A Beam Search Method to Solve the Problem of Assignment Cells to Switches in a Cellular Mobile Network

Transcription

1 A Bea Search Method to Solve the Proble of Assignent Cells to Switches in a Cellular Mobile Networ Cassilda Maria Ribeiro Faculdade de Engenharia de Guaratinguetá - DMA UNESP - São Paulo State University Av. Ariberto Pereira da Cunha 333. Pedregulho Guaratinguetá SP CEP: BRAZIL cassilda@feg.unesp.br Aníbal Tavares Azevedo Faculdade de Engenharia de Guaratinguetá - DMA UNESP - São Paulo State University Av. Ariberto Pereira da Cunha 333. Pedregulho Guaratinguetá SP CEP: BRAZIL anibal@feg.unesp.br Abstract: - Assigning cells to switches in a cellular obile networ is nown as an NP-hard optiization proble. This eans that the alternative for the solution of this type of proble is the use of heuristic ethods, because they allow the discovery of a good solution in a very satisfactory coputational tie. This paper proposes a Bea Search ethod to solve the proble of assignent cell in cellular obile networs. Soe odifications in this algorith are also presented, which allows its parallel application. Coputational results obtained fro several tests confir the effectiveness of this approach and provide good solutions for large scale probles. Key-Words: - Cobinatorial Optiization. Assignent Proble. Bea Search Method. Cellular Networ. Quadratic Integer Prograing.. Introduction The prodigious growing of telecounications in recent ties aes it ore present in odern society. The popularization of one of the ost fantastic equipents of all ties, the obile phone is a ey factor for the copetition between the operators, leading to a search for a ore efficient cellular obile networ, and with capacity to attend the increasingly deand with ore and ore quality. In a cellular obile networ the services are offered in an area called cover zone. The area of coverage is often divided into sall geographic area called cells and each cell is responsible for the coverage (service availability) of a certain nuber of assignents. Therefore, we could say that the cells are basic units of the cellular obile networ. In each cell there is a base station controller, which is used to counicate with the sae cells subscribers. Furtherore, each cell is connected to special units called switches, which are located in obile switching center (MSC), which is responsible to ae the counication between any pair of cells possible. The cells have, due to coputational reasons, the hexagonal forat, which is siilar to a honeycob structure. Each obile switching center (MSC) has a capacity for a certain nuber of assignents. So, any cells can be connected to the sae MSC, respecting the capacity of each switch. However, one cell cannot be connected to ore than one MSC at the sae tie. Figure shows an exaple where the cells A and B are connected to the MSC and the cells C and D are connected to the MSC 2. MSC A B C D MSC 2 Fig.: Cellular obile networ cell division. The base station installed in each cell use radio channels to ae the assignents between the cells and to avoid interference; two adjacent cells use a group of different radio channels. When the counication user oves fro one cell to another, the base station of the new cell is due to aintain the counication of this user; therefore it has to allocate a new radio channel. The counication transfer fro a cell to another is called handoff. The ISSN: ISBN:

2 user transfer echanis between two base stations (cells) occurs when the signal level received by the user is too low. There are two types of handoff. In the exaple of figure, when the user oves fro cell A to B, the handoff is called soft handoff, because the two cells are connected to the sae obile switching center, and the induced cost for transferring fro A to B is wea. On the other hand, when the user oves fro cell A to cell C, the handoff is called coplex. The induced cost for this transfer is high, because the two switching center ( and 2) should be active during the handoff process, and the database containing inforation about the assignent should be updated. In this paper, a Bea Search algorith to solve the Proble of Assigning Cell to Switches in Cellular Mobile Networ is presented. Bea Search is a heuristic ethod for solving optiization probles. It is a branch and bound ethod adaptation, which only soe nodes are evaluated in the search tree. In sections 2 and 3 we present the proble and its atheatical forulation. In the section 4 we present the algorith developed. Sections 5 e 6 shows respectively the results obtained and the conclusions. 