Heuristicand simulated annealing algorithms for solving
extended cell assignment problem in wireless ATM networks
Der-Rong Din
1and S. S. Tseng
2,*
,y1Department of Computer Science and Information Management, Hung-Kuang Institute of Technology,
Taichung 433 , Taiwan R.O.C.
2
Department of Computer and Information Science, National Chiao-Tung University, Hsinchu 300 , Taiwan R.O.C.
SUMMARY
In this paper, we investigate the extended cell assignment problem which optimally assigns new adding and splitting cells in Personal Communication Service (PCS) to switches in a wireless Asynchronous Transfer Mode (ATM) network. Given cells in a PCS network and switches on an ATM network (whose locations are fixed and known), we would like to do the assignment in an attempt to minimize a cost criterion. 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 modeled as a complex integer programming problem, and finding an optimal solution to this problem is NP-hard. A heuristicalgorithm and a simulated annealing algorithm are proposed to solve this problem. The heuristicalgorithm, Extended Assignment Algorithm (EEA), consists of two phases, initial assigning phase and cell exchanging phase. First, in the initial assigning phase, the initial assignments of cells to switches are found. Then, these assignments are improved by performing cell exchanging phase in which two cells are repeatedly exchanged in different switches with great reduction of the total cost. The simulated annealing algorithm, ESA (enhanced simulated annealing), generates constraint-satisfied configurations, and uses three configuration perturbation schemes to change current configuration to a new one. Experimental results indicate that EAA and ESA algorithms have good performances. Copyright # 2002 John Wiley & Sons, Ltd.
KEY WORDS: wireless ATM; PCS; optimization; heuristicalgorithm; simulated annealing; cell assignment problem
1. INTRODUCTION
The rapid worldwide growth of digital wireless communication services motivates a new generation of mobile networks to serve as infrastructure for such services. Mobile networks deployed in the next few years should be capable of smooth migration to future broadband
Received 15 December 2000 Published online 15 January 2002 Revised 19 September 2001 Copyright # 2002 John Wiley & Sons, Ltd. Accepted 17 October 2001
*Correspondence to: S. S. Tseng, Department of Computer and Information Science, National Chiao-Tung University, Hsinchu 300, Taiwan R.O.C.
yE-mail: sstseng@cis.nctu.edu.tw
services based on high-speed wireless access technologies, such as wireless asynchronous transfer mode (wireless ATM) [1]. In the architecture presented in Reference [1] (as shown in Figure 1), the base station controllers (BSCs) in traditional personal communication service (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 this paper, we address the problem that is currently faced by designers of mobile communication service and in the future, it is likely to be faced by designers of PCS.
In the designing process of PCS network systems, first, the telephone company determined the global service area according to the usage 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 connected 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 communication 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 exceed the capacities provided by the original BSs and switches. Nonetheless, the wireless ATM system must be extended so that the system can provide higher quantity of services to the mobile users. Two methods can be used to extend the capacities of the system and provide higher quantity of services. The first 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 capacity of a system can be increased. In practice, this can be achieved by using cell splitting [2] process. The cell splitting process establishes new BSs at specific points in the cellular pattern and reduces the cell size by a factor of 2 (or more) as shown in Figure 2.
In this paper, we are given a two-level wireless ATM network as shown in Figure 3. In the PCS network of Figure 3, cells are divided into two sets. One is the set of cells, which are built originally, and each cell in this set has been assigned to a switch on the ATM network (e.g. cells c1; c2 are assigned to switch s2; and cells c3; c4; and c5 are assigned to switch s4 in Figure 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 cells and
switches are fixed and known. To simplify the discussion, we assumed that the number of cells and switches are fixed. The problem is to assign new adding and splitting cells in the PCS to switches on the ATM network in an optimum manner. We would like to do the assignment in an attempt to minimize a cost criterion. The cost has two components: one is the cost of handoffs that involve two switches, and the other is the cost of cabling (or trucking) [3–5].
Consider the example shown in Figure 4, 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 s1 will
perform a handoff for this call. This handoff is relatively simple and does not involve any
Figure 2. Cell splitting.
Figure 3. Two-level wireless ATM network.
location update in the databases that record the position of the subscriber. 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 are two types of handoffs: one involves only one switch and the other involves two switches. The handoffs that occur between two cells connected to different switches consume much more network resources (therefore, are much more costly) than those between cells connected to the same switch [3–5]. Based on the discussion given in the previous paragraph, we assume that the cost of handoffs involving only one switch is negligible. Throughout this paper, we assume each cell to be connected to only one switch.
Merchant and Sengupta [5] considered the cell assignment problem which assigns 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 Reference [5], 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 two switches should be considered in calculating the location update cost. In References [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 considering the communication cost between two switches. The problem considered in References [3,4] assume that the connections between cells and switches have not yet been constructed. A three-phase heuristic algorithm and two genetic algorithms are proposed to solve the cell assignment problem in wireless ATM network, respectively. In this paper, we follow the objective function, which was formulated in References [3,4], and assume that the original set of cells in PCS have been assigned to switches on ATM network. The goal is to find an assignment of the new adding and the splitting cells to the switches on the ATM network so that the total cost can be minimized. This problem is defined as extended cell assignment problem.
The organization of this paper is as follows. In Section 2, we formally define the problem. In Sections 3 and 4, we describe details of the heuristic and the simulated annealing algorithms. The experimental results are presented in Section 5. Finally, a conclusion is given in Section 6.
