Applications of Multi-Objective Optimization Techniques ... - IEEE Xplore

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343

Applications of Multi-Objective Optimization Techniques in Radio Resource Scheduling of Cellular Communication Systems Mohammed Elmusrati, Member, IEEE, Hassan El-Sallabi, Member, IEEE, and Heikki Koivo, Senior Member, IEEE

Abstract— Novel objectives such as very low outage, high capacity, and high throughput are major challenging problems in radio resource management of mobile communication systems. More specifically, in radio resource scheduling (RRS), the aim is how to optimize available resources such as transmission power and data rate to achieve certain targeted objectives. Conventional RRS algorithms are based on optimizing one objective while keeping others as constraints. This paper proposes a novel distributed RRS algorithm based on analytic multi-objective optimization. The proposed algorithm relaxes the constraints and jointly optimizes all the required objectives. Infinity set of optimal solutions, called Pareto optimal, is obtained. Each solution in the set is optimal in a specific sense. The decision maker selects the required solution that fulfills the network requirements and conditions. Some of the conventional RRS algorithms are special cases of our multi-objective based algorithm. Detailed mathematical analysis of the proposed algorithm is given. Simulation results show the behavior of the proposed algorithm as well as its advantages over conventional algorithms. Index Terms— Multi-objective optimization, radio resource management, CDMA, cellular systems.

I. I NTRODUCTION

M

ULTI-OBJECTIVE (MO) optimization has a very wide range of successful applications in engineering and economics. Such applications can be found in optimal control systems [1], chemical engineering [2], economics [3], and engineering design [4]. The MO optimization is not widely applied in telecommunication engineering. Some papers proposed the MO optimization to handle certain problems in telecommunications, but by using genetic algorithms [5], [6] rather than analytical methods. Genetic algorithms are generally not suitable for fast real-time applications such as resource scheduling due to the high computational cost. To the best of our knowledge, our work is the first to propose analytical MO optimization for handling the RRS problems as shown in [7],[8], and recently in [26]. More general results are reported in [18]. MO formulation for the ConnectionAdmission Control using neural networks has been recently Manuscript received August 2, 2006; revised December 1, 2006 and March 21, 2007; accepted April 18, 2007. The associate editor coordinating the review of this paper and approving it for publication was G. D. Mandyam. M. Elmusrati is with the Department of Computer Science, University of Vaasa, Finland (e-mail: [email protected]). H. El-Sallabi is with the Radio Laboratory, Helsinki University of Technology, Finland (e-mail: [email protected]). H. Koivo is with the Control Engineering Laboratory, Helsinki University of Technology, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2008.060533.

proposed in [9]. The MO optimization is applied to find the optimal solution which is a compromise between multiple and contradicting objectives. It is important to refer here to the difference between joint optimization and MO optimization. In joint optimization it is not necessary for the objectives to be contradicting. For example the optimum combining between power control and beamforming could be done by joint optimization. Furthermore, in joint optimization we are usually interested in one optimal solution which could be a global or local extreme point of the combined objective function. In MO optimization we are more interested in the Pareto optimal set which contains all non-inferior solutions. The decision maker can then select the most preferred solution out of the Pareto optimal set. The weighted sum method to handle MO optimization applied in this paper is structurally similar to the joint optimization. However, we emphasize that, the RRS problem is formulated as a MO problem and other MO techniques can be used to solve it (see Appendix I). Although the weighted sum is simple and straightforward method to handle MO optimization problems, it should be noted that it is not efficient in case of nonconvex objective space [15]. One interesting application of the joint optimization between RRS and linear systems can be found in [24]. The radio resource management (RRM) is an essential component in multi-user mobile communication networks. It contains many sub-blocks such as connection admission controller, traffic classifier, radio resource scheduler, and interference and noise measurements [10]. The main operation of the RRM is to manage and optimize the different available resources to achieve a list of Quality of Services (QoS) requirements. The radio resource scheduler controls two important radio resources: transmit power and data rate. The RRS uses these two resources to achieve different objectives such as maximizing the number of simultaneous users, reducing the total transmit power, and increasing the total throughput. The conventional approach to achieve these objectives is to optimize for a selected target and keep the other objectives as constraints. The MO technique is a powerful tool to jointly compromise between all contradicting objectives. In this technique there are many non-dominated optimal solutions, which can be infinite in number, called Pareto optimal set. More discussion is given in Appendix I. Each solution of Pareto set is optimal in a given sense. The decision maker selects the most preferred solution. This paper is constructed as follows: Section II presents the

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system model discussed in this paper. Section III introduces the optimization model of the RRS problem with MO formulation. Section IV introduces a novel MO distributed power and rate control algorithm as an example of the application of MO optimization in RRM. Simulation results are presented in Section VI. Finally, the conclusions and remarks are given Section VII. II. S YSTEM M ODEL System level, multi-cell and multi-user, cellular communication system is considered in this paper. Without loss of generality, only the uplink is discussed. Real time data communication scenario is assumed. The multiple access method is the direct sequence-code division multiple access (DS-CDMA) with adaptive processing gain. General adaptive modulation techniques can be considered with minor modifications to the algorithms. Competitive environment is assumed, i.e. each user (alternatively we also use the term mobile terminal MT) demands to transmit at largest possible data rate. In addition, a MT is in outage if it cannot achieve its minimum required data rate. Each MT has a specific target of energy per bit to noise (and interference) power spectral density Eb /N0 . In this paper we refer to Eb /N0 as signal to noise ratio (SNR). At the input of the matched filter in the receiver we express the signal quality by carrier to interference ratio (CIR), which is defined as the average received power from desired transmitter divided by the average received power from other transmitters plus noise. In DS-CDMA we may express the SNR as the multiplication of the CIR and the processing gain. The target SNR depends on the allowed maximum bit error rate (BER). The target SNR has a predefined margin starting with the minimum allowed SNR; if it cannot be fulfilled, then the MT is in outage. Perfect feedback channel is assumed, i.e., each MT knows its own exact temporal SNR at the base station (BS). This assumption can be relaxed by estimating the SNR at the mobile terminal by using the quantized power control command of the current mobile systems [22]. The effects of this estimation on the MO power control algorithm are also discussed in [23]. Without loss of generality and only for sake of comparison it is assumed that all MTs use same coding properties. This assumption enables us to compare between the users’ throughputs by direct comparison between their data rates. In our model, the throughput is adapted by the processing gain only. To generalize the model to include also adaptive modulation level, one proper mapping between SNR and BER for each modulation level in the fading channel has to be considered. III. M ATHEMATICAL F ORMULATION OF THE RRS P ROBLEM IN MO F RAMEWORK This section starts by defining the main objectives to be achieved by the RRS. We present two common conventional methods to formulate the RRS. After that we show in details how to formulate the RRS problem with MO optimization. The considered RRS objectives are to: • O1 ) minimize the total transmit power • O2 ) minimize the outage • O3 ) maximize the throughput.

