A large hierarchical network star—star topology design algorithm

I Mobile Networks I

A Large Hierarchical Network Star-Star Topology Design Algorithm

JOZEF PETREK Faculty of Electrical Eng. and Information Technology of Slovak University of Technology in Bratislava, Slovakia

petrek @ e& stubask

Communication Networks, Aachen University of Technology, Germany volker @ siedr. de

VOLKER SIEDT

Abstract. The hierarchical communication network design includes solving problems like the concentrator quantity problem, the concentrator location problem and the assignment problem. Many algorithms have been proposed to solve these particular problems. The two latter problems are NP-hard and no optimal solution of these problems has been found for large networks. In this paper we present a high complexity algorithm for star-star network topology design which solves all three problems for several hierarchy levels simultaneously and finds a superior solution in a reasonable time even for large networks with a few thousand terminals due to a new and quick assignment algorithm. The computational results showed that our assignment algorithm is superior to “Simulated annealing” and ‘Tabu search”.

1 INTRODUCTION Hierarchical communication networks are able to con-

nect a lot of users with rather low demand in an economical way [ 11. Therefore many communication networks have hierarchical structures (PSTN, PCS, ISDN, ATM, computer networks etc.). It is possible to solve optimally the network topology optimization problem for small networks with several tens of nodes using “Brute force” approach. Also several analytical algorithms were invented which can solve very simple problems with several hundreds of nodes optimally. Unfortunately the optimal solution for networks with many hundreds or several thousands of nodes has not been proposed yet and with respect to the complexity of the task the invention of such an algorithm cannot be expected even in the future. The goal of this paper is not looking for the optimal solution of the design problem but looking for the way how to find a very good solution within a few days for networks with several thousands nodes. An algorithm for the hierarchical network design solves the following problems with respect to costs optimization: 1. Concentrator quantity problem - specify the number of

concentrators 2. Concentrator location problem - find optimal locations

for concentrators 3. Assignment problem - specify which terminal should

be assigned to which concentrator

4. Terminal layout problem - interconnect terminals to their associated concentrators.

To design a star-star topology network these problems have to be solved for several hierarchy levels.

To speak about different hierarchy levels it is necessary to introduce a hierarchical network terminology. In this paper we use the well-known GSM terminology even though the algorithm can be used for any hierarchical network design (see Figure 1). The lowest node hierarchy level is the BTS (Base Transmitter Station). Several BTSs are connected to a BSC (Base Station Controller) concentrator. The next hierarchy level is the MSC (Mobile Switching Centre) having muter capability.

By the design of connections between BSCs and MSCs a BSC could be treated as a terminal with traffic equal to the total sum of all traffics of BTSs connected to the BSC. A MSC can be treated as a concentrator. The highest hierarchy level used in this paper is the TSC (Transit MSC) that is a gateway connecting all MSCs that are present in the network. Problems 2-4 are well known NP-problems. Our algorithm proposed in this paper solves problems 1-3 for several hierarchy levels simultaneously. Therefore a superior solution can be found. Mostly only the design problem for lower hierarchy levels is difficult to solve because the amount of nodes in higher hierarchy levels sinks. This depends on the connection capacity of concentrators. For solving the 4-th problem the authors have proposed another algorithm that converts the star

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network topology designed by solving the problems 1-3 into a tree topology. At this point assignment decisions made in step 3 may be re-evaluated.

. BTS (Bwc Tmmmiuer Lniml

EX ( 8 - Slvmn Carenmmor)

Figure I : The GSM network topology.