2. The Assigning Cell Proble One of the ain tass about planning cellular obile networ is the optial assignent of cells to switches, i.e., find an assignent of cells to switches that respect soe constraint and iniize the total cost of the operation. With a decrease of the total cost of the operation, two factors should be considered. The first one is the cost of cabling, i.e. the lin cost. This cost depends on the distance between the cell and its switching center. The second factor is the handoff cost aong cells. The proble of assignent n cells to switches in a cellular obile networ is a NP-hard proble [], which eans that there is no exact ethod able to solve this proble in polynoial tie. Only the exhaustive search would guarantee the optial solution for this proble. But, to ae this possible, it is necessary to exaine all possible solutions, which is really not viable, because the nuber of possible solutions is enorous. Due to this difficulty, the ethods used to solve this proble are the heuristics. In ost articles, there are any types of heuristics and the ones called eta-heuristics are the ost coon, sees for exaple: [][2]. Other authors developed Iplicit Enueration heuristics, see [3] [4]. 3. Matheatical Forulation Before presenting the forulation proble, we need to clarify that in this wor it was considered that the operator would offer the costs of cabling and handoff. A ore thorough description about the handoff calculation can be seen in Alonso [5]. The forulation of the proble as a proble of quadratic integer prograing, as given below: Being n the nuber of cells to be assigned to switches. It is assued that the cells and the switching centers localization are set and nown. Being H and H according costs per unit of tie for the soft handoff and for coplex handoffs that occur between the cells i e j ( i, j =, L, n). As it was given above, it is assued that the handoff costs are nown and proportional to the frequency of handoffs that occur between cells i and j. Being C i the cabling costs per unit of tie between the cell i and the switch ( i =, L, n ; =, L, ). Being λ i the nuber of calls per unit of tie destined to cell i, and M the capacity of a switch. This proble consists of assigning cells to switches in a way that iniize the cost function. The cost function integrates the handoff cost per tie and the cabling cost between cells and switches. The optiization of this proble ust not violate the axiu capacity of each switch and ust be taen into account that each cell could be assigned to a unique switch. The atheatical forulation of the described proble above was ade using integers variables, and non-linear constraints. Therefore, the following variables were defined., if the cell i is assigned to switch x i = 0, otherwise Considering that each cell can only be assigned to a unique center, there is the following constraint: = = for i =, L, n The switches capacity constraints are given by: n λ i M for =, L, i= The total cabling cost is given by:. n i= = C i x i () (2) The following additional variables were created to represent the costs of handoff: z = x x for i, j =, L, n e =, L,. (3) i j Observe in the constraint (3) above that z = when the cells i and j are connected in the sae switch and z = 0, otherwise. Considering = = all the switches, we have z = x j = if the ISSN: ISBN:

3 cells i and j are connected to the sae switch, and z = x j = 0 if they are connected to = = different switches. The total networ cost is given by the su of the cabling costs plus the handoff costs, and can be written as: n n n n n x j ) (4) i= = i= j= = i= j= = Ci + H x j + H ( According to Hedible [2], the soft handoff cost H can be considered irrelevant when copared to the coplex handoff cost H. So, we have h = H H, replacing in (4) and not taing the constant part into account, the objective function to be iniized is: n n n Ci + h ( x j ) i= = i= j= = Therefore the proble to be solved is the following: n n n n n Min Ci + h h x j i = = i = j = i = j = = = for i =, Ln = n s.a λ = (6) i M for, L, i = x = 0 or for i =,, n and =,, i L L It ust be observed that the proble of assigning cells to switches, as stated in (6), is non linear with integers variables. Many authors considered a transfored version of this proble by replacing the constraints (3) for a set of linear constraints which leads to a linear odel with integers variables, see: Menon [4], Merchand and Sengupta [3]. The algorith proposed here does not utilize this transforation. 