It is clear that objective (O1 ) generally conflicts with objective (O3 ) since reducing the transmit power of a user without constraints leads to decreasing the data rate and/or the SNR. Objective (O3 ) is incompatible with objective (O2 ) because the users with high throughput occupy most of the available radio resources which leads to high system outage. In the literature, the RRS problem is usually formulated as a single objective optimization problem (minimizing cost function or maximizing certain utility function) and treats others as constraints. Two formulations are very widely used in the literature. The first formulation is based on finding the optimal power and rate vectors that maximize the total throughput (or some other utility function), i.e, objective (O3 ), and using the target SNR and transmit power as constraints [11]-[13], [25]. The second formulation is based on minimizing the total transmit power, i.e., objective (O1 ), and using the target SNR and data rates as constraints [14]. This work proposes a third formulation by optimizing jointly the objectives (O1 -O3 ) of the RRS using multiobjective optimization approach. This formulation leads to a more general solution than the conventional methods. Appendix I introduces some basic concepts of the MO optimization techniques. The RRS problem can be formulated using MO optimization technique as follows. Find the rate vector R = [R1 , ..., RQ ] and the power vector P = [P1 , ..., PQ ] that minimize the following vector of objectives min {f1 , f2 , −f3 } P,R

(1)

subject to P ∈ Sp and R ∈ SR where each cost function f1 to f3 corresponds to objectives (O1 ) to (O3 ), respectively, Sp is a nonempty region of feasible power solutions, i.e., the possible transmission power values, SR is a nonempty region of feasible rate solutions, i.e., the possible transmission data rate values, and Q is the number of active mobile terminals. Note that the minus sign is used to maximize the objective. The selection of proper cost functions depends on many factors such as the type of scenario, simplicity to solve, etc. Different objective functions can be used for (1), however, it is an open and rich area of research. As an application of (1), a novel radio resource scheduler is introduced in the next section. IV. M ULTI -O BJECTIVE D ISTRIBUTED P OWER AND R ATE C ONTROL A LGORITHM In this section a distributed radio resource scheduler based on MO optimization is proposed. The multi-objective distributed power and rate control (MODPRC) algorithm is distributed in the sense that each MT requires the knowledge of only its own carrier to interference ratio (CIR) to determine its optimum transmit power and data rate. This feature is important to reduce the signaling information and also in infrastructureless communication systems. The proposed algorithm has an interesting and unique feature that when the MT uses more transmit power than required, then its own QoS will be degraded, i.e. each MT suffers a penalty in using high transmit power [8],[18], [26]. Before introducing the objective functions, some important basic relations for multirate CDMA systems are presented.

ELMUSRATI et al.: APPLICATIONS OF MULTI-OBJECTIVE OPTIMIZATION TECHNIQUES IN RADIO RESOURCE SCHEDULING

A. Basic relations for multi-rate CDMA systems The relation between the transmit power and the data rate for CDMA system with variable spreading factor can be represented as δi (t) =

Rs Γi (t), t = 0, 1, .. Ri (t)

(2)

where δi (t) is the average SNR of user i at time slot t, Rs is fixed chip rate, Ri (t) is the data rate of user i during time slot t, Γi (t) is the average CIR of user i during time slot t given by Γi (t) = Q

Pi (t) Gii (t)

j=1,j=i

Pj (t) Gij (t) + σn2

(3)

where Gij is the channel gain between user j and base-station i, σn2 is the average additive white noise power at the receiver input, and Q is the number of active mobile terminals. The maximum allowed BER can be determined by specifying the target SNR value. The allowed BER depends on the application; e.g., generally a higher BER can be allowed for voice applications than for data applications. It should be noted that for time-varying fading channels the mapping between the SNR and the BER is not a trivial task, see e.g. [27] for more information. In this paper we are more concerned about the SNR. The corresponding BER can be derived for the considered channel type and the modulation level. From (2) it is clear that by fixing the SNR for user i, increasing the CIR will increase the achieved data rate Ri (t) as follows Ri (t) =

Rs Γi (t) , t = 0, 1, .. δiT

(4)

where δiT is the target SNR for user i. To achieve only objective (O3 ), i.e., maximizing total throughput, implies that all users must send at highest possible transmit power which leads to a very high outage. If a proper dropping algorithm is used, then only one or a few users (with low correlation between them) will be supported [28]. To reduce the outage probability, we define the minimum required CIR of user i at the minimum allowed data rate as Ri,min T δ (5) Γi,min = Rs i The maximum CIR of user i is defined at the maximum possible data rate such as, Γi,max =

Ri,max T δ Rs i

(6)

For a user to be in service, at least the minimum required CIR has to be achieved. This corresponds to the minimum allowed data rate. Next we explain one realization of the MO method to handle the RRS problems. In this realization we assume a competitive environment, i.e., each MT will try to get the highest possible resources and at the same time considering the other terminals. B. MO Scheduler The MO scheduler is defined as follows: We assume, without loss of generality, same maximum possible data rate Rmax for all users with target SNR vector

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  T and minimum data rate vector Rmin = δ T = δ1T , δ2T , .., δQ [R1,min , R2,min , .., RQ,min ]. Our objective is to find the optimum power vector P = [P1 , P2 , .., PQ ] and the optimum data rate vector R = [R1 , R2 , .., RQ ] that minimize the following cost function Q  N  γ N −t e2i (t), (7) J= i=1 t=1

subject to P ∈ Sp and R ∈ SR where N is the optimization time window, γ is a real-valued constant adaptation factor. The error function ei (t) is defined according to the weighted metrics method with p = 1 (see Appendix I) ei (t) = λi,1 |Pi (t) − Pmin | + λi,2 |Γi (t) − Γi,min |

(8)

