Power Allocation in Multi-Carrier Networks with Unicast and Multicast Services

Abstract

Since the provision of both unicast and multicast services is expected for the next generation of multi-carrier wireless systems, the allocation of resources for this combination of services is a relevant topic, which may have a significant impact on the system performance. The focus of this work lies on the allocation of power to the different downlink subchannels of a multi-carrier system containing both unicast and multicast users. The power allocation problem is analyzed considering on the one hand the maximization of the sum throughput and on the other hand the maximization of the minimum SNR. The solution of the former is presented, which depends on numerical optimization, and an algorithm similar to the waterfilling hypothesis testing is proposed for reducing the processing time, while for the latter a closed-form solution is demonstrated. A simplified power allocation algorithm based on multicast group quality criteria is also evaluated and shown to approximate the maximal achievable performance under certain circumstances.

Full text

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Power allocation in multi-carrier networks
with unicast and multicast services
Yuri C. B. Silva and Anja Klein
Darmstadt University of Technology, Communications Engineering Lab,
Merckstr. 25, 64283 Darmstadt, Germany, {y.silva, a.klein}@nt.tu-darmstadt.de
Abstract—Since the provision of both unicast and multicast
services is expected for the next generation of multi-carrier
wireless systems, the allocation of resources for this combination
of services is a relevant topic, which may have a significant impact
on the system performance. The focus of this work lies on the
allocation of power to the different downlink subchannels of a
multi-carrier system containing both unicast and multicast users.
The power allocation problem is analyzed considering on the one
hand the maximization of the sum throughput and on the other
hand the maximization of the minimum SNR. The solution of the
former is presented, which depends on numerical optimization,
and an algorithm similar to the waterfilling hypothesis testing
is proposed for reducing the processing time, while for the
latter a closed-form solution is demonstrated. A simplified power
allocation algorithm based on multicast group quality criteria is
also evaluated and shown to approximate the maximal achievable
performance under certain circumstances.
I. INTRODUCTION
The provision of both unicast and multicast services is
expected for the next generation of multi-carrier wireless
systems. The allocation of resources for this combination of
services is a relevant topic, which may have a significant
impact on the system performance. Multicast services, which
are specified for current 3GPP systems such as GSM/EDGE
and WCDMA networks [1], have the characteristic that the
same information has to be transmitted in the downlink to a
certain group of users. In order to improve resource efficiency,
the users of a multicast group which find themselves in
the same cell can share the same radio resources (Point-
to-Multipoint connection) [1]. In multi-carrier systems, such
as OFDMA, this means that the subchannel allocation is
done in a groupwise manner, i.e., the data transmitted on a
subchannel allocated to a multicast group is received by all
group members.
Some topics concerning the allocation of resources for mul-
ticast OFDM networks have been investigated in literature. In
[2], an adaptive modulation method is proposed for multicast
systems. In [3], a dynamic subchannel and bit allocation is
evaluated, and in [4], proportional fairness scheduling aspects
are introduced within the optimization problem.
The focus of this work lies specifically on the allocation of
power to the different subchannels of a multi-carrier system
containing both unicast and multicast users. Examples of
unicast-only power allocation algorithms can be found in [5-7].
Note that the subchannel allocation procedure is assumed to
have already been performed, since algorithms for solving this
problem have been previously proposed for both unicast [8]
and multicast users [3]. The power allocation problem does
not have a trivial solution, since the multicast users, which
share the same resources, are subject to different radio link
conditions. For this reason there are several possibilities for
performing the power allocation, e.g., according to the best or
worst user within each subchannel, or taking into account the
requirements of each individual user.
The solution and analysis of the power allocation problem
for this unicast/multicast context is the main contribution of
this paper. The proposed algorithms consider the following two
optimization criteria: the maximization of the sum throughput
and the maximization of the minimum signal-to-noise ratio
(SNR), both for a certain total transmit power constraint. It
is shown that the former can be solved through an algorithm
similar to the traditional waterfilling hypothesis testing [9, 10].
A less complex approach, which takes a group measure for
each subchannel, such as the best, worst, or average channel
quality, is evaluated as well. The latter optimization, on the
other hand, is fairness oriented, depending on the worst-user
performance of each subchannel.
The paper is organized as follows. In section II, the power
allocation problem is formulated and solved for the different
considered optimization criteria. Section III presents the nu-
merical results, which analyze the complexity and performance
of the algorithms. Finally, in section IV, the main conclusions
are drawn.
II. POWER ALLOCATION
A. System model
The system model corresponds to the downlink of a single
cell in a cellular multi-carrier system. There are K available
subchannels and N users within the cell. It is assumed
that the subchannel allocation has already been performed,
and therefore the information concerning which users are
associated to which subcarrier is available to the power
allocation algorithm. Additionally, error-tolerant hierarchical
multicast data is considered [11, 12], which means that the
perceived quality depends on the amount of correctly decoded
information. This assumption is required by the throughput
maximization algorithms, otherwise the low-quality multicast
users within a channel would eventually stall the traffic flow
due to subsequent restransmissions.
In this scenario, both unicast and multicast users are present
in the system and different multicast groups are supported.
The subchannel allocation matrix AN×K, with elements
1-4244-0353-7/07/$25.00 ©2007 IEEE
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Ai,j∈ {0,1}, determines which users are active within each
subchannel, where 0 and 1 correspond to the inactive and
active states, respectively. Since no intracell interference is
assumed, only users of the same multicast group may share
one subchannel, i.e., combinations of unicast/multicast or
multicast users of different groups are not allowed in the same
subchannel. Denoting by Kucand Kmcthe set of subchannel
indices assigned to unicast and multicast users, respectively, it
follows that:


