A genetic algorithm for solving dual-homing cell assignment problem of
the two-level wireless ATM network
D.-R. Din
a, S.S. Tseng
b,*
aDepartment of Computer Science and Information Management, Hung-Kuang Institute of Technology, Taichung 433, Taiwan, ROC
b
Department of Computer and Information Science, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC Received 3 September 2001; revised 11 March 2002; accepted 27 March 2002
Abstract
In this paper, we investigate the optimal assignment problem, which assigns cells in Personal Communication Service to switches on Asynchronous Transfer Mode network in an optimum manner. The cost has two components: one is the cost of handoffs that involve two switches, and the other is the cost of cabling. This problem is model as dual-homing cell assignment problem, which is a complex integer programming problem. Since finding an optimal solution of this problem is NP-hard, a stochastic search method, based on a genetic approach, is proposed to solve this problem. In this paper, domain-dependent heuristics are encoded into crossover operations, mutations of genetic algorithm (GA) to solve this problem. Simulation results show that GA is robust for this problem. q 2002 Elsevier Science B.V. All rights reserved.
Keywords: Cell assignment; Dual-homing; Wireless ATM; Genetic algorithm; Design of algorithm
1. Introduction
Recently there has been some interest in extending Asynchronous Transfer Mode (ATM) technology to the wireless environment [1 – 10]. The motivation behind this extension (termed wireless ATM ) includes the desire for seamless interconnection of wireless and ATM networks, and the need to support emerging mobile mutilated services. However, due to some inherent differences between these two types of networks, the introduction of ATM into the wireless environment presents many interesting challenges
[7,10]. These include support for an end-to-end ATM connection with user mobility, handling high error rate performance of wireless links, grouping cells into LAs (location area or cluster), and assigning clusters to switches in an optimum manner.
The cell assignment problem in Personal Communication Service (PCS) network was proposed by Merchant and Sengupta in Ref. [7], which assigns each cell in PCS network to only one switch on ATM network such that the total (the sum of cabling and handoff) cost can be minimized. The problem was formulated as an integer
programming problem and a heuristic algorithm was proposed to solve it. They [7] also discussed the dual-homing cell assignment problem in PCS network by considering that the calling patterns at different times of the day could be different. In the dual-homing cell assignment problem, each cell will be assigned to two switches and they allowed one to reduce the cost of handoff by increasing the cost of cables.
If the ATM backbone network is integrated with the PCS network, the handoff cost considered inRef. [7]which only depends on the frequency of handoff between two switches is not realistic. Since the switch of ATM backbone is widely spread, the communication cost between two switches should be considered in calculating the handoff cost. In
Refs. [11 – 14], the cell assignment problem considered in
Ref. [1] was extended to cell assignment problem in
wireless ATM environment. A solution model [12,14]
consists of five three-phase heuristic algorithms (NSF, LHWF, GHWF, MLCF, and MCMLCF), two genetic algorithms (GA) [11] (SGA and EGA), and a simulated annealing algorithm [13] were proposed to solve the cell assignment problem in wireless ATM environment. Exper-imental results showed that these algorithms have good efficiency.
In the designing of traditional network, robustness of networks, the ability to perform required communications www.elsevier.com/locate/comcom
0140-3664/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 4 0 - 3 6 6 4 ( 0 2 ) 0 0 0 8 6 - 5
* Corresponding author. Tel.: þ886-3-5731966; fax: þ886-3-5721490. E-mail addresses: sstseng@cis.nctu.edu.tw (S.S. Tseng),
after a specified set of components becomes unavailable
[15], may be enhanced by various technical solutions. However, it cannot be fully met without having a sufficient level of structural redundancy. The most effective and straightforward means for structural redundancy would be provided more than one path between any pair of cell and switches. There exists some network design studies in which each terminal (user) node is to be provided with redundant connections to the main hub network for backup services, this can be viewed as variants of the facility location problem. Prikul et al. [16] have investigated the concen-trator location problem where each terminal should be connected to two different concentrators. Tang et al.[17]
and Narasimhan [18] have dealt with the concentrator location problem where each terminal has multiple connec-tions to several concentrators.
In this paper, we are dealing with the topological design of the two-level hierarchical network, where the upper network is the ATM backbone network, and the lower level is the PCS network. The problem of multi-homing cell assignment problem is assigning each cell in PCS network to more than one switches on ATM network. In such a problem, each cell ci will be assigned to Ki (Ki$ 1 and
integer) switches and the corresponding Ki disjoint links
will be created. If Kiis set as 1 for all cells, the problem is
reduced to the cell assignment problem [11,12]. In this paper, we are concerned about the dual-homing cell assignment problem where Kiwas fixed at 2 for all cells.
