1

A Unified Asymptotic Analysis of Area
Spectral Efficiency in Ultradense Cellular

arXiv:1712.07733v1 [] 20 Dec 2017

Networks
Ahmad AlAmmouri, Jeffrey G. Andrews, and François Baccelli

Abstract
This paper studies the asymptotic properties of area spectral efficiency (ASE) of a downlink cellular
network in the limit of very dense base station (BS) and user densities. This asymptotic analysis relies
on three assumptions: (1) interference is treated as noise; (2) the BS locations are drawn from a Poisson
point process; (3) the path loss function is bounded above satisfying mild regularity conditions. We
consider three possible definitions of the ASE, all of which give units of bits per second per unit area.
When there is no constraint on the minimum operational SINR and instantaneous full channel state
information is available at the transmitter, the ASE is proven to saturate to a constant, which we derive
in closed form. For the other two ASE definitions, wherein either a minimum SINR is enforced or full
CSI is not available, the ASE is instead shown to collapse to zero at high BS density. We provide several
familiar case studies for the class of considered path loss models, and demonstrate that our results cover
most previous models and results on ultradense networks as special cases.

I. I NTRODUCTION
The reuse of spectrum across space, called frequency or spatial reuse, is historically the
most important attribute of terrestrial wireless networks. Installing new cellular or wireless LAN
infrastructure is expected to generate spatial reuse, whereby the time and frequency resources can
be reused across smaller spatial scales. A major question within this context is whether this will
indefinitely bring higher overall data throughputs in the network. The ASE, defined as the sum
of the maximum data rate per unit area per unit bandwidth [1], has historically increased about
The authors are with the Wireless Networking and Communications Group (WNCG), The University of Texas at Austin, Austin,
TX 78701 USA. (Email: {alammouri@utexas.edu, jandrews@ece.utexas.edu, francois.baccelli@austin.utexas.edu}). Last revised
December 22, 2017.

2

linearly with the number of BSs: doubling the number of BSs about doubles the sum throughput
the network supports in a given area. This has often been referred to as “cell splitting gain”
or “densification gain” and has been the most significant driver in increased wireless data rates
over the last few decades [2], [3], and is expected to continue to drive gains for the foreseeable
future as well [4], [5]. The aim of the present paper is to clarify what can in fact be expected
in terms of ASE under densification. This is done by the definition of a general model allowing
one to study the asymptotic behavior of ASE under broad conditions.
A. Related Work
The study of dense wireless network capacity has a rich history, in particular for the case of
ad hoc multihop networks. The seminal result of Gupta and Kumar [6] showed that the transport
√
capacity of an infrastructure-less network increases with the number of nodes n roughly as n,
assuming a given node wishes to transmit with a randomly selected other node. Many subsequent
results extended this “scaling law” approach to a wide variety of models and communication
techniques, well summarized in [7]. A main intuition from these results is that the best networkwide throughput scaling is achieved by short-range local communication coupled with multihopping towards the intended receiver, a form of spatial reuse. A key subsequent paper showed
that this square-root scaling is a fundamental property of the electromagnetic propagation [8],
thus attempts to improve the scaling law through exotic communication strategies would be
futile.
To obtain mathematically tractable results on wireless network performance without resorting
to scaling laws, stochastic geometry [9]–[12] has emerged as a key toolset, in particular for
deriving the SINR distribution and related metrics, as achieved for ad hoc networks in [13],
[14]. One of the key conclusions of early stochastic geometry results for cellular networks such
as [15], [16] was that the SINR distribution is insensitive to the BS density, once the network is
interference-limited. This indicated that the ASE should increase linearly with the BS density:
an encouraging result. Both [15], [16], as well as most prior and subsequent literature including
the aforementioned scaling laws results, use a standard power-law path loss function of the form
L(r) = c0 r −η , where r is the distance, c0 = L(1) is a constant, and η > 2 is the path loss
exponent.
However, other more recent works have shown that when adopting more realistic bounded
path loss models, different and much more pessimistic trends can be derived or observed for ASE

3

under densification [17]–[22]. The current situation can be summarized by saying that depending
on the path loss model and the ASE definition, one finds in the literature the whole spectrum
of asymptotic behaviors for ASE ranging from linear growth to collapse.
It is well understood that the pole at the origin of the standard power law model creates a
scale free cellular SIR which is the basis for the linear ASE growth alluded to above. It is also
well understood that this standard power-law path loss model has important shortcomings in the
context of dense networks: first, it does not accurately model received powers for short distances,
and second, it is intractable for line-of-sight propagation, i.e., η = 2. The more realistic bounded
path loss models alluded to above, which are the main focus of the present paper, are hence
much more natural within the context of ultra-densification. We describe several of these refined
models and results in more detail in Sect. V, and show, through a few typical examples, that
they all can be considered special cases of the asymptotic results obtained in this paper.
B. Contributions
The main contribution of the paper is a general answer to the question of how ASE scales
in a cellular network with very large density. We start by formally defining the ASE using the
Shannon rate, with the important but common caveat that interference is treated as noise. We
further provide two alternative definitions that add practical constraints regarding the minimum
operational SINR and the availability of the channel state information at the transmitters. We then
define a broad class of bounded path loss models that would appear to capture all currenly used
and physically viable propagation models. This broad class, which is assumed throughout the
paper, is defined by three simple mathematical properties, the most important being boundedness.
Then, under a Poisson point process (PPP) assumption regarding the spatial distribution of
the BSs and a general small-scale fading model, we derive the following properties on the
asymptotics of the different definitions of the ASE when the BS density grows large.
Assuming that there is no constraint on the minimum operational SINR, and that the transmitter
can send at the Shannon rate – which implies that perfect instantaneous channel state information
is available at the transmitter – we prove that the ASE saturates to a constant, which is
L0 /2π ln(2)γ, where L0 and γ are constants determined by the path loss function L(r), namely
R∞
L0 < ∞ is the path loss at r = 0, and γ = rL(r)dr. However, under the same set of
0

assumptions, when either a minimum operational SINR is required, or the CSI is not fully
available, then the ASE tends to zero when the BS density tends to infinity.