The power allocation problem consists of determining the
power vector pK×1 = [p1... pK]T, which indicates the
amount of power pk allocated to each subchannel k. The
allocation can be done according to different optimization
criteria, such as the maximization of the throughput or the
maximization of the minimum SNR. The algorithms proposed
in the following subsections, which have different characteris-
tics with regard to their complexity, capacity, and fairness,
are namely: Sum Throughput Maximization (STM), Group
Criterion for Throughput Maximization (GCTM), and Fair
Power Allocation (FPA).





N ?
n=1An,k≥ 1
n=1An,k= 1
N ?
∀k ∈ Kuc,
∀k ∈ Kmc.
(1)
B. Sum throughput maximization (STM)
In this section, the STM algorithm is described and illus-
trated. This algorithm has the purpose of maximizing the total
throughput of the system, which is here defined as the sum of
the bitrate perceived by the individual users. The throughput
of user n associated to subcarrier k is denoted by Rn,k, and
if Gaussian signalling is assumed it can be written as:
Rn,k= log2(1 + pkGn,k),
(2)
where pk is the power allocated to subcarrier k and Gn,k is
an element of matrix GN×K, which corresponds to the path
gain Hn,k(including path-loss, fading and noise) conditioned
to the subchannel allocation, i.e., Gn,k= Hn,k·An,k. In order
to compose the matrix G, information on the channel gains
of the allocated subchannels is required, which is assumed to
be available at the transmitter.
The optimization problem can be expressed as:
popt= argmax
p


K
?
k=1
N
?
n=1
log2(1 + pkGn,k),
subject to:

pk≥ 0,
K ?
∀k ∈ K,
k=1
pk= PT.
(3)
where the first constraint avoids negative power levels, PT
is the total available power, and K denotes the set of all
subchannel indices k = 1,...,K.
The application of Lagrange optimization and the Karush-
Kuhn-Tucker (KKT) necessary conditions for optimality [13]
00.10.20.30.40.5
pk
0.6 0.70.8 0.91
2
3
4
5
6
7
8
9
10
11
12
µ
µ = a1
µ = a2
µ = a3
k = 1
k = 2
k = 3
Fig. 1. Sum throughput maximization for 3 subchannels and PT= 1.
results in the following system of equations:




where µ is a Lagrange multiplier. From (4), it follows that µ
is related to the power of each subchannel k according to:


A single level µ therefore determines the power of all
subchannels. It should be noted that it is not possible to
explicitly express pkas a function of µ in (5). However, (5b)
can be rewritten as the following polynomial in pk:

















pk≥ 0,
K ?
µ ≥
?
∀k ∈ K,
k=1
pk= PT,
N ?
µ −
n=1
Gn,k
1+pkGn,k,
∀k ∈ K,
?
pk
N ?
n=1
Gn,k
1+pkGn,k
= 0,
∀k ∈ K.
(4)







