Using a genetic algorithm approach to solve the - Department of ...

Int. J. Mobile Communications, Vol. 4, No. 3, 2006

Using a genetic algorithm approach to solve the dynamic channel-assignment problem Xiannong Fu and Anu G. Bourgeois Department of Computer Science Georgia State University Atlanta, GA 30303, USA Fax: (404) 463–9912 E-mail: [email protected] E-mail: [email protected]

Pingzhi Fan Institute of Mobile Communications Southwest Jiaotong University Chengdu, Sichuan 610031, PRC, China Fax: +86 28 7600743 E-mail: [email protected]

Yi Pan* Department of Computer Science Georgia State University Atlanta, GA 30303, USA Fax: (404) 463–9912 E-mail: [email protected] *Corresponding author Abstract: The Channel Assignment Problem is an NP-complete problem to assign a minimum number of channels under certain constraints to requested calls in a cellular radio system. Examples of the many approaches to solve this problem include using neural-networks, simulated annealing, graph colouring, genetic algorithms, and heuristic searches. We present a new heuristic algorithm that consists of three stages: 1) determine-lower-bound cell regular interval assignment; 2) greedy region assignment; and 3) genetic algorithm assignment. Through simulation, we show that our heuristic algorithm achieves lower bound solutions for 11 of the 13 instances of the well known Philadelphia benchmark problem. Our algorithm also has the advantage of being able to find optimum solutions faster than existing approaches that use neural networks. Keywords: wireless cellular network; channel assignment; genetic algorithm; exhaustive search; heuristic algorithm; performance; simulation. Reference to this paper should be made as follows: Fu, X., Bourgeois, A.G., Fan, P. and Pan, Y. (2006) ‘Using a genetic algorithm approach to solve the dynamic channel-assignment problem’, Int. J. Mobile Communications, Vol. 4, No. 3, pp.333–353.

Copyright © 2006 Inderscience Enterprises Ltd.

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X. Fu, A.G. Bourgeois, P. Fan and Y. Pan Biographical notes: Xiannong Fu received his bachelor’s degree in English Language and Literature at Tsinghua University in 1998. His work as an editor for internet-based media led him to the world of software development. He entered Georgia State University in 2000 as a student of computer science and received his master’s degree in 2002. Currently, he is working as a programmer in the Card Services Operation Information Management of the Bank of America. Anu G. Bourgeois received her BS and PhD degrees in Electrical and Computer Engineering from the Louisiana State University in 1991 and 2000, respectively. During 1996–2000, she held the Board of Regents Fellowship at LSU and the Vincent A. Forte fellowship during 1998–1999. Currently, she is Assistant Professor in the Department of Computer Science at Georgia State University. Dr. Bourgeois’ research interests include parallel and distributed computing, reconfigurable networks, wireless networks and fault tolerant computing. She has served as a reviewer for several journals and as a program committee member for numerous workshops and conferences. She is a senior member of the IEEE. Pingzhi Fan is currently Professor and Director of the Institute of Mobile Communications of the Southwest Jiaotong University, PRC. Dr. Fan received his MSc degree in Computer Science from the Southwest Jiaotong University, PRC, in 1987, and his PhD degree in Electronic Engineering from the Hull University, UK, in 1994. Dr. Fan’s research interests include spread-spectrum and CDMA communications, information and coding and wireless networks. He is the inventor of 16 patents and the author of over 250 research papers including about 150 journal papers and six books, which include three books published by John Wiley and Sons Ltd., RSP (1996), IEEE Press (2003) and Springer (2004). Yi Pan is Chair and Full Professor in the Department of Computer Science at Georgia State University. He received his BEng and MEng degrees in Computer Engineering from Tsinghua University, China, in 1982 and 1984, respectively. He received his PhD degree in Computer Science from the University of Pittsburgh, USA, in 1991. Dr. Pan’s research interests include parallel and distributed computing, optical networks, wireless networks and bioinformatics. Dr. Pan has published more than 80 journal papers, with 29 papers published in various IEEE journals. He has also published over 100 papers in refereed conferences, co-edited 18 books (including proceedings) and contributed several book chapters. Dr. Pan has served as Editor-in-Chief and an editorial board member for 15 journals including five IEEE transactions.

