Two new algorithms for UMTS access network topology design

Two New Algorithms for UMTS Access NetworkTopology DesignAlp´a r J¨u ttner∗†,Andr´a s Orb´a n∗,Zolt´a n Fiala∗September17,2003AbstractPresent work introduces two network design algorithms for planning UMTS(Universal Mobile Telecommunication System)access networks.The task is to determine the cost-optimal number and location of theRadio Network Controller nodes and their connections to the Radio BaseStations(RBS)in a tree topology according to a number of planning con-straints.First,a global algorithm to this general problem is proposed,which combines a metaheuristic technique with the solution of a specificb-matching problem.It is shown how a relatively complex algorithm canbe performed within each step of a metaheuristic method still in reasonabletime.Then,another method is introduced that is able to plan single RBS-trees.It can also be applied to make improvements on each tree createdby thefirst algorithm.This approach applies iterative local improvementsusing branch-and-bound with Lagrangian lower bound.Eventually,it isdemonstrated through a number of test cases that these algorithms areable to reduce the total cost of UMTS access networks,also compared toprevious results.Keywords:Telecommunication,Metaheuristics,UMTS,Facilities planning and design,Lagrange–relaxation∗Ericsson Research Hungary,Laborc u.1,Budapest,Hungary H-1037e-mails:{Alpar.Juttner,Andras.Orban,Zoltan.Fiala}@eth.ericsson.se †Communication Networks Laboratory and Department of Operations Research,E¨o tv¨o s University,P´a zm´a ny P´e ter s´e t´a ny1/C,Budapest,Hungary H-11171IntroductionUMTS[24],[21]stands for Universal Mobile Telecommunication System;it is a member of the ITU’s IMT-2000global family of third-generation(3G)mobile communication systems.UMTS will play a key role in creating the future mass market for high quality wireless multimedia communications,serving expectedly 2billion users worldwide by the year2010.One part of the UMTS network will be built upon the transport network of today’s significant2G mobile systems[14],but to satisfy the needs of the future Information Society new network architectures are also required.Since UMTS aims to provide high bandwidth data,video or voice transfer,the radio stations communicating with the mobile devices of the end-users should be placed more dense to each other,corresponding to the higher frequency needed for the communication.In the traditional star topology of the radio stations,i.e.all the radio stations are directly connected to the central station,this would increase the number of expensive central stations as well.To solve this problem,UMTS allows a constrained tree topology of the base stations,permitting to connect them to each other and not only to the central station.This new architecture requires new planning methods as well.As shown in Figure1the UMTS topology can be decomposed into two main parts:the core network and the access network.The high-speed core network based on e.g.ATM technology connects the central stations of the access network. These connections are generally realized with high-capacity optical cables.In this paper the focus is on planning the access network,the design of the core network is beyond the scope.The reader is referred to e.g.[20]on this topic.The access network consists of a set of Radio Base Station s(RBS nodes) and some of them are provided with Radio Network Controller s(RNC nodes). The RBSs communicate directly with the mobile devices of the users,like mobile phones,mobile notebooks,etc.,collect the traffic of a small region and forward it to the RNC they belong to using the so-called access links,that are typically mi-crowave radio links with limited length(longer length connections can be realized using repeaters,but at higher cost).In traditional GSM configuration the RBSs are connected directly to an RNC station limiting the maximum number of RBSsRBS ArrayRNCFigure1:UMTS topologybelonging to a certain RNC.To overcome this limitation the UMTS technology makes it possible for an RBS to connect to another RBS instead of its RNC. However RBSs have no routing capability,they simply forward all the received data toward their corresponding RNC station,therefore all traffic from an RBS goesfirst to the RNC controlling it.For example if a mobile phone in region A on Figure1wants to communicate with a device in region B,their traffic will be sent trough the RNC R.Only the RNC station is responsible for determining the correct route for that amount of traffic.It follows that the RBSs should be connected to the RNC in a tree topology.