1
Hybrid Heuristic-Waterfilling Game Theory
Approach in MC-CDMA Resource Allocation
Lucas Dias H. Sampaio, Taufik Abrão, Bruno A. Angélico, Moisés Fernando Lima, Mario Lemes Proença Jr., and
Paul Jean E. Jeszensky
Abstract—This paper discusses the power allocation with fixed
rate constraint problem in multi-carrier code division multiple
access (MC-CDMA) networks, that has been solved through
game theoretic perspective by the use of an iterative waterfilling algorithm (IWFA). The problem is analyzed under various
interference density configurations, and its reliability is studied in
terms of solution existence and uniqueness. Moreover, numerical
results reveal the approach shortcoming, thus a new method
combining swarm intelligence and IWFA is proposed to make
practicable the use of game theoretic approaches in realistic
MC-CDMA systems scenarios. The contribution of this paper
is twofold: i) provide a complete analysis for the existence and
uniqueness of the game solution, from simple to more realist
and complex interference scenarios; ii) propose a hybrid power
allocation optimization method combining swarm intelligence,
game theory and IWFA. To corroborate the effectiveness of
the proposed method, an outage probability analysis in realistic
interference scenarios, and a complexity comparison with the
classical IWFA are presented.
Index Terms—Power-rate allocation control; SISO multi-rate
MC-CDMA; Game Theory; iterative water-filling algorithm;
QoS.
I. I NTRODUCTION
In the last years the telecommunications scenario has been
passing through a huge increase in traffic demand due to
the arrival of new devices and services. In this context, the
multiple access networks represent an important solution,
once these systems can admit more users and, at the same
time, achieve a higher throughput than other technologies.
Thus, a specific multiple access system draws the attention of
many researchers nowadays: the MC-CDMA networks. Even
with all these avails, since all users transmit at the same
time and in the same spectrum a resource allocation scheme
must be adopted in order to guarantee acceptable quality of
service (QoS) requirements, associated with minimum rates,
maximum allowed delay, maximum permitted bit error rate
(BER) and so forth.
A. Motivation
The study of resource allocation problems in wireless networks have been discussed for many years due to its impacts in
This work was supported in part by the National Council for Scientific and
Technological Development (CNPq) of Brazil under Grant 303426/2009-8.
L. H. Sampaio, Taufik Abrão, Moisés Fernando Lima, and Mario Lemes
Proença Jr. are with Computer Science Department, State University of
Londrina, PR, 86051-990, Brazil. E-mails: [email protected];
[email protected]; [email protected]; [email protected].
Bruno A. Angélico is with Federal Technological University of Paraná. PR,
Brazil. E-mail: [email protected]
Paul Jean E. Jeszensky is with Escola Politécnica of University of São
Paulo. SP, Brazil. E-mail: [email protected]
companies profits and users satisfaction. Additionally, current
technologies do not provide enough bandwidth at low operational costs for corporations which also affects their customers.
So, a practical resource allocation scheme is desirable in order
to save power1 , increase the throughput and guarantee the QoS.
B. Related Work
Several studies have been conducted in recent years in
order to find good resource allocation algorithms. Within this
context, some works may be highlighted [1]–[9].
The distributed power control algorithm (DPCA) proposed
in [1] is considered the base of many well known DPCAs. In
[2] a multi-objective resource allocation scheme is presented.
The algorithm considers three different non-linear parameters
that weight the procedure goals: minimize the power, guarantee the QoS (in terms of target rate) and maximize the rate. In
addition, a DPCA for single-rate [3] and multi-rate networks
[4] inspired on Verhulst equilibrium analytical-iterative model
is proposed in order to solve the power allocation with rate
constraints problem in DS/CDMA networks.
On the other hand, heuristic approach based on swarm
intelligence was applied to solve the power allocation with
rate constraints [5] and the rate maximization problem [6].
Besides, the total network power minimization problem subject to multi-class information rate constraints, as well as the
problem of throughput maximization constrained to power
limitation were analyzed in [7] applying swarm intelligence.
The motivation to use heuristic search algorithms is due to the
nature of the NP complexity posed by the wireless network
optimization problems. The challenge is to obtain suitable
performances in solving those hard complexity problem in a
polynomial time.
Previous results indicated that the application of heuristic
search algorithm in several wireless optimization problems
have been achieved excellent performance-complexity tradeoffs, particularly the use of genetic algorithm, evolutionary
program, particle swarm optimization (PSO), and local search
algorithm. Concerning the resource allocation issue, there
are several challenging single- or multi-objective optimization
problems associated, such as the total network power minimization subject to multi-class information rate constraints,
as well as the throughput maximization while minimizing
the total transmitted power. Multi-rate users associated with
different types of traffic can be aggregated to distinct classes
of users, with the assurance of minimum target rate allocation
per user and quality of service (QoS). In order to achieve
promising performance-complexity tradeoffs, both continuous
1 Saving
power may increase battery lifetime.
2
or discrete PSO search algorithms have been successfully
employed in the resource allocation problems [7].
A game theoretic approach for power control with rate
constraints in Gaussian parallel interference channels using
water-filling algorithm is proposed in [8], while in [9] a multilinear fractional programming approach is used to solve the
weighted throughput maximization problem in multiple access
systems. However, under strong interference density configurations, iterative water-filling algorithm (IWFA) is unable
to offer promising solutions to power allocation with fixed
rate constraint in multi-carrier code division multiple access
(MC-CDMA) networks. Hence, in this work the existence and
uniqueness of the solution are studied and shown by numerical
results that a new method combining swarm intelligence and
IWFA is more promising and suitable for realistic MC-CDMA
scenarios.
