Compared to the standard success (coverage) probability, the meta distribution of the signal-to-interference ratio (SIR) provides much more fine-grained information about the network performance. We consider general heterogeneous cellular networks (HCNs) with base station tiers modeled by arbitrary stationary and ergodic non-Poisson point processes. The exact analysis of non-Poisson network models is notoriously difficult, even in terms of the standard success probability, let alone the meta distribution. Hence we propose a simple approach to approximate the SIR meta distribution for non-Poisson networks based on the ASAPPP ("approximate SIR analysis based on the Poisson point process") method. We prove that the asymptotic horizontal gap @math between its standard success probability and that for the Poisson point process exactly characterizes the gap between the @math th moment of the conditional success probability, as the SIR threshold goes to @math . The gap @math allows two simple approximations of the meta distribution for general HCNs: 1) the per-tier approximation by applying the shift @math to each tier and 2) the effective gain approximation by directly shifting the meta distribution for the homogeneous independent Poisson network. Given the generality of the model considered and the fine-grained nature of the meta distribution, these approximations work surprisingly well.
The meta distribution of the SIR for cellular networks was proposed in @cite_24 , where the focus was on the downlink of the Poisson cellular network. Furthermore, the meta distribution of the SIR was calculated for both the downlink and the uplink of the Poisson cellular network with power control in @cite_28 , for the downlink Poisson cellular network underlaid with a device-to-device (D2D) network in @cite_16 , for the non-orthogonal multiple access (NOMA) network in @cite_20 , and with base station cooperation in @cite_2 . For general cellular networks with a multi-slope path loss model, @cite_17 gave a scaling law involving the parameters of BS and user point processes ( e.g. , the density of the point process) that keeps the meta distribution of the SIR the same. For the HIP-based @math -tier HCN, @cite_3 calculated the SIR meta distribution with cell range expansion.
@cite_28 @cite_3 @cite_24 @cite_2 @cite_16 @cite_20 @cite_17
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The meta distribution of the signal-to-interference ratio (SIR) provides fine-grained information about the performance of individual links in a wireless network. This paper focuses on the analysis of the meta distribution of the SIR for both the cellular network uplink and downlink with fractional power control. For the uplink scenario, an approximation of the interfering user point process with a non-homogeneous Poisson point process is used. The moments of the meta distribution for both scenarios are calculated. Some bounds, the analytical expression, the mean local delay, and the beta approximation of the meta distribution are provided. The results give interesting insights into the effect of the power control in both the uplink and downlink. Detailed simulations show that the approximations made in the analysis are well justified. 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 @math -tier heterogeneous cellular networks based on the homogeneous independent Poisson point process (PPP) model, with range expansion bias (offloading bias) in each tier. Expressions for the @math -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 @math -th moment curve of the conditional success probability of each tier can be tightly 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 calculation of the SIR distribution at the typical receiver (or, equivalently, the success probability of transmissions over the typical link) in Poisson bipolar and cellular networks with Rayleigh fading is relatively straightforward, but it only provides limited information on the success probabilities of the individual links. This paper focuses on the meta distribution of the SIR, which is the distribution of the conditional success probability @math given the point process, and provides bounds, an exact analytical expression, and a simple approximation for it. The meta distribution provides fine-grained information on the SIR and answers questions such as “What fraction of users in a Poisson cellular network achieve 90 link reliability if the required SIR is 5 dB?” Interestingly, in the bipolar model, if the transmit probability @math is reduced while increasing the network density @math such that the density of concurrent transmitters @math stays constant as @math , @math degenerates to a constant, i.e., all links have exactly the same success probability in the limit, which is the one of the typical link. In contrast, in the cellular case, if the interfering base stations are active independently with probability @math , the variance of @math approaches a non-zero constant when @math is reduced to 0 while keeping the mean success probability constant. The meta distribution provides fine-grained information on the signal-to-interference ratio (SIR) compared with the SIR distribution at the typical user. This paper first derives the meta distribution of the SIR in heterogeneous cellular networks with downlink coordinated multipoint transmission reception, including joint transmission (JT), dynamic point blanking (DPB), and dynamic point selection dynamic point blanking (DPS DPB), for the general typical user and the worst-case user (the typical user located at the Voronoi vertex in a single-tier network). A more general scheme called JT-DPB, which is the combination of JT and DPB, is studied. The moments of the conditional success probability are derived for the calculation of the meta distribution and the mean local delay. An exact analytical expression, the beta approximation, and simulation results of the meta distribution are provided. From the theoretical results, we gain insights on the benefits of different cooperation schemes and the impact of the number of cooperating base stations and other network parameters. We study the performance of device-to-device (D2D) communication underlaying cellular wireless network in terms of the meta distribution of the signal-to-interference ratio (SIR), which is the distribution of the conditional SIR distribution given the locations of the wireless nodes. Modeling D2D transmitters and base stations as Poisson point processes (PPPs), moments of the conditional SIR distribution are derived in order to calculate analytical expressions for the meta distribution and the mean local delay of the typical D2D receiver and cellular downlink user. It turns out that for D2D users, the total interference from the D2D interferers and base stations is equal in distribution to that of a single PPP, while for downlink users, the effect of the interference from the D2D network is more complicated. We also derive the region of transmit probabilities for the D2D users and base stations that result in a finite mean local delay and give a simple inner bound on that region. Finally, the impact of increasing the base station density on the mean local delay, the meta distribution, and the density of users reliably served is investigated with numerical results. 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. In this letter, we introduce a general cellular network model, where: 1) users and base stations (BSs) are distributed as two general point processes that may be coupled; 2) pathloss is assumed to follow a multi-slope power-law pathloss model; and 3) fading (power) is assumed to be independent across all wireless links. For this setup, we first obtain a set of contours representing the same meta distribution of signal-to-interference ratio , which is the distribution of the conditional coverage probability given the point process, for different values of the parameters of the pathloss function and BS and user point processes. This general result is then specialized to 3GPP-inspired user and BS configurations obtained by combining Poisson point process and Poisson cluster process.