Content uploaded by Lu An
Author content
All content in this area was uploaded by Lu An on Mar 06, 2017
Content may be subject to copyright.
Stochastic Geometry based Energy-Efficient Base
Station Density Optimization in Cellular Networks
Lu An, Tiankui Zhang, Chunyan Feng
SICE, Beijing University of Posts and Telecommunications
Beijing Key Laboratory of Network System Architecture and Convergence
Beijing, China
Email: {anlu, zhangtiankui, cyfeng}@bupt.edu.cn
Abstract—In the research of green networks, considering the
base station (BS) density from the perspective of energy
efficiency is very meaningful for both network deployment and
BS sleeping based power saving. In this paper, we optimize the
BS density for energy efficiency in cellular networks by the
stochastic geometry theory. First, we model the distribution of
base stations and user equipment (UE) as spatial Poisson point
process (PPP). Based on such model, we derive the closed-form
expressions of the average achievable data rate, the network
energy consumption and the network energy efficiency with
respect to the network load. Then, we optimize the BS density for
network energy efficiency maximization by adopting the Newton
iteration method. Our study reveals that we can improve the
network energy efficiency by deploying the suitable amount of
BSs or switching on/off proportion of the BSs according to the
network load. The simulation results validate the theoretical
analysis, and show that when the right amount of BSs is deployed
according to the network load, the network energy efficiency can
be maximized and the maximum energy efficiency is a fixed value
once the network parameters are given.
I. INTRODUCTION
In cellular access networks, more than 70% of the overall
network energy consumption is taken up by the base stations.
So the energy saving of BSs will be the key to designing
energy-efficient cellular networks. Except for the hardware
design of the BS, intense research efforts should be exerted on
BS density.
Within the cellular networks, the cell size which is
determined by the BS density has an impact on both network
energy consumption and system capacity. Several researches
have already proved that small cell size contributes to the
improvement of system capacity [1]-[2]. However, as the cell
size decreases, the BS density increases, which results in an
increase of overall network energy consumption. In order to
achieve greater downlink capacity while reducing power
consumption, the dynamic BS sleeping strategy [3]-[4] can be
adopted. The BS sleeping strategy which can alter the BS
density is proved to be an effective method for energy saving
via switching BS off/on. According to the BS sleeping
strategy, the researches [5]-[7] focus on the BS density
optimization from the perspective of energy efficiency, whose
optimal objects are to minimize the total network energy
consumption using the hexagonal cellular model.
However, as the network elements are becoming more and
more diverse, the traditional hexagonal cellular model cannot
reflect the network deployment in reality. A tractable
analytical modeling method for the cellular networks has been
proposed in [8], where the location distribution of BSs is
modeled as spatial Poisson point process and the cell coverage
forms the well-known Poisson Voronoi Tessellation (PVT).
What is important is that such a model allows useful
mathematical tools from stochastic geometry to trace the cell
coverage and system capacity. Based on PPP model and
stochastic geometry theory, the cell coverage with the BS
sleeping strategy is analyzed in [9]. Besides, several works
[10]-[12] begin to pay attention to the energy saving analysis
utilizing stochastic geometry theory. [10] minimizes the total
network energy consumption with the constraint of user
outage probability. And the relationship between spatially
averaged rate, user density and BS density is analyzed in [11].
[12] studies the sleeping fraction of BS in the network by
means of minimizing the network power consumption, but it
fails to provide a closed-form expression of the network
energy efficiency. The main focus of our work is the
derivation of the closed-form expression of the network
energy efficiency and the optimization of it.
Reference [8] adopts the aforementioned PPP model and
obtains the users’ achievable downlink data rate. However,
only one single user is considered in the whole system, so it
ignores the UE density that will have an impact on the system
capacity. The deficiency of the previous work is the key point
of this paper.
In this paper, we optimize the BS density for energy
efficiency in cellular networks based on PPP model and
stochastic geometry theory with network load taken into
consideration. We mainly solve the following problems. (i)
Optimal BS density for energy efficiency: Can we deploy a
certain amount of BSs according to the network load? If we
can, then what is the best BS density? Or if we want to turn off
a proportion of BSs, what is the best proportion? (ii)
Maximum network energy efficiency: Is the maximum
network energy efficiency a fixed value?