pk= 0
for
µ ≥
N ?
N ?
n=1Gn,k,
(5a)
µ =
N
?
n=1
Gn,k
1 + pkGn,k
for
µ <
n=1Gn,k.
(5b)
N
?
j=1
(pk+ G−1
j,k− Nµ−1)
N
?
i=1,i?=j
(pk+ G−1
i,k) = 0,
(6)
which has degree N and only one positive real root.
The problem now consists of finding an adequate µ such
that the resulting power vector satisfies the total power con-
straint. The optimal solution can be numerically calculated by
performing a one-dimensional search over µ [13].
In order to better illustrate the problem, Fig. 1 depicts µ as
a function of pkaccording to (5) for a system containing three
subchannels and PT= 1. This example represents a particular
system snapshot, which is characterized by the instantaneous
values of the path gain matrix G. Each curve corresponds to a
subchannel k and monotonically decreases with increasing pk.
For the considered power range, the dashed lines indicate the
maximum µ of each curve, which is achieved for pk= 0 and
is denoted by ak. From (5), it follows that ak=?N
n=1Gn,k.
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By analyzing the problem, it can be seen that a hypothesis
testing similar to that of the traditional waterfilling algorithm
[10] can also be done for this more general unicast/multicast
case, with the purpose of reducing the processing time of
the one-dimensional search for µ. The algorithm, which is
described below, assumes that for a given µ, each pk is
obtained by finding the real positive root of (6).
1) Assign the subchannel indices according to the increas-
ing order of akand set˜k = 1.
2) Set µ = a˜kand compute p˜k+1,...,pK.
K ?
otherwise set˜k =˜k + 1 and repeat step 2.
?a˜k−1,a˜k
Assume that a0= 0 for the case in which˜k = 1.
Set p1,...,p˜k−1to zero and compute p˜k,...,pK.
The algorithm does not eliminate the need for a numerical
method in order to calculate µ, but as it can be seen from step
3, it may benefit from a narrower search space and reduced
dimension (vector p with some zero elements), which may
result in significant gains in terms of processing time.
If
k=˜k+1
pk≤ PT, then proceed to step 3,
3) Find µ ∈
?
|
K ?
k=˜k
pk= PT.
C. Group criterion for throughput maximization (GCTM)
In this section, the GCTM algorithm is presented, which
also aims at the maximization of the sum throughput, but
corresponds to a simplification of the STM algorithm. It
assumes that the users of a multicast group do not have
their quality indicators (channel gains) taken into account
individually. Instead, for each subchannel, a single indicator
is considered for the whole group.
Let gkrepresent the group quality indicator for subchannel
k, then the optimization problem becomes:
popt= argmax
p


K
?
k=1
log2(1 + pkgk),
subject to:

pk≥ 0,
K ?
∀k ∈ K,
k=1
pk= PT,
(7)
which can be solved directly by the waterfilling algorithm.
The group indicator for each subchannel can be expressed
as a function of the previously defined gain G, i.e., gk =
f(Gk), where Gk is the kthcolumn of matrix G. The
functions considered in this work are the following: maximum
(GCTM-Max), minimum (GCTM-Min), and the arithmetic
mean (GCTM-Mean) of the elements of Gk. More details on
which of them are more adequate to better approximate the
solution of the STM algorithm are presented in section III-C.
D. Fair power allocation (FPA)
The algorithms considered so far have aimed at the maxi-
mization of the sum throughput, which is not a fair criterion in
terms of user performance, since the users may achieve bitrates
which largely differ from one another. In this section, the FPA
algorithm is described, which has the purpose of introducing
fairness within the power allocation procedure.
The optimization objective of the FPA algorithm is to
maximize the lowest Signal-to-Noise Ratio (SNR) within the
cell. Let the SNR perceived by user n on subchannel k be
defined as pkGn,k, then the optimization problem can be
written as:
popt= argmax
p
for
n = 1,...,N


where the min+operator is here assumed to return the mini-
mum non-zero element.
Since the allocated power does not depend on n, the
problem can be rewritten as follows:
min
{n,k}+(pkGn,k),
and
pk≥ 0,
K ?
k = 1,...,K ,
subject to:

