1

Downlink Energy Efficiency of Power
Allocation and Wireless Backhaul Bandwidth

arXiv:1710.02942v1 [] 9 Oct 2017

Allocation in Heterogeneous Small Cell
Networks
Haijun Zhang, Senior Member, IEEE, Hao Liu, Julian Cheng, Senior
Member, IEEE, and Victor C. M. Leung, Fellow, IEEE
Abstract
The widespread application of wireless services and dense devices access have triggered huge
energy consumption. Because of the environmental and financial considerations, energy-efficient design
in wireless networks becomes an inevitable trend. To the best of the authors’ knowledge, energy-efficient
orthogonal frequency division multiple access heterogeneous small cell optimization comprehensively
considering energy efficiency maximization, power allocation, wireless backhaul bandwidth allocation,
and user Quality of Service is a novel approach and research direction, and it has not been investigated.
In this paper, we study the energy-efficient power allocation and wireless backhaul bandwidth allocation
in orthogonal frequency division multiple access heterogeneous small cell networks. Different from the
existing resource allocation schemes that maximize the throughput, the studied scheme maximizes energy
efficiency by allocating both transmit power of each small cell base station to users and bandwidth for
backhauling, according to the channel state information and the circuit power consumption. The problem
is first formulated as a non-convex nonlinear programming problem and then it is decomposed into two
convex subproblems. A near optimal iterative resource allocation algorithm is designed to solve the
Haijun Zhang is with the Beijing Engineering and Technology Research Center for Convergence Networks and Ubiquitous
Services, University of Science and Technology Beijing, Beijing, 100083, China (e-mail: haijunzhang@ieee.org).
Hao Liu and Julian Cheng are with School of Engineering, The University of British Columbia, Kelowna, BC, Canada (e-mail:
hao.liu@alumni.ubc.ca, julian.cheng@ubc.ca).
Victor C. M. Leung is with the Department of Electrical and Computer Engineering, The University of British Columbia,
Vancouver, BC V6T 1Z4 Canada (e-mail: vleung@ece.ubc.ca).

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resource allocation problem. A suboptimal low-complexity approach is also developed by exploring
the inherent structure and property of the energy-efficient design. Simulation results demonstrate the
effectiveness of the proposed algorithms by comparing with the existing schemes.
Index Terms
Bandwidth allocation, energy efficiency, heterogeneous network, power allocation, small cell, wireless backhaul.

I. I NTRODUCTION
Wireless communication networks have experienced tremendous growth in the past a few
decades. It is shown that higher capacity wireless links are expected to meet the increasing
quality of service (QoS) demands of multimedia applications, and these high data rate links also
result in increasing device power consumption. The next generation communication systems
need to provide higher data rate with limited power and bandwidth due to the rapidly increasing
demands for multimedia services. Designing energy-efficient wireless communication system
becomes an emerging trend, due to rapidly increasing system energy costs and rising requirements
of communication capacity [1]–[3]. According to [4] and [5], the radio access part is a major
energy consumer in conventional wireless cellular networks, and it accounts for up to more
than 70 percent of the total energy consumption. Therefore, increasing the energy efficiency of
typical wireless networks is important to overcome the challenges raised by the rising demands
of energy consumption and communication throughput.
To offload the overloaded traffics in macrocells and enhance the capacity and energy efficiency
of the wireless networks, one proposed method is to shorten the distance between the base stations
(BSs) and the user equipments. Small cells (e.g., picocells, femtocells and relay nodes) have been
used to improve system capacity in hotspots for relieving the burden on overloaded macrocells,
which is considered as a promising technique to provide an effective solution for the challenges
in current macrocells [6], [7]. Therefore, there is no doubt that small cell has been paid much
attention in recent years from academia and industry because it can help the system spatial reuse
spectrum with low power consumption and improve the system coverage with low infrastructure
cost deployment [8]. Heterogeneous small cell networks, where small cells are overlaid within

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a macrocell to improve coverage and increase system capacity beyond the initial deployment
of macrocells, have been regarded as a promising approach to meet the increasing data traffic
demand and reduce energy consumption. Although highly promising, many important problems
related to heterogeneous small cell networks such as interference mitigation, resource allocation,
and QoS provisioning [9]–[12] should be addressed to fully reap the potential gains.
Resource allocation, such as power allocation and bandwidth allocation, has been widely
used to maximize the energy efficiency under power limitation and QoS in heterogeneous
small cell networks. Power allocation for energy efficiency has been widely studied in the
literature. The distributed power control game was studied in [13] to maximize the energy
efficiency of transmission for secondary users in cognitive radio networks and an optimal
power control problem was formulated as a repeated game. In [14], based on the hardcore
point process (HCPP), the authors investigated the maximum achievable energy efficiency of
the considered multiuser multiantenna HCPP random cellular networks with the aforementioned
minimum distance constraint for adjacent BSs. Different from the authors in [14], who took
the minimum distance in adjacent BSs into consideration to maximize the energy efficient, we
propose a suboptimal low-complexity approach of energy-efficient backhaul bandwidth allocation
by optimizing the fraction of bandwidth allocated for wireless backhauling at all small cell BSs
within a macrocell range. The authors in [15] studied energy-efficient power control and receiver
design in cognitive radio networks, and a non-cooperative power control game for maximum
energy efficiency of secondary users was considered with a fairness constraint and interference
threshold. The authors of [16] formulated the energy-efficient spectrum sharing and power
allocation in heterogeneous cognitive radio networks with femtocells as a Stackelberg game
and they proposed a gradient based iteration algorithm to obtain the Stackelberg equilibrium
solution to the energy-efficient resource allocation problem. Some works also have been done to
consider bandwidth allocation for energy efficiency. In [17], the authors studied the joint service
pricing and bandwidth allocation for energy and cost efficiency at the operator level in a multitier network where an operator deploys heterogeneous small cell networks, and they formulated
the problem as a Stackelberg game. The problem of joint link selection, power and bandwidth
allocation for energy efficiency maximization for Multi-Homing networks was investigated in