4. The Bea Search Algorith The Bea Search is an iplicit enueration ethod for solving the cobinatorial optiization probles. It is said that it is an adaptation of the Branch and Bound ethod where only a predeterined nuber of best partial solutions (nodes) are evaluated in each level of the search tree, whilst the others are discarded peranently. As a big part of the tree search nodes is discarded, it eans that only a few nodes are ept for further branching and the others (5) are pruned off peranently, the ethod execution tie is polynoial in the size of the proble. Thus a Bea Search is a tree search technique that for each level of the tree a fixed nuber of nodes is analyzed and this result in a fixed nuber of solutions. We define the bea width as the nuber of nodes analyzed for each level, denoted by β. 4. The Tree with all the Possible Solutions Before presenting the proposed algorith, the search tree structure (assignent) is described for a sall exaple, and all possible solutions are listed properly. Being n = 4 the nuber of cells that should be assigned to = 2 switches. Each cell A i is able to receive a fixed nuber of calls λ i per unit of tie. Thus, the nuber of calls covered by cell A is λ and so on. The switch C routs these calls if cell A i is connected to the switch C. Each switch has a capacity M. Each cell can only be connected to one switch. Being x i =, if cell i is connected to switch and x i = 0, otherwise. The tables, 2, 3 and 4 showed below presents the inforation about the cabling cost, handoff cost, capacity of each cell and switch, for the exaple presented. Cell Switch Switch Cell/cel Table : Cabling cost C i between cells and switches Table 2: handoff cost h between cells. Cell Nuber of call Switch Capacity Table 3: Nuber of call λ i for each cell i. Table 4:switches capacity M. Before building the search tree is necessary to establish the following definitions: (D.) The tree is constructed by level and in each level i the assignent of the i-esie cell to each one of switches is ade. (D.2) The nodes that are in the level of the tree are called seed nodes because each one of the will generate a sub-tree of decisions. (D.3) In each level i, when aing the assignent fro i-esie cell to a switch, two costs are considered: the cabling cost and the handoff cost, taing into account the (i-) attribution entioned before. To be able to find an effective solution, an assignent of n cells ust be done. In the tree representation it corresponds to go through until the ISSN: ISBN:

4 level n (D.). Figure 2: Coplete Search Tree considering the seed node C in in level N (left sub-tree) and C2 in level N (right sub-tree). The Figure 2 shows the tree with all the possible solutions for this proble. The nuber of the switch is represented in the nodes, the cell nuber is represented by the letter A, so, A represents the cell, and the levels are represented by N, for exaple N represents the level, N2 the level 2 and so on. The nodes of each level of the tree represent an assignent. It begins with the tree fixing to a switch, for exaple, to the switch C and attributing to this a cell, for exaple, A. Then, the node, of level N, eans that cell A was assigned to switch C, i.e. x =. This first node is called the seed node. Once it is done, the assignent on the seed node follows aing the other attributions, always considering the assignent that was already ade in the first nodes. Note that the nuber of seed nodes is equal to the nuber switches (). Moreover each seed node will originate one sub-tree and the coplete tree will have then sub-trees. In figure 2, the nodes ared with a cross give an unfeasible assignent and they are eliinated fro the process. It is iportant give special attention to the total nuber of solutions which is n = 2 4 = 6, but only 6 aong the are feasible, justifying the application of a search tree procedure. For each tree level the allocation of a cell for all the switches is ade, thus in the level N the allocation of the cell A to each one of the switches (seed nodes) is ade. In the level N2 the allocation of the cell A 2 for each one of the switches is ade; in the level N3 the allocation of to the cell A3 is ade, and in the level N4 the allocation of the cell A4 is ade. Then, in the Figure 2, the level N2 has 4 nodes coing fro the allocation of the cell A2 to switches and 2, in the left sub-tree and to the switch 2 and in the second right sub-tree. In the level N3 we have 6 nodes that coe fro the attribution of the cell A3 respectively to the switch 2, and 2 to the left subtree and the switch, 2 and to the right sub-tree. Beside each node in figure 2, inside the sall rectangle, the cabling costs and the partial costs of handoff related to the allocation ade in the node also are presented, respectively. For