Power-Rate Control in Multirate Multiple Access ...

Power-Rate Control in Multirate Multiple Access Networks via Heuristic Ant Colony Optimization Mateus de Paula Marques

Fernando Ciriaco

Taufik Abrão

Department of Electrical Engineering State University of Londrina Londrina, Paraná, Brasil. Email: [email protected]

Department of Electrical Engineering State University of Londrina Londrina, Paraná, Brasil. Email: [email protected]

Department of Electrical Engineering State University of Londrina Londrina, Paraná, Brasil. Email: [email protected]

Abstract—In this paper, continuous heuristic ant colony optimization (ACOR ) procedure is deployed to solve the powerrate optimization problem in multirate multi-processing gain (MPG) DS/CDMA networks. The power-rate allocation design is formulated as a special case of generalized linear fractional problem (GLFP), allowing the multiple access system to operate under best power-rate trade-off operation point. Numerical results considering realistic wireless mobile channels and system operation conditions have been shown the applicability of the ACOR heuristic approach in order to solve this hard problem with practical interest in real energy-efficient, spectral-efficient CDMA systems, as well as of paramount interest in establishing the next wireless generation green communication networks.

I.

I NTRODUCTION

Resource allocation (RA) techniques, primarily power/energy consumption minimization, are becoming increasingly important in wireless systems and networks design, since battery technology evolution has not followed the explosive demand of mobile devices. RA in wireless networks aim to maximize the sum of utilities of link rates for best-effort traffic. The usual approach consists in considering the problem jointly, i.e., optimizing the joint power control and link scheduling, which is known to be notoriously difficult to solve, even in a centralized manner. One of the most interesting ways of dealing with the power allocation problem is the energy-efficiency (EE) approach [7], [8], which aims to maximize the transmitted data per energy unit. The energy-efficient approach on CDMA system-based networks can include the joint strategies of spreading-code and receiver optimization [1], as well as the balancing of two important conflicting metrics, energy efficiency versus spectral efficiency [2]. Hence, from a wider approach, the power control problem in a multiple access system-based networks could be formulated aiming to optimize the deployment of two main resources scarcely available, i.e., spectrum and energy. Game theory, which has its roots in the economy field, has been broadly applied in wireless communications for random access and power control optimization problems. From the analysis of two conflicting metrics, namely throughput maximization and power consumption minimization, the distributed energy efficiency cost function can be formulated as a (non)cooperative game [2]. As a consequence, although the sumrate increases with the number of active users, the generated level of interference induced by the new users sharing the same bandwidth increases too. Hence, by one side the total

network power consumption enlarges in order to achieve the optimum SINR, while, on the other hand, the EE is reduced. As a solution, the best achievable EE-SE trade-off when each node allocates exactly the power necessary to attain the best SINR value, which guarantees the maximal EE while SE is determined by the attainable rate in each node given by Shannon capacity equation. Besides, the energy efficiency is normally reduced by the efficiency function, coding factor and by the circuit power consumption as well. In this work, we deal with heuristic optimization methods applied to SE maximization of CDMA systems. Thus, a cost function formulated as a generalized fractional linear program (GLFP) for weighted throughput maximization (WTM) problem is proposed, which in turn, is able to maximize the total SE of the system, while keeping the transmission powers under feasible levels. II.

S YSTEM M ODEL AND P ROBLEM F ORMULATION

Let consider a downlink (DL) multirate MPG-DS/CDMA network, in which the bit error rate (BER) is used as a QoS metric, since it is directly related to the signal to noise plus interference ratio (SNIR). Thus, the SNIR is associated to the carrier to interference ratio as follows: γi = Fi × Γi ,

i = 1, . . . , U

(1)

rc where i ∈ U is the user’s indexer, γi is the SNIR, Fi = ri,min is the processing gain, rc is the chip rate, ri,min is the base information rate, and Γi is the CIR, defined as [6], [5]:

Γi = P U

pi |gii |2

j=1,i6=j

pj |gij |2 + σ 2

, i = 1, ..., U

(2)

where pi is the transmit power bounded by pmax , |gii | the amplitude channel gain (considering the effects of path loss, shadowing and multipath fading), |gij | is the interfering channels gain and σi2 the additive white Gaussian noise (AWGN) at the i-th receiver’s input. The achievable information rate for spread spectrum systems in AWGN channel considering the gap between theoretical bound and the real information rate is defined based on Shannon channel capacity [3] as:   w bits ri = log2 (1 + θi γi ) (3) mi Fi sec where ri is the achievable information rate, mi = log2 Mi with Mi being the modulation order, θi is the inverse of the

gap between the theoretical bound and the real information rate, Fwi the user’s non spread signal bandwidth and w ≈ rc is the total system bandwidth. Usually, θi can be defined as [6]: θi = − where

BER max i

1.5 log(5 BER max i )

(4) maximize J(p) = p∈℘

is the maximum tolerable bit error rate.

