Network topology management in a mobile-switch ATM network: dynamic partition algorithms

INTERNATIONAL JOURNAL OF NETWORK MANAGEMENTInt. J. Network Mgmt 2002; 12: 99–115 (DOI: 10.1002/nem.425)

Network topology management in a mobile-switchATM network: dynamic partition algorithms

By Sheng-Tzong Cheng,Ł C. Chen, C. Li and Chia-Mei Chen

In this paper we propose two partition algorithms. The main policy of thealgorithms is finding out the area(s) in which mobile switches congregatewithin a peer group. Copyright 2002 John Wiley & Sons, Ltd.

Introduction

T he recent successful developments ofwireless and mobile communication havebrought new opportunities for potential

applications of ATM. Through rapid advancesof VLSI technology, devices could use ATMtechnology range from mobile switching systemson aircraft and ships to portable devices, such asmobile phones or personal digital assistants. Now,wireless technology changes our daily life just likethe telephone one hundred years ago. However,today’s ATM has been designed to provide acomplete networking solution for fixed networks,but the use of ATM in a mobile environment raisesmany challenging technical issues.

The ATM Forum’s Private Network-NetworkInterface (PNNI) protocol13 has the characteristicto accommodate a mobile environment. The PNNItopology favors a hierarchical structure of peergroups on different levels. The combination ofthe PNNI hierarchy and the application scenarioresults in the network structure we study in thispaper: a PNNI network structure with mobileATM switches. In this wireless ATM environment,mobile switches can move from one place toanother and cause growth or reduction of peergroup size. When the peer group grows to alarge size, it will violate the purpose of the PNNIstandard, which uses a hierarchical structure torestrict unnecessary information flooding.

Estimation of a PNNI topology for minimizingthe routing and tracking costs suggest an optimalPNNI structure with several hierarchical levelsand a few nodes on the lowest (physical) peergroups.14,15 In order to maintain the optimal net-work structure and lower the overhead, we needa peer group partition scheme to split a large peergroup. An obvious aggregation of mobile switchesin a peer group means that mobile switches inthis region seldom leave. When the peer group istoo large and needs to be partitioned, it intuitivelyseems beneficial to set this region apart from thispeer group. It is of interest to know the effect ofpeer group partition based on our hypothesis.

In this paper we propose two peer grouppartition algorithms. For a given topology and amobile switch density distribution, algorithm oneproceeds from one link with the highest movingprobability while algorithm two proceeds from thehighest and second highest ones concurrently tocluster the ATM switches. An analytical modelused to analyze the tracking and locating costs ofa PNNI topology is also proposed. This can beused to measure the gain ratio of network costsbetween the original topology and the partitionedone under a different call-to-mobility ratio.

System ModelIn this paper we assume that the entire network

is deployed on a bounded geographic plane, while

Sheng-Tzong Cheng, C. Chen and C. Li tench at the National Cheng Kung University, Tainan, Taiwan, ROC and Chia-Mei Chen at theNational Sun Yat-Sen University, Taiwan, ROC.

ŁCorrespondence to: Sheng-Tzong Cheng, Associate Professor, National Cheng Kung University, Institute of Information Engineering, Ta-Hsuen

Road, Tainan, Taiwan, ROC.

Published online 8 January 2002

Copyright 2002 John Wiley & Sons, Ltd.

100 S.-T. CHENG ET AL.

the physical network structure of ATM switchescan have an arbitrary topology. Based on the PNNIprotocol, all physical switches are grouped intopeer groups to construct a hierarchical networkstructure defined in the PNNI standard. Weassume that there are n switches in the networkthat was constructed into the PNNI hierarchy withL levels.

Each fixed switch on the ground is equippedwith wireless communication devices so a mobileswitch can connect to it. We also assume that themobile switch always connects to the ‘nearest’ fixedswitch. When a mobile switch moves around theplane, it always moves from the current switchit connects to one of the immediate neighboringswitches of the current switch, i.e., a mobile switchalways moves along the links between two adjacentfixed switches.

