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An Improved Proportional Fair Scheduling in
Downlink Non-Orthogonal Multiple Access System
Eiji Okamoto
Department of Computer Science and Engineering, Graduate School of Engineering,
Nagoya Institute of Technology
Gokiso-cho, Showa-ku, Nagoya, Aichi 466-8555, Japan.
Email: okamoto@nitech.ac.jp
Abstract—For the next-generation mobile communication system,
enlarging the system capacity by allowing the use of non-
orthogonal user multiplexing has been suggested. In a downlink
multicarrier non-orthogonal multiple access (NOMA) system,
single or multiple users are allocated to each subband. In NOMA
scheduling, a proportional fair (PF) algorithm, where users with
few allocated resources are selected on a priority basis, is
generally used. However, to reduce the calculation complexity,
only the fairness up to the previous transmitted frames is taken
into account. Therefore, to improve the performance, we propose
an improved downlink NOMA scheduling scheme that takes the
fairness of the target frame into consideration, and orders the
allocation subband based on the instantaneous fairness. Through
this scheme, a user having little resource allocation may be
assigned a good subband. Numerical results show an improved
performance in the average user throughput and fairness.
Keywords—non-orthogonal multiple access; scheduling;
proportional fair; MIMO; downlink.
I. INTRODUCTION
Along with the continuous demand for higher capacity
cellular systems, new demand for accommodating the wireless
access of Internet of things (IoT) devices and advanced
driving assistance systems is growing in wireless access
systems. As a result, increasing capacity is still needed. In the
3.9th- and fourth-generation mobile communication systems,
multicarrier transmission using orthogonal frequency division
multiple access (OFDMA) has been adopted and an enhanced
capacity has been obtained through a multiuser diversity effect
[1, 2]. For a next-generation mobile communication system,
the development of an advanced scheduling scheme is
currently being studied to further enhance the capacity, and
non-orthogonal multiple access (NOMA) is one of the strong
candidates for such realization [3-6]. In a multicarrier
orthogonal multiple access (OMA) scheme such as OFDMA,
one user is allocated to one subband, and no interference
among users occurs. In contrast, in the NOMA scheme, user
interference is allowed, and advanced scheduling in which
multiple users can be allocated into one subband is applied.
On such an occasion, users having different channel
coefficients, for example, users near and far from a base
station (BS), are selected, and the transmit power is positively
biased for a distant user signal. Then, in the receiver of a
nearby user, the interference is suppressed through successive
interference cancellation (SIC) [7]. That is, a nearby user can
correctly decode a biased distant-user signal such that it is
removed through SIC, after which the signal of the nearby
user is decoded. In the receiver of the distant user, because the
overlapped signal of the nearby user is sufficiently attenuated,
only the decoding of the distant-user signal is conducted. The
NOMA scheme is known to enhance the system capacity more
than the OMA scheme.
In downlink NOMA systems, proportional fair (PF)
scheduling is usually conducted in a BS, where one or more
users are allocated to each subband by considering the fairness
among users. When the maximizing objective function is set
to the average user throughput, which is equivalent to the
summation of the user capacity within the system, it becomes
a combination optimization problem in terms of user pair,
transmit power, and subband allocation order, which requires
an extremely large combination search. Therefore, in a
conventional NOMA scheme, among the optimization
elements, the fairness index is calculated and fixed prior to
allocation, and the calculation complexity is reduced [4-6]. In
particular, for fairness-aware scheduling and to realize a fair
allocation, a subband is allocated on a priority basis to a user
having a small number of allocated subbands. However, to
decrease the calculation complexity during scheduling, the
user fairness is fixed using up to the previous allocation
frames, and a more efficient PF scheduling will be composed
if the user fairness is updated during scheduling.
Therefore, in this paper, we propose an improved PF
scheduling scheme for a downlink NOMA, in which the
fairness index is updated during the target frame allocation,
and a better subband can be allocated to a priority user through
adaptive subband ordering. Using this scheme, more efficient
scheduling is conducted, and both the average user throughput
and the fairness are improved simultaneously. The main
contributions of this paper are as follows.