+λi,3 |Γi (t) − Γi,max | where 0 ≤  λi,k ≤ 1 (∀k = 1, 2, 3), are real-valued tradeoff 3 factors, and k=1 λi,k = 1. The advantages of combining the weighting metrics method with the least square formula (7) are • The least squares method is well known and its solution is straightforward. • General solution can be obtained by using (7) to minimize over all users for time window N . Error function (8) is one mathematical interpretation of the RRS objectives (O1 -O3 ). The first term is set to minimize the transmit power Pi (t) and to be as close as possible to Pmin . In other words, there is a penalty for using extra power. This term represents the objective (O1 ). Objective (O2 ) is achieved with the second term of the error function. In this term, the transmit power is selected so that the resultant CIR is as close as possible to the minimum required CIR. Achieving the minimum required QoS for every MT minimizes the total system outage. The third term in (8) represents objective (O3 ), where the users try to be as close as possible to the maximum possible CIR, so that they can transmit at the highest data rate. The tradeoff between these contradicting objectives is achieved by the tradeoff factors λi,1 , λi,2 , and λi,3 . By solving (7) and (8) in one-dimensional case (N = 1) we obtain the MODPRC algorithm: Pi (t + 1) =

λi,1 Pmin + λi,2 Γi,min + λi,3 Γi,max Pi (t) (9) λi,1 Pi (t) + (λi,2 + λi,3 ) Γi (t) Ri (t + 1) =

Rs Γi (t) δiT

Pmin ≤ Pi (t) ≤ Pmax ; Ri,min ≤ Ri (t) ≤ Rmax

(10) (11)

The derivation details are presented in Appendix II. Correlated channel is assumed since there is a delay of at least one time slot between the CIR measurement and the power and rate update. In other words we assume that the time slot duration is less than the coherence time of the channel. The MODPRC algorithm given by (9)-(11) has some interesting characteristics. By changing the values of the tradeoff factors λi,1 , λi,2 , and λi,3 , different optimum solutions in different senses are obtained. Let’s consider first the extreme cases. For example, for achieving only objective (O1 ) (set λi,1 =

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1, λi,2 = 0, and λi,3 = 0), it is clear that MODPRC algorithm becomes a fixed level of transmit power, i.e., no power control is used, and user i will send always at the minimum power. For λi,1 = 0, λi,2 = 1, and λi,3 = 0, the MODPRC returns to the well-known distributed power control (DPC) algorithm [20], which means that the DPC is a special case of the MODPRC. In this case, each MT tries to achieve its minimum required data rate, so that the outage is minimized. At λi,1 = 0, λi,2 = 0, and λi,3 = 1, each MT will attempt to transmit at the maximum allowed data rate. The interference will be rather high. If proper dropping algorithm is used, then most of the users will be dropped out and only one or few users with high data rate services will be supported, i.e., the outage will be high. From previous extreme conditions, one can make a tradeoff between these objectives to obtain the best possible performance according to the required specifications. The selection of the tradeoff values should be based on the communication link condition as well as the network and the user requirements. A wide range of different solutions can be obtained by changing values of the tradeoff factors. The selection of the right solution is the job of the decision maker. The tradeoff factors can be time-varying, i.e., updated with time to achieve certain objectives [18]. The rules of the decision maker are not considered here, but it is an interesting topic for future research. We will indicate one heuristic method to select the values of the tradeoff factors in the simulation section.

C. Convergence of MODPRC algorithm The convergence behavior of the MODPRC algorithm is proved in Proposition 1. Before that we define the standard interference function and then introduce two theorems from [19]. Definition 1: Interference function Ψ (.) is called standard when the following properties are satisfied for all components of the nonnegative power vector P [19]: • • •

Positivity: Ψ (P) > 0;   ¯ then Ψ (P) ≥ Ψ P ¯ >0 Monotonicity:If P ≥ P Scalability: For all real α > 1, αΨ (P) > Ψ (αP)

Definition 2: The network configuration is said to be feasible if there exists a power vector P∗ ∈ {P : Pmin ≤ P ≤ Pmax } and a rate vector R∗ ∈ {R : R ≥ Rmin } such that Γi ≥ Γi,min , for all i = 1, . . . , Q. Theorem 1: If the standard power control algorithm has a fixed point, then that fixed point is unique [19]. Theorem 2: If Ψ (P) is feasible, then for any initial power vector P0 the standard power control algorithm converges to ˆ [19]. a unique fixed point P Proposition 1: For any P (0) ≥ 0, the MODPRC algorithm (9)-(11) with λi,1 > 0 will always converge to a unique fixed ˆ At λi,1 = 0 the feasibility condition is necessary for point P. convergence. Proof : First we prove that the MODPRC is a standard power control algorithm. Then according to Theorems 1 and 2 the MODPRC algorithm converges to a unique fixed point.

The interference function Ψi (P (t)) of the MODPRC algorithm for user i is given by Pi (t + 1) = Ψi (P (t)) λi,1 Pmin + λi,2 Γi,min + λi,3 Γi,max Pi (t) (12) = λi,1 Pi (t) + (λi,2 + λi,3 ) Γi (t) Since the denominator of (3) is the total interference experienced by user i, we may define the normalized total interference of user i as follows Iˆi (t, P) =

Q 

Pj (t)

j=1,j=i

σn2 Gij (t) + Gii (t) Gii (t)

(13)

The CIR of user i in (3) becomes Γi (t) =

Pi (t) ˆ Ii (t, P)

(14)

When t is dropped for simplicity, and (14) is substituted into (12), then (12) can be rewritten as Ψi (P) =

Iˆi (P) k , i = 1, 2, . . . , Q ˆ i,2 λi,1 Iˆi (P) + λ

(15)

ˆ i,2 = λi,2 + λi,3 and k = λi,1 Pmin + λi,2 Γi,min + where λ λi,3 Γi,max . From (13) , it is clear that, for any (16) P ≥ 0, Iˆi (P) ≥ 0   ¯ = P¯1 , P¯2 , . . . , P¯Q . It follows Define another power vector P from (13) that when:   ¯ ¯ ⇒ Iˆi (P) ≥ Iˆi P (17) P≥P ˆ i,2 = 1 − λi,1 , equations (15) and Since 0 ≤ λi,1 ≤ 1 and λ (16) imply that for any P ≥ 0 ⇒ Ψi (P) ≥ 0, i = 1, 2, . . . , Q