In our system model, the probability that amobile switch moves from current switch i toits adjacent switch j, denoted by prob�i ! j�, isgiven for any switch i. By using linear algebrato solve simultaneous equations, we can calculatethe probability prob�stay i� that a mobile switchstays and connects to the fixed switch i. We definethe switch density distribution as the distributionof probability prob�stay i� of each fixed switch onthe geographic plane. Assume that the switchdensity distribution is not uniform. Because theprobabilities that mobile switches move toward aregion are high, the switch density of this regionwill be relatively higher than that of other regions.

T he Cost of tracking a moving switchwithin a large peer group is very high,

especially when the mobile switch movesfrequently.

The optimal PNNI hierarchy is studied and asuggested peer group size is also proposed. Inorder to reconstruct a PNNI hierarchy, a peer groupmust able to be reconfigured. In the PNNI networkhierarchy, a peer group identifier identifies a peergroup. In the PNNI specifications, peer groupidentifiers are specified only at configurationtime. Each node has a peer group identifier.Neighbor nodes exchange peer group identifiers

in Hello packets. They belong to the samepeer group, if they have the same peer groupidentifier. Otherwise, they belong to differentpeer groups. Hello packets are sent periodicallybetween neighbor nodes, so nodes can detect thenetwork change quickly. Suppose that peer groupidentifiers can also be assigned during run timewhen a PGL discovers that the peer group sizegrows above or falls below a threshold, it splits thelarge peer group or merges two small peer groups.A PGL can issue peer group reconfigurationcommands to some nodes in its peer group andreassigns their peer group identifiers. New peergroups can be constructed after the nodes exchangeHello packets; then the peer group leader electionprocess takes place dynamically to choose a newpeer group leader. We presume PNNI standardwill support this regrouping mechanism in a laterversion.

In this paper we propose a peer group parti-tion algorithm to guide the regrouping mecha-nism. Once the mobile switch moves, the topologyupdate messages must be flooded to inform allswitches in this peer group to modify their topol-ogy databases. The cost of tracking a moving switchwithin a large peer group is very high, especiallywhen the mobile switch moves frequently. If somemobile switches are congregated in a region of alarge peer group to be partitioned, we intuitivelythink that splitting this region to form a new peergroup would be beneficial. We believe that restrict-ing information flooding in a high switch densityregion is a good direction for our partition algo-rithm. Based on this hypothesis and system model,we will develop our partition algorithms and ana-lyze its performance.We describe the problem in mathematical

notations as follows:We have a PNNI Hierarchy H

A graphic representation of the topology forthe all peer group T

A graphic representation of the topology forthe target peer group G

G D �V, E�, 8hu, vi 2 E, The probability of hu, viPuv

is givenGiven a subset S ² V, a new H’ can be formedGraph G and Topology T won’t be changedH’ depends on S, so that H’ can be formed by H�S�Find a subset S, Min�Ctotal�H�S��

The goal of our problem is to find the minimumtotal cost of the new hierarchy.

Copyright 2002 John Wiley & Sons, Ltd. Int. J. Network Mgmt 2002; 12:99–115

NETWORK TOPOLOGY MANAGEMENT IN A MOBILE-SWITCH ATM NETWORK 101

A B

PG(A) PG(B)

A.1

A.2

A.3

A.4

B.1

B.2

PG(A.1)

PG(A.2)

PG(A.3)

PG(A.4)

PG(B.1)

PG(B.2)

PNNI hierarchy before partition

A B

PG(A) PG(B)

A.1

A.2

A.3

A.4B.1

B.2

PG(A.1)

PG(A.2)

PG(A.3)

PG(A.4)

PG(B.1)

PG(B.2)

B.3

PNNI hierarchy after partition

G

T

H H′

G

SW( d )

SW(a)

SW(b)

SW(c)SW(i)

prob(i→b)

prob(i→a)

prob(i→c)

prob(i→d)

