Hybrid heuristic-waterfilling game theory approach in MC-CDMA resource allocation

Applied Soft Computing 12 (2012) 1902–1912

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Hybrid heuristic-waterfilling game theory approach in MC-CDMA resource allocation夽 Lucas Dias H. Sampaio a , Taufik Abrão a,∗ , Bruno A. Angélico b , Moisés Fernando Lima a , Mario Lemes Proenc¸a Jr. a , Paul Jean E. Jeszensky c a

State University of Londrina, PR, 86051-990, Brazil Federal Technological University of Paraná, PR, Brazil c Escola Politécnica of University of São Paulo, SP, Brazil b

a r t i c l e

i n f o

Article history: Received 2 October 2010 Received in revised form 17 April 2011 Accepted 1 May 2011 Available online 10 May 2011 Keywords: Power-rate allocation control SISO multi-rate MC-CDMA Game theory Iterative water-filling algorithm QoS

a b s t r a c t 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 water-filling 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. © 2011 Elsevier B.V. All rights reserved.

1. Introduction In the last years the telecommunication 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. 1.1. Motivation The study of resource allocation problems in wireless networks has been discussed for many years due to its impacts in company profits and user 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 power,1 increase the throughput and guarantee the QoS. 1.2. Related work

夽 This work was supported in part by the National Council for Scientific and Technological Development (CNPq) of Brazil under Grant 303426/2009-8. ∗ Corresponding author at: Computer Science Department, State University of Londrina, PR, 86051-990, Brazil. E-mail addresses: [email protected] (L.D.H. Sampaio), taufi[email protected], taufi[email protected], taufi[email protected] (T. Abrão), [email protected] (B.A. Angélico), moisesfl[email protected] (M. Fernando Lima), [email protected] (M.L. Proenc¸a Jr.), [email protected] (P.J.E. Jeszensky). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.05.028

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

1

Saving power may increase battery lifetime.

L.D.H. Sampaio et al. / Applied Soft Computing 12 (2012) 1902–1912

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 was 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 singleor 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 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 waterfilling algorithm is proposed in [8], while in [9] a multi-linear 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 multicarrier 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. 1.3. Organization This paper is organized as follows: Section 2 presents the system model and description; the game theoretic approach is presented in Section 3. Section 3.3 discusses the iterative water-filling algorithm (IWFA); moreover, in Section 4 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 5 and conclusions in Section 6.

soft capacity of CDMA systems. The spectrum may be divided in N uncorrelated CDMA sub-channels such that each one is a flat channel, therefore, more appropriate to high transmission rates. In MC-CDMA systems the SINR at the i th user k th sub-carrier may be computed as follows [9]: ıi (k) = 



pi (k)|gii (k)|2

(1)

pj (k)|gij (k)|2 + i2 (k)

j= / i

where p is the allocated power, |g | 2 is the channel gain,  is proportional to the average channel cross-correlation among all users, and  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

N U  

pi (k)

i=1 k=1 N

s.t.



ri (k) ≥ ri∗

(2)

k=1

0 ≤ pi (k) ≤ pmax 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 i th user. 3. Game theory approach 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}

(3)

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 users’ utility without decreasing others payoffs. Therefore, problem (2) may be rewritten as follows [10]:

min

N U  

pi (k)

i=1 k=1

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:

2. System description 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

1903

ri (pi ) =

N 

log[1 + ıi (k)]

(5)

k=1

In multiple access scenarios it is important to observe that each users’ power choice interferes in all other user performance. In this

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L.D.H. Sampaio et al. / Applied Soft Computing 12 (2012) 1902–1912

case it is possible to rewrite the NE problem as generalized Nash equilibrium (GNE) problem as follows: N U  

min

pi (k) (6)

i=1 k=1

ri (pi , p−i ) ∈ Ri∗ pi ∈ P

s.t.

N U  

pi (k) (7)

i=1 k=1

ri (pi , p−i ) ∈ Ri∗ p¯ i ∈ P

s.t.

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. 3.1. 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.