C. Organization
This paper is organized as follows: Section II presents the
system model and description; the game theoretic approach is
presented in Section III. Section III-C discusses the iterative
water-filling algorithm (IWFA); moreover, in Section IV the
scenarios are characterized and further the game theoretic plus
IWFA approach are applied in orde to solve the power-rate
allocation problem. Finally, the proposed hybrid approach is
discussed in Section V and conclusions in Section VI.
where U is a total number of users sharing the channel, N
is the number of sub-carriers available, and ri∗ is the target
information rate at the ith user.
III. G AME T HEORY A PPROACH
A simple game G can be easily defined as a tuple composed by the set of players U, strategies P, and utilities F.
Mathematically:
G = {U, P, F}
In multiple access networks resource allocation games, the
players would be the active users in the system, U in (3),
the strategies would be the resources allocated to each one,
e.g. the power each user utilizes to transmit, P in (3), and
the utilities the payoff functions, F in (3), that evaluate the
strategies chosen.
Games may be played either with or without cooperation
among users. Herein a non-cooperative scenario is considered.
Hence, as it is well known, non-cooperative games may be
solved finding the Nash equilibrium(s) (NE) of the problem
[8], [10]. A NE is a set of strategies where any unilateral
change in the user strategy will not increase the user’s utility
without decreasing others payoffs. Therefore, problem (2) may
be rewritten as follows [10]:
min
II. S YSTEM D ESCRIPTION
(3)
U ∑
N
∑
pi (k)
i=1 k=1
In multiple access networks an important QoS measure is
the signal to interference plus noise ratio (SINR) since all
users transmit over the same channel at the same time causing
what is known as multiple access interference (MAI), which
is responsible for the soft capacity of CDMA systems. The
spectrum may be divided in N uncorrelated CDMA subchannels such that each one is a flat channel, therefore, more
appropriate to high transmission rates.
In MC-CDMA systems the SINR at the ith user kth subcarrier may be computed as follows [9]:
s.t.
ri (pi ) ∈ Ri∗
pi ∈ P
(4)
where Ri∗ is the set of possible achievable information rates,
P is the set of possible strategies (allocated powers) at each
sub-carrier and ri (pi ) is the rate function given the set of
power through all sub-carriers, defined as:
ri (pi ) =
N
∑
log [1 + δi (k)]
(5)
k=1
δi (k) =
ρ
∑
pi (k)|gii (k)|2
pj (k)|gij (k)|2 + σi2 (k)
(1)
j̸=i
where p is the allocated power, |g|2 is the channel gain, ρ is
proportional to the average channel cross-correlation among
all users, and σ 2 is the power noise at the respective users (i)
and sub-carriers (k).
The power allocation with rate constrains problem is a
well known telecommunications issue of non-convex nature. A
generalized statement of this problem for MC-CDMA systems
follows:
min
U ∑
N
∑
pi (k)
i=1 k=1
s.t.
N
∑
ri (k) ≥ ri∗
k=1
0 ≤ pi (k) ≤ pmax
(2)
In multiple access scenarios it is important to observe
that each choice of user power interferes in all other users’
performance. In this case it is possible to rewrite the NE
problem as generalized Nash equilibrium (GNE) problem as
follows:
min
U ∑
N
∑
pi (k)
i=1 k=1
s.t.
ri (pi , p−i ) ∈ Ri∗
(6)
pi ∈ P
where ri (pi , p−i ) is the rate allocated to the ith user given
his own power vector pi and all the other users power vectors
p−i , defined just like (5). Note that the constraints in the GNE
problem (6) are convex [10].
Furthermore, the problem in (6) may be constrained by an
average power along the N sub-channels of the i−th user, p̄i ,
3
instead of a maximum power per sub-channel pi . Thus it can
be rewritten as:
min
U ∑
N
∑
pi (k)
i=1 k=1
s.t.
ri (pi , p−i ) ∈ Ri∗
p̄i ∈ P
(7)
In order to further evaluate this approach, the conditions
in which the problem has a GNE and it is unique are
presented; hence, the existence and uniqueness of the GNE are
determined for each channel, system and interference density
scenario.
A. Existence
In order to present the existence condition of the GNE we
define matrix Zk in (8). The problem has a GNE if, and only
if, Zk is a P-matrix ∀k ∈ N [8], i.e. Zk is a complex square
matrix with every principal minor being positive.
B. Uniqueness
To evaluate the uniqueness of the GNE we must consider
the matrix B:
{ −R∗
e i,
if i = j
Bij ≡
(9)
Ri∗ max
−e β̂ij ,
otherwise
with:
max
β̂ij
(
= max
k∈N
∑
2
|gij (k)|2 σr (k) +
|gjj (k)|2
′
j ′ ̸=j |gjj (k)|
2
σi (k)
2
pej ′ (k)
)
(10)
e (k) is a column vector, such that p
e (k) =
where p
[e
p1 (k), . . . , pei (k), . . . , peU (k)]T and can be obtained as follows:

 2
∗
σ1 (k)(eR1 − 1)


..
e (k) = (Zk (R∗ ))−1 
∀k ∈ N ;
p
,
.