∀k ∈ K,
k=1
pk= PT,
(8)
popt= argmax
p
min
k
(pkg?
k),
(9)
where g?
well as the same constraints of (8), are assumed.
This means that only the worst user within each subchannel
needs to be considered. The objective is that these worst users
in the different subchannels achieve the same SNR γ for the
optimal power vector popt, which implies that pkg?
all subchannels. Assuming that dK×1represents a vector with
elements dk= g?−1
PT= ?popt?1is the total power constraint in vector form, the
following system of equations can be established:
?
PT= γ ?d?1,
whose solution is given by:
k= min
n
+Gn,k and the same range of n and k, as
k= γ for
k, ?·?1denotes the 1-norm of a vector, and
popt= γ d,
(10)
popt= PT
d
?d?1
.
(11)
III. COMPLEXITY AND PERFORMANCE RESULTS
A. Network scenario
The system consists of a single cell serving a certain number
Ngof user groups. Among these groups there are Nugunicast
groups, each containing one user, and Nmgmulticast groups,
such that Ng= Nug+Nmg. For simplicity, it is assumed that
all multicast groups have the same size, which is denoted by
Lmg, only one subchannel is allocated to each group, and the
number of available subchannels is equal to the number of
user groups, i.e., K = Ng.
The users are uniformly distributed over one hexagonal
sector of a tri-sectorized cell and a single-antenna base station
is located at the sector corner. The considered propagation
effects include the distance-based path-loss attenuation (with
exponent α = 3.5), as well as uncorrelated Rayleigh fading,
which is modelled as circularly symmetric complex Gaussian
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2468101214 16 1820
2
4
6
8
10
12
14
16
18
20
All subchannels
Non−zero subchannels
Number K of allocated subchannels
Effective length of the power allocation vector for scenario S1, STM
algorithm, Nug = Nmg = K/2, and Lmg = 3.
Average number of subchannels
Fig. 2.
random variables with variance σ2. The path-loss is modelled
by assuming that the cell border is at a distance db= 1 from
the base station and that the fading variance of a user with
distance d ≤ dbis given by σ2= 1/dα[14]. Additive white
Gaussian noise is also assumed and the transmit power is
adjusted to provide an average SNR of 10dB at the cell border.
The co-channel interference from other cells is assumed to be
Gaussian and is treated as noise. The simulation results are
obtained considering 10,000 independent channel realizations.
A simple subchannel allocation (SSA) algorithm is imple-
mented, which approximates the maximization of the sum
throughput given an equal power distribution. The considered
algorithm iteratively allocates a subchannel to each user group
according to the highest average group channel gain. After an
allocation, the corresponding user group and channel are no
longer taken into account by the further steps. The procedure
is repeated until all user groups are allocated one channel.
The evaluation of the results considers two distinct scenar-
ios. The first one, denoted as scenario S1, represents a worst-
case situation in which the users have path gains of the same
order, with σ2= 1, and no specific subchannel allocation
algorithm is employed (random allocation). This scenario can
be interpreted as all users being at the same distance from
the base station or, equivalently, it can be assumed that path
gain compensation has been performed. Scenario S2, on the
other hand, takes into account the different path-loss of the
users, with σ2= 1/dα, as well as the previously described
SSA algorithm.
B. Complexity analysis of the STM algorithm
In this section, the complexity of the STM algorithm is
analyzed. The other algorithms are not considered, because
they either have a closed-form solution, in the case of FPA, or
their complexity is the same as that of traditional waterfilling,
in the case of GCTM.
It has been shown in section II-B that the allocation of
power based on sum throughput maximization can have its
processing effort reduced by employing an algorithm similar
2468101214 161820
0
0.05
0.1
0.15
0.2
0.25
Number K of allocated subchannels
Average ratio E{(a˜k− a˜k−1)/aK}
Fig. 3.
testing for scenario S1, STM algorithm, Nug = Nmg = K/2, and Lmg = 3.
Ratio between the search space range with and without hypothesis
to the traditional waterfilling, which consists of iteratively
testing the hypothesis that a certain subchannel be allocated
zero power. The advantage of this approach is the reduction of
both the power vector dimension and the range of the search
space, which results in decreased computational effort when
searching for µ, conform section II-B.
In the following, it is analyzed to which extent it is expected
that the effective power vector length, i.e., the number of non-
zero power elements within p, and the search space be reduced
when applying the hypothesis testing of section II-B. The
simulation scenario S1 is considered and among K allocated
subchannels the same number of unicast and multicast groups
is assumed, i.e., Nug = Nmg = K/2, with each multicast
group being composed of three users (Lmg= 3).
In Fig. 2, it can be seen that the absolute difference between
the total number K of subchannels and the number Knz of
non-zero subchannels increases for larger values of K. For a
small number of subchannels the difference is negligible, but
