A Genetic Algorithm for Network Expanded Problem in Wireless
ATM Network
Der-Rong Din
Department of Computer Science and Information Management
Hung-Kuang Institute of Technology
Taichung 433, Taiwan R.O.C.
deron@sunrise.hkc.edu.tw
S. S. Tseng
Department of Computer and Information Science
National Chiao-Tung University
Hsinchu 300, Taiwan R.O.C.
sstseng@cis.nctu.edu.tw
Abstract
In this paper, we investigate thenetwork expanded prob-lem which optimally assigns new adding and splitting cells inPCS (Personal Communication Service) network to switches in an ATM (Asynchronous Transfer Mode) network. Moreover, the locations of all cells in PCS net-work are xed and known, but new switches should be installed to ATM network and the topology of the back-bone network may be changed. Given some potential sites of new switches, the problem is to determine how many switches will be added to the backbone network, the locations of new switches, the topology of the new backbone network, and the assignments of new adding and splitting cells in the PCS to switches on the new ATM network in an optimum manner. We would like to do the expansion in as attempt to minimize the to-tal communication cost under budget and capacity con-straints. This problem is modeled as a complex integer programming problem, and nding an optimal solution to this problem is NP-hard. A genetic algorithm is pro-posed to solve this problem. The genetic algorithm con-sists of three phases, Switch Location Selection Phase, Switch Connection Decision Phase, and Cell Assignment Decision Phase. First, in the Switch Location Selec-tion Phase, the number of new switches and the loca-tions of the new switches are determined. Then, Switch Connection Phase is used to construct the topology of the expanded backbone network. Final, Cell Assignment Phase is used to assign cells to switches on the expanded network. Experimental results indicate that the three-phase genetic algorithm has good performances.
keyword: Genetic algorithm, wireless ATM, network expanded problem, cell assignment problem.
Figure 1: Architecture of wireless ATM PCS.
1 Introduction
The rapid worldwide growth of digital wireless commu-nication services motivates a new generation of mobile switching networks to serve as infrastructure for such services. Mobile networks being deployed in the next few years should be capable of smooth migration to fu-ture broadband services based on high-speed wireless ac-cess technologies, such as wireless asynchronous transfer mode (wireless ATM)[1]. In the architecture presented in [1] (as shown in Fig. 1), the base station controllers (BSCs) in traditional PCS network are omitted, and the base stations (BSs or cells) are directly connected to the ATM switches. The mobility functions supported by the BSCs will be moved to the BSs and/or the ATM switches.
In the designing process of PCS network, rst, the telephone company determined the global service area
Figure 2: Cell splitting.
according to the usages of the mobile users, and divided the global service area into some smaller coverage areas which are covered by cells. Second, the cellular system and base stations are established and setup, BSs are con-nected to the switches on the ATM network to form the topology of wireless ATM. This topology may be out of date, since more and more users may use the PCS com-munication system. Some areas, which have not been covered in the original global service area, may now have mobile users to serve. The services requirement of some areas, which were originally covered by some BSs may be increased and exceeded the capacities provided by the original BSs and switches. Though, the wireless ATM system must be extended so that the system can pro-vide higher quantity of services to mobile users. Two methods can be used to extend the capacities of system and provide higher quantity of services. The rst one is: adding new cells to the wireless ATM network so that the non-covered areas can be covered by new cells. The other is: reducing the size of the cells so that the total number of channels available per unit cell and the capac-ity of a system can be increased. In practice, this can be achieved by using cell splitting[8] process. The cell splitting process establishes new BSs at speci c points in the cellular pattern and reduces the cell size by a factor of 2 (or more) as shown in Fig. 2.
In this paper, we are given a two-level wireless ATM network as shown in Fig. 3. In the PCS network, cells are divided into two sets. One is the set of cells, which are built originally, each cell in this set has been assigned to a switch on the ATM network (e:g:; cells c1,c2are assigned
to switchs1, cellsc3andc5are assigned to switchs4, and
cell c4 is assigned to switch s4 in Fig. 3). The other is
the set of cells which are newly added (e:g:; c6,c7,c8) or
established by performing the cell splitting process (e:g:; c9, c10, c11, c12, c13, and c14). Moreover, the locations
of all cells in PCS network are xed and known, but the number of switches in ATM network may be increased. Given some potential sites of new switches, the problem is to determine how many switches will be added to the backbone network, the locations of the adding switches, the connections between the adding switches and other switches, and the assignment of adding and splitting cells
Figure 3: Example of the network expanded problem in the two layers wireless ATM network.
Figure 4: Two types of hando s occurred in the wireless ATM network.
in the PCS to switches on the ATM network in an opti-mum manner. We would like to do the extension in as attempt to minimize the total communication cost under budget and capacity constraints.
