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IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, XX 2017 1
Distributed Power Allocation for Non-Orthogonal Multiple Access
Heterogeneous Networks
Zhengyu Song, Qiang Ni, and Xin Sun
Abstract—In this letter, we investigate the power allocation
problem in non-orthogonal multiple access (NOMA) heteroge-
neous networks (HetNets), where the macro cell base station
(MBS) and small cell base station (SBS) compete to maximize
their own throughput with minimum rate requirement of each
user. The throughput of MBS and SBS is formulated based on
the equivalent channel gain, and the power allocation problem is
captured as a Stackelberg game. A distributed power allocation
algorithm is proposed to achieve the Stackelberg equilibrium.
Simulation results demonstrate the convergence of the proposed
algorithm. Meanwhile, it is verified that the combination of
NOMA and HetNets possesses great potential to improve the
spectrum efficiency compared to the orthogonal multiple access
based HetNets.
Index Terms—NOMA, HetNets, Stackelberg game, equivalent
channel gain, power allocation.
I. INTRODUCTION
NOn-orthogonal multiple access (NOMA) has attracted
significant attention for its superior spectral efficiency
by allowing two or more users to multiplex in the power
domain [1]. Meanwhile, heterogeneous networks (HetNets) are
capable of meeting the explosive growth of data consumption
by deploying small cells. Thus, the combination of NOMA
and HetNets is expected to possess great potential to further
enhance the spectrum utilization.
Recently, a few work has emerged to exploit the resource
allocation problem in NOMA HetNets. In [1], a resource
allocation design is proposed for NOMA HetNets to maximize
the α-fair utility of small cell users (SUs), where the spectrum
allocation is solved by matching theory [2], [3]. However,
in the proposed model, although each small cell base station
(SBS) communicates with SUs via NOMA protocol, the macro
cell base station (MBS) still applies the conventional OMA,
and the number of NOMA SUs sharing the same resource
block is limited to two. The authors in [4] study the NOMA-
enabled traffic offloading in HetNets, which aims to minimize
the total power consumption of the MBS and SBS with
minimum rate requirement, but the MBS and SBS are allocated
non-overlapping spectrum and thus the benefit of enhancing
the spectrum utilization in HetNets cannot be fully exploited.
In HetNets, centralized control is generally impractical due
to the privacy protection and security issues. Besides, central-
ized optimization often yields significant overhead and heavy
Manuscript received September 29, 2017; revised December 6, 2017;
accepted December 27, 2017. This work was supported by the China Post-
doctoral Science Foundation under Grant 2016M600911. The associate editor
coordinating the review of this letter and approving it for publication was O.
Popescu.
Z. Song and X. Sun are with the School of Electronic and Information
Engineering, Beijing Jiaotong University, Beijing 100044, China (email:
songzy@bjtu.edu.cn, xsun@bjtu.edu.cn).
Q. Ni is with the School of Computing and Communications, Lancaster
University, Lancaster LA1 4WA, U.K. (email: q.ni@lancaster.ac.uk).
computational burden of the central controller. Consequently,
different from the centralized algorithms in [1] and [4], in
this letter, assuming that every station has an objective to
maximize its own capacity, the resulting distributed power
allocation problem fits the natural framework of a Stackelberg
game, where MBS is the leader and SBS is the follower.
To our best knowledge, there has been no related work
studying the distributed power allocation in NOMA HetNets.
In our proposed model, to fully exploit the potential benefit of
HetNets, the MBS and SBS share the same spectrum, which
is superior to the model in [4]. Besides, unlike [1], both MBS
and SBS apply NOMA protocol, and arbitrary numbers of
NOMA users can be multiplexed on the same subcarrier. A
distributed power allocation algorithm is proposed to achieve
the Stackelberg equilibrium. Simulation results illustrate that
the combination of NOMA and HetNets is capable of improv-
ing the spectrum efficiency compared to the OMA HetNets,
and the proposed Stackelberg game-based algorithm leads to
a better performance than that based on Nash game.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. System Model
Consider a downlink two-tier HetNet where the MBS pro-
vides basic coverage and the SBS is overlaid to the MBS to
enhance capability. The whole bandwidth is divided into K
subcarriers and the MBS shares the same frequency with SBS.