The concentrator quantity problem can be considered as the capacitated facility location problem [2]. The problem is capacitated because the concentrators have a maximum number of terminals which can be connected to them (CC - connection capacity) and each concentrator also has a maximum amount of traffic it can handle (FC - flow capacity). The more BSC concentrators are opened the lower are the costs for links between BTSs and BSCs. On the other hand, costs for links between BSCs and MSCs increase (and possibly between MSCs and TSC too). Also the node costs for BSCs (and possibly for MSCs too) grow. Therefore an optimal number of BSC concentrators exists. Methods often used to solve the problem are the “Add“ [4] and the “Drop” [3] algorithms. The “Drop” algorithm usually leads to better results than the “Add” algorithm [5]. Therefore it is often used in network design algorithms (e.g. [8]). The proposed algorithm uses a combination of the “Drop”, “Add“ and the clustering approach. This combination was found to generate very good solutions while consuming less runtime. How- ever, non-optimal concentrator location may cause the biggest gap between the proposed solution and the optimal one. The procedure is simplified to reduce CPU and memory complexity of the algorithm. Only very small solution degradation is therefore imported.

The concentrator quantity and location problems may be solved by clustering ( [ 5 ] and references therein) too. The algorithm [5] was modified in [6] later. A good algorithm solves problems 1-3 together. Many algorithms are based on the Lagrangean relaxation [9], [7] or the La- grangean relaxation associated with another heuristic algorithm such as the sub-gradient optimization [lo]. [13]: [22], branch-and-bound algorithm [ l l ] etc. The approaches based on the Lagrangean relaxation are not suitable for the design of very large communication networks because the number of equations grows extremely with the number of nodes in the graph. To find a lower bound for a network with thousands of nodes it is not possible to use the Lagrangean relaxation. Since the authors do not dispose of a comparable powerful heuristic algorithm, it is

not easy to egtimate the performance of the proposed heuristic algorithm [8].

The concentrator location problem can be solved without solving the assignment problem. An example of such an algorithm is the COG [15] (Center-of-gravity) at which only positions of terminals and their traffics are important for the concentrators’ location. The assignment problem may be solved using one of many assignment algorithms thereafter. The computational complexity of such an approach is very small but its disadvantage consists of a bigger gap between the proposed solution and the optimal one.

The assignment problem is often solved using different approaches [ 161, [ 171, [ 181, [ 191, [20]. The complexity of most algorithms leads to high computing times andor memory consumption for big networks. Therefore they are not suitable for solving very large problems. To store the present solution many algorithms use the incidence matrix. To store link costs often the cost matrix is used. To store just one matrix with loo00 x loo00 floats a memory of about 400 Mbytes is required. Therefore another more skilful method must be used.

The novelties of our contribution we see especially in: 0

0

a new high sophisticated algorithm for designing hierarchical network topology in one step a new, good performance and quick assignment algorithm using an advanced exchange procedure development of an economical way of network topology storing in computer memory which makes possible to design large hierarchical networks with thousands of nodes requiring only few Mbytes of RAM.

As can be seen in Section 3 the ability of the algorithm to design more hierarchy levels in one step brings savings up to 20 %. A correct concentrator location brings other savings. Testing the algorithm with 20 networks brought on average 7.3 % of savings in comparison with the COG algorithm.

2 DESCRIPTION OF THE ALGORITHM

2.1 bPUT PARAMETERS, INTRODUCTION OF VARIABLES

The input data to solve the hierarchical network design problem for the described algorithm are: 0

0

the positions of BTS stations (terminals) truficglS - the traffic generated by each BTS station. It is important to dimension the links correctly and not to exceed the traffic flow capacities of the BSCs and MSCs present in the network CCBsc (Connection Capacity of BSC) - the maximum number of BTSs that can be connected to a BSC FCBsc (Flow Capacity of BSC) - maximum amount

