Conference PaperPDF Available

# Parameter Optimization for Energy Efficient Indoor Massive MIMO Small Cell Networks

Authors:

## Figures

Content may be subject to copyright.
Parameter Optimization for Energy Efﬁcient Indoor
Massive MIMO Small Cell Networks
Chen Chen, Yan Jiang, Jiliang Zhang, Xiaoli Chu, Jie Zhang
Department of Electronic and Electrical Engineering, The University of Shefﬁeld, Shefﬁeld, S1 4ET
United Kingdom, c.chen2@shefﬁeld.ac.uk
Abstract—To better characterize indoor small cell networks
(SCN), we consider the blockages caused by interior walls and
employ the bounded path loss model to derive the expression
for energy efﬁciency (EE) of a downlink massive multiple-input
multiple-output (MIMO) SCN. Our EE expression demonstrates
that a higher penetration loss of interior walls leads to a higher
EE. For the purpose of maximizing EE, we propose a novel
genetic algorithm (GA) based scheme to jointly optimize the
number of antennas per base station (BS), the number of users
per cell, and the transmission power per antenna. Numerical
results show that our proposed scheme can achieve almost the
identical EE as the optimal greedy search algorithm, while
signiﬁcantly reducing the computational time.
Index Terms—Indoor, energy efﬁciency, massive MIMO, small
cell networks, genetic algorithm
I. INTRODUCTION
To address the requirement of thousand fold network capac-
ity increase without further increasing the power consumption,
energy efﬁciency (EE) has been one of the most important
considerations for the ﬁfth generation (5G) cellular network
[1]. Two potential techniques to improve EE are massive
MIMO and small cell networks (SCN) [2].
Massive MIMO systems can provide dramatical throughput
gains and reduction in transmission power [2]. In [3], closed-
form expressions were derived for the number of antennas per
base station (BS), the number of users per cell, and the emitted
power in a single-cell scenario with perfect CSI. The multi-
cell scenario under imperfect CSI was studied through Monte
Carlo simulations. Recently, in [5], the authors showed that
EE can be improved through adapting the number of antennas
used at a BS according to the daily load variation. However,
none of these studies has considered the effect of BS density.
SCN saves power for each BS by reducing the transmission
distance to an associated user. In [6], it was shown that EE can
be improved by increasing the BS density or the number of
antennas only when the circuit power consumption of the BS
is less than a certain threshold. Recent work [10] showed that
the network coverage probability of a SCN was overestimated
under the simpliﬁed unbounded channel model which violates
the conservation of energy. Nevertheless, the EE of the SCN
under the bounded model, especially the indoor SCN equipped
with massive MIMO, has not been studied yet.
The indoor SCN is largely characterized by the blockages of
interior walls [7]. In [8], the walls were distributed following
a Poisson point process (PPP), and the numerical results
revealed that a higher penetration loss of interior walls can
provide a higher network coverage probability and an higher
area spectral efﬁciency.
In this paper, we analyze the EE of the indoor massive
MIMO SCN considering the blockages of interior walls. To
maximize EE, we propose a novel genetic algorithm based
scheme. The contributions of this paper are summarized as
follows:
Taking into account of the penetration loss of interior
walls, we derive the average user rate and EE in an
indoor massive MIMO multi-cell scenario as functions
of the number of antennas per BS, the number of users
per cell, the transmission power per antenna, and the
number of cells. We apply the bounded channel model
to better capture the path loss of short distances. Our
analytical results show that the network densiﬁcation may
degrade the EE when the BS density is sufﬁciently large.
Furthermore, we ﬁnd that a higher penetration loss of
interior walls can provide a higher EE.
We propose a novel, low-complexity genetic algorithm
based scheme to maximize the EE, subject to the quality
of service (QoS) constraints. Our scheme jointly opti-
mizes the number of antennas per BS, the number of users
per cell and the transmission power per antenna. The
performance of our scheme is evaluated in comparison
with the optimal greedy search algorithm.
Fig. 1. Illustration of an indoor multi-cell scenario in a 30 m×30 m area,
where the cell under study is located in the centre. Each cell covers a room
and the bold lines denote the walls. The number of cells in each column is
λ.
