Will the Area Spectral Efficiency Monotonically
Grow as Small Cells Go Dense?

arXiv:1505.01920v2 [cs.NI] 26 Sep 2017

Ming Ding, National ICT Australia (NICTA), Australia {Ming.Ding@nicta.com.au}
David López-Pérez, Bell Labs Alcatel-Lucent, Ireland {dr.david.lopez@ieee.org}
Guoqiang Mao, The University of Technology Sydney and NICTA, Australia {g.mao@ieee.org}
Peng Wang, The University of Sydney and NICTA, Australia {thomaspeng.wang@sydney.edu.au}
Zihuai Lin, The University of Sydney, Australia {zihuai.lin@sydney.edu.au}

Abstract—In this paper, we introduce a sophisticated path loss
model into the stochastic geometry analysis incorporating both
line-of-sight (LoS) and non-line-of-sight (NLoS) transmissions to
study their performance impact in small cell networks (SCNs).
Analytical results are obtained on the coverage probability and
the area spectral efficiency (ASE) assuming both a general path
loss model and a special case of path loss model recommended by
the 3rd Generation Partnership Project (3GPP) standards. The
performance impact of LoS and NLoS transmissions in SCNs
in terms of the coverage probability and the ASE is shown to
be significant both quantitatively and qualitatively, compared
with previous work that does not differentiate LoS and NLoS
transmissions. Particularly, our analysis demonstrates that when
the density of small cells is larger than a threshold, the network
coverage probability will decrease as small cells become denser,
which in turn makes the ASE suffer from a slow growth or even
a notable decrease. For practical regime of small cell density, the
performance results derived from our analysis are distinctively
different from previous results, and shed new insights on the
design and deployment of future dense/ultra-dense SCNs. 1

I. I NTRODUCTION
Dense small cell networks (SCNs) are considered as the
most promising approach to rapidly increase network capacity
and meet the ever-increasing capacity demands in the 5th generation (5G) systems [1]. However, up to now, most theoretical
studies on SCNs only consider simple path loss models that do
not differentiate Line-of-Sight (LoS) and Non-Line-of-Sight
(NLoS) transmissions [2-5]. The major conclusion [2-5] is
that neither the number of small cells nor the number of cell
tiers changes the coverage probability in interference-limited
fully-loaded cellular networks. Such conclusion implies that
the area spectral efficiency (ASE) will monotonically grow
as small cells go dense. An intriguing question is: Does this
optimistic conclusion still hold when practical LoS and NLoS
transmissions are considered in SCNs?
It is well-known that LoS transmission often occurs when
the distance between a transmitter and a receiver is small,
while NLoS is more common in long-distance transmissions as
well as in office environments and in central business districts.
For a given network environment, when the distance between
1 1536-1276 c 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. Please find the final version
in IEEE from the link: http://ieeexplore.ieee.org/document/7416981/. Digital
Object Identifier: 10.1109/GLOCOM.2015.7416981

transmitter and receiver decreases, the probability that an LoS
path exists between them increases, causing a transition from
NLoS transmission to LoS transmission.
To the best of authors’ knowledge, up to now, performance
analysis considering both LoS and NLoS transmissions are [6]
and [7]. In [6], the authors assumed a multi-slope piecewise path loss function. Such multi-slope piece-wise path loss
function does not fit well with the NLoS and LoS model
defined by the 3rd Generation Partnership Project (3GPP)
standards, in which the path loss function is not a one-toone mapping to the distance [8]. In [7], the authors treated the
event of LoS or NLoS transmission as a probabilistic event for
a millimetre wave communication scenario. To simplify the
analysis, the LoS probability function was approximated by
a moment-matched equivalent step function. The single-piece
path loss model and the proposed step function for modeling
the transition from NLoS to LoS transmissions are also not
compatible with the model recommended by the 3GPP [8, 9].
In this paper, we use a general path loss model that features
piece-wise path loss functions with probabilistic LoS and
NLoS transmissions. Note that the proposed model is very
general and includes almost all existing models used to capture
LoS and NLoS transmissions [6-9] as its special cases. The
main contributions of the paper are as follows:
• Analytical results are obtained for the coverage probability and the ASE using a general path loss model
incorporating both LoS and NLoS transmissions.
• Using the above results, closed-form expressions are
further obtained for the coverage probability and the ASE
for a special case based on the 3GPP standards.
• Our theoretical analysis reveals an important finding,
i.e., the ASE will initially increase with the increase of
the small cell density, but when the density of small
cells becomes sufficiently large, the network coverage
probability will decrease as small cells become denser.
This in turn makes the ASE suffer from a slow growth
or even a notable decrease. Thereafter, when the small
cell density is very large, the ASE will then grow almost
linearly with the network densification. These results are
not only quantitatively but also qualitatively different
from previous study results [2-7]. Thus, our results shed
new insights on the design and deployment of future