The total communication cost has two components: one is the cost of hando s that involve two switches, and the other is the cost of cabling (or trucking) [3][4][5][6][10]. During the wireless environment, two types of hando s should be considered in the designing of the network, they areintra-switch hando and inter-switch hando as illustrated in Fig. 4. The intra-switch hando involves only one switch and the inter-switch hando involves two switches. The (inter-switch) hand-o s that occur between two cells, which connected to dif-ferent switches, consume much more network resources (therefore, are much more costly) than the intra-switch hando that occurs between cells, which connected to the same switch[3][4][5][6][10]. Thus, we assume that the cost of (intra-switch) hando s involving only one switch are negligible. Through this paper, we assume each cell
to be connected to only one switch. The budget con-straint used to constrain the sum of the switch setup cost, the link setup cost between two switches, and the link setup cost between cells and switches.
For the cell assignment problem, Merchant and Sen-gupta [10] considered the problem that assigns cells to switches in PCS network. They formulated the prob-lem and proposed a heuristic algorithm to solve it so that the total cost can be minimized. The total cost consists of cabling cost and location update cost. The location update cost considered in [10], which depend only on the frequency of hando between two switches, is not practical. Since switches of the ATM backbone are widely spread, the communication cost between two switches should be included in calculating the location update cost. In [3][4], this model was extended to solve the problem that grouped cells into clusters and assigned these clusters to switches on the ATM network in an optimum manner by including the communication cost between two switches. In [5] and [6], the extended cell as-signment problem has been investigated and formulated which is assigning the new adding and the splitting cells to the switches on the ATM network so that the total cost can be minimized. In [5][6], the number of adding and splitting cells was not greater than the total remain-ing capacities provided by the switches of ATM network. That is, no new switch should be added into ATM net-work, and the topology of the backbone network was not changed in this problem. A simulated annealing and a genetic algorithms have been proposed in [5][6] to solve the extended cell assignment problem, respectively.
In this paper, a more complex problem is consid-ered. Following the objective function formulated in [3][4][5][6], new cells and new switches should be intro-duced into the two-layer network. In this paper, the loca-tions of new switches, the connecloca-tions between switches, and the assignment of new and splitting cells should be determined so that the total communication cost can be minimized under budget and capacity constraints. This problem is denoted as network expanded problem in wireless ATM environment.
Since nding an optimal solution to this problem is NP-hard, in this paper, a three-phase genetic algorithm is designed to nd an approximate solution. The organi-zation of this paper is shown as follows. In Section 2, we formally de ne the problem. The backgrounds of genetic algorithms are described in Section 3. In Sections 4 and 5, we describe outline and details of the proposed ge-netic algorithm. The experimental results are presented in Section 5. Final, a conclusion is given in Section 6.
2 Problem Formulation
In this section, we give the formulation of network ex-panded problem in wireless ATM network. In what fol-lows, we introduce a number of assumptions that are
necessary for the proper modeling of the problem.
2.1 Backbone Network Assumption
Each cell is connected to a switch through a link. The switches are interconnected with a speci ed
topology through links.
The number of cells that can handled by a new
switch cannot exceedCAP.
At most one switch may be installed at a given
po-tential site.
All links of the current backbone network are kept
in place.
A switch site in the current network is also a switch
site in the expanded network.
The backbone network topologies are preserved in
the expanded backbone network.
2.2 Known Information
The location of the new cells as well as the hando
frequency between cells.
The potential switch sites.
The setup cost of switch at a particular site. The link setup cost between cells and switches. The link setup cost between switches.
Our goal is to nd the minimum-cost expanded net-work subject to all of the above assumption, facts and constraints (described later).
2.3 Mathematical Formulation
Let CG(C;L) be the PCS network, where C is a nite set of cells with jCj andL is the set of edges such that
LCC. We assume that Cnew[Cold =C, Cnew\
Cold =;,Cnewbe the set of new and splitting cells where jC
new
j=n
0, cells in Cnewhave not yet been assigned to
switches on the ATM, and Cold be the set of original
cells where jC old
j = n: Without loss of generality, we
assume that cells inCold andCnew are indexed from 1
to n and n + 1 to n + n0, respectively. If cells c
i and
cj inC are assigned to di erent switches, then an
inter-switch hando cost is incurred. Letfij be the frequency
of hando per unit time that occurs between cellsciand
cj, (i;j = 1;:::;n + n0) and is xed and known. We
assume that all edges inC are undirected and weighted; and assume cells ci and cj in C are connected by an
edge (ci;cj)2L with weight wij, wherewij =fij+fji,
wij =wji, andwii = 0[3][4][5][6]. Let Gold(Sold;Eold)
set of switches withjSoldj=m, Eold SoldSold is the
set of edges, sk, sl in Sold, (sk;sl) in Eold, and Gold is
connected. We assume that the locations of cells inCG and switches inGold are xed and known. The topology
of the ATM network Gold(Sold;Eold) is known and will
be extended toG(S;E). Let Snew is the set of potential
sites of switches. Without loss of generality, we assume that switches in Sold and Snew are indexed from 1 to
m and m + 1 to m + m0, respectively. We assume that
the expanded backbone network should be a connected network, i.e., new switches can be connected to exist switches or another new switches. Let (Xsk;Ysk) be the
coordinate of switchsk,sk2Sold[Snew,k = 1;2;:::;m+
m0, (X
ci;Yci) be the coordinate of cellci,i = 1;:2;:::;n+
n0; andd
klbe the minimal communication cost between
the switchessk andsl;sk;sl2S; k;l = 1;2;:::;m + m 0.
