A Three-Stage Heuristic Combined Genetic Algorithm Strategy to the ...

A Three-Stage Heuristic Combined Genetic Algorithm Strategy to the Channel-Assignment Problem Xiannong Fu, Yi Pan, and Anu G. Bourgeois Department of Computer Science Georgia State University Atlanta, GA 30303, USA {pan,anu}@cs.gsu.edu Pingzhi Fan Institute of Mobile Communications Southwest Jiaotong University Chengdu, Sichuan, 610031, P. R. China [email protected] Abstract The Channel Assignment Problem (CAP) is to assign a minimum number of channels to requested calls in a cellular radio system while satisfying certain constraints. It is proven to be NP-complete. Considering Sivarajan’s benchmark 21-cell system assignment problem, or the Philadelphia problem, the constraints and traffic demands are given, a lower bound of channels needed for this system is fixed, and many strategies have been provided to solve the problem. This paper presents a dynamic channel assignment algorithm consisting of three stages: 1) the determine-lower-bound cell regular interval assignment stage; 2) the greedy region assignment stage; and 3) the genetic algorithm assignment stage. Its performance is verified through the Philadelphia problem and achieves lower bound solutions on 11 of the 13 instances, which is comparable with existing algorithms [1, 2, 7]. The algorithm also has the advantage that it is able to find optimum solutions faster than approaches using neural-networks and simulated annealing. 1 INTRODUCTION The development of mobile telecommunication increased the growth of cellular users tremendously in recent years. However, the allocation of frequency spectrum for this purpose is limited. Consequently, the efficient use of channel frequencies becomes more and more significant. Based on service requirements, the allocated spectrum is divided into a number of channels. To satisfy the large demand of the mobile telephone service, channels need to be assigned and reused to minimize communication interference, and in turn, increase the traffic-carrying capacity of the system. This is called the channel assignment problem (CAP). CAP is classified as an NP-complete problem, which implies as the size of the problem increases, the time required to solve the

problem does not increase in a polynomial manner – but rather in an exponential manner. For a system with f available channels to allocate to n cells, there would be a total of nf different combinations. For example, with f = 20 and n = 10, there would be 1020 combinations. Exhausting all combinations would not be efficient and hence a method is needed to quickly find the optimal solution(s). The types of channels available in wireless communication networks vary from frequencies in FDMA, time slots in TDMA, to code channels in CDMA [5]. The frequencies, time slots, or codes in different channels must be separated far enough for them not to interfere with one another. The wireless service area is divided into a large number of hexagonal cells. Users located in these cells request calls through their portable applications, and the system assigns channels to each request and provides the service. The number of channels is usually much smaller than the number of cells times the average number of calls requested by each cell, therefore, channels need to be reused and shared by more than one cell. Due to the radio interference between frequency spectra, the channel assignment must fulfill the following three constraints [7]: 1) The Co-Channel Constraint (CCC): The same channel cannot be assigned to a pair of cells within a specified distance simultaneously. 2) The Adjacent Channel Constraint (ACC): Adjacent channels cannot be assigned to adjacent cells simultaneously. 3) The Co-Site Constraint (CSC): The distance between any pair of channels used in the same cell must be larger than a specified distance. These constraints are normally determined by the characteristics of the radio frequency propagation and spatial density of the expected traffic requirements. It can be represented by an integer, which prescribes the

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minimum gap that has to exist between channels assigned to the two cells in order to avoid interference. The goal of CAP is to find a channel assignment for every requested call using a minimum number of channels, such that each assignment satisfies the above constraints. Section 2 formulates the problem and describes the benchmark problem that will be used for comparisons. Section 3 provides and overview of channel assignment algorithms, while Section 4 presents the proposed algorithm for channel assignment using a genetic algorithm. Section 5 provides the simulation results and comparisons. Finally, in Section 6 we provide concluding remarks. 2 PROBLEM FORMULATION The basic channel assignment problem consists of the following five components [8]: 1. a set of n distinct cells; 2. a demand vector m = (mi) (1 ” i ” n); mi indicates the number of channels required for cell i); 3. an n u n symmetric compatibility matrix C = (cij); cij denotes the minimum required frequency separation between a call in cell i and a call in cell j; 4. a frequency fik, 1 d i d n, 1 d k d mi, is assigned to aik, the kth call in the ith cell. Each frequency is represented by a positive integer; 5. a set of frequency-separation constraints defined by the compatibility matrix C, |fik – fjl| t cij , for all i, j, k, l (i z j) Given the above conditions, the aim of the channel assignment procedure is to assign frequencies to each call, such that the total bandwidth of the required system is minimized. The compatibility matrix C, is used to ensure interference does not occur by providing sufficient frequency separation among subscribers. The three constrains for the channel interference can be described by matrix C as: 1. CCC is described as cij z 0. 2. ACC is described as cij t 2, if i and j are adjacent cells. 3. CSC is described as cii t k, where k is a given constant. 2.1 Sivarajan’s Benchmark Problem Sivarajan proposed a CAP model in 1989 [7] that has been widely used thereafter as a test problem in channel assignment research. It is also referred to as the Philadelphia problem. The cellular network of the Philadelphia problem is based on a regular hexagonal grid of n=21 cells, as shown in Figure 1. Each cell demands a high number of frequencies as indicated by the demand vectors. Two demand vectors, resulting in two problem instances, are provided as