2. PROBLEM FORMULATION
Let CGðC; LÞ be the PCS network, where C is a finite set of cells with jCj ¼ n and L is the set of edges such that L C C: We assume that Cnew[ Cold¼ C; Cnew\ Cold ¼|; Cnewis the set of
new adding and splitting cells where jCnewj ¼ n0; cells in Cnew have not yet been assigned to
switches on the ATM, and Cold the set of original cells where jColdj ¼ n n0: Without loss of
generality, we assume that cells in Coldand Cneware indexed from 1 to n n0and n n0þ 1 to n;
respectively. If cells ci and cj in C are assigned to different switches, then a handoff cost is
incurred. Let fij be the frequency of handoff per unit time that occurs between cells ci and
cj; ði; j ¼ 1; . . . ; nÞ and is fixed and known. We assume that all edges in CG are undirected and
weighted; and assume cells ci and cjin C are connected by an edge ðci; cjÞ 2 L with weight wij;
where wij¼ fijþ fji; wij¼ wji; and wii¼ 0 [3,4]. Let GðS; EÞ be the ATM network, where S is the
connected. We assume that the locations of cells and switches are fixed and known. The topology of the ATM network GðS; EÞ is also fixed and known. Let ðXsk; YskÞ be the co-ordinate
of switch sk; k ¼ 1; 2; . . . ; m; ðXci; YciÞ be the co-ordinate of cell ci; i ¼ 1; 2; . . . ; n; and dklbe the
minimal communication cost between the switches sk and sl: Let lik be the cost of cabling per
unit time and between cell ciswitch sk; ði ¼ 1; . . . ; n; k ¼ 1; . . . ; mÞ and assume likis the function
of Euclidean distance between cell ciand switch sk; that is,
lik¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðXci XskÞ 2þ ðY ci YskÞ 2 q ð1Þ Assume the number of calls that can be handled by each cell per unit time is equal to 1: Let Capk be the number of the remaining cells that can be used to assign cells to switch sk: Our
objective is to assign cells in Cnew to switches so that the total cost (sum of cabling cost and handoffs cost) per unit time of whole system can be minimized.
To formulate this problem, let us define the following variables. Let xik¼ 1 if cell ci2 C is
assigned to switch sk; xik¼ 0; otherwise. Since each cell should be assigned to only one switch,
we have the constraintPmk¼1 xik¼ 1; for i ¼ 1; . . . ; n: Further, the constraint on the capacity is
Xn
i¼n n0þ1
xik4Capk; k ¼1;. . . ; m ð2Þ
Also, the sum of cabling costs is
Xn i¼1
Xm k¼1
likxik ð3Þ
To formulate handoff cost, variables zijk¼ xikxjk; for i; j; ¼ 1; . . . ; n and k ¼ 1; . . . ; m are
defined in Reference [5]. Thus, zijk equals 1 if both cells ci and cj are connected to a common
switch k; otherwise it is zero. Further, let yij¼
Xm k¼1
zijk; i; j ¼ 1; . . . ; n ð4Þ
Thus, yij takes a value of 1 if both cells ci and cj are connected to a common switch and 0
otherwise. With this definition, it is easy to see that the cost of handoffs per unit time is given by Der-Rong Din and Tseng [3,4]
Xn i¼1 Xn j¼1 Xm k¼1 Xm l¼1 wijð1 yijÞxikxjldkl ð5Þ
This, together with our earlier statement about the sum of cabling costs, gives us the objective function [3,4]: minimize : X n i¼1 Xm k¼1 likxikþ a Xn i¼1 Xn j¼1 Xm k¼1 Xm l¼1 wijð1 yijÞxikxjldkl ð6Þ
where a is the ratio of the cost between cabling and handoff costs. The following assumptions will be satisfied:
(1) We assume that the number of cells in Cnewis less or equal toPmk¼1 Capk: That is, there is
(2) The structures and locations of the ATM network and the PCS network are fixed and known.
(3) Each cell in the PCS network will be directly assigned and connected to only one switch in ATM network.
(4) To simplify the discussion, we assume that Capk> 0; for k ¼ 1; . . . ; m:
Example 1
Consider the two graphs shown in Figure 3. There are 14 cells in CG which should be assigned to four switches in S: In CG; cells are divided into two sets, one is the set Cold of cells which are
built originally, and cells in Cold have been assigned to switches in the ATM network (e.g.
fc1; c2; c3; c4; c5g in Figure 3). The other is the set Cnew of cells which are new adding cells (e.g.
fc6; c7; c8gÞ or splitting cells (e.g. fc9; c10; c11; c12; c13; c14gÞ: The edge weight between two cells is
the frequency of handoffs per unit time that occurs between them. Four switches are positioned at the centre of cells c1; c2; c4; and c6: Assume that the matrix CS of the distance between cells
and switches is CS ¼ flikg144¼ s1 s2 s3 s4 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 0 1 3 pffiffiffi3 1 0 2 1 2 1 1 2 ffiffiffi 3 p 1 pffiffiffi3 0 ffiffiffi 7 p ffiffiffi 3 p 1 1 3 2 0 pffiffiffi3 4 3 1 pffiffiffi7 ffiffiffiffiffi 13 p ffiffiffi 7 p 1 2 ffiffiffi 3 p =3 2pffiffiffi3=3 2pffiffiffiffiffi21=3 pffiffiffiffiffi37=4 ffiffiffiffiffi 21 p =6 pffiffiffiffiffi21=6 pffiffiffiffiffiffiffiffi237=6 pffiffiffiffiffi17=4 2pffiffiffi3=3 pffiffiffi3=3 pffiffiffiffiffi39=3 pffiffiffi3=3 ffiffiffiffiffiffiffiffi 155 p =12 pffiffiffiffiffiffiffiffi219=12 pffiffiffiffiffiffiffiffiffiffi1115=12 pffiffiffiffiffiffiffiffi203=12 ffiffiffiffiffiffiffiffi 219 p =12 pffiffiffiffiffiffiffiffi155=12 pffiffiffiffiffiffiffiffi795=12 5pffiffiffi3=12 ffiffiffiffiffi 21 p =3 pffiffiffiffiffi21=3 5pffiffiffi3=3 2pffiffiffi3=3 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 6 6 6 6 6 6 6 6 4 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 7 7 7 7 7 7 7 7 5 3. HEURISTIC ALGORITHM
It is well known that finding an optimal solution to this problem is NP-hard. In this section, a two-phase heuristicalgorithm described below is proposed to solve this problem.
