A Tractable Approach to
Coverage and Rate in Cellular Networks Jeffrey G.Andrews,Senior Member,IEEE,François Baccelli,and Radha Krishna Ganti,Member,IEEE
Abstract—Cellular networks are usually modeled by placing the ba stations on a grid,with mobile urs either randomly scattered or placed deterministically.The models have been ud extensively but suffer from being both highly idealized and not very tractable,so complex system-level simulations are ud to evaluate coverage/outage probability and rate.More tractable models have long been desirable.We develop new general models for the multi-cell signal-to-interference-plus-noi ratio(SINR) using stochastic geometry.Under very general assumptions,the resulting expressions for the downlink SINR CCDF(equivalent to the coverage probability)involve quickly computable integrals, and in some practical special cas can be simplified to common ,the Q-function)or even to simple clod-form expressions.We also derive the mean rate,and then the coverage gain(and mean rate loss)from static frequency reu.We compare our coverage predictions to the grid model and an actual ba station deployment,and obrve that the propod model is pessimistic(a lower bound on coverage)whereas the grid model is optimistic,and that both are about equally accurate.In addition to being more tractable,the propod model may better capture the increasingly opportunistic
and den placement of ba stations in future networks.
Index Terms—Cellular systems,outage probability,SINR, stochastic geometry.
I.I NTRODUCTION
C ELLULAR systems are now nearly universally deployed
and are under ever-increasing pressure to increa the volume of data they can deliver to consumers.Despite decades of rearch,tractable models that accurately model other-cell interference(OCI)are still unavailable,which is fairly remarkable given the size of the industry.This deficiency has impeded the development of techniques to combat OCI,which is the most important obstacle to higher spectral efficiency in today’s cellular networks,particularly the den ones in urban areas that are under the most strain.In this paper we develop accurate and tractable models for downlink capacity and coverage,considering full network interference.
A.Common Approaches and Their Limitations
A tractable but overly simple downlink model commonly ud by information theorists is the Wyner model[1]–[3], which is typically one-dimensional and presumes a unit gain Paper approved by N.C.B
eaulieu,the Editor for Wireless Communication Theory of the IEEE Communications Society.Manuscript received September 2,2010;revid February21,2011,May17,2011,and August5,2011. F.Baccelli is with the Ecole Normale Superieure(ENS)and INRIA,Paris, France(e-mail:francois.baccelli@ens.fr).
J.G.Andrews and R.K.Ganti are with the Dept.of ECE,the University of Texas at Austin,1University Station,C0803Austin,TX78712USA(e-mail: jandrews@ece.utexas.edu;).
Digital Object Identifier10.1109/TCOMM.2011.100411.100541from each ba station to the tagged ur and an equal gain that is less than one to the two urs in the two neighboring cells.This is a highly inaccurate model unless there is a very large amount of interference averaging over space,such as in the uplink of heavily-loaded CDMA systems[4].This philosophical approach of distilling other-cell interference to afixed value has also been advocated for CDMA in[5]and ud in the landmark paper[6],where other-cell interference was modeled as a constant factor of the total interference.For cellular systems using orthogonal multiple access techniques such as in LTE and WiMAX,the Wyner model and related mean-value approaches are particularly inaccurate,since the SINR values over a cell vary dramatically.Nevertheless,it has been commonly ud even up to the pr
ent to evalu-ate the“capacity”of multicell systems under various types of multicell cooperation[7]–[9].Another common analysis approach is to consider a single interfering cell[10].In the two cell ca,at least the SINR varies depending on the ur position and possibly fading,but naturally such an approach still neglects most sources of interference in the network and is highly idealized.A recent discussion of such models for the purpos of ba station cooperation to reduce interference is given in[11].That such simplified approaches to other-cell interference modeling are still considered state-of-the-art for analysis speaks to the difficulty infinding more realistic tractable approaches.
