1

Stochastic Geometry Analysis of Ultra-Dense
Networks: Impact of Antenna Height and
Performance Limits
arXiv:1712.02235v1 [] 6 Dec 2017

Marcin Filo, Chuan Heng Foh, Seiamak Vahid, and Rahim Tafazolli

Abstract
The concept of Ultra Dense Networks (UDN) is often seen as a key enabler of the next generation
mobile networks. However, existing analysis of UDNs, including Stochastic Geometry, has not been able
to fully determine the potential gains and limits of densification. In this paper we study performance of
UDNs in downlink and provide new insights on the impact of antenna height and careful site selection
on the network performance. We focus our investigation on the probability of coverage, average cell rate
and average area spectral efficiency for networks with regular and random BS deployments. We show
that under a path-loss model which considers antenna height there exists an upper limit on network
performance which is dependent on the path-loss model parameters. Our analysis shows an interesting
finding that even for over-densified networks a non-negligible system performance can be achieved.

Index Terms
Ultra dense networks, stochastic geometry, regular networks, irregular networks, SINR.

I. I NTRODUCTION
The concept of Ultra Dense Networks (UDN) is often seen as a key enabler of the next
generation mobile networks [1], [2]. UDNs are built on the concept of Heterogeneous cellular
Networks (HetNet) which provide an effective way for increasing network capacity and achieving
higher data rates. In contrast to HetNets, UDNs are expected to provide full network coverage
This paper was presented in part at the IEEE International Conference on Communications Workshops (ICC Workshops),
May 2017.

2

and thus they usually consist of significantly higher number of cells densely deployed within a
region. Besides, further densification of cells also offers higher area spectral efficiency which can
lead to further increase in network capacity. This potential network capacity gain has triggered
growing interest in UDNs from research community, industry and standardization bodies.
Investigation of cellular network performance is crucial for proper assessment of different
approaches for increasing network capacity. In general, the methods for investigation of cellular
network performance (including UDN) can be subdivided into two main groups. The first
group consists of approaches based on analytical modeling with simplified assumptions to
maintain tractability whilst the second group attempts to capture system operation in detail by
performing system-level simulations. Although in general the simulation based methods provide
more accurate results (since they can capture the detail of system operation), the investigation
based on the analytical models allow for better understanding of network operation and can
provide insightful design guidelines which are often difficult to obtain from simulations. However,
due to the complexity of the wireless network behavior, capturing its characteristics by means of
analytical modeling is a challenging task. Recently, Stochastic Geometry (SG) has emerged as
one of the most prominent approaches as it captures spatial characteristics of wireless networks
for performance analysis. This feature is especially important for investigation of UDN since its
performance behavior is highly dependent on spatial information of the network. This motivated
us to use SG as our main tool for the analysis of UDN performance.
Despite the growing interest and effort from research community to further our understanding
of UDN performance behavior, several key questions still remain unanswered. One of the
fundamental questions is related to the impact of densification on the UDN performance (e.g.
area spectral efficiency, coverage)1 . Whilst efforts have been made, conclusive demonstration on
how system performance changes with Base Station (BS) density and what happens when BS
density tends to infinity remains missing. Another important question is related to the impact
of network deployment (e.g. network site planning, antenna height) on the system performance.
For instance, although it is a common belief that site planning is not suitable in the context
of UDNs due to high cost and complexity, the impact of site planning on UDN performance
have not been quantified. In the following, we review key developments of UDN performance
1

See [3] for other examples

3

investigation.
A. Related work
One of the first works investigating the relationship between network density and system
performance is due to Andrews et al. [4]. The work shows surprising results where coverage
probability as well as mean achievable data rate per cell, do not depend on BS density (also known
as SINR invariance property [5]). The constant mean achievable data rate per cell implies that
continuingly increasing the number of BSs in wireless networks could lead to limitless overall
network performance improvements. Their conclusions are based on the assumptions of simple
unbounded path-loss model, noise-less networks and BS locations following a Poisson point
process. Under these assumptions, as BS density increases, the change in aggregated interference
power is counter-balanced by the change in signal power, and thus the SINR remains unchanged
regardless of network density [4]. Following the same assumptions, Dhillon et al. [6] shows
SINR invariance property for HetNets.
The applicability of the above conclusions has been recently questioned in [5], [7], [8] where
they show that under a multi-slope path-loss model the coverage probability and mean data
rate per cell are dependent on BS density. This is in contrast to the earlier SINR invariance
property that suggested potentially infinite aggregated data rate of the network resulting from BS
densification. Moreover, it has been shown that for certain critical near-field path-loss exponents
an optimal network density exists to maximize the coverage probability and mean data rate per
cell. Beyond this optimal network density, the coverage probability decreases as BS density
increases. Similar results can be also found in [9], where Nguyen and Kountouris also show that
SINR invariance does not hold for a multi-slope path-loss model. These works indicate that the
path-loss model is a critical factor affecting the network performance of a UDN.
Further studies have also demonstrated that other features of path-loss model and network deployment may also invalidate SINR invariance property in UDN. In the aspect of path-loss model,
UDN performance was analysed with various path-loss models in the literture. For unbounded
path-loss model including Line Of Sight (LOS) and Non Line of Sight (NLOS) consideration,
Ding et al. show in simulation and analytically in [10] that the network performance depends
on BS density. In [11], Gupta et al. extend their earlier study of multi-slope path-loss model to
include a scenario in which BSs and users are located in 3-dimensional space. They came to a

4

similar conclusion that near-field path-loss exponent influences the performance of the network
and SINR invariance property may only be valid under certain conditions. Most recently, Ding
and Perez studied the impact of antenna height in UDN with results claiming that the area spectral
efficiency goes to zero as BS density goes to infinity [12], [13]. Their results also dismiss the
SINR invariance property for the considered scenario.Similar to Ding and Perez, in our previous
work we studied the impact of antenna height on the network performance. In contrast to their
work, our results hinted however that area spectral efficiency does not necessarily need to go to
zero [14]. In this paper we provide, among others, additional evidence supporting this claim.
In the aspect of network deployment, a comparative simulation-based study of irregular and
regular BS deployments has been presented in [15] where Chen et al. show that the area network
performance peaks at the certain BS density and then starts to decline showing that network
performance in their considered scenarios does depend on BS density for the deployment cases
considered. They also show an interesting finding that for certain network densities, the difference
in performance between irregular and regular BS deployments is constant approximately.
B. Contributions
Despite providing new and interesting insights, there are still a lot of questions related to UDN
performance which need to be answered. As discussed, SINR invariance property in wireless
networks allows Area Spectral Efficiency (ASE) to continue to improve as BS density increases.
This is a desirable feature for performance improvement in UDN since it maintains densification
gain as network densifies. While SINR invariance property has been shown to appear in some
scenarios and system models, as discussed above, recent studies show SINR invariance property
does not hold in many other scenarios. In this paper, we focus on the impact of antenna height
and placement on UDN performance. In particular, we attempt to answer the following questions:
•

What is the role of antenna height on the SINR invariance property in UDN? Existing papers
[10], [9], [5], [7], [8] study the impact of multi-slope feature as well as LOS/NLOS feature
in a path-loss model, and they show that SINR invariance property does not hold with these
features. In [12], Ding and Perez consider BS antenna height in the path-loss model and
show the invalidation of SINR invariance property for the scenario. In this paper, similar
to our earlier work in [14], we not only show the invalidation of SINR invariance property
for any nonzero BS antenna height, but also formulate the relationship showing how UDN

5

performance is affected by the BS antenna height. Interestingly, we found conditions where
SINR invariance property does hold for the considered path-loss model. We call this density
countering condition for maintaining the desirable SINR invariance property.
•

For our scenario, what will happen to Area Spectral Efficiency (ASE, in bps/Hz/m2 ) when
BS density approaches infinity? In the case where SINR invariance does not hold, it is
important to understand whether continuing densification will still lead to ASE improvement.
A recent work claims that ASE goes to zero when BS density approaches infinity [12], [13].
In this paper, we show that ASE does not necessarily go to zero when BS density approaches
infinity. We have presented new results to calculate the ASE.

•

Can we benefit from careful site planning in UDN? A recent paper [15] shows with
simulation that irregular and regular BS deployments do give difference ASE in UDN
for some network densities. In this paper, we show analytically the performance of regular
networks, and investigates the conditions in which regular networks can outperform irregular
networks.
II. S YSTEM M ODEL

A. Network model
We consider a single-tier cellular network utilizing a multiple access technique which ensures
orthogonal resource allocation within a cell. All BSs in the network transmit with the same
power. We further assume that a mobile user always connects to the BS with the strongest
received signal which is usually the closest BS to the mobile user. The mobile user density λu
is assumed to be much higher than the BS density λ (i.e. λu >> λ) such that each BS always
has a user to serve. Regular and irregular BS deployment models for one-dimensional (1D) and
two-dimensional (2D) Euclidean space are considered in this work.
In network of regular BS deployment, BSs are arranged in a regular geometrical structure
within the network. In the case of 1D regular deployment, BSs are deployed regularly every
fixed distance along a line. In the case of 2D, BSs are deployed in a regular hexagonal layout
with each BS placed at the center of a hexagon. In other words, the 2D locations of BSs are
√
ΦHEX = {(Υ(m + n/2), Υ(n 3/2))| m, n ∈ Z} ∈ R2

6

Fig. 1.

