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.
@cite_26: This paper considers two-dimensional interference-limited cellular radio systems. It introduces the shotgun cellular system that places base stations randomly and assigns channels randomly. Such systems are shown to provide lower bounds to cellular performance that are easy to compute, independent of shadow fading, and apply to a number of design scenarios. Traditional hexagonal systems provide an upper performance bound. The difference between upper and lower bounds is small under operating conditions typical in modern TDMA and CDMA cellular systems. Furthermore, in the strong shadow fading limit, the bounds converge. To give insights into the design of practical systems, several variations are explored including mobile access methods, sectorizing, channel assignments, and placement with deviations. Together these results indicate cellular performance is very robust and little is lost in making rapid minimally planned deployments.
@cite_4: We propose a new linear-regression model for the estimation of the path-loss exponent and the parameters of the shadowing from the propagation-loss data collected by the mobiles with respect to their serving base stations. The difficulty consists in deriving the parameters of the distribution of the propagation loss with respect to an arbitrary base station from these regarding the strongest one. The proposed solution is based on a simple, explicit relation between the two distributions in the case of infinite Poisson network and on the convergence of an arbitrary regular (in particular hexagonal) network to the Poisson one with increasing variance of the shadowing. The new approach complements existing methods, in particular the one based on COSTWalfisch-Ikegami model, which does not allow for the shadowing estimation and is not suited for indoor scenario.
@cite_20: An almost ubiquitous assumption made in the stochastic-analytic study of the quality of service in cellular networks is Poisson distribution of base stations. It is usually justified by various irregularities in the real placement of base stations, which ideally should form the hexagonal pattern. We provide a different and rigorous argument justifying the Poisson assumption under sufficiently strong log-normal shadowing observed in the network, in the evaluation of a natural class of the typical-user service-characteristics including its SINR. Namely, we present a Poisson-convergence result for a broad range of stationary (including lattice) networks subject to log-normal shadowing of increasing variance. We show also for the Poisson model that the distribution of all these characteristics does not depend on the particular form of the additional fading distribution. Our approach involves a mapping of 2D network model to 1D image of it "perceived" by the typical user. For this image we prove our convergence result and the invariance of the Poisson limit with respect to the distribution of the additional shadowing or fading. Moreover, we present some new results for Poisson model allowing one to calculate the distribution function of the SINR in its whole domain. We use them to study and optimize the mean energy efficiency in cellular networks.
The main convergence result proved in this paper can be rephrased as follows: 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 base stations are essentially invariant with respect to the exact geographic positioning of the base stations, 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. Brown @cite_26 first observed this by simulation, later confirmed independently by B aszczyszyn and Karray @cite_4 who suggested using classical convergence results of random translations of point processes; see Daley and Vere-Jones [Section 11.4] daleyPPII2008 . This approach was first adapted and applied in this setting by B aszczyszyn, Karray and Keeler @cite_20 , which forms part of the results presented here.