On the other hand,practicing systems engineers and re-archers in need of more realistic models typically model a 2-D network of ba stations on a regular hexagonal lattice,or slightly more simply,a square lattice.Tractable analysis can sometimes be achieved for afixed ur with a small number of interfering ba stations,for example by considering the “worst-ca”ur location–the cell corner–andfinding the signal-to-interference-plus-noi ratio(SINR)[12],[13]. The resulting SINR is still a random variable in the ca of shadowing and/or fading from which performance metrics like (worst-ca)average rate and(worst-ca)outage probability relative to some target rate can be determined.Naturally, such an approach gives very pessimistic results that do not provide much guid
ance to the performance of most urs in the system.More commonly,Monte Carlo integrations are done by in the landmark capacity paper [6].Tractable expressions for the SINR are unavailable in general for a random ur location in the cell and so more general results that provide guidance into typical SINR or the probability of outage/coverage over the entire cell must be arrived at by complex time-consuming simulations.In addition to being onerous to construct and run,such private simulations
0090-6778/11$25.00c⃝2011IEEE
additionally suffer from issues regarding repeatability and transparency,and they ldom inspire“optimal”or creative new algorithms or designs.It is also important to realize that although widely accepted,grid-bad models are themlves highly idealized and may be increasingly inaccurate for the heterogeneous and ad hoc deployments common in urban and suburban areas,where cell radii vary considerably due to differences in transmission power,tower height,and ur density.For example,picocells are often inrted into an existing cellular network in the vicinity of high-traffic areas, and short-range femtocells may be scattered in a haphazard manner throughout a centrally planned cellular network. B.Our Approach and Contributions
Perhaps counter-intuitively,this paper address the long-standing problems by introducing an additional source of randomness:the positions of the ba stations.Instead of assuming they are placed deterministically on a regular grid, we model their location as a homogeneous Poisson point process of densityλ.Such an approach for BS modelling has been considered as early as1997[14]–[16]but the key metrics of coverage(SINR distribution)and rate have not been determined1.The main advantage of this approach is that the ba station positions are all independent which allows substantial tools to be brought to bear from stochastic geometry;e[18]for a recent survey that discuss additional related work,in particular[19]–[21].Although BS’s are not independently placed in practice,the results given here can in principle be generalized to point process that model repulsion or minimum distance,such as determinantal and Matern process[22],[23].The mobile urs are scattered about the plane according to some independent homogeneous point process with a different density,and they communicate with the nearest ba station while all other ba stations act as interferers,as shown in Fig.1.
From such a model,we achieve the following theoretical contributions.First,we derive a general expression in Theo-rem1for the probability of coverage in a cellular network where the interference fading/shadowing follows an arbitrary distribution.The coverage probability is the probability that a ty
pical mobile ur is able to achieve some threshold SINR, i.e.it is the complementary cumulative distribution function (CCDF)of SINR.This expression is not clod-form but also does not require Monte Carlo methods.This is generalized to include an arbitrary desired received signal power distribution in Lemma1,and then simplified for a number of special cas,namely combinations of(i)exponentially distributed interference Rayleigh fading,(ii)path loss exponent of4,and(iii)interference-limited hermal noi is ignored.The special cas have increasing tractability and in the ca that all three simplifications are taken,we derive a remarkably simple formula for coverage probability that depends only on the threshold SINR.We compare the novel theoretical results with both traditional(and computationally intensive)grid-bad simulations and with actual ba station locations from a current cellular deployment in a major urban 1The paper[17]was made public after submission of this paper and contains some similar aspects to the approach in this paper.
Fig.1.Poisson distributed ba stations and mobiles,with each mobile associated with the nearest BS.The cell boundaries are shown and form a V oronoi tesllation.
area.We e that over a broad range of parameter and modeling choices our results provide a reliable lower bound to reality whereas the grid model provides an upper bound that is about equally
loo.In other words,our approach,even in the ca of simplifying assumptions(i)-(iii),appears to not only provide simple and tractable predictions of the SINR distribution in a cellular network,but also accurate ones. Next,we derive the mean achievable rate in our propod cellular model under similar levels of generality and tractabil-ity.The two competing objectives of coverage and rate are then explored analytically through the consideration of frequency reu,which is ud in some form in nearly all cellular systems2to increa the coverage or equivalently the cell edge rates.Our expressions for coverage and rate are easily modified to include frequency reu and wefind the amount of frequency reu required to reach a specified coverage probability,as well as eing how frequency reu degrades mean rate by using the total bandwidth less efficiently.
