A Jumping Gene Algorithm for Multiobjective Resource Management ...

The Computer Journal Advance Access published August 25, 2005 © The Author 2005. Published by Oxford University Press on behalf of The British Computer Society. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/comjnl/bxh135

A Jumping Gene Algorithm for Multiobjective Resource Management in Wideband CDMA Systems T. M. Chan1 , K. F. Man1 , K. S. Tang1 and S. Kwong2 1 Department of Electronic Engineering, City University of Hong Kong, Hong Kong 2 Department of Computer Science, City University of Hong Kong, 83, Tatchee Avenue,

Kowloon, Hong Kong Email: [email protected], [email protected], [email protected], [email protected] In this paper, a newly developed jumping genes evolutionary paradigm is proposed for optimizing the multiobjective resource management problem in direct sequence–wideband code division multiple access systems. This formulation enables both total transmission power and total transmission rate to be simultaneously optimized. Since these two objectives are conflicting in nature, a set of tradeoff non-dominated solutions could be obtained without violating the quality of service. This new algorithm has been statistically tested and compared with a number of various multiobjective evolutionary algorithms including the use of binary ε-indicator for classifying the capability in generating the quality of non-dominated solution sets. In addition, the capacity of finding a number of extreme solutions is an extra indication to show its ability to measure the diversity along the Pareto-optimal solution front in a unique fashion. Received 23 October 2004; revised 29 July 2005

1.

INTRODUCTION

With the increasing demand of high-speed multimedia communications, the third-generation mobile communication (3G) systems are being developed to replace the existing second-generation mobile communication (2G) systems that support only voice and low-speed data. The 3G systems, such as International Mobile Telecommunication 2000 (IMT2000) [1] and Universal Mobile Telecommunications System (UMTS) [2], are required to support various types of multimedia services including high-quality voice, data, facsimile, video and interactive applications with data rates up to 2 Mbps [3, 4]. Since the multiple access technique, code division multiple access (CDMA), is a promising technique to fulfill above requirements [4, 5] and capable of supporting high capacity (total number of users), direct sequence–wideband CDMA (DS–WCDMA) was proposed for deployment in the 3G systems [6, 7, 8, 9]. The preliminary of using WCDMA in the 3G systems was proposed and released in 1999 to meet the IMT-2000 requirements including the support of data rates up to 2 Mbps as well as both packet and circuit switched services [10, 11]. It showed that WCDMA had a high potential and tendency to become one of the multiple access techniques in the 3G systems. The main technique of CDMA is the encryption

of signal by using a unique code word (key) which forms a communication channel of a user. Each user has a unique key to encode the transmitting messages or decode the receiving messages. The detailed principle of CDMA operations can be found in [12, 13, 14]. Resource management is an important issue of the 3G systems to allocate the resources optimally while the quality of service (QoS) requirement of each connecting user can still be satisfied so that the radio spectrum can be efficiently utilized [15, 16]. QoS is a measure of signal quality of connecting users and can be represented in a number of ways, such as bit error rate, received bit energy-to-noise density ratio (Eb /No ) etc. However, Eb /No is a preferable choice [17, 18, 19] which will be used in this paper. Each media type (e.g. voice, data or video) has its own required minimum value of Eb /No . If the Eb /No value of a user is greater than the required minimum value of requested media type, it is said that the QoS requirement is satisfied. In fact, the total capacity that the WCDMA network can support depends on the requested media type of each user, Eb /No value and multiple access interference. For example, if all users demand only voice, larger total capacity can be supported because voice users consume less resource

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T. M. Chan et al.

in contrast to video users. Nevertheless, it is unlikely that this case happens since users have several choices of the media type and thus the probability that all users demand only voice is very low. Generally, the network has voice, data and video users simultaneously. Furthermore, the QoS of a user is related to the transmission power, transmission rate and multiple access interference from the mathematical point of view (See the mathematical representation of the QoS requirement in Section 3). The utilization of high power of a user can violate the QoS requirements of other users and that of large rate can violate the QoS requirement of the user. Through the power and rate control, increasing Eb /No value of a violating user can satisfy the QoS requirement and reducing Eb /No value of a satisfying user near to the required minimum value can prevent to affect others. Then, each user can adopt the sufficient power and rate in order not to violate the QoS requirement of any user. Therefore, assume that all users do not end up their calls and none of the idle users initiates a new call at a particular moment; resource management policy can remove the excessive resources consumed by satisfying users and help the violating users satisfy their QoS. Also, it can provide sufficient resources for the future connecting users if maximum capacity is not reached. In this paper, a centralized resource management in DS–WCDMA systems with the aforesaid assumptions is considered. Its implementation steps are given as follows. At the beginning of a time slot, the computer in the resource management center receives the data (i.e. measured link gain, requested media type and identification number of assigned base station of each connecting user) from the database. With these data, the computer executes the specified program written for an optimization technique and outputs the optimized power and rate values of each connecting user when the maximum allowed time is reached. Finally, each base station receives power and rate values of its connecting mobile phones from the computer and instructs each of these phones to use them by sending a command through its communication channel. Since total transmission power and total transmission rate are conflicting, the problem is formulated as a multiobjective optimization problem. The schemes, enumerative, classical optimization and stochastic methods, can be used to solve this type of optimization problem. The enumerative method can provide a set of true Pareto-optimal solutions, but it is computationally inefficient or even infeasible owing to the huge search space [20]. Owing to the searching space arbitrarily limited by using the classical optimization methods, the required solutions in the vicinity of the Paretooptimal front are not easy to come by [20]. Also, the classical methods use the point-by-point approach in which a solution obtained per iteration can be modified to a different solution in hoping that a better solution would have been obtained [21]. However, this yields only one particular Pareto-optimal solution at a time. However, the advantage of using multiobjective evolutionary algorithms (MOEAs) is multifacet. It is possible that, within a single simulation run, multiple Pareto-optimal