4

The rest of the paper is organized as follows. In Section II, we present the system model. In
Section III, we present the different definitions of the ASE and discuss the intuition behind each
one of them. Section IV is the main technical section of the paper, where we derive expressions
for the ASE. Case studies with discussions are presented in Section V, before the conclusion
and future work in Section VI.
Notation: 1{·} is the indicator function which takes the value 1 if the statement {·} is true
and takes the value 0 otherwise, R is the set of real numbers, Z is the set of integers, R+ is the
set of non-negative real numbers, and R∗+ is the set of strictly positive real numbers.
II. S YSTEM M ODEL
We consider a downlink cellular network where the BSs are spatially distributed as a homogeneous PPP Φ with intensity λ. The users are spatially distributed according to an independent
stationary point process, with intensity λu ≫ λ such that each BS has at least one user to serve.
In other words, we assume that all BSs are continually transmitting. The small scale fading h is
assumed to be distance independent with a unit mean but with arbitrary distribution. All channel
fading variables are assumed to be independent and identically distributed (i.i.d.). Closest BS
association is assumed and all BSs and users are equipped with a single omni-directional antenna.
Propagation Model. The large scale channel gain is captured by the path loss function L(r).
We focus on a broad class of path loss models that are physically reasonable. First, we assume
that L(0) = L0 is the average transmit power directly at the antenna, hence L0 is a finite constant.
Moreover, we assume that L(r) ≤ L0 ∀ r ∈ R+ , which means that at any distance, the average
received power cannot be higher than the average transmit power present at the antenna itself.
Note that assuming L(r) ≤ L0 ∀ r ∈ R+ is much weaker than requiring L(r) to be decreasing
with the distance. Hence, this assumption is quite general and includes cases where the path
loss is not deterministic with the distance, such as the common situation where depending on
a random link state (e.g. LoS or NLoS), different path loss attenuation functions are activated
[20], [23], [24].
Finally, we require that the total average received power Pavg at the origin (or any arbitrary
point in the network) in the network to be finite, which is also a mild condition that must be
satisfied in any physically realizable network. This requirement can be mathematically distilled

5

as
Pavg = E

"

X

#

hi L(ri ) = E

ri ∈Φ

where γ ,

R∞

"

X

ri ∈Φ

#

L(ri ) = 2πλ

Z∞

rL(r)dr = 2πλγ,

(1)

0

rL(r)dr. To obtain this, we used the fact that the small scale fading has a unit

0

mean, and then Campbell’s theorem [9] to get the last equality. Hence, we assume that γ is finite.
We formalize the preceding discussion a definition of physically feasible path loss models.
Definition 1. (Physically feasible path loss) Physically feasible path loss models are the family
of path loss functions L(r) with the following properties
1) L0 = L(0) ∈ R+ ,
2) L(r) ≤ L0 ∀r ∈ R+ ,
R∞
3) γ = rL(r)dr ∈ R∗+ ,
0

To summarize concisely the rationale for these three conditions: 1) insures that the transmit

power is finite, 2) insures that on average we cannot receive more power than was transmitted,
and 3) states that the average received power at any point from all BSs in the network is finite.
Hence, we conclude that it is impossible to construct an empirically verifiable path loss model
which violates any of these three conditions in Definition 1. Unsurprisingly, the path loss models
adopted in 3GPP standards, both for the conventional Sub6 GHz [25] as well as in the mmWave
bands [26], along with a great many other common path loss models, satisfy the aforementioned
requirements as we will show in Sect. V. Throughout this work, all results are for this class
of physically feasible path loss models, unless otherwise stated. Moreover, we allow for any
general small-scale fading model as long as it has finite mean, for simplicity we assume unit
mean1 .
SINR. We derive the performance of a user located at the origin. According to Slivnyak’s
theorem [10], there is no loss of generality in this assumption, and the evaluated performance
represents the average performance for all the users in the network. Based on our system model,
the signal to interference plus noise ratio (SINR) for the typical user is
SINR =

P

h0 L(r0 )
,
hi L(ri ) + N0

(2)

ri ∈Φ\B(0,r0 )
1

The analysis can be easily extended to the non-unit mean fading distributions by taking the mean as a common factor in the

SINR expression in (2). Hence, it is equivalent to considering unit mean distributions with a scaled noise power.

6

where N0 is the noise power, h0 (hi ) and r0 (ri ) are the channel small scale fading power
and the distance between the tagged user and the serving (ith interfering) BS, respectively.
Hence, we focus on the case where the network interference is treated as noise which is the
baseline practice even in 5G systems. The same assumption was made in [1], where the ASE was
proposed. Considering more advanced decoding or cooperative techniques, as in [27]–[30], is
left for future work. Before delving into the analysis, we formally introduce different definitions
of the ASE in the next section and discuss the intuition and the physical meaning behind them.
III. T HROUGHPUT

PERFORMANCE METRICS

The ASE has been widely used to study different trade-offs in cellular networks and as a
measure of the total network throughput. The definition in [1] is based on the sum of the
Shannon capacities of the users per unit area:
N

1 X
E [log2 (1 + SINRk )] ,
E=
|A| k=1

(3)

where A is the considered area, N is the number of users within A, and log2 (1 + SINRk ) is
the spectral efficiency in bps/Hz. Then by averaging over different fading realizations, we get
the expression in (3). Note that the used Shannon capacity formula log2 (1 + SINRk ) assumes
Gaussian interference, which is not true in our case since the interference is typically dominated
by near BSs. Hence, the central limit theorem does not apply. However, the Gaussian interference
is a worst case scenario for capacity calculations [1], [31].
Moreover, there is randomness in the BSs/users locations and in the small-scale fading in
our case. But since the SINR distribution seen by the typical user at the origin represents the
stationary distribution of all users in the network, and since the channels are assumed to be i.i.d,
then after averaging over different network and fading realizations, the definition (3) simplifies
to the following:
E [E(λ)] = λE0 [log2 (1 + SINR)] ,

(4)

where E0 [·] is the Palm expectation with respect to the users’ point process [10]. However, since
we are focusing on a user located at the origin, the Palm expectation is reduced to the (stationary)
expectation E[·] due to Slivnyak’s theorem [10]. Hence, we adopt the following definition of the
ASE.

7

Definition 2. (ASE) The average ASE is defined as:
E [E(λ)] = λE [log2 (1 + SINR)] .