- In a suboptimal PF NOMA scheduling scheme with a
predefined power-allocation, the fairness is increased using a
sequential update of the fairness index during the target frame
allocation process.
- Coordinating the order of subband allocation, a better
subband is allocated to a priority user having a small
allocation, and a decrease in capacity caused by the fairness
reinforcement can be avoided.
The remainder of this paper is organized as follows. In
section II, the system model of a downlink NOMA and the
proposed scheduling scheme are both described. In section III,
the numerical results are illustrated, and it is shown that the
proposed scheme can improve both the average user capacity
and the fairness. Finally, some concluding remarks are
provided in section IV.
978-1-4799-8091-8/15/$31.00 ©2015 IEEE
User equipment (UE)
Base station (BS)
UE1
f
UE1
UE2
UE3
UE4
UE
K
Number of maximum
user multiplexing
m
s
= 2
UE
K
UE3
UE4 UE2
powe r
subba nd 1 subba nd 2 subba nd 3
Fig. 1. System model of downlink NOMA.
II. SYSTEM MODEL AND SCHEDULING SCHEME
Fig. 1 shows the outline of the downlink NOMA system. We
consider the scheduling of a cellular downlink from a base
station (BS) to the user equipment (UE). The channel state
information (CSI) of the users within the target cell is informed
to the BS through feedback links, and based on the CSI, the BS
allocates the subbands to the users allowing interfered-user
multiplexing in a subband with considering the user fairness.
The objective is to maximize the average user throughput when
all subbands are allocated within the PF allocation constraint.
In the example shown in Fig. 1, the superposition allocation is
conducted in subbands 1 and 3 for UE1 and UE4, and UE2 and
UE3, respectively.
It is assumed that the number of transmit antennas at the BS
and the number of receive antennas at the UE are t
N and r
N,
respectively, and thus a multiple-input multiple-output
(MIMO) channel is used. In the target cell, there are K users,
and S subbands with a bandwidth of B each are allocated to the
users. Here, it is also assumed that the channel coefficient is
constant in a subband. If one subband consists of several
subcarriers, the average channel can be used for the subband.
In the allocation, a single-user allocation (OMA) or a non-
orthogonal multiple user allocation with s
m users at maximum
(NOMA) is conducted. At subband
s
( Ss ≤≤1), a set of
users )}(,),2(),1({ sssss miiiU "= from K users are selected,
where )(lis is the index of the l-th ( s
ml ≤≤1 ) superimposed
user. Here, the formulas are represented based on [6].
The t
N-dimensional transmit vector s
xof user )(lisat each
subcarrier in subband
s
in a BS is represented by
()()
)()(
1lilip ss
m
lsss
sdx ∑=
=, (1)
where
()
)(lip ss is the transmit power and
()
)(liss
d is an t
N-
symbol modulated vector with an average power of 1. The r
N-
dimensional receive vector in the UE is given by
() () ()
)()()( lilili sssssss wxHy += , (2)
where
()
)(liss
H is the ),( tr NN channel matrix between the
BS and user )(lis, and
()
)(liss
w is the noise plus inter-cell
interference vector. In the UE, the minimum mean square error
(MMSE) equalization is conducted with a weight of
() ()()() ()
[]
{}
1
)()()()()( −
+= r
Nssrs
H
ssss
H
sss IliNNlililili HHHW
(3)
Here,
()
)(liN ss is the average power of
()
)(liss
w and r
N
I is
an r
N-dimensional unit matrix. It is assumed that the total
transmit power at each subcarrier in subband s is fixed as
()
∑==
s
m
lss Plip
1)(. (4)
UE1
UE4
power
Near user: UE1 Far user: UE4
UE4
deco ding
UE1
decoding
-
+data
UE4
deco ding da ta
f
SIC deco ding direc t decoding
subband 1
Fig. 2. Receiver design with successive interference cancellation
for NOMA.