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[18]. A new energy-efficient scheme was presented in [19] to statistically meet the demands for
QoS during the bandwidth allocation for wireless networks.
In this paper, we define that wireless backhaul as the connection between macro BS and
small cell BSs, and it is necessary to jointly consider the design of the radio access and
backhaul network. Several related works considered the backhaul to improve energy efficiency
in wireless networks. The authors of [20] studied energy efficiency of resource allocation in
multi-cell orthogonal frequency division multiple access (OFDMA) downlink networks where
the limited backhaul capacity, the circuit power consumption and the minimum required data rate
are considered. The resource allocation problem for energy-efficient communication with limited
backhaul capacity is formulated as an optimization problem. In [21], an energy efficiency model
of small cell backhaul networks with Gauss–Markov mobile models has been proposed. In [22],
the authors maximized system energy efficiency in OFDMA small cell networks by optimizing
backhaul data rate and emission power, and they proposed a joint forward and backhaul link
optimization scheme by taking both the power consumption of forward links and the backhaul
links into consideration.
To the best of the authors’ knowledge, energy efficiency of power allocation and wireless
backhaul bandwidth allocation in heterogeneous small cell network has not been investigated. In
this work, we study the power allocation and bandwidth allocation problem in a heterogeneous
small cell network where the small cells use wireless backhauling to maximize energy efficiency
of all small cell users. Similar to the paper in [23], we also use Gradient Assisted Binary Search
(GABS) Algorithm to solve the energy-efficient power allocation problem. Reference [24] is a
conference version of this paper. Different from the conference version, we provide the detailed
proof for the theorem, complexity analysis for the proposed algorithms and more simulation
results in this paper. The key contributions of our work can be summarized as follows.
•

Design of an energy-efficient OFDMA heterogeneous small cell optimization: This is a
novel approach by considering energy efficiency maximization, power allocation, wireless
backhaul bandwidth allocation, and user QoS into the design of OFDMA heterogeneous
small cell optimization. We formulate the energy-efficient wireless backhaul bandwidth
allocation and power allocation problem in a heterogeneous small cell as a nonlinear

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programming problem, where maximum transmit power constraints of each small cell BS
to each small cell user, the downlink data rate constraint of small cell BSs and the minimum
data rate between each small cell BS and its corresponding user are considered to provide
reliable and low energy consumed downlink transmission for small cell users. The nonconvex optimization problem is then decomposed into two convex subproblems, and an
algorithm is proposed for wireless backhual bandwidth allocation and power allocation.
•

Support of the small cell backhauling in the context of designing power allocation schemes
for heterogeneous small cell networks: We study the wireless backhaul bandwidth allocation
at the small cell BS, which means a fraction of bandwidth is scheduled for backhauling
and the other is assigned for communication with corresponding users. We formulate the
bandwidth allocation problem as a convex problem and obtain the optimum solution.

•

Design of suboptimal low-complexity algorithm by decomposing the power allocation and
bandwidth allocation: The energy-efficient wireless backhaul bandwidth allocation and
power allocation problem are decomposed and are optimized separately. Correspondingly,
a suboptimal low-complexity algorithm is proposed. The effectiveness of the proposed
suboptimal algorithm is demonstrated by simulations.

The rest of this paper is organized as follows. Section II describes the system model. In Section
III, the energy-efficient resource allocation and backhauling are presented, and in Section IV,
optimization algorithms are proposed. Simulation results are discussed in Section V. Finally,
Section VI concludes the paper.
II. S YSTEM M ODEL
In this section, we formulate the problem of downlink energy efficiency of power allocation
and unified bandwidth allocation for wireless backhauling in heterogeneous small cell networks.
We consider a heterogeneous small cell network as shown in Fig. 1 with a single macro BS,
J small cells deployed within the macrocell range and K users randomly located in each small
cell.
The small cells share the same spectrum with macrocell. In this work, the unified wireless
backhaul bandwidth allocation is investigated. The unified bandwidth allocation factor β ∈ [0, 1],

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which is the fraction of bandwidth allocated for wireless backhauling at all small cell BSs within a
macrocell range. For simplicity, all small cells are assumed to have the same bandwidth allocation
factor. We assume that the multiple antenna technology is used in the macro BS and each small
cell corresponds to a beamforming group, so the interference for wireless backhaul between
different small cells can be neglected. The antenna array size at macro BS is N, which is much
greater than the beamforming group size B and the number of small cells, i.e., N ≫ B and
N ≫ J. In this work, we also assume that B ≥ J. Each small cell BS is equipped with single
antenna. OFDMA technology is used in each small cell to support the communication between
BS and users.
Let P0 be the equal transmit power of the macro BS transmit antenna targeted at corresponding
small cell and σ 2 is the additive white Gaussian noise (AWGN) power. Then the received signalto-noise ratio (SNR) in the wireless backhaul downlink of small cell j is given by

γj =

P 0 Gj
.
σ2

(1)

Let gj,k be the channel power gain between the jth small cell BS and its corresponding kth
user, where j ∈ {1, 2, ..., J}, k ∈ {1, 2, ..., K}. Let pj,k denote the transmit power from the jth
small cell BS to its corresponding kth user, and let P = [pj,k ]J×K denote the power allocation
matrix.
We assume that different users in each small cell use different subchannels and co-channel
interference between small cells as part of the thermal noise because of the severe wall penetration
loss and low power of small cell BSs [12]. The received signal-to-interference-plus-noise ratio
(SINR) of small cell user k associated with small cell j is given by
γj,k =

pj,k gj,k
σ 2 +Ij,k

(2)

where Ij,k is the interference introduced by macro BS, Ij,k = P0 Gj,k , where Gj,k is the channel
power gain between macro BS and the kth user in the jth small cell. The achievable data
transmission rate between the jth small cell BS and its corresponding kth user is determined by


