A New Technique to the Channel Assignment ... - Semantic Scholar
A New Technique to the Channel Assignment Problem in Mobile Communication Networks Carlos E. C. Vieira Military Institute of Engineering - IME Rio de Janeiro, Brazil Email: [email protected]
Jacir L. Bordim
Paulo R. L. Gondim and Carino A. Rodrigues Department of Electrical Engineering University of Bras´ılia Bras´ılia, Brazil Email:[email protected] and [email protected]
Abstract—Channel allocation has been considered a NPcomplete problem, and involves a very important issue for the adequate dimensioning of wireless communication networks. One of the possibilities for its solution involves the utilization of heuristics, able to produce near optimal solutions in an acceptable computational time. In this paper, a hybrid of GRASP (Greedy Randomized Adaptive Search Procedure) and FEA (Frequency Exhaustive Assignment) is proposed, allowing the achievement of best solutions for the Channel Allocation Problem (CAP).
I. I NTRODUCTION An important aspect in the design of a cellular radio network is the efficient use of the scarce electromagnetic spectrum, involving a channel allocation procedure, in order to satisfy the traffic demand, while electromagnetic compatibility constraints are obeyed. Three electromagnetic compatibility constraints are commonly considered: - Co-Channel Constraint (CCC): the same frequency can not be assigned to certain pair of cells simultaneosly, except a specified distance is respected; - Adjacent Channel Constraint (ACC): any pair of frequencies assigned to two adjacent cells must guard a specified distance. This distance is normally larger than the one used for CCC; - Co-Site Constraint (CSC): any pair of channels assigned to a cell must guard a specified distance. This distance is usually larger than the one used for ACC. While electromagnetic spectrum is a limited resource, the fast growing population of mobile subscribers imposes the need of an adequate treatment of the traffic demands. The distribution of the set of available channels among the base stations in the system, named channel allocation, has been extensively proposed in the literature, showing the great importance and interest related to the problem, involving static allocation ([1],[2], [3]), dynamic allocation [4]) and hybrid allocation ([5], [6]). On heavy load traffic conditions, static allocation is more efficient than dynamic allocation ([5], [7] and [8]), thus providing a more efficient reuse of the spectrum. Since heavy traffic conditions are expected in the future, this work is exclusively about static channel allocation. For the CAP, the application of meta-heuristics involves, for example, genetic algorithms [3], simulated annealing [9] and neural networks [10].
978-1-4244-2644-7/08/$25.00 © 2008 IEEE
Department of Computer Science University of Bras´ılia Bras´ılia, Brazil Email:[email protected]
This paper presents a new powerful alternative for the resolution of the Channel Allocation Problem (CAP), based on the hybridization of the metaheuristic GRASP (Greedy Randomized Adaptive Search Procedure), with a heuristic of sequential allocation, i. e., Frequency Exhaustive Assignment (FEA). The FEA strategy is incorporated into the local search phase of GRASP method, associating frequencies to the calls, leading to the GRASP-FEA technique. As a form to compare the results obtained in this work with the ones of literature, 8 problems (selected from Philadelphia benchmark) have been chosen ([11],[12]), including the most difficult ones, problems 2 and 6. The remainder of the work is organized as follows: in section II, the formulation of CAP and its instances are presented; in section III, related work is discussed; in section IV, GRASP-FEA is proposed; in section V, the results are presented. Finally, section VI contains our conclusions and suggestions for future research. II. F ORMULATION OF CAP AND I NSTANCES A. Formulation of CAP The CAP can be defined by the quintuplet P = { X, D, C, A, B } [13], where the vector X = {x1 , x2 , x3 , ...xn } is the system of cells, D = di , is the 1xn vector that informs the demand (requirements) in X; C = (cij ), nxn is the compatibility matrix in X, with cii > 0 for all i; represents the frequency separation required between a call in cell i and a call in cell j. A = (ai ) will contain the pre-assigned frequencies and B = (bi ) the blocked frequencies. F=(fi ) is the solution vector to the problem if, for all i: |Fi | = di
(1)
|fi − fj | ≥ cij
(2)
Ai ⊆ F i
(3)
Bi ∩ F i = 0
(4)
It is known that the CAP is a NP-Complete problem [14]. This class of problems does not have an exact solution nor it is possible to obtain any information about the optimal solution. We adopt the problem statement used in [12] and