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Introduction

In recent years, the number of cellular users and the popularity of mobile computing have grown tremendously. This is due to the many technological advances supporting mobile communications such as providing high-quality voice communications and high-speed data services (Andreou et al., 2005; Kendall and Mohamad, 2004; Kumar, 2004; Varshney, 2003; Lu et al., 2005). As the trend has been towards a larger number of cellular users, the reality is that there is a limited frequency spectrum available for allocation among these users. Consequently, the efficient use of channel frequencies becomes more and more significant and has become a critical research issue in recent

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years (Funabiki et al., 2000; Beckmann and Killat, 1999; Yeung and Yum, 2000; Smith et al., 2000; Agrawal and Zeng, 2002; Corne et al., 2000; Ghosh et al., 2003; Kendall and Mohamad, 2004; Sen et al., 1999). Depending upon the service requirements, the allocated spectrum is divided into a number of channels. In order to satisfy the large demand of mobile telephone services, channels need to be assigned and reused to minimise communication interference. In turn, these channels increase the traffic-carrying capacity of the system. This is referred to as the Channel Assignment Problem (CAP). CAP is classified as an NP-complete problem (Hale, 1980), which implies that 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 one. There are multiple ways to allocate the types of channels available in wireless communication networks, such as frequencies in FDMA and time slots in TDMA, to code channels in CDMA (Shan et al., 2003). The frequencies, time slots or codes in different channels must be separated far enough for them not to interfere with one another. One method to increase the frequency spectrum usage is to adopt the cellular structure approach. This approach spatially divides the wireless service area into a large number of hexagonal cell (see Figure 1). Users located in these cells request calls through their portable applications. The system then assigns channels to each request and provides the service. Figure 1

Cell division of wireless area

Consider for instance a system with f available channels to allocate to a total of n cells. This would lead to a total of nf different combinations. For example, a system with f = 20 channels and n = 10 cells would have 1020 possible combinations. Exhausting all combinations would not be efficient. Hence, a method is needed to quickly find the optimal solution(s). In addition to the complexity of the volume of possible combinations, 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 simultaneously. As the channel interference is primarily a function of frequency and distance, a channel can simultaneously be used by multiple cells if the mutual distance of these cells is beyond a certain threshold. Considering the radio interference between frequency spectra, the channel assignment must fulfill the following three interference constraints (Sivarajan et al., 1989):

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X. Fu, A.G. Bourgeois, P. Fan and Y. Pan The Co-Channel Constraint (CCC) – The same channel cannot be assigned simultaneously to a pair of cells that are not sufficiently far apart. The Adjacent Channel Constraint (ACC) – Adjacent channels cannot be assigned to adjacent cells simultaneously. The Co-Site Constraint (CSC) – The distance between any pair of channels used in the same cell must be larger than a specified distance.

The characteristics of the radio frequency propagation and spatial density of the expected traffic requirements are what normally determine the threshold values of these constraints. This threshold value, represented by an integer, is what defines the minimum gap that must exist between assigned channels to the two cells in order to avoid interference. Recall that a channel must be assigned to each call that is requested. Thus, the goal of CAP is to assign a minimum number of channels to every requested call, such that each assignment satisfies the constraints defined above. In this paper, we present an algorithm to solve the NP-complete channel assignment problem in a cellular mobile communication system. Our algorithm combines sequential heuristic methods into a genetic algorithm that consists of three stages. 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 remainder of the paper is organised as follows. Section 2 formulates the CAP problem and details the Philadelphia benchmark problem that we will use to compare our results against other existing algorithms. Section 3 describes some previous work and describes other algorithms and approaches for channel assignment. Section 4 presents our proposed channel assignment algorithm that consists of three stages, including one that utilises a genetic algorithm. Section 5 presents our simulation results and comparisons as applied to the benchmark problem. Section 6 concludes the paper.