The rest of the paper is organized as follows. In Section II,
the system model is described. In Section III, we derive the
closed form expressions of the average achievable data rate of
users, the network power consumption and the network energy
efficiency respectively, and then the optimal BS density for
This paper is supported by Beijing Natural Science Foundation
(4144079) and the National Key Technology R&D Program of China
(2012ZX03001031-004).
2015 IEEE Wireless Communications and Networking Conference (WCNC): - Track 3: Mobile and Wireless Networks
978-1-4799-8406-0/15/$31.00 ©2015 IEEE 1614
energy efficiency is given. The simulation results are
presented in Section IV. Finally, Section V concludes this
paper.
II. SYSTEM MODEL
A. Cellular Network Model
Traditional cellular networks are considered in this paper.
Only macro BSs are deployed in the system and the transmit
power of BSs is fixed without any dynamic power control. We
model the spatial distribution of the BSs in the network as
a homogeneous PPP of density
B
S
in the two dimensional
Euclidean plane. Consider an independent collection of UEs
which are located according to the independent PPP of density
UE
. We assume that each user is associated with the BS from
which the mean received signal strength is the largest. As we
consider that all BSs have the same transmit power t
P, each
user is served by the closest BS, which means that the users in
the Voronoi cell of a BS are associated with it. As a result, the
coverage areas of the BSs form a Voronoi tessellation on the
plane. Fig. 1 shows an example of the BS location distribution
and the Voronoi tessellation. For convenience, the meanings
of all the mathematical symbols used in this paper are
summarized in Table I.
B. Channel Model
In this paper, path loss and fast fading are considered when
modeling the wireless channel gain. The path loss exponent of
transmission is denoted as 2
, and the fast fading
experienced by the tagged BS and the tagged UE is assumed
to be Rayleigh fading with mean zero. Since the self-
interference is the dominate factor of noise in cellular
networks, the thermal noise is ignored in this paper, which can
also enhance tractability.
In order to keep the derivation tractable, we assume that
each BS equally allocates the resource (e.g., wireless spectrum)
among its associated users. We consider universal frequency
reuse in this paper, so except the serving BS of the tagged UE,
all the other BSs in the system are potential interferers. The
distance between the tagged UE and its serving BS o
B is l,
and the corresponding channel fading is h. The received
power of the tagged UE from its serving BS is t
Phl
, and the
received cumulative interference signal of the tagged UE from
all the other BSs in the network is
/o
tii
iB
I
Pg L
, where
i
L denotes the distance between the target UE and the ith BS
(except the serving BS o
B) over which the fading is i
g
. As
we assume that the thermal noise is ignored, the SIR (signal to
interference ratio) of the UE from its associated base station
can be expressed as
/
.
o
ii
iB
hl
SIR gL
Thus, the mean achievable downlink data rate of a typical
UE over a cell is
2
;log 1 ,
C
NA
W
RSIR
N
where, 1N is a random integer variable which denotes the
total number of UEs in a certain cell whose coverage area is
C
A
, and W indicates the system wireless bandwidth.
C. Power Consumption and Energy Efficiency
We assume that the BS uses transmit power t
P for data
transmitting, and there is still a fixed part of power
consumption caused by the air conditioning, signal processing,
power supply and other circuit power consumptions. This
fixed part of power consumption is denoted as c
P. Thus,
0 5 10 15 20 25
0
5
10
15
20
25
X coordi nate ( km)
Y coordinate (km)
Fig. 1. An example of BS location distribution and the Voronoi
tessellation.
TABLE I
SUMMARY OF THE PARAMETER NOTATIONS
B
S
BS density
UE
UE density
c
P Fixed power consumption of BS
t
P Transmit power of BS
N The number of users in a certain cell
C
A
Coverage area of a certain cell
Weighting factor of the transmit power
B
S
UE
W System bandwidth
Path loss exponent
T Maximum iterations of Newton iteration method
K
Constant value 3.575
2015 IEEE Wireless Communications and Networking Conference (WCNC): - Track 3: Mobile and Wireless Networks
1615
according to [13], the definition of the average power
consumption of a typical BS can be expressed as
,
ct
PP P
where
indicates the weight of the transmit power on power
consumption.