NOMA protocol is applied to both MBS and SBS. Denote the
set of active users on subcarrier kserved by the MBS and SBS
as Sk
1and Sk
2, respectively. Define Mk=Sk
1and Nk=Sk
2,
where |·| represents the cardinality of Sk
1and Sk
2. The channel
gains from MBS to the active macro cell user (MU) mk∈ S k
1
and the active small cell user (SU) nk∈ Sk
2on subcarrier k
are hk,mk
1and gk,nk
1, while the channel gains from SBS to the
active MU mk∈ Sk
1and active SU nk∈ Sk
2on subcarrier k
are gk,mk
2and hk,nk
2, respectively.
For NOMA systems, each user should first decode the sig-
nals of users with worse channel conditions and successively
subtract them from the composite signal to obtain its own
information. In the proposed two-tier NOMA HetNet, as the
interference is the major limiting factor for SINR, it is an
interference-limited system. Therefore, we have the following
theorem to determine the decoding order for each user.
Theorem 1: For any two MUs i, j ∈ Sk
1, assume MU i
desires to decode and subtract the signal intended for MU j
via SIC. The condition that SIC can be performed successfully
is approximated as
hk,j
1.gk,j
2≤hk,i
1.gk,i
2.(1)
Proof: For MU i, the successful decoding can be guaranteed
in SIC if MU i’s received SINR for the signal of MU j
IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, XX 2017 2
is larger than or equal to the SINR of MU jfor its own
signal. Mathematically, the condition for MU ito successfully
perform SIC is
hk,i
1pk,j
1
hk,i
1pk,i
1+gk,i
2Pk
2+σ2≥hk,j
1pk,j
1
hk,j
1pk,i
1+gk,j
2Pk
2+σ2,(2)
where pk,i
1and pk,j
1are the power allocated to MUs iand j;
Pk
2is the total transmit power of SBS on subcarrier kand σ2
is the power of additive white Gaussian noise (AWGN). The
terms gk,i
2Pk
2and gk,j
2Pk
2represent the cross-tier interference
of MUs iand jcaused by the SBS on subcarrier k.
Then, (2) can be converted as
gk,i
2Pk
2+σ2.hk,i
1≤gk,j
2Pk
2+σ2.hk,j
1.(3)
Since the NOMA HetNets are interference-limited systems,
we have gk,i
2Pk
2+σ2≈gk,i
2Pk
2and gk,j
2Pk
2+σ2≈gk,j
2Pk
2.
Therefore, (3) is approximately equivalent to
hk,j
1.gk,j
2≤hk,i
1.gk,i
2.(4)
With (4), the proof is completed.
Remark 1: In Theorem 1, hk,j
1.gk,j
2and hk,i
1.gk,i
2are
called the equivalent channel gains of MUs iand j, which
stem from the cross-tier interference in HetNets. Theorem 1
implies that for MUs applying NOMA protocol, the decoding
order in SIC depends on not only the channel gain in macro
cell, but also the cross-tier interference link gain from the
micro cell. Note that similar conclusion on decoding order for
SUs can also be drawn because of the symmetry.
Without loss of generality, for the Mkactive MUs on
subcarrier k, we assume their equivalent channel gains follow
the order as
0< hk,1
1.gk,1
2≤hk,2
1.gk,2
2≤ · · · ≤ hk,Mk
1.gk,Mk
2.(5)
Based on NOMA protocol, the achievable data rate of MU mk
on subcarrier kcan be expressed as
rk,mk
1= log2
1 + hk,mk
1pk,mk
1
hk,mk
1
Mk
P
i=mk+1
pk,i
1+gk,mk
2
Nk
P
i=1
pk,i
2+σ2
,
(6)
where pk,mk
1and pk,i
2are the power allocated to the active mk-
th MU and the active i-th SU on subcarrier k, respectively.