That is, there are two assignments of each cell in PCS network, one is the primary assignment and the other is the secondary assignment.
Under normal circumstances, each cell uses its primary assignment to communicate. If that connection between cell and the corresponding switch of the primary assignment becomes unavailable, the secondary assignment is used. Such additional switches assignment (hence, reliability and availability) comes at a cost. The increased cost arises and also due to the additional switch capacity due to the additional cabling costs of connecting the cells to their assigning switches.
In this paper, we are given a group of cells in PCS network and a group of switches in an ATM network (whose locations are fixed and known). The problem is to assign cells to switches in the ATM network in an optimum manner. We consider the topological design of a two-level
hierarchical network. The objective cost has two com-ponents. One is the inter-switch handoff cost that involves two different switches and the other is the cost of cabling that connects cells to switches of ATM network. We try to assign cells to switches such that the total cost can be minimized under some assumptions which will be described in Section 2.
We first present a mathematical programming formu-lation for the dual-homing cell assignment problem in wireless ATM network environment. Since this problem is an NP-hard problem, this implies that a solution procedure searching for the optimal solution may not terminate within a reasonable amount of computing time. In order to deal with problems of significant size, it is worthwhile develop-ing an effective GA to solve this problem.
The organization of this paper is as follows. In Section 2, we formally define the problem. In Section 3, we describe the backgrounds of GAs. In Section 4, we describe the details of solution algorithm. The experimental results are presented in Section 5. Finally, conclusions are given in Section 6.
2. Problem formulation
This section first provides an overview of various terms and notations used to explain the concepts outlined in the subsequent sections. Let n cells in PCS network CGðC; LÞ be assigned to m switches in ATM network GðS; EÞ: We assume that the location of cells and switches, the structure of ATM network GðS; EÞ are fixed and known. Assume G is connected, sk, slin S, and ðsk; slÞ in E. Let ðXsk; YskÞ be the coordinate of switch sk, k ¼ 1; 2; …; m; ðXci; YciÞ be the coordinate of cell ci, i ¼ 1; 2; …; n; and dklbe the minimal
communication cost between the switches skand sl. Let fijbe
the frequency of handoff per unit time that occurs between cells ci and cj[ C; ði; j ¼ 1; 2; …; nÞ is fixed and known.
We assume that all edges in CG are undirected and weighted; and assume cells ci and cj in C are connected
by an edge ðci; cjÞ [ L with weight wij, where wij¼ fijþ fji;
wij¼ wji; and wii¼ 0[11,12]. Let likbe the cost of cabling
per unit time and between cell ciswitch sk, ði ¼ 1; 2; …; n;
k ¼ 1; 2; …; mÞ and assume likis the function of Euclidean
distance between cell cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii and switch sk, that is, lik¼
ðXci2 XskÞ2þ ðYci2 YskÞ2
q
:
Assume the number of calls that can be handled by each cell per unit time is equal to 1. Let Capkbe the number of
cells that can be assigned to switch sk. Our objective is to
assign each cell in C to two switches so as to minimize (total cost) the sum of cabling cost and handoffs cost per unit time of whole system.
Example 1.Consider the graph shown inFig. 1. There are 10 cells in C which should be assigned to four switches in S. The weight of edge between two cells is the frequency of
Fig. 1. An example of PCS network and ATM network, (a) PCS network CGðC; LÞ; (b) ATM network GðS; EÞ:
handoffs per unit time that occurs between them. Four switches are positioned at the center of the cell: c1, c2, c4,
and c6.
Assume the matrix CS of the distance between a cell and a switch is as follows: CS ¼ {lik}10£4¼ s1 s2 s3 s4 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 0 1 2 1 1 ffiffi 3 p ffiffi 7 p ffiffi 3 p 2 ffiffi 7 p 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 ffiffi 3 p 1 0 1 2 1 1 ffiffi 3 p ffiffi 7 p ffiffi 3 p 1 3 2 1 0 ffiffi 7 p ffiffi 3 p 1 2 ffiffi 3 p 2 ffiffi 3 p 1 1 ffiffi 3 p 1 0 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 :
A possible dual-homing cell assignment of Example 1 is shown inFig. 2.