The total communication cost has two components, the rst is the cabling cost between cells and switches, the other is the hando cost which occurred between two switches. To formulate the total communication cost, let us de ne the following variables: Letlik be the cabling
cost per unit time between cellciswitchsk, (i = 1;:::;n+
n0;k = 1;:::;m + m0) and assume l
ik is the function of
Euclidean distance between cellci and switchsk.
Assume the number of calls that can be handled by each cell per unit time is equal to 1 and CAP denotes the cell handling capacity of each new switchsk2Snew,
(k = m+1, m+2, ..., m+m0). LetCap
kbe the number
of remaining cells that can be used to assigned cells to switch sk 2 Sold;(k = 1;2;:::;m): Our goal is to
deter-mine the location of the new switches, construct the new topology of the expanded backbone network, and assign cells in Cnew to switches on G so as to minimize the
overall communication cost which is the sum of cabling communication cost and hando costs per unit time un-der some constraints. Some variables are de ned here and to be used to formulated this problem, letqk=1, (k
= 1, 2, ..., m + m0) if there is a switch installed on site
sk; qk =0, otherwise (as we known, qk =0, for k= 1, 2,
...,m). Let setupkbe the setup cost of the switch at site
sk 2Snew,k= 1, 2, ..., m+m
0(as we knownsetup
k = 0,
for k=1, 2, ..., m). Let xik = 1 if cell ci is assigned to
switch sk;xik= 0, otherwise; where ci 2C, i= 1, 2, ...,
n+n0,s
k2S, s=1, 2, ..., m+m
0. Since each cell should
be assigned to only one switch, we have the constraint
Pm+m0
k=1 xik= 1, fori = 1;2;::;n + n0. Further, the
con-straints on the call handling capacity is as follows: For the new switchsk,
n+n0 X i=n+1xik CAP; k = m + 1;m + 2;:::;m + m 0 ; (1) and for the existing switchsk,
n+n0 X
i=n+1xik
Capk;k = 1;2;:::;m: (2)
If cellsciandcjare assigned to di erent switches, then
an inter-switch hando cost is incurred. To formulate hando cost, variableszijk=xikxjk; for i;j;= 1;:::;n +
n0 and k = 1;:::;m + m0 are de ned in [10]. Thus, z
ijk
equals 1 if both cellsciandcjare connected to a common
switch sk; it is zero, otherwise. Further, let
yij =m+m 0 X k=1 zijk;i;j = 1;2;:::;n + n 0 : (3)
Thus,yij takes a value of 1, if both cellsciandcj are
connected to a common switch;yij = 0, otherwise. With
this de nition, it is easy to see that the cost of hando s per unit time is given by
Handoff Cost = n+n0 X i=1 n+n0 X j=1 m+m0 X k=1 m+m0 X l=1 wij(1 ;yij)qkqlxikxjlDkl; (4)
where Dkl is the minimal communication cost between
switches sk and sl onG(S;E):
The objective of the problem is to minimize the total communication cost subject to budget constraint. Thus, together with our earlier statement about the sum of cabling cost and hando cost, the objective function is :
minimize : Total cost
= Cabling Cost + Handoff Cost = n+n 0 X j=1 m+m0 X k=1 likxik+ n+n0 X i=1 n+n0 X j=1 m+m0 X k=1 m+m0 X l=1 wij(1 ;yij)qkqlxikxjlDkl;(5)
where is the ratio of the cost between cabling commu-nication cost and inter-switch hando cost.
Letekl be the variable that represents the link status
between two switches sk andsl. If ekl=1 then there is
a link between two switches sk andsl (sk 2Snew;sl 2
Sold [Snew); ekl=0, otherwise. Let uik be link setup
cost of constructing the connection between cell ci;(i =
n+1;n+2;:::; n+n0) and switchs
k(k = 1;n+2;:::;m+
m0), and assumeu
ikis the function of Euclidean distance
between cell ciand switch sk. Let vkl be link setup cost
of constructing the connection between switch sk;(k =
m+1;m+2;:::;m+m0) and switchs
l;(k = 1;2;:::;m+
m0); and assume v
ikis the function of Euclidean distance
between switch sk and switch sl.