1−β
log2 (1 + γj,k ) .
(3)
rj,k =
K
Therefore, we have the relation between rj,k and pj,k
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Krj,k
σ 2 +Ij,k
pj,k = (2 1−β − 1)
gj,k




pj,k gj,k
1−β
log2 1 + 2
.
rj,k =
K
σ +Ij,k

(4)

Besides the transmit power during the transmission, circuit energy consumption is also incurred
by device electronics in small cell BSs [25], [26]. Circuit power represents the additional device
power consumption of devices during transmissions [27], such as digital-to-analog converters,
mixers and filters, and this portion of energy consumption is independent of the transmission
state. If we denote the circuit power as PC , the overall power assumption of the jth small cell
BS to the kth user is PC + pj,k .
For energy-efficient communication, it is desirable to send the maximum amount of data with
a given amount of energy for small cell BSs. Hence, given any amount of energy ∆e consumed
in a duration ∆t in each small cell BS to each user, ∆e = ∆t(PC + pj,k ), the small cell BSs
desire to send a maximum amount of data by choosing the power allocation vector and backhaul
bandwidth to maximize

J X
K
X
rj,k (β, pj,k )∆t
j=1 k=1

∆e

(5)

which is equivalent to maximizing
U(β, P) =

J X
K
X

Uj,k (β, pj,k )

(6)

j=1 k=1

where
Uj,k (β, pj,k ) =

rj,k (β, pj,k )
.
PC + pj,k

(7)

In (6), U(β, P) is called energy efficiency for all small cells; Uj,k (β, pj,k ) is the energy efficiency
of the kth user of the jth small cell. The unit of the energy efficiency is bits per Hertz per Joule,
which has been frequently used in the literature for energy-efficient communications [28]–[32].
When the downlink channel state information is estimated by the small cell BSs, the resource
allocation is performed by a small cell BS under the following constraints.
•

Transmit power constraint of each small cell BS to each user:
0 ≤ pj,k ≤ Pmax , ∀j, k

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where Pmax denotes the maximal transmit power of each small cell BS to each user.
•

The downlink data rate constraint of each small cell BS: The throughput of the small cell
is given by
Rj =

K
X

rj,k .

(9)

k=1

Due to the inter-user interference within the overlapping areas of macrocell and small cell
beamforming group, we use typical zero-forcing beamforming technique with equal-power
allocation for each user transmission link to eliminate the interference significantly [33],
[34], the capacity of the wireless backhaul downlink for small cell j is


N−B+1
Cj = βlog2 1 +
γj .
B

(10)

The downlink wireless backhaul constraint requires
Rj ≤ Cj

(11)

such that the downlink traffic of the jth small cell can be accommodated by its wireless
backhaul.
•

Heterogeneous QoS guarantee: The QoS requirement Rt should be guaranteed for each
user in each small cell to maintain the performance of the communication system
rj,k ≥ Rt .

(12)

Our target is to maximize the energy efficiency of power allocation and unified bandwidth
allocation for wireless backhauling in heterogeneous small cell networks under power constraint
and data rate requirements. Thus, the corresponding problem for the downlink can be formulated
as the following nonlinear programming problem
max U(β, P) = max
β,P

β,pj,k

J X
K
X

Uj,k (β, pj,k )

(13)

j=1 k=1

s.t. C1 : 0 ≤ pj,k ≤ Pmax
C2 : Rj ≤ Cj
(14)
C3 : rj,k ≥ Rt
C4 : 0 ≤ β ≤ 1.
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III. E NERGY- EFFICIENT R ESOURCE A LLOCATION

AND

BACKHAULING

Since the optimization problem formulated in (13) and (14) is non-convex and we notice that
the continuous variables β and pj,k are separable in (13). Therefore, we consider a decomposition
approach to solve the energy-efficient resource allocation problem. We decompose the nonconvex optimization problem into two convex subproblems: one for energy-efficient power
allocation and one for energy-efficient wireless backhaul bandwidth allocation. Then, we solve
the subproblems of energy-efficient power allocation and energy-efficient backhaul bandwidth
allocation individually.

A. Energy-Efficient Power Allocation
The concept of quasiconcavity will be used in our discussion and is defined in [35].
Definition 1. A function f , which maps from a convex set of real n-dimensional vectors, D, to
a real number, is called strictly quasiconcave if for any x1 , x2 ∈ D and x1 6= x2 ,
f (λx1 + (1 − λ)x2 ) > min{f (x1 ), f (x2 )}

(15)

for any 0 < λ < 1.
Given a value β for unified wireless backhaul bandwidth allocation, the optimization algorithm
begins with the power allocation subproblem P1.1 that is formulated as
P1.1 : max U(P) = max
pj,k

P

J X
K
X

Uj,k (pj,k )

(16)

j=1 k=1

s.t. C1 : 0 ≤ pj,k ≤ Pmax
C2 : Rj ≤ Cj

(17)

C3 : rj,k (pj,k ) ≥ Rt
where
rj,k (pj,k ) =



1−β
K





pj,k gj,k
log2 1 + 2
σ +Ij,k

(18)

is strictly concave and monotonically increasing in pj,k with rj,k (0) = 0, when pj,k = 0.

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The optimal energy-efficient power allocation achieves the maximum energy efficiency, i.e.
P∗ = arg max U(P).