(5)

The ASE definition in this form has been used in [18], [21], [22], [32], [33]. However, it
assumes that the system can work with any arbitrarily small SINR, which may not be feasible
in practice. Hence, a second definition was used in [20], [34], which adds a constraint on the
minimum operational SINR θ0 as in the next definition.
Definition 3. (Constrained ASE) The average constrained ASE is defined as:
h
i
˜
E E(λ)
= λE [log2 (1 + SINR)1 {SINR ≥ θ0 }] .

(6)

In addition, a third definition called the potential throughput, was proposed in [19], and then
used in [17], [18], [21], [35], [36], to study the ASE in cellular networks as the following:
Definition 4. (Potential throughput) The average potential throughput is defined as:
E [R(λ)] = λ log2 (1 + θ0 )P {SINR ≥ θ0 } .

(7)

The potential throughput captures the case where the channel state information is not available
at the transmitter, hence it transmits with a constant rate and then the messages are only decodable
at the receiver if the SINR is larger than some threshold θ0 . Which means that high SINR values
are not exploited, and if the SINR is small, then the link is in complete outage. The three
different definitions are related as follows:
E [R(λ)] = E [λ log2 (1 + θ0 )1 {SINR ≥ θ0 }]
h

≤ E [λ log2 (1 + SINR)1 {SINR ≥ θ0 }] = E Ẽ(λ)

i

≤ E [λ log2 (1 + SINR)] = E [E(λ)] .
h
i
˜
Hence E [R(λ)] ≤ E E(λ) ≤ E [E(λ)]. In all cases, the BSs are assumed to always have

messages to transmit to their users.

It was shown in [18] that adopting different definitions can lead to different insights on the
network performance. The aim of this work is to study all of them in a unified framework. In
the next section, we study the asymptotic value of the ASE under fairly general assumptions.

8

IV. P ERFORMANCE A NALYSIS
In this section, we analyze the asymptotic behavior of cellular networks in terms of the
ASE when the BS density is high. Note that due to the boundedness of L(·), the numerator
of the SINR defined in (2) is bounded on average. However, the denominator can get arbitrary
large by increasing the BS density. Hence, the SINR approaches zero for high BS density. For
completeness, we prove this in the next lemma.
Lemma 1. When λ → ∞, the SINR as defined in (2) tends to zero a.s.
Proof. Refer to Appendix A.
Next, we study the ASE as defined in (5) since it is the more general metric and then we
move to the constrained ASE (6) and the potential throughput (7). We start with the following
lemma which provides a lower bound on the ASE.
Lemma 2. The asymptotic average ASE is lower bounded by
lim E [E(λ)] ≥

λ→∞

L0
.
2π ln(2)γ

(8)

Proof. Let λ = kλ0 , where k ∈ Z+ and λ0 ∈ R∗+ . We are interested in lim E(kλ0 ) which is
k→∞

found by the following
lim E(kλ0) = lim λ0 k log2 (1 + SINR(kλ0 ))

k→∞

k→∞

λ0
kSINR(kλ0 )
k→∞ ln(2)
h0 L0
=
ln(2)2πγ
h
i
L0
,
E lim E(kλ0) =
k→∞
ln(2)2πγ
= lim

(9)
(10)
(11)

where, (9) follows because the SINR approaches zero a.s when k → ∞ as we proved in Lemma
1, (10) follows from (22), and (11) follows by taking the expectation of (10) w.r.t h0 which has
a unit mean. Then by using Fatou’s lemma [37], the following is true
h
i
L0
lim E [E(kλ0)] ≥ E lim E(kλ0 ) =
.
k→∞
k→∞
ln(2)2πγ

Note that the result is independent of λ0 , hence we can conclude that lim E [E(λ)] ≥
λ→∞

L0
.
ln(2)2πγ

9

The last lemma shows that the average ASE is lower bounded by a constant and does not
drop to zero. However, it is more interesting to show that it holds with equality. Then we will
have exact characterization of the limit and prove the existence of a densification plateau [38].
In the following theorem, we provide a sufficient condition that can be tested without dealing
with the limit of the ASE.
Theorem 1. Let I =

P

ri ∈Ψ\B(0,r0 )

L(ri )hi , where Ψ is a PPP with intensity λ0 . If the second

negative moment of I is finite for all λ0 ≥ λc , where λc ∈ R+ is a constant, then
lim E [E(λ)] =

λ→∞

L0
.
2π ln(2)γ

(12)

Proof. Refer to Appendix B.
Theorem 1 shows that the ASE converges to a finite constant, which practically means that
we cannot keep harvesting performance gains by densifying the network; at some point the
performance will saturate. Note that Theorem 1 holds as long as the second negative moment
of I as defined in the theorem is finite, which is a function of the path loss function as well as
the fading distribution. In the next corollary, we provide two sufficient conditions under which
the condition in Theorem 1 is satisfied.
Corollary 1. If any of the following is satisfied:
R∞ R∞
R∞

 
1)
2πλ0 rt exp −πλ0 r 2 − 2πλ0 r xEh 1 − e−htL(x) dx drdt is finite ∀λ0 ≥ λc ,
0 0

 

R∞ R∞
R∞
1
2
2)
2πλ0 r exp −πλ0 r − 2πλ0 xPh L(x)h ≥ √t dx drdt is finite ∀λ0 ≥ λc ,
0 0

r

where λc ∈ R+ is a constant, then the condition in Theorem 1 is satisfied.
Proof. Refer to Appendix C.