of traffic that can be handled by a BSC

0

0

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CCMsc (Connection Capacity of MSC) - the maximum number of BSCs that can be connected to a MSC FCMSC (Flow Capacity of MSC) - maximum amount of traffic that can be handled by a MSC CCTSc (Connection Capacity of TSC) - the maximum number of MSCs that can be connected to a TSC‘ F C T ~ C (Flow Capacity of TSC) - maximum amount of baffic that can be handled by a TSC’ BR - needed bit rate to transmit a communication channel. This parameter is required to dimension the transmission links (for PCM BR = 64000 bit/s). For different hierarchy levels different BRs may be required. This is also the case in the GSM system concentrator placing strategy - the algorithm places the concentrators either together with one of its terminals or using the COG algorithm for a cluster dimenfashion - the approach which can be chosen for network links dimensioning. The algorithm can do it to transmit all given node traffics or using a loss formula. The first way is suitable for mobile cellular telephone systems (GSM), in which the capacity of links between BTSs a BSCs may not restrict the available radio channels resources. At higher hierarchy level design, the traffic sources are not infinite due to the confined radio frequency resources and the number of installed TRXs (transceiver in BTS). The number of traffic sources is not much larger than the number of lines actually used in the busy hour. Therefore the use of Erlang-B formula is not an ad- visable way of links dimensioning. An appropriate dimensioning fashion for such cases is the Engset or Binomial dimensioning blocking - needed when a traffk loss formula is used LI , L h - the positions and traffics of nodes of the lower hierarchy type LI are given. The algorithm finds the cost-efficient number and positions of all nodes on the hierarchy levels level that fulfill L[ <level I Lh . All links on levels between Ll and L h will be optimised with regard to costs. Let Us as- sume that in a GSM network Ll is set to BTS and Lh is set to TSC. Then the algorithm designs the positions of all BSC, MSC and TSC stations and the connections between them CU - capacity usage parameter of concentrators. This parameter changes the initial number of concentrators placed in the network which can affect slightly the final solution.

N BTS+ BSC,

2.2 LINK DIMENSIONING IN HIERARCHICAL NETWORKS

The link dimensioning may be done most precisely using a standard procedure that consists of generating a routing table, collecting link traffics and dimensioning the links in the network. The routing table consists of paths between any two nodes. The criterion may be the shortest path, minimal number of hops, minimal delay or minimal link weights that may represent the link cost. Unfortu- nately in the process of the topology design we do not have the network topology so the routing table cannot be generated. For an economical network design we need to know the link cost and hence link dimensioning must be done. Also generating the routing table is not possible for large networks with several thousands of nodes because of excessive memory requirements.

In a star-star (or tree-tree) hierarchical network there exists only one possible path between any two nodes. Each terminal communicates with another terminal only using its concentrator. There are two possibilities of dimensioning the links: Either the link between the terminal and the concentrator is dimensioned to transmit all the traffic generated by the terminal (e.g. between BTSs and BSCc) or by using the well known Erlang or Engset loss formulas (between higher hierarchy levels). In the first case for a BTS this traffic is equal to traficgTS .

The trclfficBSC is considered to be the traffic “generated” by the “terminal” BSC in the next step.

traffic BSC = trafic BTS, +8SC (1) k

The links between MSCs and the TSC do not need to be dimensioned to the total sum of traffics of all BSCs connected to the MSC trafficMsc

because the MSC station has internal switching capabili- ties. Therefore the link between MSC and TSC is dimensioned only for the traffic

where traficmsc means the traffic that gets from the MSC to the network and

The number of BTS stations that are assigned to the BSC station BSCk we denote trafic Nm = C rraficgTs (4)

is the total network traffic of all BTSs. Network links may be dimensioned to cany all traffic

given by trafficBTs , ( I ) or (3) or using a loss formula. In the second case for the traffic given by trnfficgTs, (1) or

’ If only one TSC station is present in the network. this Panmeter should be set to infinity.

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(3) and a chosen value of the blocking parameter the required number of channels is found in traffic loss table.

If one needs to transmit a given number of channels between two nodes, one can usually do it using different link types (e.g. 64 kbitls, 2 MbiVs, 34 Mbitfs, 140 Mbit/s). Transmitting a given number of channels using different link types leads to different total link costs. Therefore to find the cost optimal solution for a given number of channels and the length of the link, different link type solutions are compared and the cheapest one is chosen. This procedure is done whenever the algorithm needs to cal- culate the link cost for a link or to dimension it.