solutions can be acquired with fewer overheads and without limiting the search space [20]. A number of MOEAs were proposed to solve this type of optimization problems e.g. the Multi-Objective Genetic Algorithms (MOGA) [22, 23], Niched Pareto Genetic Algorithm 2 (NPGA2) [24], Non-dominated Sorting Genetic Algorithm 2 (NSGA2) [25, 26], Strength Pareto Evolutionary algorithm 2 (SPEA2) [27], Pareto Archived Evolution Strategy (PAES) [28, 29] and Micro-genetic Algorithm (MICROGA) [30, 31]. Their searching performances were duly assessed through a number of benchmark test functions with various natures of the Pareto-optimal front, e.g. concave, convex etc. [32]. In this paper, a new MOEA called Jumping Gene Genetic Algorithm (JGGA) is proposed to solve this multiobjective optimization problem. This algorithm emulates a jumping gene phenomenon that was discovered by Nobel laureate, Barbara McClintock, from her work on corn plant. The main feature of JGGA is that it comprises an additional operation called transposition which is induced within the same chromosome or to another chromosome. This paper is organized as follows. The related work, mathematical formulation and the new suggested method, JGGA, will be presented in Sections 2, 3 and 4 respectively. In Section 5, the results and discussions will be provided and conclusions will be given in Section 6.

2.

RELATED WORK

Research on resource management in DS–WCDMA systems has been performed for the last 2 years and resource management algorithms using different approaches were proposed. In general, the approaches of system design are transmission power and transmission rate control [9, 17, 18, 19, 33], call admission control [4, 34, 35, 36, 37] and code assignment scheme [38, 39, 40, 41]. In this paper, we focus on the approach which is based on transmission power and transmission rate control. Thus, only the related studies will be presented here. The aim of this approach is to find an appropriate value of the transmission power and transmission rate for each connecting user while their QoS requirements can still be satisfied. The past work dealing with the transmission power and transmission rate control focused on only single objective approach i.e. the minimization of total transmission power and maximization of total transmission rate of all connecting users in a combined form of single objective function. Nevertheless, as the two objectives are conflicting with each other, solution tradeoff can thus be made. Therefore, the multiobjective approach was employed to search the nondominated tradeoff solutions for the resource management problem. Since each of these solutions can be a possible solution to the problem, the decision of choosing a solution can be made according to the preferences and the selected solution is distributed to each base station that is associated with the connecting users for implementation. The advantage of multiobjective approach is that it can give the designers


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A Jumping Gene Algorithm WCDMA Systems more flexibility as a result of a family of solutions, instead of a single solution. In addition, there were two other studies related to the multiobjective optimization in wireless networks [42, 43]. Roy et al. [42] proposed a new QoS routing algorithm developed with a non-dominated sorting based GA to determine the multicast routes on-demand by simultaneously optimizing the multiobjective QoS parameters as end-toend delay guarantee, bandwidth requirements and bandwidth utilization. The simulation results revealed that the suggested algorithm is able to find a set of QoS based near optimal, non-dominated multicast routes within a few iterations and outperforms existing scalar-optimization based algorithms. Moreover, [43] tackled the same problem as [42], except using a slightly different QoS routing algorithm. 3.

MATHEMATICAL FORMULATION

N W η gbi pi ri Pimin Pimax Rimin max R i

Eb No i γi

T λp λr λx xi

f1 = λp ·

N

f2 = λr · T − f3 = λx ·

N

(Total transmission power)

pi

i=1

N

Meaning Total number of connecting users in the system Total spread spectrum bandwidth Background noise power Link gain between the mobile user i and the base station b Transmission power of user i Transmission rate of user i Minimum allowed transmission power of user i Maximum allowed transmission power of user i Minimum allowed transmission rate of user i Maximum allowed transmission rate of user i

(Total transmission rate)

ri

i=1

(Total number of violating users),

xi

i=1

subject to the following three constraints. Constraint

As mentioned in Section 1, the aim of resource management is to allocate the resources optimally while the QoS requirement of each user can still be met. In this Section, an optimization problem based on the above objective is formulated. The resources for allocation are the transmission power and transmission rate of all connecting users. A list of notations is given as follows: Notations

added. Its purpose is to direct the search to find the solutions with minimized total power and maximized total rate while keeping the total number of violating users as a minimum. Now, the mathematical formulation is given as follows: Minimize

Mathematical representation

C1

Pimin ≤ pi ≤ Pimax , i = 1, 2, . . . , N.

C2

Rimin ≤ ri ≤ Rimax , i = 1, 2, . . . , N.

C3

Eb No i = N

≥ γi ,

The objective of the optimization problem is to minimize the total transmission power and maximize the total transmission rate of all users simultaneously. However, this straightforward formulation does not satisfy the output solutions, since a large number of users still violate the QoS requirement in the last generation of GA. The searching performance after using this formulation is therefore poor. However, an extra objective function representing the total number of violating users, f3 = λx · N x , can be i i=1

4. 4.1.