Although these conditions are general, evaluating these integrals for each path loss function
and each fading distribution is cumbersome. Hence, in pursuit of simpler sufficient conditions,
we present the following corollary for the Rayleigh fading case.
Corollary 2. For Rayleigh fading channels (i.e. hi ∼ exp(1)), if ∃ r0 ∈ R+ and ζ ∈ R∗+ such
that:
1)
2)

rL(r)
≥ ζ, ∀r ∈ [r0 , ∞),
−L′ (r)
R∞ r −πλ r2
e 0 dr is finite for
L(r)2
r0

all λ0 > λc ∈ R+ ,

10

assuming that L(r) is decreasing and differentiable ∀r ∈ [r0 , ∞), then the first sufficient condition
in Corollary 1 is satisfied.
Proof. Refer to Appendix D.
The last corollary presents simple conditions that can be easily checked for any path loss
function. Interestingly, these conditions cover most of the bounded path loss models reported in
the literature as we will show in the next section. In the next corollaries, we show that these
conditions are also sufficient for the no fading case as well as for any fading distribution as long
as it has unit mean.
Corollary 3. The sufficient conditions in Corollary 2 also hold for the no fading case.
Proof. For the no fading case, the first condition in Corollary 2 reduces to


  Z∞ Z∞
Z ∞

1
2
−tL(x)
2πλ0 rt exp −πλ0 r − 2πλ0
x 1−e
dx drdt
E 2 =
Ii
r
≤

0 0
Z∞ Z∞
0

0


Z
2
2πλ0 rt exp −πλ0 r − 2πλ0

where (13) follows since 1 − e−t ≥

t
,
1+t

∞
r


tL(x)
x
dx drdt,
1 + tL(x)

(13)

∀t ≥ 0. Then the proof follows as in the proof of

Corollary 2.
Corollary 4. The sufficient conditions in Corollary 2 hold for any fading model such that
E [h] = 1.
Proof. Refer to Appendix E.
Hence, the asymptotic behavior for the ASE is actually agnostic to the fading distribution.
In other words, fading only quantitatively affect the performance for low to moderate densities.
The conditions in Corollary 2 are easy to work with if the path loss function is deterministic
with the distance. However, if LoS/NLoS links are considered, then the path loss is no longer
deterministic and it depends on the state of the link. This case is important when the network is
operating in high frequency bands (e.g. mmWave) [26], [39]. For this specific case, we provide
the following corollary.
Corollary 5. If ∃ r0 ∈ R+ , ζ ∈ R∗+ and a differentiable decreasing function L̃ : [r0 , ∞) → R+
such that:

11

1) L̃(r) ≤ L(r) ∀r ∈ [r0 , ∞)].
2)
3)

r L̃(r)
≥ ζ, ∀r ∈ [r0 , ∞).
−L̃′ (r)
R∞ r −πλ r2
e 0 dr is finite for
L̃(r)2
r0

all λ0 > λc ∈ R+ .

Then the first sufficient condition in Corollary 1 is satisfied.
Proof. The proof is straightforward from the first sufficient condition in Corollary 2.


  Z∞ Z∞
Z ∞


1
2
−htL(x)
2πλ0 rt exp −πλ0 r − 2πλ0
xEh 1 − e
dx drdt
E 2 =
Ii
r
0

≤

0

Z∞ Z∞
0

0


Z
2
2πλ0 rt exp −πλ0 r − 2πλ0

∞

r

i 
h
−htL̃(x)
dx drdt.
xEh 1 − e

Then the proof follows as in the proof of Corollary 5.
So far we have proved that the ASE converges to a finite constant under fairly general
assumptions. As discussed earlier, the definition of the ASE as in (5) assumes that we can
work with any arbitrary small SINR. A more practical definition is the constrained ASE defined
in (6) or perhaps the potential throughput defined in (7) in some scenarios. In the following, we
leverage the analysis of the ASE to study the asymptotic behavior of the constrained ASE and
the potential throughput.
Theorem 2. If the path loss function satisfies the condition in Theorem 1 (or sufficiently Corollary
5) and θ0 ≥ ǫ > 0, then the constrained ASE defined in (6) eventually drops to zero.
Proof. Let λ = kλ0 , where k ∈ Z+ and λ0 ∈ R∗+ . Then,
h
i
lim E Ẽ(kλ0 ) = lim λ0 kE [log2 (1 + SINR)1 {SINR ≥ θ0 }] ,
k→∞
k→∞
h
i
= E lim λ0 k log2 (1 + SINR)1 {SINR ≥ θ0 } ,
k→∞

n
o
L0 h0
=E
1 lim SINR ≥ θ0 = 0,
k→∞
2π ln(2)γ

(14)
(15)

where (14) follows since λ0 kE [log2 (1 + SINR)1 {SINR ≥ θ0 }] ≤ λ0 kE [log2 (1 + SINR)] which
is uniformly integrable as proved in Theorem 1. Hence, we can push the limit inside the
expectation [40]. (15) follows from (22) and the fact that 1 {x ≥ θ0 } is continuous at x = 0
under the assumption that θ0 ≥ ǫ > 0, and the last equality follows by using Lemma 1. Finally,
h
i
˜
since the result is independent of λ0 we can conclude that limλ→∞ E E(λ)
= 0.

12

Theorem 3. The potential throughput defined in (7) eventually drops to zero if the path loss
function satisfies the conditions in Theorem 1 (or sufficiently Corollary 5) for all θ0 ∈ R+ .
Proof.
lim E [R(λ)] = lim λ log2 (1 + θ0 )P {SINR ≥ θ0 } ,

λ→∞

λ→∞

= lim λ log2 (1 + θ0 )E [1 {SINR ≥ θ0 }] ,
λ→∞

≤ lim λE [log2 (1 + SINR)1 {SINR ≥ θ0 }] ,
λ→∞
h
i
= lim E Ẽ(λ) = 0,
λ→∞

(16)

(17)

where (17) follows from Theorem 2 assuming θ0 ≥ ǫ > 0. If θ0 = 0, then it is clear from (16)
that the limit is also zero.
Hence, the constrained ASE and the potential throughput have completely different behaviors
than the ASE: we observe a densification plateau (it saturates to a constant) for the ASE and a
densification collapse (it drops to zero) for the constrained ASE and the potential throughput.
From a network throughput perspective, the ASE result means that although for very high
densities the SINR approaches zero, the sum throughput of the users can still be higher than
zero since there are many users in the network. In other words, the increase in the co-channel
interference is fully balanced by the increase in the number of users using this channel. However,
if we put a constraint on the minimum operational SINR, then many of these users will go into
outage and the sum throughput of all users approaches zero.
From the users’ application layer perspective, we can have the following two scenarios. If
the users’ share of the resources (bandwidth and time) is fixed, their rate gets very small when
the density is high. However, the sum throughput is still non-negligible. Hence, if we have
IoT devices or sensors with small data rate requirements, then the network can accommodate
densification. On the other hand, if the rate demand is high (e.g. video streaming) then the
network will not be able to satisfy their needs although their sum throughput is still relatively
high. The second scenario is when the per user share of resources scales with the intensity, simply
because more BSs means less load per BSs and bigger shares for each served user. In this case
the ASE is an indication of the per user rate. Hence, the user can benefit from densification as
long as it can work with (very) small SINRs, for example through low rate codes and/or spread
spectrum techniques.