2.3 ALGORITHM FLOW CHART

The flow chart of the algorithm is shown in Figure 2.

Initiation of the algorithm, input data filing, initiation of minimal network costs C O S ~ N ~ , cost,,,, and costpreV to infinity

The total number of BTS in the network N ~ s is found and the total network M i c rruficNm is calculated (4). The initial number of BSCs is calculated using

The algorithm consists of the following steps: 1.

2.

NBSC =

where ceil(x) rounds up the x, CU is the capacity usage, 0 c CU S 1 , CCB~C is the connection capacity of BSC and F C s s , is the traffic flow capacity of BSC. The NBSC given by (5 ) need not be sufficient. If

and such BSCk exists, that

but

NBsc given by ( 5 ) will be probably not sufficient. Therefore it is recommended to choose a smaller CU parameter (e.g. C U a . 6 ) . NBSC also presents the rnem- ory allocated to all BSC stations in the computer.

3. If NBSC > N m then N B S C : = N ~ S . Usually N 6 . y ~ c N s ~ and an initial concentrator location must be chosen. This is a well-known NP-hard problem with a complexity that grows rapidly with the number of concentrators. In this phase we use a principle similar to the “rolling snowballs” clustering approach [5]. For each BTS station BTSk its neighbour stations BTSi are

sorted according to the distance dk given by (9) (the BTSk is naturally one of BTSi stations):

ceil( CU CCBSC )

i=l dk = x d i s t ( B T S k , BTSi) (9)

where the BTSi stations assigned to the cluster of BTSk must fulfil the following conditions:

(10) C traficBTSi I FCssc . CU 1

where BTSi are the stations that are nearest to the BTSk and are not clustered to another BTS yet. The distance between the stations BTSk and BTSi can be calculated using X,Y coordinates or longitude (LNG) and latitude (LAT) node positions. With respect to the calculation speed we prefer to introduce a rectangular coordinate system. Following the sorting process the first BTSk is taken that has the smallest distance parameter dk (9). The first BSC station BSCl is placed together with B T S k . BTSk and the next BTSi stations are connected to BSCl as long as the conditions (10) and (11) are fulfilled. If this is not the case, another BSC is placed together with an unconnected BTS with smallest value of (9). This process continues until all BTS are assigned to a BSC or the number of BSCs (5) is reached.

A procedure follows that will be repeated several times. The positions of BSC stations are determined using one of the following approaches:

Using the COG (Center of Gravity) algorithm El51 but only for the BTSs assigned to the cluster of one BSC. The x,y coordinates of the BSC are

4.

N BRS 4 BSC C XBTS, . ~ ~ ~ J F C B T S ~

C ~ ~ ~ B C E T S ~

(12) i-1

8TS d 8SC ‘BSC = N

i=l

This approach is more suitable in case radio links are used to determine the approximate concentrator positions. In this case, the final placing depends on the terrain profile and the visibility between the stations.

5 1 1 ETT


costNET= cost,,, ; Return value := A store the best solution

algorithm initiation costNET= cost,Ie,= costprev = Itlfnity

7

t determine initial number of BSCs 1 2 9 3

sort BTS according to distance to nearest BTSs + clustering

I

3

1 BSC location 14 I I

remove all links 15 I I r assignment problem 1s I

add a BSC successful

I

I cost calculation I 15

< cost,,,, < c 0 s t p r e v p - + :ostpre, := cost,,," I z l A v

I I - 1 return value decision box B 1-

h

sort BSCs according to the upper fill rate parameter

I

I I try to save a BSC 1

unsavable successful

+ return value = C * I try to place new BSCs 19

I I

return v a l T k 1

9 recover the best solution 19

I I

[ change BTS order for assignment 1 Ifi I

I t return value := B

recover the best solution

designed network

Figure 2: Flow chart of the algorithm.

0 A BSC station may be placed together with any BTS station BTSk in such a way that the link costs in the cluster are minimized:

At this cik denotes the cost for the link between the i-th and k-th BTS when the link is dimensioned to transmit the traftic of the station BTSi .