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The channel assignment problem

We will now use the same model, as presented in previous papers (Sen et al., 1999; Sivarajan et al., 1989) to represent the Channel Assignment Problem (CAP). The basic CAP model consists of the following five components (Wang and Rushfort, 1996): 1 a set X of n distinct cells, with cell numbers 0, 1, …, n – 1 2 a demand vector m = (mi) (1 ≤ i ≤ n), where mi indicates the number of channels required for cell i 3

an n × 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 (0 ≤ i, j < n)

4

a frequency assignment matrix, fik, 1 ≤ i ≤ n, 1 ≤ k ≤ mi, is assigned to aik, the kth call in the ith cell. Each frequency is represented by a positive integer a set of frequency separation constraints defined by the compatibility matrix C, |fik – fjl| ≥ cij , for all i, j, k, l (i ≠ j, k ≠ l).

5

Considering the above problem definition, the channel assignment problem can be characterised by the triplet (X, m, C). The objective of this problem is to assign frequencies to each call while satisfying the frequency separation constraints as defined by component five above. These must be done while minimising the total bandwidth of the required system.

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Recall from the introduction, that the channel assignment must fulfill three interference constraints (Sivarajan et al., 1989) due to the radio interference between frequency spectra. The compatibility matrix C, is used to ensure that interference does not occur. It does so by providing sufficient frequency separation among subscribers. The three constraints preventing channel interference can be described by matrix C as follows: 1

Co-Channel Constraint (CCC) is described as cij ≠ 0.

2

Adjacent Channel Constraint (ACC) is described as cij ≥ 2, if i and j are adjacent cells.

3

Co-Site Constraint (CSC) is described as cii ≥ k, where k is a given constant.

In order to compare the performance of various algorithms designed to solve the channel assignment problem, there are some well known benchmark instances that have been defined for a 21-node cellular network. We now describe that model.

Philadelphia benchmark problem The CAP model that has set the framework for several well known benchmark instances was first proposed by Sivarajan in 1989 (Sivarajan et al., 1989). Subsequently, this model has been widely used as a test problem in channel assignment research (Funabiki et al., 2000; Sivarajan et al., 1989; Chakraborty and Chakraborty, 1999; Kendall and Mohamad, 2004). Often times, this problem model 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, numbered and positioned as shown in Figure 2. A problem instance over this network is defined by the demand vector that indicates the number of frequencies requested/demanded by each cell. Figure 2

Philadelphia problem instance 1

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 (Smith et al., 2000). Due to the fixed lower bound, the Philadelphia problem can serve well as a CAP testing example for various kinds of algorithms.

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The following two demand vectors describe two separate problem instances: 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, 20, 20, 25}. The order of the calls in each demand vector is listed in the same order of the cellular network cell numbers, such that m1i corresponds to the number demand imposed by cell i. Figure 2 depicts the problem instance corresponding to demand vector m1, such that the number shown in each cell indicates the number of frequencies demanded by that cell. As the goal of CAP is to assign frequencies to each call while minimising the total bandwidth and satisfying the compatibility matrix, we will obtain the constraints cij for the Philadelphia problem as follows. Let d(vi, vj) represent the distance between the centres of cells vi and vj, where the distance between centres of adjacent cells is assumed to be unit distance. (We assume that k = 5 for the equations below.) Then we obtain the following results for the three conditions: cij = 5, if i = j cij = 2, if 0 < d(vi, vj) ≤ 1 cij = 1, if 1 < d ( vi , v j ) < 2

3, where the cluster size is 1.