II. SY ST EM MO DE L
A. Network Model
As shown in Fig. 1, we consider the downlink of an
indoor massive MIMO SCN in a 30m×30m area consisting
of C=λ2square cells, which are denoted by the set
C={1,2,3, ..., C}. We assume that each room has a BS,
so each cell corresponds to a room. The bold lines denote
the walls with the same penetration loss z(0 < z < 1). For
each cell, the BS is equipped with Mantennas to serve K
uniformly distributed single-antenna users. We assume that
the system is completely symmetric where all the cells have
the same parameters: M,Kand the transmission power per
antenna p. A time-division duplex (TDD) protocol is employed
with perfect synchronization. The channels are assumed to be
static within a time-frequency block U=BCTC, where BC
is the coherence bandwidth and TCis the coherence time. For
analytical tractability, we assume perfect CSI. Imperfect CSI
will be studied in our future work.
B. Channel Model
Since the distances between users and the BS are much
larger than the space between antennas, the large-scale fading
is considered to be the same for the channels from a user to
all BS antennas. To better characterize the power attenuation
with small transmit distances, we model the large-scale fading
using a bounded path loss model [10], [11]:
l(d) = β0
(d2+ 1)α/2,(1)
where dis the transmit distance, αis the path-loss exponent,
and the constant β0is the channel attenuation at unit reference
distance.
We assume that the space between antennas in the BS is
adequate so that the correlation between the channels from s
BS to a single-antenna user can be omitted. The channel model
between the BS to a speciﬁc user kis denoted by the vector
hk= [hk,1, hk,2, ..., hk,M ]TCM×1, where hk,j denotes
the channel from the jth antenna to the kth user. Rayleigh
distribution with unit variance is employed to model small-
scale fading, i.e., hk CN (0M, l(dk)IM),dkis the distance
between the BS and user k,0Mand IMare the M-dimensional
null and unit column vectors.
C. User Rate Model
For simplicity, we assume that the BS can obtain perfect
CSI from uplink pilots and use zero-forcing (ZF) precoding
for downlink transmission. The ZF matrix is:
W=H(HHH)1,(2)
where H= [h1,h2, ..., hK]is the channel vector for all the
users in a cell.
The cell cin the centre is our cell under study, and all the
other cells fare interference cells, f6=cand f∈ C. The
desired signal received by user kin cell cis:
yck =hH
cck
K
X
i=1
wcisci +X
f6=c
zfchH
fck
K
X
i=1
wfisf i +nck,(3)
where hcck and hfck are the channels from the serving BS
in cell cand interfering BS in cell fto user k,wci and sci
are the ZF vector and the transmitted signal from the serving
BS in cell cto user iin cell c, respectively, wfi and sf i are
the ZF vector and the transmitted signal from the BS in cell
fto user iin cell f,zfc denotes the penetration loss of walls
between cell cand cell f,nck is the additive noise.
The tractable lower bound of average user rate in cell cis
given in [5]:
Rc=B1τK
Ulog2 1+ pM
K(MK)
σ2Gcc +pM Pf6=cGcf !,
(4)
where Bis the channel bandwidth, σ2is the variance of
additive white Gaussian noise, τis the pilot reuse factor,
1τK
Udenotes the overhead used for channel estimation, p
is the average transmission power per antenna, so pM
Kdenotes
average transmission power per user, Gcc is the average inverse
channel attenuation, while Gcf is the average ratio of the
channel attenuation between interference cell fand serving
cell c. For our model, Gcc can be calculated as:
Gcc =Zr
rZr
r
(x2+y2+ 1)α/2
4r2β0
dxdy, (5)
where ris the radius of the cell.
Pf6=cGcf is the function of λand the expression can be
found in Appendix A. As aforementioned, Rcis a function of
M,K,pand λ.
D. Power Consumption Model
The power consumption consists of two parts: power am-
pliﬁer (PA) power consumption PPA and circuit power con-
sumption PCP in Watt (Joule/second). So the total power
consumption for BS cis:
Pc=PPA,c +PCP,c,(6)
where PPA,c =BMp
η, and ηdenotes the PA efﬁciency [12].