dense/ultra-dense SCNs in realistic environments.
The remainder of this paper is structured as follows.
Section II describes the system model. Section III presents
our main analytical results on the coverage probability and
the ASE, followed by their application in a 3GPP case in
Section IV. The derived results are validated using simulations
in Section V. Finally, the conclusions are drawn in Section VI.
II. S YSTEM M ODEL
We consider a DL cellular network in which BSs are
deployed in a plane according to a homogeneous Poisson point
process (HPPP) Φ of intensity λ BSs/km2 . UEs are Poisson
distributed in the considered network with an intensity of λUE
BSs/km2 . λUE is assumed to be sufficiently larger than λ so
that each BS has at least one associated UE in its coverage
area. The distance between an arbitrary BS and an arbitrary UE
is denoted by r in km. Considering practical LoS and NLoS
transmissions, we propose to model the path loss associated
with distance r as (1), shown on the top of the next page.
In (1), the path loss function ζ (r) is segmented into N
pieces with the N -th piece ζn (r), n ∈ {1, 2, . . . , N }. For
each ζn (r), ζnL (r) is the n-th piece of the path loss function
for LoS transmission, ζnNL (r) is the n-th piece of the path loss
function for NLoS transmission and PrLn (r) is the n-th piece
of the LoS probability function. In more detail,
• ζn (r) is modeled as
(
L
for LoS
ζnL (r) = ALn r−αn ,
ζn (r) =
, (2)
−αNL
n ,
for NLoS
ζnNL (r) = ANL
r
n

•

with ALn and ANL
n , n ∈ {1, 2, . . . , N } being the path
losses at a reference distance r = 1 and αLn and
αNL
n , n ∈ {1, 2, . . . , N } being the path loss exponents
for the LoS and the NLoS cases in ζn (r), respectively.
NL
L
In practice, ALn , ANL
n , αn and αn are constants obtained
from field tests [8, 9].
PrLn (r) is the n-th piece probability function that a
transmitter and a receiver separated by a distance r ∈
[dn−1 , dn ) has an LoS path, which is usually a monotonically
decreasing function of r. For convenience,
 L
Prn (r) is further stacked into a piece-wise LoS probability function as 
PrL1 (r) , when 0 ≤ r ≤ d1




PrL2 (r) , when d1 < r ≤ d2
. (3)
PrL (r) =
..
..


.
.


 L
PrN (r) , when r > dN −1

Our model is consistent with the ones adopted in the
3GPP [8, 9]. Note that the considered path loss model in (1)
will degenerate to that adopted in [6] and [7] when PrLn (r) =
0, ∀n ∈ {1, 2, . . . , N } and N = 1, respectively.
As a common practice in the field [2-6], each UE is assumed
to be associated with the nearest BS to the UE, and the multipath fading between an arbitrary BS and an arbitrary UE is
modeled as independently identical distributed (i.i.d.) Rayleigh
fading, i.e., the channel gain is denoted by h and is modeled as

an i.i.d. exponential random variable (RV). The transmit power
of each BS and the additive white Gaussian noise (AWGN)
power at each UE are denoted by P and N0 , respectively.
III. A NALYSIS BASED

ON

G ENERAL PATH L OSS M ODEL

Using the properties of the HPPP, we study the performance
of SCNs by considering the performance of a typical UE
located at the origin o. We first investigate the coverage
probability and thereafter the ASE.
The coverage probability is defined as the probability that
the signal to interference plus noise ratio (SINR) of the typical
UE, denoted by SINR, is above a threshold γ:
pcov (λ, γ) = Pr [SINR > γ] ,

(4)

where the SINR is computed by
P ζ (r) h
,
Ir + N0
where Ir is the cumulative interference given by
X
P βi g i ,
Ir =
SINR

=

(5)

(6)

i∈Φ\bo

where bo is the BS associated with the typical UE and located
at distance r from the typical UE, and βi and gi are the path
loss and the multi-path fading channel gain associated with
the i-th interfering BS, respectively.
According to [6] and [7], the area spectral efficiency (ASE)
in bps/Hz/km2 for a given λ can be computed by
ˆ ∞
log2 (1 + x) fX (λ, x) dx, (7)
AASE (λ, γ0 ) = λ
γ0

where γ0 is the minimum working SINR for the considered
SCN, and fX (λ, x) is the probability density function (PDF)
of SINR observed at the typical UE at a particular value of λ.
Since pcov (λ, γ) can be defined as the complementary
cumulative distribution function (CCDF) of SINR, fX (λ, x)
can be expressed as
∂ (1 − pcov (λ, x))
fX (λ, x) =
.
(8)
∂x
Given the definition of the coverage probability and the ASE
presented in (4) and (7) respectively, and using the path loss
model of (1), we present our main result on pcov (λ, γ) in
Theorem 1 shown on the next page. Plugging pcov (λ, γ) from
(9) of Theorem 1 into (8), we can get the ASE using (7).
As can be seen from Theorem 1, the coverage probability,
pcov (λ, γ), is a function of the piece-wise path loss function
 L {ζn (r)} and the piece-wise LoS probability function
Prn (r) . We will investigate their impacts in the sequel.
IV. S TUDY