For the pairs of cells, i and j, that have no constraint between them, the corresponding entry of the compatibility matrix, cij, is not defined. We refer to a complete cluster Nc, as a cluster of cells, such that a constraint exists between any pair of cells. Recently, many methods to improve channel allocation have been proposed. An adaptive radio resource management in a CDMA-based hierarchical cell structure is proposed in Kwon and Cho (2002), where the macrocell can lend common frequency allocation to the microcell when microcells are busy. Shan et al. (2003) investigated a bidirectional call-overflow scheme based on the velocity of the mobile device. Lo et al. (2003) proposed a neural fuzzy control for radio resource management in hierarchical cellular systems supporting multimedia services. Yao et al. (2004) proposed virtual partitioning resource-allocation schemes for handling multiclass services with guard channels in a cellular system. A Flexible Resource Allocation (FRA) strategy with differentiated priorities and QoS and an FRA strategy with prioritised levels were proposed in Cruz-Pérez and Ortigoza-Guerrero (2004). Naik and Wei (2004) studied the concept of the call-on-hold to improve the performance of a class of dynamic channel assignment algorithms. Ferng and Tsai (2005) investigated effects on the channel allocation for the General Packet Radio Service (GPRS) system caused by priority strategy, buffering, threshold control on the buffer and channel reservation. Akl et al. (2005) presented a novel approach for designing a call-admission control algorithm for CDMA networks. The algorithm incorporates call-arrival rates and user mobility across the network and guarantees the users’ quality of service as well as pre-specified blocking probabilities. Its implementation in each cell uses the number of calls currently active in that cell. All these activities indicate that channel assignment is still a very active research area in wireless network research. We will discuss the details of some specific methods in the following section.

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Previous work

There are three broad classifications for the channel assignment problem. The first is called Fixed Channel Assignment (FCA), the second is Dynamic Channel Assignment (DCA), and the third is Hybrid Channel Assignment (HCA) (Corne et al., 2000). In FCA, channels are assigned to each cell permanently based on a predefined channel demand. In DCA, channels are dynamically assigned to each cell based on the channel requests. HCA, as the name implies, is a combination between FCA and DCA (Kendall and Mohamad, 2004). 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. FCA also provides the worst channel utilisation. However, FCA is the easiest scheme of the three. Dynamic channel assignment algorithms attempt to optimise 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. DCA, however, is much more complex than FCA. By mapping the CAP problem to the generalised graph colouring problem, the CAP problem has been proven to be NP-complete. Taking advantage of this, Sivarajan et al. (1989) proposed non-iterative algorithms using ideas from graph colouring algorithms. The eight algorithms they proposed are a combination of cell ordering and call ordering assignment methods. Yeung and Yum (2000) proposed four algorithms based on two node ordering principles: node-colour ordering (CR) and node-degree ordering (DR). These algorithms are also based on two assignment strategies: Frequency Exhaustive Assignment (FEA) and Requirement Exhaustive Assignment (REA). The four resulting algorithms are FEA/CR, FEA/DR, REA/CR and REA/DR (We will describe FEA, REA and node-degree ordering in detail in the following sections.). Another technique that has been applied to CAP in recent research work is Artificial Intelligence (AI) algorithms (Zhang and Yum, 1989). Besides AI algorithms, Chakraborty et al. (Chakraborty and Chakraborty, 1999) proposed a genetic algorithm that generates the initial population and creates near optimum solutions for large populations. They next apply a genetic mutation operation to improve the quality of the previously obtained initial solutions towards an optimum solution. A similar genetic algorithm strategy can be found in Kim et al. (1996). Beckmann and Killat (1999) 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 using the FEA strategy and evaluates the result by considering the number of blocked calls. It then executes the genetic algorithm and FEA repeatedly until the best solution is obtained. Although Chakraborty et al. (Chakraborty and Chakraborty, 1999) and Beckmann and Killat (1999) presented strategies using genetic algorithms, their methods are substantially different. First, the approach in Beckmann and Killat (1999) is for fixed channel assignments and that of Chakraborty and Chakraborty (1999) is for dynamic assignments. In Beckmann and Killat (1999), the genetic algorithm is applied to a randomly generated population, and the FEA strategy is applied in every generation to select the best of the population. While in Chakraborty and Chakraborty (1999), the initial population (with a large population size) is restricted only to genotypes representing a valid solution. The genotype in Beckmann and Killat (1999) is an array

