Table Of ContentUsing Poisson processes to model lattice cellular
networks
B. Błaszczyszyn†, M. K. Karray∗ and H.P. Keeler†
Abstract—An almost ubiquitous assumption made in the betweenthesestationsandatypicaluser),toapointprocessof
stochastic-analyticapproachtostudyofthequalityofuser-service propagationlossesonR+,whichareexperiencedbythisuser.
incellularnetworksisPoissondistributionofbasestations,often
It allows one to study all typical-user characteristics, which
completedbysomespecificassumptionregardingthedistribution
canbeexpressedintermsofitspropagationlosses(orreceived
ofthefading(e.g.Rayleigh).Theformer(Poisson)assumptionis
2 usually(vaguely)justifiedinthecontextofcellularnetworks,by powers,e.g.SINR,etc).Weobservethatthisnewprocessalso
1 variousirregularitiesintherealplacementofbasestations,which forms a Poisson process, regardless of the shadowing and/or
0
ideallyshouldformalattice(e.g.hexagonal)pattern.Inthefirst fading distribution, and is characterized only by the moment
2
partofthispaperweprovideadifferentandrigorousargument
of this distribution of order 2/β, where β is the distance-loss
l justifying the Poisson assumption under sufficiently strong log-
u normalshadowingobservedinthenetwork,intheevaluationofa exponent.
J naturalclassofthetypical-userservice-characteristics(including Inthecontextofanactualdeploymentofcellularnetworks,
1 path-loss, interference, signal-to-interference ratio, spectral effi- lattice (e.g. hexagonal) models for the base station placement
3 ciency). Namely, we present a Poisson-convergence result for a are usually thought of as more pertinent. However, perfect
broad range of stationary (including lattice) networks subject to
latticemodelsdonotseemtoallowanalytictechniquesforthe
] log-normal shadowing of increasing variance. We show also for
R study of the SINR-based characteristics. Hence, the Poisson
the Poisson model that the distribution of all these typical-user
P service characteristics does not depend on the particular form model is used and justified by positioning “irregularities” of
. of the additional fading distribution. Our approach involves a the network. It is also considered as a “worst-case” scenario
h
mapping of 2D network model to 1D image of it “perceived” by due to its complete randomness property. Although the va-
t
a thetypicaluser.ForthisimageweproveourPoissonconvergence lidity of these arguments alone may be questioned, we seek
m result and the invariance of the Poisson limit with respect to the
to support the Poisson assumption with a powerfully new
distribution of the additional shadowing or fading. Moreover, in
[ the second part of the paper we present some new results for convergenceresultwhenthereissufficientlystronglog-normal
Poissonmodelallowingonetocalculatethedistributionfunction shadowing in the network. Namely, provided a network is
1
v of the SINR in its whole domain. We use them to study and represented by a sufficiently large homogeneous point pattern
optimize the mean energy efficiency in cellular networks.
8 (whose definition will later be made precise, and includes
Index Terms—Wireless cellular networks, Poisson, Hexagonal,
0 general lattices and various “perturbed lattice” models), we
convergence, shadowing, fading, spectral/energy efficiency, opti-
2
mization show that as the variance of log-normal shadowing increases,
7
the resulting propagation losses between the stations and the
.
7 I. INTRODUCTION typical user form a stochastic process that converges to the
0 aforementioned non-homogeneous Poisson process on R, due
2 Cellular networks are being extensively deployed and up-
tothePoissonmodel.Inotherwords,theactual(largebutnot
1 gradedinordertocopewiththesteadyriseofuser-traffic.This
: necessarily Poisson) network is perceived by a typical user as
v has created the need for new and robust analytic techniques
an equivalent (infinite) Poisson network, provided shadowing
i to study the quality of user-service. The ability to tractably
X is strong enough, of logarithmic standard deviation greater
model and calculate the quality of user-service related to the
r than approximately 10dB, as shown by numerical evidence.
a signal-to-interference-and-noise ratio (SINR) will serve as the
This is a realistic assumption for outdoor and indoor wireless
motivating force behind the work presented here.
communications in many urban scenarios.
Inordertoderiveanalytictechniques,variousmathematical
This result rigorously justifies using a Poisson point pro-
modelshavebeenproposed.Acommonandsimplifyingmodel
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.
presentasimpleyetveryusefulresult,uponwhichthebulkof
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
oftheSINRinthePoissonmodel,whichremainvalidforany
stations)throughthedistance-lossfunctionandtheshadowing
distribution of the shadowing and/or fading random variables.
and/or fading variables (that model the propagation losses
We use them to investigate the spectral and energy efficiency
in cellular networks. In particular, we evaluate the mean
†Inria/Ens,23av.d’Italie75214Paris,France
∗OrangeLabs,38/40rueGe´ne´ralLeclerc,92794Issy-Moulineaux,France energy efficiency as a function of the base station transmit
2
power, which allows one to optimally tune this latter power. In this paper we do not assume any power control, i.e.,
Poisson and hexagonal networks with and without shadowing P = P for some given positive constant P > 0. In this
Xi
are compared in this context. case (as well in a more general case of iid emitted powers),
Related work: It is beyond the scope of this short intro- including P in the associated shadowing random variable,
Xi
duction to review all works with Poisson model of wireless we retrieve an equivalent model in which the shadowing
networks. Some results and further references may be found is S˜ := P S and the transmitted powers P˜ = 1.