The network energy efficiency can be defined as the ratio
of effective system capacity over total energy consumption in
the network, which can be written as
,
total
EE
total
R
P
where total
R denotes system capacity which is related to both
the mean achievable data rate of the UEs and UE density UE
.
total
P represents the total power consumption.
III. NETWORK PERFORMANCE ANALYSIS AND BS DENSITY
OPTIMIZATION
A. Average Achievable Data Rate
Since we assume that all the base stations in the cellular
network have the same transmit power, UEs are always served
by the nearest BS. For a specific base station i
B whose
coordinate in the Euclidean plane is i
b, its associated users are
located within a polygonal cell i
C which can be defined as
:min
iij
j
C x xb xb
2
R [14], where
j
b denotes
the coordinate of the jth base station j
B in the network. The
set of all these polygonal cells is defined as the well-known
Poisson Voronoi Tessellation.
Define the area of a typical PVT cell i
C as C
A
, and the
density of the BSs as
B
S
. Thus, the cell size C
A
is subject to
the Gamma distribution with a shape parameter 3.575K
[15] and its probability density function (PDF) can be
expressed as
1,
BS
C
K
BS
K
x
K
A
K
fx xe
K
in which
1
0exp
K
K
xxdx
is the gamma function.
As the users’ locations are in accordance with a PPP of
density UE
, the number of users in a PVT cell whose area is
C
A
follows the Poisson distribution and its corresponding
probability can be calculated using the following formula:
.
!
UE C
C
n
UE C A
A
A
Pn e
n
Based on [16], a striking property of PPP is Slivnyak’s
theorem which states that the number of users in a PVT cell
with cell size C
A
always follows the Poisson distribution
shown as whether we condition on having a user in the cell.
Thus, we can derive the average achievable downlink rate
of a typical UE as
2
;
200
log 1
log 1 .
1
C
CC
NA
AA
n
W
RSIR
N
W
ESIR Pnfxdx
n
According to the proof in [8], in the scenario with
Rayleigh fading, no thermal noise and PPP BS deployment,
we have
2/
2
02/
/2
21
log 1
1.
1
121 1
t
tt
ESIR
dt
dx
x
For simplicity, we set the above expression as the symbol
. It’s worth pointing out that in the special case of 4
,
has a single simple numerical integration that yields a precise
scalar 2.15 bits/s/Hz
[8]. By substituting and into
, the mean rate of a random UE in the network can be
calculated as
2
;
1
00
11
1
log 1
1!
11
1
1
1,
1
1
C
UE BS
NA
nK
UE BS
xKx
K
n
K
BS
KK
UE BS BS UE
BS
K
UE UE
BS
W
RSIR
N
xK
Wexedx
nn K
K
WK
KKK
KW
K
K
B. Energy Efficiency
Based on , the total energy consumption of the base
station is composed of the static power consumption caused by
the circuit, signal processing or some other factors and the
dynamic power consumption caused by the data transmission.
We assume that the dynamic power consumption is in
proportion to the BS’s transmit power with a weighting factor
which is related to the average number of the associated
users in the BS. As we set the BS density as
B
S
and UE
density as UE
, the average coverage area of a typical BS is
1/
B
S
, and the average area occupied by a typical UE is
1/ UE
. Thus we can derive the average number of the
associated users in a typical BS, equivalently the weighting
factor of the transmit power, as following,
1/ .
1/
B
SUE
UE BS
2015 IEEE Wireless Communications and Networking Conference (WCNC): - Track 3: Mobile and Wireless Networks
1616
By substituting into , the average energy
consumption of a typical BS can be expressed as
.
UE
ct
BS
PP P
As shown in , the network energy efficiency is the ratio
of the total throughput over the total energy consumption in
the system. The number of the BSs and UEs in the network are
both subject to the Poisson distribution of separate intensity
B
S
and UE
. Thus, we can get the expression of the network
energy efficiency as
0
0
1
!
!
1
1.