Next, we adopt the transformations ak,mk
1=PMk
i=mkpk,i
1,
ak,nk
2=PNk
i=nkpk,i
2, and let ak,Mk+1
1= 0,ak,Nk+1
2= 0.
Then, (6) is transformed as
rk,mk
1= log2 hk,mk
1ak,mk
1+gk,mk
2ak,1
2+σ2
hk,mk
1ak,mk+1
1+gk,mk
2ak,1
2+σ2!.(7)
Accordingly, the throughput of MBS is given by
RMC =
K
X
k=1
Mk
X
mk=1
log2 hk,mk
1ak,mk
1+gk,mk
2ak,1
2+σ2
hk,mk
1ak,mk+1
1+gk,mk
2ak,1
2+σ2!.
(8)
Similarly, the throughput of SBS can be expressed as
RSC =
K
X
k=1
Nk
X
nk=1
log2 hk,nk
2ak,nk
2+gk,nk
1ak,1
1+σ2
hk,nk
2ak,nk+1
2+gk,nk
1ak,1
1+σ2!.
(9)
B. Problem Formulation
In the proposed NOMA HetNet, the MBS and SBS compete
to maximize their own throughput. Therefore, the power
allocation problem can be captured as a two-stage Stackelberg
game to characterize the competition, In the Stackelberg game,
the MBS is the leader which aims to maximize its throughput
under the power and MUs’ minimum rate requirement con-
straints, while the SBS is the follower with the purpose of
finding the best response to the decision of the MBS.
Denote Rk,mk
1as the minimum rate requirement of MU mk
on subcarrier k. According to (7), rk,mk
1≥Rk,mk
1is equivalent
to
hk,mk
1ak,mk
1+gk,mk
2ak,1
2+σ2
≥2Rk,mk
1hk,mk
1ak,mk+1
1+gk,mk
2ak,1
2+σ2.(10)
Hence, the optimization problem for the MBS can be
formulated as
max
p1≻0RMC,(11a)
s.t.
K
X
k=1
Mk
X
mk=1
pk,mk
1≤PMC
max,(10),(11b)
where p1={pk,mk
1}is the power allocation vector for the
MUs and PMC
max is the maximum transmit power of MBS.
Correspondingly, the minimum rate requirement of SU nk
on subcarrier kis equivalent to
hk,nk
2ak,nk
2+gk,nk
1ak,1
1+σ2
≥2Rk,nk
2hk,nk
2ak,nk+1
2+gk,nk
1ak,1
1+σ2,(12)
and therefore, the optimization problem for the SBS can be
formulated as
max
p2≻0RSC,(13a)
s.t.
K
X
k=1
Nk
X
nk=1
pk,nk
2≤PSC
max,(12),(13b)
where p2={pk,nk
2}denotes the power allocation vector for
the SUs and PSC
max is the maximum transmit power of SBS.
III. SOLUTION TO THE STACKELBERG GAME
The objective of the proposed Stackelberg game is to find
the Stackelberg equilibrium where both MBS and SBS have no
incentive to deviate. Since the Stackelberg game captures the
sequential dependence of the decisions in the two stages, we
employ the backward induction method to find the equilibrium.