(1) Initial assigning phase: Construct an initial assignment of cells in Cnewby considering some
(2) Cell exchanging phase: Improve the initial assignment by exchanging cells in different switches.
Initial assigning phase. First, we introduce some notations required in the following. Given m sets of cells Pl; l ¼ 1; 2; . . . ; m; we assume P1[ P2[ [ Pm¼ Cold and Pi\ Pj¼ f; where
i=j; i; j ¼ 1; 2; . . . ; m: For example, in Figure 3, P2¼ fc1; c2g; P4 ¼ fc3; c4; c5g; P3¼ P1¼|:
Without loss of generality, we assume that the cells in set Pj is assigned to the switch sj; j ¼
1; 2;. . . ; m; and this definition can be extended to include the set Cnew: Let sidðc
iÞ ¼ l if ci is in
Pl; l is called the sid of cell ci: Let LUCSði; lÞ ¼Pcj2Plwijbe the sum of the location update costs
between cell ci and all cells in switch sk; and Dði; lÞ ¼ LUCSði; lÞ LUCSði; sidðciÞÞ: Therefore,
for a given partition P ; the location update (handoff) cost of the partition is aX
ci2C
X
sl2S
ðLUCSði; lÞ dsidðciÞlÞ ð7Þ
For example, as seen in Figure 3, c12 P2and c22 P2; thus sidðc1Þ ¼ sidðc2Þ ¼ 2; LUCSð2; 2Þ ¼
P
cj2P2 w2j¼ w21þ w22¼ 4 þ 0 ¼ 4; LUCSð2; 4Þ ¼
P
cj2P4 w2j¼ w23þ w24þ w25¼ 3 þ 8 þ 0 ¼ 11:
Dð2; 4Þ ¼ LUCSð2; 4Þ LUCSð2; 2Þ ¼ 11 4 ¼ 7:
To evaluate the effect of cell ci in Cnew being assigned to switch sk; we must compute the
cabling and location update costs derived from this event. By the definition described above, the cabling cost is lik: The location update cost has two components, one is the location update costs
between cells ci and cells in Cold; the other is the location update costs between cell ciand the
other cells in Cnew: Since cells in Cold have been assigned to switches on the ATM, if cell c
i in
Cnewis assigned to sk; the location update cost between the cell ciand the cell cjin Coldis fixed
and can be computed by Aik¼ a
X
sl2S; lak
ðLUCSði; lÞ dklÞ;
ði ¼ n n0þ 1; n n0þ 2; . . . ; n; k ¼ 1; 2; . . . ; mÞ ð8Þ Consider the example shown in Figure 3, if c6 is assigned to s3 and a ¼ 1; then A63¼
LUCSð6; 1Þ d31þLUCSð6; 2Þ d32þLUCSð6; 4Þ d34¼0 30 þ 0 20 þ ð2 þ 4Þ 30 ¼ 180:
Since cells in Cnewhave not yet been assigned to switches, the location update cost between cell
ciand the other cells in Cnewcannot be computed by a determinative formula. To estimate the
location update cost, let avg DISTk ¼
Pm
l¼1 dkl=ðm 1Þ be the average distance between switch
sk and the other switch, avg LUi¼
Pn0
j¼1 wij=n be the average location update cost between ci
and the other cell cjin Cnew: It is worth noting that if two cells are assigned to the same switch,
then the handoff cost between these two cells is ignored. If cell ciin Cnewis assigned to the switch
sk and the capacity of switch sk is Capk; i.e. if all cells are assigned to switches, then at most
n0 CAPk cells should be computed in considering the location update cost. Let NLi be the
number of cells in Cnew which the frequency of handoff between c
i and cj is greater than zero.
The total location update cost between ci which assigned to sk and the other cells in Cnew
assigned to other switches can be estimated by Bik¼
a ðn0 CAPkÞ avg LUi avg DISTk if Capk4NLi
0 otherwise
(
ð9Þ If Capk> NLi; then NLi cells can be assigned to the same switch sk; that is, Bik is set to 0:
For example, if c62 Cnew is assigned to switch s3 in Figure 3, then avg DIST3¼ 23:33;
avg LU6¼ ðw67þ w68Þ=9 ¼ 0:556: Assume Cap3¼ 2; at most 7 cells in Cnewshould be computed
by considering the location update cost with cell c6: Thus, B63¼ a 7 0:556 23:33 ¼ 90:728:
After computing the cabling and the location update costs between cells, it is time to assign cells in Cnewto switches according to these costs. First, we transformed the original problem to a
minimal weighted matching problem. Then the famous and efficient algorithms for the minimal weighted matching problem in the literature can be used to assign cells to switches. The transformation process is shown as follows. First, a bipartite graph BGðCnew; ScapÞ is constructed; let Scap¼ fs11; s12; . . . ; s1cap1; s21; s22; . . . ; s2cap2; . . . ; sk1; sk2; . . . ; skcapk; . . . ; sm1; sm2; . . . ; smcapmg; i.e.
for each switch skin S; Capknodes, sk1; sk2; . . . ; skcapk; are constructed and assume that each node
in Scapcan only be assigned one cell. Let Cnewbe the set of cells that have not yet been assigned to switches. Then, for each cell in Cnew; there is one-edge connected cell to each node in Scap: The
weight of edge which connected cell ci in Cnewto skp2 Scap; p ¼ 1; 2; . . . ; Capk is set to be
likþ Aikþ Bik ð10Þ
The transforming result of Example 1 is shown in Figure 5.