OF A

3GPP S PECIAL C ASE

As a special case of Theorem 1, we consider the path loss
function, ζ (r),
(
L
AL r−α ,
w/ probability PrL (r)
 , (14)
ζ (r) =
NL
ANL r−α , w/ probability 1 − PrL (r)

together with the linear LoS probability function, which is
(
1 − dr1 ,
0 < r ≤ d1
L
Pr (r) =
,
(15)
r > d1
0,

ζ (r)

(


ζ1L (r) ,


ζ
1 (r) =


ζ NL (r) ,


( 1




ζ2L (r) ,

ζ2 (r) =
ζ2NL (r) ,

..



.

(


L


ζN
(r) ,
ζN (r) =


NL
ζN (r) ,

=

with probability PrL1 (r)
 ,
with probability 1 − PrL1 (r)

when 0 ≤ r ≤ d1

with probability PrL2 (r)
 ,
with probability 1 − PrL2 (r)

when d1 < r ≤ d2
.

(1)

..
.

with probability PrLN (r)
 , when r > dN−1
with probability 1 − PrLN (r)

Theorem 1. Considering the path loss model of (1), pcov (λ, γ) can be derived as
pcov (λ, γ)

N
X

=

n=1

where TnL =

´ dn

dn−1

Pr

h

L
P ζn
(r)h
Ir +N0

i
´d
L
> γ fR,n
(r) dr, TnNL = d n

n−1

Pr

L
NL
0 and ∞. Moreover, fR,n
(r) and fR,n
(r) are represented as
L
fR,n
(r)

and

Furthermore, Pr

h

L
P ζn
(r)h
Ir +N0

=

h


TnL + TnNL ,

NL
P ζn
(r)h
Ir +N0

(9)

i
NL
> γ fR,n
(r) dr, and d0 and dN are respectively defined as


PrLn (r) × exp −πr 2 λ × 2πrλ,

(dn−1 < r ≤ dn ) ,

(10)



NL
fR,n
(r) =
1 − PrLn (r) × exp −πr 2 λ × 2πrλ, (dn−1 < r ≤ dn ) .
i
h NL
i
P ζ (r)h
> γ and Pr Irn+N0 > γ are respectively computed by
 L





P ζn (r) h
γN0
γ
Pr
>γ
= exp − L
LIr
,
Ir + N0
P ζn (r)
P ζnL (r)

and
Pr



P ζnNL (r) h
>γ
Ir + N0



=


exp −

γN0
P ζnNL (r)



LIr



γ
P ζnNL (r)



(11)

(12)

,

(13)

where LIr (s) is the Laplace transform of RV Ir evaluated at s.
Proof: See Appendix A.

both respectively recommended in the 3GPP [8, 9].
Considering the general path loss model presented in (1),
the path loss model presented in (14) and (15) can be deemed
as a special case of (1) with the following substitution: N = 2,
L
NL
ζ1L (r) = ζ2L (r) = AL r−α , ζ1NL (r) = ζ2NL (r) = ANL r−α ,
PrL1 (r) = 1 − dr1 , and PrL2 (r) = 0. For clarity, this 3GPP
special case is referred to as 3GPP Case 1 in the sequel.
According to Theorem 1, pcov (λ, γ) can be obtained as
2
X

(16)
TnL + TnNL .
pcov (λ, γ) =

Furthermore, to compute LIr
r ≤ d1 , we propose Lemma 2.



L

γr α
P AL



in the range of 0 <

n=1

In the following subsections, we investigate T1L , T1NL , T2L ,
and T2NL , respectively.
A. The Computation of

T1L

Lemma 2. LIr
calculated by



for 3GPP Case 1

T1L

From Theorem 1,
can be derived as
!
!
ˆ d1
L
L
γrα N0
γrα
L
L
LIr
fR,1
(r) dr, (17)
exp −
T1 =
L
L
P
A
P
A
0
L
where according to Theorem 1 and (15), fR,1
(r) becomes
L
fR,1 (r) =



r
1−
× exp −πr2 λ × 2πrλ, (0 < r ≤ d1 ) . (18)
d1

LIr



L

γr α
P AL



=

L

γr α
P AL



in the range of 0 < r ≤ d1 can be



 
−1

L
αL
, d1
exp −2πλ ρ1 α , 1, γr

−1 

L
αL
,r
−ρ1 α , 1, γr


 


L −1
2πλ
× exp
, d1
ρ1 αL , 2, γrα
d0




L −1
,r
−ρ1 αL , 2, γrα

!
−1
γANL αL
ρ1 α , 2,
, d1
r
AL
!!!
 NL
−1
γA
NL
αL
,r
r
−ρ1 α , 2,
AL
!!
 NL
−1
γA
NL
αL
× exp −2πλρ2 α , 1,
,
, d1
r
AL
2πλ
× exp −
d0

NL



(0 < r ≤ d1 ) , (19)
where
ρ1 (α, β, t, d) =


 (β+1) 
β+1
β+1
d
α
;1 +
; −td ,
2 F1 1,
β+1
α
α
and

!
−1
L
NL
γA
, d1
rα
exp −2πλ ρ1 αL , 1,
ANL
 L
−1 !!!
γA αNL
L
,r
r
−ρ1 α , 1,
ANL
!
 L
−1
2πλ
γA αNL
L
× exp
ρ1 α , 2,
, d1
r
d0
ANL
 L
−1 !!!
γA αNL
L
,r
r
−ρ1 α , 2,
ANL

 


−1
2πλ
NL
αNL
, d1
ρ1 α , 2, γr
× exp −
d0




NL −1
,r
−ρ1 αNL , 2, γrα





NL −1
,
, d1
× exp −2πλρ2 αNL , 1, γrα


(0 < r ≤ d1 ) , (24)
where ρ1 (α, β, t, d) and ρ2 (α, β, t, d) are defined in (20) and
(21), respectively.
Proof: See Appendix C.

(20)

ρ2 (α, β, t, d) =
 −(α−β−1) 


d
β+1
β+1
1
1,
1
−
, (21)
F
;
2
−
;
−
2 1
t (α − β − 1)
α
α
tdα
where 2 F1 [·, ·; ·; ·] is the hyper-geometric function [10].
Proof: See Appendix B.

C. The Computation of T2L for 3GPP Case 1
From Theorem 1, T2L can be derived
as
!
!
ˆ ∞
L
αL
γrα
γr N0
L
L
LIr
fR,2
(r) dr
T2 =
exp −
L
L
P
A
P
A
d1
= 0.
(25)
Note that the reason why T2L = 0 in (25) is because
according to Theorem 1 and (15), we have

L
fR,2
(r) = 0 × exp −πr2 λ × 2πrλ
= 0, (r > d1 ) .
(26)
D. The Computation of T2NL for 3GPP Case 1

B. The Computation of T1NL for 3GPP Case 1
From Theorem 1, T1NL can be derived as
!
!
ˆ d1
αNL
αNL
γr
γr
N
0
LIr
f NL (r) dr, (22)
T1NL = exp −
P ANL
P ANL R,1
0
NL
where according to Theorem 1 and (15), fR,1
(r) becomes

r
NL
2
fR,1 (r) =
× exp −πr λ × 2πrλ, (0 < r ≤ d1 ) . (23)
d1
 NL 
α
in the range of 0 <
Furthermore, to compute LIr γr
P ANL

r ≤ d1 , we propose Lemma 3.
 NL 
α
in the range of 0 < r ≤ d1 can be
Lemma 3. LIr γr
P ANL

calculated by
 NL 
α
=
LIr γr
P ANL

From Theorem 1, T2NL can be derived as
!
!
ˆ ∞
NL
NL
γrα
γrα N0
NL
NL
LIr
fR,2
(r) dr, (27)
T2 = exp −
NL
NL
P
A
P
A
d1
NL
where according to Theorem 1 and (15), fR,2
(r) becomes

NL
2
fR,2 (r) = 1 × exp −πr λ × 2πrλ

= exp −πλr2 × 2πrλ, (r > d1 ) . (28)
 NL 
α
in the range of r >
Furthermore, to compute LIr γr
P ANL

d1 , we propose Lemma 4.
 NL 
α
in the range of r > d1 can be
Lemma 4. LIr γr
P ANL

calculated
 NLby
α
LIr γr
=
P ANL


−1 

NL
αNL
, r , (r > d1 ) , (29)
exp −2πλρ2 α , 1, γr
where ρ2 (α, β, t, d) is defined in (21).
Proof: See Appendix D.

E. The Results of pcov (λ, γ) and AASE (λ, γ0 )
To sum up, p
p

cov

cov

(λ, γ) for 3GPP Case 1 can be written as

(λ, γ) = T1L + T1NL + T2NL ,

(30)

where T1L , T1NL and T2NL are computed from closed-form
expressions using (17), (22) and (27), respectively.
Plugging pcov (λ, γ) obtained from (30) into (8), we can get
the result of AASE (λ, γ0 ) from (7) for 3GPP Case 1.
V. S IMULATION

AND

D ISCUSSION

probability given by the proposed analysis decreases as λ
increases, due to the transition of a large number of interference paths from NLoS to LoS. It is important to note that the
coverage probability performance of the proposed analysis for
3GPP Case 1 peaks at a certain value λ0 . When λ increases
above λ0 , interfering BSs may be very close to the typical
UE and hence their signals may reach the UE via strong LoS
paths. Such critical point of λ0 can be readily obtained by
cov
setting the partial derivative
n cov of p (λ,
o γ) with regard to λ
∂p (λ,γ)
to zero, i.e., λ0 = arg
= 0 . The solution to this
∂λ
λ

In this Section, we use simulations to further study the
performance of SCNs and establish the accuracy of our
analysis for 3GPP Case 1 studied in Section IV. According
to [8] and [9], we use the following parameters: d1 = 0.3 km,
αL = 2.09, αNL = 3.75, AL = 10−4.11 , ANL = 10−3.29 ,
P = 24 dBm, N0 = −95 dBm.

equation can be numerically found using a standard bisection
searching [11]. In Fig. 1, the numerical results of λ0 are
15.85 BSs/km2 and 10.21 BSs/km2 for γ = 1 and γ = 10,
respectively.

A. Validation and Discussion of pcov (λ, γ)

The results of AASE (λ, 1) and AASE (λ, 10) are plotted in
Fig. 2, comparing the proposed stochastic geometry analysis
with the conventional stochastic geometry analysis in [2].
As can be seen from Fig. 2, the analysis from [2] indicates
that when the SCN is dense enough, e.g., λ ≥ 102 BSs/km2 ,
the ASE performance increases almost linearly with λ, which
is logically correct from the conclusion that pcov (λ, γ) is
invariable with respect to λ for a given γ in dense SCNs [2].
In contrast, the proposed stochastic geometry analysis reveals a more complicated trend for the ASE performance.
Specifically, when the SCN is relatively sparse, e.g., λ ≤ λ0 ,
the ASE quickly increases with λ because the network is
generally noise-limited, thus adding more small cells immensely benefits the ASE. When the SCN is extremely dense,
e.g., λ ≥ 104 BSs/km2 , the ASE exhibits a nearly linear
trajectory with regard to λ because both the signal power
and the interference power are now LoS dominated and thus
statistically stable. Note that the pace is
 lower than in [2]. As
for the practical range of λ, i.e., λ ∈ λ0 , 104 BSs/km2 , the
ASE first exhibits a slowing-down in the rate of growth (when
γ = 1) or even a notable decrease in its absolute value (when
γ = 10). This is attributed to the fast decrease of the coverage
probability at around λ ∈ [λ0 , λ1 ] BSs/km2 as shown in Fig. 1,
where λ1 is another threshold larger than λ0 . When λ ≥ λ1 ,
the ASE will pick up the growth rate as the decrease of the
coverage probability becomes very gentle. In Fig. 2, the value
of λ1 seems to be around 103 BSs/km2 .
Our new finding indicates the significant impact of the path
loss model incorporating both NLoS and LoS transmissions.
As a confirmation, noting that in Fig. 1, we can observe that
increasing γ from 1 to 10 will greatly accelerate
the

 decrease
of the coverage probability at around λ ∈ 10, 102 BSs/km2 ,
which in turn causes the notable decrease of the ASE at that
range of λ when γ = 10 in Fig. 2.
With the defined thresholds λ0 and λ1 , SCNs can be roughly
classified into 3 categories, i.e., the sparse SCN (0 < λ ≤ λ0 ),
the dense SCN (λ0 < λ ≤ λ1 ) and the very dense SCN (λ >
λ1 ). The ASEs for both the sparse SCN and the very dense
SCN grow almost linearly with the increase of λ, while the

cov

The results of p (λ, γ) with γ = 1 and γ = 10 are plotted
in Fig. 1. As can be observed from both figures, our analytical
results perfectly match the simulation results. Since the results
of AASE (λ, γ0 ) are computed based on pcov (λ, γ), we will
only use analytical results for pcov (λ, γ) in our discussion
hereafter. For comparison, we have also included analytical
results assuming a simplistic path loss model that does not
differentiate LoS and NLoS transmissions [2]. Note that in [2],
only one path loss exponent is defined and denoted by α. Here,
α is set to αNL to show the analytical results from [2].
From Fig. 1, we can observe that the coverage probability
performance given by the stochastic geometry analysis in [2]
first increases with the BS density because more BSs provide
better coverage in noise-limited networks. Then, when λ is
large enough, the coverage probability becomes independent
of λ since the network is pushed into the interference-limited
region, e.g., λ > 102 BSs/km2 for the analysis in [2]. This
observation is consistent with the conclusion in [2], which
shows that for a sufficiently large λ, the coverage probability
becomes almost a constant with the increase of the small cell
density. The intuition behind the observation is that with the
simplistic assumption on the path loss model, the increase in
interference power is counterbalanced by the increase in signal
power in a interference-limited network, and thus the coverage
probability remains the same as λ increases.
In Fig. 1, the coverage probability performance of the
proposed stochastic geometry analysis incorporating both LoS
and NLoS transmissions exhibits a significant deviation from
that of the analysis from [2], because when the distance
r decreases, or equivalently when the small cell density λ
increases, LoS transmission occurs with an increasingly higher
probability than NLoS transmission. Specifically, when the
SCN is sparse and thus noise-limited, e.g., λ ≤ 10 BSs/km2 ,
the coverage probability given by the proposed analysis grows
as λ increases for the same reason as explained in the above
paragraph, i.e., deploying more small cells is beneficial for
removing coverage holes. Then, when the network is dense
enough and all coverage holes are removed, the coverage

B. Discussion of the Analytical Results of AASE (λ, γ0 )

8

0.9

10

The proposed analysis [γ=1] (Analytical)
The proposed analysis [γ=1] (Simulation)
The analysis in [2] [γ=1] (Analytical)
The proposed analysis [γ=10] (Analytical)
The proposed analysis [γ=10] (Simulation)
The analysis in [2] [γ=10] (Analytical)

Probability of SINR>γ

0.7
0.6

The proposed analysis [γ0=1] (Analytical)

0.5
0.4
0.3
0.2
0.1
0

The analysis in [2] [γ0=1] (Analytical)

6

Area spectral efficiency [bps/Hz/km2]

0.8

10

The proposed analysis [γ0=10] (Analytical)

4

10

λ ≈10~16
0

2

10

Category 3

0

10

Category 2
Category 1

−2

10
−4

10

−2

10

0

10
BS density λ [BSs/km2]

2

10

4

−1

10

ASE of the dense SCN shows a slow growth or even a notable
decrease with the increase of λ. From Fig. 2, we can get a new
look at the ultra-dense SCN, which has been identified as one
of the key enabling technologies of the 5G networks [1]. Up to
now, there is no consensus in both industry and academia on
that at what density a SCN can be categorized as an ultra-dense
SCN. According to our study, for 3GPP Case 1, we propose
that the 5G systems should target the third category of SCNs
as ultra-dense SCNs, i.e., the SCNs with λ > λ1 , because
the associated ASE will grow almost linearly as λ increases.
Numerically speaking, λ1 is around 103 BSs/km2 from Fig. 2.
It is particularly important to note that the second category
of SCNs (λ0 < λ ≤ λ1 ) is better avoided in practical SCN
deployments due to its cost-inefficiency shown in Fig. 2.
VI. C ONCLUSION
In this paper, we have shown that a sophisticated path loss
model incorporating both LoS and NLoS transmissions has
a significant impact on the performance of SCNs, measured
by the two metrics of the coverage probability and the ASE.
Such impact is not only quantitative but also qualitative.
Specifically, our theoretical analysis have concluded that the
ASE will initially increase with the increase of the small
cell density, but when the density of small cells is larger
than a threshold λ0 , the network coverage probability will
decrease, which in turn makes the ASE suffer from a slow
growth or even a notable decrease as the small cell density
increases. Furthermore, the ASE will grow almost linearly as
the small cell density increases above another larger threshold
λ1 . According to our study, for 3GPP cases, we propose that
the 5G systems should target the SCNs with λ > λ1 as ultradense SCNs. Numerically speaking, λ1 appears to be around
several 103 BSs/km2 .
The intuition behind our conclusion is that when the density
of small cells is larger than a threshold, the interference power
will increase faster than the signal power due to the transition
of a large number of interference paths from NLoS to LoS,
and thus the small cell density matters!
As our future work, we will incorporate more sophisticated
UE association strategies and more practical multi-path fading

0

10

Fig. 1. pcov (λ, γ) vs λ for 3GPP Case 1.

3

λ1≈10

The analysis in [2] [γ0=10] (Analytical)

10

1

10

2

3

10
10
BS density λ [BSs/km2]

4

10

5

10

Fig. 2. AASE (λ, γ0 ) vs λ for 3GPP Case 1.

model into the analysis of SCNs because the multi-path fading
model is also affected by LoS and NLoS transmissions.
A PPENDIX A: P ROOF

OF

T HEOREM 1

From (4) and (5), we can derive pcov (λ, γ) straightforwardly
ˆ
as
(a)
cov
p (λ, γ) =
Pr [ SINR > γ| r] fR (r) dr

ˆr>0 
P ζ (r) h
> γ r fR (r) dr
Pr
=
Ir + N0
r>0
N
X

△
(31)
TnL + TnNL ,
=
n=1

TnL
´

where
and TnhNL are piece-wise
functions defined
i
L
d
P ζn
(r)h
n
L
>
γ
f
(r)
dr and TnNL
TnL =
Pr
R,n
Iri
+N0
n−1
h dNL
´ dn
P ζn (r)h
NL
Ir +N0 > γ fR,n (r) dr, respectively. Besides,
dn−1 Pr

as
=

d0
and dN are respectively defined as 0 and ∞. Moreover,
L
NL
fR,n
(r) and fR,n
(r) are the piece-wise PDFs of the event
that the UE is associated with the nearest BS with an LoS
path at distance r and the event that the UE is associated with
the nearest BS with an NLoS path at distance r, respectively.
L
Regarding fR,n
(r), we define two events in the following,
whose joint event is equivalent to the event that the UE is
associated with a BS with an LoS path at distance r.
•
•

Event B: the nearest BS is located at distance r
Event DL : the BS is one with an LoS path

According to [2], the cumulative density function (CDF) of
Event B with regard to r is given by

B
(r) = 1 − exp −πr2 λ , (dn−1 < r ≤ dn ) . (32)
FR,n

B
Hence, taking the derivative of FR,n
(r) with regard to r,
yields the PDF of Event B as

B
(r) = exp −πr2 λ × 2πrλ, (dn−1 < r ≤ dn ) . (33)
fR,n

B
The PDF fR,n
(r) should be further thinned by the probability
of Event DL on condition of r, which is PrLn (r), so that we
can get the PDF of the joint event of B and C NL as
L
B
fR,n
(r) = PrLn (r) × fR,n
(r) .

(34)

NL
Regarding fR,n
(r), we also define two events in the following, whose joint event is equivalent to the event that the
UE is associated with a BS with an NLoS path at distance r.
•
•

Event B: the nearest BS is located at distance r
Event DNL : the BS is one with an NLoS path

B
Similar to (34), the PDF fR,n
(r) should be further thinned
L
by the probability
of
Event
D
on condition of r, which is

1 − PrLn (r) , so that we can get the PDF of the joint event
of B and DL as
NL
fR,n
(r) =


1 − PrLn (r) ×exp −πr2 λ ×2πrλ, (dn−1 < r≤ dn ) . (35)
h L
i
P ζ (r)h
As for the calculation of Pr Irn+N0 > γ in (31), we have


 
 L
γ (Ir + N0 )
P ζn (r) h
> γ = E[Ir ] Pr h >
Pr
Ir + N0
P ζnL (r)



γ (Ir + N0 )
, (36)
= E[Ir ] F̄H
P ζnL (r)

where E[X] {·} denotes the expectation operation by taking
the expectation over the variable X and F̄H (h) denotes the
complementary cumulative density function (CCDF) of RV h.
Since we assume h to be an exponential RV, we have F̄H (h) =
exp (−h) and thus (36) can be further derived as




 L
γ (Ir + N0 )
P ζn (r) h
> γ = E[Ir ] exp −
Pr
Ir + N0
P ζ L (r)
 n


γ
γN0
LIr
, (37)
= exp − L
P ζn (r)
P ζnL (r)
of Iri evaluated at s.
where LIr (s) is the Laplace transform
h NL
P ζn (r)h
As for the calculation of Pr Ir +N0 > γ in (31), similar
to (36), we have
 



 NL
γ
γN0
P ζn (r) h
LIr
.(38)
> γ = exp − NL
Pr
Ir + N0
P ζn (r)
P ζnNL (r)
Our proof of Theorem 1 is completed by plugging (34),
(35), (37), and (38) into (31).
A PPENDIX B: P ROOF

OF

Since 0 < r ≤ d1 , E[g] {exp (−sP β (u) g)} in (39) should
consider interference from both LoS and NLoS paths. Thus,
LIr (s) can be further derived as

ˆ
LIr (s) = exp −2πλ




u
u
1−
du
d1 1 + (sP AL )−1 uαL
r


ˆ d1
u
u
du
× exp −2πλ
d1 1 + (sP ANL )−1 uαNL
r


ˆ 1∞
u
× exp −2πλ
du . (40)
NL )−1 uαNL
d1 1 + (sP A
αL

Plugging s = γr
P AL into (40), and considering the definition
of ρ1 (α, β,t, d) and
 ρ2 (α, β, t, d) in (20) and (21), we can
L

shown in (19), which concludes our proof.

A PPENDIX C: P ROOF

L EMMA 3

OF

Followingthe same
 approach in Appendix B, it is ready to
NL
γr α
derive LIr P ANL in the range of 0 < r ≤ x1 as
LIr



NL

γr α
P ANL





=
ˆ


exp −2πλ

d1



ˆ

d1



ˆ

∞


× exp −2πλ

Similarly, LIr
be calculated by
LIr



NL

γr α
P ANL





u
1−
d1

r2


× exp −2πλ



u
d1

r

1+

NL

γr α
P ANL

P AL

−1

1+



1+
NL

γr α
P ANL

uαL

u


NL

γr α
P ANL

P ANL

−1

αNL

u



NL

γr α
P ANL

P ANL

−1


du




du



u

d1



u





du . (41)

αNL

u

in the range of x1 < r ≤ d1 can

=


exp −2πλ


Based on Theorem 1, it is straightforward to derive LIr (s)
in the range of 0 < r ≤ d1 as

where (a) in (39) is obtained from [2].

γr α
P AL

obtain LIr

L EMMA 2

LIr (s) = E[Ir ] { exp (−sIr )| 0 < r ≤ d1 }






X
P βi gi  0 < r ≤ d1
= E[Φ,{βi },{gi }] exp −s


i∈Φ/bo



 Y
E[β,g] {exp (−sP βg)} 0 < r ≤ d1
= E[Φ]


i∈Φ/bo

ˆ ∞

(a)
= exp −2πλ
1 − E[g] {exp (−sP β (u) g)} udu
r

(39)
0 < r ≤ d1 ,

d1

ˆ

d1
r


× exp −2πλ

ˆ

u
d1

u
1+

∞

d1



NL

γr α
P ANL

P ANL

−1

uαNL

u
1+



NL

γr α
P ANL

P ANL

−1

uαNL




du



du . (42)

Our proof is thus completed by plugging (20) and (21) into
(41) and (42).
A PPENDIX D: P ROOF

OF

L EMMA 4

Followingthe same
 approach in Appendix B, it is ready to
NL
γr α
derive LIr P ANL in the range of r > d1 as
NL

LIr

γr α
P ANL

!

= exp −2πλ

ˆ

∞

u

!

−1 NL du
1 + γr αNL
uα





NL −1
,r
, (43)
= exp −2πλρ2 αNL , 1, γr α
r

where ρ2 (α, β, t, d) is defined in (21).
Our proof is thus completed with (43).
R EFERENCES
[1] D. López-Pérez, M. Ding, H. Claussen, and A. H. Jafari, “Towards
1 Gbps/UE in cellular systems: understanding ultra-Dense small cell
deployments,” arXiv:1503.03912 [cs.NI], Mar. 2015.
[2] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to
coverage and rate in cellular networks,” IEEE Trans. Commun., vol. 59,
no. 11, pp. 3122–3134, Nov. 2011.
[3] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M.
Franceschetti, “Stochastic geometry and random graphs for the analysis
and design of wireless networks,” IEEE J. Sel. Areas Commun., vol. 27,
no. 7, pp. 1029–1046, Sep. 2009.
[4] H. Dhillon, R. Ganti, F. Baccelli, and J. Andrews, “Modeling and
analysis of K-tier downlink heterogeneous cellular networks,” IEEE J.
Sel. Areas Commun., vol. 30, no. 3, pp. 550–560, Apr. 2012.
[5] S. Singh, H. S. Dhillon, and J. G. Andrews, “Offloading in heterogeneous
networks: Modeling, analysis, and design insights,” IEEE Trans. Wireless
Commun., vol. 12, no. 5, pp. 2484–2497, Apr. 2013.
[6] X. Zhang and J. G. Andrews, “Downlink cellular network analysis with
multi-slope path loss models,” arXiv:1408.0549 [cs.NI], 2014.
[7] T. Bai and R. W. Heath Jr., “Coverage and rate analysis for millimeter
wave cellular networks,” arXiv:1402.6430 [cs.NI], 2014.
[8] 3GPP, “TR 36.828 (V11.0.0): Further enhancements to LTE Time
Division Duplex (TDD) for Downlink-Uplink (DL-UL) interference
management and traffic adaptation,” Jun. 2012.
[9] Spatial Channel Model AHG (Combined ad-hoc from 3GPP & 3GPP2),
“Spatial Channel Model Text Description,”Apr. 2003.
[10] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products
(7th Ed.), Academic Press, 2007.
[11] R. L. Burden and J. D. Faires, Numerical Analysis (3rd Ed.), PWS
Publishers, 1985.