To formulate this problem, let us define the following
variables. Let xik1 ¼ 1 if the primary assignment of cell ciis assigned to switch sk1, sk1[ S; xik1¼ 0; otherwise. Let xik2¼ 1 if the secondary assignment of cell ciis assigned to
switch sk2, sk2[ S; xik2 ¼ 0; otherwise. It is important to note that if a cell is to be connected to the same switch in both primary and secondary assignments, its cabling cost should not be double. In order to ensure that the cabling costs are not double in the event that a cell is connected to the same switch in both assignments, variables xik, i ¼
1; 2; …; n; k ¼ 1; 2; …; m are defined as: xik¼ xik1_ xik2; for i ¼ 1; 2; …; n and k ¼ 1; 2; …; m; where the ‘ _ ‘ symbol means the ‘or’ operation [1]. Since each cell should be assigned to at most two switches, we have the constraints
Xm k1¼1 xik1¼ 1; for i ¼ 1; 2; …; n and Xm k2¼1 xik2¼ 1; for i ¼ 1; 2; …; n:
Further, since we allow the two assignments of the cell can be the same, the constraint on the call handling capacity of switch is Xn i¼1 xik1þ Xn i¼1 xik2 # Capk; for k ¼ k1¼ k2¼ 1; 2; …; m: Thus, the sum of cabling costs can be formulated as: Xn i¼1 Xm k¼1 likxik¼ Xn i¼1 Xm k¼1 likðxik1_ xik2Þ:
To formulate the handoff cost, define variables zijk1¼
xik1xjk1; for i; j ¼ 1; 2; …; n and k1¼ 1; 2; …; m: Thus, zijk1
equals 1 if both the primary assignments of cells ciand cjare
connected to a common switch sk1; otherwise it is zero.
Further, let yij¼
Xm k1¼1
zijk1; i; j ¼ 1; 2; …; n:
Thus, yijtakes a value of 1, if both the primary assignments
of cells ci and cj are connected to a common switch; 0,
otherwise. With this definition, it is easy to see that the cost of handoffs per unit time between the primary assignment of cell ciand cjis given by
Xn i¼1 Xn j¼1 Xm k1¼1 Xm l1¼1 wijð1 2 yijÞxik1xjl1dk1l1:
The formulation of handoff cost described above is directly derived from the cell assignment problem inRefs. [11,12]. In the dual-homing cell assignment problem, since each cell is assigned to two switches in ATM network, the computation of handoff cost should consider more complex model. As shown inFig. 3, assume cell ciis assigned to sk1
and sk2, cjis assigned to switches sl1and sl2, respectively. Fig. 3. The assignments two cells ciand cj:
There are four possible communication cases between cells ciand cjas shown inFig. 4. They are: (a) from sk1to sl1, (b)
from sk1to sl2, (c) from sk2to sl1, and (d) from sk2to sl2as
shown inFig. 4(a) – (d), respectively. Thus, the handoff cost should be computed by the summation (or average) of these cases. To formulate the handoff cost, several variables are introduced as follows:
† zijk1¼ xik1xjl1; for i; j ¼ 1; 2; …; n and k ¼ k1 ¼ l1¼
1; 2; …; m:
† yij¼Pmk1¼1zijk; i; j ¼ 1; 2; …; n:
† z0ijk¼ xik1xjl2; for i; j ¼ 1; 2; …; n and k ¼ k1 ¼ l2¼ 1; 2; …; m:
† y0ij¼
Pm
k¼1z0ijk; i; j ¼ 1; 2; …; n:
† z00ijk¼ xik2xjl1; for i; j ¼ 1; 2; …; n and k ¼ k2 ¼ l1¼ 1; 2; …; m:
† y00ij¼
Pm
k¼1z00ijk; i; j ¼ 1; 2; …; n:
† z000
ijk¼ xik2xjl2; for i; j ¼ 1; 2; …; n and k ¼ k2 ¼ l2¼ 1; 2; …; m:
† y000 ij ¼
Pm
k¼1z000ijk; i; j ¼ 1; 2; …; n:
With these definitions, it is easy to see that the cost of handoffs per unit time for the best case is given by
Handoff ¼a ( Xn i¼1 Xn j¼1 Xm k1¼1 Xm l1¼1 wijð1 2 yijÞxik1xjl1dk1l1 þX n i¼1 Xn j¼1 Xm k1¼1 Xm l2¼1 wijð1 2 y0ijÞxik1xjl2dk1l2 þX n i¼1 Xn j¼1 Xm k2¼1 Xm l1¼1 wijð1 2 y00ijÞxik2xjl1dk2l1 þX n i¼1 Xn j¼1 Xm k1¼1 Xm l2¼1 wijð1 2 y 000 ijÞxik2xjl2dk2l2 ) :
This, together with our earlier statement about the sum of
cabling costs, gives us the objective function:
Minimize :X n i¼1 Xm k¼1 likxikþ Handoff;
whereais the ratio of the cost between cabling and network costs. That is, our objective is to assign each cell to two switches so as to minimize (total cost) the sum of cabling costs and handoffs cost per unit time.