(18)

This proves the positivity condition. The monotonicity condition can be proven by contradiction. Assume   that there exists ¯ , i = 1, 2, . . . , Q. ¯ such that P ¯ ≤ P and Ψi (P) < Ψi P aP Then from (15), for all i = 1, 2, . . . , Q   ¯ k Iˆi P Iˆi (P) k < (19)   ˆ i,2 ˆ ˆ ˆ ¯ +λ λi,1 Ii (P) + λi,2 λi,1 Ii P    ˆ i,2 ¯ k λi,1 Iˆi (P) + λ Iˆi P (20) ⇒ Iˆi (P) k <   ˆ i,2 ¯ +λ λi,1 Iˆi P

 ˆ ¯   ˆ i,2 Ii (P) ¯ +λ λi,1 Iˆi P Iˆi (P) ⇒ Iˆi (P) k < Iˆi (P) k (21)   ˆ i,2 ¯ +λ λi,1 Iˆi P But from (17), one can conclude that

   ¯) Iˆi (P ˆ ˆ ¯ λi,1 Ii P + λi,2 Iˆ (P) i 0< 0 ⎤ ⎡ Q 2  σ (t) G ij n ⎦ + αIˆi (P) = α ⎣ Pj (t) Gii (t) Gii (t)

(24)

j=1,j=i

Q 

≥

αPj (t)

j=1,j=i

σn2 Gij (t) + = Iˆi (αP) Gii (t) Gii (t)

The equality is achieved if the additive noise is zero. From (15) we obtain αΨi (P) = α

Iˆi (P) k Iˆi (αP) k ≥ ˆ i,2 ˆ i,2 λi,1 Iˆi (P) + λ λi,1 Iˆi (P) + λ

(25)

From (13), it follows that for α > 1, Iˆi (αP) > Iˆi (P) ∀i, thus αΨi (P)

> =

Iˆi (αP) k ˆ i,2 λi,1 Iˆi (P) + λ

(26)

Ψi (αP) , i = 1, 2, . . . , Q

Then the scalability condition has been proven. From (18),(23), and (26) one can say that the MODPRC algorithm is a standard interference function. This means that the MODPRC algorithm converges to a unique fixed point. At λi,1 = 0, the MODPRC algorithm becomes the DPC algorithm. The feasibility condition is necessary for DPC algorithm to converge [20]. As stated earlier the behavior and performance of the MODPRC algorithm is determined by the values of the tradeoff factors, which determine the optimization sense. The decision maker determines the values of the tradeoff factors. Such a system can be designed using a heuristic, systematic, fuzzy, or expert system approach. Generally, a systematic approach is possible when the whole optimal Pareto set is available. In this case the decision maker may define general rules to select the most preferred solution. Radio resource scheduling problem is on-line and time varying so that it is not possible to find all optimal Pareto set before making selection decision. In this case one should use different approaches to tune the tradeoff factors. However, in most situations, the decision maker’s global disutility function is not available [30]. In a static environment or with snapshot assumption, define Piss and Γss i as steady state power and steady state CIR for user i respectively. If the system is feasible, i.e. each user establishes a steady state CIR between Γi,min and Γi,max , then from (9): Γss i

= +

λi,1 (Pmin − Piss ) (27) λi,2 + λi,3 λi,3 λi,2 Γi,min + Γi,max λi,2 + λi,3 λi,2 + λi,3

From (27) one can observe an interesting feature of the MODPRC algorithm that as the steady-state transmit power of a user is increased, the steady state CIR of this user degrades. This can enhance the fairness in distributed communication systems. To evaluate the performance of the proposed MODPRC algorithm we compare it with two centralized algorithms. First we compare it with the optimum combining of power and rate (OCPR) algorithm [18]. The optimum combining algorithm is based on solving (3) and (4) together for all possible data

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rates. Here we assume a finite set of data rates. The number of possible solutions increases dramatically with number of users and number of possible rates which makes the OCPR impractical. For example for 10 users and 7 possible data rates we will have 710 ≈ 3 × 108 different solutions! Because of the high computational load of the OCPR algorithm, we compare it with the MODPRC algorithm for a small dimension scenario as is shown in the next section. A comparison is also made with the Maximum Throughput Power Control (MTPC) algorithm [21]. The MTPC is a centralized algorithm. In centralized algorithms, the transmission power of all MTs has to be computed jointly by one organizer. The information of all MT link gains as well as the CIR values has to be available (and updated at every time slot) at that organizer. This makes the centralized algorithm unattractive in practical applications. In cellular applications, centralized algorithms have only theoretical value and they are used mainly for comparison purposes. The MODPRC algorithm is compared to MTPC algorithm for more realistic scenarios as shown in the next section. V. S IMULATION R ESULTS In this section we evaluate the performance of the proposed algorithm under different scenarios. First scenario shows the performance of the MODPRC algorithm at extreme tradeoff factor values (λi,1 , λi,2 , λi,3 ) ∈ {(1, 0, 0) , (0, 1, 0) , (0, 0, 1)}. It considers five users uniformly distributed in one cell under snapshot assumption. White Gaussian noise is added with zero mean and −63 dBm average power at the input of the receiver. The maximum transmit power is 1 W. The minimum allowed SNR is 6 dB. A user is considered in outage, if at least one of the QoS requirements, e.g., the minimum data rate or the minimum allowed SNR is not achieved. Figures 1, 2, and 3 depicts the average power, sum of the data rates and the outage probability respectively. The time slot in the x-axis is defined as the time where the transmit power as well as the data rate is updated. Figure 1 shows the average power for the three extreme cases. In the first case (λi,1 , λi,2 , λi,3 ) = (1, 0, 0) ∀i = 1, . . . , 5, the objective is to minimize the total power. Thus, the power is very small compared to other two situations. Meanwhile, the sum of data rates is (almost) zero and the outage is very high (100%) as shown in Figures 2, and 3, respectively. In the second case (λi,1 , λi,2 , λi,3 ) = (0, 1, 0) ∀i = 1, . . . , 5, where the objective is to minimize the outage. The average power and the sum of data rates are fair, and the outage converges to zero as shown in Figures 1, 2, and 3, respectively. In the third case (λi,1 , λi,2 , λi,3 ) = (0, 0, 1) ∀i = 1, . . . , 5, where the objective is to maximize the total data rates. The average power and total data rate are the highest. The outage is considerably high as shown in Figures 1, 2, and 3, respectively. From the results one can see that the performance of the MODPRC algorithm has a wide range of behavior depending on the selected values of the tradeoff factors. The tradeoff factors, used in the next part of simulation, are selected heuristically. The selection process is done by

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1

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Outage comparison of MODPRC at extreme tradeoff factors.