(19)

P

It is proved in Appendix A that U(P) has the following properties.
Lemma 1. If rj,k (pj,k ) is strictly concave in pj,k , Uj,k (pj,k ) ∈ U(P) is strictly quasiconcave.
Furthermore, Uj,k (pj,k ) is first strictly increasing and then strictly decreasing in any pj,k , i.e. the
local maximum of U(P) for each pj,k exists at a positive finite value.
For strictly quasiconcave functions, if a local maximum exists, it is also globally optimal [35].
Hence, a unique globally optimal transmission rate vector always exists and its characteristics
are summarized in Theorem 1 according to the proofs in Appendix A.
Theorem 1. If rj,k (pj,k ) is strictly concave, there exists a unique globally optimal transmission
power vector P∗ = {p∗ j,k ; (j, k) ∈ J × K} for P∗ = arg max U(P), for each element in P∗ ,
P

p∗j,k = arg max Uj,k (pj,k ) where p∗j,k is given by
pj,k

∂Uj,k (pj,k )
∂pj,k

pj,k =p∗j,k

i.e., Uj,k (p∗j,k ) =

= 0, f (pj,k ) = 0,
rj,k (p∗j,k )
PC +p∗j,k

=

∂rj,k (pj,k )
∂pj,k

pj,k =p∗j,k

.

In order to solve the problem P1.1 for power allocation, we rewrite the objective function in
(16) as
max Uj,k (pj,k ) = max
pj,k

pj,k

rj,k (pj,k )
.
PC + pj,k

(20)

If each small cell user could reach the maximum energy efficiency, all small cell users could
reach the maximum energy efficiency. The total data rate in each small cell could not exceed
the capacity of the wireless backhaul downlink for small cell j, Rj ≤ Cj , we can approximate
that the data rate for each user to be less than
for user k in small cell j is

PS
.
K

Cj
,
K

rj,k (pj,k ) ≤

and the maximum of power

Thus, P1.1 is equivalent to
P1.2 : max Uj,k (pj,k )
pj,k

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Cj
,
K

(21)

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11

PS
K
Cj
C2 : rj,k (pj,k ) ≤
K

s.t. C1 : 0 ≤ pj,k ≤

(22)

C3 : rj,k (pj,k ) ≥ Rt .
We can rewrite C2 in (22) according to (4) as


 2


P0 Gj
β
σ +Ij,k
log2 1+ N−B+1
(
2
1−β )
B
σ
2
−1 .
pj,k ≤
gj,k
We can rewrite C3 in (22) according to (4) as
 2


t
σ +Ij,k  KR
pj,k ≥
2 1−β −1 .
gj,k

(23)

(24)

Therefore, we have

Lj,k ≤ pj,k ≤ Hj,k

(25)

where


σ 2 +Ij,k  KR
t
1−β
−1
Lj,k =
2
gj,k
 2





β
N−B+1 P0 Gj
σ +Ij,k
log
1+
)
(
2
2
B
σ
= min
2 1−β
−1 , Pmax
gj,k


Hj,k

(26)
(27)

only if the following inequality is satisfied
Lj,k ≤ Hj,k .

(28)

The energy-efficient power allocation is given by
p̂∗j,k = arg max
pj,k

rj,k (pj,k )
PC + pj,k

(29)

subject to
Lj,k ≤ pj,k ≤ Hj,k .

(30)

We can solve (20) by using Theorem 1 to find the optimal power allocation solution. We
can also use the low-complexity iterative algorithms based on the GABS algorithm proposed in
[23] to realize the energy-efficient power allocation for the kth user in the jth small cell BS as
follows.
If the output p̂∗j,k satisfies the power constraint, i.e. p̂∗j,k =p∗j,k ; otherwise, we can obtain the
maximum Uj,k (pj,k ) by
p∗j,k = Lj,k
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Algorithm GABS Algorithm
1: Initialization: Each small cell BS allocates the same transmit power to each user, pj,k > 0.
∂Uj,k (pj,k )
(1)
2: Then do pj,k = pj,k , h1 ←
(1) and c > 1 (let c = 2).
∂pj,k
pj,k =pj,k
3: if h1 < 0 then
4:
repeat
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:

(2)

(1)

(1)

p

(1)

∂U

(p

)

j,k
, and h1 ← j,k
pj,k ← pj,k , pj,k ← j,k
(1)
c
∂pj,k
pj,k =pj,k
until h1 ≥ 0
else
∂U (pj,k )
(2)
(1)
pj,k ← pj,k × c and h2 ← j,k
(2)
∂pj,k
pj,k =pj,k
repeat
∂Uj (pj,k )
(1)
(2)
(2)
(2)
pj,k ← pj,k , pj,k ← pj,k × c and h2 ← ∂p
(2)
pj,k =pj,k
j,k
until h2 ≤ 0
end if
while no convergence do

p

(1)

+p

(2)

p̂∗j,k ← j,k 2 j,k , h′ ←
if h′ > 0 then
(1)
pj,k = p̂∗j,k
else
(2)
pj,k = p̂∗j,k
end if
end while
Output p̂∗j,k .

