Using Poisson processes to model lattice cellular networks

Bartłomiej Błaszczyszyn∗, Mohamed Kadhem Karray† and Holger Paul Keeler∗ ∗INRIA-ENS, 23 Avenue d’Italie, 75214 Paris, France Email: [email protected], [email protected] †Orange 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 = {LX } the process of propagation losses Finally, the convergence result presented here is in the spirit of i i∈N 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.INFINITE POISSONMODEL satisfying For motivation purposes, we first recall the usual cellular 2 E[S β ] < ∞ . (2) network model based on the Poisson process. In particular, we model the geographic locations of the base stations with an Then the process of propagation losses N experienced by the homogeneous Poisson point process Φ = {Xi} of intensity i∈N typical user is a non-homogeneous Poisson point process on λ on 2, which we refer to as the infinite Poisson model. R + with intensity measure We assume that the (typical) user is located at the origin R without loss of generality due to the stationarity of {Xi} . 2 i∈N Λ ([0, t)) := E [N ([0, t))] = at β (3) Let l (Xi) be the distance loss between a base station at Xi and the user, where l (·) is given by where β 2 l (x) = (K |x|) (1) λπE[S β ] a := (4) for two given positive constants K and β. When we incorpo- K2 rate shadowing (and/or fading), the propagation loss is Proof: The point process N may be viewed as a trans- l (Xi) LXi = formation of the point process Φ by the probability kernel SXi   where {Sx} 2 is a collection of independent and identically l (x) x∈R p(x, A) = P ∈ A , x ∈ 2,A ∈ B +
distributed (iid) positive random variables. We will sometimes S R R By the displacement theorem [1, Theorem 1.10], the point Consider the distance-loss model (1) with the constant K process N is Poisson on R+ with intensity measure replaced by the function of σ  2  Z l (x)  σ (β − 2) Λ ([0, t)) = λ P ∈ [0, t) dx K(σ) = K exp − 2 , (7) 2 S 2β R Z l (x)  where K > 0 and β > 2. + = λ 1 < t dxPS (ds) As in SectionII, we consider the point process on R of 2 + s R ×R propagation losses experienced by the typical user with respect h 2 i 2 λπE S β Z π (st) β 2 to the stations in φ β = λ 2 PS (ds) = 2 t ( β β ) + K K K(σ) |X | R N (σ) := i : X ∈ φ . (σ) i (8) which completes the proof. Si Remark 2: Note that the distribution of N is invariant with We consider also the analogous process of propagation losses respect to the distribution of the shadowing/fading S having ( ) K(σ)β|X |β 2/β N¯ (σ) := i : a < |X | < b ,X ∈ φ , same given value of the moment E[S ]. This means that (σ) σ i σ i (9) the infinite Poisson network with an arbitrary shadowing S is Si perceived at a given location statistically in the same manner where the stations in φ that are closer than aσ and farther than as an “equivalent” infinite Poisson with “constant shadowing” bσ are ignored, for all sequences 0 ≤ aσ < bσ ≤ ∞ satisfying 2/β β/2 equal to s = (E[S ]) (to have the same moment log(aσ) const → 0, (10) of order 2/β). The model with such a “constant shadowing” σ2 boils down to the model without shadowing (S ≡ 1) and the log(bσ) p → ∞. (11) constant K replaced by K˜ = K/ E[S2/β]. σ2 The above result requires condition (2), which is satisfied B. Main result by the usual models, as e.g. the log-normal shadowing or We present now our main convergence result. Rayleigh fading, and we tacitly assume it throughout the paper. Theorem 3: Given homogeneity condition (5), then N (σ) converges weakly as σ → ∞ to the Poisson point process III.CONVERGENCERESULTSUNDERLOG-NORMAL on R+ with the intensity measure Λ given by (3) with a = SHADOWING λπ/K2. Moreover, N¯ (σ) also converges weakly (σ → ∞) In this section we derive a powerful convergence result to the Poisson point process with the same intensity measure, rigorously showing that the infinite Poisson model can be used provided conditions (10) and (11) are satisfied. to analyse the characteristics of the typical user in the context The proof of Theorem3 is deferred to AppendixA. of any fixed (deterministic!) placement of base stations, meet- Remark 4: The above result, in conjunction with Lemma1, ing some empirical homogeneity condition, provided there is says that infinite Poisson model can be used to approximate the sufficiently strong log-normal shadowing. characteristics of the typical user for a very general class of homogeneous pattens of base stations, including the standard A. Model description hexagonal one. The second statement of this result says that this approximation remains valid for sufficiently large but finite Let φ = {Xi}i∈N be a locally finite deterministic point 2 patterns. pattern on R and B0(r) the ball of radius r, centered at the origin. For 0 < λ < ∞, as r → ∞ we require the empirical Remark 5: The distance-loss model (1) suffers from having homogeneity condition a singularity at the origin. This issue is often circumvented by some appropriate modification of the distance-loss function φ(B (r)) 0 → λ. (5) within a certain distance from the origin. The second statement πr2 of Theorem3 with aσ = const > 0 shows that such a Note that the above condition is satisfied by any lattice (e.g. modification is not significant in the Poisson approximation. hexagonal) pattern φ, as well as by almost any realization of To illustrate Theorem3 and obtain some insight into the an arbitrary ergodic point process. speed of convergence we used Kolmogorov-Smirnov (K-S) (σ) Let the shadowing Si between the station Xi and the test, cf [9], to compare the cumulative distribution functions origin be iid (across i) log-normal random variables (CDF) of the SIR of the typical user (see Section IV-A2) in S(σ) = exp(−σ2/2 + σZ ), (6) the infinite Poisson model versus hexagonal one consisting of i i 30 × 30 = 900 stations on a torus. We found that for 9/10 where Zi are standard normal random variables. Note that for realizations of the network shadowing the K-S test does not (σ) (σ)  (σ) 2/β such Si = S , we have E[S] = 1 and E (S ) = allow to distinguish the empirical (obtained from simulations) exp[σ2(2 − β)/β2]. 1 CDF of the SIR from the CDF of SIR evaluated in the infinite Poisson model with the critical p-value fixed to α = 10% 1 This also means that the path-loss from a given station X expressed in dB, provided σdB 10dB. On Figure1 we present a few examples i.e., dB(L (y)), where dB (x) := 10×log (x) dB, is a Gaussian random & X 10 of these CDF’s. Similar numerical study was done for the CDF variable with standard deviation σdB = σ10/ log 10, called logarithmic standard deviation of the shadowing. of the path-loss with respect to the serving station in [6]. 1 probabilities of Poisson process P{L ≥ t} = P{N ([0, t)) = 0} = exp[−Λ([0, t))]. 0.8 Corollary 6: The CDF of L is equal to P{ L ≤ t } = 1 − exp[−at2/β] where a is given by (4). Consequently, the 0.6 probability density of L is given by