m1 = {8,25,8,8,8,15,18,52,77,28,13,15,31,15,36,57,28,8,10 ,13,8}; m2 = {5,5,5,8,12,25,30,25,30,40,40,45,20,30,25,15,15,30,2 0,20,25}; The order of the calls in the demand vector is of the same order of the cellular network cell numbers. The problem instance corresponding to m1 is shown in Figure 1.

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Figure 1. Philadelphia Problem Instance 1 We obtain the constraints cij for the Philadelphia problem as follows. Let d(vi, vj) represent the distance between the centers of cells vi and vj, where the distance between centers of adjacent cells is assumed to be unit (Note: k=5) Then cij = 5, if i = j cij = 2, if 0 < d(vi, vj) d 1 cij = 1, if 1 < d(vi, vj) < 2 3 , where the cluster size is 1. For the pairs of cells that have no constraint between them, the corresponding cij is not defined. A cluster of cells, such that between any pair of cells there exists a constraint, is called a complete cluster Nc. One important characteristic of the Philadelphia problem is that the lower bound solution for this problem is fixed. It can be calculated by graph-theoretic methods, such as the minimum weight Hamiltonian paths, and Gamst bounds [10]. Due to the fixed lower bound, the Philadelphia problem can serve well as a CAP testing example for various kinds of algorithms. 3 OVERVIEW OF CHANNEL ASSIGNMENT ALGORITHMS The algorithms for channel assignments can be classified into two categories: 1) fixed channel assignment (FCA), where channels are assigned to each cell permanently and 2) dynamic channel assignment (DCA), where all unused channels are

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available for every cell. Fixed resource allocation schemes use a predetermined assignment strategy aimed at improving average case performance. However, such schemes are not able to adapt to the varying nature of user traffic. Dynamic channel assignment algorithms attempt to optimize system performance by adapting to the traffic variations. It completely removes the requirement of a static and structured frequency reuse pattern and makes available all radio channels available for every call. The CAP problem has been proven NPComplete through its transformation to the generalized graph coloring problem, and Sivarajan et al. proposed noniterative algorithms using ideas from graph coloring algorithms [7]. The eight algorithms provided in that paper are a combination of cell ordering and call ordering assignment methods. In [4], Yeung and Yum proposed four algorithms based on two node ordering principles: node-color ordering (CR), and node-degree ordering (DR) and two assignment strategies: Frequency Exhaustive assignment (FEA) and requirement exhaustive assignment (REA). The four algorithms are FEA/CR, FEA/DR, REA/CR, REA/DR. The FEA, REA and node-degree ordering will be described in detail in the following sections. In recent research work, many artificial intelligence algorithms are applied to CAP [6]. In [9], Chakraborty et al. proposed a Genetic Algorithm that generates the initial population and creates near optimum solutions when the population is large. A genetic mutation operation is then used to improve the quality of the previously obtained initial solutions to an optimum solution. A similar Genetic Algorithm strategy can be found in [3]. Beckmann and Killat [2] proposed a new genetic algorithm strategy. This algorithm first generates a list of all calls in the system, it then applies a genetic algorithm to determine an optimal call list. Next, for each generated call list, it assigns frequencies with the Frequency exhaustive assignment (FEA) strategy and evaluates the result by considering the number of blocked calls. Finally, it executes the genetic algorithm and FEA repeatedly until the best solution is found. The differences of the two genetic algorithms used in [9] and [2] are compared in details as follows. In [2], the channel number is fixed; in [9], the channel number is dynamic. In [2], the genetic algorithm is applied to a randomly generated population and the FEA strategy is applied for every generation to select the best of the population. While in [9], the initial population is restricted only to genotypes representing a valid solution, and the size of the population is large. Genotype in [2] is an array with the size of the number of cells, and the value of the array of the assigned channel number. Genotype in