for an intermediate/large amount, the reduction of the effective
power vector length may lead to significant gains in terms of
processing effort.
The average ratio between the search space range for the
cases with and without hypothesis testing, which can be
defined as E{(a˜k−a˜k−1)/aK}, where E{·} is the expectation
operator, is shown in Fig. 3. The ratio rapidly decreases as a
few subchannels are added. For more than 10 subchannels it
can be seen that the hypothesis testing is capable of reducing
the search space to less than 5% of the total range.
C. Performance in terms of throughput and fairness
This section presents the performance analysis of the pro-
posed algorithms in terms of the achievable throughput as well
as the fairness among the users. First, the relative performance
among the sum throughput maximization algorithms, namely
STM and GCTM, is compared for different scenarios, then
the FPA algorithm is included and the absolute throughput
achieved by all algorithms is analyzed, and finally the algo-
rithms are compared in terms of the worst-user SNR.
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02468101214 161820
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
S1− GCTM−Min
S1− GCTM−Max
S1− GCTM−Mean
S2− GCTM−Min
S2− GCTM−Max
S2− GCTM−Mean
Multicast group size Lmg
Avg. sum throughput ratio between GCTM and STM
Fig. 4.
criteria, for scenarios S1 and S2, Nug = Nmg = 2, and K = 4.
Sum throughput ratio between GCTM and STM for different group
In section II-C, the GCTM algorithm has been presented as
an alternative for performing the sum throughput maximiza-
tion, which consists of assuming a single quality indicator for
each subchannel and applying the waterfilling algorithm.
The performance of GCTM is shown in Fig. 4 for the
simulation scenarios S1 and S2, with Nug = Nmg = 2 and
K = 4, and for some different f(Gk) functions, which are
namely: maximum (GCTM-Max), minimum (GCTM-Min),
and arithmetic mean (GCTM-Mean). The figure depicts the
average sum throughput ratio between the GCTM and STM
algorithms, i.e., E{RGCTM/RSTM}, as a function of the
multicast group size Lmg.
For scenario S1, it can be seen that GCTM-Max is the
algorithm which best approximates the performance of STM.
The performance gets worse for an increasing group size,
but is still close to 88% for Lmg = 20. The GCTM-
Min presents the worst result, while GCTM-Mean has an
intermediate performance. For scenario S2, better results are
achieved and both GCTM-Mean and GCTM-Max present
similar performance. This performance gain is explained by
the fact that scenario S2 implements the subchannel allocation
algorithm SSA, instead of random allocation, as well as the
different path-loss perceived by the users.
The cumulative distribution function (CDF) of the sum
throughput is shown in Fig. 5 for scenario S2 and a group
size of 10 users. Note that the high sum throughput values are
a result of the large amount of multicast users, which have
resource sharing capabilities. The relative behavior among the
GCTM and STM curves with regard to Fig. 4 is maintained,
being GCTM-Max and GCTM-Mean the ones which better
approximate the STM algorithm. The GCTM-Min is a rather
inadequate criterion for GCTM, which is explained due to the
fact that the waterfilling algorithm may happen to allocate low
power to a multicast subchannel, since the power is adjusted
according to the worst user, even if there are other users with
very good channel gains which would significantly contribute
to increase the sum throughput.
In order to compare the degree of fairness of the different
708090 100110120 130140150160170
0
10
20
30
40
50
60
70
80
90
100
GCTM− Min
GCTM− Max
GCTM− Mean
STM
FPA
Sum throughput in bit/s
CDF in %
Fig. 5.
S2, Nug = Nmg = 2, K = 4, and Lmg = 10.
CDF of the sum throughput of the different algorithms for scenario
246810 1214 161820
−10
−8
−6
−4
−2
0
2
4
6
8
10
GCTM−Min
GCTM−Max
GCTM−Mean
STM
FPA
Multicast group size Lmg
Average worst-user SNR in dB
Fig. 6. Comparison of the different algorithms in terms of the average worst-
user SNR for scenario S2, Nug = Nmg = 2, and K = 4.
algorithms, the measure of the worst-user SNR is employed,
which corresponds to the lowest SNR perceived among all
users in all subchannels.
In Fig. 6, the average worst-user SNR is depicted as a
function of the multicast group size for the different power
allocation methods. The FPA algorithm presents the best
performance in terms of fairness, as already expected, and it
presents a gain of roughly 5dB with regard to the GCTM-Max
algorithm, which is maintained throughout the whole group
size range. The GCTM-Mean curve practically overlaps the
GCTM-Max, and the GCTM-Min is again the one with the
worst performance. Additionally, it is seen that the GCTM-
Mean/Max algorithms have performance similar to STM, but
for Lmg≥ 12, STM becomes worse in terms of fairness. It is
important to note that the fairness advantage of FPA comes at
the cost of reduced sum throughput when compared to GCTM
and STM, which can be seen from Fig. 5.
Fairness is an important aspect to be taken into account,
especially for users of multicast services. In the case of the
considered error-tolerant hierarchical multicast, it is probably
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.