Fig. 1. Average power comparison of MODPRC at extreme tradeoff factors. x 10

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Data rate comparison of MODPRC at extreme tradeoff factors.

defining the relative importance for the objectives. However, it should be noted that, the values of the tradeoff factors do not reflect the exact importance level of the objective, unless all objectives are normalized. For example, if the tradeoff factors are selected as (λi,1 , λi,2 , λi,3 ) = (0.01, 0.80, 0.19), it does not mean that the power consumption importance is 1%, the outage importance is 80%, and the throughput importance is 19%. However, when the tradeoff factors are selected as (λi,1 , λi,2 , λi,3 ) = (0.05, 0.80, 0.15), it means that the importance of the power consumption is higher than in the previous case and the throughput importance less. This understanding could be used when utilizing adaptive tradeoff factors. This can be an interesting topic for further research. From intensive simulations, it has been observed that keeping λ1 at a small positive value (0.001 − 0.01), gives generally good results in terms of outage and throughput compared to higher values of λ1 . Now, for the same previous scenario, we compare the MODPRC algorithm versus the MTPC algorithm and the OCPR algorithm. The optimum power and rate vectors are generated

Fig. 4. Total data rate comparison between the optimum, MTPC, and MODPRC algorithms.

by generating all possible solutions for a given situation. In the case of five user scenario with 7 possible data rates we have 75 = 16807 possible solutions! Then we select the optimum solution in a given sense, for instance the feasible solution with maximum total throughput, see [18] for more details. A performance comparison between the OCPR, the MODPRC and the MTPC algorithms is presented in Figure 4. The objective of the OCPR algorithm is to achieve the maximum total data rate with zero outage and within power constraints. In Figure 4 we can see that MODPRC algorithm has achieved the same total rate. This has been obtained with the following tradeoff factors (λi,1 , λi,2 , λi,3 ) = (0.001, 0.830, 0.169). The MTPC algorithm fails to achieve the same total data rates as can be seen in the same figure. The comparison between the considered algorithms in terms of the average power is shown in Figure 5. It is clear that the optimum average power is less than the average transmit power in the other algorithms (MTPC and MODPRC). The average power of the MTPC algorithm is less than the MODPRC algorithm. The outage probability was 0.20 for both MODPRC and MTPC algorithms.

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1

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0.9 0.8

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Fig. 7. Average data rate comparison between MTPC and MODPRC at two different tardeoff factors.

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Fig. 6. Average power comparison between MTPC and MODPRC at two different tardeoff factors.

The performance of the MODPRC algorithm is tested in a more realistic scenario in order to evaluate its behavior in dynamic channels. We consider 100 active MTs uniformly distributed in an area of 16km2 operated by 16 base stations. The DS-CDMA technique with chip rate of 3.84 Mchip/s is assumed as the multiple access method between users. Different data rates can be achieved by changing the processing gain. The minimum requested data rate for all users is 15 kb/s. The maximum data rate is 960 kb/s. The minimum allowed SNR is 6 dB. Uncorrelated log-normal shadowing is assumed with 0dB mean and 8dB variance. The MTs move randomly with a maximum speed of 30 km/h. The signal is received in multipath without dominant path (Rayleigh channel). We assume that the channel characteristics do not change during the time slot. This assumption can be justified for a short time slot duration. A competitive environment where every MT tries to get the highest possible data rates is assumed. Instantaneous and perfect handover is assumed, i.e., the MT is assigned to the base station which provides the best channel conditions. The power and data rates are updated at every time slot. The time slot duration is 0.66 ms or 1500 slot/s. The achieved

performance for every MT is determined by its tradeoff factors and its channel conditions. Same values of the tradeoff factors are selected for all users. The simulations have been done for two different sets of values of the tradeoff factors. We call these cases A and B to show the effect of selecting the proper tradeoff factors for the required objectives. In case A, our main objective is to minimize the total outage. For this reason we gave this objective 90% of the total weight. Considerable weight (9%) is also given for minimizing the transmitted power from each MT to reduce the total inter and intra interference. Finally we give small weight for maximizing the data rate (only 1%), but even with this small weight, the MTs with excellent channel conditions will send at a high data rate. From previous the tradeoff factor values for case A are (λi,1 , λi,2 , λi,3 ) = (0.09, 0.90, 0.01). As stated earlier, these values do not necessarily reflect the absolute importance of the objectives. In case B, our main objective is still to keep the total outage at a small value but we gave much higher weight for maximizing the data rate (39%). Because the objectives are contradicting we should expect worse outage than in case A, but with higher data rates. The tradeoff factors in case B are chosen to be (λi,1 , λi,2 , λi,3 ) = (0.01, 0.60, 0.39). The results of the MODPRC algorithm with these two sets of values of tradeoff factors are compared to the MTPC algorithm [21]. Figure 6 shows the average transmit power required by the two cases of the MODPRC and MTPC algorithms. In case A, the MODPRC algorithm needs very small average power compared to the MTPC algorithm. It is clear also that the MODPRC algorithm in case A uses smaller average power than case B. But the average transmitted data rate is also very small for case A as shown in Figure 7. It is clear that higher average data rate is achieved for the MODPRC algorithm in case B than in case A. Figure 8 depicts the outage probability versus time slot number. In this figure we assume instantaneous outage. This means that for every time-slot, if any MT does not achieve at least its minimum allowed SNR and its minimum data rate, it will be counted as being in outage for that time slot. Case A of

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0.5 (0.09,0.9,0.01) MTPC (0.01,0.6,0.39)

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0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

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Fig. 8. The outage probability comparison between MTPC and MODPRC at two different tardeoff factors.