The circuit power consumption is the sum of both analog
and digital parts as [3]:
PCP,c =PFIX +PTC +PCE +PZF +PC/D+PBH,(7)
where PFIX is the ﬁxed power consumption for signal control
and load-independent power, PTC is the power consumption
of transceiver chains, the typical value can be computed as
PTC =MPBS +PSYN, where PBS is the power required for
circuit components in each antenna and PSYN is the power
needed for local oscillator, PCE is the channel estimation
power, which is given by:
PCE =B
U
2τMK2
LBS
,(8)
where LBS is the computational efﬁciency for BS processing.
During downlink transmission, channel coding and modula-
tion are employed before the BS transmits signals to its served
Kusers. The power for this operation is proportional to the
user rate, i.e.,
PC/D=KRc(PCOD +PDEC ),(9)
where PCOD (Watt per bit/s) and PDEC (Watt per bit/s) are
the coding and decoding power.
PZF is the power consumption for ZF precoding, this costs:
PZF =B1τK
U2MK
LBS
+B
UK3
3LBS
+3MK2+MK
LBS ,
(10)
where the ﬁrst term is for matrix-vector multiplication and
the second one is used for the computation of ZF precoding
matrix W.
The last part of power consumption is for backhaul, which
is proportional to the average user rate, it can be computed as:
PBH =KRcPBT ,(11)
where PBT is backhaul trafﬁc power.
The expressions above allow us to rewrite PCP,c as:
PCP,c =AKRc+
3
X
i=0
CiKi+M
2
X
i=0
DiKi,(12)
where A=PCOD +PDEC +PBT,C0=PFIX +PSYN ,C1= 0,
C2= 0,C3=B
3ULBS ,D0=PBS ,D1=B
LBS (2 + 1
U),D2=
3B
ULBS .
III. ENERGY EFFIC IE NC Y OPTIMIZATION
A. Problem Formulation
We compute EE as the ratio between the sum average rate
of the users (in bit/second) and total power consumption (in
Watt) of BS c, i.e.,
EE(M, K, p) = KRc
Pc
,(13)
where EE(M, K, p)is a function with parameters M,Kand
p. Since from Fig. 4, the EE monotonically decreases with the
BS density λ, we will not optimize λ.
Then we can formulate our optimization problem as:
arg max
M,K,p
EE(M, K, p),(14)
subject to constraints:
C1 : MZ+, K Z+
C2:1KKmax
C3 : K < M Mmax
C4:0ppmax
C5 : RcRmin
C6 : pM Pmax
where, in C5,Rmin is the threshold for user rate, which guar-
antees the QoS. In C6,Pmax is the maximum transmission
power for BS and the total power can not go beyond this.
With the expressions in (4) and (6), it is intractable to derive
closed-form expressions for each parameter. In addition, from
Fig. 2, it is observed that EE is a quasi-concave function for
each parameter when the other two parameters are ﬁxed. These
make the optimization of all the parameters difﬁcult. To obtain
the near optimal parameter conﬁguration that maximizes EE,
we propose a novel genetic algorithm based scheme.
B. Proposed Scheme
Genetic algorithm (GA) is a kind of evolutionary algorithm,
which is especially suitable for solving complex optimization
problems with many parameters that are intractable to derive
the analytical expressions [13].
GA mimics the process of evolution theory. The probable
solutions are called individuals and a population of selected
individuals are called chromosomes. At ﬁrst, a random popula-
tion of individuals are selected. In each generation, the current
population will go through crossover and mutation to generate
a new population, the worst individuals will be replaced by
better individuals in order to obtain better ﬁtness.
We take use of GA to jointly optimize M,Kand p
for maximizing EE, considering the constraints in (14). The
detailed steps are given in Algorithm 1.
Algorithm1 :Joint parameter optimization
1) Input: Input Mmax,Kmax ,pmax, power constraint
Pmax, rate threshold Rmin and maximum number of
generations Imax.
2) Initialize: Generate chromosomes by combining four
parameters M,Kand p. Each chromosome has three
points. Select Nchromosomes to form initial popula-
tion, check constraints. Set iteration counter i= 1.