In order to enable the users to have minimum QoS warranty, the minimum information rate can be mapped into SNIR through the Shannon’s capacity model using the gap introduced in (4). Note that for minimum SNIR γi∗ , eq. (3) uses the minimum information rate (ri,min ) established to the ith user belong to service or user class SERV. This way, it is possible to obtain the condition needed for the minimum SNIR to be satisfied given a minimum information rate: 2 γi∗ = − ln (5 · BER ∗serv ) (2mi − 1) 3

(5)

A. Weighted Throughput Maximization (WTM) The increasing information traffic demand due to multimedia services on third generation networks (3G) and beyond, along with the need of telecommunications companies to improve their profits have motivated the works on weighted throughput maximization (WTM) problem, which aims to maximize the system throughput, been formulated as: maximize p∈℘ s.t.

f (p) ri ≥ ri,min 0 ≤ pi ≤ pmax

(6)

where f (p) is a cost function that aims to maximize the information rate of each user; ri is the i-th user’s information rate, ri,min the minimum rate needed to ensure QoS for user i, p is the power vector such that p = [p1 , p2 , . . . , pU ], and pmax is the maximum transmission power allowed in the system. Therefore, we must incorporate the multirate criterion to the WTM problem subject to maximum power allowed per user. From this, the optimization problem is formulated as a special case of generalized linear fractional programming (GLFP) [9]. This way, the problem can be described as follows: v U  Y fi (p) i maximize J(p) = p∈℘ hi (p) i=1 (7) s.t. 0 < pi ≤ pi,max , γi ≥ γi∗ , ∀i = 1, . . . , U where vi > 0 is the priority of the i-th user to transmit with satisfied QoS requirements, assumed normalized, such that PU v = 1. It is noteworthy that the second restriction in i i=1 Eq. (7) is itself the minimum information rate and maximum tolerable BER for i-th user mapped into minimum SINR. This way, the functions fi (p) and hi (p) can be readily defined as: hi (p) =

U X

pj |gij |2 + σ 2

(8)

j6=i

2

fi (p) = θFi · pi |gii | + hi (p),

∀i = 1, . . . , U.

Note that the Eq. (7) is the productory ofQlinear fractional U exponentiated functions, and the function i=1 (zi )vi is an increasing function in a nonnegative real domain [10]. Based on these properties, problem (7) can be properly rewritten as: U X

=

vi [log2 fi (p) − log2 hi (p)]

i=1 U X

−

−

vi [f i (p) − hi (p)]

(9)

i=1

s.t. (C .1) 0 < pi ≤ pi,max , ∀i = 1, . . . , U (C .2) γi ≥ γi∗ , ∀i = 1, . . . , U where, with no loss of generality, in this work the weights   −1 1 XU 1 vi = , ∀i (10) j=1 Fi Fj has been adopted. Hence, the cost function turns into a sum of logarithms, which results in a monotonic nondecreasing function. An important remark on WTM problem is regarding the multirate service demands variable SNIR target. The fixed SNIR approach to power control problem discussed in [6] is suitable for lightly loaded single-service (voice) networks. When the network gets more heavily loaded, setting the target SNIR γ ∗ to be feasible becomes challenging. More importantly, in a wireless multi-service (voice, data & video) network the SNIR assignment can be optimized according to the traffic requirements and channel conditions. Higher SNIRs imply better data rates and possibly greater reliability, while smaller SNIRs can still provide lower data rates and/or lack of reliability. Generally speaking, the CDMA cellular network operators treat differently users with high rate service, in which higher bill paying users are preferentially identified by allocating higher QoS classes; as a consequence, different SNIR target values need to be set for users under different QoS classes. Hence, defining either the rate, eq. (3) or another QoS metric as a bijective mapping to SNIR, and optimizing their allocation according to an utility function, subject to the constraint in which the SNIR assignment must be feasible, i.e., a power vector where all element values is lower than pmax , and simultaneously can realize the minimum (target) SNIR vector. Therefore, in practice, the problem becomes a joint SNIR assignment and power control problem [3]. III.