The channel power of user )(lisat each subcarrier in subband s
is given by
() ()
∑=
=min
1,)()( N
isisss liliG
λ
, (5)
where
()
rt NNN ,min
min = and
()
)(
,lisis
λ
is the i-th eigenvalue
of
()
)(liss
H. Non-orthogonal subband allocation is then
conducted according to the channel gain,
()()
)()( liNliG ssss .
Here, as shown in Fig. 2, it is assumed that using SIC at the
receiver UE, the user having a larger channel gain can correctly
remove the superimposed signals of s
m users having a smaller
channel gain at each subcarrier of subband s. In the example
shown in Fig. 2, for the decoding of subband 1 overlapped by
the UE1 and UE4 signals, UE1 close to the BS conducts SIC in
which the UE4 signal with a large power reception is first
decoded and eliminated from the received signal, and the
decoding of UE1 signal is then conducted. UE4 far from the
BS simply conducts the UE4 decoding because the overlapped
UE1 signal is sufficiently attenuated through a long-distance
propagation. Here, it is assumed that UE1 can correctly decode
and eliminate the UE4 signal. The capacity of user )(lis at
subband s is then given by [6]
()
()()
()() ()
()
()
()
()
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+
+= ∑⎥
⎦
⎤
⎢
⎣
⎡<∈ jN
jG
liN
liG
Uj sssss
ssss
sss
s
s
ss
ss
s
liNjpliG
lipliG
B
UliR
)(
)(
,
2)()(
)()(
1log
|)(
(6)
and the user selection is conducted based on the PF scheduling
metric of
()
∏
∈
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+=
Uk c
s
U
stkTt
tkR
U);()1(
;
1maxarg (7)
where );( tkT is the average user throughput of user k at time
index t, );( tkRs is the capacity of user k at subband s and time
t, which is zero when user k is not allocated at s, and c
t is the
throughput averaging parameter [4]. The number of user
candidate sets in s
U becomes
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
s
Um
K
KK
Ns
"
21 . (8)
Equation (7) becomes a combinational optimization problem
with the parameters of user K in (8), the order of already
allocated subband s at time t, and the power allocation
()
kps,
which is generally large in scale.
A. Conventional scheduling scheme
Because a high complexity is needed to obtain optimized
scheduling results, to reduce the number of combination
searches in a conventional scheme,
()
kps and );( tkT are
fixed in advance as
() () ()
()
()
()
FTPC
FTPC
α
α
−
∈
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=∑jN
jG
jNjG
P
kp
s
s
Uj ss
s
s
(9)
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−=+ ∑=
S
ss
cc
tkR
St
tkT
t
tkT 1);(
11
);(
1
1)1;( , (10)
and are eliminated as a parameter of (7). Here, FTPC
α
is the
power decay factor. By making the power and fairness in (9)
and (10) fixed, the search of (7) becomes independent on the
order of subband s, and the allocation can be conducted in
ascending order of Ss ≤≤1.
B. Proposed scheduling scheme
To improve the performance of the average user throughput
and fairness, herein we propose a new scheme allowing a slight
increase in the calculation complexity. In particular, a
successive update of );( tkT is used. The scheduling metric of
(7) is changed into
()
()
∏∑
∈=⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡+−
+=
Uk
s
isc
s
U
s
tkRtkTt
tkR
U
i
1;);()1(
;
1maxarg , (11)
where the allocation history at target time t is taken into
account, and the previously allocated capacity is included in
the metric. Here, i
s is the subband index during the i-th
subband allocation. Thus, the optimal allocation is clearly
dependent on the subband allocation order i
s, and the
calculation complexity is significantly increased the same as
through (7). Therefore, to reduce the calculation complexity,
we adopt the fixed power allocation of (9) and use a
suboptimal algorithm to conduct (11) as follows.
Proposed suboptimal algorithm:
1) Initialization
);( tkT is calculated using (10), and
()
kps is calculated by (9)
for all s. Let 1=s.
2) User selection
User )(lis having the smallest throughput of
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−= ∑=
s
is
cc
tkR
t
tkT
t
tkT i
1);(
1
);(
1
1);(' (12)
is selected from K users.