Figure 1. An example of move directions and moveprobabilities

—Partition Algorithm—

In the previous section, we discussed the overallmobility model. In this section, we look in moredetail of our model and the parameters used forpartition. In a WATM PNNI hierarchy many fixed

or mobile switches group together to form a peergroup. In the range of each fixed/mobile switchescovered, there might be many mobile switches(MSWs) communicating to the fixed/mobileswitches or handoff with other switches which arein the same peer group or not. The MSWs move-ment behavior may be a routine or a random walk,but it is hard to formulate all of them. To design themobility patterns of network areas, we first referto a single mobile switch’s behavior. The topologyused to illustrate the MSWs’ moving directionsbetween switches is the grid architecture. In thisgrid architecture, switch (i) has as neighbors switch(a), switch (b), switch (c) and switch (d), as shownin Figure 1. Even though the grid architecture isused for illustration, the same analysis applies toany two-dimensional architecture. To show themobility pattern for the arbitrary mobility model,consider the condition shown in Figure 2. Thetopology is a grid architecture and an MSW onswitch (i) can only move to switch (a), switch (b),switch (c), or switch (d). If prob�i ! a�, prob�i ! b�,prob�i ! c�, prob�i ! d�, denotes the condi-tional probability that a mobile switch moves fromswitch (i) to switch (a), switch (b), switch (c), and



SW(i)

SW(a)

SW(b) SW(c)

SW(d)

prob(a->i)

prob(b->i) prob(c->i)

prob(d->i)

Figure 2. An example for stay probability

switch (d), respectively, then prob�i ! a�Cprob�i !b� C prob�i ! c� C prob�i ! d� D 1.

Every MSW moving from one switch to otherswitches makes a contribution to the move proba-bility. In the other point of view, the MSW moveprobability is illustrated in Figure 2. The probabil-ities of visiting the switch (i) from the switch (a),switch (b), switch (c) and switch (d) are prob�a ! i�,prob�b ! i�, prob�c ! i�, prob�d ! i� respectively.So the equation of stay probability prob�stay i� iswritten as

prob�stay i� D prob�stay a�Łprob�a ! i�

C prob�stay b�Łprob�b ! i�

C prob�stay c�Łprob�c ! i�

C prob�stay b�Łprob�b ! i� �1�

Every switch (i) in this grid architecture canform an equation as shown in equation (1).According to the equations, we can solve the stayprobability of each switch (i). The mobility patternscan be constructed by summarizing the mobilityprobability between every two adjacent switchesin the architecture. The mobility probability fromswitch (i) to switch (j) equals the stay probability ofswitch (i) multiplied by the move probability fromswitch (i) to switch (j). The mobility weight betweenswitch (i) and switch (j) equals the ProbMobility(i ! j) added to ProbMobility (j ! i). The mobilityprobability and mobility weight are given in

ProbMobility�i ! j� D prob�stay i�Łprob�i ! j� �2�

WeightMobility�i, j� D ProbMobility�i ! j�

C ProbMobility�j ! i� �3�

The high mobility weight means that the MSWsmove frequently between these two switches. It isbetter to group these two switches in the same peergroup, if the mobile switches have high movementfrequency between any two of fixed or mobileswitches. This will reduce the cost of updatingrelated peer groups. For this reason, we proposetwo heuristic mobility weight partition algorithmsto the problem.