II.D OWNLINK S YSTEM M ODEL
The cellular network model consists of ba stations(BSs) arranged according to some homogeneous Poisson point pro-cess(PPP)Φof intensityλin the Euclidean plane.Consider an independent collection of mobile urs,located according to some independent stationary point process.We assume each mobile ur is associated with the clost ba station;namely the urs in the V oronoi cell of a BS are associated with it, resulting in coverage areas that compri a V oronoi tesllation on the plane,as shown in Fig.1.
2Even cellular systems such as modern GSM and CDMA networks that claim to deploy universal frequency reu still thin the interference in time or by using additional frequency bands–which is mathematically equivalent to thinning in frequency.
Fig.2.A40×40km view of a current ba station deployment by a major rvice provider in a relativelyflat urban area,with cell boundaries corresponding to a V oronoi tesllation.
An actual ba station deployment in Fig.2.The main weakness of the Poisson model is that becaus
e of the indepen-dence of the PPP,BSs will in some cas be located very clo together but with a significant coverage area.This weakness is balanced by two strengths:the natural inclusion of different cell sizes and shapes and the lack of edge he network extends indefinitely in all directions.The models are quantitatively compared to each other and the customary grid model in Section V.
The standard power loss propagation model is ud with path loss exponentα>2.As far as random channel effects such as fading and shadowing,we assume,unless otherwi noted,that the tagged ba station and tagged ur experience only Rayleigh fading with mean1,and employ a constant transmit power of1/μ.In this ca,the received power at a typical node a distance r from its ba station isℎr−αwhere the random variableℎfollows an exponential distribution with mean1/μ,which we denote asℎ∼exp(μ).We briefly describe how other distributions for the desired signalℎ(such as shadowing)can be considered after Theorem1.The interference power follows a general statistical distribution g that could include fading,shadowing,and any other desired random effects.Simpler expressions result when g is also exponential and the are given as special cas since they are likely to be of particular interest to other rearchers. Lognormal shadowing on both the desired and interfering signals is also investigated numerically,and we e it does not significantly affect the accu
racy of our analysis.Becau of the random channel effects,in our propod model not all urs will be connected to the ba station capable of providing the highest SINR.All results are for a single transmit and single receive antenna,although future extensions to multiple such antennas are clearly desirable.
The interference power at the typical receiver I r is the sum of the received powers from all other ba stations other than the home ba station and is treated as noi in the prent work.There is no same-cell interference,for example due to orthogonal multiple access within a cell.The noi power is assumed to be additive and constant with valueσ2but no specific distribution is assumed in general.The SNR=1
μσ2 is defined to be the received SNR at a distance of r=1.All analysis is for a typical mobile node which is permissible in a homogeneous PPP by Slivnyak’s theorem[23].
III.C OVERAGE
This is the main technical ction of the paper,in which we derive the probability of coverage in a downlink cellular network at decreasing levels of generality.The coverage probability is defined as
p c(T,λ,α)≜ℙ[SINR>T],(1) and can be thought of equivalently as(i)the probability that a randomly chon ur can achieve a target SINR T,(ii) the average fraction of urs who at any time achieve SINR T,or(iii)the average fraction of the network area that is in “coverage”at any time.The probability of coverage is also exactly the CCDF of SINR over the entire network,since the CDF givesℙ[SINR≤T].
Without any loss of generality we assume that the mobile ur under consideration is located at the origin.A ur is in coverage when its SINR from its nearest BS is larger than some threshold T and it is dropped from the network for SINR below T.The SINR of the mobile ur at a random distance r from its associated ba station can be expresd as
SINR=
ℎr−α
σ2+I r
,
where
I r=
∑
i∈Φ/b o
g i R−αi
is the cumulative interference from all the other ba stations (except the tagged ba station for the mobile ur at o denoted by b o)which are a distance R i from the typical ur and have fading value g i.