After the BSC location problem has been solved all links between BTS and BSC stations are removed and the algorithm finds a new assignment of BTS to BSC stations. The following algorithm was designed to run quickly without sizable memory complexity and a very small gap to the optimal solution.

5. . Nm.Bsck

costs = C C i k = min i=l

(I4)

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To speed up the design process for each BTSk the pointers to its next r (maximum 10) BSCi are stored. Let the network have NBSC stations. Then

N g s ~ for Nssc < l o 10

r = { othervise

Also the link costs for links between BTSk and each BSCi are stored. The restriction of saving only the pointers to next 10 BSCs limits the memory requirements of the algorithm and accelerates the algorithm since not every BSC must be tested as a po- tential station the BTS can be assigned to. It could introduce a gap but in real networks this gap is not significant. The number of maximum ten nodes we received by experience. We tried running the algorithm with 20 example networks and experienced the gap between the solution with stored 20 nearest stations and x stored stations. This experience has shown that for x > 8 the gap is smaller than 0.2 %. We chose to store ten nearest stations, which is not much higher than eight, not to make the algorithm slower.

The BTS stations are assigned to their nearest BSCs similar to the well known Greedy algorithm. The order of this assignment is inverse to the result of the ordering process from step 3. That is at first the BTS with the greatest distance to the nearest BTSs is assigned to its nearest BSC. A difference to the Greedy algorithm arises when BTS,,, cannot be connected to its nearest BSC because there are not enough free capacities available' (Figure 3). The Greedy algorithm would assign this BTS,,, to another BSC nearby that has enough free capacities. The described algorithm reorganizes the assignment using an exchange procedure as follows: Let the BSC stations have the connection capacity CCBsC = 5 and a new station BTSnew must be assigned to a BSC (Figure 3). Five BTS stations are already connected to the stations BSCl and BSC2. The BSCl is the nearest BSC station to the BTS,,,. The BTSs connected to BSC, are disconnected from the BSCl one after another and connected to the nearest at most nine other BSCs. The goal of this procedure is to find out the cost of assignment reorganization in other stations. Let the cost to connect

BTS,, to BSC, isA

BTS,, to the nearest station with enough spare capacity BSC, be B,

BTSi to BSC, be C,

the connection and traffic flow capacities me not exceeded aftei connecting BTS,,, to the BSC.

516

BTSi to BSC3 b e D ,

BTSi to BSC, be E,

BTSj to BSC2 be F,

BTS to BSC, be G.

\ \ \ \

- BTS

_ - - - y _ - - - -

b / \D / BTSj

\

\

'K I -BSC

Figure 3: Local exchange by solving of the assignment problem.

Then the costs to connect BTS,,,, are

(15)

To solve this problem in a program, a function is em- ployed that finds a station BTSk with minimal X-Y within the cluster of given B S C j where X are costs to connect BTSk to another station and Yare the costs to connect BTSk to SSCj. In Figure 3 the connection capacity of BSC2 is fully exploited. Therefore to cal- culate the cost E, the function is called recursively for thz BSC2 station. In this way the value of G-F is found and added to E-C. The recursion depth is lim- ited to 3, which reduces the calculation time and pre- vents the possible stack overflow. This restriction can lead to another degradation of the assignment problem solution. Since solutions with multiple recursions are usually not the best anyway, the assignment solution is usually near optimal or optimal. To speed up the algorithm for each BSC. additional memory is allocated for several structures that consists of pointers to the BTS that can be connected to another BSC with lowest

C O S ~ B T S ~ ~ ~ = = min(B,A +(D- C I A + ( E - C ) + (G- F) . . )

