for each other over a large space to establish the link. This paper focuses on analyzing the performance of the directional cell search procedure where the base stations periodically transmit signals in a set of optimal directional patterns to scan the coverage area. The mobile terminals detect the signals from the base station using the Generalized Likelihood Ratio Test (GLRT). The results show that with an appropriate scanning scheme, the use of directional antennas can outperform their omnidirectional counterparts in terms of signal detection performance as well as total time required.

The evaluation of the Signal to Interference Ratio (SIR) in cellular networks is of primary importance for network dimensioning. For static studies, which evaluate cell capacity and coverage, as well as for dynamic studies, which consider arrivals and departures of mobile stations (MS), the SIR is always an important input. Contrary to most of the analytical works evaluating SIR, we assume in this paper that the MS is attached to the best server, i.e., to the base station (BS) from which it receives the highest power. This is a policy that is more realistic than the classical assumption that considers MSs to be attached to the nearest BS. The exact formulation of the SIR is however in this case uneasy to handle and numerical methods remain heavy. In this paper, we thus propose an approximate analytical study based on truncated lognormal distributions that provides very close results to Monte Carlo simulations.

Performance analysis of coverage-centric heterogeneous cellular networks using dual-slope path loss model

To deal with the huge traffic demand in wireless networks, small base stations (SBSs) are introduced in cellular networks, not only with dense deployment, but also with caching capabilities. Due to the short distances between users and SBSs, files can be downloaded at higher transmission rate from the cache of SBSs than being downloaded over backhaul links. However, user mobility makes the file allocation in SBS caches more challenging, mostly because the user association to SBSs dynamically changes. Moreover, the amount of data that users can download depends on the sojourn time within the coverage of SBSs. In this paper, based on the assumption that the user sojourn time obeys exponential distribution, we get the upper bound of the mean download time of files. A file allocation strategy is then proposed based on the derived bounds of mean sojourn time. Optimal file allocation strategies under low and high mobility are further obtained. Simulations show that the strategy performs notably better than the popularity-based allocation strategy.