CDF 2 2a 2 β β −1 −at 0.4 P (ds) = t e dt . (13) L β This corresponds to a Frechet´ distribution with shape param- 0.2 β=3.0, σdB=8, Hexagonal β Poisson 2 2 β=4.0, σdB=9.5, Hexagonal eter and scale parameter a . Poisson β β=5.0, σdB=10.5, Hexagonal Poisson 2) Interference factor and SIR: The interference factor is 0 -10 -5 0 5 10 15 20 25 30 35 40 45 SIR [dB] defined by X 1 Fig. 1: Empirical CDF of SIR simulated in hexagonal network with shadowing and their f = L − 1 Poisson approximations. LXi i∈N where L is the path-loss factor. It may be interpreted as the IV. POISSONMODELREVISITED interference to signal ratio; that is the inverse of the SIR. P 1 We now return to the model outlined in SectionII and Introducing I = i∈ (can be interpreted as the total N LXi investigate it further. We show that Lemma1 is very useful received power) we can write f = LI − 1. Also, f = LI0 in studying random quantities of the network, independently where 1 X 1 1 of the shadowing/fading distribution. We do not pretend to I0 := I − = − develop a complete theory, which is well beyond the scope L LXi L i∈N of this paper. As a general remark let us emphasize only that may be viewed as the interference. many, already known, results (cf Related work in the Introduc- Define tion), were originally derived under specific assumptions on   −z 2 2 the distribution of shadowing and/or fading. By Lemma1 they ϕβ (z) := e + z β γ 1 − , z (14) necessarily remain valid for a general distribution, with the β appropriate specification of the value of the moment E[S2/β]. R z α−1 −t where γ(α, z) = 0 t e dt is the lower incomplete Our specific goal in this section is to present a new gamma function. Another representation of the function ϕβ(z), representation of distribution function of the SINR, which in used in evaluation of the Laplace transform of the f (cf proof consequence will allow us to study the spectral and energy of Corollary8) is given in AppendixB. efficiency — two important engineering characteristics of the Here is a key technical result for our approach. network. To the best of our knowledge, they have not yet Proposition 7: The Laplace transform of the interference been studied in the Poisson model. Moreover, as another factor conditional to the path-loss factor is equal to illustration of our convergence result, we will compare these 2  −zf  −a[ϕ (z)−1]s β characteristics in Poisson and hexagonal network, with this E e |L = s = e β . (15) latter studied by simulations. Proof: We have Unless otherwise specified all results of this section regard 0 E e−zf |L = s = E[e−zsI |L = s] . (16) Poisson model of SectionII. Observe that I0 is a shot-noise associated to the point process A. Path-loss and SIR obtained from the Poisson point process N of the propagation The novelty of our approach consists in representing the losses by suppressing its smallest (nearest to the origin) point, L SINR in terms of the path-loss to the serving station and the which is located precisely at . By the well-known property L = s respective SIR, rather than, as usual, the interference. of the Poisson process, given , this new process is also Poisson on (s, ∞) with intensity measure 1) Path loss: The weakest propagation loss denoted by M ((s, t)) = Λ ([0, t)) − Λ ([0, s]) , t ∈ (s, +∞) . L = inf LXi (12) i∈N Thus is often called path-loss factor. It may be interpreted as 2a 2 −1 M (dt) = Λ (dt) = t β dt, on (s, +∞) (17) the propagation loss with the serving base station (that is, β the one with strongest received power). Note by (3) that and by [1, Proposition 1.5]

the number of points of the process N = {LXi }i∈ is Z ∞  N h −zsI0 i − zs almost surely finite in any finite interval. Hence, the above E e |L = s = exp (e t − 1)M (dt) infimum is almost surely achieved for some base station; that is s  Z ∞  2a − zs 2 −1 L = mini∈N LXi . Moreover, Lemma1 allows us to conclude = exp − (1 − e t )t β dt . the following characterisation of the distribution of L, which β s follows immediately from the well known expression for void (18) Now we calculate the integral on the right-hand side of the Remark 9: For t ≥ 1 the (complementary) CDF of the SIR zs above equation by making the change of variable u := t , admits the following explicit expression that is t−2/β Z ∞ − zs 2 −1 P{ SIR ≥ t } = P{f ≤ 1/t} = 0 (20) (1 − e t )t β dt C (β) s Z 0 2 −1 where −u zs β −zs = (1 − e ) 2 du 2π 2Γ(2/β)Γ(1 − 2/β) z u u C0(β) = = Z z β sin(2π/β) β 2 −u − 2 −1 = (zs) β (1 − e )u β du ∞ −z t−1 0 R and Γ(z) = 0 e z dt it the complete Gamma function. 