Xi Xi Xi Xi
in [1, 2]. More specifically, our results presented in this paper Henceforth we assume, without loss of generality, that the
allow to extend to a general fading distribution the explicit transmittedpowersareallequaltoone,whilekeepinginmind
expressions for the distribution function of the SINR derived thattheshadowingrandomvariablesnowincludetheeffective
in [3], where Rayleigh fading is assumed. Moreover, using transmitted powers. This transformation will slightly simplify
the Laplace/Fourier analysis, we manage to characterize (no our notation. However, when studying the energy efficiency
longer explicitly) the SINR over its entire range, whereas in Section IV-B4 we will reintroduce emitted powers to our
the aforementioned explicit expressions are valid only for model.
SINR≥1.TheresultofLemma1appearedin[4].Theinfinite
A. Mappingofthepropagationlossesofthetypicaluserfrom
Poisson model was statistically fitted in [5] to some real data
R2 to R+
provided by an operator regarding propagation losses in order
toestimatetheparametersofthepropagationlossusingasim- DenotebyN ={L } theprocessofpropagationlosses
Xi i∈N
plelinearregressionmodel.Thespectralandenergyefficiency experienced by the typical user with respect to the stations
in hexagonal networks without shadowing was studied in [6]. in Φ. We consider N a point process on R+. Note that all
Finally,theconvergenceresultpresentedhereisinthespiritof characteristics of the typical user, which can be expressed
classical limit theorems of point processes, which are detailed in terms of its propagation losses (or received powers, under
in [7, Chapter 11]. In particular, these theorems show that the aforementioned assumption on emitted powers, e.g. SIR,
under specific conditions, the repeated superposition, thinning SINR, spectral and energy efficiency, etc) are determined by
or translation of points of a point process will result in the the distribution of N. This motivates the following simple
point process converging in the limit to a Poisson process. result that appeared, to the best of our knowledge for the first
Theremainingpartofthispaperisorganizedasfollows.In time, in [4]. In order to make this presentation more self-
section II we present the basic result on the network mapping contained we present it with a proof.
to the process of propagation losses. The main convergence Lemma 1: Assume infinite Poisson model with distance-
result is presented in Section III and its proof is given loss (1) and generic shadowing (and/or fading) variable
Appendix A. We revisit the Poisson model in order to study satisfying
the SINR distribution in Section IV. E[Sβ2]<∞. (2)
II. INFINITEPOISSONMODEL Then the process of propagation losses N experienced by the
For motivation purposes, we first recall the usual cellular typical user is a non-homogeneous Poisson point process on
networkmodelbasedonthePoissonprocess.Inparticular,we R+ with intensity measure
model the geographic locations of the base stations with an
2
homogeneousPoissonpointprocessΦ={X } ofintensity Λ([0,t)):=E[N ([0,t))]=atβ (3)
i i∈N
λ on R2, which we refer to as the infinite Poisson model.
where
We assume that the (typical) user is located at the origin 2
λπE[Sβ]
without loss of generality due to the stationarity of {X } . a:= (4)
i i∈N K2
Let l(X ) be the distance loss between a base station at X
i i Proof: The point process N may be viewed as a trans-
and the user, where l(·) is given by
formation of the point process Φ by the probability kernel
l(x)=(K|x|)β (1)
(cid:18) (cid:19)
p(x,A)=P l(x) ∈A , x∈R2,A∈B(cid:0)R+(cid:1)
for two given positive constants K and β. When we incorpo-
S
rate shadowing (and/or fading), the propagation loss is
By the displacement theorem [1, Theorem 1.10], the point
l(X )
L = i process N is Poisson on R+ with intensity measure
Xi S
Xi (cid:90) (cid:18)l(x) (cid:19)
where{Sx}x∈R2 isacollectionofindependentandidentically Λ([0,t))=λ P S ∈[0,t) dx
distributed(iid)positiverandomvariables.Wewillsometimes R2
write also S = S to simplify the notation. Let S be a (cid:90) (cid:26)l(x) (cid:27)
random variaXblie havinig the same distribution as any Sx. =λ R2×R+1 s <t dxPS(ds)
Note that the power received at the origin from the station (cid:104) 2(cid:105)
X , transmitting with power P , is equal to (cid:90) π(st)β2 λπE Sβ 2
i Xi =λ K2 PS(ds)= K2 tβ
P P S R+
p = Xi = Xi Xi .
Xi L l(X ) which completes the proof.