MODPRC algorithm has a much smaller outage than case B. When comparing the performance of the MODPRC with the MTPC algorithm, one should keep in mind that the second algorithm is a centralized and the first one is a distributed algorithm. It is clear that very different results can be obtained with different values of the tradeoff factors. The values of the tradeoff factors should be selected according to the user requirements and the link conditions. VI. C ONCLUSION In this paper we propose a novel analytical approach applying multi-objective optimization to tradeoff between the different resources in the radio resource scheduler. The MO optimization has the advantage of optimizing over different and conflicting objectives jointly. A novel algorithm has been proposed as an example of the application of MO optimization in RRS. The algorithm is called multi-objective distributed power and rate control. Simulation results have been presented to show the wide range of performance that can be obtained by adopting the MO optimization. The topic is very rich and this paper opens the doors for many future research issues. Some of them are: optimization of the RRS using different analytical MO optimization methods, cross-layer optimization using analytical MO optimization, and how to define the rules of the decision maker to select the best tradeoff factors which compromise between network as well as terminal requirements and link conditions. A PPENDIX I M ULTI -O BJECTIVE OPTIMIZATION TECHNIQUES The MO optimization is a technique to find the best solution between different and usually conflicting objectives. In the MO optimization problem we have a vector of objective functions. Each objective function is a function of the decision (variable) vector. The mathematical formulation of the MO optimization problem is [15] min{f1 (x) , f2 (x) , . . . , fm (x)}

(28)

Fig. 9. Geometric interpretation of Pareto and weakly Pareto optimal solutions.

subject to x ∈ S where we have m ( ≥ 2 ) objective functions fi : Rn → R, x is the decision (variable) vector belonging to the (nonempty) feasible region (set) S, which is a subset of the decision variable space Rn . The abbreviation min means that we want to minimize all the objectives jointly. Usually the objectives are at least partially conflicting and possibly incommensurable. This means that, in general there is no single vector x that can minimize all the objectives simultaneously. Otherwise, there is no need to consider multiple objectives. Hence, the MO optimization technique is used to search for efficient (non-inferior) solutions that can best compromise between different objectives. Such solutions are called Pareto optimal solutions. Definition A1: A decision vector x∗ ∈ S is Pareto optimal, if there does not exist any other decision vector x ∈ S such that fi (x) ≤ fi (x∗ ) for all i = 1, 2, . . . , m and fj (x) < fj (x∗ ) for at least one index j [15]. The Pareto optimal set is a set of all possible (infinite number) Pareto optimal solutions. The condition of optimal Pareto set is rather strict and many MO algorithms cannot guarantee to generate Pareto optimal solutions but only weak Pareto optimal solutions. Weak Pareto optimal solutions is defined next. Definition A2: A decision vector x∗ ∈ S is weakly Pareto optimal if there does not exist another decision vector x ∈ S such fi (x) < fi (x∗ ) for all i = 1, 2, . . . , m [15]. The set of (weak) Pareto optimum solutions can be nonconvex and nonconnected. Figure 9 shows the geometric interpretation of Pareto optimal and weakly Pareto optimal solutions. Note that all points on the line segment between points A and B are weakly Pareto optimal solutions. All points on the curve between points B and C are Pareto optimal solutions. Two different MO optimization techniques are introduced in this Appendix. The first method is called Weighting Method. The weighting method transforms the problem posed in (28) into m  λi fi (x) (29) min i=1

subject to x ∈ S, where the tradeoff factors m λi satisfy the following λi ≥ 0, ∀i = 1, 2, . . . , m, and i=1 λi = 1. Weakly

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Pareto optimal set can be obtained by solving the optimization problem (29) for different tradeoff factors values [15]. The second MO optimization technique is of special interest in the applications of MO optimization in RRS. It is the method of Weighted Metrics. If the global solutions of the objectives are known in advance, then problem (28) can be formulated as  p1 m  ∗ p λi |fi (x) − zi | (30) min i=1

Fig. 10.

The autoregressive model of power control.

zi∗

where 1 ≤ p ≤ ∞, is the optimum (or required) solution of objective i, and the tradeoff factors satisfy the following m λi ≥ 0, ∀i = 1, 2, . . . , m, and i=1 λi = 1. It is clear that (30) represents the minimization of the weighted p-norm distance. For p=2 the weighted Euclidean distance is obtained. With p = ∞ the problem (30) is called weighted Tchebycheff method. The Tchebycheff method is preferred to be applied for non-convex MO problems. The influence of value of p on the solution can be found in [1], [15], [16].

(31)

Minimizing the cost function with respect to Pi is now transformed into minimizing with respect to parameter vector w. Necessary condition for minimum N 

Suppose further that the power Pi (t) is described by a linear autoregressive model as shown in Figure 10. The transmit power is wi (k) Pi (t − k) = wi Xi (t) , t = 1, 2, . . . (33)

i (t) γ N −t ei (t) ∂e∂w =

wi = [wi (1), wi (2), . . . , wi (n)] ,  Xi (t) = [Pi (t − 1), . . . , Pi (t − n)]

(34)

w means the transpose of w. Observe that Xi (t) contains known and measured values of transmitted power. Substitute (33) into (31) to obtain ei (t)

= λi,1 (w i Xi (t) − Pmin )    X (t) w i i ˆ i,2 (t) + λ − Γi,min Iˆi (t)    X (t) w i i ˆ i,3 (t) + λ − Γi,max Iˆi (t) 

ˆ i,3 (t) ˆ i,2 (t) + λ λ αt := λi,1 + Iˆi (t)

(38)

or

t=1

wi (N ) = R−1 xx,i (N ) Rx,i (N ) Rxx,i (N ) :=

(39)

N 

Solving for wi  N N  N −t 2  γ αt Xi (t) X i (t) wi = γ N −t αt t=1 t=1   ˆ i,2 (t) Γi,min + λ ˆ i,3 (t) Γi,max Xi (t) λi,1 Pmin + λ

where

where 

∂ei (t) = 0, ∀i = 1, . . . , Q ∂w

γ N −t (αt w i Xi (t) −  ˆ i,2 (t) Γi,min − λ ˆ i,3 (t) Γi,max λi,1 Pmin −λ αt X i (t) = 0

t=1

k=1

Define

γ N −t ei (t)

∂ei (t) = αt Xi (t) ∂w Substituting (37) and (39) into (38) we obtain N 

ˆ i,2 = λi,2 sign (Γi (t) − Γi,min ) and λ ˆ i,3 = where λ λi,3 sign (Γi (t) − Γi,max ) , the sign function is defined as  +1 x ≥ 0 sign (x) = (32) −1 x < 0

n 

(37)