⎡ ⎢

Zk = ⎣

|g11 (k)|2

−(e

−(e

R∗

2

R∗

U

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 Gauss–Seidel IWFA is used as a reference in order to obtain the GNE problem solution: Algorithm 1 (IWFA).

where ri (pi , p−i ) is the rate allocated to the i th user given his own power vector pi and all the other user 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, p¯ i , instead of a maximum power per sub-channel pi . Thus it can be rewritten as: min

3.3. Water-filling solution

−(e

− 1)|g21 (k)|2 . . . − 1)|gU1 (k)|2

−(e

R∗

1

R∗

− 1)|g12 (k)|2

|g22 (k)|2 . . .

U

···

−(e

···

−(e

..

− 1)|gU2 (k)|2

R∗

1

R∗

.

···

2

− 1)|g1U (k)|2 − 1)|g2U (k)|2 . . .

⎤ ⎥ ⎦

(8)

|gUU (k)|2

3.2. Uniqueness

Output: p∗

Input: p, N; begin 1 2 3 4 5

initialize first population and set n = 0; while n ≤ I for i = 0 until U if i = n mod U pi [n + 1] = WF(pi [n], p−i [n]) else 7 pi [n + 1] = pi [n] end if end for set n = n + 1 end while p: initial power vectors; p∗ : power vector solution; I: maximum number of iterations.

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] = (i ai − bi )

Bij ≡

e

−R∗

, R∗ ˆ max , −e i ˇ i

ij

if i = j

(9)

otherwise

with:

⎛ ⎜ |gij (k)|2 ⎝ |gjj (k)|2

r2 (k) +



|gjj (k)|2 p˜ j (k)

Output: p∗i and water-level 

Input: set of pairs {(ai , bi )}, function g; begin 1 2 3 4

˜ = N; set N sort {(ai , bi )} such that ai /bi are in decreasing order; define aN+1 = bN+1 = 0; while bN˜ /aN˜ ≥ bN+1 /aN+1 or g(bN˜ /aN˜ ) ≥ 0 ˜ ˜ ˜ =N ˜ − 1; set N end while   find  ∈

j = / j

ˆ max = maxk ∈ N ⎜ ˇ ij

i2 (k)

⎟ ⎟ ⎠

+

˜ is defined below. (·) = max(·, 0); N

The following particularizations apply to the problem in (7): =1

ai (k)

U 

pj (k)|gij (k)|2 + i2 (k)

j= / i

=

R∗

1

∗

R i2 (k)(e i

− 1)

|gii (k)|2

(10)

˜ r ∗ /N

g() =  −

⎤

⎥ ⎦ , ∀k ∈ N;

(13)

∀i ∈ U and k ∈ N

˜ ˜ where p(k) is a column vector, such that p(k) = T [˜p1 (k), . . . , p˜ i (k), . . . , p˜ U (k)] and can be obtained as follows: 12 (k)(e .. ∗ −1 ⎢ ˜ p(k) = (Zk (R )) ⎣ .

1≤i≤N

+

2

i

2

⎡

bN˜ /aN˜ , bN+1 / aN+1 |g() = 0 ˜ ˜

xi = (ai − bi ) ,

bi (k)

⎞

(12)

Algorithm 2 (Practical algorithm for single water-filling solution).

6



∀i = 1, . . . , U

with (·)+ = max (0, ·), i 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 and a constraint function g, the water-level may be obtained through the practical Algorithm 2[11].

5

To evaluate the uniqueness of the GNE we must consider the matrix B:

+

(11)

˜ r ∗ /N

=

2

i

2

+ log2 ()

1 log (b (k)) 2 i ˜ N

+ log2 ()

1 log (b (k)−1) 2 i ˜ N

(14)

(15)

where  is the gap between Shannon capacity and the real infor˜ is the index of the first mation rate, considered here as 0 dB, and N ˜ < 0. sub-channel (after ordering) that satisfy g(bi (N))

− 1)

The problem has a 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].

4. Interference scenarios In this section, the game theoretic approach applicability is studied under three different scenarios and degrees of reality

L.D.H. Sampaio et al. / Applied Soft Computing 12 (2012) 1902–1912

Multi-cell: base and mobile stations location

1600

4000

d

Cell 6

Cell 5

1400

3000

distance [m]

3500

distance [m]

S

1800

S

4500

Multi-cell: base and mobile stations location

2000

5000

1905

Cell 4

2500 Cell 7

2000

1200

d1

1000 800 600

1500 Cell 3

Cell 1

400

1000

200

500

d2

Cell 2

0

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0

0

200

400

600

800

distance [m]

1000

1200

1400

1600

1800

2000

distance [m]

Fig. 1. Scenario one with d = 0.5. Forward link, interference density, I ≈ 0.24 [Interf/Km2 ].  indicates BS location, and • indicates the MT position.

