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 works in @cite_7 @cite_11 @cite_13 obtained analytically tractable results for the HIP model. For HCNs with non-Poisson deployments, it is often the case that it is hard to perform an exact mathematical analysis of key performance metrics such as the SIR distribution (sometimes called the coverage probability). Even if an exact expression of the SIR distribution exists, it is available in a complex form that does not help gain insights about the performance of the network for different network parameters @cite_22 @cite_21 @cite_23 @cite_18 @cite_10 .
@cite_18 @cite_22 @cite_7 @cite_10 @cite_21 @cite_23 @cite_13 @cite_11
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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. Due to its tractability, a multitier model of mutually independent Poisson point processes (PPPs) for heterogeneous cellular networks (HCNs) has recently been attracting much attention. However, in reality, the locations of the BSs, within each tier and across tiers, are not fully independent. Accordingly, in this paper, we propose two HCN models with inter-tier dependence (Case 1) and intra-tier dependence (Case 2), respectively. In Case 1, the macro-base station (MBS) and the pico-base station (PBS) deployments follow a Poisson point process (PPP) and a Poisson hole process (PHP), respectively. Under this setup and conditioning on a fixed serving distance (distance between a user and its nearest serving BS), we derive bounds on the outage probabilities of both macro and pico users. We also use a fitted Poisson cluster process to approximate the PHP, which is shown to provide a good approximation of the interference and outage statistics. In Case 2, the MBSs and the PBSs follow a PPP and an independent Matern cluster process, respectively. Explicit expressions of the interference and the outage probability are derived first for fixed serving distance and second with random distance, and we derive the outage performance, the per-user capacity, and the area spectral efficiency (ASE) for both cases. It turns out that the proposed Case 2 model is a more appropriate and accurate model for a HCN with hotspot regions than the multitier independent PPP model since the latter underestimates some key performance metrics, such as the per-user capacity and the ASE, by a factor of 1.5 to 2. Overall, the two models proposed provide good tradeoffs between the accuracy, tractability, and practicability. Using a multi-tier Poisson model, this paper studies the performance gain of successive interference cancellation (SIC) in the downlink of K-tier heterogeneous cellular networks (HCNs). For each tier, a fraction of base stations (BSs) is non-accessible. By using a framework based on the marked path loss process with fading and calculating the equivalent access probability, we analytically characterize the coverage probability, i.e., the probability of successfully connecting to at least one accessible BS, for a typical user equipment with finite or infinite SIC capability. The results show how the performance gain of SIC depends on many system parameters including path loss exponent, coding rate, fading distributions and BS accessibilities and densities. We show for contemporary OFDM-based HCNs, infinite SIC capability is often unnecessary. In fact, under typical system parameters, most of the gain of SIC comes from the ability of canceling only a single non-accessible BS. 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.