We consider a general heterogeneous network in which, besides general propagation effects (shadowing and or fading), individual base stations can have different emitting powers and be subject to different parameters of Hata-like path-loss models (path-loss exponent and constant) due to, for example, varying antenna heights. We assume also that the stations may have varying parameters of, for example, the link layer performance (SINR threshold, etc). By studying the propagation processes of signals received by the typical user from all antennas marked by the corresponding antenna parameters, we show that seemingly different heterogeneous networks based on Poisson point processes can be equivalent from the point of view a typical user. These neworks can be replaced with a model where all the previously varying propagation parameters (including path-loss exponents) are set to constants while the only trade-off being the introduction of an isotropic base station density. This allows one to perform analytic comparisons of different network models via their isotropic representations. In the case of a constant path-loss exponent, the isotropic representation simplifies to a homogeneous modification of the constant intensity of the original network, thus generalizing a previous result showing that the propagation processes only depend on one moment of the emitted power and propagation effects. We give examples and applications to motivate these results and highlight an interesting observation regarding random path-loss exponents.
Abstract--- This paper studies the carrier-to-interference ratio (CIR) and carrier-to-interference-plus-noise ratio (CINR) performance at the mobile station (MS) within a multi-tier network composed of M tiers of wireless networks, with each tier modeled as the homogeneous n-dimensional (n-D, n=1,2, and 3) shotgun cellular system, where the base station (BS) distribution is given by the homogeneous Poisson point process in n-D. The CIR and CINR at the MS in a single tier network are thoroughly analyzed to simplify the analysis of the multi-tier network. For the multi-tier network with given system parameters, the following are the main results of this paper: (1) semi-analytical expressions for the tail probabilities of CIR and CINR; (2) a closed form expression for the tail probability of CIR in the range [1,infinity); (3) a closed form expression for the tail probability of an approximation to CINR in the entire range [0,infinity); (4) a lookup table based approach for obtaining the tail probability of CINR, and (5) the study of the effect of shadow fading and BSs with ideal sectorized antennas on the CIR and CINR. Based on these results, it is shown that, in a practical cellular system, the installation of additional wireless networks (microcells, picocells and femtocells) with low power BSs over the already existing macrocell network will always improve the CINR performance at the MS. The Signal to Interference Plus Noise Ratio (SINR) on a wireless link is an important basis for consideration of outage, capacity, and throughput in a cellular network. It is therefore important to understand the SINR distribution within such networks, and in particular heterogeneous cellular networks, since these are expected to dominate future network deployments . Until recently the distribution of SINR in heterogeneous networks was studied almost exclusively via simulation, for selected scenarios representing pre-defined arrangements of users and the elements of the heterogeneous network such as macro-cells, femto-cells, etc. However, the dynamic nature of heterogeneous networks makes it difficult to design a few representative simulation scenarios from which general inferences can be drawn that apply to all deployments. In this paper, we examine the downlink of a heterogeneous cellular network made up of multiple tiers of transmitters (e.g., macro-, micro-, pico-, and femto-cells) and provide a general theoretical analysis of the distribution of the SINR at an arbitrarily-located user. Using physically realistic stochastic models for the locations of the base stations (BSs) in the tiers, we can compute the general SINR distribution in closed form. We illustrate a use of this approach for a three-tier network by calculating the probability of the user being able to camp on a macro-cell or an open-access (OA) femto-cell in the presence of Closed Subscriber Group (CSG) femto-cells. We show that this probability depends only on the relative densities and transmit powers of the macro- and femto-cells, the fraction of femto-cells operating in OA vs. Closed Subscriber Group (CSG) mode, and on the parameters of the wireless channel model. For an operator considering a femto overlay on a macro network, the parameters of the femto deployment can be selected from a set of universal curves. We give numerically tractable, explicit integral expressions for the distribution of the signal-to-interference-and-noise-ratio (SINR) experienced by a typical user in the downlink channel from the k-th strongest base stations of a cellular network modelled by Poisson point process on the plane. Our signal propagation-loss model comprises of a power-law path-loss function with arbitrarily distributed shadowing, independent across all base stations, with and without Rayleigh fading. Our results are valid in the whole domain of SINR, in particular for SINR <; 1, where one observes multiple coverage. In this latter aspect our paper complements previous studies reported in [1]. Cellular networks are in a major transition from a carefully planned set of large tower-mounted base-stations (BSs) to an irregular deployment of heterogeneous infrastructure elements that often additionally includes micro, pico, and femtocells, as well as distributed antennas. In this paper, we develop a tractable, flexible, and accurate model for a downlink heterogeneous cellular network (HCN) consisting of K tiers of randomly located BSs, where each tier may differ in terms of average transmit power, supported data rate and BS density. Assuming a mobile user connects to the strongest candidate BS, the resulting Signal-to-Interference-plus-Noise-Ratio (SINR) is greater than 1 when in coverage, Rayleigh fading, we derive an expression for the probability of coverage (equivalently outage) over the entire network under both open and closed access, which assumes a strikingly simple closed-form in the high SINR regime and is accurate down to -4 dB even under weaker assumptions. For external validation, we compare against an actual LTE network (for tier 1) with the other K-1 tiers being modeled as independent Poisson Point Processes. In this case as well, our model is accurate to within 1-2 dB. We also derive the average rate achieved by a randomly located mobile and the average load on each tier of BSs. One interesting observation for interference-limited open access networks is that at a given , adding more tiers and or BSs neither increases nor decreases the probability of coverage or outage when all the tiers have the same target-SINR.