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Using Poisson processes to model lattice cellular networks Bartłomiej Błaszczyszyn∗, Mohamed Kadhem Karrayy and Holger Paul Keeler∗ ∗INRIA-ENS, 23 Avenue d’Italie, 75214 Paris, France Email: [email protected], [email protected] yOrange Labs; 38/40 rue Gen´ eral´ Leclerc, 92794 Issy-les-Moulineaux, France Email: [email protected] Abstract—An almost ubiquitous assumption made in the and/or fading variables (that model the propagation losses stochastic-analytic approach to study of the quality of user-service between these stations and a typical user), to a point process of in cellular networks is Poisson distribution of base stations, often propagation losses on +, which are experienced by this user. completed by some specific assumption regarding the distribution R of the fading (e.g. Rayleigh). The former (Poisson) assumption is It allows one to study all typical-user characteristics, which usually (vaguely) justified in the context of cellular networks, by can be expressed in terms of its propagation losses (or received various irregularities in the real placement of base stations, which powers, e.g. SINR, etc). We observe that this new process also ideally should form a lattice (e.g. hexagonal) pattern. In the first forms a Poisson process, regardless of the shadowing and/or part of this paper we provide a different and rigorous argument fading distribution, and is characterized only by the moment justifying the Poisson assumption under sufficiently strong log- normal shadowing observed in the network, in the evaluation of a of this distribution of order 2/β, where β is the distance-loss natural class of the typical-user service-characteristics (including exponent. path-loss, interference, signal-to-interference ratio, spectral effi- In the context of an actual deployment of cellular networks, ciency). Namely, we present a Poisson-convergence result for a lattice (e.g. hexagonal) models for the base station placement broad range of stationary (including lattice) networks subject to are usually thought of as more pertinent. However, perfect log-normal shadowing of increasing variance. We show also for the Poisson model that the distribution of all these typical-user lattice models do not seem to allow analytic techniques for the service characteristics does not depend on the particular form study of the SINR-based characteristics. Hence, the Poisson of the additional fading distribution. Our approach involves a model is used and justified by positioning “irregularities” of mapping of 2D network model to 1D image of it “perceived” by the network. It is also considered as a “worst-case” scenario the typical user. For this image we prove our Poisson convergence due to its complete randomness property. Although the va- result and the invariance of the Poisson limit with respect to the distribution of the additional shadowing or fading. Moreover, in lidity of these arguments alone may be questioned, we seek the second part of the paper we present some new results for to support the Poisson assumption with a powerfully new Poisson model allowing one to calculate the distribution function convergence result when there is sufficiently strong log-normal of the SINR in its whole domain. We use them to study and shadowing in the network. Namely, provided a network is optimize the mean energy efficiency in cellular networks. represented by a sufficiently large homogeneous point pattern Index Terms—Wireless cellular networks, Poisson, Hexagonal, convergence, shadowing, fading, spectral/energy efficiency, opti- (whose definition will later be made precise, and includes mization general lattices and various “perturbed lattice” models), we show that as the variance of log-normal shadowing increases, the resulting propagation losses between the stations and the I. INTRODUCTION typical user form a stochastic process that converges to the Cellular networks are being extensively deployed and up- aforementioned non-homogeneous Poisson process on R, due graded in order to cope with the steady rise of user-traffic. This to the Poisson model. In other words, the actual (large but not has created the need for new and robust analytic techniques necessarily Poisson) network is perceived by a typical user as arXiv:1207.7208v2 [math.PR] 18 Jan 2013 to study the quality of user-service. The ability to tractably an equivalent (infinite) Poisson network, provided shadowing model and calculate the quality of user-service related to the is strong enough, of logarithmic standard deviation greater signal-to-interference-and-noise ratio (SINR, or SIR when the than approximately 10dB, as shown by numerical evidence. noise is neglected) will serve as the motivating force behind This is a realistic assumption for outdoor and indoor wireless the work presented here. communications in many urban scenarios. In order to derive analytic techniques, various mathematical This result rigorously justifies using a Poisson point pro- models