2 β It was proved in [3] assuming exponential distribution of S. = s β [−1 + ϕβ (z)] 2 But in the infinite Poisson model, SIR is invariant with respect where the third equality is obtained by integration by parts. to λ, and K and consequently by Lemma1 it is also invariant Combining the above equation with Equations (16), (18) gives with respect to the value of E[S2/β]. Hence the result remains the final result. valid for arbitrary distribution of S, provided E[S2/β] < ∞. The following results can be derived from Proposition7, Other results of [3], including these involving the superposition although we will not use them in the remaining part of the of independent Poisson models (called K-tier cellular network paper. model) can also be appropriately generalized using Lemma1. Corollary 8: The joint distribution of L and f is charac- terized by B. Distribution of SINR, spectral and energetic efficiency

2  −zf  1 −au β ϕ (z) + 1) SINR: In this section we are primarily interested in the E 1 {L ≥ u} e = e β , z ∈ R . ϕβ (z) signal to interference and noise ratio The unconditional Laplace transform of the interference factor 1 1 SINR = L = , (21) is P 1 1  NL + f N + i∈ L − L  −zf  1 + N Xi E e = , z ∈ R . (19) ϕβ (z) where N is the noise power. We will show how CDF of the Proof: For the first statement we have SINR can be evaluated, which opens a way for the study of Z ∞ functionals of SINR.  −zf   −zf  E 1 {L ≥ u} e = E e |L = s PL (ds) 2) Spectral efficiency: An important characteristic of a u wireless cellular network is its spectral efficiency, defined in Z ∞ 2 2 −a[ϕ (z)−1]s β 2a 2 −1 −as β = e β s β e ds the simplest case of additive white Gaussian noise (AWGN) u β and the optimal theoretical link performance as 2 1 −au β ϕ (z) = e β S := log (1 + SINR) . ϕβ (z) where for the second equality we use (13) and (15). This It tells us how many bits per second and per Hertz can be sent completes the proof of the first statement. For the second to the typical user of the network. statement, conditioning on the value of the path-loss factor 3) Energy efficiency: Up to now we have considered a unit yields transmitted power. Assume now that base stations transmit Z some power P ≥ 0. In fact, in order for a base station to  −zf   −zf  E e = E e |L = s PL (ds) be able to transmit this power to the mobile, it needs to be + R powered (i.e., consumes energy per second) at the level P 0 > where PL (·) is the distribution of L given by (13). Using the P . Following [10], assume that these two quantities are related above equation and (15), we ascertain through a simple linear relation P 0 = cP +d for some positive Z ∞ 2 2 constants c and d. The energy efficiency is defined by −zf −a[ϕ (z)−1]s β 2a 2 −1 −as β E[e ] = e β s β e ds β   0 W log 1 + 1 Z ∞ 2 NL/P +f 2a 2 β −1 −aϕβ (z)s E := E(P ) = . = s β e ds cP + d β 0 Z ∞ 2 1 2 2 β where W is the bandwidth expressed in Hz. Thus E is equal to −1 −aϕβ (z)s = aϕβ(z) s β e ds W log(1 + SINR(P ))/P 0, where SINR(P ) takes into account ϕβ(z) β 0 0 1 the transmitted power, and P is the consumed power. It tells = us how many bits per second per Watt of consumed power can ϕβ(z) be sent by a base station to the typical user. Note that E(0) = where the third equality stems from ϕβ(z) 6= 0 which follows E(∞) = 0 and thus E(P ) admits a non-trivial optimization from (40) in AppendixB and the assumption β > 2. in P . 