Regular BS deployment for 1D (top) and 2D (bottom) Euclidean space

and the 1D locations of BSs are
ΦLIN E = {Υi| i ∈ Z} ∈ R
where Z and R are sets of integer and real numbers, respectively. The quantity Υ is the intersite distance between two adjacent BSs. The spatial density of BSs in the regular deployment is
λ=

2√
Υ2 3

and λ = 1/Υ for 2D and 1D cases respectively. Mobile users are located uniformly

in the network. An illustration of both deployments is given in Figure 1.
In the irregular deployment, the locations of BSs are modelled according to a homogeneous
Poisson point process (PPP) ΦP P P with a spatial density of λ. The mean inter-site distances for
q
2
the 2D and 1D cases are Υ = λ√
and Υ = 1/λ, respectively. Mobile users are uniformly
3
distributed in the Voronoi cell of its corresponding BS.
B. Channel model
As discussed in the literature, the channel model, particularly the path-loss plays an important
role in the performance of the UDN. In this work, we consider a path-loss model function that
can describe antenna height, l1 (r) = (h2 + r2 )−α/2 , where the parameter h is the difference
between BS and mobile user antenna heights, r is the horizontal distance between the mobile
user and the BS, and α is the path-loss exponent with α > 1 for 1D and α > 2 for 2D. A
special case of this path-loss model is l0 (r) = r−α when the antenna height is set to zero. The

7

path-loss model l0 (r) is a commonly used model in network performance analysis (see e.g. [4]
and [6]). However, we shall show that l0 (r) produces overly optimistic performance results in
UDNs. In this paper, we mainly focus on l1 (r) but also use l0 (r) for comparison.
Unless otherwise stated, the random channel effects are modelled as Rayleigh fading with
mean 1, thus the received power (for the desired or interference signal) at a user is g · li (r),
where g ∼ exp(µ) is an exponentially distributed random variable with a mean value of 1/µ
which is equal to the BS transmit power, and li (r) is one of our considered two path-loss model
functions (i ∈ {0, 1}).
It is worth highlighting here that due to high density of BSs in UDNs, desired signal power as
well as interference power are significantly higher compared to Gaussian noise. We also show
that when the BS density approaches infinity, the UDN with noise in the path-loss model gives
exactly the same performance as that without noise (see Theorem 9). This allows us to neglect
noise in order to simplify expressions and conduct our investigations.
III. C OVERAGE P ROBABILITY A NALYSIS
The coverage probability is a measure of chances that a particular user can be served by
∆

the network [4]. Formally, it is defined as pc = P(SIN R > T ) which is the complementary
cumulative distribution function (CCDF) of SINR. For completeness, the results in this section are
derived for regular as well as irregular deployments (even though some of the considered cases
do not lead to tractable expressions). Based on Slivnyak’s theorem, in our considered irregular
network where BS locations are modeled as a homogeneous PPP, the coverage probabilities of
mobile users are statistically identical [16]. Thus investigating the performance of a typical user
in the irregular network is sufficient.
Assuming that a user is always associated with the BS of the strongest received signal, the
SINR of a user can be written as
SIN R =

g · l(r)
,
Ir + σ 2

(1)

where r is the distance between the considered user and its serving BS, σ 2 is constant receiver
noise power, and
Ir =

X
x∈Φ\bo

gx l(kx − u0 k),

(2)

8

is the interference power which is the sum of received power from all surrounding BSs except
the user’s serving BS at bo and kx − u0 k is the distance between each BS and the user located
at u0 .
Depending on the considered BS deployment the distance r separating a typical user from its
serving BS can have different distributions. For irregular BS deployment, as shown in [4], r is
shown to be Rayleigh distributed. The following expression presents a distribution function of
r for a d-dimensional Euclidean space
d

fr (r) = cd drd−1 λe−cd λr , r ≥ 0

(3)
∆

where d ∈ {1, 2} is the dimension of the Euclidean space and cd =

π d/2
Γ(1+d/2)

is volume of the

d-dimensional unit ball (see [17]). Given 1D and 2D space, c1 = 2, c2 = π.
In case of regular BS deployment, r is uniformly distributed leading to following expressions
(for 1D and 2D Euclidean space, respectively)
Υ
2
,r ≤
Υ
2

(4)

6 · 2r
Υ
√ ,0 ≤ r ≤
2
2 sin(θ + π3 )
Υ 3

(5)

fr (r) =
fr (r) =

where Υ is the inter-site distance and 0 ≤ θ ≤ π3 .
Definition 1: Based on Theorem 1 in [4], the coverage probability in the downlink for a
generic path-loss model function, assuming Rayleigh fading, can be represented as


Z
2T
µT
− µσ
{d,i}
pc (T, λ, α) =
e l(r) LIr
fr (r)dr,
l(r)
Rd

(6)

where LIr (s) is the Laplace transform of the random variable Ir , d is the dimension of the
Euclidean space and i ∈ {0, 1} indicates the path-loss model function.
In the following subsection, we derive formulas for coverage probability for both regular and
irregular networks.
A. Irregular network
We first study the extreme condition where BS density approaches infinity. In the following,
we show that the coverage probabilities for irregular 1D and 2D networks under l1 (r) tends
to zero when BS density approaches infinity. In case of l0 (r), the probability of coverage is

9

independent of BS density (see also [4])2 . We start by formulating the coverage probability for
irregular 1D and 2D networks in the following Lemma, assuming a generic path-loss model.
Lemma 1: The coverage probabilities for irregular 1D and 2D networks under l1 (r) path-loss
model function can be expressed as
Z

p{1,1}
(T, λ, α)
c

∞

e−2k e−µT σ

=2

2 ( k2 +h2 )α/2
λ2

· e−2ξ(T,α,k) dk

(7)

0

and
p{2,1}
(T, λ, α)
c

−λπh2 ρ2 (T,α)

Z

∞

e−µT σ

= πe

2 ( k +h2 )α/2
λ

· e−πk(1+ρ2 (T,α)) dk

(8)

0

respectively where
Z

∞

ξ(T, α, k) =
k

1 + T −1

and
ρd (T, α) = T

d/α

Z

∞

T −d/α

1
 2

t +(hλ)2
k2 +(hλ)2

 α2 dt

(9)

1
du.
1 + uα/d

(10)

Proof: The proof is based on the proof of Theorem 1 in [4]. By substituting fr (r) =
d

cd drd−1 λe−cd λr and l(r) = l1 (r) in (6), the coverage probability of d-dimensional irregular
network can be expressed as
Z
{d,1}
pc (T, λ, α) = dcd λ

∞

d

rd−1 e−cd λr e−µT σ

2 (r 2 +h2 )α/2


· LIr µT (r2 + h2 )α/2 dr

(11)

0

where

Z
LIr (s) = exp −dcd λ
r

∞


1−

µ
α
2
µ + s(h + v 2 )− 2

By assuming d = 2 and employing a change of variables u =


v

v 2 +h2
T

2
α (r 2 +h2 )

d−1


dv .

(12)

we obtain the following

expression for 2D

2
2
LIr µT (h2 + r2 )α/2 = e−λπ(r +h ) ρ2 (T,α) .

(13)

When we substitute the above result in (11) and employ a change of variables k = λr2 , we
obtain the final result for 2D.
Similarly, by assuming d = 1 and employing a change of variables k = λr we obtain following
expressions for 1D

2

The formula for 2D network is already given in [4]. Here we provide the formula for 1D network.

10


pc{d,1} (T, λ, α) = 2

Z

∞

e−2k e−µT σ

2 ( k2 +h2 )α/2
λ2

0


∞

Z

·exp −2λ

k
λ

1
1 + T −1



(vλ)2 +(hλ)2


 α2 dv  dr.

k2 +(hλ)2

(14)
Next, by employing a change of variables t = λv, we obtain the final result for 1D, thus
concluding the proof.

Based on Lemma 1, we formulate the following theorem.
Theorem 1: The coverage probability of 1D and 2D irregular networks under path-loss model
l1 (r) tends to 0 for BS density λ → ∞, that is
lim pc{d,1} (T, λ, α) = 0.