(There are research initiatives to provide some low-level routing capability for the RBSs and to allow additional links between RBSs,which may increase the reliability of the network.These developments are quite in early stage,hence we only deal with tree topology.) As a consequence,the access network can be divided into independent trees of RBSs,each rooted at an RNC.These trees are called Radio Network Subsystem s (RNS).Further advantage of the tree topology compared to the star topology is that the links can become shorter,which on one hand reduces the cost of the links,on the other hand it may require less repeaters in the network.Moreover,there are some additional links connecting the RNCs to each other (inter-RNC link s)and to the core network.The planning of these links are beyond the scope of this paper.For technical reasons the following strict topology constraints have to be takeninto account:•The limited resources of the RBS stations and the relatively low bandwidth of the access links cause considerable amount of delay in the communication.In order to reduce this kind of delay,the maximum number of the access links on a routing path is kept under a certain amount by limiting the depth of a tree to a small predefined value.This limit is denoted by l tree in our model.Currently,the usual value of l tree is3.•The degree of an RBS is also constrained.One simple reason is that the commercial devices have only limited number of ports.Another reason is that too many close RBSs can cause interference problems in their ra-dio interface if their connections are established through microwave links.Moreover the capacity of an RBS device also limits the number of con-nectable RBSs.Therefore in our model there can be at most d RBS RBSs connected directly to another RBSs on a lower level.It is typically a low value,d RBS=2in the currently existing devices.•The degree of RNCs(the number of RBSs connected directly to a certain RNCs)is also limited for similar reasons,it is at most d RNC.Generallyd RNC d RBS.The planning task investigated in the present paper is to plan cost-optimal access network,that is to determine the optimal number and location of the RNCs and tofind the connections of minimal cost between RNCs and RBSs satisfying all the topological restrictions.The cost of a UMTS access network is composed of two variable factors:the cost of the RNC stations and the cost of the links.For the exact definitions see Section2.Unfortunately,this planning problem is N P-hard.The problem offinding a minimal weight two-depth rooted tree in a weighted graph can be reduced to this problem.Moreover,the problem remains N P-hard even in the special case of planning a single tree with afixed RNC node.See Appendix A for the proof of this claim.A general UMTS network may contain even about1000RBSs,a powerful RNC device controls approximately at most200-300RBSs.This large numberof network nodes indicates that the planning algorithms have to be quite fast in order to get acceptable results.One possible approach to the problem described above would be to divide the set of input nodes into clusters and then create a tree with one RNC in each cluster.However,this“two-stage”solution has some significant drawbacks:first, it is very hard to give a good approximation to the cost of a cluster without knowing the exact connections.Second,it would be also a strong restriction to the algorithm searching for connections to work in static clusters of nodes created in the beginning.For these reasons a“one-stage”approach is introduced:an algorithm which creates a number of independent trees with connections simultaneously.The proposed method called Global is based on a widely used metaheuristic tech-nique called Simulated Annealing.However,it is not straightforward to apply Simulated Annealing to this problem,since the state space is very large and a rea-sonable neighborhood-relationship is hard tofind.To overcome these problems a combination of the Simulated Annealing and a specific b-matching algorithm is proposed.To make this method efficient it will be shown how the relatively complex and time-consuming b-matching method can be performed within each step of a metaheuristic process in still acceptable time.Then,a Lagrangian relaxation based lower bound computation method is presented for the problem of designing a single tree with predefined RNC node. Using this lower bound a branch-and-bound method is proposed to compute the theoretical optimal solution to this problem for smaller but still considerable large number of nodes.For bigger single-tree design tasks,a second heuristic method called Local is proposed based on an effective combination of a local search technique and the branch-and-bound procedure above.This Local algorithm can be used effectively either in circumstances when only a single tree should be designed or to improve each trees provided by the Global algorithm.Related work.Similar problems have already been examined by several au-thors,using different notations according to the origin of their optimization tasks.By adding a new virtual root node r to the underlying graph and connectingthe possible RNC nodes to r the problem transforms to the planning problem of a single rooted spanning tree with limited depth and with inhomogeneous degree constraints.There are some papers in the literature dealing with planning a minimal cost spanning tree with either of the above constraints(see e.g.