4) Evaluation of the CDF of the SINR: Proposition7 in 1 conjunction with Corollary6 completely characterizes the joint 0.8 distribution of L and f. It does not, however, allow for an explicit expression for the CDF of the SINR in the whole 0.6 domain (cf Remark9). In this section we describe a practical 0.4 way for numerical computation of this CDF. We will use it CDF to study the spectral and energy efficiency. In fact, we will 0.2 compute the CDF of the random variable Y := NL + f, 0 which is sufficient, in view of (21). Hexagonal, σ =0dB (simulation) σ dB Hexagonal dB=12dB (simulation) Poisson (simulation) Proposition 10: The cumulative distribution function of Y -0.2 Infinite Poisson (Laplace inversion) Infinite Poisson (Explicit, valid for SINR[dB]>0) is given by -10 -5 0 5 10 15 20 25 30 35 40 Z ∞ SINR [dB] P (Y < x) = Fs(x − Ns)PL (ds) (22) Fig. 2: CDF of the SINR for (finite) hexagonal network with and without (σdB = 0dB) 0 shadowing, finite Poisson as well as for the infinite Poisson model. The last in the legend curve corresponds to (20). where PL (ds) is given by (13), and Fs (y) := 120000 Hexagonal, σ =0dB σ dB P (f < y|L = s) may be expressed by Hexagonal, dB=12dB Infinite Poisson γt ∞ 2e Z 100000 F (y) = 1 − R L (γ + iu)
cos ut du , (23) s π Fs 0 80000 where γ > 0 is an arbitrary constant, R is the real part if it complex argument and 60000 2 1 1 β −a[ϕβ (z)−1]s 40000 L (z) = − e (24) Energy efficiency [bits/s/W] Fs z z with ϕβ (·) is given by (14). 20000 Proof: The expression (22) is trivial. The expression (23) 0 of the (conditional) CDF Fs is based on the Bromwich contour 0 10 20 30 40 50 60 70 80 90 Transmitted power [dBm] inversion integral of the Laplace transform LFs of 1−Fs. The Fig. 3: Mean energy efficiency of finite hexagonal model with and without (σdB = 0dB) expression (24) for this latter follows from (15). shadowing as well as the infinite Poisson in function of the transmitted power. Remark 11: As shown in [11], the integral in (23) can be numerically evaluated using the trapezoidal rule, with the for finite hexagonal network on the torus with and without γ parameter allowing control of the approximation error. The shadowing as well as the finite Poisson network on the same F function s may be also retrieved from (15) using other torus. For the infinite Poisson model we use our method based inversion techniques. on the inversion of the Laplace transform. For comparison, we Remark 12: For t ≥ 1 the (complementary) CDF of the plot also the curve corresponding the explicit expression (20), SINR admits the following expression which is valid only for SINR > 1. −2/β Z ∞ 2t 2 −β/2 β Figure3 shows the expected energy efficiency E(E) of the P{SINR ≥ t} = re−r Γ(1−2/β)−Na r dr 2 finite hexagonal model with and without shadowing as well Γ(1 + β ) 0 (25) as the infinite Poisson, assuming the affine relation between with a given by (4). It follows from [3, Theorem 1] (where the consumed and emitted power with constants c = 21, 45 and exponential distribution is assumed for S) and our Lemma1. d = 354.44W. Note on Figure2 that the expression is not valid for t < 1. On both figures we see that the infinite Poisson model gives 5) Numerical examples: In what follows we assume the a reasonable approximation of the (finite) hexagonal network following parameter values. If not otherwise specified we take provided the shadowing is high enough. In particular, the value log-normal shadowing with logarithmic standard deviation of the transmitted power at which the hexagonal network with σdB = 12dB (cf the footnote on page3). The parameters the shadowing attains the