(15)

λ→∞

Proof: By applying the Monotone Convergence Theorem to formula (7) for 1D network,
and by using basic properties of limits, we can rewrite (7) as
Z ∞
2 α/2
2 k2
{1,1}
lim pc (T, λ, α) = 2
e−2k lim e−µT σ ( λ2 +h )
λ→∞
λ→∞
0


Z ∞
1


· exp −2
lim

 α2 dt dk. (16)
2
2
λ→∞
k
1 + T −1 kt 2+(hλ)
+(hλ)2
Simplifying the above, yields
lim

λ→∞

p{1,1}
(T, λ, α)
c

∞

Z

−2k −µT σ 2 hα

e

=2

e

 Z
· exp −2
k

0

∞


1
dt dk.
1 + T −1

We finally conclude the proof for 1D network by exploiting the fact that
Z ∞
1
dt = ∞,
1 + T −1
k

(17)

(18)

which brings the coverage probability to 0 as λ → ∞.
Similar to the 1D case, by applying the Monotone Convergence Theorem to formula (8) for
2D network, and by using basic properties of limits, we have
Z ∞
2 α/2
2 k
{2,1}
−λπh2 ρ2 (T,α)
lim pc (T, λ, α) = lim πe
·
lim e−µT σ ( λ +h ) e−πk(1+ρ2 (T,α)) dk. (19)
λ→∞

λ→∞

0

λ→∞

From the above results, we can see that
2ρ

lim πe−λπh

λ→∞

2 (T,α)

= 0,

(20)

11

and
lim e−µT σ

2 ( k +h2 )α/2
λ

λ→∞

= e−µT σ

2 hα

,

(21)

thus concluding the proof for 2D.

It is worth noting that Theorem 1 shows the dependence of coverage probability on BS density
for the considered path-loss model. This is in line with the recent works [5], [7], [8], [10] which
dismiss the SINR invariance property for other path-loss models. In other words, for h > 0, the
increase in the interference power is not counter-balanced by the increase in the signal power.
Interestingly, we also observe that changes in the BS density λ can be counter-balanced by
the adjustments of the path-loss model parameter h to maintain the same coverage probability.
We call this the density countering condition which leads to maintenance of SINR invariance
property. We describe this condition in the following theorem.
Theorem 2: The coverage probability of noise-less irregular networks for path-loss model
l1 (r) is constant, when λ hd = c and c is constant.
Proof: The proof is based on Lemma 1. By substituting σ 2 → 0 and λhd into (7) for 1D
and (8) for 2D, we obtain the following expressions which are constant for a constant c where


Z ∞
Z ∞
1
(22)
p{1,1}
(T, λ, α) = 2
e−2k · exp −2
 α dt dk
c
t2 +c2 2
−1
0
k
1+T
2
2
k +c
and
p{2,1}
(T, λ, α) =
c

e−cπρ2 (T,α)
.
1 + ρ2 (T, α))

(23)

The coverage probabilities for irregular 1D and 2D networks under l1 (r) can be computed
numerically using Lemma 1. The results permit simple expressions for noise-less condition. For
the case of 2D network, we can derive a closed form expression. In case of 1D network, an
expression which requires a single numerical integration can be obtained for some specific α
values, such as α = 2. Both expressions are presented in the following proposition.
Proposition 1: The coverage probability under l1 (r) path-loss model function for noise-less
1D (given α = 2) and 2D networks can be expressed as follows
√
Z ∞
T ((λh)2 +k2 )(arctan(k−1 T ((λh)2 +k2 )+(λh)2 ))
√
2
T ((λh)2 +k2 )+(λh)2
dk
p{1,1}
(T, λ, 2) = 2
e−2k · e
c
0

(24)

12

2

(T, λ, α)
p{2,1}
c

e−λπh ρ2 (T,α)
=
.
1 + ρ2 (T, α))

(25)

Proof: By assuming σ 2 → 0, the proof follows directly from Lemma 1.
Notice that formulas for l1 (r) reduce to the coverage probability for l0 (r) = r−α in [4], when
h = 0. For completeness, the formula for noise-less 1D and 2D networks under l0 (r) = r−α is
presented below
p{d,0}
(T, λ, α) =
c

1
.
1 + ρd (T, α))

(26)

B. Regular 1D network
We consider a network where the locations of BSs are generated using a deterministic points
process ΦLIN . We study the coverage probability of a user located at a particular point in this
network. Without loss of generality we assume that this user is located in the origin and thus
the BS locations form the following point process on R
Φ̂LIN
= {x − y | x ∈ ΦLIN },
r

(27)

where r = kyk is a distance between a user and its serving (closest) BS at point y.
We now formulate two lemmas for the coverage probability in 1D regular network under the
path-loss models l0 (r) and l1 (r).
Lemma 2: The coverage probability for 1D regular network under l0 (r) path-loss model
function can be expressed as
p{1,0}
(T, λ, α)
c

2
=
Υ

where Υ = λ1 , and
LIr (s) = 2

Υ
2

Z

e−µT σ

2 rα

LIr (µT rα ) dr,

(28)

0

∞
Y

1
1 + (r + i Υ)−α µs

i=1

!
.

(29)

Proof: The proof of Lemma 2 is based on the proof of Theorem 1 provided in [4]. As
the desired signal is assumed to be exponentially distributed, by substituting fr (r) =

2
Υ

in (6),

the coverage probability in 1D regular network under a generic path-loss model function can be
expressed as
p{1,i}
(T, λ, α)
c

2
=
Υ

Z

Υ
2

e
0



2

σ
− µT
l(r)

L Ir

µT
l(r)


dr.

(30)

13

Using the definition of the Laplace transform we can show that


X
(a)
LIr (s) = EIr [e−sIr ] = E{gx } exp(−s
gx l(kxk)) =
\{bo }
x∈Φ̂LIN
r

Y

Eg [exp(−sg l(kxk))]

\{bo }
x∈Φ̂LIN
r
(b)

=

µ
l(kxk)
µ
+
l(kxk)
LIN
\{bo }
x∈Φ̂r

Y

s

(31)

where (a) follows from the i.i.d distribution of gx and its independence from the point process
Φ̂LIN
and (b) from the Laplace transform of an exponential random variable with mean
r

µ
.
l(kxk)

Using the above result, and taking into account that locations of BSs in 1D regular network
follow ΦLIN , the Laplace transform can be further expressed as
!
∞
Y
1
,
LIr (s) =
1 + l(kr + i Υk) µs
i=−∞

(32)

i6=0

where kr + i Υk is the distance from the typical user to the i-th base station.
By substituting l(r) = l0 (r) into (30) and removing the absolute value expression from the
product in the expression for the Laplace transform presented above, we immediately obtain the
final result.
Lemma 3: The coverage probability for 1D regular network under l1 (r) path-loss model
function can be expressed as
p{1,1}
(T, λ, α)
c

2
=
Υ

Υ
2

Z

e−µT σ

2 (r 2 +h2 )α/2


· LIr µT (r2 + h2 )α/2 dr,

(33)

0

where Υ = λ1 , and
LIr (s) = 2

∞
Y
i=1

1
1 + ((r + i Υ)2 + h2 )−α/2 µs

!
.

(34)

Proof: The proof follows that of Lemma 2.
{1,1}

It is worth noting here that setting h = 0 in l1 (r) reduces pc

{1,0}

(T, λ, α) to pc

(T, λ, α).

Based on Lemma 2 and Lemma 3, we have the following theorems (i.e. Theorem 3 - 5).
Theorem 3: The coverage probability of noise-less 1D regular network under the standard
path-loss model function l0 (r) does not depend on BS density λ.
Proof: The proof is based on Lemma 2. By substituting t =
obtain
(T, λ, α) = 4
p{1,0}
c

Z
0

1
2

∞
Y
i=1

1
tα
1 + T (t+i)
α

r
Υ

and assuming σ 2 → 0 we

!
dt.

(35)

14

From the above formula it can be easily seen that the probability of coverage is independent
of λ, thus concluding the proof.
Theorem 4: The coverage probability of noise-less 1D regular networks under path-loss model
l1 (r) tends to 0 as BS density λ → ∞, that is
lim pc{1,1} (T, λ, α) = 0.

(36)

λ→∞

Proof: The proof is based on Lemma 3. By substituting t =

r
,
Υ

and given that Υ =

1
λ

we

then obtain
p{1,1}
(T, λ, α) = 4
c

1
2

Z

e−µT σ

2 (( t )2 +h2 )α/2
λ

0

·

∞
Y







i=1

1
1+T




α/2  dt

t2 +(hλ)2

(37)

(t+i)2 +(hλ)2

By taking λ → ∞ and applying the Monotone Convergence Theorem to above formula, we
further get
lim pc{1,1} (T, λ, α) = 4

Z

λ→∞

0

1
2

lim e−µT σ

2 (( t )2 +h2 )α/2
λ

λ→∞

·

∞
Y
i=1





lim 

λ→∞

1
1+T



t2 +(hλ)2


α/2  dt.

(t+i)2 +(hλ)2

(38)
Simplifying the above expression yields
lim

λ→∞

Next, given that T > 0,

p{1,1}
(T, λ, α)
c

Z
=4

1
2

e

−µT σ 2 hα

0

∞
Y
i=1

∞ 
Y
i=1

1
1+T


dt.