Xi i
3
Remark 2: NotethatthedistributionofN isinvariantwith We consider also the analogous process of propagation losses
respect to the distribution of the shadowing/fading S having (cid:40) (cid:41)
K(σ)β|X |β
same given value of the moment E[S2/β]. This means that N¯(σ) := i :a <|X |<b ,X ∈φ , (9)
the infinite Poisson network with an arbitrary shadowing S is S(σ) σ i σ i
i
perceived at a given location statistically in the same manner
wherethestationsinφthatarecloserthana andfartherthan
σ
as an “equivalent” infinite Poisson with “constant shadowing”
b areignored,forallsequences0≤a <b ≤∞satisfying
σ σ σ
equal to s = (E[S2/β])β/2 (to have the same moment
const
log(a )
of order 2/β). The model with such a “constant shadowing” σ →0, (10)
σ2
boils down to the model without shadowing (S ≡ 1) and the
constant K replaced by K˜ =K/(cid:112)E[S2/β]. log(bσ) →∞. (11)
The above result requires condition (2), which is satisfied σ2
by the usual models, as e.g. the log-normal shadowing or B. Main result
Rayleigh fading. Henceforth, when not specifying any par-
We present now our main convergence result.
ticular distribution of S we tacitly assume that condition (2) Theorem 3: Given homogeneity condition (5), then N(σ)
holds true.
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 → ∞)
to the Poisson point process with the same intensity measure,
In this section we derive a powerful convergence result
provided conditions (10) and (11) are satisfied.
rigorouslyshowingthattheinfinitePoissonmodelcanbeused
The proof of Theorem 3 is deferred to Appendix A.
to analyse the characteristics of the typical user in the context
Remark 4: Theaboveresult,inconjunctionwithLemma1,
of any fixed (deterministic!) placement of base stations, meet-
saysthatinfinitePoissonmodelcanbeusedtoapproximatethe
ing some empirical homogeneity condition, provided there is
characteristics of the typical user for a very general class of
sufficiently strong log-normal shadowing.
homogeneous pattens of base stations, including the standard
A. Model description
hexagonal one. The second statement of this result says that
Let φ = {Xi}i∈N be a locally finite deterministic point thisapproximationremainsvalidforsufficientlylargebutfinite
pattern on R2. For 0 < λ < ∞, as r → ∞ we require the patterns.
empirical homogeneity condition Remark 5: Thedistance-lossmodel(1)suffersfromhaving
φ(B (r)) asingularityattheorigin.Thisissueisoftencircumventedby
0 →λ. (5) some appropriate modification of the distance-loss function
πr2
withinacertaindistancefromtheorigin.Thesecondstatement
Note that the above condition is satisfied by any lattice (e.g.
of Theorem 3 with a = const > 0 shows that such a
hexagonal) pattern φ, as well as by almost any realization of σ
modification is not significant in the Poisson approximation.
an arbitrary ergodic point process.
Let the shadowing S(σ) between the station X and the To illustrate Theorem 3 and obtain some insight into the
i i speed of convergence we used Kolmogorov-Smirnov (K-S)
origin be iid (across i) log-normal random variables
test, cf [8], to compare the cumulative distribution functions
Si(σ) =exp(−σ2/2+σZi), (6) (CDF) of the SIR of the typical user (see Section IV-A2) in
the infinite Poisson model versus hexagonal one consisting of
where Z are standard normal random variables. Note that for
i
such S(σ) = S(σ), we have E[S] = 1 and E(cid:2)(S(σ))2/β(cid:3) = 30×30 = 900 stations on a torus. We found that for 9/10
i realizations of the network shadowing the K-S test does not
exp[σ2(2−β)/β2].
allow to distinguish the empirical (obtained from simulations)
Consider the distance-loss model (1) with the constant K
CDFoftheSIRfromtheCDFofSIRevaluatedintheinfinite
replaced by the function of σ
Poisson model with the critical p-value fixed to α = 10%
K(σ)=Kexp(cid:18)−σ2(β−2)(cid:19), (7) providedσdB (cid:38)10dB.OnFigure1wepresentafewexamples
2β2 oftheseCDF’s.SimilarnumericalstudywasdonefortheCDF
where K >0 and β >2. of the path-loss with respect to the serving station in [5].
As in Section II, we consider the point process on R+ of
IV. POISSONMODELREVISITED
propagationlossesexperiencedbythetypicaluserwithrespect
We now return to the model outlined in Section II and
to the stations in φ
(cid:40) (cid:41) investigate it further. We show that Lemma 1 is very useful
N(σ) := K(σ)β|Xi|β :X ∈φ . (8) in studying random quantities of the network, independently
S(σ) i of the shadowing/fading distribution. We do not pretend to
i
develop a complete theory, which is well beyond the scope
Thisalsomeansthatthepath-lossfromagivenstationXexpressedindB, of this paper. As a general remark let us emphasize only that
i.e.,dB(LX(y)),wheredB(x):=10×log10(x)dB,isaGaussianrandom many,alreadyknown,results(cfRelatedworkintheIntroduc-
variable with standard deviation σdB = σ10/log10, called logarithmic
standarddeviationoftheshadowing. tion), were originally derived under specific assumptions on
4
1 This corresponds to a Fre´chet distribution with shape param-
eter 2 and scale parameter aβ2.