exaple, in the level N3, we have the attribution of cell A3 to switch 2. In this case the cabling cost between A3 and C2 is 4 and the partial cost of handoff is 6 because there is a handoff between A3A, AA3, A3A2 and A2A3. In each level, only the partial costs of the solution that is being obtained are calculated because as it is said that the solution will only be coplete when the ultiate level of the tree nodes is generated. Thus, in the nodes of level N, there is only the cabling cost (linear cost). In the N2 level, the linear costs plus the handoff costs of the seed node until the nodes of the level N2 are already calculated, and therefore, in the ultiate level, there are linear costs plus the handoff costs of the seed nodes until the nodes of level N n. As we go down in the tree, the cost is being copleted by the decision taing. Therefore, the branches of the tree containing the nodes A C, A 2 C, A 3 C 2, and A 4 C 2 indicate that the following allocation was ade: x =, x 2 =, x 32 =, x 42 = The Tree Generated by Bea Search To avoid the exponential growth of the tree, the proposed algorith does not create all the nodes of the tree, as it has been shown in figure 2. The nodes are created according to soe rules. Theses rules also ai to prevent the creation of unfeasible solutions. In each node of the following inforation tree are stored: () Switch Identification; (2) Cell Identification; (3) Fixed cabling costs and partial handoff costs related to the allocation ade in the node;(4) Reaining switch capacity that it is representing. The reaining switch capacity M is coputed by decreasing the switch initial capacity M, the nuber of calls λ i referring to the cell deands that were already allocated to this switch. (5) Partial cost of the solution, i.e., cabling cost plus handoff of the seed node until the node were entioned. During the construction process of the solution tree, the first criterion whether to create or not a node in the level i, we ust chec the reaining capacity M of a switch. One node will only be created (assignent cell Ai) in the level i if λ i, M, i.e., if the call nuber λ i covered by the cell A i is less or equal to the reaining capacity ISSN: ISBN:

5 M of the switch C. If M < λ i, all the branches that will be originated fro this node are not going to exist, because the cell i cannot be attributed to the center. After going through the test of the switch capacity, the node will only reain in the solution tree if it passes in the search width criterion β. For exaple, if the search width chosen by the user is β =2, in each level, only the nodes that generate the two inors solutions, coputed by the technique of the greedy algorith, reain in the tree. Thus, for β =2, each node will only generate two branches, and they will be those that the greedy solution coputation produces the least cost. The algorith used for constructing the tree is the following: Begin level = 0 While (level < nuber of cells n) do: º Step: Do level = level + S... Create the nodes of this new level according to the switch capacity rule. S..2. For each node created in the level i, follows: Store the identification of the cell and the switch; Calculate and store the node cost, i.e., the fixed cabling costs and the partial handoff costs related to the attribution ade in the node. Calculate and store the solution partial cost, i.e., the cabling cost plus the handoff cost of the seed node until the created node; Calculate the reaining capacity M of all the switches ; If (level < nuber of switch) Then 2º Steps. For each node created in the º. Step (S.) and considering the calculated costs until then, find a greedy solution for the proble, through a Greedy Best-First Search (GBFS)[6] in the tree, starting fro this node. 3 o. Steps Keep in the tree, in this level, the nodes that generated the best β greedy solutions. Discard the rest of the nodes. End While Initially the algorith is in the level zero of solution, because the allocations will start to be ade now. Then do level= and start creating nodes (S.) in this level. The nodes of level are the seed nodes and are created one seed node for each switch. In (S.2), for each node created in (S.) is assigned a cell of the vector of cells considering the capacity of the switch. Follows it is calculated the cabling and handoff costs of each one of these nodes. The greedy best-first search perfored in the 2º Step, ais to choose, in the 3 o. Step, which nodes, of the level i, should reain in the tree of feasible solutions, to respect the width of search β. For exaple if β=2, in each level will reain only