1
= 0,
1+T

(39)

(40)

which brings the expression in (39) to zero.

Theorems 3 and 4 show that similar to 1D irregular network, the 1D regular network does
not exhibit SINR invariance property for l1 (r) path-loss model. However, it does exhibit SINR
invariance property (similar to noise-less irregular networks) for l0 (r) path-loss model.
In the previous subsection, we showed the density countering condition for irregular 1D and
2D networks under l1 (r). As shown in the following theorem, this condition also applies to 1D
regular network.

15

Theorem 5: The coverage probability of noise-less 1D regular networks under path-loss model
l1 (r) is constant when λ h = c and c is constant.
Proof: The proof is based on the proof of Theorem 4. By substituting σ 2 = 0 and λh = c
in (37) we obtain the following expression which is constant for a constant c


Z 1Y
∞
2
1


p{1,1}
(T, λ, α) = 4


α/2  dt.
c
t2 +c2
0 i=1
1 + T (t+i)
2 +c2

(41)

Lemma 2 allows us to numerically calculate coverage probability for arbitrary α > 1 and
a generic path-loss model function l(r). However, by considering some integer α values we
can obtain a simpler expression which allow us to gain additional insight. In the following
propositions, we derive coverage probability expressions for α = 2 for the considered path-loss
model functions.
Proposition 2: The coverage probability for 1D regular network using the standard path-loss
model function l0 (r) = r−α , when α = 2 is
Z
σ 2 x2
(1 + T ) π − µT
cos(x) − 1
{1,0}
√ dx.
e (2πλ)2 ·
pc (T, λ, 2) =
π
cos(x) − cosh(x T )
0
Proof: Using Lemma 2, we start by expressing the Laplace transform of Ir as


∞
Y
1

.
LIr (s) =
s
µΥ2
1 + ( r +i)2
i=−∞
i6=0

(42)

(43)

Υ

By using the following expression for an infinite product
√

∞ 
Y
cos(2πa) − cos(2π −b2 − z)
z
,
=
1+
(k + a)2 + b2
cos(2πa) − cosh(2πb)
k=−∞
and with the setting of b = 0, z =

−s
,
µΥ2

a=

r
Υ

(44)

and the exclusion of k = 0,we obtain

(r2 + µs )(cos(2π Υr ) − 1)
√
.
(45)
r2 (cos(2π Υr ) − cosh(2π Υr T ))
Plugging in s = µT r2 and substituting x = 2πλr to the above result concludes the proof.
LIr (s) =

Proposition 3: The coverage probability for 1D regular network using the path-loss model
function l1 (r), when α = 2 can be expressed as
Z
σ 2 x2
1 + T π − µT
cos(x) − cosh(2πλh)
{1,1}
p
pc (T, λ, 2) =
e (2πλ)2 ·
dx. (46)
π
cos(x) − cosh( (2πλh)2 + T (x2 + (2πλh)2 ))
0
Proof: The proof follows that of Proposition 2 but in (44), the substitution b =
instead.

h
d

is used

16

C. Regular 2D network
Similar to the 1D regular network, we consider a network generated by the deterministic point
process ΦHEX , and we study the coverage probability of a user located at the origin. We first
see that the BS locations form the following point process on R2
= {(x − r cos θ, y − rsinθ) | (x, y) ∈ ΦHEX },
Φ̂HEX
r,θ

(47)

where r and θ are the distance and angle between the user, and its serving BS, respectively.
Following a similar approach as in 1D regular network, we first provide the following lemma
for the coverage probability in 2D regular network, assuming a generic path-loss model.
Lemma 4: The coverage probability for 2D regular network under l0 (r) path-loss model
function can be expressed as
p{2,0}
(T, λ, α)
c
where Υ =

q

2
√
,
λ 3

12
√
=
Υ2 3

0

Υ
2 sin(θ+ π
3)

Z

e−µσ

2 T rα

· LIr,θ (µT rα ) rdr dθ,

(48)

0

and

"
LIr,θ (s) =

π
3

Z

Y

1+

(n,m)∈
Z2 \{(0,0)}



#
2  √ 3
2 − α2 s −1
n
Υ(m + ) − r cos θ + Υn
− r sin θ
. (49)
2
2
µ

Proof: The proof of Lemma 4 is analogous to that of Lemma 2 and is based on the proof
of Theorem 1 provided in [4]. As the desired signal is assumed to be exponentially distributed,
by substituting (5) in (6), the coverage probability can be expressed as
p{2,i}
(T, λ, α)
c

12
√
=
Υ2 3

Z

π
3

Υ
2 sin(θ+ π
3)

Z

0



2

T
− µσ
l(r)

e

· LIr,θ

0

µT
l(r)


rdr dθ.

(50)

Similar to Lemma 1, by using the definition of the Laplace transform we can show that



LIr,θ (s) = EIr,θ [e−sIr,θ ] = E{gu } exp(−s


X

 (a)
gu l(kuk)) =

u∈Φ̂HEX
\{bo }
r,θ

Y

Eg [exp(−sg l(kuk))]

u∈Φ̂HEX
\{bo }
r,θ
(b)

=

µ
l(kuk)
µ
+
l(kuk)
u∈Φ̂HEX
\{b
}
o
r,θ

Y

s

(51)

17

where (a) follows from the i.i.d distribution of gu and it is independence of the point process
Φ̂HEX
, and (b) follows from the Laplace transform of an exponential random variable with mean
r,θ
µ
.
l(kuk)

Using the above, and taking into account that locations of BSs in 2D regular network

follow ΦHEX , the Laplace transform can be further expressed as
"
#−1
√
Y
3
n
s
LIr,θ (s) =
1 + l(kΥ(m + ) − r cos θ, Υn
− rsinθ)k)
.
2
2
µ
(n,m)∈
Z2 \{(0,0)}

By substituting l(r) = l0 (r) into (50), and removing the absolute value expression from the
product in the expression for the Laplace transform presented above, we immediately obtain the
final result which concludes the proof.
Lemma 5: The coverage probability for 2D regular network under l1 (r) path-loss model
function can be expressed as
pc{2,1} (T, λ, α)

12
√
=
Υ2 3

Z

π
3

0

Z

Υ
2 sin(θ+ π
3)

e−µσ

2 T (r 2 +h2 )α/2


· LIr,θ µT (r2 + h2 )α/2 rdr dθ, (52)

0

where
"
LIr,θ (s) =

Y
(n,m)∈
Z2 \{(0,0)}

1+



− α2 #−1
2  √ 3
2
s
n
− r sin θ + h2
.
Υ(m + ) − r cos θ + Υn
2
2
µ
(53)

Proof: The proof follows that of Lemma 4.
Theorem 6: The coverage probability for noise-less 2D regular network under the standard
path-loss model function l0 (r) = r−α does not depend on BS density λ.
Proof: The proof is based on Lemma 4. By substituting t =
coverage probability of 2D regular network can be expressed as

1
π)
Y 
√ Z π3 Z 2 sin(θ+
3
{2,0}

pc (T, λ, α) = 2 3

0
0
1+ 
(n,m)∈
√
Z2 \{(0,0)}

r2
Υ2

and assuming σ 2 → 0, the


1
T

α
t2

(m+ n
− t cos θ)2 +(n
2

α
√
√
3
− t sin θ)2 2
2


 dtdθ.


From the above formula it can be easily seen that the coverage probability is independent of
λ.

18

Theorem 7: The coverage probability of 2D regular networks under the path-loss model l1 (r)
tends to 0 as BS density λ → ∞, that is
lim pc{2,1} (T, λ, α) = 0.

(54)

λ→∞

Proof: The proof is based on Lemma 5. We start by substituting t =

r2
Υ2

and Υ2 =

2
√
λ 3

into

(52). Next, by applying the Monotone Convergence Theorem to the obtained formula and some
algebraic manipulation, we obtain the following expression

lim

λ→∞

pc{2,1} (T, λ, α)

√ Z
=2 3
0

·

Y
(n,m)∈
Z2 \{(0,0)}

π
3

1
2 sin(θ+ π
3)

Z

−µσ 2 T (

2
√
t+h2 )α/2
λ 3

λ→∞

0



lim 
λ→∞ 

lim e


1
√


1+T

(m+ n
−
2

√

3
λh2
2√
√
√
2
t cos θ) +(n 23 − t sin θ)2 + 23 λh2

t+



 α2  dtdθ. (55)


The above formula can be further simplified to

lim

λ→∞

p{2,1}
(T, λ, α)
c

√ Z
=2 3

π
3

Z

0

1
2 sin(θ+ π
3)

−µσ 2 T hα

e

0

·

Y



(n,m)∈
Z2 \{(0,0)}

1
1+T


dtdθ.