TABLE I CAP I NSTANCES Problem
1
2
3
4
5
6
7
8
ACC
1
2
1
2
1
2
1
2
CSC
5
5
7
7
5
5
7
7
Demand Vector D1 D1 D1 D1 D2 D2 D2 D2 Lower Bound
381 427 533 533 221 253 309 309
[14] and . Given • n: the number of cells in the system; • di , 1 ≤ i ≤ n: the number of channels required in cell i; • cij , 1 ≤ i, j ≤ n: the frequency separation; Find • fik , 1 ≤ i ≤ n, 1 ≤ k ≤ di : the frequency separation assigned to the k th call in the ith cell, such that the maximum frequency is the minimum possible respecting all the constraints. The frequencies are represented by positive integers. In this way, we must find the minimum number of frequencies that solve the problem, while respecting the mentioned constraints. B. CAP Instances For comparison among the techniques, 8 problems [11], presented in Table 1, were considered. The problems are derived assuming 2 demand vectors (D1 e D2) and 4 different matrices of electromagnetic compatibility (C1, C2, C3 e C4) [10]. These instances were applied to a 21-cells system. The 8 problems differ in the possible presence of ACC, where, for example, a value of 2 in the corresponding column implies that the neighboring frequencies cannot be simultaneously used in adjacent cells. It is assumed a seven-cells cluster-size for all problems. This means that a frequency can only be associated to 2 BSs(BS - Base Station), if these BSs are separated by, at least, 2 other cells, thus respecting the constraints related to co-channel interference. III. R ELATED W ORK In [13] it was proposed a methodology to calculate some Lower Bounds (LB), while respecting all the channel constraints. Basically, three lemmas (7, 9 and 10), formulated by [13], have been extensively used to test algorithms proposed to solve the CAP ([12], [14], [11] and [15]). Some basic strategies to solve the CAP include the Frequency Exhaustive Assignment (FEA) and Requirement Exhaustive Assignment (REA) heuristics. Considering a crescently ordered set of the available frequencies, the FEA strategy assigns calls to the minimum available frequencies, while respecting the constraints. On the other hand, for the use of REA, the lowest available frequency is assigned to
the maximum number of calls that is possible respecting the constraints. Several works used combinations of FEA or REA together with other techniques, in order to find near optimal solutions. In this sense, [12] proposed algorithms based on node-color and node-degree orderings, combined with FEA and REA heuristics.[14] developed three local search algorithms to solve the CAP, named CAP1, CAP2 and CAP3. The best result was obtained by CAP3 and will be used(with some adaptation) in this paper, as the local search phase of the GRASP. Many researchers explored the use of evolutionary strategy and genetic algorithms (GA) to solve the CAP. [19] presented an evolutionary approach for frequency assignment problem in cellular radio networks. The results of this strategy were compared to simulated annealing and constraint programming, with a different set of instances from the one used in this paper. In [5] evolutionary strategy is used to solve the CAP for a hybrid channel allocation, considering a smaller set of constraints (for example, the authors do not consider co-site interference constraints). [11] developed a three-stage genetic algorithm, combined with FEA strategy, to solve the CAP, for some of the problems defined in section II above, to test its solution, but it did not reach the LB for the problem number 2. These problems will be studied in this work by using the GRASP-FEA approach and, as we will see, all the LBs for the eight problems will be reached, including the most difficult problems, 2 and 6. The problems analyzed by [11] were the same as proposed by [12] which used 7 and 12-cells clusters. But the problem 2 is the same in [12] and [11]; therefore, our comparison is correct. [9] applied simulated annealing meta-heuristic to solve the CAP, but the compatibily matrix, used in that work, treats all interference relations between base stations with equal strength. We also use a compatibily matrix, but closer to the practical problem, it deals in a different way with adjacent and co-channel constraints. In [9], a different set of instances and a graph coloring approach were used. Some other papers (see, e.g., [16], [17] and [18]) also adopted the same approach. [15] proposed the Viterbi-like algorithm (VLA) to solve the CAP. The VLA is based on the original Viterbi algorithm used for information decoding in digital communication