3) Subband selection for user )(lis
The subband i
s of
()
)(maxarg min lis sN
s
i
λ
= is selected.
4) NOMA or OMA user allocation at i
s
For subband i
s, the user(s) are allocated based on (11). Let
1+→ ss and return to 2). If all S subbands are allocated, the
p
rocess is finished.
First, in 1), the initial average throughput of all users is
calculated and the power allocation is conducted in advance.
Next, in 2), user )(lis having the minimum average throughput
is selected, and subband i
s, in which user )(lis can increase
the throughput at maximum, is set as the subband to be
allocated in 3). Then, for i
s, based on the NOMA scheduling
of (11), the search of user set s
U is conducted for all candidate
combinations. By selecting i
s, a large improvement in
throughput will be obtained for user )(lis when selected in
(11). This algorithm does not guarantee that user )(lis will
always be selected in subband i
s. However, when this user is
selected, their throughput is increased the most compared with
the other subbands, and the average user throughput and
fairness of the system can be improved.
The drawback of the proposed scheme is an increase in the
calculation complexity. An update of (12) is needed for every
subband allocation, and this iterative update increases the
calculation complexity compared with a conventional scheme.
However, the complexity of (11) is almost the same as that of
(7), and the increase in the total calculation complexity is
within the acceptable range.
III. NUMERICAL RESULTS
The performance of the proposed scheme was evaluated
through numerical simulations. For the performance criteria,
the user capacity and Jain’s fairness index (JFI) [8] were
calculated. JFI is the fairness among users in terms of the
allocated resources, as defined by
∑∑
== ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
K
k
k
K
k
kCKCJFI
1
2
2
1, ∑=
=S
ssk tkRC 1);( , (13)
TABLE I. SIMULATION PARAMETE RS
Cell layout No
n
-sectorizedhexagonal
19-cell model
Cell radius 500
m
Freq. reuse facto
r
1
Frequency band 2 GHz
No. of Tx and Rx antennas (
N
t
,
N
r
) = (2,2
)
No. of use
r
s
/
cell
K
1
6
Max. user multiplexing
m
s
1, 2,
3
Power decay factor
FTPC
α
0.4 [6]
No. of subbands
S
= 1024
FFT siz
e
1024
S
ubband spacing 15 KHz
Channel 16 path 1 dB decay,
quasi-static Rayleigh
Path loss exponent
3.5
Standard deviation
of shadowing loss 7 dB
Channel estimatio
n
ideal
Scheduling algorith
m
proportional fai
r
No. of simulation iteratio
n
s
of user distribution 200
No. of simulation iteratio
n
s
per user distribution 100
Throughput avera
g
ing
factor,
t
c
20
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10203040
Cumulative distribution function
User throughput Mbps
Proposed
Conventional
Proposed w/o subband ordering
Fig. 3. CDF of user capacity for NOMA at ms = 2.
where Ck is the total channel capacity of user k. JFI is between
zero and 1, and is closer to 1 when the difference in the
capacity of each user channel is small and fairness is achieved.
Table I shows the simulation parameters. A non-sectorized
hexagonal 19-cell model is used, and the users in the center
target cell receive an interference signal from the adjacent 18-
cell BSs. It is assumed that the CSI and the interference
channel coefficients of all users from the adjacent BSs are
perfectly known to the target BS. The number of subbands S
and the size of fast Fourier transform (FFT) are both 1,024,
which indicates that one subband is identical to one subcarrier.
In addition, FTPC
α
is set to 0.4 [6], and the number of maximum
user multiplexes at each subband s
m is one (OMA) to three
(NOMA). In the simulation, the users were randomly
distributed within the target cell, and the capacity and JFI were
calculated based on 100 time channel generations. The user
distribution was then iteratively changed 200 times.