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representing the size of the number of cells and the value of the array of the assigned channel number. The genotype in Chakraborty and Chakraborty (1999) is an M × N matrix, where M is the number of channels and N the number of cells. The final results obtained by the two algorithms differ as well. An optimal call list is the end product of Beckmann and Killat’s algorithm, whereas an optimum frequency assignment solution is the end product of the algorithm of Chakraborty et al. Funabiki and Okutani (Funabiki et al., 2000) proposed a three-stage neural-network algorithm. The first stage of this algorithm assigns channels first to the cell that determines the lower bound at an interval of CSC. In the second stage, the cluster of cells whose centre cell has the largest node-degree (which is called the greedy region), is assigned channels by the REA strategy. Each time the overall 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 simultaneously assigned channels by a binary neural network. Each of the algorithms presented in this section has a unique approach to solve the CAP problem. Some of the methods combined various techniques to achieve the end result. Our proposed algorithm that we will present in the next section is for the dynamic channel assignment problem. It utilises genetic algorithms as well.

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Proposed three-stage genetic algorithm

We propose solving the channel assignment problem by combining a DCA heuristic with genetic algorithms. The proposed algorithm is derived by composing the algorithms of Funabiki et al. (2000) and Beckmann and Killat (1999). The design was motivated by detecting portions of existing solutions that could be improved to achieve the results more efficiently. We were not able to obtain the simulation results for the general GA approach. Thus, we focus on the existing algorithms mentioned above. We will compare our method against the existing algorithms later in this section. Our algorithm consists of the following three channel assignment stages: 1

regular interval assignment

2

greedy assignment

3

genetic algorithm assignment.

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. Each channel assigned thereafter is at an equal distance away from the previously assigned channel as determined by the CSC. The second stage determines a greedy region of the network and assigns channels within the region. 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 that are adjacent to the existing greedy region. Then, it reattempts to assign channels to 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. The following list of steps describes the proposed algorithm:

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1

Use the following items as input: compatibility matrix C, demand vector D, cluster size Nc, CCC, ACC, and CSC.

2

Initialise the total number of channels M by the lower bound.

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Apply the regular interval assignment stage to the cell that determines the lower bound of M.

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Compute the degree of cell i, di, where 1 ≤ i ≤ N, as follows: N

di = (∑ d j Cij ) − Cii 1

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Initialise the greedy region by including the cell with the maximum degree and its six adjacent cells.

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Fix the assignment in Step 3, and apply the greedy assignment stage to the greedy region determined in Step 5.

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If this assignment fails, expand the greedy region by including all other cells adjacent to the cells already in the greedy region. Go back to Step 6.

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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 continue.

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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 the Frequency Exhaustive Assignment (FEA) as evaluation.

10 If b = 0, then the algorithm is successful. As mentioned above, our proposed algorithm is a composition of the algorithms presented by Funabiki et al. (2000) and Beckmann and Killat (1999). As in the algorithm by Funabiki et al., our new algorithm has three stages; the first stages are the same. The second stage of the proposed algorithm uses an FEA strategy for the greedy region assignment instead of the REA strategy. This is because test results show that the probability of finding an optimum solution by FEA is much higher than that of REA. Our third stage applies a genetic algorithm similar to the algorithm by Beckmann and Killat to find an optimum call list for the cells left unassigned by the first two stages. Some of the parameters used in the new algorithm differ from that in Beckmann and Killat (1999), based on the test result. We will now provide details regarding each of the three stages of our proposed algorithm. Stage 1

Regular interval assignment

The first stage of the algorithm, called the regular interval assignment stage, assigns channels to calls in the cell that determines the lower bound on the number of channels M. This is typically the cell that has the largest call demand. The assignments made are spaced apart at regular intervals satisfying the CSC in order to minimise the total number of channels M that are used.

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Stage 2

Greedy assignment

Prior to discussing the details of the second stage, we will first describe some techniques that it uses. Specifically, we will consider node-degree ordering and two exhaustive assignment strategies.

4.1 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 follows: N

di = ∑ m j cij − cii , if mi ≠ 0, 1 ≤ i ≤ N

(1)

j =1

else, di = 0. We can simplify the equation as follows since the term cij on the right hand side of Equation (1) is the same for all cells: N

di = ∑ m j cij ,

1 ≤ i ≤ N.