Fig. 2. A more realistic scenario, d1 = 0.2 and d2 = 0.1. Reverse link, interference density, I ≥ 0.75 [Interf/Km2 ].  indicates BS location, and • indicates the MT position.

resemblance. From the first to the third the reality and the multiple access interference increase 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  2 = d2 , where d is the normalized distance between transmitter-receiver link. For the interfering links, it was assumed the normalized distance between the interfering user and receiver.

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 ␥ in Eq. (16) represents different  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 i th BS (from i th cell) in relation to the other j th BS (or cell). So, for adjacent cells it was assigned a  = 4; for non-adjacent cells  = 6 and for the links in each cell  = 2 was adopted.

4.1. Scenario one

⎡

The first scenario is characterized by a seven-hexagonal-cell 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 1. 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 Table 1 Parameter values for scenario 1. Parameters MC-CDMA power-rate allocation system Noise power Chip rate Min. signal-noise ratio Max. power per user per sub-carrier Time slot duration # Mobile terminals # Base station Cell geometry # Sub-carriers Interference density per sub-carrier Channel gain Path loss exponent, dist. Fading

User types User target rates

Adopted values Pn = − 63 dBm Rc = 3.84 × 106 SNRmin = 4 dBm Pmax = 7.8 dBm Tslot = 666.7 ␮s or Rslot = 1500 slots/s K=7 BS = 7 Hexagonal, with xcell = ycell = 1 Km 32 I ≈ 0.24 [Interf/Km2 ]

 = − 2 or −6, or ␥ defined as Eq. (16) Rayleigh with 6 paths (equal gain combining rule),  2 = d2 (for each link),  2 = (dist./S)2 (for the interfering base stations) 1 or 2 bits/symb/subch ∀U users

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)

4.2. Scenario two 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, meaning I ≥ 0.75 [Interf/Km2 ]. The simulation parameter values for scenario two are presented in Table 2. 4.3. 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 ≈ 3.75 [Interf/Km2 ]. The analysis is done in the reverse link, such that each BS receives interfering in cell and out cell MT signals. Fig. 3 shows the mobile terminal and base station 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

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L.D.H. Sampaio et al. / Applied Soft Computing 12 (2012) 1902–1912 Table 3 Parameter values for scenario 3.

Table 2 Parameter values for scenario 2. Parameters

Adopted values

MC-CDMA power-rate allocation system Noise power Chip rate Min. signal-noise ratio Max. power per user per sub-carrier Time slot duration # Mobile terminals # Base station Cell geometry # Sub-carriers Interference density per sub-carrier Channel gain Path loss exponent, dist. Fading

User types User target rates

Parameters

Pn = − 63 dBm Rc = 3.84 × 106 SNRmin = 4 dBm Pmax = 7.8 dBm Tslot = 666.7 ␮s or Rslot = 1500 slots/s K=4 BS = 4 Rectangular, with S = 1 Km 32 I ≈ 0.75 [Interf/Km2 ]

 =−6 Rayleigh with 1 path,  2 = d22 (for each link),  2 = (dist./S)2 (for each interfering MT)

Adopted values

MC-CDMA power-rate allocation system Noise power Chip rate Min. signal-noise ratio Max. power per user per sub-carrier Time slot duration # Mobile terminals # Base station Cell geometry # Sub-carriers Interference density per sub-carrier Channel gain Path loss exponent, dist. Fading User types User target rates

1

Tslot = 666.7 ␮s or Rslot = 1500 slots/s K = 16 BS = 4 Quadratic, with xcell = ycell = 1 Km 32 I ≈ 3.75 [Interf/Km2 ]

 =−6 Rayleigh, 1 path,  2 = d2 (for each link),  2 = (dist./S)2 (for each interfering MT) 1 bit/symb/subch ∀U users

1 bit/symb/subch ∀U users

Scenario 1 – Existence and Uniqueness Probability; γ = −6; ρ = 1

0.9 0.8 0.7 0.6

P (d)

two scenarios. Table 3 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 probability of existence and uniqueness of the GNE was 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 3. Numerical results are shown in the following subsections.