Transmission power of each user must lie on the range between the minimum and maximum allowed power. Transmission rate of each user must lie on the range between the minimum and maximum allowed rate.

gbi ·pi /ri

j =1 j =i

gbj ·pj +η

W

i = 1, 2, . . . , N.

Received bit energy-to-noise density ratio of user i Required minimum value of the received bit energyto-noise density ratio of user i A constant is considered to be larger than the total transmission rate of N users The fixed cost per unit power in W The fixed cost per unit rate in Kbps The fixed cost per user who violates the QoS requirement The integer variable is limited to be 0 or 1 only. xi = 1 if user i violates the QoS requirement and 0 otherwise

Physical meaning

This constraint represents the QoS requirement of each user. Each user’s received bit energy-to-noise density ratio must be greater than or equal to a required minimum value, γi , so as to satisfy the QoS.

JUMPING GENE GENETIC ALGORITHM Biological background

The phenomenon of jumping genes, also known as transposons, was first discovered by Barbara McClintock [44, 45]. It was found that the corn chromosome (number 9) had two parts dissociated from each other and the breaking place is always the same from generation to generation. It was later revealed that there are non-autonomous transposable elements called dissociation (Ds) elements, which could transpose (jump) from one position to another within the same chromosome or to another chromosome under the presence of autonomous transposable elements called


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FIGURE 1. Schematic representation of transposable element evolution in relation to the host.

activator (Ac) elements. Autonomous transposable elements are transposons that are capable of jumping by themselves, whereas non-autonomous transposable elements are those which can jump only under the activation of autonomous transposable elements. These two different forms of transposons, Ds and Ac, imply that the genes can jump in the genome to cause the breakage. Further experimental observation also indicated that there were two ways in which the jumping genes could move around the genome. The first one was called cut-and-paste, which means a piece of DNA is cut and pasted somewhere else. The second one was known as copy-and-paste. This means that the genes remain at the same location whereas the message in the DNA is copied into RNA and then copied back into DNA at another place in the genome. Although vertical transmission (VT) is a gene transmission from generation to generation, i.e. from parent to children, the ‘jump’ of genes is a kind of horizontal transmission (HT). This type of gene transmission is a lateral movement of genes within a chromosome or even to other individuals. It turns out that functional genes have an alternative way to survive through HT and then VT. The process of implanting a foreign set of genes is not streamlined, nor is it planned in advance as natural selection tends to be opportunistic, not foresighted. Most genetic takeover, acquisition, mergers and fusions ensue under conditions of environmental hardship [46]. A schematic representation of jumping gene (transposable element) evolution in relation to the chromosome (host) is

shown in Figure 1 [47]. Transposable elements invade new species through HT. A balance between forces increasing and decreasing copy number by transposition must then be established. High or low copy numbers of the transposable element can be acquired during the transition period. Finally, an excessive transposition rate could bring the population to extinction, and the transposable elements can be lost from the population or they can gain a selective function that benefits the organism [47]. In the same analogy, HT can be adopted in evolutionary computing to provide an extra mobility of genes other than the usual operations of crossover, mutation and selection. Based on these phenomena, a jumping gene transposition is proposed for the JGGA and discussed below. 4.2.

Computational jumping gene paradigm

The JGGA introduces a new operation named jumping gene transposition with the cooperation of NSGA2 [25, 26]. The associated sorting strategy, crowding mechanism and elitism strategy are applied in JGGA. As mentioned before, there exist two types of transposons (i) cut-and-paste transposon and (ii) copy-and-paste (replicate) transposon. In order to emulate the jumping behavior (transposition process), the following two issues must be addressed in the design and implementation of the JGGA: (i) Each chromosome has some consecutive genes known as transposons. The number of transposons


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(a)

(b)

FIGURE 2. Cut-and-paste transposition operation.

in a chromosome can be >1 and the length of each transposon can be more than one unit (e.g. one bit for binary code, one integer for integer code etc). Also, the locations of the transposons can be randomly assigned, but their contents can be freely transferred within the same chromosome or even to another chromosome in the pool. (ii) Cut-and-paste and copy-and-paste transposition operations are depicted in Figures 2 and 3 respectively. The actual manipulation of the former operation is that the element is cut from an original position and pasted into a new position of a chromosome. As for the later operation, the element replicates itself and the copy is inserted into a new location of the chromosome, whereas the original one remains unchanged. The procedures for the transposition operation are implemented between the selection process and the crossover mechanism. As previously stated that natural selection tends to be opportunistic, not foresighted, the jumping process is not streamlined, nor is planned in advance. Thus, the transposition operation is similar to other genetic operations (i.e. crossover and mutation) that are operated on the

basis of opportunity. Furthermore, in the transposition operation, the cut-and-paste and copy-and-paste operations are chosen randomly. The transpositions made within the same chromosome or to a different chromosome are also selected randomly and there is no restriction to the chromosome choice. The flowchart of a complete evolutionary cycle of JGGA is shown in Figure 4. Moreover, the flowcharts of transposition, cut-and-paste operation and copy-and-paste operation are shown in Figures 5–7. 4.3.

Why cut-and-paste and copy-and-paste transposition?