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cost. The pointer to the other BSC and the cost of reorganization are also stored in the structure. Without the restrictions of the recursion depth and the maximum number of considered parents the algorithm would perform like the algorithm presented in [ 161 that solves the assignment problem optimally but requires much more CPU time and too much computer memory for very large networks. In case the algorithm is not able to connect the new station BrSnew it will remove all existing links between BTSs and BSCs, adds a BSC together with the first not-assigned BTS and step 5 is repeated again. The network costs C U S Z , ~ , ~ are calculated. If custnew < costp,, than costprev := cost,,,, and continue with step 4. Else continue with step 6. The criterion above could lead the algorithm into an infinite cy- cle because the last solutions might start oscillating between two solutions. Therefore another criterion is required. If the network costs oscillate between some values, the algorithm will continue with step 6 after the minimal value has been reached. The algorithm is started recursively for higher hierar-

chy levels. Using known positions of BSCs and their traffics (1) the positions of all MSCs are found and fi- nally the position of the TSC is found, the assignment problem is solved and the links are dimensicned to transmit the required traffics. If the TSC station were in reality not present in the network it would be reasonable to introduce such a fictitious station that would have infinite connection and traffic flow capacities. Then the algorithm places the MSC stations closer to each other and the link cost of mesh topology between MSCs would be lower. The higher hierarchy level design in this step is important because the costs of all nodes and links on higher hierarchy levels are considered in the cost calculation. The existence of links between BSC, MSC and TSC stations affects the final number and positions of BSCs.

If the network costs are smaller, the last network solution with minimal costs C U s t N m will be stored in the following memory efficient way: The x,y coordinates of all BSC stations are stored in a matrix NBSC x 2 . Each BTS remembers just an integer that represents the number of BSC the BTS is assigned to. The link dimensioning is not stored. This is done later after recovering the best solution. Also the location and assignment of nodes on higher hierarchy levels are not stored. This can be done again later without big CPU complexity because usually CCBSC >> 1 .

The BSC stations are sorted according to the upper fill rate parameter [8]

6.

7.

8.

If no BSC station exists the upper fill rate of which is smaller than the chosen value (e.g. 85 %) or if the algorithm has checked unsuccessfully all BSCs with frBsc

smaller that the chosen value3, the algorithm will continue with step 9. Otherwise the BSCk with the smallest value of fill rate (16) is chosen to be removed from the network. From the principle of the “Drop” algorithm the concentrator that brings the biggest cost reduction should be removed first. This is not done because i t would be necessary to check all possibilities and it would make the algorithm much slower. It is not guar- anteed that the final solution would be optimal for the “Drop” algorithm anyway. If two different BSC stations have the same value of the upper fill rate parameter, first the BSC with a bigger distance to the nearest BTSs will be chosen. This results in a better final solution especially for small values of CCBSC. If the algorithm succeeds in connecting all BTSs, which were connected to BSCk , to other BSCs, the station BSCk will be removed from the network and the steps 4-8 are repeated. If the solution without BSCk is more expensive than with it, BSCk is added back into the network and marked as non-removable (is stored in short-term mem-

9. The biggest drawback of the algorithm described above is the possibility to stay in a local optimal solution. To reduce this possibility in case the node traffics of BTSs are much different, another test is proposed: If a BTSk is assigned to a BSC that is placed together with BTSj and the traffic flow of BTSk is significantly bigger than the one of BTSj ‘, a BSC will be placed together with BTSk and steps 3-8 are repeated. If the described’BTSk does not exist or all such BTS are marked as tested, continue with step 10.

10. Since the assignment algorithm is non-optimal, its results depend on the order in which the BTS stations are assigned. Therefore the order of BTSs is changed and steps 5-7 are repeated again. Then continue with

1 1. If C U S ~ N ~ > custnew the cheapest solution will be re- covered.

12. Due to restrictions in step 5 the assignment solution is allowed to be non-optimal. Therefore the algorithm tries to find a better solution using a local exchange procedure. This method rarely leads to a better solution.

ory).

step 11.