Pn = − 63 dBm Rc = 3.84 × 106 SNRmin = 4 dBm Pmax = 7.8 dBm

0.5 0.4

4.4. Scenario one – applicability The probability of existence and uniqueness was 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  = 0.35, and  = 1. Fig. 4 shows the simulation results. As expected, the probability of existence and uniqueness for the same d ∈ [0.1, 0.2] is higher for a low rate profile, i.e. R = 1 bit/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,

0.3

Existence – R =1

0.2

Uniqueness – R =1 Uniqueness – R =2

0.1

Existence – R =2

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d Fig. 4. Probability of the GNE existence and uniqueness with path loss exponent  = − 6 and  = 1. Scenario one.

Scenario 1 – Existence and Uniqueness Probability; γ = −6; ρ = 0.35 1

Multi-cell: base and mobile stations location

2000

0.9 1800 0.8 1600 0.7 0.6

d

1200

P (d)

distance [m]

1400

1000 S

0.5 0.4

800 0.3

Existence – R =1

0.2

Uniqueness – R =1

600 400

Uniqueness – R =2

0.1

Existence – R =2

200 0 0.1

0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d

distance [m] Fig. 3. A more realistic scenario with higher user and interference density. Reverse link, I ≈ 3.75 [Interf/Km2 ].  indicates BS location, and • indicates the MT position.

Fig. 5. Probability of the GNE existence and uniqueness with path loss exponent  = − 6 and  = 0.35. Scenario one.

L.D.H. Sampaio et al. / Applied Soft Computing 12 (2012) 1902–1912

Scenario 1 – Existence and Uniqueness Probability; γ = −2; ρ = 1

1

1907

Scenario 2 - Existence and Uniqueness;

γ = − 2; ρ = 1

1 0.8

P (d1 , d2 )

0.9 0.8 0.7

P (d)

0.6

Existence Uniqueness

0.6 0.4 0.2

0.5

0 1

0.4

0.8 Existence – R = 1

0.3

d1

Uniqueness – R = 1

0.2

0.4

Uniqueness – R = 2

0.1

0.6

0.2

Existence – R = 2

0

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.7

0.8

0.9

0.9

0.6

that for a R = 1 bit/symb/subchannel rate profile, the GNE existence and uniqueness probability have the same values. From Figs. 4 and 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-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  = − 6, the channel gains achieve values in the range of −270 dB, 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  = − 2, representing higher multiple access interference than in previous situations (Figs. 4 and 5), Fig. 6 indicates that the probability of GNE existence and uniqueness is more restrictive. On the other hand, the simulation results for ␥, as in (16); i.e.  = − 2 for each link while  = − 4 for adjacent interference signals and  = − 6 for non-adjacent interference, were not shown herein since for any  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 ( ≥ − 4), a favorable situation for GNE existence 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. 4.5. Scenarios two and three – applicability For the second scenario, only one rate profile was considered (R = 1 bit/symb/subchannel). Also the simulations were done with combinations of the two different parameters, d1 and d2 . Both  = 1 and  = 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. Note in Fig. 7 that for a more realistic scenario the probability of existence and uniqueness of the GNE is 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 slight improvement in the existence and a substantial increase in the uniqueness probability.

0.4

0.3

0.2

0.1

Fig. 7. Probability of the GNE existence and uniqueness with  = − 2 and  = 1. Scenario two.

Scenario 2 — Existence and Uniqueness; ρ = 0 .35; γ = − 2; R = 1 bit/symb 1 0.8

P (d1 , d2 )

Fig. 6. Probability of the GNE existence and uniqueness with path loss exponent  = − 2. Scenario one.

0.5

d2

d

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

0.4

0.5

0.3

0.2

0.1

d2

Fig. 8. Probability of the GNE existence and uniqueness with  = − 2 and  = 0.35. Scenario two.

Furthermore, for the third and most realistic scenario, where the interference density per sub-carrier achieves I ≈ 3.75 [Interf/Km2 ], there was none d for which the problem would have a GNE, even for  = 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. From these simulations we can infer that the non-cooperative 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

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.