β
0.8 2) Interference factor and SIR: The interference factor is
defined by
0.6 (cid:88) 1
f =L −1
CDF i∈N LXi
0.4
where L is the path-loss factor. It may be interpreted as the
0 .02 β=β=5β4.=0.30, .,σ0 σd,B dσ=Bd=1B09=..855,,, HHHeeePPPxxxaaaooogggiiisssooosssnnnoooaaannnlll iIrnnecttereorifvdeeurdecninpcgoewIteor=)siwg(cid:80)enaic∈laNrnaLtwi1Xori;ite(thcaafnt =ibseLthiInet−eirnp1vr.eertAesedlsooa,sfftthh=ee tSoLItRIa(cid:48)l.
-10 -5 0 5 10 15 20 25 30 35 40 45
SIR [dB] where
1 (cid:88) 1 1
Fig. 1. Empirical CDF of SIR simulated in hexagonal network with I(cid:48) :=I− = −
shadowingandtheirPoissonapproximations. L L L
i∈N Xi
may be viewed as the interference.
Define
thedistributionofshadowingand/orfading.ByLemma1they
(cid:18) (cid:19)
napecperossparriialtye srepmecaiifincavtaiolindoffotrheavgaelnueeraolfdthisetrmiboumtioennt, Ew[iSth2/tβh]e. ϕβ(z):=e−z+zβ2γ 1− β2,z (14)
Our specific goal in this section is to present a new
representation of distribution function of the SINR, which in where γ(α,z) = (cid:82)0ztα−1e−tdt is the lower incomplete
consequence will allow us to study the spectral and energy gammafunction.Anotherrepresentationofthefunctionϕβ(z),
efficiency — two important engineering characteristics of the used in evaluation of the Laplace transform of the f (cf proof
network. To the best of our knowledge, they have not yet of Corollary 8) is given in Appendix B.
been studied in the Poisson model. Moreover, as another Here is a key technical result for our approach.
illustration of our convergence result, we will compare these Proposition 7: The Laplace transform of the interference
characteristics in Poisson and hexagonal network, with this factor conditional to the path-loss factor is equal to
latter studied by simulations.
Unless otherwise specified all results of this section regard E(cid:2)e−zf|L=s(cid:3)=e−a[ϕβ(z)−1]sβ2 . (15)
Poisson model of Section II.
Proof: We have
A. Path-loss and SIR
E(cid:2)e−zf|L=s(cid:3)=E[e−zsI(cid:48)|L=s]. (16)
The novelty of our approach consists in representing the
SINR in terms of the path-loss to the serving station and the Observe that I(cid:48) is a shot-noise associated to the point process
respective SIR, rather than, as usual, the interference. obtainedfromthePoissonpointprocessN ofthepropagation
1) Path loss: The weakest propagation loss denoted by lossesbysuppressingitssmallest(nearesttotheorigin)point,
which is located precisely at L. By the well-known property
L= infL (12)
i∈N Xi of the Poisson process, given L=s, this new process is also
Poisson on (s,∞) with intensity measure
is often called path-loss factor. It may be interpreted as
the propagation loss with the serving base station (that is, M((s,t))=Λ([0,t))−Λ([0,s]), t∈(s,+∞) .
the one with strongest received power). Note by (3) that
the number of points of the process N = {L } is Thus
Xi i∈N
almost surely finite in any finite interval. Hence, the above 2a
infimumisalmostsurelyachievedforsomebasestation;thatis M(dt)=Λ(dt)= tβ2−1dt, on (s,+∞) (17)
β
L=mini∈NLXi. Moreover, Lemma 1 allows us to conclude
the following characterisation of the distribution of L, which and by [1, Proposition 1.5]
follows immediately from the well known expression for void (cid:104) (cid:105) (cid:20)(cid:90) ∞ (cid:21)
probabilities of Poisson process P{L ≥ t} = P{N([0,t)) = E e−zsI(cid:48)|L=s =exp (e−zts −1)M(dt)
0}=exp[−Λ([0,t))]. s
(cid:20) 2a(cid:90) ∞ (cid:21)
Corollary 6: The CDF of L is equal to P{L ≤ t} = =exp − (1−e−zts)tβ2−1dt .
1−exp[−at2/β] where a is given by (4). Consequently, the β s
probability density of L is given by (18)
2a 2 Now we calculate the integral on the right-hand side of the
PL(ds)= β tβ2−1e−atβdt. (13) above equation by making the change of variable u := zs,
t
5
that is Remark 9: Fort≥1the(complementary)CDFoftheSIR
(cid:90) ∞ admits the following explicit expression
(1−e−zts)tβ2−1dt
t−2/β
s P{SIR≥t}=P{f ≤1/t}= (20)
(cid:90) 0 (cid:16)zs(cid:17)2−1 −zs C(cid:48)(β)
= (1−e−u) β du
u u2
z where
(cid:90) z
= (zs)β2 (1−e−u)u−β2−1du 2π 2Γ(2/β)Γ(1−2/β)
C(cid:48)(β)= =
0 βsin(2π/β) β
β
2
= sβ 2 [−1+ϕβ(z)] and Γ(z) = (cid:82)∞e−zzt−1dt it the complete Gamma function.