Fig. 3 shows the cumulative distribution function (CDF) of
the average user throughout (capacity) at 2=
s
m. For
comparison, the performance of the conventional scheme
using (7) through (10), and that of the proposed scheme
without subband ordering, in which only (11) is used and the
subbands are allocated in ascending order, are plotted. The
proposed scheme was shown to have a better performance at
5% (0.05) and 50% (0.5) user throughput for the conventional
scheme. This is because the probability of users with a small
resource allocation is decreased using (11). However, as a
tradeoff, the user probability for a throughput of over 7 Mbps
decreases at 60% of the CDF. In comparison with the
proposed method without subband ordering (proposed w/o
subband ordering in Fig. 4), a slight improvement in
performance is obtained through adaptive subband ordering.
This is because a large increase in throughput is obtained for
the user having the minimum average throughput from the
subband ordering during the allocation. However, in the
simulation, the selection probability of user )(lis of 2) for the
proposed algorithm is about 6.5% for 4), which is not large.
On the other hand, the probability of the proposed scheme
without ordering is 8.7%. Therefore, it can be concluded that
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Cumulative distribution function
Jain's fairness index (JFI)
Proposed
Conventional
Proposed w/o subband ordering
Fig. 4. CDF of Jain’s fairness index at ms = 2.
the intended combination of user )(lis and subband i
s is not
always selected for the proposed scheme, but its effect is large
when selected, and the total average user throughput is
improved.
Next, the CDF of JFI was calculated during the same
simulation. The results of Fig. 4 show that around a 0.071 point
improvement is obtained at a 50% CDF for the proposed
scheme as compared with a conventional scheme, which is a
large improvement in fairness. This effect is achieved by (11).
Compared with the proposed scheme without ordering, almost
the same performance is obtained in fairness. Hence, the
proposed scheme improves both the average user throughput
and the fairness, the latter of which is largely increased.
The number of maximum user multiplexes s
m was then
changed to 1=
s
m through 3, and the performances were
confirmed. Fig. 5 shows the average user throughput versus
s
m at a value of 50% CDF. In all schemes, the throughput was
shown to increase according to an increase in s
m, that is, an
increase in non-orthogonal user multiplexing. Among them, the
proposed scheme has the highest throughput. It can be stated
0.0
1.0
2.0
3.0
4.0
5.0
6.0
123
User throughput Mbps
No. of multiplexing users u_max
Proposed
Conventional
Proposed w/o subband ordering
CDF=50%
Fig. 5. User throughput versus s
m at 50% CDF.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
123
Jain's fairness index (JFI)
No. of multiplexing users u_max
Proposed
Conventional
Proposed w/o subband ordering
CDF=50%
Fig. 6. Jain’s fairness index versus s
m at 50% CDF.
the proposed algorithm effectively works for an average user
with a 50% CDF.
Fig. 6 shows the JFI versus s
m at a value of 50% CDF.
Although a slight improvement was obtained in the
conventional scheme for an increase in s
m, in the proposed
scheme with or without subband ordering, JFI was decreased as
s
m increased. This means that, for the non-orthogonal
allocation, the proposed algorithm contributes more to an
enhancement in the average throughput than to an
improvement in fairness. Comparing the results of Figs. 5 and
6, it can be concluded that the one-user overlapping of 2=
s
m
is the best and balanced configuration in the proposed scheme
in terms of the improvements in the average user throughput
and fairness.
IV. CONCLUSIONS
We proposed an improved PF scheduling scheme in a
downlink multicarrier NOMA system, in which both the
average user throughput and the fairness were improved. In a
conventional scheme, the fairness of the previous frames is
taken into account to reduce the calculation complexity during
scheduling. In contrast, for the proposed scheme, the fairness
index is updated during the target frame scheduling, and the
subband ordering is conducted to select a better channel for a
priority user having minimum throughput. The numerical
results show that the proposed scheme improves both the
average user throughput and the fairness simultaneously. In
particular, because the fairness is largely improved compared
with the conventional scheme, the proposed scheme was shown
to be effective for a throughput enhancement of the cell-edge
users.
In future studies, a performance evaluation that takes the
user mobility into account, and a link-level simulation beyond
a capacity evaluation, will be conducted. An adaptive
transmission power allocation scheme of
()
kps will also be
considered.
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