∂Uj,k (pj,k )
pj,k =p̂∗j,k
∂pj,k

if p̂∗j,k < Lj,k , or we can get the maximum Uj,k (pj,k ) by
p∗j,k = Hj,k

(32)

if p̂∗j,k > Hj,k , since Uj,k (pj,k ) is first strictly increasing and then strictly decreasing in any
positive finite pj,k .
B. Energy-Efficient Wireless Backhaul Bandwidth Allocation
Once the optimal solution P∗ = {p∗j,k ; (j, k) ∈ J × K} is obtained for the convex subproblem
P1.2 parameterized by β, it can be used in the following subproblem P1.3 for the unified wireless
backhaul bandwidth allocation
P1.3 : max U(β, P∗ ) = max
β

October 10, 2017

β

J X
K
X

Uj,k (β, p∗j,k )

(33)

j=1 k=1

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13

s.t. C1 : 0 ≤ β ≤ 1
(34)

C2 : Rj (β, P∗) ≤ Cj (β, P∗)
C3 : rj,k (β, p∗j,k ) ≥ Rt

where Rj (β, P∗) is the function value of Rj evaluated at P∗ , Cj (β, P∗) is the function value of
Cj evaluated at P∗ . In order to obtain the solution to the original problem in (13) and (14), the
two subproblems P1.2 and P1.3 are solved iteratively until convergence.
Maximizing the objective function of P1.3 with respect to β is equivalent to maximizing
(1 − β) only, since (33) is a monotonically decreasing function of β. Problem P1.3 reduces to
a feasibility problem whose solution is the smallest feasible value of β given constraints (34).
According to C2, Rj (β, P∗) ≤ Cj (β, P∗), we have


K
P
p∗ gj,k
log2 1 + σ2j,k+Ij,k
k=1
β≥


 P
K
P 0 Gj
Klog2 1 + N −B+1
+
log
2 1+
B
σ2
k=1

According to C3,

rj,k (β, p∗j,k )

≥ Rt , we have

KRt

β ≤ 1−



log2 1 +
So we have

p∗j,k gj,k
σ2 +Ij,k

p∗j,k gj,k
σ2 +Ij,k

.

.

(36)

β = max {φj }
where
φj =

K
P

k=1



Klog2 1 +


log2 1 +

N −B+1 P0 Gj
B
σ2

only if Rt satisfies the following condition



+

(37)

p∗j,k gj,k
σ2 +Ij,k
K
P

k=1




log2 1 +

(38)
p∗j,k gj,k
σ2 +Ij,k



Rt ≤ min {ϕj }
where

(35)





p∗j,k gj,k
N −B+1 P0 Gj
log2 1 + σ2 +Ij,k
log2 1 + B
σ2
ϕj =


 P
.
K
p∗j,k gj,k
N −B+1 P0 Gj
Klog2 1 + B
+
log2 1 + σ2 +Ij,k
σ2

(39)

(40)

k=1

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IV. A LGORITHM D ESIGN
According to the analysis of power allocation and wireless backhaul bandwidth allocation
discussed above, we propose an iterative optimization algorithm and a suboptimal low-complexity
algorithm.

A. Iterative Resource Allocation Algorithm
The proposed iterative resource allocation algorithm is shown in Algorithm 1.
Algorithm 1 Iterative Resource Allocation Algorithm
1: Initialization: Each small cell BS allocates the same transmit power to each user, pj,k > 0
and set l = 1.
2: repeat
3:
Backhaul Bandwidth Allocation
4:
Compute optimum β according to (37).
5:
Macro BS broadcasts the updated wireles backhaul bandwidth allocation factor to all small
cell BSs.
6:
for each small cell BS do
7:
for each small cell user do
8:
Power Allocation
9:
a) find p̂∗j,k = arg max Uj,k (pj,k ) according to GABS;
10:
b) check power constraint;
11:
if Lj,k ≤ p̂∗j,k ≤ Hj,k then
12:
p∗j,k = p̂∗j,k
13:
end if
14:
if p̂∗j,k < Lj,k then
15:
p∗j,k = Lj,k
16:
end if
17:
if p̂∗j,k > Hj,k then
18:
p∗j,k = Hj,k
19:
end if
20:
end for
21:
end for
22:
l = l + 1.
23: until total energy efficiency convergence or l = Lmax

In Algorithm 1, each small cell BS calculates φj according to (38) and then sends φj to macro
BS. The macro BS chooses the maximum φj to be the optimal bandwidth allocation factor β
and broadcasts β to all small cell BSs.

October 10, 2017

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15

B. Low-Complexity Optimization Algorithm
To reduce the complexity of Algorithm 1, we propose a low-complexity optimization algorithm
where bandwidth allocation factor is calculated from the equal power allocation and we fix β
to calculate the power allocation according to the scheme proposed in Section IV. This lowcomplexity optimization algorithm is shown in Algorithm 2.
Algorithm 2 Fixed β and Optimum Power Allocation Algorithm
1: Initialization: Each small cell BS allocates the same transmit power to each user, pj,k > 0.
2: Backhaul Bandwidth Allocation
3: Compute optimum β according to (37).
4: Macro BS broadcasts the wireles backhaul bandwidth allocation factor to all small cell BSs.
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:

for each small cell BS do
for each small cell user do
Power Allocation
a) find p̂∗j,k = arg max Uj,k (pj,k ) according to GABS;
b) check power constraint;
if Lj,k ≤ p̂∗j,k ≤ Hj,k then
p∗j,k = p̂∗j,k
end if
if p̂∗j,k < Lj,k then
p∗j,k = Lj,k
end if
if p̂∗j,k > Hj,k then
p∗j,k = Hj,k
end if
end for
end for

C. Complexity Analysis
Since the problem formulated in (13) and (14) is not convex, the only way to obtain the
optimal solution is to use the method of exhaustion. If we assume that it costs P operations to
calculate rj,k and it costs Q operations to calculate Cj , the complexity of checking C2 and C3
in (14) entails KP + K + Q operations and P + 1 operations, respectively. If we assume that
it costs S operations to calculate Uj,k , the complexity of obtaining the total energy efficiency of
all small cell users entails JKS + (J − 1) (K − 1) operations. The total complexity of getting
the value of objective function in (13) under the constraints in (14) entails KP + K + P + 1 +
October 10, 2017

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16

JKS + (J − 1) (K − 1) operations for specific pj,k and β values. If we assume the value step
JK
size for pj,k is a and the value step size for β is b, there are 1b Pmax
choices for the values
a