0
It was proved in [3] assuming exponential distribution of S.
where the third equality is obtained by integration by parts.
ButintheinfinitePoissonmodel,SIRisinvariantwithrespect
CombiningtheaboveequationwithEquations(16),(18)gives
to λ, and K and consequently by Lemma 1 it is also invariant
the final result.
withrespecttothevalueofE[S2/β].Hencetheresultremains
The following results can be derived from Proposition 7,
valid for arbitrary distribution of S, provided E[S2/β]<∞.
although we will not use them in the remaining part of the
Otherresultsof[3],includingtheseinvolvingthesuperposition
paper.
ofindependentPoissonmodels(calledK-tiercellularnetwork
Corollary 8: The joint distribution of L and f is charac-
model) can also be appropriately generalized using Lemma 1.
terized by
E(cid:2)1{L≥u}e−zf(cid:3)= 1 e−auβ2ϕβ(z), z ∈R+. B. Distribution of SINR, spectral and energetic efficiency
ϕβ(z) 1) SINR: In this section we are primarily interested in the
signal to interference and noise ratio
TheunconditionalLaplacetransformoftheinterferencefactor
is 1 1
E(cid:2)e−zf(cid:3)= 1 , z ∈R+. (19) SINR= N +(cid:16)(cid:80) L 1 − 1(cid:17) = NL+f, (21)
ϕβ(z) i∈N LXi L
Proof: For the fist statement we have where N is the noise power. We will show how CDF of the
SINR can be evaluated, which opens a way for the study of
(cid:90) ∞
E(cid:2)1{L≥u}e−zf(cid:3)= E(cid:2)e−zf|L=s(cid:3)P (ds) functionals of SINR.
L
u 2) Spectral efficiency: An important characteristic of a
=(cid:90) ∞e−a[ϕβ(z)−1]sβ2 2asβ2−1e−asβ2ds wireless cellular network is its spectral efficiency, defined in
β the simplest case of additive white Gaussian noise (AWGN)
u
1 2 and the optimal theoretical link performance as
= e−auβϕβ(z)
ϕ (z)
β S :=log(1+SINR) .
where for the second equality we use (13) and (15). This
IttellsushowmanybitspersecondandperHertzcanbesent
completes the proof of the fist statement. For the second
to the typical user of the network.
statement, conditioning on the value of the path-loss factor
3) Energyefficiency: Uptonow wehaveconsideredaunit
yields
transmitted power. Assume now that base stations transmit
(cid:90)
E(cid:2)e−zf(cid:3)= E(cid:2)e−zf|L=s(cid:3)P (ds) some power P ≥ 0. In fact, in order for a base station to
L
R+ be able to transmit this power to the mobile, it needs to be
powered (i.e., consumes energy per second) at the level P(cid:48) >
where P (·) is the distribution of L given by (13). Using the
L P. Following [9], assume that these two quantities are related
above equation and (15), we ascertain
throughasimplelinearrelationP(cid:48) =cP+dforsomepositive
(cid:90) ∞ 2 2a 2 constants c and d. The energy efficiency is defined by
E[e−zf]= e−a[ϕβ(z)−1]sβ sβ2−1e−asβds
β (cid:16) (cid:17)
20a(cid:90) ∞ 2 W log 1+ NL/1P+f
= β sβ2−1e−aϕβ(z)sβds E :=E(P)= cP +d .
0
= ϕ 1(z)(cid:90) ∞aϕβ(z)β2sβ2−1e−aϕβ(z)sβ2ds wWheloreg(W1+isStIhNeRb(aPnd))w/iPdt(cid:48)h, wexhperreessSeIdNiRn(HPz).tTahkuessEintioseaqcucoalutnot
β 0
1 the transmitted power, and P(cid:48) is the consumed power. It tells
=
ϕ (z) ushowmanybitspersecondperWattofconsumedpowercan
β
be sent by a base station to the typical user. Note that E(0)=
wherethethirdequalitystemsfromϕβ(z)(cid:54)=0 whichfollows E(∞) = 0 and thus E(P) admits a non-trivial optimization
from (40) in Appendix B and the assumption β >2. in P.
6
4) Evaluation of the CDF of the SINR: Proposition 7 in 1
conjunctionwithCorollary6completelycharacterizesthejoint
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 Remark 9). In this section we describe a practical
way for numerical computation of this CDF. We will use it CDF 0.4
to study the s pectral and energy efficiency. In fact, we will 0.2
computetheCDFoftherandomvariableY :=NL+f,which
is sufficient, in view of (21). 0
Proposition 10: The cumulative distribution function of Y -0.2 InHHfienexixtaeag gPooonniasals lσ,o σdnPBd (=BoL=1ias20spddolaBBnc e(((sss iiiimmmnvuuuelllaaarstttiiiioooonnnn))))
is given by Infinite Poisson (Explicit, valid for SINR[dB]>0)
-10 -5 0 5 10 15 20 25 30 35 40
(cid:90) ∞ SINR [dB]
P(Y <x)= F (x−Ns)P (ds) (22)
s L Fig. 2. CDF of the SINR for (finite) hexagonal network with and without
0 (σdB = 0dB) shadowing, finite Poisson as well as for the infinite Poisson
where P (ds) is given by (13), and F (y) := model.Thelastinthelegendcurvecorrespondsto(20).