maximal expected energy efficiency of the distance-loss model are K = 4250Km−1, β = 3.52 is very well predicted by the Poisson model. (which corresponds to the COST-Hata model for urban en- vironment with the BS height 30m, mobile height 1.5m, CONCLUSIONS carrier frequency 805MHz and penetration loss 19dB). For the We present a powerful mathematical convergence result hexagonal network model simulations, we consider 900 base rigorously justifying the fact that any actual (including regular stations (30x30) on a torus, with the cell radius R = 0.26Km hexagonal) network is perceived by a typical user as an (i.e., the surface of the hexagonal cell is equal to this of the equivalent (infinite) Poisson network, provided log-normal disk of radius R). System bandwidth is W = 10MHz, noise shadowing variance is sufficiently high. Good approximations power N = −93dBm. are obtained for logarithmic standard deviation of the shad- Figure2 shows the CDF of the SINR assuming the base owing greater than approximately 10dB, which is a realistic station power P = 58.5dBm. We obtain results by simulations assumption in many urban scenarios. Moreover, the equivalent infinite Poisson representation is invariant with respect to Noting that G(−t) = 1 − G(t) := G¯(t), apply integration by an additional fading distribution. Using this representation parts on the above integral we study the distribution of the SINR of the typical user. √ √ Z ∞ −2t n Z ∞ 2t n In particular, we evaluate and optimize the mean energy exp G(t)dt = exp G¯(t)dt efficiency as a function of the base station transmit power. −∞ β −∞ β  √  ∞ β 2t n ¯ APPENDIX = √ exp G(t) 2 n β √ −∞ A. Proof of Theorem3 β Z ∞ 2t n √ n := σ2 + exp g(t)dt, In order to simplify the notation we set . Moreover, 2 n −∞ β without loss of generality we assume that n takes positive integer values. Also, it is more convenient to study the point where g(t) is the normal probability density, and, via the process of propagation losses on the logarithmic scale, which inequality we do in what follows. In this regard denote by Λlog the image ∞ −y2 Z 2 e of the measure Λ given by (3) with a = λπ/K2 through the e−x dx < , y ≥ 0 , (27) logarithmic mapping y (1 + y) Z π 2s (cf [12, Section 7.8]) the final integrated term on the left β Λlog((−∞, s]) := 1(log(t) ≤ s) Λ(dt) = 2 e R+ K vanishes. Hence √ for s ∈ R. Z ∞ −2t n For a given i, we first observe by (6) that exp G(t)dt −∞ β " ! # √ K(n)β|X |β β Z ∞  t2 2t n dt ν (s, |X |) := P log i ≤ s = √ exp − + √ n i (n) 2 n −∞ 2 β 2π Si " √ 2#  s − β log(K|X |) − n/β  β 2n Z ∞ 1  2 n dt i √ √ = P Zi ≤ √ = exp 2 exp − t − n 2 n β −∞ 2 β 2π s − β log(K|X |) − n/β  (28) = G √ i , (26) n β 2n = √ exp , where G is the CDF of the standard Gaussian random variable. 2 n β2 Let B (r) = {x ∈ 2 : |x| < r}. We now need to derive 0 R which gives two results. ∞   Lemma 13: Z s − β log(Kr) − n/β π 2s √ β Z Z 2π rG dr = 2 e . 0 n K lim νn(s, |x|)dx = lim νn(s, |x|)dx n→∞ n→∞ 2 B0(bn)\B0(an) R We proceed similarly with the other integral in Lemma 13 = Λ ((−∞, s]) log Z provided that an and bn satisfy (10) and (11). νn(s, |x|)dx B (b )\B (a ) Proof: We start with the integral over R2 in polar form 0 n 0 n Z bn s − β log(Kr) − n/β  Z Z ∞ s − β log(Kr) − n/β  = 2π rG √ dr. ν (s, |x|)dx = 2π rG √ dr n an n 2 n R 0 s−β log(Kr)−n/β √ The change of variables t = n gives Introduce a change of variables Z bn s − β log(Kr) − n/β  s − β log(Kr) − n/β β dr 2π rG √ dr t = √ , dt = −√ , n n n r an √ √ Z un 1  2 √  n 1  1 √  nr = 2π exp s − t n − n/β
G(t) dt r = exp s − t n − n/β
, dr = − dt, 2 vn K β β K β β h i √ 2 u √ hence n exp β (s − n/β) Z n −2t n = 2π exp G(t)dt Z ∞ s − β log(Kr) − n/β  β K2 β 2π rG √ dr vn 0 n Z ∞   √ where 1 2 √