A.Distance to Nearest Ba Station
An important quantity is the distance r parating a typical ur from its tagged ba station.Since each ur communi-cates with the clost ba station,no other ba station can be clor than r.In other words,all interfering ba stations must be farther than r.The probability density function(pdf)of r can be derived using the simple fact that the null probability of a2-D Poisson process in an area A is exp(−λA).
ℙ[r>R]=ℙ[No BS clor than R]
=e−λπR2.
Therefore,the cdf isℙ[r≤R]=F r(R)=1−e−λπR2and the pdf can be found as
f r(r)=
d F r(r)
d r
=e−λπr22πλr.
Meanwhile,the interference is a standard M/M shot noi [22],[24],[25]created by a Poisson point process of intensity λoutside a disc at center o and of radius r,for which some uful results are known and applied in the quel.
B.General Ca and Main Result
We now state our main and most general result for coverage probability from which all other results in this ction follow.Theorem 1:The probability of coverage of a typical ran-domly located mobile ur in the general cellular network model of Section II is
p c (T,λ,α)=πλ
∫∞
e −πλvβ(T,α)−μT σ2v α/2
d v,where
β(T,α)=
2(μT )2αα
E [g 2
α(Γ(−2/α,μT g )−Γ(−2/α))],and the expectation is with respect to the interferer’s channel distribution g .Also,Γ(a,x )=∫∞
x t a −1e −t
d t denotes th
e incomplete gamma function,and Γ(x )=∫∞0t x −1e −t d t the standard gamma function.
Proof:Conditioning on the nearest BS being at a distance r from the typical ur,the probability of coverage averaged over the plane is
p c (T,λ,α)=E r [ℙ[SINR >T ∣r ]]
=
∫
r>0
ℙ[SINR >T ∣r ]f r (r )d r (a )=∫r>0
ℙ[ℎr −ασ2+I r
>T
r ]e −πλr 22πλr d r
=∫r>0
e −πλr 2
ℙ[ℎ>T r α(σ2+I r )∣r ]2πλr d r.
thank you是什么意思The distribution f r (r )and hence (a )follows from Sub-ction III-A.Using the fact that ℎ∼exp(μ),the coverage
probability can be expresd as
ℙ[ℎ>T r α(σ2+I r )∣r ]=E I r
[
ℙ[ℎ>T r α(σ2+I r )∣r,I r ]
]
=E I r
[
exp(−μT r α(σ2+I r ))∣r ]=e −μT r
α
σ2
ℒI r (μT r α),
where ℒI r (s )is the Laplace transform of random variable I r evaluated at s conditioned on the distance to the clost BS from the origin.This gives a coverage expression
p c (T,λ,α)=∫
r>0
e −πλr 2e −μT r ασ2
ℒI r (μT r α)2πλr d r.(2)
De fining R i as the distance from the i th interfering ba station to the tagged receiver and g i as the interference channel coef ficient of arbitrary but identical distribution for all i ,using the de finition of the Laplace transform yields
ℒI r (s )=E I r [e −sI r ]=E Φ,g i [exp(−s
∑
i ∈Φ∖{b o }
g i R −αi )]=E Φ,{g i }⎡
⎣
∏
i ∈Φ∖{b o }
exp(−sg i R −αi )
⎤
⎦(a )
=E Φ⎡
⎣
∏
i ∈Φ∖{b o }
E g [exp(−sgR −αi )]⎤
⎦=exp (−2πλ
∫
∞
r
(1−E g [exp(−sgv −α)])
v d v )
,
(3)
where (a )follows from the i.i.d.distribution of g i and its
further independence from the point process Φ,and the last step follows from the probability generating functional (PGFL)[23]of the PPP,which states for some function f (x )that E [∏x ∈Φf (x )]=exp (−λ∫ℝ2(1−f (x ))d x ).The integra-tion limits are from r to ∞since the clost interferer is at least at a distance r .Let f (g )denote the PDF of g .Plugging in s =μT r α,and swapping the integration order gives,ℒI r (μT r α)=
exp (−2πλ
∫∞
(∫
∞
r
(1−e −μT r
αv −α
g
)v d v )f (g )d g ).