Cut-and-paste transposition. The cut-and-paste transposition opens a path through the landscape of an incomprehensibly larger number of possible chromosome changes [45]. It is a more efficient strategy for creating and trying out new genes than a chromosome that can make random element changes in random places. However, it guarantees neither a selective advantage nor fitter progeny. Furthermore, in the context of cut-and-paste transposons and host interaction, the environment that natural selection


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(a)

(b)

FIGURE 3. Copy-and-paste transposition operation.

must concern consists of not only the physical environment and other species, but also the microenvironment of the chromosome itself. The cut-and-paste transposons only have a semi-autonomous existence within the chromosome in that they are largely responsible for their own replication and may not have the same evolutionary interests as their host. As a matter of fact, transposition is indeed often harmful to the host, and the natural selection will favor host organisms that can devise ways to mitigate the deleterious consequences of element insertion [48]. Copy-and-paste transposition. A transposon can not only jump, but can also carry information for encoding protein pieces for shaping the genome. It was suggested that at least 50 human genes were directly derived from transposons [49]. These movements create places in the genome where stretches of DNA can pair and exchange information. Genes with similar sequences of the family do recombine. Repetitive sequences that are present throughout the genome may enable the exchange of information between unrelated genes by recombination. Hence, copy-and-paste transposition, which can enhance the possibility of merging the various types of genes together, eventually benefits the phenotypic shaping of chromosomes. Moreover, an increased probability of recombination at repeat sequences provides a more focused strategy for genetic exploration

rather than wandering the vast landscape of random base change [45]. 4.4.

Effect of jumping gene on multiobjective functions

The mimicking of jumping gene phenomenon requires some thoughts, as the optimal value of a gene in a chromosome cannot be easily obtained. Complex systems, life included, tend to arise in order to bring their gradient-rich surrounding to equilibrium. The same analogy is applied for evolutionary computing. Acceptable solutions can only be obtained when the population converges. A complicated engineering problem or design is not much different from a live being and thus it depends on environmental gradients. The solutions obtained through the stressful exploration and exploitation of the fitness landscape (phenotype space) would be a better goal for reducing the gradients to reach the desirable solutions. When the genome senses the stress, the genes jump [45]. It has been identified genetically from the corn genome that the jumping genes move around under the ultraviolet light [50]. Similarly, the use of Pareto optimization procedures can be considered as a form of ‘stress’ that causes the pressure on the chromosome to enhance the genetic exploration and



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FIGURE 4. Flowchart of jumping gene paradigm.

FIGURE 5. Flowchart of transposition.

exploitation. Therefore, the jumping gene transposition is an additional operator to the already well-inspired crossover, mutation and selection operators. All these operators manipulate each genotype individual appropriately as the search proceeds through the phenotype landscape [20].

4.5.

Chromosome representation

The genes are encoded in the real floating-point format for optimizing the transmission power and transmission rate of each connecting user. A gene represents a power or rate


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T. M. Chan et al. value of a user. The chromosome encoding method is shown in Figure 8. Moreover, more than one connecting user can utilize the same transmission power or transmission rate, or even both, because the power and rate values of each user are independent, although these two values can affect the QoS of all users. 4.6.

Initial population

After the chromosome encoding method is known, a population of initial chromosomes (solutions) is required. The power and rate values of each user in these chromosomes can be generated by using the following regulations.

FIGURE 6. Flowchart of cut-and-paste operation.

4.6.1. Power generation In the power generation scheme, we employ two new specified power ranges with respect to the link gains instead of the range between minimum and maximum allowed power without concerning them. The reason is that the users with larger link gains should utilize lower powers in order not to increase the multiple access interference of other users. This avoids the reduction of signal-to-noise ratios and hence the chance of violating their QoS constraints is decreased. Also, those with smaller link gains should utilize higher powers so as to increase their chance of satisfying their QoS constraints. However, the blind random generation method fails to do so and thus always results in a very large number of violating users. The generation rule is now given as follows. If the link gain of a user is greater than or equal to a threshold value, γ , the power value is generated randomly between the initial minimum power (pimin ) and the borderline power (pborder ). Otherwise, it is generated randomly between pborder and the initial maximum power (pimax ). A suitable set of the values of γ , pimin , pborder and pimax , which produces better results (i.e. the output solutions with zero violating user can be found in a larger number of occurrences), can be selected after testing and comparing their different combinations. In the simulations, these values used are γ = 100, pimin = 0.001, pborder = 0.005 and pimax = 0.01. Also, in the real-world environment, the chosen values of γ , pimin , pborder and pimax may vary with the two following factors: (i) the range between the maximum and minimum measured link gain, and (ii) the range between the maximum and minimum transmission power of the connecting users in the realistic systems. 4.6.2. Rate generation Since the transmission rate of a user does not affect the signalto-noise ratio of other users, blind random generation of rates between the guaranteed and peak rate of the requested media type for each user is adopted. Note that even though the above power and rate generation method may not be the best, it is a possible way to improve the search performance. 4.7.

FIGURE 7. Flowchart of copy-and-paste operation.

Jumping gene transposition

Since there are four types of genes (power gene, voice-user rate gene, data-user rate gene and video-user rate gene) in



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FIGURE 8. Chromosome encoding method.

a chromosome, four corresponding types of transposons can be devised. Each type of transposons can jump only in the specified range. For example, as shown in Figure 8, assuming that there are six connecting users in the system, the ranges of the power gene, voice-user rate gene, data-user rate gene and video-user rate gene are between gene 1 and gene 6, between gene 7 and gene 8, between gene 9 and gene 10, and between gene 11 and gene 12 respectively. A chromosome will be infeasible if any of transposons jumps to any position outside its corresponding range. Therefore, their jumping positions are appropriately conformed. To carry out the jumping gene transposition, we generate (Np × T ) random numbers ranged between 0 and 1 where Np is the population size (total number of chromosomes in the population pool) and T is the total number of transposons in each chromosome. If a random number corresponding to a transposon in the chromosome is less than the jumping probability pj , the transposon will jump to another randomly generated valid position through the cut-and-paste operation or copy-and-paste operation. 4.8.