(56)

Next, given that T > 0,
Y



(n,m)∈
Z2 \{(0,0)}

1
1+T


= 0,

(57)

and thus bring the expression (56) to zero.
Similar to the previous scenarios, the following theorem shows that the density countering
condition also applies to 2D regular network.
Theorem 8: The coverage probability of a noise-less 2D regular network for the path-loss
model function l1 (r) does not change when λh2 = c and c is constant.
Proof: The proof is based on the proof of Theorem 7. By substituting σ 2 = 0 and λh2 = c
in (52), we obtain the following expression which is constant for a constant c

1
π)
Y 
√ Z π3 Z 2 sin(θ+
1
3
{2,1}
lim pc (T, λ, α) = 2 3
dtdθ,
λ→∞
1 + T β(n, m)
0
0
(n,m)∈
Z2 \{(0,0)}

19

TABLE I
S UMMARY OF CONDITIONS WHEN SINR INVARIANCE PROPERTY HOLDS FOR REGULAR / IRREGULAR 1D/2D

Path-loss

Irregular

Regular

models 1D

2D

1D

2D

l0 (r)

holds [4]

holds

holds

λh = c

λh2 = c

holds

l1 (r)

2

λh = c

NETWORKS

λh = c

Note: c is constant

where

√



3
c
2
√
(n 23

 α2

t+

β(n, m) =
(m +

n
2

−

√
t cos θ)2 +

−

√

√

t sin θ)2 +

3
c
2

.

In contrast to Lemma 2, the Laplace transform of the interference power LIr,θ (s) in Lemma 4
does not have a closed form expression. If desirable, the coverage probability under l0 (r), l1 (r)
may be calculated numerically.
In Table I, we summarize the conditions when SINR invariance property holds. For the
considered scenarios when the SINR invariance property does not hold, we show that the coverage
probability tends to zero.
IV. AVERAGE ACHIEVABLE R ATE A NALYSIS
In the following section, we focus on the analysis of the mean achievable data rate over a cell.
More specifically, we compute the ergodic capacity which measures the long-term achievable
rate averaged over all channel and network realizations [18].
Definition 2: The average ergodic rate achievable over a cell in the downlink, assuming
Rayleigh fading for desired signal, can be represented as (see Theorem 3 in [4])


Z
Z ∞
µσ 2 (et −1)
1
µ(et − 1)
1
∆
−
{d,i}
l(r)
E[ln(1 + SIN R)] =
fr (r)
e
LIr
dtdr
τ
(λ, α) =
ln 2
ln 2 Rd
l(r)
γ0
Z ∞
1
=
p{d,i}
(et − 1, λ, α)dt (58)
c
ln 2 γ0
where d is the dimension of the Euclidean space and i ∈ {0, 1} indicates the path-loss model
function. The quantity γ0 is the minimum working SINR. In this paper, we set γ0 = 0 to
investigate the performance upper bound.

20

The average achievable rate per cell can be computed numerically using Definition 2 with
formulas for the coverage probability. However, since we have derived some closed form expressions for the coverage probability under certain conditions, further insights can be obtained
for the per cell average achievable rate. In the following, we focus on deriving the average
achievable rate per cell for these conditions.
A. Irregular network
The average ergodic rate in the downlink of noise-less irregular network for path-loss model
l1 (r) for 1D and 2D can be respectively expressed as
2λ
τ {1,1} (λ, 2) =
ln 2

∞

Z

Z

(et −1)(h2 +r 2 )(arctan r((et −1)(h2 +r 2 )+h2 )−1/2 −π/2)

∞

e
0

−2λr

√

2λ

· e

)

dr dt (59)

0

and
τ

(

(et −1)(h2 +r 2 )+h2

{2,1}

1
(λ, α) =
ln 2

Z

∞

0

2

t

e−πλh (ρ2 (e −1,α))
dt.
1 + ρ2 (et − 1, α)

(60)

The above results are obtained by substituting (24) into (58) and (25) into (58), as well as
setting σ 2 = 0.
Note that by simply setting h = 0 in the equations above, we can obtain the average ergodic
rate for path-loss l0 (r) as
τ

{d,0}

1
(λ, α) =
ln 2

Z
0

∞

1
dt.
1 + ρd (et − 1, α)

(61)

As we have learned from the previous section the SINR invariance property holds for the
path-loss model l0 (r) but not for l1 (r). From the above, we can see that the average rate per
cell is unaffected by the BS density for l0 (r) but it is affected by the BS density for l1 (r) (see
also Figs. 2-3). Based on Definition 2 and Theorem 1, we further know that the average rate per
cell approaches zero eventually. Interestingly, it is possible to counter this decay by tuning the
antenna height according to BS density in order to maintain per cell average rate. This can be
observed by the countering condition described in Theorem 2 in conjunction with Definition 2.
B. Regular 1D network
In the previous section, we have derived a closed form coverage probability expression for
noise-less 1D network under l0 (r) path-loss model with α = 2. By substituting (42) into (58)

21

Average Achievable Rate: 1D network, noise-less, α = 2
3.5
reg, l1 (r), h = 0 (i.e. l0 (r))
reg, l1 (r), h = 5
reg, l1 (r), h = 10
irr, l1 (r), h = 0 (i.e. l0 (r))
irr, l1 (r), h = 5
irr, l1 (r), h = 10

Average Achievable Rate [bps/Hz]

3

2.5

2

1.5

1

0.5

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

BS density λ

Fig. 2.

Impact of BS density λ and h on average ergodic rate τ of noise-less 1D network for different path-loss models with

α = 2. The path-loss parameter h affects the rate of decay.
Average Achievable Rate: 2D network, noise-less, α = 4
3.5
reg, l1 (r), h = 0 (i.e. l0 (r))
reg, l1 (r), h = 5
reg, l1 (r), h = 10
irr, l1 (r), h = 0 (i.e. l0 (r))
irr, l1 (r), h = 5
irr, l1 (r), h = 10

Average Spectral Efficiency [bps/Hz]

3

2.5

2

1.5

1

0.5

0
0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

BS density λ

Fig. 3. Impact of BS density λ and h on average ergodic rate τ of noise-less 2D network for different path-loss models withS
α = 4. The path-loss parameter h affects the rate of decay.

and employing a change of variables x = 2πλr, k =
cell as
τ

{1,0}

2
(λ, 2) =
ln 2 π

Z
0

∞

Z
0

π

√
et − 1, we obtain the average rate per

k(cos(x) − 1)
dxdk.
cos(x) − cosh(xk)

(62)

The above expression indicates a constant per cell average achievable rate regardless of BS
density which is similar to what was shown in [4] for irregular networks. This expression also
allows us to numerically compute the per cell average rate. We have found that regular network

22

does indeed give a higher per cell average rates than that of the irregular network. For regular
network, the approximate average rate per cell is ≈ 3.037bits/sec/Hz. For irregular network, the
approximate average rate per cell can be computed by (61) which gives ≈ 2.148bits/sec/Hz. By
comparing the above results of regular and irregular networks, performing site planning results
in an approximate gain of 1.414 times higher achievable rate.
Next, for path-loss model l1 (r), by substituting (42) into (58) and employing a change of
variables k = et − 1, the average achievable rate per cell with α = 2 can be expressed as
Z ∞Z π
cos(x) − cosh(2πλh)
1
{1,1}
p
·
dxdk.
(63)
τ
(λ, 2) =
2
2
ln 2 π 0
0 cos(x) − cosh( ((2πλh) (1 + k) + x k)
From the last result, we can see that the average rate per cell for l1 (r) with α = 2 depends
on BS density as also seen in Figure 2. However, by setting h =

c
λ

where c is some constant

positive value, we can maintain a constant average rate per cell regardless of BS density. With
Theorem 5, the above argument can be extended to a general condition of α > 1.
From the last result, by considering l1 (r) with α = 2, we see that the average achievable rate
per cell depends on BS density (see also Fig. 2). Similar to the irregular network, the average
achievable rate per cell approaches zero when BS density goes to infinity. However, by setting
h=

c
λ

where c is some constant positive value, we can maintain a constant average achievable

rate per cell regardless of BS density. This argument is also valid for general condition of α > 1
by observing Theorems 4 and 5 in conjunction with Definition 2.
C. Regular 2D network
As the probability of coverage for 2D regular network does not yield any closed form
expression, we seek numerical computation to calculate its average achievable rate per cell.
We present the numerical results in Figure 3 illustrating the impact of λ and h on the average
ergodic rate per cell τ {2,1} (λ, α). As can be seen, the rate of decay of τ {2,1} (λ, α) for an increasing
λ depends on h and τ {2,1} (λ, α) tends to zero. This is in line with the results for other network
configurations discussed in the previous subsections.
Additionally, based on Theorem 8 and Definition 2, it can be easily shown that the average
achievable rate per cell in the noise-less 2D regular network under path-loss model l1 (r) does
not change when λh2 = c and c is constant. In other words, it is possible to maintain per cell