n = 2π 2 exp s − t n − n/β G(t) dt s − β log(Kan) − n/β −∞ K β β un = √ , h i n √ exp 2 (s − n/β) Z ∞  √  n β −2t n s − β log(Kbn) − n/β = 2π 2 exp G(t)dt vn = √ . β K −∞ β n Moreover limit of n → ∞, the first summation in (31) disappears X √ νn(s, |Xi|) Z un   −2t n X ∈φ∩B (r ) exp G(t)dt i 0 k0 β   vn √ X s − β log(K|Xi|) − n/β −vn   = P Z ≤ √ Z 2t n n = exp G¯(t)dt X ∈φ∩B (r ) β i 0 k0 −un   √ −v √ X s − β log(K|Xi|) − n/β   n Z −vn   β 2t n β 2t n = G √ = √ exp G¯(t) + √ exp g(t)dt n Xi∈φ∩B0(rk0 ) 2 n β −u 2 n −un β √ n     −vn s − β log(K|X∗|) − n/β β 2t n ≤ φ(B (r ))G √ = √ exp G¯(t) 0 k0 2 n β n −un " √ 2# → 0 (n → ∞). β 2n Z −vn 1  2 n dt + √ exp exp − t − √ . h s−β log(K|X|)−n/β i 2 where X∗ gives the maximum of G √ over 2 n β −un 2 β 2π n

X ∈ φ∩B0(rk0 ) which exists since φ is (by our assumption) a locally finite point measure. For the second summation in (31) We now derive the conditions of an and bn which ensure |Xi| we write νn(s, |Xi|) = νn(s, |x| ), hence that the above integral converges and the integrated term |x| disappears. The latter can be achieved, given inequality (27), νn(s, |Xi|) Z in the limit as −un and√ −vn both approach infinity,√ or equiv- 1 |Xi| β log(Kb ) n β log(Ka ) n = ν (s, |x| )dx. alently √ n + → ∞ and √ n + → ∞ n n β n β |Ak| Ak |x| as n → ∞, which agrees with conditions (10) and (11). Then the bounds Further conditions are revealed after the change of variable √ rk |Xi| rk+1 w = t − 2 n/β, yielding the integral e− = ≤ ≤ = e, rk+1 |x| rk  √ and form of νn, which implies νn(s, |x|e ) = νn(s − β, |x|), √ 2 n Z −vn  2 n 2 Z −vn− β 2 − 1 t− dt w dw lead to the lower bound 2 β − 2 e √ = √ e √ ∞ ∞ 2π 2 n 2π φ(A ) Z −un −un− β X X X k νn(s, |Xi|) ≥ νn(s−β, |x|)dx, |Ak| Ak k=k0 Xi∈φ∩Ak k=k0 whose limits of integration imply, in light of the earlier integral (32) β log(Kbn) and the upper bound (28), that as n → ∞ the following n > 1/β and β log(Kan) ∞ ∞ Z < 1/β are required, of which both conditions (10) X X X φ(Ak) n ν (s, |X |) ≤ ν (s+β, |x|)dx. and (11) satisfy. n i n |Ak| Ak k=k0 Xi∈φ∩Ak k=k0 (33) Lemma 14: Assume (5), (10) and (11), then Moreover, we write φ(A ) φ(B (r )) − φ(B (r )) k = 0 k+1 0 k X |Ak| |B0(rk+1)| − |B0(rk)| lim νn(s, |Xi|) (29) n→∞ φ(B0(rk+1)) φ(B0(rk)) |B0(rk)| X ∈φ∩(B (b )\B (a )) − i 0 n 0 n = |B0(rk+1)| |B0(rk)| |B0(rk+1)| X |B0(rk)| = lim νn(s, |Xi|) = Λlog((−∞, s]). (30) 1 − n→∞ |B0(rk+1)| X ∈φ i φ(B0(rk+1)) − φ(B0(rk)) e−2 = |B0(rk+1)| |B0(rk)| , 1 − e−2 k Proof: For k ≥ 0 and a fixed  > 0, let rk = e and φ(Ak) and requirement (5) yields limk→∞ = λ. Hence, for |Ak| Ak = B0(rk+1) \ B0(rk), and write the summation in (30) as any fixed δ > 0, there exists a k0(δ) such that for all k ≥ k0, the bounds X φ(A ) νn(s, |Xi|) (1 − δ)λ ≤ k ≤ (1 + δ)λ, Xi∈φ |Ak| ∞ X X X hold. Lower bound (32) becomes = ν (s, |X |) + ν (s, |X |), n i n i ∞ ∞ Z X ∈φ∩B (r ) k=k X ∈φ∩A X X X i 0 k0 0 i k νn(s, |Xi|) ≥ (1 − δ)λ νn(s − β, |x|)dx, Ak (31) k=k0 Xi∈φ∩Ak k=k0 Z = (1 − δ)λ νn(s − β, |x|)dx k ≥ 0 for some 0 , whose value will be fixed later on. In the |x|≥rk0 Finally, Lemma 13 allows us to set an = rk0 and bn = ∞, REFERENCES hence [1] F. Baccelli and B. Błaszczyszyn, Stochastic Geometry ∞ X X πλ 2s−β and Wireless Networks, Volume I&II, ser. Foundations lim νn(s, |Xi|) ≥ (1 − δ) e β , n→∞ K2 and Trends in Networking. 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Introducing the modified incomplete gamma function γ∗ defined by (38) (which is holomorphic on C × C; see [13, 6.5.4]) we obtain (37). The expansion (39) is given in [13, 6.5.29]. This expansion shows that if α > −1 then for + ∗ any z ∈ R , γ (α, z) > 0, and therefore by (37) ϕβ (z) > 0 for any β > 2, z ∈ R+.