Crossover

To begin with, we are required to generate Np random numbers ranged between 0 and 1 to decide which pair(s) of chromosomes will be chosen for performing crossover. If a random number corresponding to a chromosome is less than the crossover probability pc , that chromosome will be selected to perform crossover with the others. Three types of crossover, one-point crossover, two-point crossover and uniform crossover are tested and compared. Since using uniform crossover produces better results, it is employed in the simulations and only its operation will be introduced as follows. Figure 9 demonstrates the operation of uniform crossover. Assume that 25 users are connecting to the system; there exist 2 × 25 = 50 genes in the chromosome shown in the figure. Then, a mask containing the elements of a binary sequence restricted to 0 or 1 only is generated. Each binary number corresponds to one gene of the chromosomes, parents 1 and 2. If a binary number corresponding to a gene is 1, the genes of parents 1 and 2 are exchanged. Otherwise, they remain

unchanged. As shown in the figure, genes 1, 24, 27 and 49 of parents 1 and 2 are exchanged because the corresponding values of the mask are 1. 4.9.

Mutation

For the standard operation of mutation, we produce (Np ×Q) random numbers ranged between 0 and 1 where Q is the total number of genes in each chromosome. In case a random number corresponding to a gene in the chromosome is less than the mutation probability pm , the gene will be mutated. Nevertheless, a new mutation operation is proposed and utilized in this optimization problem instead of the standard one. To begin with, we tried to use the standard operation but found that the performance is slightly improved by comparing the final and initial solutions. Although the number of users violating their QoS constraints is decreased by using the rule stated in Section 4.6, in the first generation of GA, there is still a large number of violating users (over a half) from our investigated results. Moreover, since the mutation rate is usually set as a low value to prevent the solutions from falling in local optima, the probability that the power and rate genes of the violating users can be selected is very low. As a result, only a small number of violating users in the first generation of GA may change to the satisfying users in the last generation. A similar performance is obtained even if the mutation rate is set to a medium or a large value. It is because the changes of violating users to satisfying users and vice versa occur at the same time. Therefore, the net improvement is small after reaching the last generation of GA. Also, this makes the search behavior of GA totally random but not guided random. Then, the new mutation operation is introduced to overcome the aforementioned problems and improve the searching performance. It is divided into three following cases and the decision of how the power and rate values change for each case depends on the mathematical representation of (Eb /No ) as indicated in Section 3. Case (i): (Eb /No )i > γi for any user i, i = 1, 2, 3, . . . , N (i.e. satisfy the QoS constraint). The power gene of user i


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FIGURE 9. The operation of uniform crossover.

will be mutated if the corresponding random number is 0. The binary ε-indicator Iε is defined as Iε (A, B) = inf ∀f 2 ∈ B, ∃f 1 ∈ A : f 1 ε f 2

Case III: Iε (A, B) > 1 and Iε (B, A) > 1 ⇒ The non-dominated sets A and B are incomparable. This statistical test requires two MOEAs for a comparison. For each MOEA, 50 simulation runs will produce 50 nondominated solution sets. Hence, there will be 2500 pairs of non-dominated solution sets for comparisons. Tables 8 and 9 show the statistical results of the binary ε-indicator in terms of the number of occurrences and the corresponding percentage value of three cases for both scenarios (a) and (b) respectively. To further illustrate the effectiveness of JGGA as compared with other algorithms, the following terms are defined:

ε∈

for any two non-dominated solution sets A, B. It can be calculated as Iε (A, B) = max min max

f 2 ∈B f 1 ∈A 1≤i≤n

fi1 fi2

.

There are three cases which can classify whether the nondominated solution set A is better than B, or vice versa, or they are incomparable, i.e. equal performance.

(i) JGGA favorable: NI > NII and NI > NIII (ii) JGGA unfavorable: NII > NI and NII > NIII (iii) JGGA inconclusive: NIII > NI and NIII > NII where NI is the number of occurrences of case I, NII is the number of occurrences of case II and NIII is the number of occurrences of case III. On the basis of the above definitions, the results obtained in Tables 8 and 9 can be summarized as follows. (A) Scenario (a): 25 users..

Case I: Iε (A, B) ≤ 1 and Iε (B, A) > 1 ⇒ The non-dominated set A is better than B. Case II: Iε (A, B) > 1 and Iε (B, A) ≤ 1 ⇒ The non-dominated set B is better than A.

(i) As compared with MOGA, JGGA is more favorable, except sets 5 and 6. (ii) As compared with NPGA2, JGGA is more favorable, except sets 5 and 6.