1
1
UE
BS
n
UE
total n
EE m
total BS
m
BS
K
cBS tUE UE
BS
Ren
Rn
PPem
m
KW
KP P
K
C. Optimal BS Density Analysis
In this subsection, we give an optimal BS density which
can maximize the energy efficiency of the whole network.
According to , the optimal BS density can be obtained
from solving the energy efficiency optimization problem as
the following,
1
1
max 1 .
1
1
BS
BS
EE K
cBS tUE UE
BS
KW
KP P
K
As proved by theorem 1 in [10], the optimal BS density in
the interference-limited cellular network is linear with the UE
density, i.e.
*
B
SUE
. Assuming that
B
SUE
, the
optimization problem can be rewritten as
*
1
1
arg max 1 .
1
11
K
t
c
KW
P
KP
K
The above problem has a unique solution, because the
right side of the equation which can be seen as
f
is a
strictly convex function. Therefore, the optimal result can be
achieved numerically through the binary search algorithm.
However, the calculation of the optimal
is very
complicated, since the right side of the equation is a nonlinear
function of
.
To solve the maximum value and stationary point of
f
, we set the value of the first partial derivative of
as
zero,
0.gf
The first-order approximation of Taylor expansion of
g
at any point
x
is shown as
.
xxx
gg g
From and , we can derive the approximate value
of
as
.
x
x
x
g
g
Then, we can utilize the Newton-iterative method to gain
the root of the nonlinear equation , where the iterative
formula is presented as
1,0,1,2,.
t
tt
t
gt
g
We choose the initial value 0
as 1
K
, and plug it in .
With multiple iterations, we can get the near-optimum BS/UE
density ratio *
1
T
, where T is the maximum number of
the iterations.
Equation shows that the derivative of ()f
does not
depend on the density of UEs UE
, which means that the
optimal energy efficiency of the cellular network is
independent of the UE density which can also be deemed as
network load because we assume that the BS allocates
resources equally to its associated users. No matter how the
density of UEs in the network fluctuates, we can deploy base
stations or adopt BS sleeping strategy with a corresponding
density which can provide the optimal network energy
efficiency. And the optimal network energy efficiency is a
fixed value once the network parameters are given. In this
paper, these network parameters include the fixed power
consumption of BS c
P, the transmit power of BS t
P, system
bandwidth W, and the path loss exponent
.
00.1 0.2 0. 3 0.4 0. 5 0.6 0. 7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X co ordinate (km)
Y coordinate (km)
Fig. 2. An example of the coverage region and location distribution of
BSs and UEs.
2015 IEEE Wireless Communications and Networking Conference (WCNC): - Track 3: Mobile and Wireless Networks
1617
IV. SIMULATION RESULTS
In the simulation, we consider the traditional cellular
network in which only macro-cell BSs are present. The system
bandwidth is 10 MHz and the frequency reuse factor is 1. For
simplicity, the wireless channel fading only includes the fast
fading with free space path loss. The power path loss exponent
is considered as 4, and the fast fading of the wireless
channel is a complex Gaussian channel with the mean value of
zero and the variance of one. The transmit power of the
macro-cell BSs is 46 dBm. The coverage of the whole cellular
network is an area with 200 km radius and the locations of
BSs are generated by PPP.
In the simulation, we investigate the network energy
efficiency under the condition of changeable BS density and
UE density. Fig. 2 shows an example of the coverage regions
in the network and the locations of BSs and UEs. The density
of macro-cell BSs in this figure is 0.00003
BS
, which
means that the coverage area of each BS is about 30000 m2
and the inter-site distance is about 200 m. The density of UEs
is set as 0.0003
UE
. In Fig. 2, the red circles are locations
of macro-cell BSs whose coverage borders are drawn using
blue lines and the black triangles represent the locations of
UEs.
Since we have developed the expressions for the energy
efficiency and optimal BS density for cellular network, it is
very important to verify whether these analytical results could
or not capture the simulation results of the practical scenario.
We use Monte Carlo simulation method to obtain the network
energy efficiency for different BS density by 10000 drops of
BSs distribution following PPP. Fig. 3 and Fig. 4 show the
comparison between the simulation results and the theoretical
numerical results obtained from with respective UE
density 0.0003
UE
and 0.01
UE
. In these two figures,
horizontal axis represents the value of BS density in the
network, and the vertical axis represents the corresponding
value of network energy efficiency in units of bit/Joule. The
simulation results show that, whether the UE density is low or
high, the curves of the network energy efficiency obtained via
can capture the statistical behavior of the network energy
efficiency in practical scenario accurately.