L s
P(f <y|L=s) may be expressed by
F (y)=1− 2eγt (cid:90) ∞R(cid:0)L (γ+iu)(cid:1)cosutdu, (23) 120000 HHeexaxaggoInonfnainal,i lt,σe σd PBd=Bo=1is02sddoBBn
s π 0 Fs 100000
where γ >0 is an arbitrary constant, R is the real part if it
W] 80000
complex argumLeFnst(azn)d= z1 − z1e−a[ϕβ(z)−1]sβ2 (24) Energy efficiency [bits/s/ 4600000000
with ϕ (·) is given by (14).
β
20000
Proof:Theexpression(22)istrivial.Theexpression(23)
ofthe(conditional)CDFF isbasedontheBromwichcontour
s 0
0 10 20 30 40 50 60 70 80 90
inversionintegraloftheLaplacetransformLFs of1−Fs.The Transmitted power [dBm]
expression (24) for this latter follows from (15).
Fig.3. Meanenergyefficiencyoffinitehexagonalmodelwithandwithout
Remark 11: As shown in [10], the integral in (23) can
(σdB = 0dB) shadowing as well as the infinite Poisson in function of the
be numerically evaluated using the trapezoidal rule, with the transmittedpower.
parameter γ allowing control of the approximation error. The
function F may be also retrieved from (15) using other
s
inversion techniques. Figure 2 shows the CDF of the SINR assuming the base
Remark 12: For t ≥ 1 the (complementary) CDF of the stationpowerP =58.5dBm.Weobtainresultsbysimulations
SINR admits the following expression for finite hexagonal network on the torus with and without
shadowing as well as the finite Poisson network on the same
2t−2/β (cid:90) ∞
P{SINR≥t}= re−r2Γ(1−2/β)−Na−β/2rβdr torus.FortheinfinitePoissonmodelweuseourmethodbased
Γ(1+ 2)
β 0 ontheinversionoftheLaplacetransform.Forcomparison,we
(25)
plot also the curve corresponding the explicit expression (20),
withagivenby(4).Itfollowsfrom[3,Theorem1](wherethe
which is valid only for SINR>1.
exponential distribution is assumed for S) and our Lemma 1.
Figure 3 shows the expected energy efficiency E(E) of the
Note on Figure 2 that the expression is not valid for t<1.
finite hexagonal model with and without shadowing as well
5) Numerical examples: In what follows we assume the
as the infinite Poisson, assuming the affine relation between
followingparametervalues.Ifnototherwisespecifiedwetake
consumed and emitted power with constants c = 21,45 and
log-normal shadowing with logarithmic standard deviation
d=354.44W.
σ = 12dB (cf the footnote on page 3). The parameters
dB OnbothfiguresweseethattheinfinitePoissonmodelgives
of the distance-loss model are K = 4250Km−1, β = 3.52
a reasonable approximation of the (finite) hexagonal network
(which corresponds to the COST-Hata model for urban en-
providedtheshadowingishighenough.Inparticular,thevalue
vironment with the BS height 30m, mobile height 1.5m,
ofthetransmittedpoweratwhichthehexagonalnetworkwith
carrierfrequency805MHzandpenetrationloss19dB).Forthe
the shadowing attains the maximal expected energy efficiency
hexagonal network model simulations, we consider 900 base
is very well predicted by the Poisson model.
stations (30x30) on a torus, with the cell radius R=0.26Km
(i.e., the surface of the hexagonal cell is equal to this of the CONCLUSIONS
disk of radius R). System bandwidth is W = 10MHz, noise We present a powerful mathematical convergence result
power N =−93dBm. rigorouslyjustifyingthefactthatanyactual(includingregular
7
hexagonal) network is perceived by a typical user as an hence
equivalent (infinite) Poisson network, provided log-normal (cid:90) ∞ (cid:20)s−βlog(Kr)−n/β(cid:21)
2π rG √ dr
shadowing variance is sufficiently high. Good approximations
n
are obtained for logarithmic standard deviation of the shad- 0 (cid:90) ∞ 1 (cid:20)2 (cid:0) √ (cid:1)(cid:21) √n
owing greater than approximately 10dB, which is a realistic =2π exp s−t n−n/β G(t) dt