Assume Cap1¼ 4; Cap2¼ 2; Cap3¼ 4; and Cap4 ¼ 1: After transformation, the resulting
bipartite graph of Example 1 is shown in Figure 5. Two dummy cells c15 and c16 are introduced
into the bipartite graph. For example, the weight of edge which connected cell c6 to s3 * (e.g.
s31; s32) is l63þ A6þ B6¼ 0 þ 180 þ 90:728 ¼ 270:728: Obviously, the subproblem can be
formulated as a minimal weighted matching problem on bipartite graph, which is known as the assignment problem [6]. Therefore, Hitchcoch Algorithm [6] can be applied to find the optimal solution of assignment problem in OðmaxfjCnewj; jScapjg3Þ ¼ Oðmaxfn0;Pm
k¼1 Capkg3Þ ¼
OððPmk¼1 CapkÞ3Þ time.
Cell exchanging phase. The goal of cell exchanging phase is to select two cells in Cnewwhich have been assigned to different switches and exchange them in order to reduce the total cost. The basic idea of cell exchanging phase is borrowed from Kernighan–Lin [7] algorithm which only works on traditional 2-way graph partition problems. In our two-level cell-switch wireless ATM network environment, location update and cabling costs must be considered simultaneously. Hence, we modified Kernighan–Lin’s algorithm to exchange cells in different switches by selecting the ‘most preferable’ cells to exchange instead of arbitrarily exchange two cells.
We have the following lemmas: Lemma 1[4]
Ignoring the capacity restriction, if cell ci2 Cnew in k is moved to l; then
(1) the cabling cost is reduced by dik dil;
(2) the location update cost is reduced by aðLUCSði; lÞ LUCSði; kÞÞ dkl:
If cell ci and cj are exchanged, the gain can be computed from the following lemma:
Lemma 2[4]
If cell ci2 Cnew in switch sk and cj2 Cnewin switch slare exchanged, then
(1) The reduced cabling cost is: RCði; jÞ ¼ dik dilþ djl djk
(2) The reduced location update cost is
RLUði; jÞ ¼ ðDði; lÞ Dðj; kÞ 2wijÞ dkl a þ ðdkz dlzÞ
a ðLUCSði; sÞ LUCSðj; zÞÞ (3) The reduced total cost is: exchangeði; jÞ ¼ RCði; jÞ þ RLUði; jÞ:
After cells ci2 Cnewand cj2 Cnewhave been selected and exchanged, the matrix LUCS should
be updated so that the algorithm can run effectively and avoid sequential searching. Lemma 3[4]
After cells ci2 Cnewin switch sk and cj2 Cnewin switch slare exchanged, the LUCS values can
be updated as follows:
(1) LUCSnewði; kÞ ¼ LUCSði; kÞ þ wij
(2) LUCSnewði; lÞ ¼ LUCSði; lÞ wij
(3) LUCSnewðj; kÞ ¼ LUCSðj; kÞ wij
(4) LUCSnewðj; lÞ ¼ LUCSðj; lÞ þ wij
For any cell a in switch sz
(5) LUCSnewða; kÞ ¼ LUCSða; kÞ þ waj wai; if ðsz¼ kÞ and ða=iÞ
(6) LUCSnewða; lÞ ¼ LUCSða; lÞ wajþ wai; if ðsz¼ kÞ and ða=iÞ
(7) LUCSnewða; kÞ ¼ LUCSða; kÞ þ waj wai; if ðsz¼ lÞ and ða=jÞ
(8) LUCSnewða; lÞ ¼ LUCSða; lÞ wajþ wai; if ðsz¼ lÞ and ða=jÞ
(9) LUCSnewða; kÞ ¼ LUCSða; kÞ þ waj wai; if ðsz=kand sz=lÞand ða=i and a=jÞ
(10) LUCSnewða; lÞ ¼ LUCSða; lÞ wajþ wai; if ðsz=kand sz=lÞand ða=i and a=jÞ
Given an initial assignment, the total cost can be reduced by reassigning a cell in current switch to another switch, or exchanging two cells which can be assigned to different switches. A set DM of Pmk¼1Capk jCnewj dummy cells are introduced into the CG to ensure that
reassigning one cell in one step is possible. Cell exchanging phase consecutively selects two cells in Cnew[ DM which can be assigned to different switches to exchange. At each iteration, two
cells caand cb 2 Cnew[ DM are selected which maximize the reduced exchanging cost exchange
ða; bÞ where
exchangeða; bÞ ¼ max
ðci;cjÞ2fCnew[DMgfCnew[DMg
exchangeði; jÞ ð11Þ The iteration continues if exchangeða; bÞ50:
Algorithm: Cell exchanging
Step 1: For each cell in CG [ DM and each switch in G; compute values of matrices LUCS ði; lÞ and Dði; lÞ: Let ¼ Cnew[ DM:
Step 2: Find two cells ca and cb in B assigned different switches such that
exchangeða; bÞ ¼ max
ðci;cjÞ2fCnew[DMgfCnew[DMg
exchangeði; jÞ
Step 3: If exchangeða; bÞ > 0; then exchange cells ca and cb; and delete ca and cb from B:
Step 4: Update LUCSði; lÞ and Dði; lÞ for each ci2 B and sl in L:
Step 5: If B is go to Step 2.