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A Jumping Gene Algorithm WCDMA Systems TABLE 8. Statistical test using the binary ε-indicator for scenario (a): 25 users. Number of occurrences JGGA/algorithms

MOGA

NPGA2

NSGA2

Set 1 Case I JGGA is better than Case II JGGA is worse than Case III Incomparable

2006 (80.24%) 194 (7.76%) 300 (12%)

1317 (52.68%) 495 (19.8%) 688 (27.52%)

1144 (45.76%) 641 (25.64%) 715 (28.6%)

1376 (55.04%) 524 (20.96%) 600 (24%)


1881 (75.24%) 327 (13.08%) 292 (11.68%)

1013 (40.52%) 705 (28.2%) 782 (31.28%)

1243 (49.72%) 676 (27.04%) 581 (23.24%)

982 (39.28%) 682 (27.28%) 836 (33.44%)


2147 (85.88%) 97 (3.88%) 256 (10.24%)

1148 (45.92%) 529 (21.16%) 823 (32.92%)

1403 (56.12%) 586 (23.44%) 511 (20.44%)

1183 (47.32%) 656 (26.24%) 661 (26.44%)


1807 (72.28%) 273 (10.92%) 420 (16.8%)

1305 (52.2%) 534 (21.36%) 661 (26.44%)

1184 (47.36%) 663 (26.52%) 653 (26.12%)

1210 (48.4%) 635 (25.4%) 655 (26.2%)


643 (25.72%) 690 (27.6%) 1167 (46.68%)

364 (14.56%) 1108 (44.32%) 1028 (41.12%)

1280 (51.2%) 541 (21.64%) 679 (27.16%)

896 (35.84%) 637 (25.48%) 967 (38.68%)


643 (25.72%) 872 (34.88%) 985 (39.4%)

398 (15.92%) 1007 (40.28%) 1095 (43.8%)

1159 (46.36%) 715 (28.6%) 626 (25.04%)

452 (18.08%) 992 (39.68%) 1056 (42.24%)


2411 (96.44%) 28 (1.12%) 61 (2.44%)

2036 (81.44%) 213 (8.52%) 251 (10.04%)

1378 (55.12%) 594 (23.76%) 528 (21.12%)

1911 (76.44%) 261 (10.44%) 328 (13.12%)


1641 (65.64%) 312 (12.48%) 547 (21.88%)

1056 (42.24%) 747 (29.88%) 697 (27.88%)

1506 (60.24%) 507 (20.28%) 487 (19.48%)

1300 (52%) 500 (20%) 700 (28%)


1974 (78.96%) 71 (2.84%) 455 (18.2%)

1134 (45.36%) 515 (20.6%) 851 (34.04%)

1100 (44%) 645 (25.8%) 755 (30.2%)

1616 (64.64%) 339 (13.56%) 545 (21.8%)


1325 (53%) 586 (23.44%) 589 (23.56%)

898 (35.92%) 770 (30.8%) 832 (33.28%)

(iii) As compared with NSGA2, JGGA is more favorable for all 10 sets. (iv) As compared with SPEA2, JGGA is more favorable, except sets 5, 6 and 10. (B) Scenario (b): 50 users. (i) As compared with MOGA, JGGA is more favorable, except set 5. (ii) As compared with NPGA2, JGGA is more favorable, except set 5, 6 and 7.

913 (36.52%) 763 (30.52%) 824 (32.96%)

SPEA2

637 (25.48%) 878 (35.12%) 985 (39.4%)

(iii) As compared with NSGA2, JGGA is more favorable, except set 1. (iv) As compared with SPEA2, JGGA is more favorable, except set 5. On the basis of the above comparisons, JGGA is found to score 33 favorable cases, 6 inconclusive cases and only 1 unfavorable case for scenario (a). As for scenario (b), JGGA scores 34 favorable cases, 5 inconclusive cases and only 1 unfavorable case. Therefore, these findings through


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T. M. Chan et al. TABLE 9. Statistical test using the binary ε-indicator for scenario (b): 50 users. Number of occurrences JGGA/Algorithms

MOGA

NPGA2

NSGA2


2316 (92.64%) 102 (4.08%) 82 (3.28%)

1334 (53.36%) 416 (16.64%) 750 (30%)

815 (32.6%) 734 (29.36%) 951 (38.04%)

2089 (83.56%) 233 (9.32%) 178 (7.12%)


1667 (66.68%) 423 (16.92%) 410 (16.4%)

859 (34.36%) 811 (32.44%) 830 (33.2%)

1127 (45.08%) 680 (27.2%) 693 (27.72%)

1099 (43.96%) 703 (28.12%) 698 (27.92%)


2108 (84.32%) 105 (4.2%) 287 (11.48%)

1575 (63%) 389 (15.56%) 536 (21.44%)

1102 (44.08%) 659 (26.36%) 739 (29.56%)

1750 (70%) 414 (16.56%) 336 (13.44%)


2333 (93.32%) 70 (2.8%) 97 (3.88%)

1662 (66.48%) 451 (18.04%) 387 (15.48%)

1243 (49.72%) 626 (25.04%) 631 (25.24%)

1801 (72.04%) 278 (11.12%) 421 (16.84%)


719 (28.76%) 488 (19.52%) 1293 (51.72%)

354 (14.16%) 1089 (43.56%) 1057 (42.28%)

1142 (45.68%) 676 (27.04%) 682 (27.28%)

363 (14.52%) 1033 (41.32%) 1104 (44.16%)


1642 (65.68%) 285 (11.4%) 573 (22.92%)

478 (19.12%) 902 (36.08%) 1120 (44.8%)

1175 (47%) 671 (26.84%) 654 (26.16%)

1277 (51.08%) 527 (21.08%) 696 (27.84%)


1343 (53.72%) 428 (17.12%) 729 (29.16%)