23

average achievable rate by countering the increase in BS density through lowering the antenna
height accordingly.
Furthermore, Theorem 6 and Definition 2 indicate that the 2D noise-less regular network under
l0 (r) exhibits SINR invariance property for all conditions. Consequently, the average rate per
cell for l0 (r) does not depend on BS density.
V. P ERFORMANCE L IMITS IN UDN
In the previous section, we have derived the coverage probability and average achievable rate
per cell for various scenarios. In this section, we investigate how extreme densification affects
gains from site planning and its impact on ASE.
A. Deployment gain
We define deployment gain as the ratio of average achievable rate per cell for regular network
to that of the irregular network. In Section IV, we have presented the average achievable rate per
cell for different scenarios in Figures 2 and 3. When comparing the rate performance between
the regular and irregular networks, we made the following observations: (i) for the case where
SINR invariance property holds, the deployment gain remains constant regardless of BS density,
(ii) for the case where SINR invariance property does not hold, the deployment gain depends
on BS density and no gain is observed when BS density tends to infinity, (iii) in general, we
achieve a performance gain when the deployment of BSs follows a regular pattern. To illustrate
our observations, we further plot the deployment gain in Figure 4 for 1D network with α = 2
and Figure 5 for 2D network with α = 4. To focus on the performance in ultra dense region,
our plots use average inter-site distance on x-axis.
From Figure 4, we first see a constant deployment gain of approximately 1.414 for 1D
network when SINR invariance property holds. When SINR invariance property does not hold,
the deployment gain varies based on BS density. We see from the figure that the deployment
gain increases as the average inter-site distance decreases. After peaking at a certain point, the
gain appears to decrease to zero when the inter-site distance approaches zero. In other words,
there appears to be no gain when BS density is extremely high. Interestingly, there is a small
region of inter-site distance when deployment gain value falls below one, indicating negative

24

Deployment Gain: 1D network, noise-less, α = 2

1.5

1.4
l1 (r), h = 0, (i.e. l0 (r))
l1 (r), h = 5

Deployment Gain

1.3

l1 (r), h = 10

1.2

1.1

1

0.9
0

Fig. 4.

10

20

30

40
50
60
Inter-site distance [m]

70

80

90

Deployment gain for 1D noise-less network with α = 2.
Deployment Gain: 2D network, noise-less, α = 4

1.5

1.4

l1 (r), h = 0, (i.e. l0 (r))
l1 (r), h = 5
l1 (r), h = 10

Deployment Gain

1.3

1.2

1.1

1

0.9
0

Fig. 5.

5

10

15

20
25
30
Inter-site distance [m]

35

40

45

50

Deployment gain for 2D noise-less network with α = 4.

impact of site planning. In Figure 5, we observe similar behavior for the 2D network. In the
following, we show that deployment gain vanishes when BS density approaches infinity.
Proposition 4: System performance under l1 (r) does not depend on BS deployment strategy
(i. e. no gain from site planning) as λ → ∞.
Proof: We first study the distance between a user at a particular point and an arbitrary
interfering BS. In regular network, since the location of the interfering BS is deterministic, the
distance between the two is also deterministic. Let this distance be R. In irregular network, the
location of the BS follows the PPP. The distance is a random variable, say R̃, which has the

25

following statistical properties [17] for 1D network
n
2λ
n
var(R̃) = 2 ,
4λ
E[R̃] =

and 2D network

1 Γ(n + 12 )
E[R̃] = √
λπ Γ(n)

2 !
Γ(n + 12 )
1
var(R̃) =
n−
.
λπ
Γ(n)

(64)
(65)

(66)
(67)

Considering an interfering BS located at a particular distance away from the user. For regular
network, this distance is a fixed value R. For irregular network, this distance is R̃ with mean value
of R. Notice that when λ → ∞, the variance of R̃ tends to zero. In other words, when λ continues
to increase, the variance of this distance continues to decrease which reduces the deployment
difference between irregular and regular networks. For a regular network, the interference caused
by this BS is g · l(R) where g is a random variable describing the channel fading and l(·) is the
path-loss expression. For an irregular network, the interference caused by this BS is g · l(R̃).
Given that l1 (r) path-loss model is bounded, we have l1 (r) < ∞ for an arbitrary r ≥ 0. When
λ → ∞, the variance of R̃ decreases to zero, we see that the interference characteristic caused by
the BS in the irregular network converges to that of the regular network, that is gl(R̃) → gl(R).
This shows that system performance under l1 (r) for irregular network converges to that of the
regular network when λ → ∞.
The above proposition can be extended to any bounded path-loss model. For the unbounded
path-loss model, we show in Section IV that a constant performance gain is achieved when the
deployment of BSs follows a regular pattern.
B. Average Area Spectral Efficiency
The ASE is a measure of the overall rate over a network area and is defined by
ASE = λ · τ {d,i} (λ, α).

(68)

ASE may increase or decrease as network densifies. This depends on whether the decay of rate
per cell as BS density increases, can be countered by spatial reuse, and therefore it is of interest

26

to investigate whether continued network densification will still lead to ASE improvement. The
finding may help network operators to optimize their investments in the infrastructure and identify
when further densification may not be beneficial.
In the previous section, we showed that when SINR invariance does not hold, the average
achievable rate per cell goes to zero as λ → ∞. However, as we show in the following, the
average ASE does not necessarily go to zero. To support this, we formulate the following
Theorem.
Average Area Spectral Efficiency: 1D network, noise-less, α = 2

Average Area Spectral Efficiency [bps/Hz/m]

0.1

reg, l1 (r), h = 0 (i.e. l0 (r))
reg, l1 (r), h = 5
reg, l1 (r), h = 10
irr, l1 (r), h = 0 (i.e. l0 (r))
irr, l1 (r), h = 5
irr, l1 (r), h = 10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01
0

10

20

30

40

50

60

70

80

90

100

Inter-site distance [m]

Fig. 6.

Average Area Spectral Efficiency for noise-less 1D networks for α = 2
Average Area Spectral Efficiency: 2D network, noise-less, α = 4
0.02
reg, l1 (r), h = 0 (i.e. l0 (r))
reg, l1 (r), h = 5
reg, l1 (r), h = 10
irr, l1 (r), h = 0 (i.e. l0 (r))
irr, l1 (r), h = 5
irr, l2 (r), h = 10

Average Area Spectral Efficiency [bps/Hz/m2 ]

0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0

5

10

15

20

25

30

35

40

Inter-site distance [m]

Fig. 7.

Average Area Spectral Efficiency for noise-less 2D networks for α = 4

45

50

27

Theorem 9: Average ASE of cellular networks under l1 (r) for h > 0 as λ → ∞, given α > 1
for 1D or α > 2 for 2D converges to a non-zero value.
Proof: Proof is given in Appendix A.
As can be seen in the proof of Theorem 9, the ASE of UDN under l1 (r) has the following
lower bounds when λ → ∞, for 1D network
1


2 ln 2 h 1 +
and 2D network

Γ(α−1)
Γ(α)

,

√
2 3

.
12 ln 2 h2 1 + 2 Γ(α−2)
Γ(α−1)

(69)

(70)

These lower bounds depend on h and α. They show that by lowering h, the ASE can be
improved. This is very encouraging as network operators could theoretically provide services
even with over-densified networks. In the extreme case when h → 0, we have ASE approaching
infinity. This is true since when h = 0, the path loss model l1 (r) reduces to l0 (r), and SINR
invariance property holds for this scenario.
It is also worth noting here that the lower bounds also depend on γ0 which is the minimum
working SINR (see Definition 2). The above results consider γ0 = 0 setting. A higher γ0 will
lower the per cell rate and thus will reduce the ASE. With some sufficiently high value of γ0 ,
the ASE reduces to zero as indicated in [13].
In the following, we show ASE performances for some specific scenarios using the above
developed results. We derive exact ASE performance for 1D network when λ → ∞ for various
α values.
Proposition 5: Average area spectral efficiency (in terms of bits/Hz/m2 ) of 1D network for
α = {2, 4, 6} under l1 (r) path-loss model when λ → ∞ is
lim λ τ {1,1} (λ, 2) =

1
,
ln 2 πh

(71)

lim λ τ {1,1} (λ, 4) =

2
,
ln 2 πh

(72)

8
.
3 ln 2 πh

(73)

λ→∞

λ→∞

lim λ τ {1,1} (λ, 6) =

λ→∞

28

Proof: Proof is given in Appendix B.
For 2D network, as a closed form expression for Laplace transform of Ir (s) is unavailable,
we seek to obtain a tighter bound for specific α values as follows.
Proposition 6: The upper and lower bounds on the average area spectral efficiency (in terms
of bits/Hz/m2 ) of 2D network for α = 4 under l1 (r) path-loss model, when λ → ∞ is
√
√
2 3
2 3
{2,1}
≥ lim λ τ
(λ, 4) ≥
.
9 ln 2 h2 λ→∞
12 ln 2 h2

(74)