3) Fitness: Compute the ﬁtness of each chromosome ac-
cording to the EE function.
4) Sorting: Sort the chromosomes in initial population in
terms of ﬁtness.
5) Iteration: Repeat the next steps until convergence.
a) Crossover: Decide whether to apply crossover
according to pre-deﬁned crossover probability ρc.
If applied, choose two parent chromosomes with
higher ﬁtness ranking, then randomly select an ex-
change point and exchange the information. Check
constraints.
b) Mutation: For each chromosome in current pop-
ulation, decide whether to apply mutation to a
chromosome according to mutation probability ρm.
If applied, choose a point and change its value
randomly. Check constraints.
c) Accepting: Compute ﬁtness for current population
and sort the values. The chromosomes with lower
ﬁtness will be replaced by those having higher
ranking.
d) Check termination: If i=Imax, select the best
chromosome in current population, terminate the
algorithm. Otherwise, continue the iteration.
41 50 60 70 80 90 100 110 120
Number of Antennas
0
10
20
30
40
50
60
70
Energy Efficiency [Mbit/Joule]
z = -20 dB
z = -10 dB
z = 0 dB
(a)
1 10 20 30 40 50 60 70 79
Number of Users
0
10
20
30
40
50
60
70
Energy Efficiency [Mbit/Joule]
z = -20 dB
z = -10 dB
z = 0 dB
(b)
0 0.005 0.01 0.015 0.02
Transmission Power Per Antenna (W)
0
10
20
30
40
50
60
Energy Efficiency [Mbit/Joule]
z = -20 dB
z = -10 dB
z = 0 dB
0246
10-4
56
58
1 2
10-4
34
36
5 10 15
10-5
12.5
13
13.5
14
(c)
Fig. 2. Energy efﬁciency versus: (a) The number of antennas per BS. (b) The number of users per cell. (c) Transmission power per antenna.
41 50 60 70 80 90 100 110 120
Number of Antennas
0
50
100
150
200
Average User Rate [Mbit/s]
z = -20 dB
z = -10 dB
z = 0 dB
(a)
1 10 20 30 40 50 60 70 79
Number of Users
0
50
100
150
200
250
300
Average User Rate [Mbit/s]
z = -20 dB
z = -10 dB
z = 0 dB
(b)
0 0.2 0.4 0.6 0.8 1
Transmission Power Per Antenna (W) 10-3
0
50
100
150
200
Average User Rate [Mbit/s]
z = -20 dB
z = -10 dB
z = 0 dB
(c)
Fig. 3. Average user rate versus: (a) The number of antennas per BS. (b) The number of users per cell. (c) Transmission power per antenna.
In Algorithm 1, the constraints in (14) are checked for
all the chromosomes in initialization and for new generated
chromosomes every time crossover and mutation are applied.
If the constraints are not satisﬁed, the chromosome will repeat
mutation operation until it satisﬁes all the constraints.
IV. SIMULATION RESULTS
This section presents simulation results to validate our pro-
posed scheme. The simulations are performed on a laptop with
Intel i7 CPU and 16G memory. The corresponding simulation
parameters are summarized in Table I [3].
In Fig. 2 and Fig. 3, we analyze the inﬂuence of M,Kand
prespectively when the other two parameters are ﬁxed. We
set λ= 3 and assume an initial combination of M,Kand
pto be 80, 40 and 0.002 W. From Fig. 2, it is observed that
EE is a quasi-concave function of M,Kand p, respectively.
While in Fig. 3, the average user rate increases monotonically
with M,pand decreases monotonically with K. In addition,
it is obvious that higher penetration loss of the walls can lead
to both higher EE and higher average user rate.
In Fig. 4, we show the inﬂuence of network densiﬁcation.
The BS density can be computed as λ2/900 BSs/m2. Different
from [2], we observe that under the bounded path loss model,
the system EE will decrease with the increasing BS density
when the BS density is sufﬁciently large.
