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_18: 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.
@cite_11: The large system limit of the Random Energy Model (REM) and generalized Random Energy Model (GREM) of Derrida is investigated, and found to be universal. This permits systematic calculations of relevance in particular to Parisi's solution of the Sherrington-Kirkpatrick spin-glass model.
@cite_22: 
@cite_24: 
@cite_2: The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter θ, introduced by Kingman, is PD(0, θ). Known properties of PD(0, θ), including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of PD(α, θ) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 < α < 1, PD(α, 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index α. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is PD(1 2, 0), and the corresponding distribution for a Brownian bridge is PD(1 2, 1 2). The PD(α, 0) and PD(α, α) distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index α.
@cite_5: We consider stochastic cellular networks where base stations locations form a homogenous Poisson point process and each mobile is attached to the base station that provides the best mean signal power. The mobile is in outage if the SINR falls below some threshold. The handover decision has to be made if the mobile is in outage for some time slots. The outage probability and the handover probability is evaluated in taking into account the effect of path loss, shadowing, Rayleigh fast fading, frequency factor reuse and conventional beamforming. The main assumption is that the Rayleigh fast fading changes each time slot while other network components remain static during the period of study.
@cite_31: This paper analyzes the carrier-to-interference ratio of the so-called shotgun cellular system (SCS). In the SCS, base-stations are placed randomly according to a two-dimensional Poisson point process. Such a system can model a dense cellular or wireless data network deployment, where the base station locations end up being close to random due to constraints other than optimal coverage. The SCS is a simple cellular system where we can introduce several variations and design scenarios such as shadow fading, power control features, and multiple channel reuse groups, and assess their impact on the performance. We first derive an analytical expression for the characteristic function of the inverse of the carrier-to-interference ratio. Using this result, we show that the carrier-to-interference ratio is independent of the base station density and further, we derive a semi-analytical expression for the tail-probability. These results enable a complete characterization of the cellular performance of the SCS. Next, we incorporate shadow fading into the SCS and demonstrate that it merely scales the base station density by a constant. Hence, the cellular performance of the SCS is independent of shadow fading. These results are further used to analyze dense cellular scenarios.
@cite_10: New results about Poisson-Dirichlet point processes and Derrida-Ruelle cascades allow us to express Guerra's interpolation entirely in the language of Derrida-Ruelle cascades and to streamline Guerra's computations. Moreover, our approach clarifies the nature of the error terms along the interpolation.
@cite_17: The interference factor, defined for a given location in the network as the ratio of the sum of the path-gains form interfering base-stations (BS) to the path-gain from the serving BS is an important ingredient in the analysis of wireless cellular networks. It depends on the geometric placement of the BS in the network and the propagation gains between these stations and the given location. In this paper we study the mean interference factor taking into account the impact of these two elements. Regarding the geometry, we consider both the perfect hexagonal grid of BS and completely random Poisson pattern of BS. Regarding the signal propagation model, we consider not only a deterministic, signal-power-loss function that depends only on the distance between a transmitter and a receiver, and is mainly characterized by the so called path-loss exponent, but also random shadowing that characterizes in a statistical manner the way various obstacles on a given path modify this deterministic function. We present a detailed analysis of the impact of the path loss exponent, variance of the shadowing and the size of the network on the mean interference factor in the case of hexagonal and Poisson network architectures. We observe, as commonly expected, that small and moderate shadowing has a negative impact on regular networks as it increases the mean interference factor. However, as pointed out in the seminal paper [16], this impact can be largely reduced if the serving BS is chosen as the one which offers the smallest path-loss. Revisiting the model studied in this latter paper, we obtain a perhaps more surprising result saying that in large irregular (Poisson) networks the shadowing does not impact at all the interference factor, whose mean can be evaluated explicitly in a simple expression depending only on the path-loss exponent. Moreover, in small and moderate size networks, a very strong variability of the shadowing can be even beneficial in both hexagonal and Poisson networks.
This convenient property has been observed independently in physics models @cite_11 @cite_10 and network models @cite_31 @cite_17 @cite_5 ; see @cite_18 and references therein for further details. For example, compare [Corollary 10] equivalence2013 and [Lemma 2] panchenko2007guerra , which effectively both give the same equivalence result for, in our setting, the marked propagation process of the typical user. In the physics context, these invariance results imply Bolthausen-Sznitman invariance property (cf [Eq. (2.26)] panchenko2013sherrington ) for the Poisson-Dirichlet process, which is used to study the Sherrington-Kirkpatrick model for spin glasses (types of disordered magnets). This latter process It should not be confused with the better known Poisson-Dirichlet process of Kingman @cite_22 . In fact both processes are special cases of two-parameter Poisson-Dirichlet process extensively studied in @cite_2 . is exactly the so-called signal-to-total-interference ratio (STIR) process, which in turn is trivially related to the SIR process in (interference-limited) Poisson networks @cite_18 @cite_24 .