13

TABLE I: Path loss functions.
lim E [E (λ)]

Path loss function

λ→∞

L1 (r) = A min(c0 , r

−η

)

L2 (r) = A(c0 + r)−η
L3 (r) = A(c0 + r η )−1

2
η
(η−2)c0

ηπ ln(2)
η 2 −3η+2
2π ln(2)c2
 0
η sin 2π
η
2π 2

L4 (r) = A(c20 + r 2 )
L5 (r) = Ae−αr

−η
2

β

2
η
ln(2)c0

η−2
2πc2
0 ln(2)
2
βα β

 
2
2π ln(2)Γ β

Parameters

L̃(r)

r0


−1
η
max 1, c0

−η

A > 0, c0 > 0, η > 2

Ar

A > 0, c0 > 0, η > 2

A(c0 + r)−η

1

A > 0, c0 > 0, η > 2

A(c0 + r η )−1

c0 (η−2)
2

A > 0, c0 > 0, η > 2

A(c20 + r 2 )

1

A > 0, 2 ≥ β > 0, α > 0

Ae−αr

β

−η
2

1

Another insight from the previous theorems is that channel state information has to be available
at the transmitter to be able to harvest gains from densifying the network. Theorem 3 shows
that transmitting with a constant rate will drop the potential throughput to zero regardless of
how small is the rate. The previous three theorems also explain the different conclusions in the
literature regarding the asymptotic ASE: adopting different definitions with the same network
model leads to totally different conclusions.
Note that the analysis in this work is based on the assumption that all BSs always have users
to serve. The other case where we have a finite user density and some BSs will be idle and silent
can be handled as in [21], [41]. After establishing the analytical part of this work, we move to
the next section where we apply the presented theorems to the commonly used bounded path
loss models in the literature and we show that the framework is general enough to cover most
the used models.
V. C ASE S TUDIES
A. Single-slope models
We start with the famous power-law path loss model where the power attenuation is captured by
L(r) = Ar −η , where A and η are positive parameters [42]. This function is clearly unbounded,
since L(r) → ∞ as r → 0 which is not physically feasible. Hence it is not included in the
class of functions we are considering. In fact, it was shown in [15] that the SIR distribution is
independent of the BS density, a property referred to as the SIR-invariance, for Rayleigh fading
channels, and extended later for more general fading channels [43], [44]. Hence, by checking the
definitions in (5), (6), and (7), it is clear that all of these metrics will increase linearly with the

14

BS density. Which is a totally different scaling law than the ones we have. However, by slightly
modifying this path loss function, we can get path loss functions that have the desired properties
in Definition 1. Examples of such functions are L1 (r) = A min(c0 , r −η ), L2 (r) = A(c0 + r)−η ,
and L3 (r) = A(c0 + r η )−1 , where c0 , η, and A are positive parameters. These functions are
bounded and satisfy the desired properties mentioned in Section II. Moreover, these functions
also satisfy the conditions in Corollary 5 as shown in Table I, where we provide the asymptotic
ASE along with L̃(·) and r0 needed to verify the conditions in Corollary 5. Note that both the
constrained ASE and the potential throughput decay to zero for these functions according to
theorems 2 and 3.
It was shown in [45] that the potential throughput -called the ASE in that paper- decays to
zero assuming path loss functions in the form of L2 (·) and L3 (·) with c0 = 1 and Rayleigh
fading channels. Hence, it agrees with our results and we further proved that this observation
holds for any fading distribution. Analysis of asymptotic ASE assuming path loss functions in
the form of L1 (·), L2 (·), and L3 (·) were previously absent from the literature to the best of our
knowledge.
Another interesting path loss function is L4 (r) = A(c20 + r 2 )

−η
2

which captures the case where

the elevation difference between the BSs and users is c0 > 0 [17]. This model has the desired
properties in Definition 1 and also satisfies the conditions in Corollary 5 as shown in Table I,
which also includes the asymptotic ASE under this model in a simple form that exactly shows
how the ASE scales with the elevation difference c0 .
B. Multi-slope models
Moving to a more general path loss model, the multi-slope path loss has been widely used in
the literature [19], [46], [47] and it is the baseline model in 3GPP standards [25], [26]. It can
be represented as the following:
L6 (r) =

n
X
i=1

Ai r −ηi 1 {ri−1 ≤ r < ri } ,

where Ai , ηi , n, and ri are positive parameters. Multi-slope path loss captures the case where the
path loss exponent η is also a function of the distance. The number of slopes (regions) is n with
ηi the effective path loss exponent in the ith region that lies between ri−1 and ri . More details
about this path loss function are provided in [19]. Without loss of generality, we set r0 to zero
and rn to ∞ to cover the whole positive real line and we assume that n is at least 2. Moreover,

15

we set η1 to zero to make it bounded and assume that ηn > 2, otherwise the aggregate network
interference will have an infinite average power [19]. Under these assumptions, this function has
the desired properties in Definition 1, and by choosing L̃(r) = An r −ηn and r0 = max (1, rn−1),
the conditions in Corollary 5 are satisfied and the asymptotic ASE is given by the following:
!!−1
n−1
−η +2
2−η
A1 r12 An rn−1n X
ri−ηi +2 ri−1i
A1
.
+
Ai
+
+
lim E [E(λ)] =
λ→∞
2π ln(2)
2
ηn
2 − ηi
ηi − 2
i=2