The following constraints must be satis ed: EC = m+m 0 X k=m+1qk setupk+ n+n0 X i=n+1 m+m0 X k=1 uik xikqk
+(m+m 0 X k=m+1 m+m0 X l=1 eklvklqkql)=2 Budget (6) xikqk; for k = 1;2;:::;m: (7) wklqk and wklql; for k = 1;2;:::;m: (8)
Example 1.
Consider the graph shown in Fig. 3. There are 14 cells in CG which should be assigned to switches in ATM network. InCG, cells are divided into two sets, one is the setColdof cells which are builtorig-inally, and cells in Cold have been assigned to switches
in the ATM network (e:g:; fc1,c2,c3,c4,c5gin Fig. 3).
The other is the setCnew of cells which are new adding
cells (e:g:;fc6,c7,c8g) or splitting cells (e:g:;fc9;c10;c11,
c12,c13,c14g). The edge weight between two cells is the
frequency of hando s per unit time that occurs between them. Ten potential site of switches can be chosen to be the new switches. We assume that the capacity of each switch is 2.
3 Background of Genetic
Algo-rithms
The Genetic Algorithm (GA) was developed by John Holland at the University of Michigan[7]. Genetic Al-gorithms 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 ex-pected improved performance. The concept of GA is based on the theory of adoption in natural and arti -cial systems[7]. In arti arti -cial adaptive systems, adapta-tion starts with an initial set of structures (possible so-lutions). These initial structures are modi ed according to the performance of their solution by using an adap-tive 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 [7]. The traditional methods of optimization and search do not fare well over a broad spectrum of problem domains[2]. Some are limited in scope because they employ local search techniques (e.g., calculus based methods). Others, such as enumerative schemes, are not ecient 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 repre-sented 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 tness. A good chromosome is the one that has a high/low tness value, depending upon the nature of the problem (maximization/minimization). The strength of a chromosome is represented by its tness value. Fitness values indicate which chromosomes are to be carried to the next generation. A set of chromosomes and associ-ated tness values is called the population. This popula-tion at a given stage of GA is referred to as a generapopula-tion. The general GA proceeds as follows:
Genetic Algorithm()
Begin
Initialize population;
while (not terminal condition) do Begin
choose parents from population; /* Selection */
construct o spring by combining parents; /* Crossover */
optimize (o spring); /* Mutation */ if suited (o spring) then
replace worst t (population) with better o -spring;
/*Survival of the ttest */ End;
End.
There are three main processes in the while loop for GA:
(1) The process of selecting good strings from the cur-rent generation to be carried to the next generation. This process is called selection/reproduction.
(2) The process of shuing two randomly selected strings to generate new o spring is called crossover. Sometimes, one or more bits of a chromosome are com-plemented to generate a new o spring. This process of complementation is called mutation.
(3) The process of replacing the worst performing chromosomes based on the tness value.
The population size is nite in each generation of GA, which implies that only relatively t chromosomes in generation (i) will be carried to the next generation (i + 1). The power of GA comes from the fact that the algorithm terminates rapidly to an optimal or near opti-mal solution. The iterative process terminates when the solution reaches the optimum value. The three genetic operators, namely, selection, crossover and mutation, are discussed in the next section.
3.2 Selection / Reproduction
Since the population size in each generation is lim-ited, only a nite number of good chromosomes will be copied in the mating pool depending on the tness value. Chromosomes with higher tness values contribute more copies to the mating pool than do those with lower tness values. This can be achieved by assigning proportion-ately a higher probability of copying a chromosome that has a higher tness value[2]. Selection/reproduction uses the tness values of the chromosome obtained after eval-uating the objective function. It uses a biased roulette wheel[2] to select chromosomes, which are to be taken in the mating pool. It ensures that highly t chromosomes (with high tness value) will have a higher number of o spring in the mating pool. Each chromosome (i) in the current generation is allotted a roulette wheel slot sized in proportion (pi) to its tness value. This
propor-tionpi can be de ned as follows. Let Ofi be the actual
tness value of a chromosome (i) in generation (j) of g chromosomes, Sumj =Pg
i=1Ofi be the sum of the
t-ness values of all the chromosomes in generation j, and letpi=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: rst, from the mating pool, two chromosomes are selected at random for mat-ing, and second, crossover sitec is selected uniformly at random in the interval [1;n]. Two new chromosomes, called o spring, are then obtained by swapping all the characters between positions c + 1 and n. This can be shown using two chromosomes, say P and Q. each of length n = 6 bit positions
chromosome P: 111j000;
chromosome Q: 000j111.
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 o spring can be ob-tained as follows:
chromosome R: 111j111;
chromosome S: 000j000.
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 comple-menting a bit (0 to 1 and vice versa) at random. This ensures that good chromosomes will not be permanently lost.
Figure 5: Outline of the genetic algorithm for solving the complex extended cell assignment problem.