[4] is an M u N matrix where M is the number of channels and N the number of cells. In [2], the genetic algorithm is used to find the optimal call list. In [4], the genetic algorithm is used to find an optimum frequency assignment solution. Funabiki and Okutani [1] proposed a threestage neural-network algorithm. In this algorithm, the cell that determines the lower bound is assigned channels first at an interval of CSC in the first stage. In the second stage, the cluster of cells whose center cell has the largest node-degree, which is called the greedy region, is assigned channels the by REA strategy. Every time the whole assignment fails, the greedy region expands by additionally including the cells adjacent to the original region. In the third stage, the calls in the remaining cells are assigned channels by a binary neural network in parallel. 4 PROPOSED ALGORITHM In this paper, we introduce an approach that combines a DCA heuristic with genetic algorithms. There are three stages in this algorithm 1) the Regular Interval Assignment Stage; 2) the Greedy Assignment Stage; and 3) the Genetic Algorithm Assignment Stage. The first stage assigns channels at regular intervals to the cell whose calls determine the lower bound on the total number of channels. This cell is the one with the largest demand of calls. The assignment starts with the first channel, and each channel assigned after is at an equal distance away determined by CSC. The second stage composes a greedy region and assigns channels. The greedy region consists of the cell that has a largest degree and its adjacent cells. If the second stage fails, then the algorithm expands the greedy region to include all other cells adjacent to all cells already in the greedy region. If the second stage succeeds, then the algorithm applies a genetic algorithm to assign channels for the remaining cells. If the second stage fails, then the algorithm executes the third stage based on the result of the first stage. This proposed algorithm is essentially a composition of the algorithms provided by [1] and [2]. As in [1], the new algorithm has three stages. The first stage is the same. The second stage uses an FEA strategy for the greedy region assignment instead of the REA strategy because test results show the probability of finding an optimum solution by FEA is much higher than that of REA. The third stage applies a genetic algorithm similar to [2] to find an optimum call list in the unassigned cells after the first two stages. Some of the parameters used in the new algorithm differ from that in [2], based on the test result.

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The overall procedure is described as follows: 1) Input a compatibility matrix C, demand vector D, Cluster Size Nc, CCC, ACC, and CSC. 2) Initialize the total number of channels M by the lower bound. 3) Apply the regular interval assignment stage to the cell that determines the lower bound of M. 4) Compute the degree of cell #i, di for i = 1, …, N N

(¦ d j C ij )  C ii

di

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5) Initialize the greedy region by a cell with the maximum degree and its adjacent cells. 6) Apply the greedy assignment stage to the greedy region while the assignment by step 3 is fixed. 7) If this assignment is failed, expand the greedy region by including all other cells adjacent to the cells already in the greedy region. Go back to step 6. 8) If step 7 succeeds, go to step 9. If all cells are included in the greedy region but the greedy assignment still fails, clear the greedy region and go to step 9. 9) Randomly generate different call orders for the calls in the remaining cells. Apply the genetic algorithm on these cells. Use the blocked calls b as the fitness, and use Frequency Exhaustive Assignment (FEA) as evaluation. 10) If b = 0, then successful. Regular Interval Assignment Stage The regular interval assignment stage assigns channels to calls in the cell that determines the lower bound on the number of channels M. Usually it is the cell with the largest demand of calls. The assignments are at a regular interval satisfying CSC in order to minimize the total number of channels M. Prior todiscussing the second stage, we describe node-degree ordering and two exhaustive assignment strategies. Node-degree Ordering Using node-degree ordering, cells are ordered in descending values of di. The first cell has the highest priority of getting a channel first. Let the degree di be a measure of the difficulty of assigning a channel to cell i. It is defined as

di

N

¦m c

j ij

 cii , 1 d i d N

j 1

--- Equation (1)

if mi z 0; otherwise, di = 0. Since the term cij on the right hand side of Equation (1) is the same for all cells, we propose to remove it for a simple form.