In Section III, we have mathematically analyzed that the
optimal energy efficiency of the cellular network is a fixed
value which is independent of UE density once the network
parameters are given. We now provide the results of scenario
simulation in Fig. 5 as validation. As shown in Fig. 5, the
theoretical optimal value of the network energy efficiency
keeps invariant no matter what the UE density is. The fixed
optimal network energy efficiency is about
4
2.2258 10 bit/Joule. There is at most 3% difference
between the simulation results and the theoretic value which is
an acceptable statistic error. Thus, the simulation results reveal
that the proposed energy efficiency model coincides with the
statistical behavior of energy efficiency in cellular network
very well.
00.01 0.02 0.03 0.04 0. 05 0.06 0.07 0.08 0.09 0. 1
0
0.4
0.8
1.2
1.6
2
2.4
x 10
4
BS Densi ty
Network Energy Efficiency (bit/Joule)
UE
=0.01 experimental data
UE
=0.01 theoretical data
Fig. 4. The network energy efficiency obtained with different BS
densities as 0.01
UE
.
00.006 0.012 0. 018 0.024 0.03
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
x 10
4
UE Densi ty
Optimal Energy Efficiency (bit/Joule)
Experimental optimal energy efficiency
Theoretic al optimal energy ef ficiency
At most 3 % diffe renc e here
Fig. 5. Optimal energy efficiency in network for different UE densities.
00.1 0.2 0.3 0.4 0. 5 0.6 0.7 0.8 0. 9 1
x 10-3
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
x 104
BS Densit y
Network Energy Efficiency (bit/Joule)
UE
=0.0003 experimental data
UE
=0.0003 theoretical data
Fig. 3. The network energy efficiency obtained with different BS
densities as 0.0003
UE
.
2015 IEEE Wireless Communications and Networking Conference (WCNC): - Track 3: Mobile and Wireless Networks
1618
We have also obtained the optimal BS/UE density ratio via
Newton iteration method. Then, we need to verify the
performance of the Newton iteration method with different
maximum iterations by comparing its performance with both
the theoretical optimal BS density and the statistical result of
the practical network scenario. The simulation results are
shown in Fig. 6, where the horizontal axis represents the
density of UEs in the network, and the vertical axis represents
the corresponding optimal BS density. In Fig. 6, to present a
clearer vision, we zoom in part of this figure at the point
where the UE density UE
is 0.018. From the simulation
results, we can see that the results obtained via Newton
iteration method with either large maximum iterations or small
maximum iterations are basically the same, which means that
the Newton iteration method has a fast convergent rate which
is suitable for practical scenario. Also, the results obtained by
Newton iteration method coincide with the theoretical optimal
BS density and the statistical simulation results very well.
V. CONCLUSION
In this paper, we have modeled the distribution of the BSs
and UEs in the cellular network as spatial PPP, and derived
the closed form expressions of users’ average achievable
downlink data rate and network energy efficiency. By
adopting the Newton Iteration Method, we have optimized the
BS density for maximum network energy efficiency and
analyzed its impact on network energy efficiency with
different network loads. Based on our analysis, the best
proportion of the base stations which can be turned off to
achieve high network energy efficiency according to the BS
sleeping strategy can be obtained. Finally, we have validated
our theoretic analysis with both numerical and practical
scenario simulations. This work can provide theoretical basis
for energy-efficient dynamic operation control and network
planning in cellular networks.
REFERENCES
[1] B. Badic, T. O'Farrrell, P. Loskot, and J. He, "Energy Efficient Radio
Access Architectures for Green Radio: Large versus Small Cell Size
Deployment," in Vehicular Technology Conference Fall (VTC 2009-
Fall), 2009 IEEE 70th, 2009, pp. 1-5.