systems. The VLA uses as metric, instead of the Hamming distance, the excess frequency factor which is defined by the adjacent and co-channel constraints, compared with the difference of the frequencies assigned to a pair of cells. [1] described a hyper-heuristic to find the solution to the CAP. The hyper-heuristic is a problem independent strategy (high level), that manages a set of low level heuristics (LLH), defining the borderline of the data flow. This methodology combines a greedy constructive heuristic to generate the initial solution, then a group of LLH is used to improve the solution quality. An algorithm chooses which heuristic will be used each time. [17] and [18] used GRASP approach to solve the CAP, both using a graph coloring model. [17] combined GRASP with
simulated annealing in the local search phase. [18] made a combination of GRASP and Path Relinking as a strategy for local search. Those techiques were not tested in Philadelphia benchmark. In the GRASP-FEA hybrid meta-heuristic we modelled the CAP as a sequential allocation, applying a distinct greedy function(different from [17] and [18]) and other strategy to assign frequencies to calls respecting all interference constraints. IV. T HE P ROPOSAL OF THE GRASP-FEA A PPROACH The meta-heuristic GRASP[20] is composed of two phases: the phase of construction of the initial solution and the local search phase. In the construction phase, a set of initial solutions is constructed and afterward, a local search is carried through in the neighborhood of each constructed solution. FEA Strategy, which associates frequencies to the lists of calls, will be executed during the local search phase composing the hybridization with the GRASP. A. Solution Construction Phase The solution construction phase is responsible for the denomination of the GRASP, where a set of initial solutions is constructed iteratively, in a greedy, random and adaptive way. The procedure of construction GRASP-FEA uses the following a greedy function, which is executed on each Base Station, as the following calculation [12]: gi = (
n X
dj cij ) − cii , 1 ≤ i ≤ n
(5)
j=1
Equation (5) is the measure of the difficulty to allocate a frequency to a call in a cell i. When it is used a purely greedy procedure, it is iteratively constructed only a combination of possible solutions, without testing other possibilities, through an optimization criteria defined by the greedy function. In contrast, the method of construction GRASP generates diversity of solutions applying a semi-greedy heuristic [21]. The Restrict Candidate List (RCL) is constructed by the elements for which the constraints are more severe, and the cell that will be part of the solution is selected randomly from RCL. While the solution is incomplete, new calculations are carried out taking into account the choice of the previous element. RCL will be constructed again and a new element will randomly be chosen (amongst those cells whose constraints are more severe) to compose the solution. A parameter α(0 ≤ α ≤ 1) is used to control the RCL. The RCL includes all those elements that are not yet part of the solution, whose impact in the value of the objective function is in the interval, [cmin , α(cmax − cmin ) + cmin ], where cmin corresponds to the greedy selection and cmax implicates in the highest cost increase in the objective function. It is observed that α = 0 implies a purely greedy choice and α = 1 implies a random choice (considering a minimization problem). In the same way, in the case of a maximization problem, the RCL will be defined by the
interval [cmax + (1 − α)(cmin − cmax ), cmax ]. In this in case, α = 0 implies a random choice from a RCL that contains all the possible elements, while α = 1 consists in a pure greedy choice [22]. Algorithm for the Solution Construction Phase, adapted from [20] begin xi = 0; i = 1; Initialize candidates C; While (C != empty) do begin cmin
Jacir L. Bordim
Paulo R. L. Gondim and Carino A. Rodrigues Department of Electrical Engineering University of Bras´ılia Bras´ılia, Brazil Email:[email protected] and [email protected]
Abstract—Channel allocation has been considered a NPcomplete problem, and involves a very important issue for the adequate dimensioning of wireless communication networks. One of the possibilities for its solution involves the utilization of heuristics, able to produce near optimal solutions in an acceptable computational time. In this paper, a hybrid of GRASP (Greedy Randomized Adaptive Search Procedure) and FEA (Frequency Exhaustive Assignment) is proposed, allowing the achievement of best solutions for the Channel Allocation Problem (CAP).