[17,18, 2]),butfinding an algorithm handling both requirements is still a challenge.On the other hand,our problem can be considered as a version of facility location problems.The problem of facility location is to install some“facilities”at some of the possible places,so that they will be able to serve a number of “clients”.The objective is usually to minimize the sum of distances between the clients and the corresponding facilities while satisfying some side constraints,e.g.the number or capacity of facilities is limited.See e.g.[9,19]for more details on facility location problems and on the known algorithmic approaches.Facility location problems give a good model to the planning task of tra-ditional GSM access network topology,for it requires the design of one-level concentrators.The case of multilevel concentrators is also studied in the liter-ature,though less extensively.[22]examined the problem similar to ours,but without depth and degree constraint and with capacity dependent cost functions.[5]discusses the case of two-level concentrators,also without degree constraints.Finally,our problem can also be considered as an extension of the so-called hub location problem.In this scenario we have a given set of nodes communicating with each other and the amount of the traffic between the node pairs is given by a traffic matrix.Instead of installing a link between every pair of nodes,each node is connected to only a single special node called hub.The hubs are fully interconnected,so the trafficflow between two nodes is realized on a path of length three,where the two intermediate nodes are hubs.The task is tofind an optimal placement of hubs,where the cost is composed of the cost of the hubs and the capacity dependent cost of the links.There are several papers examining hub location problems with various side constraints and optimization objectives.A good review on this topic can be found e.g.in[16].A previous method that is able to a give solution to the problem presented in this paper can be found in[13].This algorithm called TreePlanfinds the number and the places of the RNC devices by Simulated Annealing metaheuristic and connects the nodes to the RNSs using an extension of Prim’s spanning tree algo-rithm,which respects the topological requirements on the tree.This algorithm was used as a reference method in our experimental tests.The rest of the paper is organized as follows.First,in Section 2the exactly defined planning problem is introduced.Then,in Section 3the Global algorithm for the general problem is described in detail.In Section 4the Local method is introduced for planning a single tree with one RNC.Section 5shows the re-sults of test cases of both algorithms also compared to former solutions.Finally,Appendix A gives a short proof of the N P -completeness of the problem in the special case when only a single tree with a given RNC node is planned.2Problem Definition and NotationsThe access network is modeled as a directed graph G (N,E ),where N is the set of the RBSs.For each feasible solution there exists a natural one to one mapping of the set of links to the set of edges.Each edge in E corresponds to a link between its ends and directed toward the corresponding RNC.On the other hand,this set E of directed edges determines the set of RNCs as well:a node is RNC if and only if it has no outgoing edge.In order to illustrate the logical structure of the network the notion of the level of nodes is introduced (Figure 2).Let all RNC nodes be on level 0,and let the level of an RBS station be the number of edges of the path that connects it to its controlling RNC.The level of a link is defined as the level of its end on the greater level-number.Some other important notations used in this paper is shown in Table 1.RBSRNC0. level1. level2. level3. levelFigure 2:Logical structure of the networkN the set of RBSsE the set of linksn the number of network nodesl tree the maximal depth of the treesL E,L i the set of nodes on the i-th level of the graphiE i the set of links on the i-th level of the graphl E(v),l(v)the level of the node vl E(e),l(e)the level of the edge ed RBS the maximal degree of an RBSd RNC the maximal degree of an RNCcost RNC the cost of an RNCTable1:Some important notationsThe input of the examined planning problem consists of the set N of RBSs, the cost function c link of the links(described later),the installation cost cost RNC of the RNCs and the constraints d RNC,d RBS and l tree.Moreover we are given the set R R of places of required RNCs and the set R P of places of possible RNCs. (It is useful since in many practical cases already existing networks should be extended.)The set E of links is a feasible solution to this input if•E forms a set of disjoint rooted trees,which cover the whole set N,•the depth of each tree is at most l tree,•the degree of the root of each tree is at most d RNC,•the in-degree of each other node is at most d RBS,⊆R P.•R R⊆L EThe total cost of the network is composed of the following factors.•The cost of the RNC stations.The cost of one RNC,cost RNC,means the installation cost of that particular RNC.This constant may contain otherfactors as well,e.g.the cost of the links between the RNCs can be included, assuming that it has a nearly constant additional cost for every RNC.•The cost of the access links.In this model the cost c link(i,j,l)of a link depends on its endpoints i and j and its level l.A possible further simpli-fication is to assume that c link(i,j,l)=f l·c link(i,j),where f l is a constant factor representing the weight of level l.This kind of link-cost function has two applications.First,since access links closer to the RNC aggregate more traffic,this is an elementary way to model the capacity dependent costs by giving higher cost on the lower levels.Second,it can be used to prohibit the usage of some links on some levels by setting their cost to an extremely large value.Also,it makes it possible to force a node to be on a predefined level.So,the task is tofind a feasible connection E minimizing the total cost|L E0|·cost RNC+(ij)∈Ec linki,j,l E(ij)(1)3The Global AlgorithmThe aim of this algorithm is tofind the optimum places of the RNC nodes(i.e.to decide which RBSs should be equipped with RNC devices)and to connect each RBS to an RNC directly or indirectly as described in Section2.The basic idea of the proposed approach is the following.Assuming that the level l(v)is known for each node v,the theoretically min-imal cost can be determined for that given level-distribution(see Section3.2). The algorithm considers a series of such distributions,determines the cost for each of them and uses some metaheuristic method to reach thefinal solution. From among the wide range of metaheuristic methods existing in the literature, Simulated Annealing was chosen for this purpose,but some other local search methods could also be used,e.g.the so-called Tabu Search method[11,1].3.1Using Simulated AnnealingIn this section the application of Simulated Annealing([15,1])to the problem is illustrated.To use Simulated Annealing to a specific optimization problem, an appropriate state space S corresponding to the possible feasible solutions, a neighborhood-relation between the states and a cost function of each state should be selected.The role of neighborhood-relation is to express the similarity between the elements of the state space.The neighborhood of a state s is typically defined as the set of the states that can be obtained by making some kind of local modifications on s.Then the Simulated Annealing generates a sequence of feasible solutions s0,s1,···,approaching to a suboptimal solution as follows.It starts with an arbitrary initial state s0.In each iteration it chooses a random neighbor s i+1of the last solution s i and calculates its cost c(s i+1),then decides whether it accepts this new solution or rejects it(i.e.s i+1:=s i).This iteration is repeated until a certain stop condition fulfills.If the cost of the new state c(s i+1)is lower than c(s i),the new state is always accepted.If it is higher then it is accepted with a given probability P accept de-termined by the value of the deterioration and a global system variable,the so called temperature T of the system.In this case P accept is also positive,however, it is an exponentially decreasing function of cost deterioration.In general,the state s i+1is accepted with probabilityP accept=min1,exp−c(s i+1)−c(s i)T i.The temperature decreases exponentially during the execution,i.e the tem-perature T i in the i-th iteration is given by:T i=T i−1∗fact,T0=const.where fact is the so called decreasing factor,which is a number close to1, typically0.99-0.9999.The values T0and fact are declared in the beginning of the algorithm.The short pseudo-code in Figure3illustrates the framework of Simulated Annealing.procedure Sim Ann(steps,factor,temp,s0){curcost:=cost(s0);oldcost:=curcost;while(steps>0){Choose a random neighbor s i+1of s i;curcost:=cost(s i+1);if(acceptable(curcost,oldcost,temp))oldcost:=curcost;else s i+1:=s