Step 6: If exchangeðca; cbÞ > 0 then go to Step 1; otherwise terminate the algorithm.
4. SIMULATED ANNEALING ALGORITHM
Simulated annealing is a stochastic computational technique derived from statistical mechanics for finding near globally-minimum-cost solutions to large optimization problems. Kirkpatrick et al. [8] were the first to propose and demonstrate the application of simulation techniques from statistical physics to the problem of combinatorial optimization. Due to the complexity of the extended cell assignment problem in a two-level wireless ATM network, the provision of an optimal solution in reasonable time is not guaranteed. In this respect, the usual step is to devise an approximate algorithm for solving this problem. The simulated annealing (SA) technique is applied to solve the extended cell assignment problem in this section.
In the design of simulated annealing algorithm, if the traditional-SA approach is used to solve the extended cell assignment problem, it may generate a significant number of configurations, but only a small fraction of these are indeed constraint-satisfied (10 per cent or 20 per cent). Thus, the performance of the traditional-SA algorithm is not promising. In this paper, we attempt to develop an enhanced-SA approach to solve the extended cell assignment problem by generating configurations, which are constraint-satisfied. The key elements in a simulated annealing algorithm are a cost function, a configuration space, a perturbation mechanism, and a cooling schedule. In our case, the solution method is shown as follows.
4.1. Configuration space and perturbation mechanism
The objective of the extended cell assignment problem in a two-level wireless ATM network is to find an optimal assignment of new adding and splitting cells to switches so that the object function value is minimized. To do this, the configuration space is designed to be the set of possible solutions, which is defined as a binaryPmk¼1 Capk m matrix X : Define xikin X to be 1
if the cell ci is assigned to switch sk; xik¼ 0; otherwise. Consider a possible assignment of the
example shown in Figure 3, the configuration matrix of the assignment is shown in Figure 6(a). In this example, two dummy cells c15 and c16 are introduced into the configuration matrix
(configuration).
It is worth noting that the configuration can be divided into two parts, fixed and variable parts. The first part of matrix, which represents the assigning status of cells in Cold; is fixed in the
running of SA. Thus, the first part of the matrix can be ignored since it is never changed during experiments. For the purpose of easy understanding, the fixed part of the matrix is still kept in configuration in the rest of the paper.
If the initial configuration is randomly generated or new configuration is formed by changing the assigning status of cells by way of randomly choosing cells and switches, then, there is a large chance that the generated configuration is not a constraint-satisfied one. To avoid this, algorithms, which are guaranteed to generate constraint-satisfied configurations and perturba-tion schemes, must be constructed. To generate the constraint-satisfied initial configuraperturba-tion, we propose the following algorithm:
Algorithm: Initial configuration generating algorithm(ICGA) Step 1. Let A ¼ S; B ¼ Cnew[ DM:
Step 2. Repeat Steps 2.1, 2.2 and 2.3 until B is empty. Step 2.1. Randomly select a switch sa from A:
Step 2.2. Randomly assign jCapkj cells in B to switch sa and remove these cells form B:
Step 2.3. Remove sa from A:
In the traditional-SA algorithm, the search process may get stuck at a local minimum due to the small change moves, particularly as the barrier is high and the temperature is low. Hence, we introduce an idea of large perturbation schema whose function is the same as the mutation operation in genetic algorithm. The main idea of large perturbation schema is to leap over the barrier during a search and to explore another region of the search space. This can be achieved by applying a certain number of moves, consecutively, special problem domain perturbations, or local search heuristic perturbation. Three types of perturbations are introduced to the enhanced-SA algorithm, which are shown as follows.
* Cells exchanging schema: First, the cells exchanging schema randomly selects two cells c i
and cjin Cnew[ DM; which have been assigned to different switches skand sl; respectively.
Then, exchange the cell ci to the switch sl and the cell cj to the switch sk: For example,
assume cells c6 and c10 are randomly selected; after performing cells exchanging
perturbation, the configuration matrix is shown in Figure 6(b).
* Multiple cells exchanging schema: First, the multiple cells exchanging schema randomly
selects two switches sk and slfrom S: Then, reassigns cells in the switch sk to the switch sl
and vice versa. Since the original configuration is constraint-satisfied, after perturbation,
Figure 6. Possible feasible configuration for Example 1; (b) Cell exchanging for Example 1 and (c) Multiple cells exchanging for Example 1:
the resulting configuration must be a feasible one. If we reassign cells in two switches directly, the resulting configuration may violate the constraints. It can be proven that several cases need to be considered as shown in Table I (contradiction means that this case will not appear). As seen in Table I; we found that the following strategies can be used to guarantee that the final configuration is constraint-satisfied. Let COSk be the set of cells
which are currently assigned to switch sk: In case 1, if nl> Capk and nk5Capl; then Capk
cells are randomly selected from the set COSland reassigned to switch sk; at the same time,
all cells in COSkare reassigned to the switch sl: In case 2, if nk> Capland nl5Capk; Capl
cells are randomly selected from the set COSkand reassigned to the switch sk; at the same
time, all cells in COSkare reassigned to the switch sl: Otherwise, all cells being assigned to
the switch sk are directly reassigned to the switch sl and vice versa. For example, assume
switches s1 and s2 are randomly selected; after performing multiple cells exchanging, the
configuration matrix is shown in Figure 6 (c) (assume cells c6and c7are randomly selected
from the switch s1).