(4)
The inside integral can be evaluated by using the change of variables v −α→y ,and the Laplace transform is
ℒI r (μT r α)=exp (λπr 2
−
2πλ(μT )2
αr 2α
⋅
∫∞
g 2
α[Γ(−2/α,μT g )−Γ(−2/α)]f (g )d g ).Combining with (2),and using the substitution r 2→v ,we obtain the result.
Theorem 1can be further generalized to allow the desired signal to experience an arbitrary fading distribution, e.g.lognormal shadowing,in the following lemma.
Lemma 1:If the PDF of the desired signal fad-ing/shadowing is square integrable,the probability of coverage of a typical randomly located mobile ur in the general cellular network model of Section II is
p c (T,λ,α)=∫r>0
2πλre −πλr 2∫∞
−∞
e −2πσ2
js ⋅ℒI r (2πjs )
ℒℎ(−2π(T r α)−1
js )−1
2πjs
d s d r,(5)
where ℒℎ(s )is the Laplace transform of the desired signal fading/shadowing and ℒI r (s )denotes the Laplace transform of I r provided in (4).
Proof:The proof is given in Appendix A.
Becau of the incread generality,Lemma 1los an ad-ditional layer of tractability,so in the quel we focus on the ca of Rayleigh fading for the desired signal.We now turn our attention to a few relevant special cas where signi ficant simpli fication is possible and quite simple and intuitive coverage probability expressions can be found.
C.Special Cas:Interference Still Experiences General Fad-ing
The main simpli fications we will now consider in var-ious combinations are (i)allowing the path loss exponent α=4,(ii)an interference-limited 1/σ2→∞,which we term “no noi”and (iii)interfer
ence fading power g ∼exp(μ)rather than following an arbitrary distribution 3.In this subction we continue assume the interference power follows a general distribution,so we consider two special cas corresponding to (i)and (ii)above.
3The
加拿大新总理interference power is also attenuated by the path loss so the mean
interference power for each individual ba station is less than the mean desired power,by de finition,even though the fading distributions have the same mean μ,which is a proxy for the transmit power.
1)General Fading,Noi,α=4:First,ifα=4,Theorem 1admits a form that can be evaluated according to轻轻松松背单词
∫∞
0e−ax e−bx2d x=
√
π
b
exp
(
a2
4b
)
Q
(
a
√
2b
)
,(6)
where Q(x)=1√
2π∫∞
x
exp(−y2/2)d y is the standard Gaus-
sian tail probability.Setting a=πλβ(T,α)and b=μTσ2= T/SNR gives
p c(T,λ,4)=π3 2λ
√
T
SNR exp
(
(λπβ(T,4))2
4T
SNR
)
Q
⎛
⎝λπβ(T,4)
√
2T
SNR
⎞
⎠.
(7)
Therefore,given the numerical calculation ofβ(T,4)for a chon interference distribution,the coverage probability can be found in quasi-clod form since Q(x)can be evaluated nearly as easily as a basic trigonometric function by modern calculators and software programs.
2)General Fading,No Noi,α>2:In most modern cel-lular networks thermal noi is not an important consideration. It can be neglected in the cell interior becau it is very small compared to the desired signal power(high SNR),and also at the cell edge becau the interference power is typically so much larger(high INR).Ifσ2→0(or transmit power is incread sufficiently),then using Theorem1it is easy to e
that
p c(T,λ,α)=
1
β(T,α)
.(8)
In the next subction,we show that(7)does in fact reduce to(8)asσ2→0,which is not obvious by inspection.