N UMERICAL R ESULTS

In this subsection, the WTM problem is investigated through ACOR metaheuristic [11]. Results including information rate, transmission power and run time are compared with the analytical optimization approach performed by CVX tools [4]. In all numerical results, the same initial power-vector and quasi-static Rayleigh channel coefficients configuration have been adopted for both ACOR and CVX approaches. Typically, the quasi-static power channel losses in dB for a system with U users results in a relative power loss matrix of the interfering signals ranging from [8; 40] dB, representing a high interference regarding the direct line-of-sight links; besides, high cross-correlation among the spreading sequences has been adopted. Finally, for the WTM problem, the ACOR input parameters where adjusted in a non-exhaustive way.

1 Corresponds

to the time interval in which the channel characteristics do not suffer expressive variations.

ΣPACO U=12

26

ΣPopt U=20

24

ΣPACO U=20

17 16 15

ΣPopt U=30 ΣPACO U=30

22

13

ΣRopt U=12

12

ΣRACO U=12

11

ΣRopt U=20

Σ Powers [W]

14

ΣRACO U=20

10

20 18 16 14

ΣRopt U=30

9

ΣRACO U=30

12

8 10 7 8

6 5 0 10

1

2

10

3

10

10

6 0 10

1

Figure 1.

2

10

Iterations, N

3

10

10

Iterations, N

ΣR and ΣP evolution versus ACOR iterations.

Moreover, when dimension problem increases (U ≥ 30 users), the ACOR optimization approach is not able to converge completely under N ≤ 1000 iterations; for instance, in Fig. 1, despite the ΣRACO achieves optimal ΣRCVX , ∀U ∈ [2; 30] users, the ΣPACO > ΣPCVX for U = 30 users. Table II.

A CHIEVED MEAN POWER PER USER CLASS . U 12

20

30

Users - Class

Avg. Power [W]

6 - VOICE 4 - VIDEO 2 - DATA 9 - VOICE 6 - VIDEO 5 - DATA 16 - VOICE 9 - VIDEO 5 - DATA

0.0791 1.3730 1.9791 0.1900 1.2423 1.5619 0.0200 0.2022 1.0438

4

x 10 12

Rmin i

10

Ri

8

Ri

VOICE USERS

ACO

Rates [b/s]

CVX

6 4 2 2

4

6

8 10 User, u ∈ U

12

14

16

6

5

x 10

x 10

DATA USERS

VIDEO USERS

12

3

10 Rates [b/s]

Fig. 1 depicts the overall rate (ΣR) and overall system power (ΣP) evolution as a function of the ACOR iterations. Optimal solution ("opt") is obtained through analytical CVX approach. One can see the monotonic ACOR evolution for ΣR optimization, while ΣP is not monotonic, since the objective is to maximize the total system throughput. Besides, ΣP level is smaller for U = 30 than U = 12 and U = 20 due to two reasons: firstly, the average power level per user class decreases as the system loading increases, due to the need for QoS warranty to every user; secondly, the total power level depends on the number of users on each service class, since VIDEO and DATA class users use much higher power levels than VOICE users. The average power level per user class can

ΣPopt U=12

28

18

M ULTI - SERVICE MACRO - CELL DS/CDMA SYSTEM , CHANNEL AND ACO ALGORITHM INPUT GENERAL PARAMETERS

Parameters Adopted Values DS/CDMA Power-Rate Allocation System Noise Power Pn = −63 [dBm] Chip rate rc = 5 × 106 Max. Power per user pmax = 2 [W] Min. Power per user pmin = 0 [W] Time slot duration Tslot = 666.7µs # mobile terminals U ∈ {5; 10; 20; 30} users # Users per Class {U VOICE ; U VIDEO ; U DATA } U = [12, 20, 30]: [{6; 4; 2}, {9; 6; 5}, {16; 9; 5}] # base station BS = 1 Cell geometry Rectangular, xcell = ycell = 5 Km Mobile terminals distrib. ∼ U [xcell , ycell ] Fading Channel Type Path loss ∝ d−2 Shadowing Log-normal, σ2 = 6 dB Fading Rayleigh Time selectivity slow User Features and QoS User Services [VOICE; VIDEO; DATA] rc rc rc rc Min. User Rates ( F ) ri,min = [ 256 ; 16 ; 8 ] [bps] i ∗ −3 User BER target, BERserv [5 · 10 ; 5 · 10−5 ; 5 · 10−8 ] Basic (lowest) user-rate r = 19.5 [kbps] RA-ACO Algorithm File Size F s ∈ [8, 25] Diversity Factor q ∈ [0, 1]; Pheromone Evapor. Rate ξ ∈ [0, 1]; Population Size m ∈ [7, 35]; Max. # iterations N = 1000