∗
σi2 (k)(eRi
Algorithm 1 IWFA
Input: p, N ;
Output: p∗
begin
1. initialize first population and set n = 0;
2.
while n ≤ I
3.
for i = 0 until U
4.
if i = n mod U
5.
pi [n + 1] = WF(pi [n], p−i [n])
else
7.
pi [n + 1] = pi [n]
end if
end for
8.
set n = n + 1
end while
—————————————p = initial power vectors;
p∗ = power vector solution;
I = maximum number of iterations;
Algorithm 2 Practical algorithm for single water-filling solution
Input: set of pairs {(ai , bi )}, function g;
Output: p∗i and waterlevel µ
begin
e = N;
1. set N
2. sort {(ai , bi )} such that ai /bi are in decreasing order;
3. define aN +1 = bN +1 = 0;
4. while bNe /aNe ≥ bNe +1 /aNe +1 or g(bNe /aNe ) ≥ 0
e =N
e − 1;
set N
end while
5. find µ ∈ (bNe /aNe , bNe +1 /aNe +1 ]|g(µ) = 0
6. xi = (µai − bi )+ ,
1≤i≤N
—————————————(·)+ = max(·, 0);
e is defined below;
N
and a constraint function g, the water-level may be obtained
through the practical Algorithm 2 [11].
The following particularizations apply to the problem in (7):
ai (k)
=
− 1)
(11)
The problem has an unique GNE if, and only if, it satisfy
the existence conditions, i.e. if Zk is a P-matrix ∀k ∈ N , and
B is a P-matrix [8].
bi (k)
=
1
U
∑
(13)
pj (k)|gij (k)|2 + σi2 (k)
j̸=i
|gii (k)|2
∀ i ∈ U and k ∈ N
ri∗
C. Water-Filling Solution
g(µ)
As shown in [8] the water-filling algorithm is the simplest
solution for the GNE problem in (7) when the level of
interference is low or moderate. Therein, the following GaussSeidel IWFA is used as a reference in order to obtain the GNE
problem solution:
The water-filling operator in Algorithm 1 is applied to each
subcarrier of each user considering the interference of U − 1
users, and is defined as [11]:
+
WF (pi [n], p−i [n]) = (µi ai − bi )
∀i = 1, . . . , U (12)
with (·) = max(0, ·), µi is the water-level that satisfies
the rate constraints, and ai , bi are arbitrary positive numbers.
Given a set of pairs {(ai , bi )} for each user in the system
+
2 Nf + log2 (Γ)
=
µ−
=
2 Nf + log2 (Γ)
1
2 Nf log2 (bi (k))
(14)
ri∗
µ
(15)
1
−1
2 Nf log2 (bi (k) )
where Γ is the gap between Shannon capacity and the real
e is the index of
information rate, considered here as 0dB, and N
e )) < 0.
the first sub-channel (after ordering) that satisfy g(bi (N
IV. I NTERFERENCE S CENARIOS
In this section, the game theoretic approach applicability
is studied under three different scenarios and degrees of
reality resemblance. From the first to the third the reality and
4

∗
∗
|g11 (k)|2
 −(e − 1)|g21 (k)|2

Zk = 
..

.
−(eR1 − 1)|g12 (k)|2
|g22 (k)|2
..
.
···
···
..
.
−(eR1 − 1)|g1U (k)|2
∗
−(eR2 − 1)|g2U (k)|2
..
.
−(eRU − 1)|gU 1 (k)|2
−(eRU − 1)|gU 2 (k)|2
···
|gU U (k)|2
R2∗
∗
∗
the multiple access interference increases while the global
performance decreases. For each one scenario, a table with
simulation parameters values is offered. Note that in all the
three simulations scenarios, results were obtained assuming
flat Rayleigh fading channels with zero mean and σ 2 = d2 ,
where d is the normalized distance between transmitterreceiver link. For the interfering links, it was assumed the
normalized distance between the interfering user and receiver.
A. Scenario One
The first scenario is characterized by a seven-hexagonalcell with only one link per cell and a low multiple access
interference density per sub-carrier, as described in Fig. 1.
The signal is sent from the base station (BS) to the mobile
terminals (MT) (direct link) such that each BS signal interferes
on the MTs that do not belong to that cell. Parameters for this
scenario are shown in Table I.
Multi-cell: base and mobile stations location
5000





(8)
TABLE I
PARAMETER VALUES FOR S CENARIO 1.
Parameters
Adopted Values
MC-CDMA Power-Rate Allocation System
Noise Power
Pn = −63 [dBm]
Chip rate
Rc = 3.84 × 106
Min. Signal-noise ratio
SN Rmin = 4 dB
Max. power per user per sub-carrier
Pmax = 7.8 [dBm]
Time slot duration
Tslot = 666.7µs
or Rslot = 1500 slots/s
# mobile terminals
K=7
# base station
BS = 7
cell geometry
hexagonal,
with xcell = ycell = 1 Km
# sub-carriers
32
[
]
Interference density per sub-carrier
I ≈ 0.24 Interf
Km2
Channel Gain
path loss exponent, dist.γ
γ = −2 or −6, or
γ defined as Eq. (16)
fading,
Rayleigh with 6 paths (equal
gain combining rule), σ 2 = d2
(for each link), σ 2 = (dist./S)2
(for the interfering base stations)
User Types
User Target Rates
1 or 2 bits/symb/subch ∀ U users
S
4500
4000
d
Cell 6
Cell 5

distance [m]
3500
3000




γ =




Cell 4
2500
Cell 7
2000
1500
Cell 3
Cell 1
1000
2
4
6
6
6
4
4
4
2
4
6
6
6
4
6
4
2
4
6
6
4
6
6
4
2
4
6
4
6
6
6
4
2
4
4
4
6
6
6
4
2
4
4
4
4
4
4
4
2










(16)
500
Cell 2
0
0
500
1000
1500
2000
2500
3000
B. Scenario Two
3500
4000
4500
5000
distance [m]
Fig. 1. Scenario
[
] One with d = 0.5. Forward link, interference density,
I ≈ 0.24 Interf
. ∆ indicates BS location, and • indicates the MT position.