It is interesting to note that in this ca the probability of coverage does not depend on the ba station densityλ.It follows that both very den and very spar networks have a positive probability of coverage when noi is negligible.Intu-itively,this means that increasing the number of ba stations does not affect the coverage probability,becau the increa in signal power is exactly counter-balanced by the increa in interference power.This matches empirical obrvations in interference-limited urban networks as well as predictions of traditional,less-tractable models.In interference-limited networks,increasing coverage probability typically requires interference management techniques,for example frequency reu,and not just the deployment of more ba stations.Note that deploying more ba stations does allow more urs to be simultaneously covered in a given area,both in practice and under our model,becau we assume one active ur per cell.
3)General Fading,Small but Non-zero Noi:A poten-tially uful low noi approximation of the succ
ess proba-bility can be obtained that is more easily computable than the constant noi power expression and more accurate than the no noi approximation forσ2=0.Using the expansion exp(−x)=1−x+o(x),x→0it can be found after an integration by parts of(1)that
p c(T,λ,α)=1
β(T,α)−μTσ
2(λπ)−α/2
β(T,α)
Γ
(
1+α
2
)
+o(σ2)
(9)
For the special ca ofα=4,it is not immediately obvious
that(7)is equivalent to(8)asσ2→0,but indeed it is true.
It is possible to write(7)as
p c(T,λ,4)=
π32λ
√
2
a
xQ(x)exp
(
x2
2
)
(10)
where x=a√
2b
and a,b as before.The ries expansion of
Q(x)for large x is
Q(x)=
1√
2π
exp
(
−
x2
2
)[
1
x
−
1
x2
+o(x−4)
]
which means that
lim
x→∞
xQ(x)exp
(
x2
2
)
=
1
√
2π
,
which allows simplification of(10)to(8)for the ca of no
noi.
D.Special Cas:Interference is Rayleigh Fading
Significant simplification is possible when the interference
power follows an exponential interference
experiences Rayleigh fading and shadowing is neglected.We
give the coverage probability for this ca as Theorem2.
Theorem2:The probability of coverage of a typical ran-
domly located mobile ur experiencing exponential interfer-
ence is
p c(T,λ,α)=πλ
∫∞
e−πλv(1+ρ(T,α))−μTσ2vα/2d v,(11)
where
ρ(T,α)=T2/α
∫∞
T−2/α
1
1+uα/2
d u.(12)
Proof:The proof is a special ca of Theorem1,but
however lends to much simplification.The proof is provided
in Appendix B.
We now consider the special cas of no noi andα=4.
1)Exponential Fading,Noi,α=4:Whenα=4,using
the same approach as in(6),we get
jx2锄大地
p c(T,λ,4)=
π32λ
√
T
SNR
exp
(
(λπκ(T))2
4T
SNR
)
Q
⎛
⎝λπκ(T)
√
2T
SNR
⎞lo you
⎠,
(13)
whereκ(T)=1+ρ(T,4)=1+
√
T(π/2−arctan(1/
√
T)).
This expression is quite simple and is practically clod-
form,requiring only the computation of a simple Q(x)value.
2)Exponential Fading,No Noi,α>2:In the no noi
ca the result is very similar to general fading in appearance,
<
p c(T,λ,α)=
1
1+ρ(T,α)
,
withρ(T,α)being faster and easier to compute than the more
general expressionβ(T,α).When the path loss exponentα=
4,the no noi coverage probability can be further simplified
to
p c(T,λ,4)=
1
1+
√
T(π/2−arctan(1/
√
T))
.(14)
This is a remarkably simple expression for coverage probabil-ity that depends only on the SIR threshold T ,and as expected it goes to 1for T →0and to 0for T →∞.For example,if T =1(0dB,which would allow a maximum rate of 1bps/Hz),the probability of coverage in this fully loaded network is 4(4+π)−1=0.56.This will be compared in more detail to classical models in Section V.A small noi approximation can be performed identically to the procedure of Section III-C3with 1+ρ(T,α)replacing β(T,α)in (9).