(2)

j =1

4.2 Frequency exhaustive strategy and requirement exhaustive strategy Once the cells have been ordered by either node-degree ordering (sorted based on node degree) or node-colour ordering (sorted based on node colouring) and all channels have been ranked according to their carrier frequencies, two channel assignment strategies, Frequency Exhaustive Strategy (FEA) and Requirement Exhaustive Strategy (REA) are 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, shown in Figure 3, 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. Figure 3

Frequency exhaustive assignment strategy Loop 1: for each call i in the call list. Loop 2: for each channel j within the lower bound IF i is not assigned AND j can be assigned to i Assign j to i Break Loop2 End If End Loop 2 End Loop 1

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On the other hand, 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). Figure 4 shows the pseudocode for the REA strategy. As with the FEA strategy, the calls are first ordered within the cells and then ordered by their cell number. Figure 4

Requirement exhaustive assignment strategy Loop 1: for each channel j within the lower bound Loop 2: for each call i in the call list IF i is 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 cell-to-cell channel requirements, it is more important to maximise the number of co-channel cells. Therefore, 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 (Yeung and Yum, 2000). In the Philadelphia problem, the channel requirements are high and non-uniform. Therefore, we use FEA in our algorithm.

4.3 Greedy assignment stage Now that we have explained the node-ordering technique and the exhaustive assignment strategies, we can now describe the procedure for the second stage of our algorithm. The greedy assignment stage assigns channels to cells that form the greedy region. The greedy region is regarded as the region of cells with the highest assignment difficulties. The procedure of this stage is as follows: 1

Initialise the greedy region to include the cell with the largest node-degree and all six of its adjacent cells.

2

Make a list of calls from the greedy region in the ascending order of the cell index number.

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Apply FEA to the channels in the call list.

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If every call in the list is assigned a channel, then terminate the procedure as successful and proceed to the genetic algorithm stage. Else, if the greedy region did not cover all calls, expand the greedy region by including all cells adjacent to the existing greedy region. Then, go back to Step 3. If the greedy region assignment fails, go to Stage 3 for the genetic algorithm stage. Stage 3

Genetic algorithm assignment

Once the first two stages of our algorithm are completed, calls in the remaining cells are randomly ordered. Now, 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. The algorithm starts by randomly generating an initial population of possible solutions. For our problem, 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. Afterwards, we process a loop 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, producing new individuals to maintain the population size. The algorithm continues to iterate the loop until we achieve convergence or we exceed the maximum number of generations. In this genetic algorithm, not only are calls in neighbouring cells 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 (Beckmann and Killat, 1999), which may occur in many fixed channel assignment algorithms. Figure 5 illustrates the genetic algorithm used in Beckmann and Killat (1999). It also illustrates the third stage of our proposed algorithm. The differences are, in Beckmann and Killat (1999), the input parameters are demand D, interference matrix C and number of frequencies z. In our new algorithm, the input parameters also include the assignments made by the first two stages. In Beckmann and Killat (1999), 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. The flowchart of the general genetic algorithm is shown in Figure 6. Figure 5

Genetic algorithm – FEA channel assignment

Using a genetic algorithm approach Figure 6

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Genetic algorithm flowchart

4.4 Evaluation phase The first phase of the genetic algorithm is to generate a population of individuals. Next, 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 as the measure. We can also refer to it as blocked call, b, as the fitness, which results from the attempt to solve the CAP using FEA. 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 is a good measure of the quality of the solutions we are searching for.

4.5 Selection phase After the individuals have been evaluated according to their respective fitness measures, the ones with better fitness will be selected, and the others will be eliminated. This will help improve the total fitness of the population. There are many different strategies available to implement the procedure of selection. We use the quality measure q of an individual, q = b, which is the number of blocked calls after FEA, to determine which individuals should be selected. To do so, we apply a method where the probability Pi of an individual Li to be selected is proportional to the quality value q(Li):

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q( Li ) z

∑ q( Li )

.