Km2
Note in Fig. 1 that the distance d is the normalized distance
from the border of the cell. Increasing d brings the mobile
terminals closer to the respective base station; thus, signal to
multiple access interference levels are heightened and, as a
consequence, the channel conditions between transmitter and
receiver link are improved.
Considering a more realistic assignment for the path loss
exponents, matrix γ in Eq. (16) represents different γ for intercellular interference which takes into account the distances of
the interfering base stations, such that each row represents the
path loss exponents for the ith BS (from ith cell) in relation to
the other jth BS (or cell). So, for adjacent cells it was assigned
a γ = 4; for non-adjacent cells γ = 6 and for the links in each
cell γ = 2 was adopted.
This is a more realistic scenario based on the scenario
described in [9]. This case is a four-cell each one with one
link, where the information is sent from the mobile terminals
to their respective base station (reverse link). Additionally, two
parameters d1 and d2 reshape the scenario such that d1 brings
all 4 cells closer/farther to each other and d2 decrease/increase
the distance between each MT and its BS. Note that both d1
and d2 are normalized distances by S, as described in Fig.
2. As a result, the MAI
density
can be classified as medium,
]
[
.
The
simulation parameter values
meaning I ≥ 0.75 Interf
2
Km
for Scenario Two are presented in Table II.
C. Scenario Three
Scenario three describes a situation with higher user density with four-quadratic-cells each one containing four users,
resulting
[ in] high interference density per sub-carrier, I ≈
. The analysis is done in the reverse link, such that
3.75 Interf
Km2
5
Multi-cell: base and mobile stations location
2000
S
1800
1600
1600
1400
1400
1200
distance [m]
distance [m]
1800
d1
1000
800
1000
S
800
600
400
400
200
200
d2
0
200
400
600
800
1000
d
1200
600
0
Multi-cell: base and mobile stations location
2000
1200
1400
1600
1800
2000
0
0
200
400
600
distance [m]
Fig. 2. A more realistic scenario,
[ d1 =
] 0.2 and d2 = 0.1. Reverse link,
interference density, I ≥ 0.75 Interf
. ∆ indicates BS location, and •
Km2
indicates the MT position.
800
1000
1200
1400
1600
1800
2000
distance [m]
Fig. 3. A more realistic[scenario
] with higher user and interference density.
Reverse link, I ≈ 3.75 Interf
. ∆ indicates BS location, and • indicates
Km2
the MT position.
TABLE III
PARAMETER
TABLE II
PARAMETER VALUES FOR S CENARIO 2
Adopted Values
MC-CDMA Power-Rate Allocation System
Noise Power
Pn = −63 [dBm]
Chip rate
Rc = 3.84 × 106
Min. Signal-noise ratio
SN Rmin = 4 dB
Max. power per user per sub-carrier
Pmax = 7.8 [dBm]
Time slot duration
Tslot = 666.7µs
or Rslot = 1500 slots/s
# mobile terminals
K=4
# base station
BS = 4
cell geometry
rectangular, with S = 1 Km
# sub-carriers
32
]
[
Interference density per sub-carrier
I ≈ 0.75 Interf
Km2
Channel Gain
path loss exponent, dist.γ
γ = −6
fading
Rayleigh with 1 path, σ 2 = d22
(for each link), σ 2 = (dist./S)2
(for each interfering MT)
User Types
User Target Rates
1 bit/symb/subch ∀ U users
Parameters
S CENARIO 3
Adopted Values
MC-CDMA Power-Rate Allocation System
Noise Power
Pn = −63 [dBm]
Chip rate
Rc = 3.84 × 106
Min. Signal-noise ratio
SN Rmin = 4 dB
Max. power per user per sub-carrier
Pmax = 7.8 [dBm]
Time slot duration
Tslot = 666.7µs
or Rslot = 1500 slots/s
# mobile terminals
K = 16
# base station
BS = 4
cell geometry
quadratic,
with xcell = ycell = 1 Km
# sub-carriers
32
]
[
Interference density per sub-carrier
I ≈ 3.75 Interf
Km2
Channel Gain
path loss exponent, dist.γ
γ = −6
fading
Rayleigh, 1 path, σ 2 = d2
(for each link), σ 2 = (dist./S)2
(for each interfering MT)
User Types
User Target Rates
1 bit/symb/subch ∀ U users
VALUES FOR
Parameters
D. Scenario One – Applicability
each BS receives interfering in cell and out cell MT signals.
Fig. 3 shows the mobile terminals and base stations placement
through each cell coverage. The parameter d is the normalized
(by the cell size) distance that each user is from its cell border.
This scenario characterizes cellular communication systems
in a more realist way than the previous two scenarios. Table
III shows the parameters used in the simulations for Scenario
Three.
Aiming to evaluate the non-cooperative game theoretic
approach under the three scenarios, in the sequel, the probabilities of existence and uniqueness of the GNE were evaluated
by simulation through the mean over 500 different channel
realizations. The three previous scenarios were tested using
the conditions for the existence and uniqueness presented in
Section III. Numerical results are shown on the following
subsections.
