Geographic locations of cellular base stations sometimes can be well fitted with spatial homogeneous Poisson point processes. In this paper we make a complementary observation: In the presence of the log-normal shadowing of sufficiently high variance, the statistics of the propagation loss of a single user with respect to different network stations are invariant with respect to their geographic positioning, whether regular or not, for a wide class of empirically homogeneous networks. Even in perfectly hexagonal case they appear as though they were realized in a Poisson network model, i.e., form an inhomogeneous Poisson point process on the positive half-line with a power-law density characterized by the path-loss exponent. At the same time, the conditional distances to the corresponding base stations, given their observed propagation losses, become independent and log-normally distributed, which can be seen as a decoupling between the real and model geometry. The result applies also to Suzuki (Rayleigh-log-normal) propagation model. We use Kolmogorov-Smirnov test to empirically study the quality of the Poisson approximation and use it to build a linear-regression method for the statistical estimation of the value of the path-loss exponent.
Recently several characteristics related to the propagation process in the Poisson model have been studied. For example, closed and semi-closed expressions for the coverage probability in so-called multi-tier network models @cite_35 @cite_6 @cite_39 @cite_1 @cite_25 , with the concept of @math -coverage being later introduced @cite_7 @cite_18 . More recently, these models have been advanced by studying the coverage probability under signal coordination @cite_19 @cite_15 and interference cancellation schemes @cite_0 @cite_38 , with a recent contribution being the joint probability density of the order statistics of the process formed from the SINR values of the typical user @cite_18 . The recently observed relation between the SINR values and a type of Poisson-Dirichlet process (cf @cite_24 ) can potentially bring some further progress to this subject.
@cite_35 @cite_18 @cite_38 @cite_7 @cite_1 @cite_6 @cite_39 @cite_0 @cite_19 @cite_24 @cite_15 @cite_25
2098045019 1640283668 2086605530 2049496443
Based on a stationary Poisson point process, a wireless network model with random propagation effects (shadowing and or fading) is considered in order to examine the process formed by the signal-to-interference-plus-noise ratio (SINR) values experienced by a typical user with respect to all base stations in the down-link channel. This SINR process is completely characterized by deriving its factorial moment measures, which involve numerically tractable, explicit integral expressions. This novel framework naturally leads to expressions for the k-coverage probability, including the case of random SINR threshold values considered in multi-tier network models. While the k-coverage probabilities correspond to the marginal distributions of the order statistics of the SINR process, a more general relation is presented connecting the factorial moment measures of the SINR process to the joint densities of these order statistics. This gives a way for calculating exact values of the coverage probabilities arising in a general scenario of signal combination and interference cancellation between base stations. The presented framework consisting of mathematical representations of SINR characteristics with respect to the factorial moment measures holds for the whole domain of SINR and is amenable to considerable model extension. 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. We define and analyze a random coverage process of the @math -dimensional Euclidian space which allows one to describe a continuous spectrum that ranges from the Boolean model to the Poisson-Voronoi tessellation to the Johnson-Mehl model. Like for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidian space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulated effect of all marks. This cumulated effect is defined as the shot noise process associated with the marked point process. In addition to analyzing and visualizing this continuum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyze convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The obtained results allow one to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows one to characterize handover strategies etc. 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].