 
Pmax JK
.
of pj,k and β. Therefore, the complexity for the method of exhaustion is O JKS
b
a
In Algorithm 1, the worst-case complexity of calculating bandwidth allocation factor β from

(37) entails J operations in each iteration. If we assume that it costs Ω operations in each
GABS to search the optimum power allocation without power constraint, then the worst-case
complexity of finding the power allocation for every user in each small cell entails JK (Ω+4)
operations in each iteration. Suppose the Algorithm 1 needs ∆ iterations to converge, so the
total complexity of Algorithm 1 is O (JKΩ∆). Since iteration is not applied in Algorithm 2,
the total complexity of Algorithm 2 is O (JKΩ), which is less than that of Algorithm 1. In the
simulation, the typical value for ∆ is around 16, the typical value for Ω is less than 500, and the
typical values for

1
b

and

Pmax
b

are both 100. So the complexities of Algorithm 1 and Algorithm

2 are always less than that of the method of exhaustion. When the number of small cells J and
the number of users in each small cell K increase, the complexity of the method of exhaustion
increases exponentially, so the complexity of the method of exhaustion is much larger than the
complexities of two proposed algorithms.
V. S IMULATION R ESULTS
Simulation results are given in this section to evaluate the performance of the proposed power
allocation and backhaul bandwidth allocation algorithms. In the simulations, it is assumed that
small cells are uniformly distributed in the macrocell coverage area, and small cell users are
uniformly distributed in the coverage area of their serving small cell. AWGN power σ 2 =3.9811×
10−14 W. The coverage radius of the macrocell is 500 m, and that of a small cell is 10 m. Small
cell has a minimal distance of 50 m from the macro BS. The minimal distance between small
cell BSs is 40 m. We assume that the channel fading is composed of path loss, shadowing fading,
and Rayleigh fading. The pathloss model for small cell users is based on [36]. The lognormal
shadowing between small cell BS and small cell users is 10 dB. At the macro BS, we assume
that the transmit power is 33 dBm, the antenna array size N = 100 and beamforming group size
is B = 20. We consider that all the small cell users have the same QoS requirement.

October 10, 2017

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17

Figure 2 shows the convergence in terms of the energy efficiency of all small cell users for
the proposed Algorithm 1 versus the number of iterations, where J = 5, Rt = 0.01 bps/Hz,
Pmax = 20 dBm. It can be observed that the proposed resource allocation algorithm takes nearly
16 iterations to converge to stable solutions. This result, together with the previous analysis,
ensures that the proposed Algorithm 1 is applicable in heterogeneous small cell networks.
Figure 3 shows the total energy efficiency of all small cell users when the number of users per
small cell is increased from 2 to 10, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10 dBm and
Pmax = 20 dBm compared with Algorithm 1 under Pmax = 20 dBm. The simulation parameters
are set as J = 5, Rt = 0.01 bps/Hz. Fig. 3 shows that the energy efficiency performance of
Algorithm 1 is 20% more superior to that of Algorithm 2. It also can be seen from Fig. 3 that
the more number of users in small cell is, the better performance is obtained because of the
multi-user diversity.
Figure 4 shows the total energy efficiency of all small cell users when the number of small
cells is increased from 3 to 15, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10 dBm and
Pmax = 20 dBm when compared with Algorithm 1 under Pmax = 20 dBm. The simulation
parameters are set as K = 5, Rt = 0.01 bps/Hz. Fig. 4 indicates that more number of small cell
is, the better performance is obtained. It can also be seen from Fig. 4 that the energy efficiency
performance of Algorithm 1 is always better than that of Algorithm 2 and the gap between them
becomes larger when the number of small cells increases. The energy efficiency performance of
Algorithm 1 is 30% superior to that of Algorithm 2 when the number of small cells is 10.
Figure 5 shows the total downlink capacity of all small cell users when the number of users
per small cell is increased from 2 to 10, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10
dBm and Pmax = 20 dBm compared with Algorithm 1 under Pmax = 20 dBm. The simulation
parameters are set as J = 5, Rt = 0.01 bps/Hz. Fig. 5 shows that the total downlink capacity of
Algorithm 1 is more than 3 bps/Hz higher than that of Algorithm 2. It can also be seen from
Fig. 5 that the more number of users in small cell is, the better performance is obtained due to
the multi-user diversity. The total downlink capacity of Algorithm 1 is 21% higher than that of
Algorithm 2 when the number of users in each small cell is over 10.
Figure 6 shows the total downlink capacity of all small cell users when the number of small

October 10, 2017

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18

cells is increased from 3 to 15, for Algorithm 2 under Pmax = 7 dBm, Pmax = 10 dBm and
Pmax = 20 dBm compared with Algorithm 1 under Pmax = 20 dBm. The simulation parameters
are set as K = 5, Rt = 0.01 bps/Hz. Fig. 6 illustrates that Algorithm 1 is superior to Algorithm
2 in terms of the total downlink capacity and the gap between them becomes larger when the
number of small cells increases. The total downlink capacity of Algorithm 1 is 29% larger than
that of Algorithm 2 when there are 14 small cells in the heterogeneous network.
Figure 7 shows the total energy efficiency of all the small cell users when using Algorithm
2 for power constraint Pmax ranging from 0 dBm to 12.79 dBm where the number of users
in each small cell is 3, 4, 5. The simulation parameters are set as J = 5, Rt = 0.01 bps/Hz.
Fig. 7 presents that the more users in each small cell are, the higher total energy efficiency can
be obtained, which has already been shown in Fig. 3. It can also be seen from Fig. 7 that the
larger power constraint is, the better performance is obtained. This is because the larger power
constraint leads to the larger region of the optimizing variable.
Figure 8 shows the total energy efficiency of all the small cell users when the number of users
per small cell is increased from 2 to 10, for different algorithms. Algorithm 1 and Algorithm 2 are
the iterative optimization algorithm and the low-complexity optimization algorithm, respectively,
which we have proposed in Section IV. Algorithm 3 is an existing energy efficiency optimization
algorithm with equal power allocation and Algorithm 4 is an algorithm that uses the optimum
power allocation we proposed given a random β to optimize energy efficiency. All the algorithms
are under the constraint Pmax = 20 dBm. Fig. 8 indicates that more users in each small cell are,
the better performance can be obtained, which has already been shown in Fig. 3. It also can be
seen from Fig. 8 that Algorithm 1 has the best performance, and then it follows by Algorithm
2, Algorithm 3 and Algorithm 4. The energy efficiency performance of Algorithm 1 is 30.5%
and 56.6% higher than that of Algorithm 3 and Algorithm 4, respectively.
Figure 9 shows the total energy efficiency of all the small cell users when the number of small
cells is increased from 2 to 5, for the optimal solution and the two proposed algorithms. Since
the complexity of the method of exhaustion is high, we only consider the situation with small
dimension where there are two users located in each small cell, K = 2. All the algorithms are
under the setting of Pmax = 20 dBm and Rt = 0.01 bps/Hz. From Fig. 9, we can observe that the