di

N

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j ij

,

1d i d N

j 1

--- Equation (2) FEA and REA Once the cells are ordered by either node-degree ordering or node-color ordering [1] and all channels are ranked according to their carrier frequencies, two channel assignment strategies, frequency exhaustive strategy (FEA) and requirement exhaustive strategy (REA) can be used to assign channels to cells. FEA starts with the first cell in the ordered list. Each cell with unsatisfied channel requirements is assigned a channel with the lowest rank, such that the assignment satisfies all constraints with previous assignments. The following pseudocode illustrates the Frequency Exhaustive Assignment strategy. In the pseudocode, the calls are first ordered in each cell, and are then ordered by the cell number. Loop 1: for each call i in the call list. Loop 2: for each channel j within lower bound IF i not assigned AND j can be assigned to i Assign j to I Break Loop2 End If End Loop 2 End Loop 1 REA attempts to assign the first channel to all cells that have unsatisfied channel requirements starting from the first cell. Next, follow the same procedure to assign the rest of the calls. Repeat this procedure until all channel requirements are satisfied (or exhausted). The following is the pseudocode. Again, the calls are first ordered within the cells and then ordered by their cell number. Loop 1: for each channel j within lower bound Loop 2: for each call i in the call list IF i not assigned AND j can be assigned to i Assign j to i Break Loop2 End If End Loop 2 End Loop 1 For systems with uniformly distributed channel requirements and smaller variations in cellto-cell channel requirements, it is more important to maximize the number of cochannel cells. Therefore

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REA tends to give a better performance. For systems with highly non-uniformly distributed channel requirements, it is more critical to satisfy the channel requirements of the most difficult-to-assign cells first. Therefore FEA tends to perform better. [4] In the Philadelphia problem, the channel requirements is high and non-uniformly, therefore, we use FEA in our algorithm. Greedy Assignment Stage The greedy assignment stage assigns channels to cells that form the greedy region. The greedy region is regarded as a region of cells with the highest assignment difficulties. The procedure of this stage is as follows. 1) Make a list of calls from the greedy region in the ascending order of the cell index number. 2) Initialize the greedy region as the cell with the largest node-degree and all six of its adjacent cells. 3) Apply FEA on the channels to the call list. 4) If every call in the list is assigned a channel, then terminate the procedure as successful and go to the genetic algorithm stage, else if the greedy region did not include all cells, expand the greedy region by including all adjacent cells of all cells already in the greedy region, and go to step 3. 5) If the greedy region assignment fails, go to stage three for the genetic algorithm stage. Genetic Algorithm Stage After the regular interval assignment stage and the greedy assignment stage, calls in the remaining cells are randomly ordered. Next, the genetic algorithm generates a call list by using the evaluation resulting from the FEA strategy. Generally, the genetic algorithm provides an efficient manner to search for an optimum solution. It starts by randomly generating an initial population of possible solutions. In this paper, the population is the randomly ordered call list. Then the fitness of each individual is evaluated. We use FEA to assign channels to the call list. The total number of blocked calls for each list is used as the fitness. We process a loop afterwards to simulate the generations. In each generation, we select the individuals probabilistically according to their fitness. The genes of selected individuals will mutate and crossover and producing new individuals to maintain the number of the population. Continue the loop until convergence is achieved or the maximum number of generations is exceeded. In this genetic algorithm, not only are calls in neighboring cells are investigated, but solutions in completely different areas are visited due to the application of the crossover process. This

significantly reduces the danger of getting deadlocked in a local minimum [2], which may occur in many fixed channel assignment algorithms. Comparing the third stage of our proposed algorithm, the differences are: in [2], the input parameters are demand D, interference matrix C, number of frequencies z; in our new algorithm, the input parameters also include the assignments made by the first two stages. In [2], FEA is applied to the whole call list; in our new algorithm, FEA is only applied to the call lists that have not been assigned channels. EVALUATION In the genetic algorithm, once we generate individuals of a population, we need to find a quality measure to decide how fit one individual is among the whole generation. This is called the fitness. In our application, an individual is a randomly generated call list. We use the number of unassigned calls, or we can call it blocked call, b, which results from the attempt to solve the CAP using FEA, as the fitness. This is because our goal is to find an optimum solution for the assignment of channels to the calls. The fittest call list will have 0 blocked calls. Using blocked calls as the fitness well reflects the goodness of the solutions we are searching for. SELECTION In the genetic algorithm, after the individuals are evaluated by their respective fitness, the ones with better fitness will be selected and the others will be eliminated, in order to improve the total fitness of the population. There are many different strategies to implement the procedure of selection. In our application, the quality measure q of an individual is q = b, which is the number of blocked calls after FEA, can be used to determine which individuals should be selected. We apply a method where the probability Pi of an individual Li to be selected is proportional to the quality value q(Li). q( Li ) Pi z ¦ q( Li ) j 1