A. Solution to the SBS
Since the SBS is the follower, the cross-tier interference
term gk,nk
1ak,1
1in (9) can be treated as a constant. Therefore,
problem (13) is actually a convex optimization problem of
p2[5]. We relax the constraint (13b) and the Lagrangian
IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, XX 2017 3
LSC =
K
P
k=1
Nk
P
nk=1
log2hk,nk
2ak,nk
2+gk,nk
1ak,1
1+σ2
hk,nk
2ak,nk+1
2+gk,nk
1ak,1
1+σ2+µ2PSC
max −
K
P
k=1
Nk
P
nk=1
pk,nk
2
+
K
P
k=1
Nk
P
nk=1
βk,nk
2hk,nk
2ak,nk
2+gk,nk
1ak,1
1+σ2−2Rk,nk
2hk,nk
2ak,nk+1
2+gk,nk
1ak,1
1+σ2,
(14)
function of problem (13) is obtained as (14), where µ2and
β2=nβk,nk
2oare Lagrangian multipliers associated with
constraint (13b). Then, by taking the derivation of (14) with
respect to pk,1
2and setting it to zero, we have
∂LSC
∂pk,1
2
=hk,1
2
hk,1
2ak,1
2+gk,1
1ak,1
1+σ2ln 2
−µ2+βk,1
2hk,1
2= 0.
(15)
Thus, the optimal transmit power of SBS on subcarrier kis
ak,1
2=1
µ2−βk,1
2hk,1
2ln 2
−gk,1
1
hk,1
2
ak,1
1−σ2
hk,1
2
,(16)
where 1
(µ2−βk,1
2hk,1
2)ln 2 is the water-filling level. µ2and βk,1
2
can be obtained by the iterative water-filling algorithm. With
(16), the optimal power allocation for each NOMA user on
subcarrier kcan be obtained by the method proposed in [5].
Remark 2: It can be observed from (16) that given the values
of Lagrangian multipliers µ2and βk,1
2, for any subcarrier k,
the optimal transmit power of the follower, ak,1
2, is a linear
function of the leader’s transmit power ak,1
1. With (16), we
next solve the optimization problem (11) for the MBS.
B. Solution to the MBS
As the leader, the MBS aims to find its best power allocation
strategy by considering the response of the follower on each
subcarrier. Hence, the cross-tier interference term gk,mk
2ak,1
2in
(8) cannot be treated as a constant. By substituting the linear
representation (16) into (8), the objective function in (11) can
be rewritten as
RMC =
K
X
k=1
Mk
X
mk=1
log2 hk,mk
1ak,mk
1+Ak
mkak,1
1+Bk
mk
hk,mk
1ak,mk+1
1+Ak
mkak,1
1+Bk
mk!,
(17)
where Ak
mk=−gk,mk
2gk,1
1
hk,1
2
,Bk
mk=gk,mk
2
(µ2−βk,1
2hk,1
2)ln 2 +
1−gk,mk
2
hk,1
2σ2.
The objective function (17) is non-convex with respect
to p1, and accordingly, problem (11) is a non-convex op-
timization problem. Here, we employ D.C. programming to
solve problem (11). D.C. programming transforms (17) as a
difference of two convex functions, and establishes successive
convex approximations to convert non-convex problem (11)
into convex subproblems [6]. Specifically, we define
F(p1) =
K
X
k=1
Mk
X
mk=1
log2hk,mk
1ak,mk
1+Ak
mkak,1
1+Bk
mk,
(18)
G(p1) =
K
X
k=1
Mk
X
mk=1
log2hk,mk
1ak,mk+1
1+Ak
mkak,1
1+Bk
mk.
(19)
Algorithm 1 D.C. Programming for Power Allocation of MBS
1. Initialize the power allocation vector p(0)
1and set the
iteration number l= 0.
2. while RMC p(l+1)
1−RMC p(l)
1> ε do
3. Define convex approximation of RMC (p1)at the
point p(l)
1as
b
R(l)
MC (p1) = F(p1)−Gp(l)
1
−∇GTp(l)
1p1−p(l)
1.
4. Solve the convex optimization problem
p(l+1)
1= arg max
p1≻0b
R(l)
MC (p1),s.t.(11b).(21)
5. l←l+ 1
6. end while
Then, optimization problem (11) can be rewritten as
max
p1≻0RMC (p1) = max
p1≻0F(p1)−G(p1),s.t.(11b).(20)
Apparently, F(p1)and G(p1)are both convex with respect
to p1, which enables the application of D.C. programming for
power allocation of MBS as presented in Algorithm 1, where
∇GTp(l)
1is the gradient of G(p1)at the point p(l)
1.