Proof: Proof is given in Appendix C.
VI. C ONCLUSION
This paper investigated the impact of the relative antenna height between BS and UE antennas
on the performance of UDNs. We showed that for any nonzero relative antenna height the
highly desirable SINR invariance property does not hold, indicating that the UDN performance
is dependant on BS density. Moreover, the per cell average achievable rate performance of
UDN decays to zero, as BS density increases. In order to manage this decay, we showed that
decreasing the relative antenna heights across the network can counter the decay of per cell
average achievable rate. We explicitly derived the relationship between BS density and relative
antenna height showing how this can be achieved. Interestingly, appropriate adjustment of BS
antenna heights enables the network to retain SINR invariance property.
We further studied the ASE performance of the network. Despite the pessimistic conclusion
related to the per cell performance shown in the literature, in this paper we showed that ASE
does not necessarily decay to zero as BS density approaches infinity. A non-negligible ASE
can be achieved which is dependent of the path-loss exponent and relative antenna height. This
result is optimistic as it indicates that theoretically even in a highly over-densified network, the
network can continue to serve with a reduced performance.
We finally investigated the deployment gain which studies the performance improvement due to
careful site planning in UDNs. We showed that, in general, we can always achieve a performance
gain when the deployment of BSs follows a regular pattern. This gain remains constant as BS
density increases when SINR invariance property holds and deteriorates to zero otherwise. The
insights provided in this work may help network operators to optimize their investments in the

29

infrastructure and identify when further densification or careful site selection may no longer be
beneficial due to low cost-effectiveness.
A PPENDIX A
P ROOF OF T HEOREM 9
We start by stating the following Lemmas which are then used in the proof of the theorem.
Lemma 6: Coverage probability performance of regular and irregular networks under l1 (r)
with Rayleigh fading converge to that of networks without fading as λ → ∞.
Proof: Consider an irregular network of density λ0 with Rayleigh fading and another irregular
network of density λ without fading. Using findings in [19], [20], Baccelli et al. show in [21]
that distributions of SINR for both networks are the same if λ0 = λ/Γ(δ + 1) where δ =

d
.
α

Since the distributions of SINR are the same, both networks give the same coverage probability
performance. Given that the path-loss model l1 (r) is bounded, with λ → ∞, the coverage
probability of the network with Rayleigh fading simply converges to that of the network without
fading as λ → ∞.
Using Proposition 4, the above conclusion can be extended to the case of regular networks,
thus concluding the proof.
Lemma 7: Given a path-loss model l1 (r) and h > 0, the ASE performance of a network when
λ → ∞ converges to the ASE performance of a network where each user is collocated with its
serving BS.
Proof: Let τ̂ {d,i} (λ, α) be the average achievable rate per cell of a network without fading
{d,i}

(i.e. gx ≡ 1). Let τ̂0

(λ, α) be the achievable rate per cell of a network without fading given

both a typical user and its serving BS are located in the origin.
For 1D regular network, τ̂ {1,i} (λ, α) can be expressed as


Z Υ
2 2
l(x)
{1,i}
P∞
dx
· log2 1 + P∞
τ̂
(λ, α) =
2
Υ
0
i=1 l(i Υ + x) +
i=1 l(i Υ − x) + µσ
where Υ = λ1 . By substituting x = k Υ2 to the above result, we obtain


Z 1
l(Υ k2 )
{1,i}
P∞
τ̂
(λ, α) =
log2 1 + P∞
dk.
k
k
2
0
i=1 l(Υi + Υ 2 ) +
i=1 l(Υi − Υ 2 ) + µσ

(75)

(76)

30

Next, by assuming a bounded path-loss model function (i.e. l(0) < ∞)) and applying the
Monotone Convergence Theorem, we derive the following limit expression


Z 1
l(0)
{1,i}
P∞
dk
lim τ̂
(λ, α) = lim
log2 1 + P∞
i
i
2
λ→∞
λ→∞ 0
i=1 l( λ ) +
i=1 l( λ ) + µσ


l(0)
P∞
= lim log2 1 + P∞
. (77)
i
i
2
λ→∞
l(
)
+
i=1
i=1 l( λ ) + µσ
λ
{d,i}

Now we express τ̂0

(λ, α) for 1D regular network under a bounded path-loss model function

as
{1,i}
τ̂0 (λ, α)


= log2 1 + P∞

i=1

l(i Υ) +

l(0)
P∞

i=1

l(i Υ) + µσ


.
2

(78)

Using (77) and (78), we can easily obtain the following
{d,i}

lim λτ̂ {d,i} (λ, α) = lim λτ̂0

λ→∞

λ→∞

(λ, α).

(79)

We can repeat the above approach for 2D networks, thus concluding the proof.
In the following we provide the proof of Theorem 9.
Proof: We begin the proof by focusing on the 1D network. Using Lemma 7, we write the
limit of ASE for 1D regular network without fading when λ → ∞ as


l(0)
{1,i}
P∞
. (80)
lim λτ̂0 (λ, α)
=
lim λ log2 1 + P∞
2
λ→∞
λ→∞
i=1 l(i Υ) + µσ
i=1 l(i Υ) +
To derive the lower bound of the above expression for l1 (r), we introduce a new path-loss
model function l2 (r) = (max(h, r))−α where l2 (r) ≥ l1 (r), for all r ≥ 0, h ≥ 0. By substituting
l(r) = l1 (r) and l(r) = l2 (r) in (80) it can be easily seen that
{1,1}

lim λτ̂0

λ→∞

{1,2}

(λ, α) ≥ lim λτ̂0
λ→∞

(λ, α).

(81)

The above inequality can be further simplified into the following expression


h−α
{1,1}
 , (82)
lim λτ̂0 (λ, α)
≥
lim λ log2 1 +
λ→∞
λ→∞
µσ 2 + 2 nh−α + Υ−α ζ(α, 1 + n)
where

∞
X
1
ζ(α, 1 + n) =
iα
i=n+1

is the Hurwitz Zeta function and n = b Υh c.

(83)

31

We substitute t =

h
Υ

and without loss of generality, as λ → ∞ (or Υ → 0), we can safely

assume t ∈ N which leads to the following expression


t
1
lim log2 1 +
.
t→∞ h
µσ 2 hα + 2 (t + tα ζ (α, 1 + t))

(84)

By applying Watson’s Lemma to the integral representation of the Hurwitz Zeta function we
obtain the asymptotic expansion of ζ(α, 1 + t) for t → ∞ as

1
ζ(α, 1 + t) =
Γ(α)

Z

∞

0

1
xα−1 e−(1+t)x
Γ(α − 1)
(1 + t)−α+1 + (1 + t)−α + O(t−α−1 ).
dx ∼
−x
1−e
Γ(α)
2
(85)

By substituting ζ(α, 1 + t) with it’s asymptotic expansion and by applying the L’Hopital rule
we can rewrite (84) and simplify it to obtain the following result
"
#


−1
t
Γ(α
−
1)
1
lim log2 1 + µσ 2 hα + 2 t + tα
· (1 + t)−α+1 + (1 + t)−α
t→∞ h
Γ(α)
2
"
#

−1

1
Γ(α − 1)
k
2 α
−α+1
−α
= lim+
log2 1 + k · kµσ h + 2 1 +
· (1 + k)
+ (1 + k)
k→0 kh
Γ(α)
2
=

Γ(α)
.
2 ln 2 h (Γ(α) + Γ(α − 1))

(86)

From the above result it can be easily seen that the lower bound of ASE as λ → ∞ for 1D
regular network without fading under l1 (r) is greater than zero and the lower bound depends
on h and α. Next, using Proposition 4 and Lemma 6 it can be shown that this result can be
extended to 1D irregular network and 1D regular network with Rayleigh fading, thus concluding
the proof for 1D network.
For 2D network, similar to 1D network, we use Lemma 7 and write the limit of ASE for 2D
regular network without fading when λ → ∞ as

lim

λ→∞

{2,i}
λτ̂0 (λ, α)

= lim λ log2
λ→∞

1+

µσ 2 +

l(kb k)
P 0

ΦHEX \bo

!
l(kbi k)

.

(87)

Comparing the limit of ASE under the path-loss models between l1 (r) and l2 (r), we get
{2,1}

lim λτ̂0

λ→∞

{2,2}

(λ, α) ≥ lim λτ̂0
λ→∞

(λ, α).