* Local search heuristic schema: The local search heuristic schema used here is trying to find
the local optimum by exchanging cells in different switches. The cell exchanging heuristic described in Section 4.2 can be used to find the local optimum of current configuration. Algorithm: Local Search heuristic perturbation (LSHP)
Step 1: For each cell in CG and each switch in G; compute values of matrices LUCSði; lÞ and Dði; lÞ: Let B ¼ Cnew[ DM :
Step 2: Find two cells ca and cb in B in different switches such that
exchangeða; bÞ ¼ max
ðci;cjÞ2BB
exchangeði; jÞ Step 3: If exchangeða; bÞ > 0; then exchange cells ca and cb:
It is worth noting that if the probabilistic decision mechanism of SA is disabled, and three perturbation schemes are used to perturb the initial configuration to generate a new one, then, it is easy to prove that all possible feasible configurations can be reached by applying a sequence of perturbations. In the experiment, let p1; p2; and p3 be the probabilities of transforming current
configuration to a new one by applying cell exchanging schema, multiple cells exchanging schema, and local search heuristic perturbation, respectively. We assume p1þ p2þ p3¼ 1
and the values of p1; p2; and p3 will be empirically determined and described in later
sections.
4.2. Cooling schedule
One of the most important problems involved in the simulated annealing algorithm implementation is the definition of a proper cooling schedule, which is based on the choice of
Table I. Nine cases for consideration in multiple cells exchanging.
nk> Capl nk¼ Capl nk5Capl
nl> Capk Contradiction Contradiction Case 1
nl¼ Capk Contradiction Directed exchange Directed exchange
the following parameters: starting temperature, final temperature, length of Markov chains, the way of decreasing temperature. A correct choice of these parameters is crucial because the performances of the algorithm strongly depend on it. In the literature many cooling schedules are maintained [9]. These cooling schedules lead to a polynomial time execution of the simulated annealing, but it cannot give any guarantee for the deviation in cost between the final solution obtained by the algorithm and the optimal cost [9,10]. The different parameters of the cooling schedule are determined based on the statistics calculated during the search. In the following we describe these parameters.
Initial value of the control parameter: The rule used in our enhanced-SA is that the starting temperature c0 is determined by calculating the average increase in cost, DCþ; for 50 random
transitions and solve c0from c0¼ DCþ=lnðw 10 Þ; where accepted ratio w0is defined as the number
of accepted transitions divided by the number of proposed transitions. In this paper, the accepted ratio w0 is empirically set to 0:99:
Decrement of the control parameter: The decreasing rate of the temperature needs to be small enough to reach thermal equilibrium for each temperature value. As the temperature is decreased, the accepted ratio is lowered. When no solution that increases the objective function can be found, the system is ‘frozen’ and converged to a certain solution. The speed of coverage of the simulated annealing algorithm depended on the decreasing rate of the temperature and the length of the Markov chain. As mentioned in Reference [9], the decrement is chosen such that small Markov chain lengths suffice to re-establish quasi-equilibrium after the decrement of the temperature is applied. The decrement rule in enhanced-SA is defined as follows: Tkþ1¼ gTk;
where g is empirically set to 0:99:
The final value of the control parameter: The iterative procedure is terminated when there is no significant improvement in the solution after a pre-specified number of iterations. It can also be terminated when the maximum number of iterations is reached.
The length of Markov chains: In References [9,10], it is concluded that the decrement function of the control parameter requires only a ‘small’ number of trial solutions to rapidly approach the stationary distribution for a given next value of the control parameter. In general, a chain length of more than 100 transitions is reasonable. In this paper, the chain length is empirically set to a value of n:
Enhanced-SA algorithm of extended cell assignment problem: The details of the simulated annealing is described as follows:
Algorithm: Enhanced-SA algorithm
Step 1. For a given initial temperature T ; perform Initial Configuration Generating Algorithm to generate initial configuration IC. The currently best configuration (CBC) is IC, i.e. CBC ¼ IC; and the current temperature value (CT) is T ; i.e. CT ¼ T :
Step 2. If CT ¼ 0 or the stop criterion is satisfied then go to Step 7.
Step 3. Generate a random number p in ½0; 1Þ; if p4p1 then new configuration (NC) is
generated by applying cells exchanging schema; if p15p4p1þ p2 then NC is generated by
applying multiple cells exchanging schema; otherwise NC is generated by applying local search heuristicschema.
Step 4. The difference of the costs of the two configurations, CBC and NC is computed, i.e. DC ¼ EðCBCÞ EðNCÞ:
Step 5. If DC50 then the new configuration NC becomes the currently best configuration, i.e. CBC ¼ NC: Otherwise, if e ðDC=CTÞ> random½0; 1Þ; the new configuration NC becomes the currently best configuration, i.e. CBC ¼ NC: Otherwise, go to Step 2.
Step 6. The cooling schedule is applied, in order to calculate the new current temperature value CT and go to Step 1.
Step 7. End.