Before we propose our algorithm, we mustfirst show how to generate prob�i ! j�. Theprob�i ! j� means the move probability fromswitch (i) to switch (j), as we describe above. Onthe architecture, we may randomly assign manypeaks to reflect a trend of mobility density. Thesepeaks will affect the move probability of eachswitch in the area. In Figure 3, we present anexample of the grid architecture. Switch (i) hasfour neighbors, switch (a), switch (b), switch (c)and switch (d), where Pk, k D 1, 2, 3, 4, denotesthe move probability from switch (i) to switch (a),switch (b), switch (c) and switch (d). We dividethe architecture into four regions, N, W, E, S bythe bisector of move probability vectors P1, P2, P3,and P4. The bisector lines are L1 and L2 becausethe four move probability vectors are orthogonalto each other. Each region has a base score, forexample region N’s base score is 1, and then findthe peaks in each region if they exist. If we find apeak in the region, we then add the result of peakintensity divided by the distance from the peak to

SW(a)

SW(c)SW(i)

SW(b)

SW(d)

N

W E

S

Peak 1

Peak 2

Peak 3

L2 L1

r1

r2

r3

P2

P4

P3

P1

Figure 3. An example for calculating the moveprobability



E�G� Edge set of graph G

V �G� Vertex set of graph G

VE A set of visited edges; Initialized to �

S1 A set of vertices; The final result of PARTITION ONE; Initialized to �

e Initial selected edge

CE A set of considered edges; Initialized to �

NE�S� A set of neighbor edges of the nodes in the set S, but not theedges between nodes in the set S

INPUT: A set of vertices;

OUTPUT: A set of neighbor edges

Max edge(E) An edge with the maximum weight among all edges in edgeset E

INPUT: A set of edges;

OUTPUT: The max edge of the edge set

DECOMPOSITION(S) Procedure for partition starts with vertex set S


OUTPUT: SUCCESS or FAILED

SUCCESS and FAIL Boolean type; Return from the DECOMPOSITION procedure

R Fraction (ratio); 0 < r � 0.5

L A vertex incident to the candidate edge

Candidate edge A Max edge of considered edges

Threshold R Ł jV �G�j; Fraction multiple to the number of vertices in graph G

Table 1. Notations used in PARTITION ONE()

switch (i) as a score to the region. In Figure 3, thereare two peaks in the W region. We assume thatthe intensity of Peak1 is T1, the intensity of Peak2

is T2, and the intensity of Peak3 is T3, where T1,T2, T3 are given. By calculation, the (T1/r1 C T2/r2)bonus scores are added to the W region. There is apeak on the bisector line of the P1 and P3 vectors.So we both add the bonus score (T3/r3) to the Nregion and the E region. Now the total scores of 4regions are

The total score of the N region D 1 C �T3/r3�.

The total score of the W region D 1 C �T1/r1 C T2/r2�.

The total score of the E region D 1 C �T3/r3�.

The total score of the S region D 1.

We then we calculate Pk, k D 1, 2, 3, 4, by thefollowing equations.

P1 D The total score of N region∑The total score of all regions

,

P2 D The total score of W region∑The total score of all regions

,

P3 D The total score of E region∑The total score of all regions

,

P4 D The total score of S region∑The total score of all regions

Each switch in the graph can obtain moveprobabilities of its own. If the switch has manyneighbors, we can also find the bisector of eachmove probability vector and perform the samesteps as mentioned above. Finally, we constructthe mobility pattern of the graph and start thepartition algorithm. In the following we proposetwo partition algorithms of a given graph.



—Partition I—

The basic approach of our algorithm is simi-lar to the Greedy Algorithm. Algorithm PARTI-TION ONE starts with finding the maximum edgeof the graph G, and adds both the incident nodesof maximum edge into RESULT VERTEX set andthe maximum edge into RESULT EDGE set. Thenchoose the maximum edge adjacent to the RESULTEDGE set. This edge is called the candidate edge.Before adding the vertices that are incident to the

PARTITION ONE(G, R, S1, VE)f//INPUT: A graph G, fraction R, initial vertex set S1, and initial visited edge set//OUTPUT: Bi-partition RESULT VERTEX SET S1 D fv1, v2, v3, . . .g

let iniS1 D S1;while �E�G� � VE 6D ��f

e D max edge �E�G� � VE�;let e be h vi, vji;let S1 D S1 [ fvi, vjg;if (DECOMPOSITION�S1� DD SUCCESS�

return S1 // DONE SUCCESSFUL!elsef // FAILED this time

VE D VE [ fh vi, vjig;S1 D iniS1; //Restore the iniS1 into S1

ggreturn NULL

// FAILED ! find no solutiongBoolean DECOMPOSITION(S1)f

CE D NE�S1�while (CE 6D �)f

L D fv j h u, vi D max edge(CE) & v /2 S1 & u 2 S1gif �V�G� � �S1 [ L� is CONNECTED)f