i;i:=i+1;steps:=steps-1;temp:=temp*factor;}return s n;}Figure3:Structure of the Simulated Annealing algorithmFor an effective Simulated Annealing the following criteria should be met.The state space to be searched should be possibly small;each state should have lot of (meaningful)neighborhoods,allowing to reach the optimum in a low number of steps;the cost of a state should be determined relatively fast.None of these criteria can be fulfilled easily in case of this planning problem. The most obvious idea for the state space would be the set of all feasible con-nections.Two such connections would be neighbors,if they can be reached from each other by changing one edge.This solution violates thefirst two criteria:the state space is very large,and because of the topological constraints,each state has only few neighbors and there are a lot of local minima.Although the cost of a state can be calculated easily,since the exact connections are given in every state,this solution is not usable.Instead,we propose following idea.The state-space of Simulated Annealing is the set of all possible distributions of the nodes on the different levels of the graph.Thus a given distribution can be represented as an n-dimensional integer vector s in S.(Note that the distribution of an arbitrary state i is denoted by s i.)D={0,1,2,..,l tree},S=D n,s∈SThe state space with this selection will be much smaller than in the previous case.Two level-distributions are neighbors if they can be reached from each other trough one of the following slight modifications of the current distribution:•moving an arbitrary node onto an adjacent level upwards or downwards•swapping an arbitrary RNC with an arbitrary RBS,that is,moving an RNC to another siteThe price for the smaller state space is that the calculation of the cost of a state becomes more difficult.Section3.2shows how the cost c(s i)of a given state s i∈S can be calculated.The initial state s0of the Simulated Annealing algorithm can be any feasible state.Such a state can be constructed easily by setting each possible nodes to RNC and connecting the other nodes arbitrarily fulfilling the criteria defined in Section2.The fact that there exists no feasible solution at all can be detected, as well.3.2Finding the Exact Cost for a GivenLevel-distribution s iLet us assume that the distribution of the nodes on different levels is known.In order tofind the optimal connections for this given distribution,the connections between the adjacent levels have to be determined.The main observation is that the connections of the different levels to their parents are independent,so the task of connecting the nodes of a certain level can be performed separately for each level.(As there are l tree adjacent level-pairs,the algorithm described now has to be run l tree times in each step of the Simulated Annealing process.)Generally,for each adjacent level-pair(L i and L i+1,i=0,···,l tree−1)a connection has to be found so that•all nodes in set L i+1are covered•the maximal degree k of the nodes in set L i is given.As already mentioned, k=d RNC if i=0,k=d RBS if i>0.This can be formulated as a special b-matching problem,which can be solved in strongly polynomial time[3,7].3.2.1Bipartite b-matchingA bipartite b-matching problem is the general minimal-cost matching problem, where there is a predefined lower bound low(v)and upper bound upp(v)for the degree of each node v in a bipartite graph.Definition3.1Let G(V,E)be a bipartite graph and let low,upp:V−→N be two predefined functions on the set of nodes.A subgraph M of G is called b-matching if low(v)≤deg M(v)≤upp(v)for each v∈V.Obviously,our case is a special b-matching problem with low(v)=1and upp(v)=1for each node v∈L i+1and low(w)=0and upp(w)=k for each node w∈L i.3.2.2The Solution of the b-matching ProblemAs it was mentioned above,the problem offinding the cost of a given level-distribution can be reduced to a specific b-matching problem.In this section the solution of this b-matching problem is outlined.The detailed description of this method is skipped and only its main idea and the definition of the used notions is sketched in order to show how the b-matching algorithm can be accelerated significantly when it is called with a series of inputs such that each input only slightly differs from the previous one.Let a bipartite graph G=(L i,L i+1,E i+1),a cost function c:E i+1−→R≥0 and a degree bound k of the nodes L i be given.Definition3.2A subset M of E i+1is called partial matching if the degree of each node in L i is at most k and in L i+1at most1.A partial matching is called full matching if the degree of each node in L i+1is exactly1.Thus,the aim is tofind a c-minimal full matching.Of course,it can be supposed that k·|L i|≥|L i+1|,otherwise there cannot be a full matching.Definition3.3For