3. Background of genetic algorithms
The Genetic Algorithm was developed by John Holland at the University of Michigan [19]. GAs are search techniques for global optimization in a complex search space. As the name suggests, GA employs the concepts of natural selection and genetic. Using past information, GA directs the search with expected improved performance. The concept of GA is based on the theory of adoption in natural and artificial systems [19]. In artificial adaptive systems, adaptation starts with an initial set of structures (possible solutions). These initial structures are modified according to the performance of their solution by using an adaptive plan to improve the performance of these structures. It has been proved by Holland that repeatedly applying this adaptive plan to input structures results in optimal or near optimal solutions[19]. The traditional methods of optimization and search do not fare well over a broad spectrum of problem domains [20]. Some are limited in scope because they employ local search techniques (e.g. calculus-based methods). Others, such as enumerative schemes, are not efficient when the practical search space is too large. 3.1. Concept of GA
The search space in GA is composed of possible solutions to the problem. A solution in the search space is represented by a sequence of 0s and 1s. This solution string is referred as a chromosome in the search space. Each chromosome has an associated objective function called the fitness. A good chromosome is the one that has a high/low fitness value, depending upon the nature of the problem (maximization/minimization). The strength of a chromo-some is represented by its fitness value. Fitness values indicate which chromosomes are to be carried to the next generation. A set of chromosomes and associated fitness values is called the population. This population at a given stage of GA is referred to as a generation. The general GA proceeds as follows:
Genetic Algorithm( ) Begin
Initialize population;
while (not terminal condition) do Begin
choose parents from population; / p Selection p / construct offspring by combining parents; / p Crossover p /
optimize (offspring); / p Mutation p / if suited (offspring) then
replace worst fit (population) with better offspring; / p Survival of the fittest p /
End; End.
There are three main processes in the while loop for GA: 1. The process of selecting good strings from the current generation to be carried to the next generation. This process is called selection/reproduction.
2. The process of shuffling two randomly selected strings to generate new offspring is called crossover. Sometimes, one or more bits of a chromosome are complemented to generate a new offspring. This process of complementa-tion is called mutacomplementa-tion.
3. The process of replacing the worst performing chromo-somes based on the fitness value.
The population size is finite in each generation of GA, which implies that only relatively fit chromosomes in generation (i ) will be carried to the next generation ði þ 1Þ: The power of GA comes from the fact that the algorithm terminates rapidly to an optimal or near optimal solution. The iterative process terminates when the solution reaches the optimum value. The three genetic operators, namely, selection, crossover and mutation, are discussed in Sections 3.2 – 3.4.
3.2. Selection/reproduction
Since the population size in each generation is limited, only a finite number of good chromosomes will be copied in the mating pool depending on the fitness value. Chromo-somes with higher fitness values contribute more copies to the mating pool than those with lower fitness values do. This can be achieved by assigning proportionately a higher probability of copying a chromosome that has a higher fitness value [20]. Selection/reproduction uses the fitness values of the chromosome obtained after evaluating the objective function. It uses a biased roulette wheel[20] to
select chromosomes, which are to be taken in the mating pool. It ensures that highly fit chromosomes (with high fitness value) will have a higher number of offspring in the mating pool. Each chromosome (i ) in the current generation is allotted a roulette wheel slot sized in proportion ( pi) to its
fitness value. This proportion pican be defined as follows.
Let Ofibe the actual fitness value of a chromosome (i ) in
generation ( j ) of g chromosomes, Sumj¼P g
i¼1Ofi be the
sum of the fitness values of all the chromosomes in generation j, and let pi¼ Ofi=Sumj:
When the roulette wheel is spun, there is a greater chance that a better chromosome will be copied into the mating pool because a good chromosome occupies a larger area on the roulette wheel.
3.3. Crossover
This phase involves two steps: first, from the mating pool, two chromosomes are selected at random for mating, and second, crossover site c is selected uniformly at random in the interval [1,n ]. Two new chromosomes, called offspring, are then obtained by swapping all the characters between positions c þ 1 and n. This can be shown using two chromosomes, say P and Q, each of length n ¼ 6 bit positions
chromosome P: 111l000; chromosome Q: 000l111.
Let the crossover site be 3. Two substrings between 4 and 6 are swapped, and two substrings between 1 and 3 remain unchanged; then, the two offspring can be obtained as follows:
chromosome R: 111l111; chromosome S: 000l000. 3.4. Mutation
Combining the reproduction and crossover operations may sometimes result in losing potentially useful infor-mation in the chromosome. To overcome this problem, mutation is introduced. It is implemented by complement-ing a bit (0 to 1 and vice versa) at random. This ensures that good chromosomes will not be permanently lost.