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Iterations Fig. 9. Power-rate allocation under scenario one,  = − 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.

scenarios, due to the absence of guarantee of GNE existence in those scenarios.

The first step in this approach consists of applying the PSO algorithm to solve the average-power allocation with average rate constraint:

5. Heuristic plus IWFA approach min 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.

U 

p¯ i

i=1

s.t.

(17)

r¯ i = r¯ i∗ 0 < p¯ i ≤ p¯ max ∀i ∈ U

In order to minimize the average power of all users, the following cost function is employed [2,5]:



p¯ i 1  th F 1− p¯ max U U

¯ = J(p)

i=1

 (18)

L.D.H. Sampaio et al. / Applied Soft Computing 12 (2012) 1902–1912 Table 4 PSO input parameters.

where Fth is defined as:



F th =

1909

1, if r¯ i ≥ rˆi∗ 0, otherwise

(19)

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 p¯ i = 0 are removed from the system in that time slot (they come back in the next time slot if p¯ i > 0). 5.1. Particle swarm optimization The particle swarm optimization (PSO) algorithm was designed by Kennedy and Eberhart [12,13] based on bird 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] + 1 · Up1 [n](bbest [n] − bp [n]) p [n] − bp [n]) + 2 · Up2 [n](bbest g

(20)

where ω[n] is the inertia at the n th iteration, 1 is the local solution acceleration coefficient, 2 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 [n] and ∼U ∈ [0, 1] generated for the p th particle at iteration n, bbest p bbest [n] are the local best candidate and the global best candidate g 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 Eqs. (20) and (21). A pseudo-code for power minimization with rate constraints PSO algorithm is presented in Algorithm 3. Algorithm 3 (SOO continuous PSO algorithm for the power allocation problem). Output: p∗ Input: M, I, ω, 1 , 2 , Vmax ; begin 1. initialize first population: n = 0; bp [0]∼U[pmin ; pmax ]∀p ∈ M best best bp [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 best if J(bp [n]) < J(bp [n]) ∧ Rp [n] ≥ rp,min , best bp [n + 1] ← bp [n] best best else bp [n + 1] ← bp [n] end   best

if ∃bp [n] such that J(bp [n]) < J(bg



∧ J(bp [n]) ≤

 J(bp [n]),

[n]) ∧ Rp [n] ≥ rp,min



∀ p = / p ,

best

bg [n + 1] ← bp [n] best best else bg [n + 1] ← bg [n] d. Evolve to a new swarm population bp [n + 1], using (21); e. set n = n + 1. end best 3. p∗ = bg [I]. end M, population size; I, maximum number of swarm iterations; pmax and pmin , maximum and minimum allowed power, respectively.

Parameter

Value

1

2

2 2 0.01 · (pmax − pmin ) −vmax 1 U

vmax vmin m M

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]: w[n] = (winitial − wfinal ) ·

 I − n m I

+ wfinal

(23)

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 4 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 and 3.3.2]. 5.2. 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. 5.3. 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  ∈ 1, 0.35,  = − 2, ri∗ = 1 bit/symb/sub-channel and an average value over 500 realizations. Fig. 12 shows the simulation 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  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 Fig. 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. 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

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Iterations Fig. 10. Typical power allocation through PSO plus IWFA in scenario three. (a) Allocated power using PSO for d = 0.6, R = 1 bit/symb/subchannel,  = − 2; (b) sum power and sum rate allocation through IWFA after PSO average power allocation in (a).

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 Fig. 13 should be carefully analyzed. Firstly, once the target rate cannot 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. Note that the performance considering full interference, i.e.  = 1, is better. This is a consequence of the high outage probability when compared to  = 0.35, which means that there are fewer users in the system.

5.4. 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, subchannels and iterations) instead of two in single carrier cases (users

Table 5 Number of operations per iteration. DPCA

Eq.

Sum

Multipl.

Exponen.

IWFA

Algorithm 1 Algorithm 2 (1) (18) (20) (21)

1 N2 + N + 1 U2 + U 2U U×M U×M

1 N2 + 2N + 1 2U2 + 2U 2U + 1 (3U + 2) × M 0

0 4 0 0 0 0

PSO

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1911

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(a)

1

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d Fig. 12. Scenario 3,  = − 2, ri∗ = 1 bit/symb/sub-channel: (a)  = 1; (b)  = 0.35.