a partial matching M a path P={e1,e2,...,e2t+1}is called M-alternating,if e2i∈M for all i=1,2,...,t and e2i+1∈M for all i= 0,1,...,t.Definition3.4A node v∈L i is called saturated if the degree of v in M is maximal,that is if deg M(v)=k.A node v∈L i+1is saturated if M covers it, that is if deg M(v)=1.The most important property of M-alternating path is that if there exists an M-alternating path P between two non-saturated nodes,then a partial matching with one more edge can be found by“flipping”the edges of the path P.More exactly M :=(M\P)∪(P\M)is again a partial matching and|M |=|M|+1. It also holds that if a partial matching is not maximal,then it can be extended through alternating paths.Definition3.5A real functionπ(u)defined for each node u∈L i∪L i+1is called a node potential:π:L i∪L i+1→RA givenπis called c-feasible or simply feasible ifπ(v)≤0∀v∈L i(2)and the conditionπ(x)+π(y)≤c xy(3) holds for every edge(x,y)∈E i+1.If inequality(3)is actually an equation,the⊆E i+1denote the set of edge(x,y)∈E i+1is called an equality edge.Let Eπi+1all equality edges with respect to the potentialπ.The following theorem is also fundamental in matching theory.Theorem3.6A full matching M is c-minimal if and only if there exists a fea-sible potentialπfor which all edges in M are equality edges.The algorithm is based on this theorem.It stores a feasible potentialπand a partial matching M⊆Eπ,i.e.a partial matching having only equality edges.i+1If M is a full matching then it is optimal.If it is not,then in each iteration the algorithm eitherfinds a partial matching having one more edges or“improves”the potentialπ.The algorithmfinds an optimal full matching in O(n3)steps in full bipartite graphs.3.2.3Acceleration of the AlgorithmThe b-matching algorithm described in Section3.2should be run in each step of the Simulated Annealing for all adjacent level-pairs.Because the complexity of the b-matching algorithm is O(n3)this process is quite time-consuming.This section introduces an idea to accelerate the whole algorithm significantly,so that it can solve even large inputs in acceptable time.Considering that in each transition only a minor part of the level distribution changes,therefore a significant part of the former connections can be reused in the next step.The possible changes are:•a node is deleted•a new node is added•a node is moved from the set L i to L i+1or vice versa.The idea is,that after making a modification to the level-distribution,a better initial potentialπand initial partial matching M can be used instead of the zero potential and the empty matching.If a node is deleted,the previous potential resulted by the algorithm remains feasible so it does not need to be modified.If a new node v is added,it is easy tofind an appropriate potentialπ(v)for this new node in such a way that(2) and(3)hold.Moving a node is a combination of a deletion and an addition.Furthermore,a significant part of M can be reused,too,only the edges which ceased to be equality edges after the modification must be deleted.These improved initial values make it possible to reduce the running time efficiently,since there is no need to calculate all connections again from the be-ginning.Section5shows that this acceleration makes the algorithm significantly faster,nearly square-wise to the number of input nodes.It enables us to run the quite difficult b-matching algorithm in each step of the Simulated Annealing process.4The Single-Tree ProblemIn practical planning problems it is often the case that—for geographical,polit-ical or economical reasons—the exact number and location of the RNC nodes and the set of RBSs belonging to them is already known in the beginning.For these reasons in this section we discuss the special problem where only one fixed RNC and a set of RBS nodes are given.Of course the Global algorithm can solve this special case,as well,but an alternative method called Local algorithm, whichfinds remarkably better solutions,is introduced.Although this algorithm is slightly slower than the Global algorithm,it is efficient for about200-300nodes, which is the typical number of RBSs controlled by one RNC.The Local algorithm is based on a branch-and-bound method thatfinds the exact optimal solution for smaller inputs(40-50nodes).To sum up,the Local algorithm can be used for the following purposes.•Planning a tree of RBSs belonging to afixed RNC.•Improving the trees created by an arbitrary previous algorithm.•Determining the real optimum for smaller inputs.The simplified single-tree problem can be formulated as follows.A single RNC and n−1RBSs are given by their coordinates.Note that the problem of positioning the RNC is omitted in this model.The task of the algorithm is to connect all the RBSs directly or indirectly to the RNC.These。