4 Outline of Solution Algorithm
In general, the network expanded problem is a multi-constraints optimization problem. The designs are used to nd optimal location of the switches, topological con-nections and cells assignment such that the total commu-nication cost is minimized and yet satis es the budget constraint and maximum capacity constraint. In fact, the problem is NP-hard and for the practical problem with a modest number of nodes, only approximate solu-tions can be obtained through heuristic algorithms. In this paper, considering that the problem is largely gov-erned by the speci cation that is formulated in Section 2, we divide the problem into three subproblems. Thus, the whole complexity of the problem is broken down and driven only by the constraints and the requirements of cost. This can be simpli ed into three sets of design variables which corresponds to the number of optimiza-tion phases. In this way, each optimizaoptimiza-tion level has the main core and a GA cycle, with similar architecture. This similarity can reduce the complexity of the system design.The genetic algorithm proposed to solve the network expanded problem consists of three phases and the out-line is shown in Fig. 5. They are Switch Site Selection Phase, Switch Connection Decision Phase, andCell As-signment Decision Phase.
A detail description for each level of optimization is given in the following subsections.
4.1 Switch Site Selection Phase
In the Switch Site Selection Phase, we use genetic algo-rithm to determine the number of new switches to be setup and the locations of the new switches. During this phase, we assume that the connections between switches and the assignments of cells to switches are randomly generated. The subproblem can be formulated as fol-lows:
Given
CG, Gold,CAP, Capk, Setupk,uik,vkl,Budget, fij, and
potential sites Snew
Minimize
Total cost of the two-layerwireless ATM network
Subject to
ECBudget; Pn+n 0 i=n+1xikCAP, k= m + 1, m + 2, ..., m + m0; Pn+n 0 i=n+1xikCapk, k=1, 2,...,m; Pm+m 0 k=1 xik= 1; i=1, 2, ..., n + n0; andG(S;E) must correspond to a connected topology.
Determine
qk, k=m+1, m+2, ..., m + m04.2 Switch Connection Decision Phase
In the Switch Connection Decision Phase, we assume the number and the locations of the new switches are xed and known, genetic algorithm is used to determine the connections between switches, and expanded the cur-rent exist network to a connected network. During this phase, we assume that the assignments of cells are ran-domly generated. The subproblem can be formulated as follows:
Given
CG, Gold[Snew,CAP, Capk,uik,vkl,Budget, fij, and location of new switches
Minimize
Total cost of the two-layerwireless ATM network
Subject to
ECBudget Pn+n 0 i=n+1xikCAP, k= m + 1, m + 2,..., m + m0; Pn+n 0 i=n+1xikCap k, k = 1, 2, ..., m; Pm+m 0 k=1 xik= 1, i=1, 2, ..., n + n0;andG(S;E) must correspond to a connected topology.
Determine
ekl, k=m+1, m+2, ..., m + m0;l=1,2, ...,m + m0.
4.3 Cell Assignment Decision Phase
In the Cell Assignment Decision Phase, we assume the topology of the expanded backbone network is xed and known, genetic algorithm is used to determine the as-signment of cells inCnew to switches inS.
Given
CG, G;CAP;Capk;uik;vkl;Budget;fijMinimize
Total cost of the two-layer wireless ATMnetwork
Subject to
ECBudget Pn+n 0 i=n+1xikCAP, k= m + 1, m + 2, ..., m + m0 Pn+n 0 i=n+1xikCap k,k = 1;2;:::;m Pm+m 0 k=1 xik= 1; i=1, 2, ..., n + n0G(S;E) must correspond to a connected topology
Determine
xik, i=n+1, n+2, ...,n + n0;k=m+1, m+2, ...,m + m0
5 Genetic Algorithm for network
expanded Problem
In this section, we discuss the details of GA developed to solve the network expanded problem. The develop-ment of GA requires: (1) a chromosomal coding scheme, (2) initial population generation, (3) a chromosome ad-justment procedure, (4) a genetic crossover operator, (5) mutation operators, (6) a tness function de nition, (7) a replacement strategy, and (8) termination rules.
5.1 Chromosomal coding
To solve this problem, two two-level genes are intro-duced as illustrated in Figs. 6 and 7. In these encoding schemes, the activation of the low-level gene is governed by the value of the high-level gene. To indicate the acti-vation of the high-level gene, an integer "1" is assigned for each high-level gene that is being ignited were "0" is for turning o . When "1" is signaled, the associated low-level gene due to that particular active high-level gene is activated in the lower level structure. It should be noticed that the inactive gene always exist within the chromosome even when "0" appears. This architec-ture implies that chromosome contains more information than that of the conventional GA structure. Hence, it is called Hierarchical Genetic Algorithm (HGA)[9].
To solve the network expanded problem in wire-less ATM network, three types of genes, known as
switch-location gene, switch-connection gene, and cell-assignment gene, are introduced. The switch-location gene in the form of bits decide the activation or deacti-vation of the corresponding new switches. The switch-connection gene de nes the link switch-connections between new switches and another (new or old) switches. The cell-assignment gene de nes the assignment of cells to switches. For example, in Fig. 6, the switch-connection s5 and s6, with switch-location gene signi ed as "0" in
the corresponding sites, are not being activated. Fur-thermore, the switch s5 and s6 will not appear in the
Figure 6: Two level genes used to represent the relation between the locations of switches and the connections between switches.
Figure 7: Two level genes used to represent the relation between the locations of switches and the assignments of cells.
The detail information of three types of genes are de-scribed as follows:
Switch-location gene (SL): Since there are m 0
po-tential sites for the choosing of news switches, a bi-nary encoding method is used to represent whether the site is selected or not. A binary array SL[m + 1;:::;m + m0] is used to represent the choose. If
SL[k] = 1 (m +1km +m
0) then a new switch
is located at potential sitesk;SL[k] = 0, otherwise.
For example, assumes8,s9, ands13 are selected be
the new switches, the switch-location gene of the example shown in Fig. 3 is shown in Fig. 8(a).
Switch-connection genes (SC): A positive integral
encoding method is used to describe the connections between switches. Since the existing ATM network is connected. Thus, the only thing that we have to do is to keep the information of how the new switch is connected to another switches. A integral array SL[m+1;:::; m+m0] with maximum positive integer
numberm+m0is used to represent the connections
of new switches to another switches. If SC[k] =
Figure 8: Three types of genes used to encode the ex-ample shown in Fig. 1, (a) switch-location genes, (b) switch-connection gene, and (c) cell-assignment genes.
l(m + 1km + m 0;1
l m + m
0) then there
is a link between switch sk and sl. For example,
assume s8, s9, and s13 are selected to be the new
switches, a possible switch-connection gene of the example shown in Fig. 3 is shown in Fig. 8(b).
Cell-assignment gene (CA): Since the cell
assign-ment subproblem involves representing connections between cells and switches, we employ a coding scheme that use positive integer numbers. The cell{ assignment gene is shown in Fig. 8(c), where theith cell belongs to theCA[i]-th switch. For example, a possible cell-assignment gene of the example shown in Fig. 3 is shown in Fig. 8(c). It should be noticed that, the cell-assignment gene can be divided into two sets, the rst set of cells which represents the assignment of cells inColdis xed in running of GA.
Thus, the rst set of cells can be ignored since it is unchanged during experiments.
Furthermore, switch-location, switch-connection and cell-assignment genes are used in Switch Site Selection Phase; the value of each gene is randomly generated by a random number in the evolution-ary process. In running Switch Connection Deci-sion Phase, we assume that Switch Location genes are xed, and the other two genes are variable. In running Cell Assignment Decision Phase, switch-location gene and switch-connection gene are xed, and cell-assignment gene is variable.
5.2 Initial Chromosome Generating
Procedure
As shown in Figs. 6 and 7, since switch-location gene (SL) be the high-level gene, the content of SL does e ect the contents of the switch-connection gene (SC) and cell-assignment gene (CA). Once the location of new switches have been selected, the connections of switches should be updated according to the selection of new switches. To do this, let switch-pool for con-nection (SPC) be the set of numbers be the indices of the switch which will be used to determine SC. Thus SPC =f1;2:::;mg[fijSL[i] = 1;m+1im+m
0
g.
To generate the connection of new switches, the value of array SC are randomly selected from SPC. This pro-cess will guarantee that new switch will be connected to an exist switch in S. Similarly, the assignments of cells to switches should be updated according to the selection of new switches. Let switch-pool for assign-ment (SPA)be the set of numbers be the indexes of the switch can be used in choosing of CA. For switch sk,
1 k m, Capk \k" are inserted into set SPA. For
switchck, ifm + 1km + m
0andSL[k] = 1, CAP
\k" are inserted into set SPA. For example, assume s8,
s9, and s13 are selected be the new switches, the SPC
is f1;2;3;4;8;9;13g(assume CAP=2) and the SPA is f2;2;3;8;8;9;9;13;13g: To assign cells to switches, the
value of element in arrayCA is randomly selected a num-ber fromSPA and removed it from SPA.
5.3 Chromosome Adjustment
Proce-dure
Since we assume that the expanded backbone network must be a connected network, but from the observa-tion of the initial chromosome generating procedure de-scribed in previous subsection, there is no guarantee to generate a connected network. Moreover, the switch ca-pacity constraint may be violated. These events are ad-justed by means of the chromosome adjustment proce-dure described below.
For the switch-connection gene: we have to test
whether the expanded backbone generated by initial chromosome generating procedure is a connected network or not. If yes then there is no need to change; otherwise, a modi ed algorithm should be performed to change it to a connected network. Since the original existing network is connected, all we have to do is to test whether each new switch can reach to one of switches in Sold by repeatedly
traversing the path to next switch as indicated in the switch-connection gene. If the expanded backbone network was not a connected network then there exist a cycle of new switches. To modi ed the back-bone network to a connected one, for each cycle, arbitrarily select a new switch in this cycle and
con-nect it to a randomly selected switch inSold, then
this will break the cycle and connect all switches in this cycle to a switch in the original existing back-bone network. This test and modify process can be done inO(m0) time.
For the cell-assignment gene: we have to test
whether the cell assignment violates the capacity constraints or not. The Chromosome Adjustment Procedure[5] can be used to generate a constraint-satis ed assignment. Since the initial population of our solution method is randomly generated and the operator of GA sometimes generates a chromo-some which does not represent a feasible assign-ment. This event is adjusted by means of the chro-mosome adjustment procedure described below: Let nkbe the number of cells assigned to switchsk,k=1,
...,m + m0; three types of switches are de ned:
(1)over-switch: ifnk > Capk;
(2)saturated-switch: ifnk =Capk;
(3) poor{switch: if nk< Capk:
Switches are grouped into setsSover,Ssat, andSpoor
for over-switch, saturated-switch and poor-switch, respectively. To change infeasible chromosomes into feasible ones, chromosome adjustment procedure is repeatedly used to reassign the cells from over-switches to poor-over-switches until all over-over-switches be-come saturated-switches. We have following algo-rithm:
Algorithm: Chromosome Adjustment
Proce-dure.
Step 1:
Switches are grouped into setsSover,Ssat,and Spoor according to the number of cells
being assigned to it; without loss of general-ity, switches are renumbered such that nk
nk+1,k = 1;:::;m + m0 ;1.
Step 2:
Construct a set SP (switch pool) of num-ber of switches by putting Capk-nk \k" intoSP, if nk < Capk, fork = 1;2;:::;m + m0.
Step 3:
Randomly generate a number as the ad-justment pointz in [n+1;n+n0], whileSover
is nonempty do Step 4.
Step 4:
Ifl = vz2Sover, then randomly select andremove a number (sayq) from SP; reassign cell cz to switch sq, i.e., set the value of vz
toq; decrease the nl by 1; if nl=Capl then
move switchslfromSover toSsat. Otherwise,
increasez by 1, if z > n + n0thenz = n + 1.
5.4 Genetic crossover operator
The traditional single point crossover was used in the ge-netic algorithm. The details are described in the follows:
In the Switch Site Selection Phase, the single point
genes (saySL1 and SL2) for crossover from
previ-ous generations and then by using a random num-ber generator, an integer valuei is generated in the range [m + 1;m + m0]. This number is used as the
crossover site. To create new o spring, rst, all characters between i and m + m0 of two parents
are swapped and childrenC1 andC2are generated.
Then, the following SL gene and CA gene are re-generated according to the contents of the C1 and
C2.
In the Switch Connection Decision Phase,
switch-location gene is xed. The single point crossover is randomly selecting two switch-connection genes (saySC1 andSC2) for crossover. After performing
crossover operation, the resulting genes may repre-sent a disconnected networks. Thus, the chromo-some adjust procedure as illustrated in Section 4.3 must be applied to change the children SC genes into feasible genes. It should be notice that the cell-assignment gene does not change.
In the Cell Assignment Decision Phase,
switch-location gene and switch-connection gene are xed. The single point crossover is randomly selecting two cell-assignment genes (say CA1 and CA2) for
crossover. After performing crossover operation, the resulting genes may violate the capacity constraint, Thus, chromosome adjust procedure proposed in previous subsection should be applied to change the CA genes into an feasible one.
5.5 Mutation
The traditional single cell mutating operation, which mutates a cell in genes at a time, is used to develop of this algorithm.
In the Switch Site Selection Phase, the mutating
operator changes a randomly selected cell in switch-location gene from "0" to "1" or from "1" to "0". The value of the mutated cell in theSL is randomly assigned to a switch. If the resulting backbone net-work is not connected then the chromosome adjust-ment procedure should be applied. Final, the Cell-assignment gene is regenerated according to the con-tent of the SL:
In the Switch Connection Decision Phase, the
mu-tating operator changes a randomly selected cell in switch-connection gene from current value to an in-teger which is randomly selected from theSPC. If the resulting backbone network is not a connected network then the chromosome adjustment proce-dure should be applied. It should be noticed that the switch-location gene and the cell-assignment gene need not change.
In the Cell Assignment Decision Phase,
switch-location gene and switch-connection genes are xed. Four types of mutations can be applied toCA gene. It is worth noting that After mutation, the chromo-some may became a infeasible one, thus, the Chro-mosome Adjustment Procedure must be applied to the chromosome.
(1)Traditional Mutation (
TM
): randomly select a cell of vectorvi, wherei in [n+1;n+n0] andtrans-form to a random number between 1 tom + m0.
(2)Multiple Cells Mutation (
MCM
): randomly se-lect two random numbersk, l between 1 and m+m0,transform the value of cells inCnewwhich value is
k to l and l to k.