4. Genetic algorithm for dual-homing cell assignment problem
In this section, we discuss the details of GA developed to solve the dual-homing assignment problem of optimum assignment of cells in PCSs to switches in the ATM network. The development of GA requires: (1) a chromo-somal coding scheme, (2) genetic crossover operators, (3) mutation operators, (4) fitness function and penalty function
Fig. 5. (a) oriented representation of chromosome structure. (b) Cell-oriented representation of Example 1.
definitions, (5) a replacement strategy, and (6) termination rules.
4.1. Chromosomal coding
Since our problem involves representing connections between cells and switches, we employ a coding scheme that use positive integer numbers. Cells are labeled from 1 to n (the total number of cells), and switches are labeled from 1 to m (the total number of switches). The cell-oriented representation of chromosome structure shown inFig. 5(a)
consists of two parts. The first part is the primary assignment of cells, where the ith cell belongs to the vith
switch, and the second part is the secondary assignment, where the ith cell belongs to the vðiþnÞth switch. Since each
cell will be assigned to two switches on ATM network, we use 2 £ n array to represent the assignment of cells. If cell ci
is assigned to switches sk1, and sk2, then vi¼ k1and vðiþnÞ¼
k2: For example, the chromosome of the Example 1 shown
inFig. 2is shown inFig. 5(b). It is worth noting that, the cell-oriented representation of chromosome structure can be divided into two sets which represent the primary assign-ment and the secondary assignassign-ment of cells, respectively. 4.2. Genetic crossover operator
Four types of genetic operators were used to develop this algorithm:
1. partial single point crossover, 2. global single point crossover, 3. partial cell-exchanging operator, and 4. global cells-exchanging operator. 4.2.1. Partial single point crossover
The partial single point crossover is randomly selecting two chromosomes (say P1 and P2) for crossover from
previous generations and then by using a random number generator, an integer value i is generated in the range ½1; 2n: This number i is used as the crossover site. All characters between i þ 1 and 2n of two parents are swapped and offspring chromosomes O1 and O2 are generated. The
following example provides the detailed description of partial single point crossover operation: (assume crossover site i ¼ 5),
parent P1:
2 2 3 3 3 l 4 3 2 4 4 k 1 4 4 1 2 1 4 4 1 2;
parent P2:
1 4 4 1 2 l 1 4 4 1 2 k 2 2 3 2 2 4 3 2 4 4:
Two substrings between 6 and 20 are swapped, we have:
offspring O1:
2 2 3 3 3 l 1 4 4 1 2 k 2 2 3 2 2 4 3 2 4 4; offspring O2:
1 4 4 1 2 l 4 3 2 4 4 k 1 4 4 1 2 1 4 4 1 2:
4.2.2. Global single point crossover
The global single point crossover is slightly different from the partial single point crossover, which randomly selects two chromosomes (say P1and P2) for crossover from
previous generations and then by using a random number generator, an integer value i is generated in the range ½1; n: Two numbers i and ði þ nÞ are used as the crossover sites. All characters between i þ 1 and n, ði þ n þ 1Þ and 2n of two parents are swapped and offspring chromosomes O1and
O2 are generated. The following example provides the
detailed description of global single point crossover operation: (assume crossover sites are i ¼ 5 and 15),
parent P1:
2 2 3 3 3 l 4 3 2 4 4 k 1 4 4 1 2 l 1 4 4 1 2;
parent P2:
1 4 4 1 2 l 1 4 4 1 2 k 2 2 3 2 2 l 4 3 2 4 4:
First, two substrings between 6 and 10 are swapped, then two substrings between 16 and 20 are swapped, we have:
offspring O1:
2 2 3 3 3 l 1 4 4 1 2 k 1 4 4 1 2 l 4 3 2 4 4;
offspring O2:
1 4 4 1 2 l 4 3 2 4 4 k 2 2 3 2 2 l 1 4 4 1 2:
4.2.3. Partial cell-exchanging operator
In partial cell-exchanging operator, two sites of chromo-some in range ½1; 2n are randomly selected, and the primary assignment of two cells are exchanged.
For example, before exchanging, we have: 2 2 3p3 3 4 3 2p4 4 k 1 4 4 1 2 1 4 4 1 2:
(Assume that the two cells c3 and c8 are selected) After
exchanging:
2 2 2p3 3 4 3 3p4 4 k 1 4 4 1 2 1 4 4 1 2:
4.2.4. Global cell-exchanging operator
In global cell-exchanging operator, two cells in C of a chromosome ½l; n (say ciand cj) are randomly selected, the
primary and secondary assignments of two cells ci, and cj
are exchanged. That is, vi and vj, vðiþnÞ and vðjþnÞ are
exchanging. For example, before exchanging, we have: 2 2 3p3 3 4 3 2p4 4 k 1 4 4p1 2 1 4 41 2:
(Assume that the two cells c3 and c8 are selected) After
exchanging:
2 2 2p3 3 4 3 3p4 4 k 1 4 4p1 2 1 4 4p1 2:
4.3. Mutations and heuristic mutations
Five types of mutations are used to develop this algorithm and the active probabilities of mutations are the same.