October 10, 2017

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19

difference between the optimal solution and Algorithm 1 in terms of energy efficiency is small,
which ensures the effectiveness of the proposed algorithms. The energy efficiency performance
of the optimal solution is only about 7% and 24% higher than that of Algorithm 1 and Algorithm
2 respectively when the number of small cells is 3. The difference between the optimal solution
and the proposed Algorithm 1 is mainly caused by the approximation of C2 in (17). We can also
observe that the performance of Algorithm 1 is slightly better than that of the existing algorithm,
which is the backhaul bandwidth allocation in conjunction with the resource allocation algorithm
in [14]. This phenomenon can be explained as follows. As the QoS requirement of small cell
users increases, the more power is required to meet the higher QoS requirement.
VI. C ONCLUSION
In this paper, we investigated the energy-efficient wireless backhaul bandwidth allocation and
power allocation in a heterogeneous small cell network. We demonstrated the existence of a
unique globally optimal energy efficiency solution and provided an iterative algorithm to obtain
this optimum. For the downlink scenario, we first found the near optimal energy-efficient resource
allocation approach, and then developed a low-complexity suboptimal algorithm by exploring the
inherent structure of the objective function and the feature of energy-efficient design. From the
simulation results, we observed that energy efficiency is improved by increasing the number of
small cells and the number of users per small cell, and the capacity is also improved by increasing
the number of small cells and the number of users per small cell. Simulation results showed great
energy efficiency improvement of the proposed iterative optimization algorithm than that of the
low-complexity optimization algorithm and the existing schemes. The proposed low-complexity
algorithms can achieve a promising tradeoff between performance and complexity. If the future,
we will investigate the nonunified backhaul bandwidth allocation and inter-small-cell interference
in heterogeneous small cell networks.
A PPENDIX A
P ROOF

OF

L EMMA 1

We first focus on Uj,k (pj,k ) and then we can obtain the properties of U(P). If every user
in each small cell can reach the maximum energy efficiency, all small cell users can reach the
October 10, 2017

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20

maximum energy efficiency.
Denote the α–superlevel sets of Uj,k (pj,k ) as
Sα = { pj,k ≥ 0| Uj,k (pj,k ) ≥ α}

(41)

where pj,k is nonnegative. Based on the propositions in [35], Uj,k (pj,k ) is strictly quasiconcave if
and only if Sα is strictly convex for any real number α. In this case, when α < 0, no points exist
on the contour Uj,k (pj,k ) = α. When α = 0, only pj,k = 0 is on the contour Uj,k (0) = α. Hence,
Sα is strictly convex when α ≤ 0. Now, we investigate the case when α > 0. We can rewrite
the Sα as Sα = { pj,k ≥ 0| αPC + αpj,k − rj,k (pj,k ) ≤ 0}. Since rj,k (pj,k ) is strictly concave in
pj,k , −rj,k (pj,k ) is strictly convex in pj,k ; therefore, Sα is strictly convex. Hence, we have the
strict quasiconcavity of Uj,k (pj,k ).
Next, we can obtain the partial derivative of Uj,k (pj,k ) with pj,k as
f (pj,k )
∂Uj,k (pj,k )
(PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k )
=
=
2
∂pj,k
(PC + pj,k )
(PC + pj,k )2

(42)

where f (pj,k ) = (PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k ), r ′ j,k (pj,k ) is the first partial derivative of
rj,k (pj,k ) with respect to pj,k . If p∗j,k exists such that

∂Uj,k (pj,k )
∂pj,k

pj,k =p∗j,k

= 0, it is unique, i.e. if

there is a p∗j,k such that f (p∗j,k ) = 0. In the following, we investigate the conditions when p∗j,k
exists.
The derivative of f (pj,k ) is
f ′ (pj,k ) = (PC + pj,k )r ′′ j,k (pj,k )

(43)

where r ′′ j,k (pj,k ) is the second partial derivative of rj,k (pj,k ) with respect to pj,k . Since rj,k (pj,k )
is strictly concave in pj,k , so r ′′ j,k (pj,k ) < 0, f ′ (pj,k ) < 0. Hence, f (pj,k ) is strictly decreasing.
lim f (pj,k ) = lim ((PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k ))

pj,k →∞

pj,k →∞

(44)
′

′

= lim (PC r j,k (pj,k ) + pj,k r j,k (pj,k ) − rj,k (pj,k ))
pj,k →∞

where
r ′ j,k (pj,k ) =



1−β
K



gj,k
σ 2 + Ij,k



1
ln 2



1
1+

pj,k gj,k
σ2 +Ij,k

!