IV.A VERAGE A CHIEVABLE R ATE
In this ction,we turn our attention to the mean data rate
achievable over a cell.Speci fically we compute the mean rate in units of nats/Hz (1bit =ln(2)=0.693nats)for a typical ur where adaptive modulation/coding is ud so each ur can t their rate such that they achieve Shannon bound for their instantaneous ln(1+SINR ).Interference is treated as noi which means the true channel capacity is not achieved,which would require a multiur receiver [26]–[28],but future work could relax this constraint within the random network framework,[29],[30].In general,almost any type of modulation,coding,and receiver structure can be easily treated by adding a gap approximation to the rate τ→ln(1+SIN
R /G )where G ≥1is the gap.Naturally,multi-antenna transmission could further increa the rate.The technical tools and organization are similar to Section III so the discussion will be more conci.The results are all for exponentially distributed interference power but general distributions could be handled as well following the approach of Theorem 1and techniques from [31].
A.General Ca and Main Result
We begin by stating the main rate theorem that gives the
ergodic capacity of a typical mobile ur in the downlink.Theorem 3:The average ergodic rate of a typical mobile ur and its associated ba station in the downlink is
τ(λ,α)≜E [ln(1+SINR )]
=
∫
r>0
e −πλr 2∫t>0
e −σ
2
μr α(e t −1)
⋅ℒI r (μr α
(e t
−1))d t 2πλr d r,
where
ℒI r (μr α(e t −1))=
exp (−πλr 2(e t −1)2/α
英语 翻译∫
∞
(e t −1)−2/α
1
1+x α/2
d g )
.
Proof:The proof is provided in Appendix C.
The computation of τin general requires three numerical integrations.
B.Special Ca:α=4
For α=4the mean rate simpli fies to
τ(λ,4)=
∫t>0∫
r>0
e −σ2μr 4(e t
−1)⋅e −πλr 2
(1+√e t
−1(π/2−arctan(1/√
e t −1)))
2πλr d r d t.
=∫t>0∫
receptionv>0
e −σ2μv 2(e t −1)/(πλ)
2
⋅e
−v (1+√e t −1(π/2−arctan(1/√
e t −1)))
d v d t.
Using (6),
τ(λ,4)=∫t>0音乐英语
六级时间安排√
π
b (t )
exp (a (t )24b (t )
)
Q (
a (t )
√
2b (t )
)d t,(15)where a (t )=1+√e t −1(π/2−arctan(1/√e t −1))
新通国际and b (t )=σ2μ(e t −1)/(πλ)2.The final expression (15)be eval-uated numerically with one numerical integration (presuming an available look up table for Q (x )).
C.Special Ca:No Noi
When noi is neglected,σ2→0,so τ(λ,α)
=
∫
r>0
∫t>0
e
−πλr 2
(
1+(e t −1)
2/α
∫∞
(e t −1)−2/α1
1+x α/2
d x
)
⋅2πλr d r d t.
Using the substitution πλr 2→v ,
τ(λ,α)
=∫t>0∫r>0
e −v (1+(e t −1)2/α∫∞(e t −1)−2/α11+x α/2
d x )d v d t =∫t>011+(
e t −1)2/α∫∞(e t −1)−2/α11+x α/2d x d t,(16)a quantity that again does not depend on λ.As in the ca of
coverage,increasing the ba station density does not increa the interference-limited ergodic capacity per ur in the down-link becau the distance from the mobile ur to the nearest ba station and the average distance to the nearest interferer both scale like Θ(λ−1/2),which cancel.Note,however,that the overall sum throughput and area spectral ef ficiency of the network do increa linearly with the number of ba stations since the number of active urs per area achievin
g rate τis exactly λ,assuming that the ur density is suf ficiently large such that there is at least one mobile ur per cell.
In the particular ca of α=4in conjunction with no noi,
(e t −1)2/α
∫∞
(e t −1)−2/α11+x α/2
d x =√
e t −1(π
2−arctan(1√e t −1
))
,so the mean rate is expresd to a single simple numerical integration that yields a preci scalar
τ(λ,4)=
∫
t>01
1+
√e t −1(π/2−arctan(1/√e t −1))d t ≈1.49nats /c /Hz .(17)