5. EXPERIMENTAL RESULTS
From the previous sections, a heuristic algorithm and a simulated annealing algorithm are proposed to solve the extended cell assignment problem. In order to evaluate their performances, we have implemented the algorithms and applied them to a number of examples with randomly positioned cells and switches. The results of these experiments are reported below. For all 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 are configured as an H-mesh. The handoff frequency fijof two
cells was generated by a normal random number generator with mean 100 and variance 20: To examine effects of different number of cells, Cell Graph CG with n ¼ 50; 100; 150; and 200 cells were tested. jCnewj ¼ 3n=4; jColdj ¼ n=4; m ¼ 10; a ¼ 1 and the Cap=n value of each
problem is 0:2: The proposed heuristicalgorithm consists of two phases (initial assigning phase and cell-exchanging phase), termed EAA, which is a heuristic approach to a rather complex problem. To measure the performance of each phase of the EAA algorithm, we constructed a two-phase heuristicalgorithm, termed NSF, as the reference. The first phase of the NSF algorithm is assigning each cell to the nearest switch; if the nearest switch is full then find the next nearest switch. The second phase of the NSF algorithm is the cell exchanging phase same as in EEA. For all experiments, the CPU time in seconds of the heuristic algorithm and the objective cost reduction ratio are two major concerns. First, the cost reduction ratio of the cell exchanging phase is evaluated. To test the effect of cell exchanging phase, we compare the costs of the algorithm by running this phase. Let CA; and CB be the costs resulting from running cell
initial assigning and cell exchanging phases, respectively. The cost improvement ratio ðCA
CBÞ=CA is shown in Table II. As seen in Table II; after running cell exchanging phase, the total
cost of EAA algorithm is reduced by 8:4 per cent on average.
To know the efficiency of the enhanced-SA algorithm (ESA) to the traditional-SA algorithm, we also implement a traditional-SA algorithm that does not guarantee to generate constraint-satisfied configurations. The experimental results shown in Figure 7 explain that the enhanced-SA algorithm have better performance than the traditional-enhanced-SA algorithm. In other words, constraint-satisfied configurations and perturbation schemes of enhanced-SA algorithm are indeed more efficient than the traditional-SA algorithm and enhanced-SA algorithm has better convergent behaviours.
Table II. Cost reduction ratio of cell exchanging phase.
# of cells NSF (%) EEA (%) 50 4.2 6.5 100 3.2 9.3 150 3.7 9.7 200 3.1 7.9 Average 3.6 8.4
To evaluate the effect of the probabilities of different perturbation schemes of enhanced-SA algorithm described in Section 5, we test the enhanced-SA algorithm with different values of probabilities p1; p2 and p3: The experiments consist of three parts:
(1) Assume p3 is fixed, the effect of different values of p1 and p2 is tested,
(2) Assume p1 is fixed, the effect of different values of p2 and p3 is tested,
(3) Assume p2 is fixed, the effect of different values of p1 and p3 is tested.
In the first part of the experiments, assume p3¼ 0; p2¼ 1 p1; and the value of p1 is in
f1:00; 0:99; 0:95; 0:90; 0:01; 0:00g: Figures 8 and 9 show the results of the experiment, where x-axis represents the number of acceptances when the configuration is perturbed in enhanced-SA algorithm and y-axis represents the total cost of the problem instance. When p1¼ 1:00
(p2¼ 0:00), i.e. the multiple cells exchanging perturbation does not activate in the experiment.
We found that the enhanced-SA algorithm converges very slowly and traps into local minima, since the cell exchanging perturbation only exchanges two cells at one time. When p1¼
0:99; p1¼ 0:95; or p1¼ 0:90; that is, only a small chance that the multiple cells exchanging
perturbation may activate in the experiment, the enhanced-SA algorithm converges faster than
Figure 7. Comparison of the result of traditional- and enhanced-SA algorithms.
the case with p1¼ 0 and has very good performance as shown in Figure 8. When p1¼ 0:00 or
p1¼ 0:01; the single cell exchanging perturbation does not activate or has very small chance to
activate. As seen in Figure 9, the current configuration up and down rapidly and hardly converges to the global minima. Thus, we can conclude that if the probability of cells exchanging perturbation is higher (p1¼ 0:90–0:98), the enhanced-SA algorithm has very good
performance.
In the second part of experiments, assume p2is fixed and is equal to 0:01; the value of p1is in
f0:99; 0:98; 0:97; 0:96; 0:95g; and p3¼ 0:99 p1: The experimental results of the running time
and the total cost of different problem instance are shown in Figures 10(a) and (b), respectively. As seen in Figure 10, we found that the execution time of ESA is increasing at a higher rate as the value of p3 increases; but the result of total cost does not reach the best.
In the third part of the experiments, assume p1is fixed and is equal to 0:95; the value of p2is
in f0:00; 0:01; 0:02; 0:03; 0:04; 0:05g; and p3¼ 0:05 p1: The experimental results of the running
time and the total cost of different problem instance are shown in Figures 11(a) and (b), respectively. As seen in Figure 11, we found that the execution time of ESA is increasing at a higher rate as the value of p3 increases; but the result of total cost does not reach the best.
Therefore, we can conclude that the activation probabilities of multiple cells exchanging perturbation and local search heuristic perturbation do effect the result of ESA algorithm. If the values of p2 and p3are kept smaller (0.01 or 0.02) then ESA algorithm has better performance
and behaviour.
Figure 9. Comparison of the convergence of different probabilities of p1and p2:
The results of the executions of the NSF, the EEA and the ESA for the above networks are given in Table III. Since the NSF and the EEA are deterministic algorithms, the corresponding columns show the solutions found in the single run of the NSF and the EEA on each example problem. However, the ESA was run with 50 random seeds on each problem in order to get some statistical information about the quality of their solutions. The ESA column of Table III shows the minimum, mode, average, maximum, and standard deviation of the cost of solutions for 50 runs. The most repeated solution or mode is close to the minimum cost solution for all example problems. Take the minimum of the total cost of EAA as a reference, we compute the result of the EAA to the ESA, the ratio is 103 per cent on average.