I. I NTRODUCTION An important aspect in the design of a cellular radio network is the efficient use of the scarce electromagnetic spectrum, involving a channel allocation procedure, in order to satisfy the traffic demand, while electromagnetic compatibility constraints are obeyed. Three electromagnetic compatibility constraints are commonly considered: - Co-Channel Constraint (CCC): the same frequency can not be assigned to certain pair of cells simultaneosly, except a specified distance is respected; - Adjacent Channel Constraint (ACC): any pair of frequencies assigned to two adjacent cells must guard a specified distance. This distance is normally larger than the one used for CCC; - Co-Site Constraint (CSC): any pair of channels assigned to a cell must guard a specified distance. This distance is usually larger than the one used for ACC. While electromagnetic spectrum is a limited resource, the fast growing population of mobile subscribers imposes the need of an adequate treatment of the traffic demands. The distribution of the set of available channels among the base stations in the system, named channel allocation, has been extensively proposed in the literature, showing the great importance and interest related to the problem, involving static allocation ([1],[2], [3]), dynamic allocation [4]) and hybrid allocation ([5], [6]). On heavy load traffic conditions, static allocation is more efficient than dynamic allocation ([5], [7] and [8]), thus providing a more efficient reuse of the spectrum. Since heavy traffic conditions are expected in the future, this work is exclusively about static channel allocation. For the CAP, the application of meta-heuristics involves, for example, genetic algorithms [3], simulated annealing [9] and neural networks [10].
978-1-4244-2644-7/08/$25.00 © 2008 IEEE
Department of Computer Science University of Bras´ılia Bras´ılia, Brazil Email:[email protected]
This paper presents a new powerful alternative for the resolution of the Channel Allocation Problem (CAP), based on the hybridization of the metaheuristic GRASP (Greedy Randomized Adaptive Search Procedure), with a heuristic of sequential allocation, i. e., Frequency Exhaustive Assignment (FEA). The FEA strategy is incorporated into the local search phase of GRASP method, associating frequencies to the calls, leading to the GRASP-FEA technique. As a form to compare the results obtained in this work with the ones of literature, 8 problems (selected from Philadelphia benchmark) have been chosen ([11],[12]), including the most difficult ones, problems 2 and 6. The remainder of the work is organized as follows: in section II, the formulation of CAP and its instances are presented; in section III, related work is discussed; in section IV, GRASP-FEA is proposed; in section V, the results are presented. Finally, section VI contains our conclusions and suggestions for future research. II. F ORMULATION OF CAP AND I NSTANCES A. Formulation of CAP The CAP can be defined by the quintuplet P = { X, D, C, A, B } [13], where the vector X = {x1 , x2 , x3 , ...xn } is the system of cells, D = di , is the 1xn vector that informs the demand (requirements) in X; C = (cij ), nxn is the compatibility matrix in X, with cii > 0 for all i; represents the frequency separation required between a call in cell i and a call in cell j. A = (ai ) will contain the pre-assigned frequencies and B = (bi ) the blocked frequencies. F=(fi ) is the solution vector to the problem if, for all i: |Fi | = di
(1)
|fi − fj | ≥ cij
(2)
Ai ⊆ F i
(3)
Bi ∩ F i = 0
(4)
It is known that the CAP is a NP-Complete problem [14]. This class of problems does not have an exact solution nor it is possible to obtain any information about the optimal solution. We adopt the problem statement used in [12] and
TABLE I CAP I NSTANCES Problem
1
2
3
4
5
6
7
8
ACC
1
2
1
2
1
2
1
2
CSC
5
5
7
7
5
5
7
7
Demand Vector D1 D1 D1 D1 D2 D2 D2 D2 Lower Bound
381 427 533 533 221 253 309 309