(45)

and
lim r ′ j,k (pj,k ) = 0.

pj,k →∞

October 10, 2017

(46)
DRAFT

21

So we have
lim PC r ′ j,k (pj,k ) = 0.

(47)

pj,k →∞

According to the L’Hopital’s rule, it is easy to show that
!




g
1
p
1
−
β
j,k
j,k
lim pj,k r ′j,k (pj,k ) = lim
p g
pj,k →∞
pj,k →∞
K
σ 2 + Ij,k
ln 2
1 + σ2j,k+Ij,k
j,k
!




1−β
gj,k
1
1
= lim
gj,k
2
pj,k →∞
K
σ + Ij,k
ln 2
σ2 +Ij,k



1
1−β
= lim
pj,k →∞
K
ln 2
 



1−β
pj,k gj,k
lim (−rj,k (pj,k )) = lim −
log2 1 + 2
= −∞.
pj,k →∞
pj,k →∞
K
σ +Ij,k

(48)

(49)

So we have
lim f (pj,k ) < 0.

(50)

pj,k →∞

Besides,
lim f (pj,k ) = lim ((PC + pj,k )r ′ j,k (pj,k ) − rj,k (pj,k ))

pj,k →0

pj,k →0

(51)

(0)
PC r ′ j,k (pj,k )

−

1−β
K



gj,k
2
σ + Ij,k



1−β
K



gj,k
σ 2 + Ij,k



=

(0)
rj,k (pj,k )

(0)

where pj,k denotes pj,k = 0

r ′ j,k (pj,k ) =



=



(0)

(0)

1
ln 2



1
ln 2



1
1+

pj,k gj,k
σ2 +Ij,k

!

rj,k (pj,k ) = 0




1−β
PC gj,k
1
lim f (pj,k ) =
> 0.
pj,k →0
K
σ 2 + Ij,k
ln 2
So, together with

pj,k =0

(52)

(53)
(54)

lim f (pj,k ) < 0, we see that p∗ j,k exists and Uj,k (pj,k ) is first strictly

pj,k →∞

increasing and then strictly decreasing in pj,k .
Lemma 1 is readily obtained.

October 10, 2017



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22

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...

small cell
base station

small cell

small cell
user

small cell

macro cell
user

macrocell
base station
small cell

small cell
macrocell

Fig. 1.

Topology of a heterogeneous small cell network.

October 10, 2017

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25

Energy efficiency of all small cell users (bits/Hz/Joule)

200
180
160
140
120
100

Pmax=20 dBm, K=5

60
40
20
0
0

Fig. 2.

Pmax=20 dBm, K=7

80

2

4

6

8

10
12
Iteration index

14

16

18

20

The convergence in terms of energy efficiency of all small cell users over the number of iterations.

Energy efficiency of all small cell users (bits/Hz/Joule)

185
180
175
170

Algorithm 1, Pmax=20 dBm

165

Algorithm 2, Pmax=20 dBm
Algorithm 2, Pmax=7 dBm

155
150
145
140
135
130
2

Fig. 3.

Algorithm 2, Pmax=10 dBm

160

3

4

5
6
7
Number of users per small cell

8

9

10

Energy efficiency versus the number of users per small cell.

October 10, 2017

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26

500
Energy efficiency of all small cell users (bits/Hz/Joule)

Algorithm 1, P

max

450
400

Algorithm 2, Pmax=20 dBm
Algorithm 2, Pmax=10 dBm
Algorithm 2, Pmax=7 dBm

350
300
250
200
150
100
50
2

Fig. 4.

=20 dBm

4

6

8
10
Number of small cells

12

14

16

Energy efficiency versus the number of small cells.

Total downlink capacity of all small cell users (bps/Hz)

20

19
Algorithm 1, Pmax=20 dBm
Algorithm 2, P

max

=10 dBm

Algorithm 2, Pmax=7 dBm

17

16

15

14
2

Fig. 5.

Algorithm 2, Pmax=20 dBm

18

3

4

5
6
7
Number of users per small cell

8

9

10

Capacity versus the number of users per small cell.

October 10, 2017

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27

Total downlink capacity of all small cell users (bps/Hz)

55
50

Algorithm 1, P

max

Algorithm 2, Pmax=20 dBm
45

Algorithm 2, Pmax=10 dBm

40

Algorithm 2, Pmax=7 dBm

35
30
25
20
15
10
5
0
2

Fig. 6.

=20 dBm

4

6

8
10
Number of small cells

12

14

16

Capacity versus the number of small cells.

Energy efficiency of all small cell users (bits/Hz/Joule)

150

145

140

135

125

120

115
0

Fig. 7.

Algorithm 2, K=5
Algorithm 2, K=4
Algorithm 2, K=3

130

0.002

0.004

0.006

0.008 0.01 0.012 0.014
Power constraint Pmax (W)

0.016

0.018

0.02

Energy efficiency versus the power constraint.

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Energy efficiency of all small cell users (bits/Hz/Joule)

180

170

160

150

140

130

120

110
2

Fig. 8.

Algorithm 1
Algorithm 2
Algorithm 3
Algorithm 4

3

4

5
6
7
Number of users per small cell

8

9

10

Energy efficiency comparison for different algorithms.

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Energy Efficiency of all small cell users (bits/Hz/Joule)

240
220

Algorithm 1, Pmax=20 dBm
Algorithm 2, Pmax=20 dBm

200

Existing Algorithm, Pmax=20 dBm

180
160
140
120
100
80
60

Fig. 9.

Optimal solution, Pmax=20 dBm

2

2.5

3

3.5
4
4.5
Number of small cells

5

5.5

6

Energy efficiency comparison for the optimal solution and proposed algorithms.

October 10, 2017

DRAFT