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d Fig. 13. Probability of QoS assurance, averaged over 500 trials, ri∗ = 1 bit/symb/subchannel,  = − 2. Scenario 3. (a)  = 1; (b)  = 0.35.

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Mathematical Operations for PSO and IWFA

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# Mathematical Operations

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e.g. in scenario three a interference density of 3.75 interf/Km2 is observed, which is considered still low, and 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 sub-channels of the system, reducing even more the computational complexity. Besides other simpler heuristic methods and a reformulation of the game may be tested. Acknowledgement

Fig. 14. PSO and IWFA computational complexity, assuming U = N.

and iterations). Table 5 contains the number of operations for each algorithm. 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 5 and the last statement, one may imply: CIWFA (U)

= 4U(3U 2 + 3U + 8) = 12U 3 + 12U 2 + 32U

(24)

and, considering M = U: CPSO (U)

= 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 Fig. 14. 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. 6. Conclusion The paper pointed out the main flaws of the game theoretic approach alone. In most cases the game does not have a GNE 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,

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. References [1] G. Foschini, Z. Miljanic, A simple distributed autonomous power control algorithm and its convergence, IEEE Transactions on Vehicular Technology 42 (November (4)) (1993) 641–646. [2] M. Moustafa, I. Habib, M. Naghshineh, Genetic algorithm for mobiles equilibrium, in: MILCOM 2000. 21st Century Military Communications Conference Proceedings, October 2000. [3] T.J. Gross, T. Abrão, P.J.E. Jeszensky, Distributed power control algorithm for multiple access systems based on verhulst model, AEÜ. International Journal of Electronics and Communications 37 (November (5)) (2010) 631–643. [4] L.D.H. Sampaio, M.F. Lima, B.B. Zarpelão, M.L. Proenc¸a, T. Abrão Jr., Power allocation in multirate ds/cdma systems based on verhulst equilibrium, in: IEEE ICC 2010 – Communication QoS, Reliability and Modeling Symposium, May 2010, pp. 1–6. [5] H. Elkamchouchi, H. EIragal, M. Makar, Power control in cdma system using particle swarm optimization, in: 24th National Radio Science Conference, March 2007, pp. 1–8. [6] L.D.H. Sampaio, M.F. Lima, B.B. Zarpelão, M.L. Proenc¸a, T. Abrão Jr., Swarm power-rate optimization in multi-class services ds/cdma networks, in: XXVIII Simpósio Brasileiro de Redes de Computadores e Sistemas Distribuídos, May 2010, pp. 615–628. [7] T. Abrão, L. D. H. Sampaio, B. A. Angélico, M. L. P. Jr., and P. J. E. Jeszensky Multiple Access Wireless Networks Optimization via Heuristic Search Algorithms, in: Search Algorithms, 1st ed., vol. 1, Intech, Vienna, Austria, April 2011, pp. 261–298. [8] J.-S. Pang, G. Scutari, F. Facchinei, C. Wang, Distributed power allocation with rate constraints in gaussian parallel interference channels, IEEE Transactions on Information Theory 54 (August (8)) (2008) 3471–3489. [9] J.H. Li Ping Qian, Ying Jun, Zhang, Mapel: Achieving global optimality for a non-convex wireless power control problem, IEEE Transactions on Wireless Communications 8 (March (3)) (2009) 1553–1563. [10] G. Scutari, D.P. Palomar, F. Facchinei, J.-S. Pang, Convex optimization, game theory, and variational inequality theory, IEEE Signal Processing Magazine May (2010) 35–49. [11] D.P. Palomar, J.R. Fonollosa, Practical algorithms for a family of water-filling solutions, IEEE Transactions on Signal Processing 53 (February (2)) (2005) 686–695. [12] J. Kennedy, R. Eberhart, Particle swarm optimization, in: IEEE International Conference on Neural Networks, 1942–1948. [13] J. Kennedy, R.C. Eberhart, Swarm Intelligence, 1st ed., Morgan Kaufmann, March 2001. [14] A. Chatterjee, P. Siarry, Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization, Computers & Operations Research 33 (March (3)) (2006) 859–871.