[14] and . Given • n: the number of cells in the system; • di , 1 ≤ i ≤ n: the number of channels required in cell i; • cij , 1 ≤ i, j ≤ n: the frequency separation; Find • fik , 1 ≤ i ≤ n, 1 ≤ k ≤ di : the frequency separation assigned to the k th call in the ith cell, such that the maximum frequency is the minimum possible respecting all the constraints. The frequencies are represented by positive integers. In this way, we must find the minimum number of frequencies that solve the problem, while respecting the mentioned constraints. B. CAP Instances For comparison among the techniques, 8 problems [11], presented in Table 1, were considered. The problems are derived assuming 2 demand vectors (D1 e D2) and 4 different matrices of electromagnetic compatibility (C1, C2, C3 e C4) [10]. These instances were applied to a 21-cells system. The 8 problems differ in the possible presence of ACC, where, for example, a value of 2 in the corresponding column implies that the neighboring frequencies cannot be simultaneously used in adjacent cells. It is assumed a seven-cells cluster-size for all problems. This means that a frequency can only be associated to 2 BSs(BS - Base Station), if these BSs are separated by, at least, 2 other cells, thus respecting the constraints related to co-channel interference. III. R ELATED W ORK In [13] it was proposed a methodology to calculate some Lower Bounds (LB), while respecting all the channel constraints. Basically, three lemmas (7, 9 and 10), formulated by [13], have been extensively used to test algorithms proposed to solve the CAP ([12], [14], [11] and [15]). Some basic strategies to solve the CAP include the Frequency Exhaustive Assignment (FEA) and Requirement Exhaustive Assignment (REA) heuristics. Considering a crescently ordered set of the available frequencies, the FEA strategy assigns calls to the minimum available frequencies, while respecting the constraints. On the other hand, for the use of REA, the lowest available frequency is assigned to
the maximum number of calls that is possible respecting the constraints. Several works used combinations of FEA or REA together with other techniques, in order to find near optimal solutions. In this sense, [12] proposed algorithms based on node-color and node-degree orderings, combined with FEA and REA heuristics.[14] developed three local search algorithms to solve the CAP, named CAP1, CAP2 and CAP3. The best result was obtained by CAP3 and will be used(with some adaptation) in this paper, as the local search phase of the GRASP. Many researchers explored the use of evolutionary strategy and genetic algorithms (GA) to solve the CAP. [19] presented an evolutionary approach for frequency assignment problem in cellular radio networks. The results of this strategy were compared to simulated annealing and constraint programming, with a different set of instances from the one used in this paper. In [5] evolutionary strategy is used to solve the CAP for a hybrid channel allocation, considering a smaller set of constraints (for example, the authors do not consider co-site interference constraints). [11] developed a three-stage genetic algorithm, combined with FEA strategy, to solve the CAP, for some of the problems defined in section II above, to test its solution, but it did not reach the LB for the problem number 2. These problems will be studied in this work by using the GRASP-FEA approach and, as we will see, all the LBs for the eight problems will be reached, including the most difficult problems, 2 and 6. The problems analyzed by [11] were the same as proposed by [12] which used 7 and 12-cells clusters. But the problem 2 is the same in [12] and [11]; therefore, our comparison is correct. [9] applied simulated annealing meta-heuristic to solve the CAP, but the compatibily matrix, used in that work, treats all interference relations between base stations with equal strength. We also use a compatibily matrix, but closer to the practical problem, it deals in a different way with adjacent and co-channel constraints. In [9], a different set of instances and a graph coloring approach were used. Some other papers (see, e.g., [16], [17] and [18]) also adopted the same approach. [15] proposed