Traditional Mutation (TM). Randomly select a cell of vector vi; where i in ½1; 2n and transform it to a random
number q between 1 and m. The following example provides the detailed description of traditional mutation: (assume cell c8 is randomly selected and q ¼ 4) before mutating, we have
2 2 3 3 3 4 3 2p4 4 k 1 4 4 1 2 1 4 4 1 2: After mutating, we have:
2 2 2 3 3 4 3 4p4 4 k 1 4 4 1 2 1 4 4 1 2:
Multiple Cells Mutation (MCM). Randomly select two random numbers k, l between 1 and m, transform the value of cells value is k to l and l to k. The following example provides the detailed description of multiple cells mutation: (assume random number k ¼ 3 and l ¼ 2) before mutating, we have
2þ2þ3p3p3p4 3p2þ4 4 k 1 4 4 1 2þ1 4 4 1 2þ: After mutating, we have
3p3p2þ2þ2þ4 2þ3p4 4 k 1 4 4 1 3p1 4 4 1 3p:
Heaviest Weight First Preference (HWFP) [11]. Since the handoff cost involving only one switch is negligible, two cells can be assigned to the same switch so as to reduce the handoff cost between these cells. Two cells with higher weight wij should have a higher probability of being
assigned to the same switch. Thus, if we consider two connected cells ci and cj[ C; then the probability of
mutation from vi of cell ci to the value vj of cell cj is as
follows Pði;jÞ¼ wij Xn i¼1 X degreeðciÞ j¼1 wij ; for i; j ¼ 1; 2; …; n;
where degreeðciÞ is the number of cells connected to cell ci
in CG:
Minimal Cabling Cost First Preference (MCCFP)[11]. To reduce the cabling costs between cells and switches, we
prefer to assign each cell to the nearer switch rather than the farther one. Cell ciand switch skwith lower cabling cost lik
should result in higher probability that ciwill be assigned to
sk: Thus, if we consider the randomly selected cell ci; then
the probability of mutation from viof cell cito the value vkis
Pði;kÞ¼ Lmax2 lik Xm l¼1 ðLmax2 lilÞ ;
where Lmax¼ maxml¼1{lil}:
Unique Switch First Mutation (USFM). It is important to note that if a cell is to be connected to the same switch in both primary and secondary assignments, its cabling cost should not be double. That is, if the two assignments of the cell are assigned to the same switch, the cabling cost will be reduced. If we randomly selected cell ci; then the value of vi
can be mutated to vðiþnÞ or vðiþnÞ can be mutated to viwith
equal probabilities. The following example provides the detailed description of unique switch first mutation: (assume cell c8is randomly selected) before mutating, we have
2 2 3 3 3 4 3 2p4 4 k 1 4 4 1 2 1 4 4p1 2:
After mutating, assume the primary assignment is mutated to the secondary assignment, we have
2 2 2 3 3 4 3 4p4 4 k 1 4 4 1 2 1 4 4p1 2:
4.4. Fitness function definition
Generally, GAs use fitness functions to map objectives to costs to achieve the goal of an optimally designed two-level wireless ATM network. If cell ciis assigned to switch sk1; and sk2; then viin the chromosome is set as k1and vðiþnÞis set as k2: Let dðvi;vjÞ; dðvi;vðjþnÞÞ; dðvðiþnÞ;vjÞ; and dðvðiþnÞ;vðjþnÞÞ be the minimal communication cost between switches skl and sl1; sk1 and sl2; sk2 and sl1; sk2 and sl2 in G, respectively. An objective function value is associated with each chromo-some, which is the same as the fitness measure. If vi¼ vðiþnÞ
then qi¼ 0; otherwise qi¼ 1: We use the following
objective function: minimize
OBJ ¼X n i¼1 ½liviþ qilivðiþnÞ þa Xn i¼1 Xm k¼1 wij{dðvi;vjÞþ dðvi;vðjþnÞÞ þ dðvðiþnÞ;vjÞþ dðvðiþnÞ;vðjþnÞÞ}: Note that if assumptionsPmk1¼1xik1 ¼ 1;
Pm
k2¼1xik2 ¼ 1; for i ¼ 1; 2; …; n; andPmk1¼1xik1þ
Pm
k2¼1xik2¼ 1 # Capk; k ¼ 1; 2; …; m are considered with this objective function, we have a complete problem formulation. While breeding chromosomes, the GA does not require the chromosome to reflect a feasible solution. Thus, we need to attach a penalty to the fitness function in the event the solution is infeasible. Let nkbe the number of cells assigned to switch sk; define
rewrite the formulation above in an unconstrained form: minimize cost ¼ OBJ þP;
whereP¼bðPm
k1¼1pkÞ is the penalty measure associated with a chromosome, and,bis the penalty weight. Since the best-fit chromosomes should have a probability of being selected as parents that is proportional to their fitness, they need to be expressed in a maximization form. This is done by subtracting the objective from a large number Cmax.