TABLE I
SIMULATION PARAMETERS
Parameter Value
Transmission bandwidth: B20 MHz
Channel coherence bandwidth: BC180 kHz
Channel coherence time: TC10 ms
Total noise power: 296 dBm
Path loss at reference distance: βL
0103.85
Path loss exponent: α3
Pilot length: 1
Computational efﬁciency at BS: LBS 12.8Gﬂops/W
PA efﬁciency at BS: η0.39
Fixed power consumption: PFIX 18 W
Power for local oscillator: PSYN 2W
Power for circuit components at BS: PBS 1W
Power required for coding: PCOD 0.1W/(Gbit/s)
Power required for decoding: PDEC 0.8W/(Gbit/s)
Power required for backhaul: PBT 0.25 W/(Gbit/s)
Next, we compare our proposed joint optimization scheme
with the greedy search algorithm in Fig. 5. We set λ= 3 so
that the BS density is 0.01 BSs/m2. For the constraints in (14),
we assume Kmax = 80,Mmax = 120,pmax = 0.001 W, the
user rate threshold Rmin = 30 Mbit/s, maximum power for
the BS Pmax = 0.1W. For GA, the number of generations is
50, the size of population is 100, ρc= 0.8and ρm= 0.04.
After the algorithm converges, we get the optimal EE: 14.48
Mbit/Joule, 38.78 Mbit/Joule and 66.5 Mbit/Joule when zis
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Density (BSs/m2)
5
10
15
20
25
30
35
40
45
50
55
60
Energy Efficiency [Mbit/Joule]
z = -20 dB
z = -10 dB
z = 0 dB
Fig. 4. Energy efﬁciency versus BS density.
5 10 15 20 25 30 35 40 45 50
Generations
10
20
30
40
50
60
70
Energy Efficiency [Mbit/Joule]
Proposed scheme, z = -20 dB
Proposed scheme, z = -10 dB
Proposed scheme, z = 0 dB
Greedy search, z = -20 dB
Greedy search, z = -10 dB
Greedy search, z = 0 dB
Fig. 5. Comparison of our proposed scheme and the greedy search algorithm.
0dB, 10 dB and 20 dB, respectively. For z= 0 dB, the
optimal parameters are: M= 108,K= 46,p= 0.1mW. For
z=10 dB, M= 89,K= 57,p= 0.3mW and for z=
20 dB, M= 81,K= 61,p= 0.8mW. Therefore, higher
penetration of walls can provide higher EE with less antennas
and higher transmission power while serving more users. As
shown in Fig. 5, our proposed scheme can obtain nearly the
same EE with greedy search. The average computational time
for our proposed scheme is 21.89 s while for greedy search is
1288.83 s. These demonstrate that our proposed scheme is a
nearly optimal algorithm with lower time complexity.
V. CONCLUSIONS
In this paper, we formulate the network energy efﬁciency of
an indoor Massive MIMO network with blockages caused by
interior walls. We employ the bounded path loss model and
show that the network will suffer from lower EE with higher
BS density. Furthermore, we maximize the EE by jointly
optimizing the number of antennas per BS, the number of users
per cell and the transmission power per antenna, taking into
account the user rate and total transmission power constraints.
To solve this joint optimization problem, we propose a low-
complexity GA based scheme to obtain the optimal EE. The
results show that the penetration loss of walls is beneﬁcial for
both EE and average user rate. In the future, we will extend our
research to millimeter wave massive MIMO indoor scenarios
while considering the impact of pilot contamination.
APPENDIX A
DERIVATION OF INTE R- CE LL INTERFERENCE
The average ratio of the channel attenuation between inter-
ference cell fand serving cell cdepends on the value of λ,λ
is an odd number and λ3. Deﬁne two odd numbers iand
j,i= 1,3...λ 2,j=1,1...λ 2. Deﬁne two functions
f(x, y) = x2+y2+ 1,g(x, y) = [x(i+ 1)r]2+ [y(j+
1)r]2+ 1, then Pf6=cGcf can be computed as follows:
X
f6=c
Gcf =4
λ2
X
i=1
λ2
X
j=1Rr
rRr
r(f(x, y))α/2zi+j+2
2dxdy
Rr
rRr
r(g(x, y))α/2dxdy,
(15)
ACKNOWLEDGMENT
This work was funded by the European Union’s Horizon
2020 research and innovation programme under grant agree-
ment No. 766231.