The probabilities of existence and uniqueness were calculated for two different rate profiles (R = 1 and R = 2
bit/symb/subchannel), for a set of parameters d ∈ [0.1, 0.9]
and for two different values of interference cross-correlation
ρ = 0.35, and ρ = 1. Fig. 4 shows the simulations results.
As expected, the probabilities of existence and uniqueness for
the same d ∈ [0.1, 0.2] are higher for a low rate profile, i.e.
R = 1bit/symb/subchannel. Besides, P (d) → 1 as d → 0.9.
For the lower cross-correlation case, the interference assumes
smaller values, so the results for the P (d) are better than the
ones for the higher cross-correlation case, as one can see in
Fig. 5. Observe, also, that for a R = 1 bit/symb/subchannel
rate profile, the GNE existence and uniqueness probability
have the same values.
From Fig. 4 and Fig. 5 one would ask if the power-rate
GNE problem in Eq. (7) could be efficiently solved using
IWFA approach if a more realistic multiple access multi-
6
Scenario 1 – Existence and Uniqueness Probability; γ = −2; ρ = 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
P (d)
P (d)
Scenario 1 – Existence and Uniqueness Probability; γ = −6; ρ = 1
1
0.5
0.5
0.4
0.4
0.3
Existence – R = 1
0.3
0.2
Uniqueness – R = 1
0.2
Uniqueness – R = 1
Uniqueness – R = 2
0.1
Existence –R = 1
Uniqueness – R = 2
0.1
Existence – R = 2
Existence – R = 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
Fig. 4. Probability of the GNE existence and uniqueness with path loss
exponent γ = −6 and ρ = 1. Scenario One.
0.6
0.7
0.8
0.9
d
d
Fig. 6. Probability of the GNE existence and uniqueness with path loss
exponent γ = −2. Scenario One.
Scenario 1 – Existence and Uniqueness Probability; γ = −6; ρ = 0.35
1
and uniqueness for a given random d is characterized, when
compared to bucolic or LOS scenarios. However, the larger is
the path loss exponent the higher is the power level needed to
surpass the channel attenuation.
0.9
0.8
0.7
P (d)
0.6
E. Scenario Two and Three – Applicability
0.5
0.4
0.3
Existence – R = 1
0.2
Uniqueness – R = 1
Uniqueness – R = 2
0.1
0
0.1
Existence – R = 2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
d
Fig. 5. Probability of the GNE existence and uniqueness with path loss
exponent γ = −6 and ρ = 0.35. Scenario One.
For the second scenario, only one rate profile was considered (R = 1bit/symb/subchannel). Also the simulations were
done with combinations of the two different parameters, d1
and d2 . Both ρ = 1 and ρ = 0.35 average cross-correlation
values were considered. Reverse link is analyzed under a
higher interference density that generated in Scenario One.
This characterizes a more realistic scenario and, thus, has a
more appealing importance.
Scenario 2 - Existence and Uniqueness; γ = −2; ρ = 1
1
0.8
P (d1 , d2 )
carrier systems is considered, meaning interference density I
increasing substantially. As already stated, Scenario One does
not represent a totally realistic system configuration. Also, for
a path loss exponent γ = −6, the channel gains achieve values
in the range of −270dB, which characterizes an unfeasible
situation once too much power is needed to equalize the
channel state.
Under Scenario One, and assuming now a line-of-sight
(LOS) link, i.e., a path loss exponent γ = −2, representing
higher multiple access interference than in previous situations
(Fig. 4 and Fig. 5), Fig. 6 indicates that the probability of
GNE existence and uniqueness is more restrictive. On the other
hand, the simulation results for γ , as in (16); i.e. γ = −2 for
each link while γ = −4 for adjacent interference signals and
γ = −6 for non-adjacent interference, were not shown herein
since for any γ and d values combination the GNE always
exist and it is unique.
In conclusion, for this scenario the difference in the path loss
exponent results only shift of the probability values. Besides,
in this case of high number of reflections and strong channel
attenuation (γ ≥ −4), a favorable situation for GNE existence
Existence
Uniqueness
0.6
0.4
0.2
0
1
0.8
d1 0.6
0.4
0.2
0
0.9
0.8
0.7
0.6
d2
0.5
0.4
0.3
0.2
0.1
Fig. 7. Probability of the GNE existence and uniqueness with γ = −2 and
ρ = 1. Scenario Two.
Note in Fig. 7 that for a more realistic scenario the probabilities of existence and uniqueness of the GNE are even more
restricted (in terms of feasible combinations of (d1 , d2 )) than
the previous scenario. Besides, Fig. 8 confirms that the main
effect of considering a lower average cross-correlation is a
7
Scenario 2 — Existence and Uniqueness; ρ = 0.35; γ = −2; R = 1 bit/symb
1
Existence
60
40
ΣN
k=1 pi (k)
20
0
10
20
30
40
50
60
70
80
90
100
ii) Channel Capacity for each user at each iteration
0.4
0.2
0
1
0.8
d1
80
Uniqueness
0.6
Sum Rate [bits/symb]
P (d1 , d2 )
0.8
i) Sum power for each user at each iteration
100
Sum Power [mW ]
slight improvement in the existence and a substantial increase
in the uniqueness probability.
0.6
30
20
Ropt
10
R (xik , xi−k )
0
10
20
30
40
50
Iterations
0.4
60
70
80
90
100
(a)
0.2
0.8
0.7
0.6
0.5
d2
0.4
0.3
0.2
Fig. 8. Probability of the GNE existence and uniqueness with γ = −2 and
ρ = 0.35. Scenario Two.