However, both the potential throughput and the constrained ASE eventually drop to zero,

which agrees with the results in [19], where the authors showed that the potential throughput
drops to zero in the case of η1 = 0. Note that in the case of n = 2 and η1 = 0, the multi-slope
model reduces to L1 (·) given in Table 1.
C. Stretched exponential model
The limiting case of the multi-slope model with large number of slopes is captured by the
stretched exponential path loss proposed in [18] to capture the power attenuation in urban and
dense cellular networks where the signal loss is mainly due to obstructions. This path loss
β

function takes the following form: L5 (r) = Ae−αr , where A, α, and β are positive parameters.
For more details about the intuition behind this model, refer to [18]. As shown in Table I, this
model also satisfies the conditions in Corollary 5. It was proven in [18] that the ASE saturates
to a constant and the potential throughput drops to zero for high BSs densities. Our results in
this work confirm that both of these observations also hold for any general fading model. In
addition, we prove that the constrained ASE also drops to zero.
D. LoS/NLoS models
We have focused so far on path loss models that are deterministic with the distance. However,
in the case of mmWave transmission, the power attenuation strongly depends on the link state
(Los or NLoS) [39]. In fact, the path loss function adopted recently by the 3GPP standard
[26] consider this scenario not only for mmWave, but for the whole frequency range from 0.5
GHz to 100 GHz. This path loss has three ingredients: multi-slope path loss, LoS/NLoS link
states, and non-zero elevation difference between the BSs and the users. Mathematically it can
be represented by the following:
L(r, c0 ) = PLoS (r)LLoS (r, c0 ) + (1 − PLoS (r))LN LoS (r, c0 ),

16

with
LLoS (r, c0 ) =
LN LoS (r, c0 ) =

n
X

i=1
n
X
i=1

ηi

Ai (r 2 + c20 )− 2 1 {ri−1 ≤ r < ri } ,
η̄i

Āi (r 2 + c20 )− 2 1 {r̄i−1 ≤ r < r̄i } ,

where, PLoS (r) is the LoS probability, c0 is the elevation difference between the users and BSs,
Ai depends on the operating frequency, and the parameters associated with NLoS attenuation
are distinguished by the bar above them. The LoS probability and all other parameters depend
on the considered environment (e.g. RMa, UMa, UMi). However, regardless of the considered
environment, we have the following characteristics of this path loss function:
1) The LoS probability is decreasing with the distance. Hence, L(0, c0 ) = LLoS (0, c0 ). Moreover, LLoS (r, c0 ) and LN LoS (r, c0 ) are decreasing functions with the distance r.
2) Due to the elevation difference between the users and the BSs, LLoS (0, c0 ) = A1 (c20 )

−η1
2

is

bounded regardless of the value of η1 .
3) For a given distance r, we always have LLoS (r, c0 ) ≥ LN LoS (r, c0 ). Hence, L(0, c0 ) ≥
L(r, c0 ) ∀r ∈ R+ .
4) The values of ηn and η̄n are always larger than 2 in all cases. In other words, the long
distance path loss exponent is always larger than 2, which is the path loss exponent for free
R∞
space propagation. This ensures that γ = rL(r)dr is finite.
0

Based on the previous points, this path loss model satisfies the three properties in Definition
−η̄n
1. Moreover, by choosing L̃(r) = A¯n (r 2 + c2 ) 2 along with r0 = max (1, r̄n−1 ), it is straight0

forward to see that the conditions in Corollary 5 are also satisfied. Hence, we can conclude that
the ASE saturates to a constant, while the constrained ASE and the potential throughput drop
to zero for high BS densities under this path loss model.
The models in [17], [34] consider similar path loss function. In [17], it was shown that the
potential throughput drops to zero eventually, and in [34] it was shown that the constrained ASE
also drops to zero for high BSs densities. Our results agree with both of these works, and adds
to them the fact that the ASE saturates to a constants and does not drop to zero, even for general
small-scale fading models.
Overall, we have shown in this section that the conditions we have on the path loss functions
are simple to check and yet general enough to capture the practical path loss models used in
the literature and the standards. Moreover, the unified framework allows us to draw conclusions

17

regarding the ASE, constrained ASE, and the potential throughput at the same time by checking
the same conditions.
VI. C ONCLUSION
We have studied the asymptotic area spectral efficiency (ASE) in a downlink cellular network
under ultradensification. Assuming a Poisson point process for the spatial distribution of the base
stations, a general small-scale fading model, and a class of physically feasible path loss models,
we provided a general framework to analyze the ASE. Our results show that the average ASE
saturates to a constant when the BS density is large, while the constrained ASE and potential
throughput both collapse to zero. These results show that there are fundamental physical limits
to the throughput gains that can be harvested with densification, with the important caveats that
interference is treated as noise and the active user density scales with the base station density.
The results are also all asymptotic with BS density and we do not study when these asymptotics
kick in. These caveats point to interesting future work for dense networks, such as characterizing
at what density the ASE saturation or collapse manifest. Considering the effects of advanced
interference suppression techniques such as joint (over multiple BSs) transmission or decoding,
or successive interference cancellation, would also be of interest.

18

A PPENDIX A
P ROOF

OF

L EMMA 1

Let λ = kλ0 , where k ∈ Z+ and λ0 ∈ R∗+ . We are interested in lim kλ0 SINR(kλ0 ) which is
k→∞

found by the following
lim kλ0 SINR(kλ0 ) = lim kλ0

k→∞

P

k→∞

h0 L(r0 )
hi L(ri ) + N0

ri ∈Φ\B(0,r0 )

= lim kλ0 P
k→∞

ri ∈Φ

h0 L(r0 )
hi L(ri ) − h0 L(r0 ) + N0

= lim λ0 1 P
k→∞

k

ri ∈Φ

h0 L0
hi L(ri ) −

h 0 L0
k

+

(18)

N0
k

h0 L0
= lim λ0 1 P
k→∞
hi L(ri )
k

(19)

ri ∈Φ

= lim

k→∞

1
k

k
P

(20)

i=1 ri,j ∈Ψj

λ0 h0 L0

=
E

"

P

ri,1 ∈Ψ1

=

λ0 h0 L0
P
hi,j L(ri,j )

hi,1 L(ri,1 )

(21)

#

λ0 h0 L0
h0 L0
=
,
R∞
2πγ
2πλ0 rL(r)dr

(22)

0

h 0 L0
k

N0
k

→ 0 and
P
→ 0 a.s, (20) by exploiting the superposition property of the PPP [9], where Φ = ki=1 Ψi

where, (18) follows because L(r0 ) → L0 a.s. as k → ∞, (19) since as k → ∞,

and Ψi ∀i ∈ Z∗+ are i.i.d. PPPs each with density λ0 , (21) from the law of large numbers,
and (22) from Campbell’s theorem [9]. Note that since the result is independent of λ0 , we can
conclude that lim λSINR(λ) =
λ→∞

completes the proof.