K2 β β
assumptioninmanyurbanscenarios.Moreover,theequivalent −∞
(cid:104) (cid:105)
infinite Poisson representation is invariant with respect to √nexp β2 (s−n/β) (cid:90) ∞ (cid:20)−2t√n(cid:21)
an additional fading distribution. Using this representation =2π exp G(t)dt
β K2 β
we study the distribution of the SINR of the typical user. −∞
In particular, we evaluate and optimize the mean energy Noting that G(−t)=1−G(t):=G¯(t), apply integration by
efficiency as a function of the base station transmit power. parts on the above integral
√ √
(cid:90) ∞ (cid:20)−2t n(cid:21) (cid:90) ∞ (cid:20)2t n(cid:21)
APPENDIX exp G(t)dt= exp G¯(t)dt
β β
−∞ −∞ √
A. Proof of Theorem 3 (cid:20) (cid:21) (cid:12)∞
= √β exp 2t n G¯(t)(cid:12)(cid:12)
In order to simplify the notation we set n:=σ2. Moreover, 2 n β (cid:12)
√ −∞
without loss of generality we assume that n takes positive β (cid:90) ∞ (cid:20)2t n(cid:21)
integer values. Also, it is more convenient to study the point + √ exp g(t)dt,
2 n β
process of propagation losses on the logarithmic scale, which −∞
wedoinwhatfollows.InthisregarddenotebyΛ theimage where g(t) is the normal probability density, and, via the
log
of the measure Λ given by (3) with a = λπ/K2 through the inequality
logarithmic mapping (cid:90) ∞ e−y2
e−x2dx< , y ≥0, (27)
(cid:90) π (1+y)
Λlog((−∞,s]):= 1(log(t)≤s)Λ(dt)= K2e2βs y
R+ (cf [11, Section 7.8]) the final integrated term on the left
for s∈R. vanishes. Hence
√
For a given i, we first observe by (6) that (cid:90) ∞ (cid:20)−2t n(cid:21)
exp G(t)dt
ν (s,|X |):=P(cid:34)log(cid:32)K(n)β|Xi|β(cid:33)≤s(cid:35) −∞β (cid:90) ∞ β (cid:20) t2 2t√n(cid:21) dt
n i S(n) = √ exp − + √
=P(cid:20)Z ≤ s−βilog(K√|Xi|)−n/β(cid:21) 2βn −∞(cid:20)2n(cid:21)(cid:90) ∞2 (cid:34)β 1(cid:18) 2π2√n(cid:19)2(cid:35) dt
i n = √ exp exp − t− √
(cid:20)s−βlog(K|X |)−n/β(cid:21) 2 n β2 −∞ 2 β 2π
=G √ i , (26) (28)
n
(cid:20) (cid:21)
β 2n
= √ exp ,
whereGistheCDFofthestandardGaussianrandomvariable. 2 n β2
Let B (r) = {x ∈ R2 : |x| < r}. We now need to derive
0 which gives
two results.
Lemma 13: 2π(cid:90) ∞rG(cid:20)s−βlog√(Kr)−n/β(cid:21)dr = π e2βs.
(cid:90) (cid:90) n K2
0
lim ν (s,|x|)dx= lim ν (s,|x|)dx
n n We proceed similarly with the other integral in Lemma 13
n→∞ B0(bn)\B0(an) n→∞ R2 (cid:90)
=Λ ((−∞,s])
log ν (s,|x|)dx
n
provided that a and b satisfy (10) and (11). B0(bn)\B0(an)
Proof: Wenstart winth the integral over R2 in polar form =2π(cid:90) bnrG(cid:20)s−βlog√(Kr)−n/β(cid:21)dr.
n
(cid:90) (cid:90) ∞ (cid:20)s−βlog(Kr)−n/β(cid:21) an
R2νn(s,|x|)dx=2π 0 rG √n dr The change of variables t= s−βlog√(Knr)−n/β gives
(cid:90) bn (cid:20)s−βlog(Kr)−n/β(cid:21)
2π rG √ dr
n
Introduce a change of variables an √
(cid:90) un 1 (cid:20)2 (cid:0) √ (cid:1)(cid:21) n
s−βlog(Kr)−n/β β dr =2π exp s−t n−n/β G(t) dt
t= √n , dt=−√n r , vn K2(cid:104) β (cid:105) β
1 (cid:20)1 (cid:0) √ (cid:1)(cid:21) √nr √nexp β2 (s−n/β) (cid:90) un (cid:20)−2t√n(cid:21)
r = exp s−t n−n/β , dr =− dt, =2π exp G(t)dt
K β β β K2 β
vn
8
where for some k ≥ 0, whose value will be fixed later on. In the
0
limit of n→∞, the first summation in (31) disappears
s−βlog(Ka )−n/β
u = √ n ,
n n (cid:88) ν (s,|X |)
n i
s−βlog(Kb )−n/β
vn = √nn . X=i∈φ∩B(cid:88)0(rk0) P(cid:20)Z ≤ s−βlog(K√|Xi|)−n/β(cid:21)
n
Moreover Xi∈φ∩B0(rk0)
(cid:90) un (cid:20)−2t√n(cid:21) = (cid:88) G(cid:20)s−βlog(K√|Xi|)−n/β(cid:21)
exp G(t)dt n
vn β√ Xi∈φ∩B0(rk0)
=(cid:90) −vnexp(cid:20)2t n(cid:21)G¯(t)dt ≤φ(B (r ))G(cid:20)s−βlog(K√|X∗|)−n/β(cid:21)
β 0 k0 n
−un √ √
= √β exp(cid:20)2t n(cid:21)G¯(t)(cid:12)(cid:12)(cid:12)−vn+ √β (cid:90) −vnexp(cid:20)2t n(cid:21)g(t)dt →0 (n→∞).