[2] L. Hansung, B. Seon Yeob, and S. Dan Keun, "The Effects of Cell Size
on Energy Saving, System Capacity, and Per-Energy Capacity," in
Wireless Communications and Networking Conference (WCNC), 2010
IEEE, 2010, pp. 1-6.
[3] L. Chiaraviglio, D. Ciullo, M. Meo, and M. A. Marsan, "Energy-
efficient management of UMTS access networks," in Teletraffic
Congress, 2009. ITC 21 2009. 21st International, 2009, pp. 1-8.
[4] L. Saker, S. E. Elayoubi, and T. Chahed, "Minimizing Energy
Consumption via Sleep Mode in Green Base Station," in Wireless
Communications and Networking Conference (WCNC), 2010 IEEE,
2010, pp. 1-6.
[5] M. A. Marsan, L. Chiaraviglio, D. Ciullo, and M. Meo, "Optimal Energy
Savings in Cellular Access Networks," in Communications Workshops,
2009. ICC Workshops 2009. IEEE International Conference on, 2009,
pp. 1-5.
[6] O. Eunsung and B. Krishnamachari, "Energy Savings through Dynamic
Base Station Switching in Cellular Wireless Access Networks," in
Global Telecommunications Conference (GLOBECOM 2010), 2010
IEEE, 2010, pp. 1-5.
[7] W. Yiqun and N. Zhisheng, "Energy efficient base station deployment in
green cellular networks with traffic variations," in Communications in
China (ICCC), 2012 1st IEEE International Conference on, 2012, pp.
399-404.
[8] J. G. Andrews, F. Baccelli, and R. K. Ganti, "A Tractable Approach to
Coverage and Rate in Cellular Networks," Communications, IEEE
Transactions on, vol. 59, pp. 3122-3134, 2011.
[9] H. S. Dhillon, R. K. Ganti, and J. G. Andrews, "Load-Aware Modeling
and Analysis of Heterogeneous Cellular Networks," Wireless
Communications, IEEE Transactions on, vol. 12, pp. 1666-1677, 2013.
[10] C. Dongxu, Z. Sheng, and N. Zhisheng, "Optimal Combination of Base
Station Densities for Energy-Efficient Two-Tier Heterogeneous Cellular
Networks," Wireless Communications, IEEE Transactions on, vol. 12,
pp. 4350-4362, 2013.
[11] V. Suryaprakash, A. Fonseca dos Santos, A. Fehske, and G. P. Fettweis,
"Energy consumption analysis of wireless networks using stochastic
deployment models," in Global Communications Conference
(GLOBECOM), 2012 IEEE, 2012, pp. 3177-3182.
[12] S. Yong Sheng, T. Q. S. Quek, and M. Kountouris, "Dynamic sleep
mode strategies in energy efficient cellular networks," in
Communications (ICC), 2013 IEEE International Conference on, 2013,
pp. 3131-3136.
[13] O. Arnold, F. Richter, G. Fettweis, and O. Blume, "Power consumption
modeling of different base station types in heterogeneous cellular
networks," in Future Network and Mobile Summit, 2010, 2010, pp. 1-8.
[14] F. P. Schoenberg, C. Barr, and J. Seo, "The distribution of Voronoi cells
generated by Southern California earthquake epicenters,"
Environmetrics, vol. 20, pp. 159-171, 2009.
[15] E. Pineda, V. Garrido, and D. Crespo, "Domain-size distribution in a
Poisson-Voronoi nucleation and growth transformation," Physical
Review E, vol. 75, p. 040107, 2007.
[16] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M.
Franceschetti, "Stochastic geometry and random graphs for the analysis
and design of wireless networks," Selected Areas in Communications,
IEEE Journal on, vol. 27, pp. 1029-1046, 2009.
00.006 0.012 0. 018 0.024 0.03
0
1
2
3
4
5
6
x 10
-3
UE Density
Opt imal BS Dens ity
0.018
3.4
3.45
3.5
x 10
-3
Th eor et ica l op tim al BS d ens ity
Newton iteration method with T =1000
Newton iteration method with T =5
Experimental optimal BS density
Fig. 6. The optimal BS density obtained with different UE densities.
2015 IEEE Wireless Communications and Networking Conference (WCNC): - Track 3: Mobile and Wireless Networks
1619