h 0 L0
2πγ

which is finite a.s, hence lim SINR(λ) = 0 a.s, which
λ→∞

19

A PPENDIX B
P ROOF

OF

T HEOREM 1

Since we proved the lower bound in Lemma 2, it is sufficient to prove that the asymptotic
ASE is upper bounded by the constant given in (12). The proof is as follows:
lim E [E(kλ0)] = lim E [λ0 k log2 (1 + SINR(kλ0 ))]
k→∞


λ0
kSINR(kλ0 )
≤ lim E
k→∞
ln(2)


k→∞

 λ0
k
= lim E 
k→∞
ln(2)

P

=





h0 L(r0 )


hi L(ri ) + N0

ri ∈Φ\B(0,r0 )



 λ0 L0
≤ lim E 
k
k→∞
ln(2)

(23)

h0
P

hi L(ri )

ri ∈Φ\B(0,r0 )

λ0 L0

lim E k
ln(2) k→∞





(24)




,

(25)

1
P

hi L(ri )

ri ∈Φ\B(0,r0 )

where, (23) follows since ln(1 + x) ≤ x ∀x ∈ R+ , (24) follows because L(r) <= L0 ∀r ∈ R+
and N0 > 0, and (25) by taking the expectation with respect to h0 . Next, assuming that the RV
inside the expectation in (25) is uniformly integrable (which we will prove next if I has a finite
second negative moment), then we can push the limit inside the expectation2 [40].




λ0 L0

lim E  1
ln(2) k→∞
k

1

P

ri ∈Φ\B(0,r0 )

 λ0 L0 
E  lim
=
k→∞
ln(2)
hi L(ri )
=

1

1
k

L0
,
ln(2)2πγ

P

hi L(ri )

ri ∈Φ\B(0,r0 )




(26)

where (26) follows from the law of large numbers and Campbell’s theorem similar to the proof
in Lemma 1. Hence, the asymptotic average ASE is upper bounded by

L0
ln(2)2πγ

and we already

proved in Lemma 2 that it is lower bounded by this same constant. Moreover, since the results
hold for all λ0 ≥ λc ∈ R+ , then we can conclude that lim E [E(λ)] =
λ→∞

2
1

L0
.
ln(2)2πγ

Precisely, if the RV inside the expectation in (25) is uniformly integrable, then a.s convergence implies convergence in the

L norm [40, Theorem 5.5.2].

20

The final step is to prove that the RV inside the expectation in (25) is uniformly integrable if
I has a finite second negative moment, which is proved as follows:
1
1
k

P

hi L(ri )

≤

ri ∈Φ\B(0,r0 )

1
1
k

k
P

P

1
k

hi,j L(ri,j )

i=1 ri,j ∈Ψi \B(0,ri,0 )

1

=

(27)

k
P

i=1

(28)
Ii

k

1X 1
≤
,
k i=1 Ii

(29)

where, (27) follows because in the LHS we exclude the closest node to the origin from the
interference and in the RHS we exclude the closest point in each Ψi , where Ψi ∀i are i.i.d.
PPPs with intensity λ0 , so we remove the closest BS in Φ and additional ones, (28) by defining
P
Ii =
hi,j L(ri,j ), and (29) follows from the convexity of 1/x. Note that in (29),
ri,j ∈Ψi \B(0,ri,0 )

the random variables

1
Ii

are i.i.d. and independent from the indexing k. Hence, if

1
Ii

has a finite

second moment, then the sequence of RVs (indexed by k) in (29) has a finite second moment
and is uniformly integrable [40, Theorem 5.5.2], which implies that the sequence of RVs in the
LHS of (27) is also uniformly integrable and this concludes the proof.
A PPENDIX C
P ROOF

OF

C OROLLARY 1

The second negative moment can be written as:
  Z∞
1
E 2 = tMI (−t)dt,
Ii
0

where MI (·) is the moment generating function of the interference I. Using classical stochastic
geometry analysis [15], it can be represented as the following
h P
i
 
MI (t) = E etI = E et ri ∈Ψ\B(0,r0 ) L(ri )hi


Z ∞
Y
2
=
E
etL(ri )hi |r0  2πλ0 e−πλ0 r0 dr0
0

=

Z

0

∞

ri ∈Ψ\B(0,r0 )



exp −2πλ0

Z

r

∞


 htL(x) 
2
x 1 − Eh e
dx 2πλ0 e−πλ0 r0 dr0 ,

(30)

21

where (30) follows from the probability generating functional (PGFL) of a PPP [10]. Hence, the
second negative moment of I is given by


  Z∞ Z∞
Z ∞
 −htL(x) 
1
2
2πλ0 rt exp −πλ0 r − 2πλ0
x 1 − Eh e
dx drdt.
E 2 =
Ii
r
0

(31)

0

From (31), we get the first sufficient condition. The second condition can be found by the
following:






1
E 2 = E 
Ii
=

Z∞

X

ri ∈Ψ\B(0,r0 )



P

max

ri ∈Ψ\B(0,r0 )

0

=

0
Z∞

L(ri )hi   ≤ E

L(ri )hi

Z∞ 
= P

−2 

−2

1
max L(ri )hi < √
ri ∈Ψ\B(0,r0 )
t

 

E 1

0

max

ri ∈Ψ\B(0,r0 )

"

max

ri ∈Ψ\B(0,r0 )

L(ri )hi

−2 #

!

> t dt


1
L(ri )hi < √

dt

t



dt






Z∞
 n
o
Y
1
1
log 1 L(ri )hi < √
 dt
t
e
= E
1 L(ri )hi < √  dt = E 
t
ri ∈Ψ\B(0,r0 )
ri ∈Ψ\B(0,r0 )
0
0


∞
∞
∞



Z Z
Z
1
dx drdt
(32)
=
2πλ0 r exp −πλ0 r 2 − 2πλ0 xEh 1 − 1 L(x)h < √
t
0 0
r


∞
∞


Z Z
Z∞ 
1
dx drdt,
(33)
=
2πλ0 r exp −πλ0 r 2 − 2πλ0 x 1 − Ph L(x)h < √
t


Z∞

0

Y

0

r

where (32) follows from the PGFL of a PPP [9].