3 COMPUTATIONAL RESULTS

The algorithm has been programmed in the object- oriented C++ language. It has been tested with a lot of

’ to store such BSCs uses the algorithm short time memory like the

the link that transmits traffic of BTSk is more expensive than a link ‘Tabu search” algorithm [231.[181. [241.

that would transmit the tnFfic of STS, . 4

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J. Petrek, V. Siedt

networks differing in size and input parameters. It is as- certained that the gap between solutions with different order of nodes is not bigger than 1 %. The computational tests have revealed the calculation time superiority of the algorithm to “Simulated annealing” and ‘Tabu search”. Thereby the algorithm requires much less CPU time and little memory (about 500 Byteshode). In this section we present the advantage of the high complexity of the algorithm.

In the following examples the “Deutsche Telecom” tariff link functions are used. The costs are a non-linear function of the link lengths. The connection capacities are set to 10 ( CCasc = C C M ~ C = lo). The traffic flow capacities of the concentrators are chosen high enough not to affect the final solution.

Figure 4: Network NET,. node traffic dimensioning. I step running: BTS-TSC, costs = 1.028. lo7 .

The first comparison presents the advantage of designing several hierarchy levels in one step. In Figure 4

and Figure 5 the links are dimensioned to transmit all traffic of terminal nodes. The difference between the networks is that all hierarchy levels were designed in one step (ti = B T S , L h =TSC ) for network in Figure 4. In Figure 5 the algorithm has been run in three steps:

1. Li = BTS, Lh = BSC 2. L/ =BSC, Lh =MSC 3. L, =MSC, L , =TSC

The costs of the network in Figure 4 are about 2.4% higher compared to the costs of the solution in Figure 5 because in the first step of more steps design the link and node costs for higher hierarchy levels are not taken into account.

Figure 5: Network NET,, node traffic dimensioning, 3 steps running: BTS-BSC, BSC-MSC, MSC-TSC.

The behaviour of the algorithm was tested on many networks with different number of nodes, sizes and node

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traffics. In this paper only three sample networks are presented. In the Table l"NET," is the network in Table 4 and Table 5, are network costs in lo6 D W month for one design step and COS~~,,~,,~ are the costs for the same network designed in three steps. Each BTS carries the traffic of 30 Erlangs. The gap between solutions proposed in one step and three steps depends on many parameters,

which depend on random BTS stations arrangement. Therefore only statistical processing is possible. For 20 experimental networks with the same BTS traffics 30 Er- langs (a 2 Mbitfs link between each BTS and i ts BSC) the average gap was 3.46 % and the solution with three-step design was newer better than those with one step design.

Table I: Costs are in I $ DM/Month.

Figure 6: Erlang dimensioning, 1 step running: 7 BTS-TSC, Figure 7: Erlang dimensioning, 3 steps running: BTS-BSC, BSC-MSC, MSC-TSC.costs= 1 .689.107. Li = BTS, Lh = TSC , costs= 1.532.10 1.532.

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Table 2: Costs are in lo“ DM/hfot~tl~.

Net\placing NET, NET7 NETT

The problem of not-one-step design causes even greater gap if link costs are higher in comparison to costs of BSC nodes. The networks in Figure 6 and Figure 7 are dimensioned for equal node traffic values (30 Erlangs) using the Erlang loss formula, blocking 0.001, This means the link capacities and costs are higher than in the previ- ous case. Therefore in the first step of three-steps design process it is cheaper to place more concentrators rather than having longer links. Hence CC,, are often unused and the algorithm places a lot of concentrators. Therefore the network created in three steps (Figure 7) is 10.242 % more expensive than the solution created in one step design ( f, = BTS. 4, = TSC ,Figure 6 ) . This is a difference of 1,568,800 DM per month. In Table 2 the results for three networks are compared. The node traffic for all BTSs was set to 30 Erlangs, blocking parameter was set to 0.001. The average gap for 20 tested networks was 9.4 % and the solution after three steps of design was newer superior to our one step design.

Table 3 presents the importance of a correct concentrator location. Only one hierarchy level is designed here ( L ! = B T S , L h = S S C ) . For each of the considered networks three concentrator placing strategies are regarded: 1 . concentrators are placed together with some terminals.