Theorem 2: Algorithm 1 monotonically increases the value
of RMC (p1)at each iteration and finally converges.
Proof: Denote p(l)
1as the optimal solution to b
R(l)
MC (p1)
after the l-th iteration of Algorithm 1. Then, we have the
following inequalities:
RMC p(l)
1=b
R(l)
MC p(l)
1≤b
R(l)
MC p(l+1)
1≤RMC p(l+1)
1.
(22)
The first inequality holds because p(l+1)
1is the optimal so-
lution to problem (21). The second inequality follows from
b
R(l)
MC (p1)≤RMC (p1),∀p1. As a result of (22), the
objective function RMC (p1)is increased after each iteration.
Since RMC (p1)is upper-bounded, the proposed Algorithm 1
must converge.
C. Algorithm for the Stackelberg Equilibrium
From the solutions to the MBS and SBS, it is observed
that any changes in the power allocation of MBS can cause a
change in the power allocation of SBS. Hence, an iterative
method is proposed to find the Stackelberg equilibrium as
given in Algorithm 2. For the two-player game model in
this letter, the Stackelberg equilibrium is unique. Note that
Algorithm 2 is a distributed power allocation algorithm, which
shifts part of the computational burden of the MBS to the
SBS. Besides, although perfect CSI is assumed available in
this letter, the proposed network model and game-theoretical
approach can be extended to the imperfect CSI case.
IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, XX 2017 4
Algorithm 2 Distributed Iterative Power Allocation for the
Stackelberg Equilibrium
1. Initialize the Lagrangian multipliers µ(0)
2and βk,1(0)
2
and set the iteration number t= 0.
2. while p(t+1)
1−p(t)
1> δ or p(t+1)
2−p(t)
2> δ do
3. Solve optimization problem (20) by Algorithm 1
to obtain the power allocation p(t+1)
1for the MBS.
4. Obtain the optimal power allocation p(t+1)
2and
the Lagrangian multipliers µ(t+1)
2and βk,1(t+1)
2
for the SBS by the iterative water-filling algorithm.
5. t←t+ 1
6. end while
TABLE I
SIMULATION PARAMETE RS
Parameters Values Parameters Values
Macro cell radius 1000 m PMC
max 46 dBm
Small cell radius 200 m PSC
max 26 dBm
Noise power -174 dBm/Hz K16
Path-loss exponent 3.8 Mk, Nk3
IV. SIMULATION RESULTS
In the section, we evaluate the performance of the proposed
distributed power allocation algorithm for NOMA HetNets by
simulations. The simulation parameters are listed in Table I.
Fig. 1 demonstrates the convergence evolution of the pro-
posed Algorithm 2 for achieving the Stackelberg equilibrium
based on one random sample. It can be seen that the iterative
algorithm converges fast. Thus, Algorithm 2 is cost-efficient
for the distributed power allocation in NOMA HetNets.
In Fig. 2, we compare our proposed distributed power
allocation algorithm with the following power control schemes
in HetNets: 1) distributed power allocation by Nash game in
spectrum sharing NOMA HetNets, where both MBS and SBS
water-fill the whole band by treating the cross-tier interference
as noise until the generalized Nash equilibrium is achieved [7];
2) distributed power allocation in NOMA HetNets with dedi-
cated spectrum allocation (DSA) where each tier is allocated
non-overlapping spectrum as presented in [4]; 3) distributed
power allocation in OMA HetNets [8].
As illustrated in Fig. 2, the proposed distributed power
allocation algorithm in NOMA HetNets with spectrum sharing
outperforms the one with dedicated spectrum allocation and
OMA HetNets in terms of system throughput, which verifies
the capability of the combination of NOMA and HetNets to
improve the spectrum efficiency. Also, the proposed algorithm
based on Stackelberg game is superior to the one by Nash
game. This is because in Stackelberg game, the MBS considers
the reaction of SBS when maximizing its throughput, while in
Nash game, both players treat the action of the other as fixed.