Furthermore, for the third and most realistic scenario,
where[ the interference
density per sub-carrier achieves I ≈
]
3.75 Interf
,
there
was
none d for which the problem would
2
Km
have a GNE, even for ρ = 0.35.
With these results one could ask: i) what does happen with
the algorithm convergence when does not exist GNE? ii) is
there another tool that could efficiently and pragmatically solve
the power allocation with rate constraints problem in real
multiple access networks with low-moderate computational
complexity? For the first question, simulation results demonstrate that the algorithm convergence is not guaranteed2 when
a GNE does not exist, as shown in Fig. 9(b).
From these simulations we can infer that the noncooperative game theoretic approach is not totally efficient to
implement power-rate resource allocation policies in realistic
multiple access multi-carrier systems under medium or high
interference scenarios, due to the absence of guarantee of GNE
existence in those scenarios.
V. H EURISTIC PLUS IWFA A PPROACH
Due to the limitations of the game theoretic approach in
more realistic scenarios, an approach combining a heuristic
and IWFA is proposed. The main idea is to select the users
with the best channel conditions using particle swarm optimization (PSO) and then use the IWFA to solve the resource
allocation problem. Under realistic channel and system operation scenarios, it is necessary to remove some users from the
system, mainly the ones with bad channel conditions, since
the IWFA has no guarantee of convergence when there is
not a GNE for some determined system configuration, i.e. for
some combination of high interference density condition, QoS
requirements, and deeply fading (sub-)channel states.
2 The algorithm convergence is guaranteed when the number of maximum
iterations tends to infinity. However, in real systems the convergence must be
achieved within a shorter number of iterations as possible, due to lack of time
and energy.
i) Sum power for each user at each iteration
160
Sum Power [mW ]
0.9
0.1
140
120
100
ΣN
k=1 pi (k)
80
60
10
20
30
40
50
60
70
80
90
100
ii) Channel Capacity for each user at each iteration
Sum Rate [bit/symb]
0
Ropt
30
R(xik , xi−k )
20
10
0
10
20
30
40
50
60
70
80
90
100
Iterations
(b)
Fig. 9. Power-rate allocation under Scenario One, γ = −2 and R = 1
bit/symb/subchannel: (a) Existent and Unique GNE: typical power and rate
allocation, d = 0.6; (b) Non-existent GNE: typical power and rate allocation,
d = 0.1.
The first step in this approach consists of applying the PSO
algorithm to solve the average-power allocation with average
rate constraint:
min
U
∑
p̄i
i=1
s.t.
r̄i = r̄i∗
(17)
0 < p̄i ≤ p̄max
∀i ∈ U
In order to minimize the average power of all users, the
following cost function is employed [2], [5]:
)
(
U
1 ∑ th
p̄i
J(p̄) =
F
1−
(18)
U i=1
p̄max
where F th is defined as:
{
1,
th
F =
0,
if r̄i ≥ r̂i∗
otherwise
(19)
8
The mean power minimization problem in (17) is solved
through a heuristic method, considering the mean channel
conditions over the N sub-channels. Once PSO has completed
its procedure the users with p̄i = 0 are removed from the
system in that time slot (they come back in the next time slot
if p̄i > 0).
A. Particle Swarm Optimization
The particle swarm optimization (PSO) algorithm was designed by Kennedy and Eberhart [12], [13] based on birds
social behavior in which the principle is the movement of
particles, distributed in the search space, each one with its
position and velocity. Basically, the algorithm consists in
updating the velocity of each particle and applying it to
their position. Given a particle bp , also denominated solution
candidate, its velocity at iteration t can be computed as:
vp [n + 1]
=
ω[n] · vp [n] + ϕ1 · Up1 [n](bbest
p [n] − bp [n])
+ϕ2 · Up2 [n](bbest
g [n] − bp [n])
(20)
where ω[n] is the inertia at the nth iteration, ϕ1 is the local
solution acceleration coefficient, ϕ2 is the global solution
acceleration coefficient, Up1 [n] and Up2 [n] are diagonal matrices with dimension U whose elements are random variables
with uniform distribution ∼ U ∈ [0, 1] generated for the pth
best
particle at iteration n, bbest
p [n] and bg [n] are the local best
candidate and the global best candidate at iteration n.
Once the velocity is computed, the position of each particle
should be updated through:
bp [n + 1] = bp [n] + vp [n + 1]
(21)
where bp [n + 1] and bp [n] are the particle position at iteration
n + 1 and n, respectively. The PSO algorithm consists in
applying equations (20) and (21). A pseudo-code for power
minimization with rate constraints PSO algorithm is presented
in Algorithm 3.
Aiming to reduce the possibility that the particles might
leave the search space, a maximum velocity factor, Vmax , is
introduced and will be responsible for limiting each particles
velocity in the range ±Vmax , such that:
vp [n] = min {Vmax ; max {−Vmax ; vp [n]}}
(22)
Besides, in order to improve convergence rate, an adaptive
inertia value was implemented such that [14]:
(
)m
I −n
w[n] = (winitial − wfinal ) ·
+ wfinal
(23)
I
where winitial and wfinal are the initial and final weight inertia,
respectively, winitial > wfinal , I is the maximum number of
iterations, and m ∈ [0.6; 1.4] is the nonlinear modulation index
[14].