S1 D S1 [ Lif (j S1j < R Ł j V�G�j)

CE D NE�S1� // A new considered edge set is createdelse

return SUCCESSgelse // not CONNECTED;

// drop the fh u, vig from the candidate edge set CECE D CE � fh u, vig

g //end of whilereturn FAILED

g

Table 2. Pseudo-code of the algorithm PARTITION ONE

candidate edge to the RESULT VERTEX set, wemust check whether the vertices are connected toeach other. The vertices we check are in graph Gbut not in the RESULT VERTEX set or incident tothe candidate edge.

If connected, we add the vertices to theRESULT VERTEX set and the candidate edge to theRESULT EDGE set. Then, we choose the maximumedge adjacent to the RESULT EDGE set repeatedly.We check connections of the graph and then addthe incident vertices to the RESULT VERTEX set.



Until the number of vertices in the RESULT VER-TEX set equals RŁjV�G�j, the algorithm stops. If theresult of checking connections is not connected, wefind the maximum edge adjacent to the RESULTEDGE set in the rest of the candidate edge set. Ifall the edges adjacent to the RESULT EDGE set letthe graph besides of the RESULT EDGE set andcandidate edge itself to be not connected. Thenwe choose next candidate edge of the graph Gto be the initial maximum edge and perform thePARTITION ONE algorithm again.

The S1 group will contain the total number oftarget groups multiplied by to the ratio R. ThePARTITION ONE algorithm can handle an unbal-anced bi-partition according to the given ratio R.The ratio R we set is 0.5 because we want tomaintain the symmetrics of the PNNI hierarchystructure. In a balanced PNNI hierarchy structure,the peer groups in the same hierarchy level arealmost the same size. The ratio R can be changeddynamically and arbitrarily, depending on the den-sity distribution behavior of the target peer group.It can be up to 0.5, because the density distributionspreads normally on average. If there is only asmall percentage of high density in the distribu-tion graph, the ratio R can be set a small value.Table 1 summarizes the notations used in the algo-rithm PARITION ONE. The pseudo-code of thealgorithm PARTITION ONE is shown in Table 2.

—Partition II—

This approach is similar to PARTITION ONE.But it starts with finding the maximum edge ofgraph G for group A, adds the incident verticesof the maximum edge to the RESULT VERTEX setand the maximum edge to the RESULT EDGE set.In addition, we find the second maximum edge ofgraph G for group B, and perform the same steps asfor group A. We choose a maximum edge adjacentto the RESULT EDGE set of group A. The selectededge is called the candidate edge of group A.

Before we add the vertices incident to thecandidate edge to RESULT VERTEX set, we mustcheck whether the vertices are connected. Thevertices are in graph G but not in the RESULTVERTEX SET of groups A and B or incident to thecandidate edge. We perform the same step to addthe vertices which are incident to the candidateedge of group B to its RESULT VERTEX set. The

following steps of group A and group B are similarto Algorithm I. We add the vertex to the RESULTVERTEX set, after checking the connections of thegraph. If the candidate edge of groups A and Bare the same, we must check the total number ofvertices in groups A and B.

If the summation of vertices in group A andgroup B is less than the threshold, we will combinegroup A and group B into one group S1 and useAlgorithm I with the initial edge set S1. If thesummation exceeds the threshold value, group Bmust find the other candidate edge. The algorithmwill be stopped until the number of vertices ofgroup A reaches the threshold. Finally the solutionis RESULT VERTEX set of group A.