616 (24.64%) 854 (34.16%) 1030 (41.2%)

1230 (49.2%) 559 (22.36%) 711 (28.44%)

945 (37.8%) 698 (27.92%) 857 (34.28%)


2060 (82.4%) 130 (5.2%) 310 (12.4%)

1239 (49.56%) 646 (25.84%) 615 (24.6%)

974 (38.96%) 761 (30.44%) 765 (30.6%)

1398 (55.92%) 441 (17.64%) 661 (26.44%)


1965 (78.6%) 226 (9.04%) 309 (12.36%)

1022 (40.88%) 719 (28.76%) 759 (30.36%)

1049 (41.96%) 667 (26.68%) 784 (31.36%)

1513 (60.52%) 462 (18.48%) 525 (21%)


2289 (91.56%) 62 (2.48%) 149 (5.96%)

1094 (43.76%) 534 (21.36%) 872 (34.88%)

993 (39.72%) 666 (26.64%) 841 (33.64%)

1419 (56.76%) 420 (16.8%) 661 (26.44%)

the vigorous testing of binary ε-indicator show that the performance of JGGA is much better than other MOEAs. 5.4.

Extreme solutions

One of the unique performances of JGGA is its ability to find the extreme solutions near both ends of the Pareto-optimal front. To show that JGGA is capable of achieving good performance on diversity, i.e. spread of the non-dominated

SPEA2

solutions, counting the number of extreme solutions is another way in addition to using the diversity performance metric. These extreme solutions sometimes are important in decision making. In this context, the decision maker may wish to reach a condition that while a maximum (minimum) total transmission power (f1 ) is given, the total transmission rate (f2 ) must be the maximum (minimum). Under normal circumstances, these types of solutions are not easily reachable. In the case of JGGA, an attempt was



(a)

(b)

(c)

(d)

(e)

(f)

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FIGURE 12. Performance of different MOEAs. (a) Typical landscape of solutions for scenario (a): 25 users. (b) Typical landscape of solutions for scenario (b): 50 users. (c) 1% deviation from the extreme reference solutions for scenario (a): 25 users. (d) 2% deviation from the extreme reference solutions for scenario (a): 25 users. (e) 2% deviation from the extreme reference solutions for scenario (b): 50 users. (f) 3% deviation from the extreme reference solutions for scenario (b): 50 users.

sought to verify its capability in this domain. The measure of this maxmax (or minmin) hypothesis is to determine the obtained number of the non-dominated solutions as compared with the most extreme solutions at both ends of the reference front. Its computational procedures will be discussed as follows. Assuming that the coordinates of the two extreme points in the reference front are xf 1_ min , yf 2_ min

and xf 1_ max , yf 2_ max , the radii of the two corresponding circles with the center at these points are

2 2 r1 = and r2 = x1 − xf 1_ min + y1 − yf 2_ min

2 2 x − x + y2 − yf 2_ max respectively where 2 f 1_max

x1 , y1 and x2 , y2 are any two points lying on the two circles respectively. Consider that a scaling factor (α) is given for controlling the size of the circle; x1 , y1 , x2 and y2 are


18 of 20

T. M. Chan et al. TABLE 10. Averages and standard deviations of running times for four different cases.

therefore x1 = (1 − α)xf 1_ min , y1 = (1 − α) yf 2_ min , x2 = (1 − α) xf 1_ max and y2 = (1 − α) yf 2_ max .

Cases

Case (i) Case (ii) Case (iii) Case (iv)

Average (ms) 283.64 Standard deviation (ms) 17.81

Should an extreme non-dominated solution lie within the prescribed circle e.g. 10% away from the coordinates xf 1_ min , yf 2_ min and xf 1_ max , yf 2_ max , this implies α = 0.1. Based on this simple calculation, the total number of extreme solutions can be found by counting the accumulative number of obtained solutions that fall into the vicinity of coordinates xf 1_ min , yf 2_ min and the two extreme xf 1_ max , yf 2_ max for a number of runs. To count the number of extreme solutions falling within the circles, it can proceed as follows: Step 1: Set the counter C of the number of extreme solutions to zero. Step 2: For each considered non-dominated solution i with coordinates (xi , yi ), calculate the distances d1 between solution i and the extreme solution xf 1_ min , yf 2_ min as well as i and the extreme d2 between solution solution xf 1_ max , yf 2_ max by using the formulae

2 2 d1 = xi − xf 1_ min + yi − yf 2_ min and d2 =

2 2 xi − xf 1_ max + yi − yf 2_ max . Step 3: If either one of the following two conditions is satisfied, (i) d1 ≤ r1 , (ii), d2 ≤ r2 , C = C + 1. Step 4: If i ≤ N where N is the total number of nondominated solutions found, go to step 2. Otherwise, output the number of extreme solutions C. Figure 12a and b show a landscape of a typical performance of various methods in terms of the non-dominated solutions as well as the reference front based on the results obtained in 50 simulation runs for the scenarios (a) and (b) respectively. Moreover, for scenario (a), Figure 12c and d show the results of the total number of extreme solutions acquired from all 10 sets and falling into the vicinity of both ends of the reference front on the basis of a radius equivalent to a distance 1 and 2% of the most extreme solution at the reference front for a total of 50 simulation runs respectively. Figure 12e and f show the same result for 2 and 3% for scenario (b) respectively. The reason why 1 and 2% are chosen for scenario (a) but 2 and 3% for scenario (b) is that the distance between the non-dominated fronts of the MOEAs and the reference front is larger in the later scenario than the former. The use of 1% in scenario (b) caused that the total number of extreme solutions found by each MOEA is zero and hence the results cannot be compared. Referring to the results shown from the figures, JGGA obtained a larger total number of extreme solutions than other MOEAs. On the basis of the massive number of data, JGGA is therefore considered as an effective and reliable algorithm for searching extreme solutions.