For all the schemes in Fig. 2, the throughput decreases as
each user’s minimum rate requirement increases. It is further
observed that when the minimum rate requirement is lower
(less than 0.25 bits/s/Hz in Fig. 2), the throughput of NOMA
HetNets almost remains the same with the increase of the
minimum rate requirement, while for the OMA HetNets, the
throughput continuously decreases. Such observations imply
that NOMA HetNets possess greater potential to satisfy the
0 5 10 15 20 25 30
Number of iterations
13
14
15
16
17
18
19
20
21
22
Individual throughput of MBS and SBS (bits/s/Hz)
32
33
34
35
36
37
38
39
40
41
System throughput of HetNets (bits/s/Hz)
Throughput of MBS
Throughput of SBS
Throughput of HetNets
Fig. 1. Convergence of the proposed distributed power allocation algorithm.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Minimum rate requirement (bits/s/Hz)
28
30
32
34
36
38
40
42
System throughput of HetNets (bits/s/Hz)
Proposed algorithm
NOMA HetNets by Nash game
NOMA HetNets with DSA
OMA HetNets
Fig. 2. System throughput comparisons for distributed power allocation.
minimum rate requirement of each user thanks to the wider
capacity region of NOMA compared to OMA.
V. CONCLUSION
In this letter, the power allocation problem in NOMA
HetNets has been studied. The throughput of MBS and SBS
was formulated based on the equivalent channel gain, and the
power allocation problem was captured as a Stackelberg game.
We have proposed a distributed power allocation algorithm. It
has been shown that the proposed algorithm converges fast,
and the combination of NOMA and HetNets is able to improve
the spectrum efficiency compared to the OMA based HetNets.
REFERENCES
[1] J. Zhao, Y. Liu, K. K. Chai et al., “Spectrum allocation and power
control for non-orthogonal multiple access in HetNets,” IEEE Trans.
Wirel. Commun., vol. 16, no. 9, pp. 5825–5837, Sep. 2017.
[2] L. Xu, C. Jiang, Y. Shen, T. Q. S. Quek et al., “Energy efficient D2D
communications with pricing mechanism and matching methodology,”
IEEE Trans. Wirel. Commun., vol. 15, no. 11, pp. 7272–7285, Nov. 2016.
[3] Y. Gu, W. Saad, M. Bennis, M. Debbah, and Z. Han, “Matching theory
for future wireless networks: Fundamentals and applications,” IEEE
Commun. Mag., vol. 53, no. 5, pp. 52–59, May 2015.
[4] Y. Wu and L. P. Qian, “Energy-efficient NOMA-enabled traffic offloading
via dual-connectivity in small-cell networks,” IEEE Commun. Lett.,
vol. 21, no. 7, pp. 1605–1608, Jul. 2017.
[5] Z. Yang, W. Xu, C. Pan, Y. Pan, and M. Chen, “On the optimality of
power allocation for NOMA downlinks with individual QoS constraints,”
IEEE Commun. Lett., vol. 21, no. 7, pp. 1649–1652, Jul. 2017.
[6] H. H. Kha, H. D. Tuan, and H. H. Nguyen, “Fast global optimal power
allocation in wireless networks by local D.C. programming,” IEEE Trans.
Wirel. Commun., vol. 11, no. 2, pp. 510–515, Feb. 2012.
[7] G. Bacci, E. V. Belmega, P. Mertikopoulos et al., “Energy-aware competi-
tive power allocation for heterogeneous networks under QoS constraints,”
IEEE Trans. Wirel. Commun., vol. 14, no. 9, pp. 4728–4742, Sep. 2015.
[8] S. Guruacharya, D. Niyato, D. I. Kim, and E. Hossain, “Hierarchical com-
petition for downlink power allocation in OFDMA femtocell networks,”
IEEE Trans. Wirel. Commun., vol. 12, no. 4, pp. 1543–1553, Apr. 2013.