two nodes. The value of the inor greedy solution calculated in the 2 o. Step, is also used as upper bound (cutting) for generating or not the other nodes of the tree, in the º. Step. This upper bound should be updated when the inor greedy solutions are found. With this we have two cutting criterion for creating or not the nodes. The first of the is the switch capacity and the second is the cost of the inor greedy solution, i.e., if the partial cost of the solution in the node that has been just created is bigger than the best greedy solution cost this node should be excluded fro the solution. After finishing the greedy best-first search it should be verify if the level of the search tree is equal the nuber of cells. If it is the case, the algorith finishes, because the end of the tree is reached. If not, the algorith goes bac to the o. Step to generate the nodes of the next level. The Figures 3 (a)-(f), presented the detailed solution of the Exaple of the Section 3. by the Bea Search algorith. In the exaple is considered the width equal to β =2. The steps to be followed for solving the exaple are: Construct the nodes of level N (one node for each switch). The cabling and handoff costs, at this level are the sae as zero (see the costs in Table and 2). (ii) In level N2 initially are created 4 nodes, because fro the assignent ade in the level N, there are 4 possible attributions for the cell 2 as follows: A 2 C, A 2 C 2, in the sub-tree of the left and A 2 C 2, A 2 C in the sub-tree of the right. This situation is described in Figure 3(a) and corresponds to Bea Search º. Step. As the search width is two, only two of these nodes will reain in the tree. (iii) For choosing which nodes of level N2 will reain in the tree, a greedy best-first search is ade, until the last level, for each one of the 4 attribution ade in level N2. Suppose for exaple the node 2 of the level N2, on the left sub-tree, of Figure 3(b). The assignents ade until this node are: AC in level N and A2C2 in level N2 with cabling cost 4 and partial handoff 6. Now, should be ade the assignent of the cells A3 in level N3 and A4 in level N4. The cell A3 can be assigned to the switch C with cabling and handoff costs respectively of 4 e ISSN: ISBN:

6 2 (node of level N3), and assigned to the switch C2 with cabling costs and handoff, respectively of 4 and 4 (node 2 of level N3). As it regards of a greedy algorith, it is chosen to assign the cell A3 the switch C because the partial cost of this solution is less. It can be discard then the node 2 of level N3. Continuing the attributions fro the node of level N3, due to a capacity constraint, the cell A4 can only be attributed to the switch C2 and the cost of this attribution is zero for cabling and 2 for handoff. Applying this procedure for all the 4 nodes of level N2, will be found 4 greedy solutions, where will be choose the two with less cost. This situation corresponds to the 2º Step of Bea Search. The Figure 3c shows that the nodes choose to stay in the tree, level N2, (3 o Step) were: node 2, in the left sub-tree and node in the right sub-tree. Below are created the nodes of level N3 (º Step), and later a greedy search fro these nodes (Fig.3d), for choosing which nodes of level N3 will reain in the tree. The Fig.3e shows the nodes choose of level N3 (3 o. Step). In the end, it is obtained two solutions: A C, A 2 C 2, A 3 C, A 4 C 2 with cost of 28 and A C 2, A 2 C, A 3 C 2, A 4 C with cost of 36. It is then adopted the solution with less cost between the two. Fig. 3d Steps of Bea Search for the Exaple of section 3.. Fig. 3e - Steps of Bea Search for the Exaple of section 3.. Fig. 3f - Steps of Bea Search for the Exaple of section 3.. Fig. 3a Steps of Bea Search for the Exaple of section 3.. Fig. 3b Steps of Bea Search for the Exaple of section 3.. Fig. 3c Steps of Bea Search for the Exaple of section Changes in the Bea Search It is possible to iprove the perforance of Bea Search aing soe changes, such as establishing an order for aing the attribution of the cells to switches, according to a heuristic criterion. The exaple presented in section 4. followed for the nueric order of the cells. The descending order of the estiated cost of handoff and of cabling was chosen as a criterion to ae the attribution of the cells, i.e., for each cell A i was calculated the estiated cost of handoff and cabling CEHC given by Eq. (7). The Table 5 shows these calculations for the exaple of Section 4.. n CEHCi = h + ci j= = Antena A A A A CEHC Orde Table 5: Allocation order of cells (7) Another odification consists in applying the Bea Search algorith into each sub-tree separately. The advantage is that instead of solving a big proble, we solve saller sub probles and so it is possible to decrease the ris of eliinating a coplete sub-tree, when using the width search β, eliinating the region where the optial solution is found. To better illustrate, iagine a ore coplex proble, than the one presented in Section 4., with ISSN: ISBN:

7 4 switches. Reeber that each switch corresponds to a seed node, i.e, a coplete tree. If we tae the sae width search used in Section 4. exaple, β = 2, then only two seed nodes will reain in level N. Thus, two entire sub-trees will be cut without any Greedy Best-First Search Coputation. To avoid this proble we enforce the solution coputation for each sub-tree by iposing the Bea Search application for each seed node. This procedure has two advantages: avoid the eliination of an entire tree, without any node evaluation of this tree; (ii) could be ipleented in parallel and each for tree a Bea Search is coputed independently by one processor. In this way, if the proble has four switches, it would have 4 processors and each one of these processors would execute one Bea-Search in one sub-tree. This ipleentation perits to obtain a big reduction of coputational tie, because in each sub-tree we have a search proble totally independent, which disisses the counication between the processors providing a speed-up and an elevated efficiency. 5. Coputational Results We tested two versions of the bea search. The Bea Search considering the aendent of subsection 4.2 BSHCSP and the Bea Search (BS) considering the nuerical order for the allocation of cells as in the exaple of section instances were run with varied diensions. It has started with considered inor probles (45 cells and 2 switches), until probles considered big (250 cells and 4 switches). The instances are the sae of [3]. The algoriths were ipleented in Matlab 7.0, a achine with processor Intel 2.0 GHz, G of RAM. The Table 6 shows, for each one of the instances, the values of objective function obtained with the two versions of Bea Search: BS and BSHCSP. Instan Cells/ ce Switch BSHCSP BS Instan ce Cells/ Switch BSHCSP BS 45/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Table 6 Results obtained with the Bea Search ethods 6. Conclusion Heuristics type Bea Search does not require an initial solution and always find a feasible solution. One of the advantages of theses heuristics is that they do not generate all branches of the tree; their execution tie is polynoial. The result obtained with the two heuristics showed that they are relatively equivalents, with a sall gaining for those that used the order of allocation considering the handoff and cabling costs. The Bea Search subprobles (BSHCSP) can be easily ipleented in parallel. Therefore, it is needed to put a processor to wor on each sub-tree. Each processor will then search the best solution of each one of these subtrees. The efficiency of a parallel algorith depends on a balance distribution of the tass that are perfored on each processor and the volue of counications ade between the processors. In the BSHCSP the sub-trees have the sae size and are copletely independent, thus the tass to be executed by the processors are balanced and require no counication between the. Then is possible to obtain a reduction of coputational tie, which is of about 80% of the tie spent in the sequential algorith. References: [] AbuAara, M. H.; Sait, S. M.; Subhan, A., A, Heuristics Based Approach for Cellular Mobile Networ Planning, The International Wireless Counications and Mobile Coputing Conference (IWCMC06), July, 2006,Vancouver, Canada. [2] Hedible, C.; Pierre, S., Genetic algorith for the assignent of cells to switches in personal counication networs, Electrical and Coputer Engineering, Canadian Conf. on, vol. 2, 2000, pp [3] Merchant,A.; Sengupta, B., Assignent of cells to switches in PCS networs, IEEE/ACM Trans. Networing, vol. 3, Oct, 995, pp [4] Menon, S.; Gupta, R., Assigning cells to switches in cellular networs by incorporating a pricing echanis into siulated annealing, IEEE Transactions on Systes, Man, and Cybernetics, Part B, vol. 34, n., 2004, pp [5] Alonso, E., Meier-Hellstern, S., Pollini, G.P., Influence of Cell Geoetry on Handover and Registration Rates in Cellular and Universal Personal Telecounications Networs, 8 th ITC Specialist Seinar on universal Counication, Genova, Italy, 992. [6] Russell, S. Norvig, P. Artificial Intelligence: A Modern Approach., pp.94-95, Prentice Hall series in artificial intelligence. Prentice Hall, Upper Saddle River, 2nd edition, ISSN: ISBN: