647
Heuristic Algorithm for Optimal Design of the Two-Level
Wireless ATM Network
DER-RONGDIN ANDS. S. TSENG Department of Computer and Information Science
National Chiao Tung University Hsinchu, Taiwan 300, R.O.C.
In this paper, we investigate the optimal assignment problem of cells in PCS (Per-sonal Communication Service) to switches in a wireless ATM network. Given cells and switches on an ATM network (whose locations are fixed and known), the problem is grouping cells into clusters and assigning these clusters to the switches in an optimum manner. This problem is modeled as a complex integer programming problem and find-ing an optimal solution of this problem is NP-complete. A three-phase heuristic algo-rithm MCMLCF (Maximum cell and minimum local communication first) consisting of Cell Pre-Partitioning Phase, Cell Exchanging Phase, and Cell Migrating Phase, is pro-posed. First, in the Cell Pre-Partitioning Phase, a three-step procedure (Clustering Step, Packing Step, and Assigning Step) is proposed to group cells into clusters. Second, Cell Exchanging Phase is proposed to greatly improve the result by repeatedly exchanging two cells in different switches. Finally, Cell Migrating Phase is proposed to reduce cost by repeatedly migrating all cells in a used switch to an empty switch. Experimental re-sults indicate that the proposed algorithm runs efficiently. Comparing the rere-sults of the algorithm to a naive heuristic called NSF, we have shown that the computation time is reduced by 30.1%. Experimental results show that Cell Exchanging and Cell Migrating phases can reduce the total cost by 34.1% on average. By comparing the results of the proposed algorithm to the genetic algorithm, the heuristic method came close to opti-mum - on average within 5%.
Keywords: wireless ATM, design of algorithms, assignment problem, clustering problem,
graph partitioning problem, personal communication services
1. INTRODUCTION
Recently there has been much interest in extending ATM technology into the wireless environment [1-11]. The motivation behind this (termed wireless ATM) includes the desire for seamless interconnection of wireless and ATM networks, and the need to support emerging mobile multimedia services. However, due to inherent differences in these two types of networks, the introduction of ATM into the wireless environment pre-sents many interesting challenges [3, 10]. These include supporting an end-to-end ATM connection with user mobility, handling the high error rate performance of wireless links and grouping cells into LAs (location areas or clusters).
In this paper, we investigate the optimal assignment of cells in PCSs (Personal Communication Services) to switches in an ATM network. The system area of PCSs is
Received December 1, 1999; revised June 1 & July 10, 2000; accepted August 9, 2000. Communicated by Norio Shiratori.
divided into several LAs (location areas). In general, an LA consists of an aggregation of cells forming a contiguous geographical region. When a subscriber enters a cell that be-longs to a different LA, a location update (LU) procedure that informs the network about the subscriber’s new location is performed. This will generate network traffic overhead in PCS networks and consume scarce radio resources. Moreover, LU also increases the load on distributed location databases and, thus, increases the complexity of implement-ing the databases [7, 9].
Fig. 1. Two-level hierarchical network. The handoff from B to C is more expensive than from B to A.
Consider the example shown in Fig. 1, where cells A and B are connected to switch
s1, and cells C and D are connected to switch s2. If the subscriber moves from cell B to
cell A, switch s1will perform a handoff for this call. This handoff is relatively simple and
does not involve any location update in the databases that record the position of the sub-scriber. The handoff also does not involve any network entity other than switch s1. Now,
let us suppose that the subscriber moves from cell B to cell C. Then the handoff involves the execution of a fairly complicated protocol between switches s1and s2. In addition, the
location of the subscriber in the databases must be updated. There is actually one more fact that makes this type of handoff difficult. If switch s1is responsible for keeping the
billing information about the call then, switch s1cannot simply remove itself from the
connection as a result of the handoff. In fact, the call continues to be routed through switch s1(for billing purposes). In this case, the connection is from cell C to switch s2,
then to switch s1, and finally to the telephone network [10].
Merchant and Sengupta [10] considered the problem of assigning cells to switches in PCS network. They formulated the problem and proposed a heuristic algorithm to solve it so that the total cost can be minimized. The total cost consists of cabling and location update. The location update cost considered in [10], which depends only on the
frequency of handoff between two switches, is not practical. Since the switch of the
ATM backbone is wide spread, the communication cost between the two switches should be considered in calculating the location update cost.
net-work (whose locations are fixed and known). The problem is to group cells into LAs and assign LAs to switches in the ATM network in an optimum manner. We consider the topological design of a two-level hierarchical network. The upper-level of the network is a connected ATM network, while the lower-level is a cell network which is configured as
an H-mesh (see Fig. 1). The objective cost has two components. One is the location
update cost, which involves two different switches; the other is the cost of cabling which connected cells to switches of the ATM network. We try to assign cells to switches so
that the total cost is minimized under some assumptions described later. The
organiza-tion of this paper is shown as follows. In Secorganiza-tion 2, we formally define the problem. In Section 3, we outline the proposed algorithms. In Sections 4, 5 and 6, we describe details of the algorithms. The experimental results are presented in Section 7. Finally, a conclu-sion is given in Section 8.
2. PROBLEM FORMULATION
This section first provides an overview of various terms and notations used to ex-plain the concepts outlined in the subsequent sections. Then, we introduce the definitions of various parameters used in this paper, and list assumptions made in representing the network topology.
Notation:
n = total number of cells in the PCS network
m = total number of switches in the ATM network
G(S, E) = ATM network, where S is the set of switches and E⊆ S × S
CG(C, L) = cell network, where C is the set of cells and L⊆ C × C (sk, sl) = edge between switches skand slin S
(ci, cj) = edge between cell ciand cjin C
(Xsk, Ysk,) = coordinate of switch sk∈ G, k = 1, 2,..., m
(Xci, Yci) = coordinate of cell ci∈ C, i = 1, 2,..., n
dkl = minimum communication cost between switches skand slin G
fij = frequency per unit time of the handoffs that occur between cells ciand cjin
CG, i, j = 1,..., n
lik = cost of cabling per unit time between cell ci∈ CG and switch sk∈ G, i =
1, ..., n; k = 1,..., m; assume 2 2 ( ) ) ( k i k i s c s c ik X X Y Y l = − + −
wij = weight of edge (ci, cj)∈ CG where wij= fij+ fji, wij= wji, and wii= 0; i, j =
1, ..., n
Cap = cell handling capacity of the switch
= ′ Cap n
m = number of switches that need to be assigned
Assumptions:
(1) The structures and positions of the ATM network and PCS networks are known. (2) We assume that the cost of handoffs involving only one switch is negligible.
(3) Each cell in the cell network will be directly assigned and connected to only one switch in the ATM network.
(4) The capacity of a switch, defined as the number of cells that it can handle, is limited
to a maximumCap.
(5) The cost has two components, the cost of handoffs that involve two switches, and the cost of cabling (or trucking).
(6)minimal switches assumption: It is worth noting that if the number of assigned
switches is minimized, then the location update (handoff) costs will be reduced. (7) load balance assumption: The load on assigned switches is assumed to be balanced.
If the load balance assumption is satisfied,
= ′ Cap n
m switches need to be assigned,
and the number of cells assigned to switches is ′ m n or ′ m n . Decision variables: Problem:
Find variablesxijwhich minimize
(1) subject to
∑
≤
=
= n i ikm
k
Cap
x
1;
,...,
1
,
(2)∑
m= = = k ik , i ,...,n; x 1 1 for 1 (3) ; ,..., 1 , 1 m k m n x m n n i ik = ′ ≤ ∑ ≤ ′ = (4) . 1,..., for ; 1,..., {0,1}for i n k m xik∈ = = (5) = = = = = = = =∑
= otherwise 0 switch common a to connected are and cells both if 1 1 otherwise 0 switch common a to connected are cells both if 1 1 and 1 for otherwise 0 switch to assigned is cell if 1 1 j i ij m k ijk ij k i ijk jk ik ijk k i ik c c y ., e . i n ,..., j , i , z y s c z ., e . i m ,..., k n ,..., , j , i , x x z s c x∑ ∑
=n = +α∑ ∑ ∑ ∑
= = = = − i m k n i n j m k m l lk jl ik ij ij ik ikx w ( y )x x d l 1 1 1 1 1 1 1If cells ci and cjare assigned to different switches, then a cost is incurred. Every
time a handoff occurs; if fijis the frequency per unit time of a handoff that occurs
be-tween cells ciand cj, (i, j = 1,..., n), which we assume, is fixed and known. Our objective
is to assign each cell to a switch so as to minimize the sum of the cabling and handoff costs per unit time, i.e., the total cost.
Thus, our goal is to minimize objective function (1). The first part of (1) is the to-tal cabling cost between cells and switches, the second part is the cost of handoffs per unit time, andαis the ratio cabling cost to handoff cost. Constraint (2) ensures that the number of calls is limited to the capacity Cap. Constraint (3) ensures that each cell is assigned to exactly one switch. Constraint (4) ensures that the minimal used switches assumption and load balance assignment assumption are satisfied. Constraint (5) ensures that xikis a binary value.
3. OUTLINE OF HEURISTIC ALGORITHM
It is well known that finding an optimal solution to this problem is NP-hard. In this paper, a three-phase heuristic algorithm is proposed to solve this problem, which can be described as follows:
(1) Cell Pre-Partitioning Phase: Construct a pre-partition of CG by considering some basic constraints.
(2) Cell Exchange Phase: Take the ATM network environment into account, and im-prove of pre-partition by exchanging cells in different switches.
(3) Cell Migration Phase: Migrate cells in one switch to an empty switch such that the total cost can be reduced.
The algorithm’s input are CG, G and parameters (α, CAP,...). The goal of Cell
Pre-Partitioning Phase is to group cells in CG into clusters and to assign clusters to switches of ATM network G. It is worth noting that if two cells are connected to the same switch, the handoff cost between two cells can be ignored. Thus, cells are grouped into clusters in Clustering Step, based on some threshold values defined later. Since threshold constraints exist, after Clustering Step completing, the number of clusters may be greater than m′, which is further merged and adjusted in Packing Step. In Assignment Step, there are m′clusters of CG that will be assigned to m switches of the ATM switch network N. This subproblem can be formulated as an assignment problem [13, 14].
After executing Cell Pre-Partitioning Phase, it can be easily seen that each cell is connected to one switch and the minimal switches and load balance assumptions are sat-isfied. Obviously, during Clustering Step, the transmission cost of the ATM network is ignored. The Assigning Step assigned clusters to switches and did not consider the ca-bling cost between switches and individual cells. As a result, Cell Exchanging Phase and Cell Migrating Phase were developed to improve the result of Cell Pre-Partitioning Phase. The main objective of Cell Exchanging Phase is to reduce the total cost by considering the 2-layer cell-switch network. Two cells in different switches which provide the great-est improvement are selected, and the assigned switches are exchanged. This process continuously runs until no more improvement can be made. In Cell Migrating Phase, a
used switch and an empty switch which provide the greatest improvement are selected, and all cells the switch being used are migrated to the empty switch. This process also runs continuously until no more improvement can be made.
4. CELL PRE-PARTITIONING PHASE
In this section, the three steps that make up the Cell Pre-Partitioning Phase are de-scribed. They are:
(1) Clustering Step: Group cells into clusters so as to reduce the handoff cost between cells.
(2) Packing Step: Reduce the number of clusters to m′.
(3) Assignment Step: Assign clusters to switches so that the cabling cost is minimized.
Example 1.Consider the two graphs shown in Fig. 2. There are ten cells in CG which should be assigned to four switches in S. The edge weight between two cells is the fre-quency per unit time of the handoffs that occur between them. Four switches are posi-tioned at the center of cells: c1, c5, c7, and c9. Assume the matrix CS of the distance
be-tween cells and switches is
Fig. 2. Example of cell and ATM networks.
= = 1 1 3 7 0 3 1 2 1 7 1 3 3 0 2 7 1 1 1 3 1 2 0 1 3 3 1 1 2 1 3 2 3 3 1 1 2 7 1 0 } {lik CS
4.1 Clustering Step
In Clustering Step, three thresholds are introduced to constrain the clustering process. (1) Average load (avgLoad =
′ m
n ): This is used to avoid generating extremely large
and load inbalanced clusters.
(2) Average edge cost ( =∑ ∑n= = −
i n j wij n n t avgEdgeCos 1 1 2 )
/( ): This is used to avoid
merging very low-cost edges.
(3) Average cell to cell distance =∑ ∑n= = − + − −
i n j Xc Xc Yc Yc n n t avgCellDis j i j i 1 1 2 2 2 )) /( ) ( ) ( ( :
This is used to avoid merging distant cells.
Let W(ci) be the load on ciwith initial value 1, and let dist(i, j) be the Euclidean
dis-tance of two cells ci and cj. To decide whether or not two cells can be merged, three
merging constraints are defined. If cells ciand cjare merged, the following constraints
must be satisfied:
(1) W(ci) + W(cj)≤ avgLoad.
(2) wij> avgEdgeCost.
(3) dist(i, j)≤ avgCellDist.
Initially, set A is formed as the set of n cells. The clustering procedure is to
itera-tively merge a pair of cells which satisfy the merging criterion. A new cell is formed
with its location at the center-of-mass (COM) of the two cells being merged, where the weights used in the COM calculation are simply the weights assigned to each cell. The group of real cells, which represent merging the two old cells, forms the new cell. The old cells are then removed from set A. The new cell will be added to set A later. The weight of the new cell is simply the sum of the weights of the old cells. Note that with the initial weights all set equal to one, the weight is simply the number of real cells rep-resented by the new cell. The location update cost between the new cell and the old cells is the sum of the location update costs between the two old cells and the other cells.
Algorithm: Clustering
Step 1.1Repeat Steps 1.2, 1.3 and 1.4 until no more cell can be merged.
Step 1.2 Find the cell activecell with maximal edge sum (edge radiated from it) in the cell graph.
Step 1.3Find the cell condcell and the edge connected to activecell with minimum ‘radi-ating sum’ which satisfies the three merge constraints.
Step 1.4 If no suitable condcell is found, repeat Step 1.5 until no more cells can be merged. Otherwise, merge condcell to activecell, if (condcell < activecell); or vice versa.
Step 1.5Find the next cell activecell with maximal edge sum (edge radiated from it) in the cell graph. Find the cell condcell and the edge connected to activecell with minimal ‘radiating sum’ which satisfies three merge constraints. If no suitable
condcell found then algorithm terminate; otherwise merge condcell to activecell,
An example of applying Algorithm Clustering to Example 1 is shown in Fig. 3. First, select c5as activecell and c8as condcell. In Fig. 3(b), the two cells are merged into a new
cell c*5 in Step 1.4, and cell c5* is then selected as the newactivecell. Repeating this process, the cell network can finally be clustered into the two groups shown in Fig. 3(i).
Fig. 3. Result of applying clustering algorithm to Example 1.
4.2 Packing Step
To satisfy the load balance assignment and minimal switches assumption, this
step merges some small cells (clusters) left into the cell graph to larger cells in order to reduce the number of clusters tom′.
Algorithm: Packing
Step 2.1Repeat Steps 2.2, 2.3 and 2.4 until the number of clusters remaining inCGism'.
Step 2.2 Sort the remaining clusters in nondecreasing order according to the weight of each cluster (or cell).
Step 2.3 Select the minimal weight cluster ci and find the cluster cjwith the greatest
weight such that W(ci) + W(cj) < avgLoad. If found, merge the two clusters
which decreases the number of clusters by 1. Otherwise, go to Step 2.4.
Step 2.4 For each cell in the minimal weight clusterci, find the heaviest weight edge
connected to it and the corresponding cluster; merge the cell into this cluster. This process is repeated until all cells in this cluster are adjusted. Then, de-creases the number of clusters by 1.
4.3 Assignment Step
After obtainingm′clusters, the goal of this step is to assign clusters to switches so as to minimize cabling costs between clusters and switches. LetP= {Pl},l= 1, …,m′,
(m′<m) be anm′-way partition obtained from previous steps ofCG. DefineF(k, l) as the sum of cabling costs of cells which are clustered intoPkand assigned to switchl, 1≤k≤
. 1 , 1 , ) , (k l l k m l m F k i P c il ≤ ≤ ∑ ≤ ≤ ′ = ∈
Obviously, the subproblem can be formulated as a minimal weighted matching problem on a bipartite graph which is known as the assignment problem. Therefore, the Hitchcoch Algorithm [14] can be applied to find the optimal solution of the assignment problem in O(m3) time. The result after running Assigning Step is shown in Fig. 4. Cells
c1, c2, c4, c5and c8are connected to switch s2with cabling cost 4, while the others are
connected to switch s4with cabling cost 5.732.
5. CELL EXCHANGING PHASE
The goal of Cell Exchanging Phase is to select two cells in different switches and exchange them in order to reduce the total cost. The basic idea of Cell Exchanging Phase is borrowed from Kernighan-Lin [15] algorithm which only works on traditional 2-way graph partition problems. In our 2-layer cell-switch network environment, location up-date costs and cabling costs must be considered simultaneously. Hence, we modified Kernighan-Lin’s procedure to exchange cells in different clusters by selecting the “most preferable” cells to exchange instead of exchanging two arbitrary cells. We first intro-duce some notations required in the following. Given m′ nonempty sets of cells P = {Pl},
l = 1, 2,..., m′, P is called a m′-way partition of CG, if P1∪ P2∪...∪ Pm′= C and Pi∩ Pj
=φ, where i≠ j. Without loss of generality, we assume the cells in set Pjare assigned to switch sj, j = 1, .., m′. Let sid(ci) = l if ciis in Pl, l is called the sid of cell ci. Let LUCS(i, l)
= ∑j∈Plwij and D(i, l) = LUCS(i, l)−LUCS(i, sid(ci)). Then for a given partition P, the
location update cost of the partition is:
∑ ∑ ⋅ α ∈C ∈ ci sl S sidcil d l i LUCS(, ) ) ( ( )
Fig. 5. Cell ciis moved from switch skto sl.
If cell ciwhich is currently assigned to switch sk, is reassigned to switch sl, then the
gain of reduced objective saving cost can be computed from the following lemma:
Lemma 1. Ignoring the size restriction, if cell ciwhich is currently assigned to sk, is
moved to switch sl, then
(1) the cabling cost is reduced by dik− diland
(2) the location update cost is reduced byα(LUCS(i, l)− LUCS(i, k)) ⋅ dkl
Proof:The proof of the cabling cost can be easily obtained. Let z be the sum of weights of all edges except those between cell ciand all cells of Pl. Then from Fig. 5, we can see
old cost = (z + LUCS(i, l))× dkl×α new cost = (z + LUCS(i, k))× dkl×α The reduced location update cost (handoff cost)
= (LUCS(i, l)− LUCS(i, k)) × dkl×α
= D(i, l)× dkl×α.
Now, if cell ci and cj are exchanged, the gain can be computed from the following
lemma:
Lemma 2.If cell ciin switch skand cjin switch slare exchanged, then
(1) The reduced cabling cost is: RC(i, j) = dik− dil+ djl− djk
(2) The reduced location update cost is:
RLU(i, j) = (D(i, l)− D(j, k) − 2wij)× dkl×α+ (dkz− dlz)×α× (LUCS(i, s) − LUCS(j, z))
(3) The reduced total cost is: exchange(i, j) = RC(i, j) + RLU(i, j).
Proof:The proofs of (1) and (3) are straightforward. From Lemma 1 and Fig. 6, the
re-duced cost of cell ciis:
(LUCS(i, l)− LUCS(i, k) − wij)× dkl×α; the reduced cell cjcost is:
(LUCS(j, k)− LUCS(j, l) − wij)× dkl×α;
the reduced cost of switch szto cell ciis (z≠ k and z ≠ l):
(dkz− dlz)×α× LUCS(i, z).
the reduced cost of switch szto cell cjis (z≠ k and z ≠ l):
(dlz− dkz)×α× (LUCS(i, z) − LUCS(i, z)).
The reduced location update cost is:
(D(i, l)− D(j, k) − 2wij)× dkl×α+ (dkz− dlz)×α× (LUCS(i, z) − LUCS(j, z)).
After cells ciand cjhave been selected and exchanged, the matrix LUCS should be
updated so that the algorithm can run effectively and avoid sequential searching.
Lemma 3.After cells ciin switch skand cjin switch slare exchanged, the LUCS values
can be updated as follows:
(1) LUCSnew(i, k) = LUCS(i, k) + wij
(2) LUCSnew(i, l) = LUCS(i, l)− wij (3) LUCSnew(j, k) = LUCS(j, k)− wij (4) LUCSnew(j, l) = LUCS(j, l) + wij
For any cell a in switch sz
(5) LUCSnew(a, k) = LUCS(a, k) + waj− wai, if (sz= k) and (a≠ i)
(6) LUCSnew(a, l) = LUCS(a, l)− waj+ wai, if (sz= k) and (a≠ i)
(7) LUCSnew(a, k) = LUCS(a, k) + waj− wai, if (sz= l) and (a≠ j)
(9) LUCSnew(a, k) = LUCS(a, k) + waj− wai, if (sz≠ k and sz≠ l) and (a ≠ i and a ≠ j)
(10) LUCSnew(a, l) = LUCS(a, l)− waj+ wai, if (sz≠ k and sz≠ l) and (a ≠ i and a ≠ j)
Proof:The proofs of (1)-(4) are trivial. For a cell ca, which is neither with sid cinor with
sid cj(see Fig. 7), LUCS(a, k) is increased by− wai+ waj(9), and LUCS(a, l) value is
in-creased by−waj+ wai(10).
Fig. 7. Cell ciin switch skand cjin switch slare exchanged.
If sid(ca) = sid(ci) = k(see Fig. 8) then LUCS(i, k) and LUCS(i, l) are increased by waj− waiand wai− waj, respectively. Thus, (5) and (6) hold. For cell cawith sid l, a
simi-lar argument shows that (7) and (8) hold.
Given an initial assignment, Cell Exchanging Phase consecutively selects two cells in different switches to exchange. At each iteration, two cells caand cbare selected which
maximize the reduced exchanging cost exchange(a, b) where ). , ( max ) , ( ) , ( j i exchange b a exchange C C c ci j∈ × =
The iteration continues if exchange(a, b)≥ 0.
Algorithm: Cell Exchanging
Step 1 For each cell in CG and each switch in G, compute values of matrices LUCS(i, l) and D(i, l). Let B = C.
Step 2 Find two cells caand cbassigned to B in different switch such that
). , ( max ) , ( ) , ( j i exchange b a exchange C C c ci j∈ × =
Step 3 If exchange(a, b) > 0, then exchange cells caand cb; and delete caand cbfrom B.
Step 4 Update LUCS(i, l), and D(i, l) for each ci∈ B and slin L.
Step 5 If B is nonempty go to Step 2.
Step 6 If exchange(ca, cb) > 0 then go to Step 1; otherwise terminate the algorithm.
6. CELL MIGRATION PHASE
The basic idea of Cell Migration Phase is to assign cells of a used switch to an-other empty switch in order to reduce the total cost. Without loss of generality, let Pi, i =
1, ..., m′ < m be the partition of CG. Cells of partition Piare assumed to be assigned to switch si, i = 1, ..., m′. Define Cabling(k) as the sum of the cabling cost associated with
switch sk. That is, ∑ = ∈k i P c ik l k Cabling( ) .
Let Cabling(k, l) be the reduced cost of all cells after reassigning them from switch
skto switch sl, assuming switch slis empty, i.e.,
∑ − = ∈Pk i ik il l l l k Cabling( , ) ( ). Let Cabling(k, k) = Cabling(k) and
∑ = ∈k i P c k i LUCS l k LUSS( , ) (, ).
Lemma 4. If switch slis empty and the cells in switch skare migrated to switch sl, then
(1) The reduce location update cost is
∑ − α = ≠ ≠ ∈Sz kz l sz zl zk d d k z LUSS l k Mig , , ). )( , ( 2 ) , (
(2) The reduce total cost is
). , ( ) , ( ) , (k l Cabling k l Mig k l migrate = +
de-Fig. 9. Cells in switch skare migrated to switch sl.
crease to 0, and LUSS(z, l) and LUSS(k, z) increase the original value of LUSS(k, z). Thus, it is easy to prove that the reduced location update cost is
). ) , ( ) , ( ( 2α∑sz∈S,z≠k,z≠l LUSS z k ×dzk−LUSS zl ×dzl Since ), , ( ) , (z k LUSS zl LUSS = ). ( ) , ( ( 2 ) , (kl s S,z k,z l LUSS z k dzl dzk Mig z − ∑ α = ∈ ≠ ≠
The following lemmas can be easily derived:
Lemma 5.After migrating cells in switch skto switch sl, LUSS can be updated in linear
time.
Lemma 6.After migrating cells in switch sk to switch sl, Mig can be updated in O(n2)
time.
Given an initial assignment, Cell Migration Phase sequentially selects two switches to migrate. At each iteration, a used switch sa and an empty switch sb are selected to
maximize the reduced migrated cost migrate(a, b), where ). , ( max ) , ( , l k migrate b a migrate S s S sk∈ l∈ =
The iteration continuous while migrate(a, b) > 0.
Algorithm: Cell Migration
Step 1 Let a group of switches S be represented by B. Calculate LUSS(k, l), Cabling(k,
l), and Mig(k, l) for switches sk, sl∈ S;
Step 2 Find a used switch saand an empty switch sbin B such that
). , ( max ) , (a b migratek l migrate S s S sk∈ ∧l∈ =
Step 3 If migrate(a, b) > 0, migrate cells in switch sato switch sb.
Step 4 Update LUSS(k, l), Mig(k, l), and Cbl(k, l) for sk, slin L.
Step 5 If migrate(a, b) > 0 then go to Step 2; otherwise terminate.
7. EXPERIMENTAL RESULTS
The proposed algorithm consists of three phases (Cell Pre-Partitioning Phase, Cell-Exchanging Phase, and Cell-Migrating Phase), termed MCMLCF(Maximum cell and minimum local communication first), which is a heuristic approach to a rather com-plex problem. In order to evaluate its performance, we have implemented the algorithm and applied it to a number of examples with randomly positioned cells and switches. The results of these experiments are reported below.
In all the experiments, the implementation language is C, and some experiments have been made on Windows NT with a Pentium II 450MHz CPU and 256MB RAM. We simulated a hexagonal system in which cells were configured as an H-mesh. The
handoff frequency fijfor each border was generated by a normal random number
genera-tor with mean 100 and variance 20. To examine the effects of a different number of cells, Cell Graph CG with n = 100, 200, 300, 400, and 500 cells were tested. We used m = 100,
α= 1/100 and Cap/n = 0.1 for each problem. The ratio of the number of switches used to
the total number of switches is 1/10.
To measure the performance of each phase of the three-phase heuristic MCMLCF algorithm, we simply construct a heuristic algorithm termed NSF (Nearest Switch First) as the Cell Pre-partition Phase by assigning a cell to the nearest switch. If the nearest switch is full then find the next nearest. Then, switches are sorted in nondecreasing order according to their load of switches, and run Packing Step to reduce the number of switches being assigned to m′. It is clear that no further assignment process is needed.
For all experiments, the CPU time in seconds of the heuristic algorithm and the ob-jective cost reduction ratio are the major concerns. First, the CPU time for running the three-phase algorithms NSF and MCMLCF by simulating with different number of cells
are shown in Table 1. TNSF and TMCMLCFare the CPU times for the problem by running
NSF and MCMLCF algorithms in three-phase, respectively. Taking the CPU time for the
algorithm NSF as the reference, the CPU time reduction ratio is computed by (TNSF −
TMCMLCF)/TNSF and shown in Table 1. We found that the CPU time reduction ratio of
MCMLCF to NSF is 30.1% on average.
Table 1. CPU time in seconds for algorithm.
# of cells NSF MCMLCF (TNSF− TMCMLCF)/TNSF 100 14.9 8.8 41.3 200 102.1 68.9 32.5 300 458.1 324.1 29.3 400 1336 1023 23.7 500 3030 2320 23.4 average 30.1
Second, not only is the cost reduction ratio of the whole algorithm evaluated, the cost reduction ratio of the Cell Exchanging Phase, and Cell Migrating Phase are evalu-ated as well. To test the effects of Cell Exchanging Phase and Cell Migrating Phase, we
compare the costs of the algorithm running the two phases. Let CA, CBand CCbe the
costs resulting from running Cell Pre-Partitioning, Cell Exchanging, and Cell Migrating Phases, respectively. The cost improvement ratios (CA− CB)/CA, (CB− CC)/CB, and (CA− CC)/CAare shown in Tables 2, 3 and 4, respectively. As seen in Table 4, after running
Cell Exchanging and Cell Migrating Phases, the total cost of the NSF algorithm is re-duced by 22.1% and the total cost of MCMLCF algorithm is rere-duced by 34.1%.
Table 2. Reduction in cost ratio of Cell Exchange Phase.
# of cells NSF MCMLCF 100 4.2% 6.5% 200 3.2% 9.3% 300 3.7% 9.7% 400 3.1% 7.9% 500 3.3% 19.0% average 3.5% 11.1%
Table 3. Cost reduction ratio of Cell Migration Phase.
Algorithms NSF MCMLCF 100 16.8% 28.8% 200 15.8% 26.4% 300 16.8% 28.3% 400 18.8% 28.1% 500 25.1% 34.6% average 18.7% 29.2%
Table 4. Cost reduction ratio of Cell Exchange and Cell Migration Phases.
Algorithms NSF MCMLCF 100 16.8% 36.8% 200 17.8% 26.8% 300 22.0% 40.8% 400 27.8% 28.4% 500 26.1% 37.6% average 22.1% 34.1%
The integer programming method failed to find any solution when the problem size grew beyond 30 because the constraint set grew too large for the memory available. For comparison we also implemented a genetic algorithm [16] to solve the same problem. The experimental parameters of the genetic algorithm are: population size is 50, cross-over probability is 1.0, mutation probability is 0.05, the number of generations is 1000. The solution found nearly optimal in the objective value. We compare the performance
of the NSF and MCMLCF heuristic algorithms against the genetic algorithm. The results in Table 5 show that the heuristic method MCMLCF came close to the optimum - on average within 5%.
Table 5. Comparison of the Heuristic algorithms and GA.
# of cells NSF/GA MCMLCF/GA
100 133% 103% 200 122% 110% 300 124% 100% 400 114% 110% 500 117% 103% average 122% 105%
8. CONCLUSIONS
In this paper, we have investigated the problem of obtaining the optimal assign-ment of cells in PCS (Personal Communication Service) to switches in a wireless ATM network. Given cells and switches on an ATM network (whose locations are fixed and known), the problem is to group cells into clusters and to assign these clusters to switches in an optimum manner.
This problem has been modeled as a complex integer programming problem, and the optimal solution of this problem have been found to be NP-hard. A three-phase heu-ristic algorithm, MCMLCF, consisting of Cell Pre-Partitioning Phase, Cell Exchanging
Phase, and Cell Migrating Phase is proposed. First, in the Cell Pre-partitioning phase, a
three-step procedure (Clustering Step, Packing Step, and Assigning Step) is proposed to group cells into clusters. Second, Cell Exchanging Phase is proposed to improve the re-sult by repeatedly exchanging two cells in different switches so as to provide the greatest improvement. Finally, Cell Migrating Phase is proposed to reduce cost by repeatedly migrating all cells in a used switch to an empty switch.
Further, we evaluate the performance of the Cell Exchanging Phase, Cell Migrat-ing Phase, and the whole algorithm (MCMLCF). Experimental results indicate that the proposed algorithm runs efficiently. Comparing the results of the algorithm to the NSF algorithm, the computation time is reduced by 30.1%. The results show that Cell Ex-changing Phase and Cell Migrating Phase reduce the total cost as well, and by compari-son to the result of the genetic algorithm, the heuristic method MCMLCF came close to the optimum -on average within 5%. The computation time of algorithm MCMLCF to NSF is reduced by 30.1%.
ACKNOWLEDGMENT
This work was supported in part by the MOE Program of Excellence Research under Grant 89-E-FA-04-1-4.
REFERENCES
1. M. Cheng, S. Rajagopalan, L. F. Chang, G. P. Pollini, and M. Barton, “PCS mobility support over fixed ATM networks,” IEEE Communication Magazine, Vol. 18, No. 2, 1997, pp. 82-91.
2. D. C. Cox, “Personal communication−a viewpoint,” IEEE Communication Magazine,
Vol. 11, No. 8, 1990, pp. 8-20.
3. U. Varshney, “Internet watch: supporting mobile using wireless ATM,” IEEE
Com-puter, Vol. 30, No. 1, 1997, pp. 131-133.
4. T. Hattori, “Personal communication-concept and architecture,” in International
Conference on Communications (ICC’90), 1990, pp. 333.4.1-335.4.7.
5. R. Jain, Y. B. Lin, C. Lo, and S. Mohan, “A caching strategy to reduce network im-pacts of PCS,” IEEE Journal on Selected Areas in Communications, Vol. 12, No. 8, 1994, pp. 1434-1444.
6. Y. B. Lin, “Determining the user locations for personal communications services networks,” IEEE Transactions on Vehicular Technology, Vol. 43, No. 3, 1994, pp. 466-473.
7. Y. B. Lin and S. Y. Hwang, “Comparing the PCS location tracking strategies,” IEEE
Transactions on Vehicular Technology, Vol. 45, No. 2, 1996, pp. 114-121.
8. S. Mohan and R. Jain, “Two user location strategies for personal communications services,” IEEE Personal Communications, Vol. 1, No. 1, 1994, pp. 42-50.
9. D. Raychaudhuri and N. Wilson, “ATM based transport architecture for multiservices wireless personal communication network,” IEEE Journal on Selected Areas in
Communicaions, Vol. 12, No. 8, 1994, pp. 1401-1403.
10. A. Merchant and B. Sengupta, “Assignment of cells to switches in PCS networks,”
IEEE/ACM Transactions on Networking, Vol. 3, No. 5, 1995, pp. 521-526.
11. R. Steele, “Deploying personal communication networks,” IEEE Communication
Magazine, Vol. 4, No. 1, 1990, pp. 6-11.
12. M. D. Yacoub, Foundations of Mobile Radio Engineering, CRC Press, Boca Raton, FL, 1993.
13. E. L. Laeler, Combinatorial Optimization: Networks and Matroids, Rinehart & Winston, New York, Holt, 1976.
14. H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Research
Logistics Quarterly, Vol. 2, No. 1, 1955, pp. 83-97.
15. B. W. Kernighan and S. Lin, “An efficient heuristic procedure for partitioning graphs,” Bell Systems Technical Journal, Vol. 49, No. 3, 1970, pp. 291-307.
16. D. R. Din and S. S. Tseng, “Genetic algorithms for optimal design of two-level wire-less ATM network,” to appear in Proceeding of NSC.
Der-Rong Din () received the M.S. degree in
Com-puter and Information Science from National Chiao Tung Uni-versity, Taiwan, R.O.C., in 1994, and is currently a doctoral
can-didate at National Chiao Tung University. His research interests
include wireless ATM, parallel processing, computer algorithm, and parallelizing compiler design.
Shian-Shyong Tseng () received his Ph.D. degree in
Computer Engineering from the National Chiao Tung University
in 1984. Since August 1983, he has been on the faculty of the
Department of Computer and Information Science at National
Chiao Tung University, and is currently a Professor there. From
1988 to 1991, he was the Director of the Computer Center at
Na-tional Chiao Tung University. From 1991 to 1992 and 1996 to
1998, he acted as the Chairman of Department of Computer and
Information Science. Form 1992 to 1996, he was the Director
of the Computer Center at Ministry of Education and the
Chair-man of Taiwan Academic Network (TANet) Chair-management committee. In December
1999, he founded Taiwan Network Information Center (TWNIC) and is now the
Chair-man of the board of directors of TWNIC. His currently research interests include