The random selection strategy assures that sometimes solutions of lower quality are also occasionally chosen to produce new generations. This will reduce the danger of converging too fast against one single solution that might correspond to a local minimum [2]. MUTATION In our application, we first select a fixed mutation rate, which refers to the probability of a call to mutate in the call list. Next, we iterate through the

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The mutation will be applied to all call lists selected after the selection process. CROSSOVER Crossover is the most important step in the context of genetic algorithms. After the selection step, the eliminated individuals are added by applying crossover to the selected individual, and the number of the added individuals will be equal to the size of the population minus the selected individual number. We first randomly select two parent call lists p1 and p2, then we randomly generate a binary vector whose length is equal to the length of the unassigned call list, and then we create a blank new call list. Thereby, each element of this binary vector is chosen independently with the same probability for the two possible values zero and one. Afterwards, we copy the elements of p1 to the same positions within the new list, if the element of the binary vector at the corresponding position has a value of one. The remaining elements of p1 are permuted so that they appear in the same order as in p2. Then, we enter these resorted elements in the empty gaps of the new call list [2]. An example of crossover is illustrated in Figure 3.

CHOICE OF PARAMETERS There are several parameters that are important in genetic algorithms, namely population size, mutation rate, number of select parents, and crossover rate. In our application of genetic algorithms, the population size, mutation rate, and number of selected parents need to be set. We have made several tests of different parameters. The convergence behavior of genetic algorithms is given in Figure 4 and Figure 5. Our observations show that the adjustment of different parameters would most probably improve the convergence speeds presented in the following section. Unfortunately, the optimal choice of parameters depends significantly on the given problem and there does not exist exact rules for the determination of the best parameters [2]. Since we have a given problem and a fixed lower bound, we select fixed parameters according to our test result, and only change it whenever the given parameter cannot find an optimal solution. Actually, the new algorithm is not very sensitive to parameter tuning and obtains the same result for all moderately selected parameters. This can be an important advantage in comparison to existing parametersensitive algorithms like simulated annealing.

Population Size & GA Convergence 60 50 Series1

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unassigned call list and randomly determine the calls that will mutate according to the mutation rate. When the call is to mutate, its two neighbors exchange positions, and its two two-cell away neighbors exchange positions also. The exchange will not be executed if calls on either side are out of the index bound. In the following example, call 8 is mutated, but if 6 is to mutate, only 2 and 3 will exchange positions.

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Figure 4. Population Size & GA Convergence

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Sorting rest calls of parent 1 according to parent 2 3,1,4,9,2 o 2, 9, 3, 4, 1 Parent 1: 0 6 2 5 9 3 8 4 7 1 Figure 3. Crossover Example.

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5 SIMULATION The new algorithm has been implemented using Java on a Pentium IV 1500. For comparison, we use the results of Sivarajan [7], Funabiki [1], and Beckmann [2]. Table I summarizes the specifications. In the compatibility matrix of Table I, Nc is the cluster size for CCC. Cij is the minimum channel distance between any pair of cells in the same cluster except for adjacent cells for CCC. ACC is the minimum channel distance between adjacent cells. Cii is the minimum distance within the same cell for CSC. In the demand vector, Call Case 1 and 2 represent the corresponding channel requirements m1 and m2. m1 = {8,25,8,8,8,15,18,52,77,28,13,15,31,15,36,57,28,8,10 ,13,8}; m2 = {5,5,5,8,12,25,30,25,30,40,40,45,20,30,25,15,15,30,2 0,20,25}; Total calls is the total number of requested channels in each instance.

Table I: Specifications for Sivarajan’s Benchmark Problem. Compatibility Matrix Demand Vector Instance Nc Cij ACC Cii Call TotalCalls Case #1 12 1 2 5 1 481 #2 7 1 2 5 1 481 #3 12 1 2 7 1 481 #4 7 1 2 7 1 481 #5 12 1 1 5 1 481 #6 7 1 1 5 1 481 #7 12 1 1 7 1 481 #8 7 1 1 7 1 481 #9 12 1 2 5 2 470 #10 7 1 2 5 2 470 #11 12 1 2 7 2 470 #12 7 1 2 7 2 470 #13 12 1 2 12 2 470 Table II is the simulation results of different channel assignment algorithms. As we can see, the three-stage genetic algorithm proposed in this paper successfully found the lower bound solution in 11 of the 13 instances. Our observation shows that for the easy cases, #3 - #8 and #10 - #13, using the first two stages of our new algorithm, sometimes with just one or two generations of the third algorithm, the optimum solution can be easily found. The computation time is just a few seconds. For cases #1, #2, #9, #10, the third stage participates much more to achieve the optimum solution, and the computation time can be as long as 10 hours. This is comparable to other algorithms based on the fact that this algorithm is implemented using JAVA. This reflects the combination of the heuristic algorithm and genetic algorithm. It has a very practical advantage of finding an optimum solution faster than neural-networks and simulated annealing, which both need a lot of computation even with simple cases, and it always finds a optimum or near-optimum solution with the power of the genetic algorithm.