(3)

j =1

The random selection strategy, given by Equation (3), assures that solutions of lower quality are also occasionally chosen to produce new generations. This reduces the danger of converging too fast against one single solution that might correspond to a local minimum (Beckmann and Killat, 1999), resulting in deadlock.

4.6 Mutation phase 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 unassigned call list and randomly determine the calls that will mutate according to the mutation rate. When the call is to mutate, its two neighbours exchange positions. Its two neighbours that are two cells away exchange positions as well. The exchange will not be executed if calls on either side are out of the index bound. Refer to Figure 7 below for an example of mutation. The first array is a call list prior to the mutation of Call 8. The second array shows the call list after the mutation of call 8. Notice that the two neighbours of 8, Calls 5 and 7, have exchanged positions. The calls that are distance two away, Calls 4 and 9, have also exchanged positions as shown in the second array. However, if Call 6 is to mutate, only 2 and 3 will exchange positions since the other pair of calls will be outside the index bound. Figure 7

Example of mutation

The mutation will be applied to all call lists selected after the selection process as shown in the flowchart of Figure 6.

4.7 Crossover phase 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. 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. After doing this, we create a blank new call list. Thereby, each element of this binary vector is chosen independently with the same

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probability for the two possible values zero and one. Afterwards, if the element of the binary vector at the corresponding position has a value of one, then we copy the elements of p1 to the same positions within the new list. The remaining elements of p1 are permutated so that they appear in the same order as in p2. Next, we enter these resorted elements in the empty gaps of the new call list (Beckmann and Killat, 1999). An example of crossover is illustrated in Figure 8. Figure 8

Example of crossover

4.8 Choice of parameters In the context of genetic algorithms, there are several parameters that are important. Specifically, these parameters are population size, mutation rate, number of select parents and crossover rate. In our application of applying genetic algorithms to solve the channel assignment problem, we set the population size, mutation rate and number of selected parents. Based upon these set parameters, we have made several tests of the remaining parameters. The convergence behaviour of the genetic algorithms is given in Figures 9 and 10. We can see that by adjusting different parameters, the convergence speeds would most probably improve, as presented in the following section. Unfortunately, the optimal choice of parameters depends significantly on the given problem. There exists no exact set of rules for determining the best parameters (Beckmann and Killat, 1999). In the context of the CAP problem, we have a given problem and a fixed lower bound. Also, we are able to 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 parameter-sensitive algorithms like simulated annealing.

348 Figure 9

X. Fu, A.G. Bourgeois, P. Fan and Y. Pan Population size and GA convergence

Figure 10 Mutation rate and GA convergence

Using a genetic algorithm approach

5

349

Simulation results

We simulated our proposed three-stage genetic algorithm using Java on a Pentium IV 1500. For comparison, we solved the Philadelphia problem and compared our results with the results of Sivarajan (Sivarajan et al., 1989) and Funabiki (Funabiki et al., 2000). As we could not obtain simulation results for the general GA approach, we focused on three similar approaches as given in Table 2. Table 1 summarises the obtained results. In the compatibility matrix of Table 1, 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 Cases 1 and 2 represent the corresponding channel requirements m1 and m2, respectively: 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, 20, 20, 25}. The term total calls, is the total number of requested channels in each instance. Table 1

Specifications for Sivarajan’s benchmark problem Compatibility matrix

Demand vector

Number of instance

Nc

Cij

ACC

Cii

Call case

Total calls

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 2 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. From our simulations, we see that for the easy cases, Number 3 – Number 8 and Number 10 – Number 13, we can find an optimal solution by using the first two stages of our new algorithm and sometimes with just one or two generations of the third stage. The performance is satisfying. Also, the computation time took just a few seconds. Cases Number 1, 2, 9 and 10, on the other hand, usually require over 200,000 generations of the third stage to achieve the optimum solution. The computation time can be as long as 10 hours. This is comparable to other algorithms based on the fact that this

350

X. Fu, A.G. Bourgeois, P. Fan and Y. Pan

algorithm is implemented using JAVA. Neither the language used nor the hardware are the best candidates for producing an efficient computation timing. The efficiency of this algorithm is evident, however, from the number of generations required as described above. Table 2

Simulation results for Sivarajan’s, Funabiki’s three-stage neural-network algorithm, and the three-stage genetic algorithm

Number of instance

Lower bound

Sivarajan’s

Funabiki’s

Three-stage GA

1

427

460

427

432

2

427

447

427

432

3

533

536

533

533

4

533

533

533

533

5

381

381

381

381

6

381

381

381

381

7

533

533

533

533

8

533

533

533

533

9

253

283

258

253

10

253

269

258

253

11

309

310

309

309

12

309

310

309

309

13

529

529

529

529

To briefly illustrate our results, we will use Instance Number 1 as an example. Figure 11 shows a partial output obtained for Instance Number 1 (Only 12 rows following the header are shown.). The header line in the figure lists the 21 points and corresponds to Call Case 1 and channel requirements m1. The program is running two loops. The first loop will stop only after all 21 points are assigned the required number of channels. The second loop will try to assign a channel to each of the 21 points from Point 1 to Point 21 that does not conflict with any of the constraints: NC, ACC, Cij and Cii. The number of the channels assigned is printed out in each row. If a point is assigned the required number of channels, then 0 is printed out in each of the following rows. For example, Point 1 requires eight channels, therefore, starting from row nine, all numbers listed under Point 1 is 0. The genetic algorithm will generate tens of thousands solutions. However, only the solution that fits all constraints is printed out. Our results reflect the combination of the heuristic algorithm and genetic algorithm. It has a very practical advantage of finding an optimum solution faster than neural-networks, which needs a lot of computation even with simple cases. Also, this algorithm combination always finds an optimum or near-optimum solution due to the power of the genetic algorithm used in the third stage of our algorithm. All of the above evidences indicate that our new algorithm is an efficient and powerful tool for solving the channel assignment problem.

Using a genetic algorithm approach

351

Figure 11 Partial output results for instance Number 1

6

Conclusion

Although the number of cellular users continues to increase, the resources available in terms of frequency spectrum available for allocation among these users is limited. As a result, it is imperative to design a method that efficiently assigns the use of channel frequencies (Funabiki et al., 2000; Beckmann and Killat, 1999; Yeung and Yum, 2000; Smith et al., 2000; Agrawal and Zeng, 2002; Corne et al., 2000; Ghosh et al., 2003; Kendall and Mohamad, 2004; Sen et al., 1999). In order to satisfy the large demand of the mobile telephone services, channels need to be assigned and reused to minimise communication interference, resulting in an increase in the traffic-carrying capacity of the system. We presented an algorithm to solve the NP-complete channel assignment problem in a cellular mobile communication system. Our algorithm combines sequential heuristic methods into a genetic algorithm that consists of three stages. 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 first stage assigns channels at regular intervals to the cell whose calls determine the lower bound on the total number of channels. The second stage determines a greedy region of the network and assigns channels within the region. If the second stage fails, then the algorithm executes the third stage based on the result of the first stage. We verify the performance of our algorithm by solving the Philadelphia problem and by comparing our results with the results of Sivarajan et al. (1989) and Funabiki et al. (2000). Our algorithm achieves the lower bound solution in 11 of the 13 instances. The results we obtained indicate that our new algorithm is an efficient and powerful tool for solving the channel assignment problem. With the increase of mobile communication demands, the hierarchical cellular system will be the next generation of cellular communication. In Shan et al. (2003), 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 that it can be applied to further reduce call-blocking probability and increase channel utilisation. Our results could also have a large impact on many mobile applications that affect mobile commerce. For instance, future applications will

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X. Fu, A.G. Bourgeois, P. Fan and Y. Pan

include mobile advertising based upon user and/or location specific data (Varshney, 2003; Wang and Wang, 2005). If we are able to increase the bandwidth of information delivered by assigning channels as efficiently as possible, then many of the location based mobile commerce applications will be able to reach a larger span of consumers.

Acknowledgement Yi Pan and Pingzhi Fan would like to acknowledge the support provided by the National Natural Science Foundation of China (NSFC) under Grant No. 60440420451 (‘two base’ project).

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