The solutions of NSF, EEA, and ESA (with probabilities p1¼ 0:98; p2¼ p3 ¼ 0:01) are
shown on the bar chart of Figure 12(a). As seen in Figure 12(a), ESA’s advantage for the quality of solutions increases as the network gets larger. Run times of a single run of the NSF, the EEA, and the ESA for five example problems are compared in Figure 12(b). The ESA algorithm is
Figure 11. Comparison of the different probabilities of p2and p3: (a) Running time and (b) Total cost.
Table III. Comparison of NSA, EEA, and ESA solutions.
NSF EAA ESA
jnj jmj Cap Min. Ratio (%) Min. Ratio (%) Min. Mode Avg. Max. Std. Dev. 50 10 10 2676.4 107 2552.5 103 2476.7 2478.2 2480.4 2485.7 352.26 100 10 20 6516.5 108 6154.7 102 6011.3 6026.9 6022.9 6035.7 1141.1 150 10 30 14 806.2 104 14 823.4 104 14 219.9 14 236.8 14 237.0 14 257.0 4266.5 200 10 40 24089.9 109 22740.7 103 22052.6 22075.8 22075.9 22099.8 7289.7
Figure 12. Comparison of the NSF, EEA and ESA solutions for 5 different example problems. (a) Total cost and (b) Running time.
quicker than the NSF and EEA algorithms for all these five examples. Moreover, the execution times of the NSF and EAA increase at a higher rate than that of the ESA algorithm as the cell size increases. The ESA algorithm is superior to the NSF and EAA algorithms in both run time and quality.
6. CONCLUSIONS
In this paper, we have investigated the extended cell assignment problem which optimally assigned new adding and splitting cells in PCS to switches in a wireless ATM network. Given cells in PCS and switches on ATM network (whose locations are fixed and known), we would like to do the assignment in as attempt to minimize a cost criterion. 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 modelled as a complex integer programming problem and it is easy to recognize that finding an optimal solution of this problem is NP-hard.
A heuristicalgorithm EAA (extended assignment algorithm) and a simulated annealing algorithm ESA (enhanced-SA) are proposed to solve this problem. The heuristic algorithm EEA consists of two phases, initial assignment phase, and cell exchanging phase. First, in the initial assigning phase, the initial assignments of cells to switches are found. Then, these assignments are improved by performing cell exchanging phase, which repeatedly exchanges two cells in different switches with great reduction of the total cost. Experimental results indicate that EEA algorithm has good performance and shows that cell exchanging phase can really reduce total cost near 8:4 per cent on average.
Owing to the inability of SA to generate solutions that always satisfy all the constraints, the performance of a traditional-SA approach is not so promising. The SA technique is, however, easy to implement, requires little expert knowledge and is not memory intensive. Hence, in this paper, we have developed an enhanced-SA algorithm to solve the extended cell assignment problem. The enhanced-SA algorithm constructs constraint-satisfied configurations and perturbation mechanism to ensure that the candidate configurations produced are constraint-satisfied. The performance of the enhanced-SA algorithm is demonstrated through simulation. The results are compared with previous reported solution method and indicate that the proposed algorithm runs efficiently.
ACKNOWLEDGEMENTS
This work was supported in part by the MOE Program of Excellence Research under Grant 90-E-FA04-1-4.
REFERENCES
1. Cheng M, Rajagopalan S, Chang LF, Pollini GP, Barton M. PCS mobility support over fixed ATM networks. IEEE Communication Magazine 1997; 18(2):82–91.
2. Macario RCV. Cellular Radio. McGraw-Hill: New York, 1993.
3. Der-Rong Din, Tseng SS. GeneticAlgorithms for Optimal design of two-level wireless ATM network. Proceeding of NSC 2001; 25(3):151–162.
4. Der-Rong Din, Tseng SS. Heuristicalgorithm for optimal design of two-level wireless ATM network. Journal of Information Science Engineering 2001; 17:674–665.
5. Merchant A, Sengupta B. Assignment of cells to switches in PCS networks. IEEE/ACM Transaction on Networking 1995; 3(5):521–526.
6. Kuhn HW. The Hungarian method for the assignment problem. Naval Research Logistics Quarterly 1955; 2:83–97. 7. Kernighan BW, Lin S. An efficient heuristic procedure for partitioning graphs. Bell System Technique Journal 1970;
49:291–307.
8. Kirkpatrick S, Gelatt CD, Vecchi MP. Optimization by simulated annealing. Science 1983; 220:671–680. 9. Laarhoven V, Arts E. Simulated Annealing: Theory and Application. D. Reidel Publishing Company: Holland, 1987. 10. Arts E, Korst J. Simulated Annealing and Boltzman Machines. Wiley: New York, 1989.
AUTHORS’ BIOGRAPHIES
Der-Rong Din received his BS degree in Computer Science from Chinese Culture University in 1991; the MS and the PhD in Computer Science from National Chiao-Tung University in 1993 and 2001, respectively. Now he is on the faculty of Department of Computer Science and Information Management and the Director of the Computer Center at Hung-Kuang Institute of Technology. His current research interests are in mobile communication, WDM networking, parallel compiler and algorithms.
Shian-Shyong Tseng received his PhD 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 National 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 Chairman of Taiwan Academic Network (TANet) management committee. In December 1999, he founded Taiwan Network Information Center (TWNIC) and is now the Chairman of the board of directors of TWNIC. His currently research interests include parallel processing, expert systems, computer algorithm, and internet-based application.