Table IV shows the PSO input parameters used in the
simulations. These input parameters were optimized according
to the methodology developed in [7]. In that work, both
multiuser detection and resource allocation problems were
analyzed under the PSO heuristic optimization perspective,
with emphasis on the input parameters optimization. For more
details on the choice of input parameters values for resource
allocation problems, see [7, Sections 3.2.1, 3.3.2].
Algorithm 3 SOO Continuous PSO Algorithm for the Power
Allocation Problem
Input: M , I, ω, ϕ1 , ϕ2 , Vmax ;
Output: p∗
begin
1. initialize first population: n = 0;
bp [0] ∼ U [pmin ; pmax ] ∀p ∈ M
best
bbest
p [0] = bp [0] and bg [0] = Pmax ;
vp [0] = 0: null initial velocity;
2. while n ≤ N
a. calculate J(bp [n]), ∀bp [n] ∈ B[n] using (18);
b. update velocity vp [n], p = 1, . . . , P, through (20);
c. update best positions:
for p = 1, . . . , P
if J(bp [n]) < J(bbest
p [n]) ∧ Rp [n] ≥ rp,min ,
bbest
p [n + 1] ← bp [n]
best
else bbest
p [n + 1] ← bp [n]
end
]
[
if ∃ bp [n] such that J(bp [n]) < J(bbest
g [n]) ∧ Rp [n] ≥
rp,min
[
]
∧ J(bp [n]) ≤ J(b′p [n]), ∀ p′ ̸= p ,
bbest
g [n + 1] ← bp [n]
best
else bbest
g [n + 1] ← bg [n]
d. Evolve to a new swarm population bp [n + 1], using (21);
e. set n = n + 1.
end
3. p∗ = bbest
g [I].
end
−−−−−−−−−−−−−−−−−−−−−−−−−
M : population size.
I: maximum number of swarm iterations.
pmax and pmin : maximum and minimum allowed power, respectively.
TABLE IV
PSO I NPUT PARAMETERS .
Parameter
ϕ1
ϕ2
vmax
vmin
m
M
Value
2
2
0.01 · (pmax − pmin )
−vmax
1
U
B. Power-Rate Allocation Comparison
In order to characterize the power allocation optimization
over the iterations using PSO-IWFA approach, the numerical
results for allocated power, sum power and sum rate versus
PSO number of iterations are presented in Fig. 10. Note in
Fig. 10 that after reducing the number of active users in the
system the QoS could be achieve for all users. On the other
hand, Fig. 11 shows the difference, in terms of guaranteed
QoS and power allocation, if IWFA is used without PSO. It
is worthy of note that using jointly PSO and IWFA does not
guarantee 100% times QoS satisfaction as show in the next
subsection.
C. Existence, Uniqueness and Outage Probability
In order to evaluate the improvement of the proposed
approach the existence and uniqueness probability tests were
conducted. Also, since the PSO algorithm decides which users
are dropped of the system, the outage probability was calculated. Simulations were carried out considering ρ ∈ 1, 0.35,
γ = −2, ri∗ = 1bit/symb/sub-channel and an average value
over 500 realizations.
9
Allocated Power, K = 16, Swarm Population, M = 16 particles φ1 = 2 φ2 = 2
0
Allocated Power [mW ]
10
−1
10
Figure 12 shows the simulations results for different average interference cross correlation coefficients. Note that, the
perturbation over the existence and uniqueness probability is
caused mainly by the stochastic characteristic of the channel
conditions. Moreover, a substantial decrease in outage probability caused by a reduction of ρ can be observed. However, a
decrement in outage probability does not have a simple relation
to the GNE existence probability. In fact, one may infer from
Figure 12 that for a d ≃ 0.3 the GNE existence and uniqueness
probability achieve its peak value with a 90% outage chance,
i.e. only 10% of the users still are on the system.
Scenario 3 —- Existence, Uniqueness and Outage — γ = −2; ρ = 1
−2
10
1
bbest
g
Existence — R = 1
0.9
0
100
200
300
400
500
600
700
800
900
Uniqueness — R = 1
1000
Iterations
0.8
(a)
0.6
P (d)
Sum Power [mW ]
0.7
i) Sum power for each user at each iteration
150
100
0.5
0.4
50
0.3
ΣN
k=1 pi (k)
0.2
0
Sum Rate [bits/symb]
Outage — R = 1
10
20
30
40
50
60
70
80
90
100
0.1
ii) Channel Capacity for each user at each iteration
40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
d
30
(a)
20
Scenario 3 — Existence, Uniqueness and Outage — γ = −2; ρ = 0.35
1
10
R(xik , xi−k )
Existence — R = 1
0.9
0
0
10
20
30
40
50
60
70
80
90
Uniqueness — R = 1
100
0.8
Iterations
0.7
Fig. 10. Typical power allocation through PSO plus IWFA in Scenario Three.
(a) Allocated power using PSO for d = 0.6, R = 1bit/symb/subchannel,
γ = −2; (b) Sum Power and Sum Rate allocation through IWFA after PSO
average power allocation in Fig. 10(a).
0.6
P (d)
(b)
Outage — R = 1
0.5
0.4
0.3
i) Sum power for each user at each iteration
Sum Power [mW ]
120
0.2
110
0.1
100
0
0.1
90
0.2
0.3
0.4
70
0.6
0.7
0.8
0.9
(b)
ΣN
k=1 pi (k)
60
50
0.5
d
80
10
20
30
40
50
60
70
80
90
100
Fig. 12. Scenario 3, γ = −2,
ρ = 0.35.
ri∗
= 1bit/symb/sub-channel: (a) ρ = 1; (b)
Sum Rate [bit/symb]
ii) Channel Capacity for each user at each iteration
30
20
Ropt
10
R(xik , xi−k )
0
10
20
30
40
50
60
70
80
90
100
Iterations
Fig. 11. Typical power and rate allocation in Scenario Three using IWFA
without PSO, d = 0.6, ρ = 0.35.