5.5.

314.06 45.16

545.2 36.70

591.68 64.39

Discussion of real-time implementation

The integrated scheme comprised a GA and fast closed loop power control is proposed in [19] to make the implementation possible in real time. This power control adopted by the Interim Standard 95 (IS-95) [56] has been suggested by 3G proposals [7, 8]. Furthermore, it is indicated that adjusting the control period of the proposed scheme to 0.1 s is very affordable on current microprocessors [19]. Therefore, our JGGA can work in real-time environments with rapid changes. The factors affecting the computational complexity of JGGA are the total number of connecting users in the network, which increases the total chromosome length and the transposition operations, which in turn, increases the computational load. To test how fast the JGGA can work with the variation of these factors, the statistical results of running times by carrying out 50 simulation runs in a Pentium IV 1.3 GHz computer for the four following different cases are obtained: (i) The total number of connecting users is 25 and without transposition operations (i.e. transposition rate is zero), (ii) the total number of connecting users is 25 and with transposition operations, (iii) the total number of connecting users is 50 and without transposition operations (i.e. transposition rate is zero), and (iv) the total number of connecting users is 50 and with transposition operations. Note that the parameters used for this test are the same as in Table 3. The averages and standard deviations of running times are shown in Table 10. The differences of average between cases (i) and (iii), and (ii) and (iv) are 261.56 and 277.62 ms respectively. This indicates that the time used increases with the total number of connecting users as expected. Also, the differences of average between cases (i) and (ii), and (iii) and (iv) are 30.42 and 46.48 ms respectively. The time increase owing to the transposition operations is therefore small. The small standard deviations of the four cases obtained imply that the running times of JGGA are quite stable. Undoubtedly, the JGGA will work with more generations to hopefully obtain the better performance if faster microprocessor is adopted. In the real implementation, JGGA can be forced to give the optimal power and rate values for each connecting user when the maximum allowed time is reached, no matter how the total number of connecting users varies.


A Jumping Gene Algorithm WCDMA Systems 6.

CONCLUSIONS

This paper considers the multiobjective resource management optimization for the DS–WCDMA systems. The objective of optimization is to have a minimum total transmission power, and at the same time, to yield a maximum total transmission rate without violating the requirement of QoS. A new evolutionary scheme JGGA was developed for the optimization. It has been verified and tested on the basis of convergence and diversity performance metrics to guarantee the quality of non-dominated solution sets. A binary ε-indicator was also employed for the quality evaluation in optimization by comparing with other MOEAs. In addition, an extra advantage of using JGGA was its ability to gain extreme solutions along the Pareto-optimal front. ACKNOWLEDGEMENTS This work is supported by research grant no: CityU 1124/01E, City University of Hong. REFERENCES [1] Callendar, M. H. (1994) Future public land mobile telecommunication systems. IEEE Pers. Commun., 1, 18–22. [2] Nijhof, J. A. M., Dewantara, I. S., Roovers, Ad. J. M. and Prasad, R. (1994) Base station system configurations for future universal mobile telecommunication systems. IEEE Trans. Veh. Technol., 43, 659–665. [3] Andermo, P. G. and Ewerbring, L. M. (1995) A CDMAbased radio access design for UMTS. IEEE Pers. Commun., 2, 48–53. [4] Kim, Y. W., Kim, D. K., Kim, J. H., Shin, S. M. and Sung, D. K. (2001) Radio resource management in multiple-chiprate DS/CDMA systems supporting multiclass services. IEEE Trans. Veh. Technol., 50, 723–736. [5] Schilling, D. L. (1994) Wireless communications going into the 21st century. IEEE Trans. Veh. Technol., 43, 645–652. [6] ARIB IMT-2000 Study Committee (1998) Japan’s proposal for candidate radio transmission technology on IMT-2000: WCDMA. Association of Radio Industries and Businesses, Japan. [7] ETSI/UTRA (1998) The ETSI UMTS Terrestrial Radio Access (UTRA) ITU-R RTT candidate submission. European Telecommunications. Standard Institute. [8] TIA/TR45.5.4 (1998) The CDMA2000 ITU-R RTT candidate submission (0.18). Telecommunications Industry Association. [9] Soleimanipour, M., Zhuang, W. H. and Freeman, G. H. (2002) Optimal resource management in wireless multimedia wideband CDMA systems. IEEE Trans. Mobile Comput., 1, 143–160. [10] Dahlman, E., Beming, P., Knutsson, J., Ovesjo, F., Persson, M. and Roobol, C. (1998) WCDMA—the radio interface for future mobile multimedia communications. IEEE Trans. Veh. Technol., 47, 1105–1118. [11] Furuskar, A., Parkvall, S., Persson, M. and Samuelsson, M. (2002) Performance of WCDMA high speed packet data. In Proc. IEEE 55th Vehicular Technology Conf., Birmingham, Alabama, May 6–9, pp. 1116–1120. IEEE, USA. [12] Yang, S. C. (1998) CDMA RF System Engineering. Artech House, Boston.

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