30

19

(11)

−1 where Tslot = Rslot is the time slot duration, Rslot is the updating rate for the RA parameters, such as transmitted power vector and user symbol information. (∆t)c is the channel coherence time1 . As part of the SNIR estimation process, the channel is assumed constant in each optimization window, herein admitted Tslot = 667µs. Thus, the ACOR algorithm must converge to the solution within each 667µs interval. Numerical results are obtained by Monte-Carlo simulation (MCS) procedure over T = 1000 realizations.

Table I.

20

Σ Rates [Mbps]

Tslot < (∆t)c

be seen in table II, along with the number of users on each service class for each system loading evaluated.

Rates [b/s]

The DS/CDMA resource allocation simulations were carried out within the MatLab 7.0 platform; the main scenario parameters are presented in Table I. A rectangular cell with one base station in the center and users uniformly distributed across all the cell extension have been adopted; furthermore, all mobile terminals experience slow fading channels:

8 6

2.5 2 1.5

4

1

2

0.5 18

20 22 User, u ∈ U

24

26

27

28 29 User, u ∈ U

30

Figure 2. Minimum and optimized rates for CvX and ACOR tools in a system with U = 30 users.

the information rate for voice users decreases as the algorithm evolves since voice users have the smaller contribution in terms of overall system spectral efficiency (SE) regarding the other service classes. Hence, the cost function value decreases when the power level of voice users goes beyond ≈ 200 [mW]. On the other hand, individual user information rate increases for users of VIDEO and DATA classes, since they contribute much more for SE than VOICE users. This convergence characteristic is due to the weighting approach adopted in this work, eq. (10). Furthermore, accordingly, the dual behaviour for power levels convergence can be seen on Fig. 4.b. Next, the normalized mean square error (NMSE) figure of merit, regarding the analytical optimization (CVX) is evaluated in order to check the quality of solution achieved by the proposed ACOR approach. In this context, NMSE is expressed as: T 1 X ||xt [n] − x∗ ||2 NMSE[n] = · (12) T t=1 ||x∗ ||2

Fig. 2 and 3 depicts the individual Rates and Power levels for each user in a system with U = 30 users, respectively. It can be seen that the difference between the two algorithms results becomes more evident in terms of individual power levels. Besides, from Fig. 1 it is clear that for U = 30 users, the ACOR algorithm reaches a solution which spends more power for an overall throughput level marginally lower than CVX solution, i.e for U = 30 users condition, heuristic ACOR approach is not able to achieve total convergence (global optimum) within N ≤ 1000 iterations. Since the achievable information rate is a monotonic increasing function of power, one can argue invoking the trivial solution for the problem of throughput maximization, which consists of setting every user with the maximum transmission power level available. Nevertheless, the MAI would be very strong in this case, and the achievable sum rate generally is smaller than the optimum. Hence, under WTM problem formulation given by cost function (9), the heuristic optimization procedure proposed herein is able to find the best system throughput while keeping transmission power under feasible levels.

where || · ||2 denotes the squared Euclidean distance between vector xt at the t-th realization and the optimum solution vector x∗ given by the CVX solution; in this problem vector x can be the power vector p, or information rate vector, r = [r1 , r2 , . . . ri , . . . rU ]; T is the number of realizations, assumed herein T = 1000. Fig. 5 shows the NMSE evolution for systems with U ∈ [12; 20; 30] users. It is clear that ACOR converges superlinearly in direction to the optimum transmission power vector p∗ . Besides, the heuristic algorithm is able to find good near-optimum solutions (NMSE < 10−3 ) for systems with U ≤ 20 and a maximum of N = 1000 iterations, while for U = 30 the convergence is degraded slowly, although acceptable, NMSE ≈ 2 · 10−2 .

35 PACO i

30

CVX

Pi

Power [dBm]

25 20 15 10

1

10

5 0

0

10

0

5

10

15 User, u ∈ U

20

25

30 −1

10

a)

NMSE

Figure 3. Optimized power levels for CvX and ACOR tools in a system with U = 30 users.

b)

−2

10

−3

10 6

10

0

10

NMSEU=12

−4

10

U=20

NMSE

Rates [b/s]

Power [W]

NMSEU=30 −5

10

10 5

10

−2

0

1

10

2

10 Iterations, N

3

10

10

1

10

2

ACO Iterations

10

3

10

Figure 5. ACOR NMSE evolution regarding transmit power vector p for U = [12; 20; 30] users.