Considering the results, one may rise some questions: i) if
the objective was to improve the GNE existence and uniqueness probability why the results for PSO-IWFA in scenario
three are so different from the ones for scenario one and two
using only IWFA? ii) even without GNE, is the QoS satisfied
or partially satisfied? and iii) if partially satisfied what is the
probability that a fraction of the QoS is satisfied?
In order to answer those questions, results in Figure 13
should be carefully analyzed. Firstly, once the target rate can-
10
not be achieved by the maximum allowed power, the algorithm
considers better to transmit a fraction of the minimum rate
instead of no rate. Hence, Fig. 13 shows the probability of
achieving – fully or partially (90% and 70%) – the QoS for
different normalized distance parameters. That explains why
GNE existence and uniqueness probability in Scenario Three
are not as good as their parameters values for scenarios one
and two.
Probability of QoS Assurance — γ = −2; ρ = 1
1
0.9
0.8
DPCA
IWFA
Eq.
Algorithm 1
Algorithm 2
(1)
(18)
(20)
(21)
PSO
Sum
1
N2 + N + 1
U2 + U
2U
U ×M
U ×M
90%
70%
0.6
0.5
0.4
0.3
0.2
CIW F A (U ) =
0.1
=
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CP SO (U )
(a)
Probability of QoS Assurance — γ = −2; ρ = 0.35
0.8
90%
70%
0.7
0.6
P (d)
4U (3U 2 + 3U + 8)
12U 3 + 12U 2 + 32U
(24)
= 1000(8U 2 + 9U + 1)
= 8000U 2 + 9000U + 1000
(25)
where C(U ) is the total number of operations for a given number of users U . Moreover, the total number of mathematical
operations for different system loadings is presented in Figure
14.
100%
0.9
Exponen.
0
4
0
0
0
0
and, considering M = U :
d
1
Multipl.
1
N 2 + 2N + 1
2U 2 + 2U
2U + 1
(3U + 2) × M
0
For the IWFA we might consider that the number of subchannels and users are in the same magnitude, such that
U ≡ N . Also, note that for the PSO algorithm the number
of iterations I = 1000 and for IWFA a non-exhaustive
search revealed that a I ≡ 4 U is enough for the algorithm
to converge. Besides, each Algorithm 1 iteration executes a
sorting algorithm through the N sub-channels. As it is known,
the worst case complexity for sorting algorithms (such as
bubble sort) is O(N 2 ). Therefore, according to Table V and
the last statement, one may imply:
100%
0.7
P (d)
TABLE V
N UMBER OF OPERATIONS PER ITERATION .
Mathematical Operations for PSO and IWFA
7
10
0.5
6
10
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
d
(b)
Fig. 13. Probability of QoS Assurance, averaged over 500 trials, ri∗ =
1bit/symb/sub-channel, γ = −2. Scenario 3. (a) ρ = 1; (b) ρ = 0.35.
# Mathematical Operations
0.4
5
10
4
10
3
10
2
10
IWFA
PSO
1
10
Note that the performance considering full interference, i.e.
ρ = 1, is better. This is a consequence of the high outage
probability when compared to ρ = 0.35, which means that
there are fewer users in the system.
D. Complexity Analysis
A computational complexity analysis was conducted taking
into account the number of mathematical operations. Note that
in MC-CDMA system we must consider three variables (users,
sub-channels and iterations) instead of two in single carrier
cases (users and iterations). Table V contains the number of
operations for each algorithm.
2
4
6
8
10
12
14
16
Users, U
Fig. 14.
PSO and IWFA computational complexity, assuming U = N .
The computational complexity asymptotic behavior of the
PSO algorithm is about O(U 2 ) and the IWFA O(U 3 ). So, in
asymptotically, the proposed approach complexity would be
O(U 3 + U 2 ) which means O(U 3 ), i.e. there is no increase in
the complexity of the proposed method in asymptotic terms.
VI. C ONCLUSION
The paper pointed out the main flaws of the game theoretic
approach alone. In most cases the game does not have a GNE
11
and, thus, the IWFA does not truly converge, either by using
more than the maximum allowed transmitted power or by not
satisfying QoS requirements. Therefore, a method combining
PSO and IWFA is proposed in order to find a solution in the
situations that there are no GNE present.
Tested through different scenarios, the game theoretic approach does not have a GNE for scenarios with low interference density, e.g. in scenario three a interference density
of 3.75 interf.
is observed, which is considered still low, and
Km2
yet simulations results manage to find no GNE. A huge
improvement was achieved combining the heuristic approach
with IWFA, such that for more realist scenarios the IWFA was
able to find solutions to the power control problem.
Moreover, it is important to highlight that, asymptotically,
the computational complexity of the proposed method did not
increased. Since IWFA can be implemented in a totally distributed way, this is an important result once mobile terminals
are restricted in terms of computational power.
Finally, future work includes finding a way to redistribute
the average power found by the PSO algorithm into the N subchannels of the system, reducing even more the computational
complexity. Besides other simpler heuristic methods and a
reformulation of the game may be tested.
ACKNOWLEDGEMENT
The authors would like to express gratitude to the anonymous reviewers for their thorough reviews and for the thoughtful comments and suggestions, which have enhanced the
readability and quality of the manuscript.
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