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Table II Simulation Results for Sivarajan’s, Funabiki’s Three-Stage Neral-Network Algorithm, Genetic Algorithm, and the Three-Stage Genetic Algorithm. Instance

Lower Bound

Sivarajan’s

Funabiki’s

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13

427 427 533 533 381 381 533 533 253 253 309 309 529

460 447 536 533 381 381 533 533 283 269 310 310 529

427 427 533 533 381 381 533 533 258 258 309 309 529

ThreeStage GA 432 432 533 533 381 381 533 533 253 253 309 309 529

[2]

[3]

[4]

[5]

[6] 6 CONCLUSIONS This paper presents a three-stage algorithm of combining sequential heuristic methods into a genetic algorithm for the NP-complete channel assignment problem in cellular mobile communication system. The three-stage algorithm seeks the optimum solution by using the regular interval assignment stage, the greedy assignment stage, and the genetic algorithm assignment stage. The performance is verified through solving the Philadelphia problem and compared with the results of Sivarajan et al [7], Funabiki et al. [1], and Beckmann and Killat [2]. Our algorithm achieves the lower bound solution in 11 of the 13 instances. With the increase of mobile communication demands, the hierarchical cellular system will be the next generation of cellular communication. In [5], a bi-directional call-overflow scheme based on the velocity of the mobiles in the cells of the hierarchical cellular system is proposed. Because of the staging characteristic of our algorithm, we believe it can be applied to improve this scheme by achieving better performance in call-blocking prevention.

[7]

[8]

[9]

[10]

in Cellular Mobile Systems” IEEE Transactions on Vehicular Technology, vol. 49, no. 2, pp. 397-403, March 2000. D. Beckmann and U. Killat, “A New Strategy for the Application of Genetic Algorithms to the Channel-Assignment Problem” IEEE Transactions on Vehicular Technology., vol. 48, no. 4, pp. 1261-1269, July 1999. J.S. Kim, S.H. Park, P.W. Dowd, and N.M.Nasrabadi, “Channel assignement in cellular radio using genetic algorithms,” Wireless Personal Commun., vol.3, no. 3, pp. 273-286, Aug. 1996. K. Yeung, T.P.Yum, “Fixed Channel Assignment Optimization for Cellular Mobile Networks”, IEICE Trans. Commun., vol. E83-B, No.8 August 2000 W. Shan, P. Fan, and Yi Pan, “Performance Evaluation of a Hierarchical Cellular System with Mobile Velocity-based Bidirectional Call-overflow Scheme,” IEEE Transactions on Parallel and Distributed Systems, accepted, to appear. M. Zhang, T.P. Yum, “Comparisons of Channel-Assignment Strategies in Cellular Mobile Telephone Systems”, IEEE Trans. Vehicular Technology. Vol.38. No.4. November 1989. K. N. Sivarajan, R. J. McEliece, and J. W. Ketchun, “Channel Assignment in Cellular radio”, in Proc. 39th IEEE Vehicular Technology Conf., May 1989, pp. 846-850. Wei Wang and Craig K. Rushfort, “An Adaptive Local-Search Algorithm for the Cannehl-Assignment Problem (CAP),” IEEE Transactioin on Vehicular Technology, pp. 459-466, vol. 45, no. 3, August 1996. G. Chakraborty, B. Chakraborty, “A Genetic Algorithm Approach to Solve Channel Assignment Problem in Cellular Radio Networks”, IEEE Midnight-Sun Workshop on Soft Computing Methods in Industrial Applications, Kusamo, Finland, June 16-18, 1999. D.H.Smith, S. Hurley, S.M.Allen, “A New Lower Bound for the Channel Assignment Problem”, IEEE Transactions on Vehicular Technology, Vol 49, No.4, July 2000.

REFERENCES [1] N. Funabiki, N. Okutani and S. Nishikawa, “A Three-Stage Heuristic Combined NeuralNetwork Algorithm for Channel Assignment

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