have been proposed. A common and simplifying model cess to model typical-user characteristics in a wide range of assumption is that the base stations are located according to a network models, which includes the hexagonal model and a Poisson process in the plane. In the first section of this paper large class of perturbed lattice models, thus adding theoretical we recall such a model with shadowing and/or fading, and weight to the work being done under the Poisson assumption. present a simple yet very useful result, upon which the bulk of After stating this convergence result, we derive important the remaining work here hinges. This result involves mapping complementary analytic tools for the study of the distribution the point process on R2 (that models the locations of base of the SINR in the Poisson model, which remain valid for any stations) through the distance-loss function and the shadowing distribution of the shadowing and/or fading random variables. We use them to investigate the spectral and energy efficiency write also SXi = Si to simplify the notation. Let S be a in cellular networks. In particular, we evaluate the mean random variable having the same distribution as any Sx. energy efficiency as a function of the base station transmit Note that the power received at the origin from the station power, which allows one to optimally tune this latter power. Xi, transmitting with power PXi , is equal to Poisson and hexagonal networks with and without shadowing are compared in this context. PXi PXi SXi pXi = = : Related work: It is beyond the scope of this short intro- LXi l (Xi) duction to review all works with Poisson model of wireless In this paper we do not assume any power control, i.e., networks. Some results and further references may be found P = P for some given positive constant P > 0. In this in [1,2]. More specifically, our results presented in this paper Xi case (as well in a more general case of iid emitted powers), allow to extend to a general fading distribution the explicit including P in the associated shadowing random variable, expressions for the distribution function of the SINR derived Xi we retrieve an equivalent model in which the shadowing in [3], where Rayleigh fading is assumed. Our representation is S~ := P S and the transmitted powers P~ = 1. of the network via the process of propagation losses is similar Xi Xi Xi Xi Henceforth we assume, without loss of generality, that the tho this considered in [4] for other purposes, mostly to study transmitted powers are all equal to one, while keeping in mind the effect of shadowing/fading on connectivity. Moreover, that the shadowing random variables now include the effective using the Laplace/Fourier analysis, we manage to characterize transmitted powers. This transformation will slightly simplify (no longer explicitly) the SINR over its entire range, whereas our notation. However, when studying the energy efficiency the aforementioned explicit expressions are valid only for in Section IV-B4 we will reintroduce emitted powers to our SINR ≥ 1. The result of Lemma1 appeared in [5]. The infinite model. Poisson model was statistically fitted in [6] to some real data provided by an operator regarding propagation losses in order to estimate the parameters of the propagation loss using a sim- A. Mapping of the propagation losses of the typical user from ple linear regression model. The spectral and energy efficiency R2 to R+ in hexagonal networks without shadowing was studied in [7]. Denote by N = fLX g the process of propagation losses Finally, the convergence result presented here is in the spirit of i i2N experienced by the typical user with respect to the stations classical limit theorems of point processes, which are detailed + in [8, Chapter 11]. In particular, these theorems show that in Φ. We consider N a point process on R . Note that all under specific conditions, the repeated superposition, thinning characteristics of the typical user, which can be expressed or translation of points of a point process will result in the in terms of its propagation losses (or received powers, under point process converging in the limit to a Poisson process. the aforementioned assumption on emitted powers, e.g. SIR, The remaining part of this paper is organized as follows. In SINR, spectral and energy efficiency, etc) are determined by sectionII we present the basic result on the network mapping the distribution of N . This motivates the following simple to the process of propagation losses. The main convergence result that appeared, to the best of our knowledge for the first result is presented in Section III and its proof is given time, in [5]. In order to make this presentation more self- AppendixA. We revisit the Poisson model in order to study contained we present it with a proof. the SINR distribution in SectionIV. Lemma 1: Assume infinite Poisson model with distance- loss (1) and generic shadowing (and/or fading) variable II.
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