REFERENCES
[1] J. G. Andrews et al., “What will 5G be?,” IEEE J. Sel. Areas Commun.,
vol. 32, no. 6, pp. 1065-1082, Jun. 2014.
[2] E. Bjrnson, L. Sanguinetti and M. Kountouris, “Deploying dense net-
works for maximal energy efﬁciency: small cells meet massive MIMO,”
IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 832-847, Apr. 2016.
[3] E. Bjrnson, L. Sanguinetti, J. Hoydis and M. Debbah, “Optimal design
of energy-efﬁcient multi-user MIMO systems: is massive MIMO the
answer?,” IEEE Trans. Wireless Commun., vol. 14, no. 6, pp. 3059-3075,
Jun. 2015.
[4] W. Xu, S. Li, S. Wang, Z. Feng, J. Lin and A. V. Vasilakos, “Joint
parameter selection for massive MIMO: an energy-efﬁcient perspective,”
IEEE Access, vol. 4, pp. 3719-3731, 2016.
[5] M. M. A. Hossain, C. Cavdar, E. Bjrnson and R. Jntti, “Energy saving
game for massive MIMO: coping with daily load variation,IEEE Trans.
Veh. Technol., vol. 67, no. 3, pp. 2301-2313, Mar. 2018.
[6] C. Li, J. Zhang and K. B. Letaief, “Throughput and energy efﬁciency
analysis of small cell networks with multi-antenna base stations,” IEEE
Trans. Wireless Commun., vol. 13, no. 5, pp. 2505-2517, May. 2014.
[7] H. Zheng, J. Zhang, H. Li, Q. Hong, H. Hu and J. Zhang, “Exact line-of-
sight probability for channel modeling in typical indoor environments,
IEEE Antennas Wireless Propag. Lett., vol. 17, no. 7, pp. 1359-1362,
Jul. 2018.
[8] M. K. Mller, S. Schwarz and M. Rupp, “Investigation of area spectral
efﬁciency in indoor wireless communications by blockage models,
International Symposium on Modeling and Optimization in Mobile, Ad
Hoc, and Wireless Networks (WiOpt), Shanghai, 2018, pp. 1-6.
[9] T. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wave
cellular networks,” IEEE Trans. Wireless Commun., vol. 14, no. 2, pp.
1100-1114, Feb. 2015.
[10] J. Liu, M. Sheng, L. Liu, and J. Li, “Effect of densiﬁcation on cellular
network performance with bounded pathloss model,” IEEE Commun.
Lett., vol. 21, no. 2, pp. 346-349, Feb. 2017.
[11] A. AlAmmouri, J. G. Andrews, and F. Baccelli, “A uniﬁed asymptotic
analysis of area spectral efﬁciency in ultradense cellular networks,IEEE
Trans. Inf. Theory, vol. 65, no. 2, pp. 1236-1248, Feb. 2019.
[12] Y. Yan, H. Gao, T. Lv and Y. Lu, “Energy-efﬁcient resource allocation in
ultra-dense networks with massive MIMO,IEEE Globecom Workshops
(GC Wkshps), Singapore, 2017, pp. 1-7.
[13] N. Sharma and A. S. Madhukumar, “Genetic algorithm based propor-
tional fair resource allocation in multicast OFDM systems,” IEEE Trans.
Broadcast., vol. 61, no. 1, pp. 16-29, Mar. 2015.
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
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.
Article
Full-text available
In this paper, we investigate how network densification influences the performance of downlink cellular network in terms of coverage probability (CP) and area spectral efficiency (ASE). Instead of the simplified unbounded pathloss model (UPM), we apply a more realistic bounded pathloss model (BPM) to model the decay of signal power caused by pathloss. It is shown that network densification indeed degrades CP when the base station (BS) density $\lambda$ is sufficiently large. This is inconsistent with the result derived using UPM that CP is independent of $\lambda$. Moreover, we shed light on the impact of ultra-dense deployment of BSs on the ASE scaling law. Specifically, it is proved that the cellular network ASE scales with rate $\lambda e^{-\kappa\lambda}$, i.e., first increases with $\lambda$ and then diminishes to be zero as $\lambda$ goes to infinity.