(3)Heaviest Weight First Preference (
HWFP
)[3]: Since the hando cost involving only one switch is negligible, two cells can be assigned to the same switch so as to reduce the hando cost between these cells. Two cells with higher weightwij should havea higher probability of being assigned to the same switch. Thus, if we consider two connected cells ci
andcj 2C, then the probability of mutation from
viof cell ci to the valuevj of cell cj is as follows:
P(i;j)=Pn wij
i=1
Pdegree(c i)
j=1 wij;
wheredegree(ci) is the number of cells connected to
cellci in CG.
(4) Minimal Cabling Cost First Preference
(MCCFP)
[3]: To reduce the cabling costs be-tween cells and switches, we prefer to assign each cell to the nearer switch rather than the farther one. Cell ci and switch sk with lower cabling costlikshould result in higher probability thatciwill be
assigned tosk. Thus, if we consider the randomly
selected cell ci, then the probability of mutation
fromvi of cellci to the valuevk is :
P(i;k)= Lmax;lik Pm
l=1(Lmax;lil);
whereLmax = maxml=1flilg.
5.6 Fitness function de nition
Generally, genetic algorithms use tness functions to map objectives to costs to achieve the goal of an op-timally designed two-level wireless ATM network. If cell ci is assigned to switch sk, thenvi in the CA genes is
set tok. Let d(vi;vj)be the minimal communication cost
between switches sk and slin G. An objective function
value is associated with each chromosome, which is the same as the tness measure. We use the following ob-jective function:
minimize n+n0 X j=1 m+m0 X k=1 livi+ n+n0 X i=1 n+n0 X j=1 m+m0 X k=1 m+m0 X l=1 wij (1;yij)qkqlxikxjld(v i;vj)+ ; (9) where = Pm+m 0 k=1 jnk;Capkj + maxf(EC;
Budget);0g is the penalty measure associated with a
chromosome, and assume nk be the number of cells in
Cnew be assigned to switch sk, and and are the
penalty weights (see [7] for a further discussion of penalty measures).
Since the best- t chromosomes should have a proba-bility of being selected as parents proportional to their tness, they need to be expressed in a maximization form. This is done by subtracting the objective from a large number Cmax. Hence, the tness function
becomes:
maximize
Cmax ; [ n X i=1livi+ n X i=1 n X j=1wijd(vi;vj) + m+m0 X k=1(nk ;Capk) + (maxfEC;Budget;0g)] (10)where Cmax denotes 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, andCmax
can be calculated by the following iterative equation: Cmax=maxfCmax;costg
whereCmaxis initialized to zero.
5.7 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. Each o spring 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 o spring is better than that of only one of the parents, then we select a chromosome randomly from the better parent and the o spring. If the o spring is worse than both parents then each of the parents is selected at ran-dom 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.
5.8 Termination rules
Execution of GA can be terminated when the number of generations exceeds an upper bound speci ed by the user.
Figure 9: The e ect of the GA with di erent population size.
6 Experimental Results
In order to evaluate its performance, we have imple-mented the algorithm and applied it to solve problems that were randomly generated. The results of these ex-periments are reported below. In all the exex-periments, the implementation language was conducted in C, and all experiments were run on a Windows NT with a Pen-tium II 450MHZ CPU and 256MB RAM. We simulated a hexagonal system in which the cells were con gured as an H-mesh. The hando frequency fij for each border
was generated from a normal random number with mean 100 and variance 20. To examine the e ect of the di er-ent population size of genetic algorithms, we setn = 400, n0 = 200, m = 20, CAP = 25, = 1, population size
(popsize) is in setf20;40;50;60;80;100g, crossover
prob-ability (Pc)=1:0, maximum number of generations of
each phase is 1000, and mutation probability is 0.05. The coverage behaviors of the three-phase GA were shown in Fig. 9 and Fig. 10.
7 Conclusions
In this paper, we investigate thenetwork expanded prob-lem which optimum assignment new and splitting cells in PCS network to switches on an ATM network. This problem is currently faced by designers of mobile commu-nication service and in the future, it is likely to be faced by designers of personal communication service (PCS).
Since nding an optimal solution of the network ex-pended problem is NP-hard, a stochastic search method based on a genetic approach is proposed to solve it. Sim-ulation results showed that genetic algorithm is robust
Figure 10: The e ect of the GA with di erent population size.
for this problem. In our methods, three types of genes (switch-location, switch-connection, cell-assignment) are used to encode chromosome. Chromosome adjustment method is proposed to adjust chromosome to represent a feasible solution and nd the tness of chromosome. The traditional single point crossover and single cell mu-tation are employed in our method. Experimental results indicate that the algorithm run eciently.
8 Acknowledgment
This work was supported in part by the MOE Program of Excellence Research under Grant 90-E-FA04-1-4.
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