−1

10

0

10

0

10

1

10

2

10 Iterations, N

3

10

Figure 4. ACO convergence for a) individual user rates; b) individual user power levels in a system with U = 12 users.

The ACOR evolution through convergence for the individual power levels and information rate are depicted in Fig. 4.a. Indeed, the proposed heuristic approach is able to find suitable information rate under feasible power levels due to

Figs. 6 depicts the cost function value as a function for the transmission power allocation of the a) first user, p1 and b) the last user, p30 , while the others users hold individually their best power allocation given by ACOR solution computed after the optimization process, i.e., after N = 1000 iterations. One can see the evolution of the WTM-ACOR algorithm towards the optimum value of the cost function across the N = 1000 iterations. Indeed, in fig. 6-a (a VOICE user), it can be seen that cost function decreases when the allocated power level goes greater than ≈ 100 [mW]. Besides, when the power is smaller than this level, cost function varies just marginally, what shows that VOICE users do not contribute considerably to the spectral efficiency. On the other hand, for a DATA user Fig.6-b shows

that DATA (or even VIDEO) users contribute more effectively to improve remarkably the SE, since cost function value results much smaller when users belong to this class of service are not operating in its optimal power levels.

ACOR regarding CVX for the three system loading considered. One can note that for U = 30 users, the meta-heuristic is able to reach 99.5% of the optimum throughput, expending 27% more power, with 43.3% less run time than CVX. Hence, if we consider circuitry power, the solution achieved by ACOR is very close to the optimal SE (5% smaller, in the worst case); on the other hand, heuristic ACOR optimization approach can achieve significant EE even under the overall SE maximization perspective due to the lower circuitry power usage to run the heuristic optimization procedure.

6.4 Cost Function ACO p

6.2

ev

Cost Function [bit / (s ⋅ Hz)]

6

Max. Cost Function

5.8 5.6

Table III.

5.4 5.2

U

5

12 20 30

4.8 4.6 4.4 4.2

b)

a) −4

−2

10

10 p [W]

0

10

−4

−2

10

p

1

30

10 [W]

0

10

Figure 6. Cost function behavior for the optimal vector p∗ , except to a) p1 and b) p30 user, in a system with U = 30 users.

A. Run Time Analysis Run time analysis corroborates that ACOR ’s computational complexity is remarkably smaller than the CVX approach, what makes the metaheuristic ACOR an interesting procedure if the circuitry power consumption along with transmission power levels should be considered. Hence, as a consequence, the energy efficiency of the system can be improved, even under the spectral-energy efficiencies trade-off perspective suggested herein. Furthermore, since the ACOR computational 700

[1]

600

[2] 500

Run Time [s]

[3] 400

[4] 300

[5] 200

[6]

100

20 Users, U

30

[7]

Figure 7. Run time for the WTM optimization problem as a function of the number of the multirate users.

complexity is lower regarding the CVX, and aiming to provide an efficient optimization method in terms of quality of solution versus computational complexity trade-off, a performance analysis regarding robustness and run time is addressed in the following. The run time for ACOR and CVX as a function of the number of users is depicted in Fig. 7. It is clear that the complexity of ACOR is remarkably smaller than CVX, at the cost of lower convergence quality when the system loading is high enough (U ≤ 30 users).Notwithstanding, Table III shows P P the achieved percentages of P, R and Run Time for

P

R [%]

100 100 99.5

P

P [%] 100 100 127

Run Time [%] 22 12.12 8.91

IV. C ONCLUSIONS In this contribution, the metaheuristic ACOR optimization approach has been successfully applied to the weighted throughput maximization problem in a multirate code-division multiple access network under realistic wireless mobile channels and system operation conditions. Numerical results have demonstrated that the metaheuristic ACO-based WTM method is very competitive regarding the analytical CVX approach in terms of both suitable performance metrics and reduced run-time complexity. It is clear that ACOR is a promising approach in solving the WTM optimisation problem, and more importantly, the developed metaheuristic optimization procedure has demonstrated to be useful in order to obtain energy and spectral efficient systems for the next wireless generation green communication networks. R EFERENCES

ACO CVX

12

P P A CHIEVED PERCENTAGES OF P, R AND RUN T IME FOR ACO R REGARDING ANALYTICAL CVX .

[8]

[9]

[10]

[11]

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