Hence, the fitness function becomes maximize Cmax2 ½OBJ þP;
where Cmaxdenotes the maximum value observed, so far, of
the cost function in the population. Let cost be the value of
the cost function for the chromosome; Cmax can be
calculated by the following iterative equation Cmax¼ max{Cmax; cost};
where Cmaxis initialized to zero.
4.5. Replacement strategy
This subsection discusses a method used to create a new generation after crossover and mutation is carried out on the chromosomes of the previous generation. Several replace-ment strategies have been proposed in the literature, and a good discussion can be found inRef. [20]. The algorithm developed here combines to both the concepts maintained above. Each offspring generated after crossover is added to the new generation if it has a better objective function value than both of its parents. If the objective function value of an offspring is better than that of only one of the parents, then we select a chromosome randomly from the better parent
and the offspring. If the offspring is worse than both parents, then each of the parents is selected at random for the next generation. This ensures that the best chromosome is carried to the next generation, while the worst is not carried to the succeeding generations.
4.6. Termination rules
Execution of GA can be terminated using any one of the following rules:
R1: when the average and maximum fitness values exceed a predetermined threshold;
R2: when the average and maximum fitness values of strings in a generation become the same; or
R3: when the number of generations exceeds an upper bound specified by the user.
The best value for a given problem can be obtained from a GA when the algorithm is terminated using R2[20].
5. Experimental results
In this section, we present the experimental results of the GA. The description of the experimental results is divided into four subsections. First, in Section 5.1, we describe that the effect of the mutation probability. In Section 5.2, we describe the effect of the crossover probability. In Section 5.3, we describe the effect of population size. Finally in Section 5.4, the effect of the heuristic mutations is examined.
In all the experiments, the implementation language is C, and all experiments were run on Windows NT with a Pentium II 450 MHZ CPU and 256MB RAM. We simulated a hexagonal system in which cells were configured as an H-mesh as shown inFig. 6. We assumed that the antenna for each cell was at the center of the cells, and was also assumed to be at the center of the cells. Switches are located at the same position of cells which are randomly selected from the
Fig. 6. H-mesh of the experiments. Fig. 7. Experimental results of different values of mutation probability for Set 1 ðn ¼ 16Þ:
Table 1
Test sets of the wireless ATM network
Set No. No. of cells (n ) No. of switches Cap a b
1 16 8 8 1.0 100 000
cells. The cabling cost between a switch and a cell was taken to be proportional to the geometric distance between the two. The topology of the backbone network is assumed to be a complete graph and the communication cost between two switches is assumed to be proportional to the geometric distance. The handoff frequency fij for each border was
generated from a normal random number with mean 100 and variance 20. Two different set of cells and switches shown in Table 1 are used to test the performance of the algorithm.
5.1. Effect of mutation probability
To examine the effect of the mutation probability of GAs, we set population size (popsize) ¼ 100, crossover
prob-ability ðPcÞ ¼ 1:0; maximum number of
generations ¼ 3000, and mutation probability is selected from {0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.2, 0.25}. The results shown in Figs. 7 and 8 are the best solution of different values of mutation probability after 10 runs of Set 1 and Set 2, respectively. InFig. 7, we found that when the mutation probability is lower than (0.15), GA can find the best solution. If the mutation probability is higher (0.20 or 0.25), the GA may get trapped in local minima. InFig. 8, for
the Set 2, we found that when the mutation probability is (0.05), GA can find the best solution.
Let Cpm be the average result of the GA running in
mutation probability Pmin 10 runs, and let the known best
solution total cost be the OPT. The total cost ratio computed by Cpm/OPT% of Set 2 is shown inFig. 9. The CPU time of
different values of mutation probability for Set 2 is shown in
Fig. 10. FromFigs. 9 and 10, we can conclude, for the Set 2, if Pmis set as 0.05, GA have the best result and efficiency.