From (37)

The error function can be modified such as

Pi (t) =

ˆ i,2 (t) Γi,min ei (t) = αt wi Xi (t) − λi,1 Pmin − λ ˆ i,3 (t) Γi,max −λ

t=1

A PPENDIX II D ERIVATION OF MODPRC A LGORITHM ˆ i,2 (Γi (t) − Γi,min ) ei (t) = λi,1 (Pi (t) − Pmin ) + λ ˆ i,3 (Γi (t) − Γi,max ) , t = 0, 1, . . . +λ

and using this in (35), ei (t) becomes

N 

γ N −t αt2 Xi (t) Xi (t)

(40)

(41)

(42)

(43)

t=1 N  γ N −t αt × Rx (N ) := t=1   ˆ i,2 (t) Γi,min + λ ˆ i,3 (t) Γi,max X (t) λi,1 Pmin + λ i

(44)

Formulae (42)-(44) are well-known from least squares techniques. Equation (42) can be solved using the Recursive Least Square (RLS) method. To avoid the matrix inversion, Rxx,i (N ) may be computed recursively as (35)

 (36)

2 Rxx,i (N ) = γRxx,i (N − 1) + αN Xi (N ) Xi (N )

(45)

The adaptation factor γ sometimes called forgetting factor because it indicates how fast the old samples would be decayed as shown in (45). Since the inverse of Rxx,i (N ) is needed we can use the matrix inverse identity to obtain (46) [29]. Also Rx,i (N ) can be computed recursively as (47). The power

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R−1 xx,i (N )

  −1  2 R−1 1 xx,i (N − 1)αN Xi (N )Xi (N )Rxx,i (N − 1) −1 Rxx,i (N − 1) − = 2 X (N )R−1 (N − 1)X (N ) γ γ + αN i i xx,i

  ˆ i,2 (t) Γi,min + λ ˆ i,3 (t) Γi,max Xi (N ) Rx,i (N ) = γRx,i (N − 1) + αN λi,1 Pmin + λ

control algorithm should be easy to implement as well as computationally light to be applicable for existing wireless communication systems. The simplest case is obtained when n = 1 and also N = 1. Solving (41) for the simplified case we obtain: ˆ i,2 Γi,min + λ ˆ i,3 Γi,max λi,1 Pmin + λ   wi (t + 1) = (48) ˆ i,2 + λ ˆ i,3 Γi (t) λi,1 Pi (t) + λ From (33), the transmit power of MT i at time t is given by Pi (t + 1) =

ˆ i,2 Γi,min + λ ˆ i,3 Γi,max λi,1 Pmin + λ   Pi (t) ˆ i,2 + λ ˆ i,3 Γi (t) λi,1 Pi (t) + λ

(49) ˆ i,3 , the ˆ i,2 + λ Due to the sharp changes in the signs of λ transmit power in (49) may take negative values as well as very large power values which are not part of the feasible power subspace. To overcome these problems only the positive ˆ i,3 are considered, i.e. λ ˆ i,2 = λi,2 and ˆ i,2 + λ values of λ ˆ λi,3 = λi,3 . This simplification has considerably reduced the complexity of the MODPRC algorithm at slight degradation in the convergence speed [18]. ACKNOWLEDGMENT First author would like to thank Garyounis University Libya and Nokia Foundation - Finland for their financial support of this work. R EFERENCES [1] G. Liu, J. Yang, and J. Whidborne, Multiobjective optimization and control, RSP LTD, 2003. [2] V. Bhaskar, S. K. Gupta, and A. K. Ray, “Applications of multiobjective optimization in chemical engineering,” Reviews Chem. Eng., vol. 16, pp. 1–54 2000. [3] E. Tsoi, K. Wong, and C. Fung. “Hybrid GA/SA algorithms for evaluating trade-off between economic cost and environmental impact in generation dispatch,” in Proc. Second IEEE Conference on Evolutionary Computation (ICEC’95), pp. 132–137, 1995. [4] J. Andersson, A Survey of Multiobjective Optimization in Engineering Design, Technical Report No. LiTH-IKP-R-1097, Department of Mechanical Engineering, Linkoping University, 2000. [5] R. Kumar, P. Parida, and M. Gupta, “Topological design of communication networks using multiobjective genetic optimization,” in Proc. Congress on Evolutionary Computation (CEC’2002), vol. 1, pp. 425– 430, May 2002. [6] A. Roy, N. Banerjee, and K. Das, “An efficient multi-objective QoSrouting algorithm for wireless multicasting,” in Proc. IEEE Vehicular Technology Conference-Spring, vol. 3, pp. 1160–1164, 2002. [7] M. Elmusrati and H. Koivo, “Multi-objective distributed power control algorithm,” in Proc. IEEE Vehicular Technology Conference-Fall, vol. 2, pp. 812–816, 2002. [8] M. Elmusrati and H. Koivo, “Multi-objective distributed power and rate control for wireless communications,” in Proc. IEEE International Conference on Communication 2003. [9] C. Ahn and R. Ramakrishna, “QoS provisioning dynamic connectionadmission control for multimedia wireless networks using a Hopfield neural network,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 105– 117, Jan. 2004.