the Viterbi-like algorithm (VLA) to solve the CAP. The VLA is based on the original Viterbi algorithm used for information decoding in digital communication systems. The VLA uses as metric, instead of the Hamming distance, the excess frequency factor which is defined by the adjacent and co-channel constraints, compared with the difference of the frequencies assigned to a pair of cells. [1] described a hyper-heuristic to find the solution to the CAP. The hyper-heuristic is a problem independent strategy (high level), that manages a set of low level heuristics (LLH), defining the borderline of the data flow. This methodology combines a greedy constructive heuristic to generate the initial solution, then a group of LLH is used to improve the solution quality. An algorithm chooses which heuristic will be used each time. [17] and [18] used GRASP approach to solve the CAP, both using a graph coloring model. [17] combined GRASP with
simulated annealing in the local search phase. [18] made a combination of GRASP and Path Relinking as a strategy for local search. Those techiques were not tested in Philadelphia benchmark. In the GRASP-FEA hybrid meta-heuristic we modelled the CAP as a sequential allocation, applying a distinct greedy function(different from [17] and [18]) and other strategy to assign frequencies to calls respecting all interference constraints. IV. T HE P ROPOSAL OF THE GRASP-FEA A PPROACH The meta-heuristic GRASP[20] is composed of two phases: the phase of construction of the initial solution and the local search phase. In the construction phase, a set of initial solutions is constructed and afterward, a local search is carried through in the neighborhood of each constructed solution. FEA Strategy, which associates frequencies to the lists of calls, will be executed during the local search phase composing the hybridization with the GRASP. A. Solution Construction Phase The solution construction phase is responsible for the denomination of the GRASP, where a set of initial solutions is constructed iteratively, in a greedy, random and adaptive way. The procedure of construction GRASP-FEA uses the following a greedy function, which is executed on each Base Station, as the following calculation [12]: gi = (
n X
dj cij ) − cii , 1 ≤ i ≤ n
(5)
j=1
Equation (5) is the measure of the difficulty to allocate a frequency to a call in a cell i. When it is used a purely greedy procedure, it is iteratively constructed only a combination of possible solutions, without testing other possibilities, through an optimization criteria defined by the greedy function. In contrast, the method of construction GRASP generates diversity of solutions applying a semi-greedy heuristic [21]. The Restrict Candidate List (RCL) is constructed by the elements for which the constraints are more severe, and the cell that will be part of the solution is selected randomly from RCL. While the solution is incomplete, new calculations are carried out taking into account the choice of the previous element. RCL will be constructed again and a new element will randomly be chosen (amongst those cells whose constraints are more severe) to compose the solution. A parameter α(0 ≤ α ≤ 1) is used to control the RCL. The RCL includes all those elements that are not yet part of the solution, whose impact in the value of the objective function is in the interval, [cmin , α(cmax − cmin ) + cmin ], where cmin corresponds to the greedy selection and cmax implicates in the highest cost increase in the objective function. It is observed that α = 0 implies a purely greedy choice and α = 1 implies a random choice (considering a minimization problem). In the same way, in the case of a maximization problem, the RCL will be defined by the
interval [cmax + (1 − α)(cmin − cmax ), cmax ]. In this in case, α = 0 implies a random choice from a RCL that contains all the possible elements, while α = 1 consists in a pure greedy choice [22]. Algorithm for the Solution Construction Phase, adapted from [20] begin xi = 0; i = 1; Initialize candidates C; While (C != empty) do begin cmin