5.2. Effect of crossover probability
To examine the effect of the crossover probability of GAs, we set population size (popsize) ¼ 100, mutation
probability (Pm) is 0.05, maximum number of
generations ¼ 3000, and crossover probability is selected from {1.00, 0.95, 0.90, 0.85, 0.80, 0.70, 0.60, 0.50}. The results shown in Figs. 11 and 12 are the best solutions of different values of crossover probability after 10 runs of Set 1 and Set 2, respectively. InFig. 11, we found that when the crossover probability is greater than (0.85), GA can find the best solution. If the crossover probability is lower (0.50), the GA may get trapped in local minima. In Fig. 12, we
Fig. 9. The total cost ratio of different values of mutation probability for Set 2 ðn ¼ 100Þ:
Fig. 11. Experimental results of different values of crossover probability for Set 1 ðn ¼ 16Þ:
Fig. 8. Experimental results of different values of mutation probability for Set 2 ðn ¼ 100Þ:
Fig. 10. The CPU time ratio of different values of mutation probability for Set 2 ðn ¼ 100Þ:
found that when the crossover probability is (1.00), GA can find the best solution.
The comparison of different values of crossover probability for Set 2 is shown in Fig. 13. The CPU time of different values of crossover probability for Set 2 is shown inFig. 14. We found that when Pc¼ 1:0; GA can
find the best solution. From Figs. 13 and 14, we can conclude, for the Set 2, if Pcis set as 1.00, GA have the best
result and efficiency.
5.3. Effect of population size
To examine the effect of population size of GAs, we set crossover probability (Pc) is 1.00, mutation probability (Pm)
is 0.05, maximum number of generations ¼ 3000, and population size is selected from {100, 200, 300, 400, 500,
600, 700, 800, 900, 1000 }. The result shown inFigs. 15 and 16are the best solution of different values of population size after 10 runs of Set 1 and Set 2, respectively. InFig. 15, we found when the population size is greater than 100 (except 500), GA can find the best solution. InFig. 15, for Set 2, we found that when the population size is (300), GA can find the best solution.
The comparison of different values of population size for Set 2 is shown inFig. 17. The CPU time of different values of population size for Set 2 is shown inFig. 18. We found that when popsize ¼ 300, GA can find the best solution. FromFigs. 17and 18, we can conclude, for the Set 2, if Pcis
set as 300, GA have the best result and efficiency.
Fig. 13. The total cost ratio of different values of crossover probability for Set 2 ðn ¼ 100Þ:
Fig. 14. The CPU time ratio of different values of crossover probability for Set 2 ðn ¼ 100Þ:
Fig. 16. Experimental results of different values of population size for Set 2 ðn ¼ 100Þ:
Fig. 17. The total cost ratio of different values of population size for Set 2 ðn ¼ 100Þ:
Fig. 12. Experimental results of different values of crossover probability for
Set 2 ðn ¼ 100Þ: Fig. 15. Experimental results of different values of population size for Set 1 ðn ¼ 16Þ:
5.4. Effect of heuristic mutation
To evaluate the effect of the heuristic mutations described in Section 4.3, we simply constructed a trivial GA called TGA (trivial GA). The crossover operations of TGA were the same as our genetic algorithm (OGA) but the mutation operations were not. The mutation operation used in TGA is only the traditional mutation (TM) described in Section 4.3. The network Set 2
shown in Table 1 was used in experiments and the
parameter of two GAs are set as Pm¼ 0:05; Pc¼ 1:0;
and popsize ¼ 300. Fig. 19 shows the best results in 10 runs of two GA (OGA and TGA). Observe the results shown in Fig. 19, the total cost of our GA decreases rapidly and is better than TGA. That is, heuristic mutations in OGA can decrease the total cost and keep the cost from getting trapped a local minima.
6. Conclusions
In this paper, we investigate the dual-homing cell assignment problem which optimally assigns each cell in PCS to two switches on ATM network. This problem is currently faced by designers of mobile communi-cation service and in the future, it is likely to be faced by designers of PCS. Since finding an optimal solution of this problem is NP-hard, a stochastic search method based on a genetic approach is proposed to solve it. Simulation results showed that GA is robust for this problem. In our methods, cell-oriented representation is used to represent the cell assignment; three general genetic operators, selection, crossover, and mutation were employed. Four types of operators (partial single point crossover, global single point crossover, partial cell-exchanging crossover and global cell-exchanging crossover) and five types of mutations (TM, MCM, LHWFP, MCCFP, and USFM) are employed in our method. Experimental results indicate that the algorithm can run efficiently.
Acknowledgments
This work was supported in part by MOE program of Excellence Research under Grant 90-E-FA04-1-4.
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