A PPENDIX D
P ROOF OF C OROLLARY 2


tL(x)
. By substituting this in the first
For the Rayleigh fading case, Eh 1 − e−htL(x) = 1+tL(x)

condition in the last corollary we get

Z∞ Z∞
Z
2
2πλ0 rt exp −πλ0 r − 2πλ0 t
0

0

∞
r


xL(x)
dx drdt = T1 + T2 ,
1 + tL(x)

22

with
T1 =

T2 =

Z∞ Zr0


Z
2
2πλ0 rt exp −πλ0 r − 2πλ0 t

∞


xL(x)
dx drdt
1 + tL(x)

∞

0 r0


Z
2
2πλ0 rt exp −πλ0 r − 2πλ0 t


xL(x)
dx drdt.
1 + tL(x)

0 0
Z∞ Z∞

r

r

(34)

Then if T1 and T2 are finite, we prove the corollary. Let y0 = L(r0 ), then

T1 =

≤
=

≤
=

Z∞ Zr0

0 0
Z∞ Zr0

0 0
Z∞ Zr0

0 0
Z∞ Zr0

0 0
Z∞ Zr0
0

=

0

Z∞ Zr0
0

∞


xL(x)
dx drdt
1 + tL(x)

Z

∞


xL(x)
dx drdt
1 + tL(x)


Z
2
2πλ0 rt exp −πλ0 r − 2πλ0 t

y0


Z
2
2πλ0 rt exp −πλ0 r − 2πλ0 t

r



2πλ0 rt exp −πλ0 r − 2πλ0 t

r0

0



2

2πλ0 rt exp −πλ0 r − 2πλ0 t

Z

0

y0


L−1 (y)y
1
dy drdt
−L′ (L−1 (y)) 1 + ty

(35)


ζ
dy drdt
1 + ty

(36)


2πλ0 rt exp −πλ0 r 2 − 2πλ0 ζ log(1 + ty0 ) drdt
2πλ0 rt exp −πλ0 r 2

0

−πλ0 r02

= (1 − e

)

Z∞
0

=

2



1
(1 + ty0 )2πλ0 ζ

t
(1 + ty0 )2πλ0 ζ

drdt

dt,

−πλ0 r02

1−e
,
− 1)(2πλ0 ζ − 2)

(38)

y02 (2πλ0 ζ

where, (35) follows by the substitution y = L(r), (36) since
means that

(37)

L−1 (y)y

−L′ (L−1 (y))

rL(r)
′
−L (r)

≥ ζ, ∀r ∈ [r0 , ∞) which

≥ ζ, ∀y ∈ (0, y0 = L(r0 )]. Finally, (38) holds as long as λ0 >

T1 is finite. Moving to T2 defined in (34),

1
.
πζ

Hence

23

T2 =

Z∞ Z∞


Z
2
2πλ0 rt exp −πλ0 r − 2πλ0 t

Z∞ Z∞

L(r)

2πλ0 rt exp −πλ0 r 2 − 2πλ0 t

0 r0

Z

Z∞ Z∞

L(r)

2πλ0 rt exp −πλ0 r 2 − 2πλ0 t

Z

Z∞ Z∞

2πλ0 rt exp −πλ0 r 2

r

0 r0

=

≤

0 r0

=

0 r0

=

Z∞

2πλ0 r exp −πλ0 r

Z∞

2πλ0 r exp −πλ0 r 2

2

r0

=

r0

2πλ0
=
b

Z∞

r0

1
πζ






Z∞
0


xL(x)
dx drdt
1 + tL(x)

0

0

!
L−1 (y)y
1
dy drdt
−L′ (L−1 (y)) 1 + ty
!

ζ
dy drdt
1 + ty

1
(1 + tL(r))2πλ0 ζ
t

(1 + tL(r))2πλ0 ζ

(39)

drdt

dtdr

1
dr
bL(r)2


r
2
exp
−πλ
r
dr,
0
L(r)2

where (39) follows because
λ0 ≥

∞

L−1 (y)y
−L′ (L−1 (y))

(40)

≥ ζ, ∀y ∈ (0, y0 = L(r0 )] and (40) holds as long as

with b = (2πλ0 ζ − 2)(2πλ0 ζ − 1). The last integral is finite by assumption. Hence, T2

is also finite which proves the corollary.
A PPENDIX E
P ROOF
Define Ĩ = I/2 =

P

ri ∈Ψ\B(0,r0 )

OF

C OROLLARY 4

L(ri )h̃i , where h̃i = hi /2, Clearly, if Ĩ has a finite second

negative moment, then I will have a finite second negative moment.

  Z∞ Z∞
Z ∞  h
i 
1
2
−h̃tL(x)
dx drdt
=
2πλ0 rt exp −πλ0 r − 2πλ0
x Eh̃ 1 − e
E
Ĩ 2
r
0 0
# !
Z∞ Z∞
Z ∞ "
h̃L(x)
≤
2πλ0 rt exp −πλ0 r 2 − 2πλ0 t
Eh̃ x
dx drdt,
1 + h̃tL(x)
r
0

0

(41)

24

t
, ∀t ≥ 0. Moreover,
where (41) follows since 1 − e−t ≥ 1+t
"
# Z∞
xh̃L(x)
xh̃L(x)
Eh̃
fH̃ (h̃)dh̃
=
1 + h̃tL(x)
1 + h̃tL(x)
0

≥

Z1

xh̃L(x)
fH̃ (h̃)dh̃
1 + h̃tL(x)

Z1

n
o
xh̃L(x)
xL(x)
fH̃ (h̃)dh̃ =
P h̃ ≤ 1
1 + tL(x)
1 + tL(x)

0

≥

0

≥

(42)

1 xL(x)
,
2 1 + tL(x)

(43)

where fH̃ (·) is the probability density function (PDF) of h̃. (42) follows since the integrand is
n
o
positive and (43) since by Markov’s inequality P h̃ ≤ 1 ≥ 1 − E[h̃] = 21 . Hence,

 Z∞ Z∞

Z ∞
1
xL(x)
2
E
dx drdt.
≤
2πλ0 rt exp −πλ0 r − πλ0 t
1 + tL(x)
Ĩ 2
r


0

0

Then the proof follows as in the proof of Corollary 2.
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