= 22√βnnexp(cid:20)2tββ√n(cid:21)G¯(t)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)−−vunn 2 n −un β wXh∈ereφX∩B∗0g(irvke0s)thwehmichaxeixmisutmssoinfcGeφ(cid:104)si−sβ(bloyg(oK√urn|Xa|s)s−unm/βpt(cid:105)ioonv)ear
+ √β exp(cid:20)2n(cid:21)(cid:90) −vn−euxnp(cid:34)−1(cid:18)t− 2√n(cid:19)2(cid:35)√dt . lwoecawllyritfienνitne(ps,o|iXntim|)e=asνunre(.sF,|oxr|t|hX|xei||s)e,choenndcesummationin(31)
2 n β2 2 β 2π
−un ν (s,|X |)
n i
We now derive the conditions of a and b which ensure 1 (cid:90) |X |
n n = ν (s,|x| i )dx.
that the above integral converges and the integrated term |A | n |x|
k Ak
disappears. The latter can be achieved, given inequality (27),
in the limit as −u and −v both approach infinity, or equiv- Then the bounds
n √ n √
aaslenntly→βlo∞g√(K,nbwnh)i+ch βangr→ees∞withancdonβdiltoigo√(nKnsan(1)0+) aβnnd→(11∞). e−(cid:15) = rk ≤ |Xi| ≤ rk+1 =e(cid:15),
r |x| r
Further conditions are revealed after the change of variable k+1 k
√
w =t−2 n/β, yielding the integral and form of ν , which implies ν (s,|x|e(cid:15))=ν (s−β(cid:15),|x|),
n n n
lead to the lower bound
√
(cid:90)−−uvnne−12(cid:16)t−2√βn(cid:17)2 √d2tπ =(cid:90)−−uvnn−−2√2ββnn e−w22 √d2wπ (cid:88)∞ (cid:88) νn(s,|Xi|)≥ (cid:88)∞ φ|(AAk|)(cid:90) νn(s−β(cid:15),|x|)dx,
k=k0Xi∈φ∩Ak k=k0 k Ak
whoselimitsofintegrationimply,inlightoftheearlierintegral (32)
(28), that as n → ∞ the following βlog(Kbn) > 1/β and and the upper bound
n
βlog(Kan) <1/β are required, of which both conditions (10)
and (n11) satisfy. (cid:88)∞ (cid:88) ν (s,|X |)≤ (cid:88)∞ φ(Ak)(cid:90) ν (s+β(cid:15),|x|)dx.
n i |A | n
k=k0Xi∈φ∩Ak k=k0 k Ak
Lemma 14: Assume (5), (10) and (11), then (33)
Moreover, we write
(cid:88)
lim ν (s,|X |) (29)
n i φ(A ) φ(B (r ))−φ(B (r ))
n→∞ k = 0 k+1 0 k
Xi∈φ∩(B0(bn)\B0(an)) |A | |B (r )|−|B (r )|
(cid:88) k 0 k+1 0 k
= lim νn(s,|Xi|)=Λlog((−∞,s]). (30) φ(B0(rk+1)) − φ(B0(rk)) |B0(rk)|
n→∞Xi∈φ = |B0(rk+1)| |B0(rk)| |B0(rk+1)|
1− |B0(rk)|
Proof: For k ≥ 0 and a fixed (cid:15) > 0, let r = e(cid:15)k and |B0(rk+1)|
k φ(B0(rk+1)) − φ(B0(rk))e−2(cid:15)
Ak =B0(rk+1)\B0(rk), and write the summation in (30) as = |B0(rk+1)| |B0(rk)| ,
1−e−2(cid:15)
(cid:88)
ν (s,|X |)
n i and requirement (5) yields lim φ(Ak) = λ. Hence, for
Xi∈φ any fixed δ >0, there exists a kk→(δ∞) su|Achk|that for all k ≥k ,
∞ 0 0
(cid:88) (cid:88) (cid:88)
= ν (s,|X |)+ ν (s,|X |), the bounds
n i n i
Xi∈φ∩B0(rk0) k=k0Xi∈φ∩Ak (31) (1−δ)λ≤ φ(Ak) ≤(1+δ)λ,
|A |
k
9
hold. Lower bound (32) becomes function γ∗ defined by (38) (which is holomorphic on C×C;
∞ ∞ (cid:90) see [12, 6.5.4]) we obtain (37). The expansion (39) is given
(cid:88) (cid:88) (cid:88)
ν (s,|X |)≥ (1−δ)λ ν (s−β(cid:15),|x|)dx, in [12, 6.5.29]. This expansion shows that if α>−1 then for
n i n
k=k0Xi∈φ∩Ak k=k0 Ak any z ∈R+, γ∗(α,z)>0, and therefore by (37) ϕβ(z)>0
(cid:90) for any β >2,z ∈R+.
=(1−δ)λ ν (s−β(cid:15),|x|)dx
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Proof: Using (14) and recurrence formula [12, 6.5.22]
we obtain (36). Introducing the modified incomplete gamma