Hence there is one link saved per concentrator. Therefore the solution is usually beneficial

2. concentrators are placed using the COG algorithm for known clusters. Hereby the coordinates and traffics of the terminals belonging to a cluster are taken into account

3. the positions of all concentrators are determined using the COG algorithm [ 151 first. Afterwards for objectiv- ity the assignment problem is solved using the assignment approach of the described algorithm.

In the last column the gap of the 3rd to the 2nd approach is shown. Obviously the concentrator placing is crucially important for an economical network design.

The computation time depends on many factors. The CPU times’ for CC,sc = 10 and one hierarchy level design L, = BTS. Lh = BSC are shown in Table 4. For multiple hierarchy levels the CPU times are longer: for CCBSC = C C M ~ C = 10 and L, = BTS, Lh = TSC the CPU times shown in Table 4 are for large networks to be multiplied by a coefficient of about 2-3.

1 2 3 3-2[%] 5.423 5.689 6.104 7.287 2.088 2.142 2.315 8.065 2.457 2.587 2.762 6.730

Table 3: Costs are in 106 DM/Month.

By smaller values of CC of concentrators the coefficient could be higher and vice versa since the CPU time is approximately proportional to N&$ . N ~ ~ ~ . The presented assignment problem solution is compared with the one generated by “Simulated annealing”, “Tabu search” and different heuristics. The algorithm described here found a better or equal solution in about 95 ’% of all cases. The CPU times were much shorter than those of “Simulated annealing” or “Tabu search”. The network costs and CPU times for the described algorithm and “Simulated annealing” are compared in Table 5 for networks NETA..NETc with 60, 40 and 30 BTSs, and traffics 30Erlangs per node. The CPU time and quality of solution of the “Simulated annealing” algorithm depend on the cooling schedule factor c [25] . Therefore the results of “Simulated annealing” were calculated for four different factors. The time is in seconds, the costs in lo6 DWmonth.

4 CONCLUSION The algorithm described in this paper is suitable for

the design of large hierarchical networks with a very low gap to the optimal solution. It has been integrated into “COMPOSIS”’ - the communication networks planning tool developed at Comnets, University of Aachen. Other similar algorithms for the design of hierarchical networks are:

a network expansion algorithm (connecting new nodes to an existent network) that uses the assignment principle described here

an algorithm that converts the star topology, designed by the described algorithm, to cost optimized tree to-

In Figure 8 is shown a designed network with 3700 PO10,oY

BTS using the described algorithm.

are measured on SUN Ultn I

520 ETT


NBTS 55 125 300 1500

Table 4: Algorithm running times.

2200 3700 4500 . Comp. time 1.48 s 6.3 s 48.08 s

Eile Edit giew ~ l g a i t h m s Analysis Preferences Help II

130.75 min 9.45 hour 74.75 hour 191.5 hour

I Delete Node/Lin k I ( -26,52, 70.26)

Figure 8: A network with 3700 BTS designed by the algorithm.

Vol. 12, No. 6, November-December 2001 52 I

J . Petrek, V. Siedt

Algorithm

NETA N E T R

*

Tdde 5: Assigriinmt solrr tiori compiireil to “Simulated Annenlirig ” olgoritlim.

Proposed algorithm SA 4 . 9 SA d . 9 9 9 SA d . 9 9 9 9 SA -0.99999 time costs time costs time costs time costs time Costs 0.76 6.6961 5.8 6.9441 9.72 6.9415 40.3 6.6961 254 6.6965 0.6 2.7209 5.3 2.7947 8.1 2.8116 63.3 2.7351 267 2.7250

ACKNOWLEDGMENT

The authors would like to thank Mr. Alexander Speetzen and other members of ‘TND’ group, which is concerned with developing of Network topology design tool at Communication Networks - Aachen University of Technology, for cooperation and all advice.

Manuscript received on Febrnar), 2, 1999.

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A large hierarchical network star—star topology design algorithm