(88)

32

It can be also easily seen that I0 =

P

ΦHEX \bo

l(kbi k) in (87) is the cumulated interference

from BSs deployed according to ΦHEX given that a typical user and its serving BS are located
at the origin. As this expression does not have a closed-form we use the expressions for lower
and upper bounds as presented in [17]
∞
∞
X
X
√
6k · l(Υk) < I0 < 6 l(Υ) +
6k · l(Υ 3/2 k)
k=1

(89)

k=2

By substituting l(r) = l2 (r) and using the definition of Hurwitz zeta function, we can simplify
the expressions for lower and upper bound as presented below
n1
X

6k · h−α + 6Υ−α ζ(α − 1, 1 + n1 ) < I0

k=1
n2
X

√
6k · h−α + 6Υ−α ( 3/2)−α ζ(α − 1, 1 + n2 ) > I0

k=1

where n1 =

b Υh c,

n2 = b √2h
c and n1 ≥ 0, n2 ≥ 1.
3Υ

By taking the upper bound of the cumulated interference we derive the lower bound of
{2,2}

limλ→∞ λτ̂0

(λ, α) as presented below

{2,2}

(λ, α) >


h−α
2
√ log2 1 +
√
lim
=
Υ→0+ Υ2 3
µσ 2 + 3(n2 + 1)n2 h−α + 6Υ−α ( 3/2)−α ζ (α − 1, 1 + n2 )


1
2

 .
lim+ √ log2 1 +
α √
Υ→0 Υ2 3
−α ζ α − 1, 1 + b √2h c
√2h c + 6 h
(
µσ 2 hα + 3(b √2h
c
+
1)b
3/2)
Υ
3Υ
3Υ
3Υ
lim λ τ̂0

λ→∞

(90)
{2,1}

Note that it is also the lower bound of limλ→∞ λτ̂0

(λ, α). We substitute k =

√2h
3Υ

in (90).

As λ → ∞, we have Υ → 0 and hence we can safely assume that k ∈ N which leads to the
following expression

{2,1}

lim λ τ̂0

λ→∞

{2,2}

(λ, α) ≥ lim λ τ̂0 (λ, α) >
λ→∞
√


k2 3
1
lim
. (91)
· log2 1 +
k→∞ 2h2
µσ 2 hα + 3k(k + 1) + 6k α ζ (α − 1, 1 + k)

33

Next, by applying Watson’s Lemma to the integral representation of the Hurwitz Zeta function
we obtain the asymptotic expansion of ζ(α − 1, 1 + k) for k → ∞.

1
ζ(α−1, 1+k) =
Γ(α − 1)

∞

Z
0

xα−2 e−(1+k)x
1
Γ(α − 2)
(1+k)−α+2 + (1+k)−α+1 +O(k −α ).
dx ∼
−x
1−e
Γ(α − 1)
2
(92)

By substituting ζ(α − 1, 1 + k) for its asymptotic expansion in (91) and by applying the
L’Hopital rule limx→c

f (x)
g(x)

= limx→c

f 0 (x)
g 0 (x)

we provide the lower bound for l1 (r) as

"
√


k2 3
Γ(α − 2)
2 α
α
log2 1 + µσ h + 3k(k + 1) + 6k
lim
> lim
2
λ→∞
k→∞ 2h
Γ(α − 1)(1 + k)α−2
"
#
√


−1
3
Γ(α − 2)
1
2
2 α 2
1
+
t
·
µσ
h
t
+
3(1
+
t)
+
6
+
log
=
lim
2
α−1
2
2
t→0+ 2t h
2(1 + k)
Γ(α − 1)(1 + t)α−2
#
√
−1
t
2 3
.

(93)
+
=
2(1 + t)α−1
12 ln 2 h2 1 + 2 Γ(α−2)
Γ(α−1)
{2,1}
λ τ̂0 (λ, α)

From the above result it can be easily seen that the lower bound of ASE as λ → ∞ for 2D
regular network without fading under l1 (r) is greater than zero and depends on h and α. Next,
similar to 1D network, by using Proposition 4 and Lemma 6 it can be shown that this result
can be extended to 2D irregular network and 2D regular network with Rayleigh fading, thus
concluding the proof for 2D network.
A PPENDIX B
P ROOF OF P ROPOSITION 5
The proof follows that of Theorem 9. We start by substituting l(r) = l1 (r) into (80) and
rearranging the terms to obtain the following expression for the ASE limit for 1D regular network
without fading

lim λ τ

λ→∞

{1,1}

1
(λ, α) = lim+ log2 1 +
Υ→0 Υ

Then, by substituting k →

h
Υ

we get

µσ 2 + 2Υ−α

h−α
P 
∞
i=1

!
i2 +

 −α/2
h 2
d

.

34

∞
X
k=1

 k
 n X
n−2
Γ(n − 12 )
1
(n)k j
j
·
= √
(π c coth(πc) − 1) +
(c2 + k 2 )n
2c
k!
2c
2 π Γ(n) c2n
k=0


(−1)n ζ (n − k, 1 − jc) + (−1)k ζ (n − k, 1 + jc)
(97)

lim λ τ

{1,1}

λ→∞

h−α
k
(λ, α) = lim log2 1 +
P
2
2 −α/2
k→∞ h
µσ 2 + 2h−α k α ∞
i=1 (i + k )

!
.

(94)

By using (97), given n = α/2, and assuming α = 2, we obtain the following expression
∞
X
i=1

(i2

1
π coth(πk)
1
=− 2 +
.
2
+k )
2k
2k

(95)

We use the above expression to simplify (94). By further applying the L’Hopital rule, we
obtain the ASE limit for α = 2 as follows





−2

k
h

log2 1 +
k→∞ h
µσ 2 + 2 h−2 k 2 − 2k12 +

lim λ τ {1,1} (λ, α) = lim

λ→∞

π coth(πk)
2k

 =

1
. (96)
ln 2 πh

Following a similar approach, and by using (97) for n = α/2 we can prove ASE limits for
α = 4 and α = 6.
Next, by using Proposition 4 and Lemma 6 it can be shown that the results can be extended to
1D irregular network and 1D regular network with Rayleigh fading, thus concluding the proof.
A PPENDIX C
P ROOF OF P ROPOSITION 6
The proof follows that of Theorem 9. We start by substituting l(r) = l1 (r) into (87) and
rearranging the terms to obtain the following limit expression

lim

λ→∞

{2,1}
λτ̂0 (λ, α)

= lim λ log2
λ→∞

1+

µσ 2 +

l (kb k)
P1 0

ΦHEX \bo l1 (kbi k)

!
.

(98)

35

Next, similar to the proof of Theorem 9, we use the expressions for lower and upper bounds of
cumulated interference in hexagonal (triangular) lattice given that a typical user and its serving
BS are located at the origin
∞
X

6k · l1 (Υk) < I0 < 6 l1 (Υ) +

k=1

∞
X

√
6k · l1 (Υ 3/2 k)

(99)

k=2

Assuming α = 4 and using the following formula for infinite series
∞
X
k=n+1

(k 2

k
j
= (ψ (1) (n + jc) − ψ (1) (n − jc))
2
2
+c )
4c

(100)

we obtain inequalities presented below
I0 > 6Υ−4

I0 < 6Υ−4

"
1+

 h 2 −2
Υ

j Υ (1)
(ψ (j h/Υ) − ψ (1) (−j h/Υ))
4h

(101)

#





√
Υ
2h
j
2h
− ψ (1) 1 − j √
+ ( 3/2)−3 ·
ψ (1) 1 + j √
4 h
3Υ
3Υ
(102)

where ψ (1) (x) is the Polygamma function of the first order, and j =

√

−1.

By taking the lower bound of the cumulated interference and substituting k =
{2,2}

the upper bound of limλ→∞ λτ0

lim

λ→∞

{2,1}
λ τ0 (λ, 4)

h
Υ

we obtain

(λ, 4) in the form presented below

2k 2
1
= lim √ · log2 1 +
j
2
4
3
(1)
k→∞ h2 3
µσ h + 6k 4 (ψ (jk) − ψ (1) (−jk))

!
. (103)

Next, by applying Watson’s Lemma to the integral representation of the Polygamma function
of the first order we obtain the asymptotic expansion of ψ (1) (jk) in the following form
Z ∞
xejkx
1
(1)
ψ (jk) =
dx ∼ (jk)−1 + (jk)−2 + O((jk)−3 ).
−x
1−e
2
0

(104)

By substituting ψ (1) (jk) and ψ (1) (−jk) with their asymptotic expansions and then applying
the L’Hopital rule, we obtain the upper bound for 2D regular network without fading.

36

!
2
2k
1
{2,1}
lim λ τ0 (λ, 4) = lim √ · log2 1 +
λ→∞
k→∞ h2 3
µσ 2 h4 + 6k 3 4j ((jk)−1 − (−jk)−1 )
√


2
2 3
t2
√ · log2 1 +
= lim+
=
(105)
t→0 t2 h2 3
µσ 2 h4 t2 + 3
9 ln 2 h2
{2,2}

Following a similar approach the lower bound of limλ→∞ λτ0

(λ, 4) can also be obtained.

Next, using Proposition 4 and Lemma 6 it can be shown that the results can be extended to
2D irregular networks and 2D regular networks with Rayleigh fading, thus concluding the proof.
R EFERENCES
[1] J. Zander and P. Mahonen, “Riding the data tsunami in the cloud: myths and challenges in future wireless access,”
Communications Magazine, IEEE, vol. 51, no. 3, pp. 145–151, March 2013.
[2] E. Hossain, M. Rasti, H. Tabassum, and A. Abdelnasser, “Evolution toward 5g multi-tier cellular wireless networks: An
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