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[10] M. Moustafa, I. Habib, M. Naghshineh, and M. Guizani, “QoS-enabled broadband mobile access to wireline networks,” IEEE Commun. Mag., vol. 40, no. 4, pp. 50–56, April 2002. [11] K. Chawla and X. Qiu, “Throughput performance of adaptive modulation in cellular systems,” in Proc. IEEE ICUPC, pp. 945–950, 1998. [12] L. Song and N. Mandayam, “Hierarchical SIR and rate control on the forward link for CDMA data users under delay and error constraints,” IEEE J. Select. Areas Commun., vol. 19, no. 10, pp. 1871–1882, Oct. 2001. [13] S. Ulukus and L. Greenstein, “Throughput maximization in CDMA uplinks using adaptive spreading and power control,” in Proc. IEEE ISSSTA, pp. 565–569, Sept. 2000. [14] A. Sampath, P. Kumar, and J. Holtzman, “Power control and resource management for a multimedia CDMA wireless system,” in Proc. IEEE PIMRC, pp. 21–25, 1995. [15] K. Miettinen, Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, 1998. [16] C. Coello, “A short tutorial on evolutionary multiobjective optimization,” in Proc. First International Conference on Evolutionary Multi-Criterion Optimization, Springer-Verlag, Lecture Notes in Computer Science No. 1993, pp. 21–40, March 2001. [17] J. Zander, “Distributed cochannel interference control in cellular radio systems,” IEEE Trans. Veh. Technol., vol. 41, no. 3, pp. 305–311, Aug. 1992. [18] M. Elmusrati, “Radio resource scheduling and smart antennas in CDMA cellular communication systems,” Ph.D. thesis, Control Engineering Laboratory, Helsinki University of Technology, Finland, 2004 (can be downloaded from http://lib.tkk.fi/Diss/2004/isbn9512272202/). [19] R. Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Select. Areas Commun., vol. 13, pp. 1341–1347, Sept. 1995. [20] S. Grandhi and J. Zander, “Constrained power control in cellular radio systems,” in Proc. IEEE Vehicular Technology Conference, vol. 2, June 1994, pp. 824–828. [21] X. Qiu and K. Chawla, “On the performance of adaptive modulation in cellular systems,” IEEE Trans. Commun., vol. 47, no. 6, pp. 884–895, June 1999. [22] M. Elmusrati, M. Rintamaki, I. Hartimo, and H. Koivo, “Fully distributed power control algorithm with one bit signaling and nonlinear error estimation,” in Proc. IEEE Vehicular Technology Conf., Oct. 2003. [23] M. Elmusrati and H. Koivo, “Multi-objective totally distributed power and rate control for CDMA mobile communication systems,” in Proc. IEEE Vehicular Technology Conference, Spring 2003. [24] L. Xiao, M. Johansson, H. Hindi, S. Boyd, and A. Goldsmith, “Joint optimization of communication rates and linear systems,” IEEE Trans. Automatic Control, vol. 48, pp. 148–153, Jan. 2003. [25] S. Oh, D. Zhang, and K. Wasserman, “Optimal resource allocation in multiservice CDMA networks,” IEEE Trans. Wireless Commun., vol. 2, pp. 811–821 July 2003. [26] M. Elmusrati, R. Jantti, and H. Koivo, “Multi-objective distributed power control algorithm for CDMA wireless communication systems,” IEEE Trans. Veh. Technol., vol. 56, no. 2, pp. 779–788, Mar. 2007. [27] M. Simon and M. Alouini, Digital Communication over Fading Channels, 2nd ed. John Wiley and Sons, 2005. [28] D. Zhao, M. Elmusrati, and R. Jantti, “On the throughput maximization in downlink of DS-CDMA Systems,” in Proc. IEEE Vehicular Technology Conference, Spring 2005. [29] J. Proakis, Digital Communications. McGraw-Hill, 3rd ed. 1995. [30] J. Yang and D. Li, “Normal vector identification and interactive tradeoff analysis using formulation in multiobjective optimization,” IEEE Trans. Syst., Man, Cybernetics, vol. 32, no. 3, May 2002.

ELMUSRATI et al.: APPLICATIONS OF MULTI-OBJECTIVE OPTIMIZATION TECHNIQUES IN RADIO RESOURCE SCHEDULING

Mohammed Elmusrati received the B.Sc. (with honors) and M.Sc. (with high honors) degrees in telecommunication engineering from the Department of Electrical and Electronic Engineering, Garyounis University, Benghazi, Libya, in 1991 and 1995, respectively, and the Licentiate of Science in technology (with distinction) and Doctor of Science in Technology degrees in control engineering from Helsinki University of Technology (TKK), Espoo, Finland, in 2002 and 2004, respectively. He was a Lecturer with the Department of Electrical and Electronic Engineering, Garyounis University, during 1995-1999. He is currently an Acting Professor of telecommunication engineering of the Telecommunication Group - Department of Computer Science, University of Vaasa, Vaasa, Finland. He is also a Docent (Adjunct Professor) with the Control Engineering Laboratory - Automation Department, TKK. He has also been a Visiting Researcher with the Radio Communication Systems Laboratory (now WirelessKTH), Royal Institute of Technology (KTH), Stockholm, Sweden. His research interests include radio resource management, wireless automation, sensor networks, smart antennas, 3G and beyond mobile systems, ultrawideband, and data fusion.

Hassan El-Sallabi received the B.Sc. (with honors) and M.Sc. degrees in electrical engineering, both from Garyounis University (GA), Benghazi, Libya, and the Licentiate and D.Sc. (with distinction) degrees, both from Helsinki University of Technology (TKK), Espoo, Finland. He worked as Telecommunication Engineer in a general electric company, Benghazi, where he held several positions. He also worked as an Assistant Lecturer with the Faculty of Engineering at GA. He held positions as a Senior Research Engineer and a Project Manager with Radio Laboratory at TKK, Finland, from September 2003 to September 2005. He also served as a Task Leader with the WINNER European Commission project from January 2004 until the end of September 2005. Since October 2005, he has been the Academy of Finland Postdoctoral Researcher at the Information Systems Laboratory, Stanford University, Stanford, CA. He has published more than 45 journal and conference papers, contributed to the organization of various international conferences as a member of the Technical Program Committee and as a session chairman, and served as a reviewer for many international journals. His research interests include physical and stochastic channel modeling for wireless communications, diffraction theory, wireless system designs, and multiantenna techniques.

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Heikki Koivo (S’67-M’71-SM’86) received the B.S.E.E. degree from Purdue University, West Lafayette, IN, and the M.S. degree in electrical engineering and Ph.D. degree in control sciences from the University of Minnesota, Minneapolis. He is a Professor of control engineering with the Control Engineering Laboratory, Helsinki University of Technology (HUT), Espoo, Finland. Before joining HUT in 1995, he served in various academic positions at the University of Toronto, Toronto, ON, Canada, and Tampere University of Technology, Tampere, Finland. He has been the Principal Investigator in more than 100 research projects. He has authored more than 300 scientific publications. His research interests include the study of complex systems, adaptive and learning control, mechatronics, microsystems, and wireless communication systems. Dr. Koivo is a member of the Editorial Board of the Journal of Intelligent and Fuzzy Systems, Intelligent Automation and Soft Computing, and the Journal of Systems and Control Engineering. He was an Associate Editor of the IEEE Transactions on Robotics and Automation and a member of the Administrative Council of the IEEE Robotics and Automation Society. He is a Fellow of the Finnish Academy of Technology.