The PARTITION TWO algorithm is applied tothe graph presented in Figure 4. We show howit works step by step. The notations used in thealgorithm are summarized in Table 3. Accordingto the first step of the PARTITION TWO algorithm,we select the maximum and the second edge in thegraph, but they cannot be adjacent to each other.Then we can assign the value eA D hi, ji, eB D hg, ki,SA D fi, jg, iniSA D �, SB D fg, kg, and iniSB D �.Second, the algorithm runs the sub-procedureDECOMPOSITION2 to find the candidate vertexLA with maximum mobility weight between thevertices for SA, and LB with maximum mobilityweight between the vertices for SB. We find that

a b c d

e f g h

i l

p

4 6 1 0

3

7

11

6

14

10

9

9

6

8

6

4

5

8

12

10

13

5

11

7

3

j

m n o

k

Figure 4. An example for PARTITION TWO



E�G� Edge set of graph G

V �G� Vertex set of graph G

VEA A set of visited edges of group A; Initialized to �

VEB A set of visited edges of group B; Initialized to �

SA A set of vertices in group A; The final result of the PARTITION TWO;Initialized to �

SB A set of vertices in group B; Initialized to �

eA Initial selected edge of group A

eB Initial selected edge of group B

CEA A set of Considered edges of group A; Initialized to �

CEB A set of Considered edges of group B; Initialized to �

NE�S� A set of neighbor edges of the nodes in the set S, but not theedges between nodes in the set S


OUTPUT: A set of neighbor edges

Max edge(E) An edge with the maximum weight among all edges in edgeset E

INPUT: A set of edges;

OUTPUT: The max edge of the edge set

DECOMPOSITION2(S) Procedure for partition starts with vertex set S


OUTPUT: SUCCESS or FAILED

Check(S1, S2) Check whether the number of the vertices in S1 and S2 is lessthan the threshold, and the graph V �G� � �S1 [ S2� is connected;

If satisfied with some conditions in Check function, then do thePARTITION ONE�G, R, S1, VE� function

Check Connected(S1, S2)

Check whether the graph V �G� � �S1 [ S2) is connected

SUCCESS and FAIL Boolean type; Return from the DECOMPOSITION procedure

CFTS Boolean type; It means Can’t Find The Solution with the initial SA,SB edge pairs

R Fraction (ratio); 0 < r � 0.5

LA A vertex incident to the candidate edge of group A

LB A vertex incident to the candidate edge of group B

Candidate edge A Max edge of considered edges

Threshold R Ł jV �G�j; Fraction multiple to the number of vertices in graph G

Table 3. Notations used in PARTITION TWO()

LA D n, and then check whether the graph V�G�-(SA [SB [LA) is connected. Unfortunately, the resultof checking is not connected. We must find the

next candidate L, L D m, and the result of checkingis connected. Then the vertices set SA D fi, j, mgnow, but SA has not yet reaches the threshold.



Next we find that LB D c, and then check whetherthe graph V�G�-(SA [ SB [ LA [ LB) is connected.If not, we find the next candidate LB, LB D f .The vertex j is not considered, because the vertexis in SB. We then check the connection statusof the graph V�G�-(SA [ SB [ LA [ LB), i.e. V�G�-fg, k, f , i, j, mg. If it is connected, we can performthe sub-procedure DECOMPOSITIION2 again tofind the next candidate vertex LA and LB untilthe number of the vertices set SA reaches thethreshold. In order, we find the vertices fn, o, pgare qualified to add to the set SA. At the sametime, the vertices fe, a, lg are qualified to add tothe SB. Finally, the SA vertices set cannot findthe candidate LA to join in. In this situation, thePARTITION TWO algorithm must run again withthe same SA but finding another SB for starting. Westart with the value eA D hi, ji, eB D hd, ni, SA D fi, jg,iniSA D �, SB D fd, ng, and iniSB D �. Performingthe algorithm steps in order, then we can find thatthe vertices fm, n, k, e, o, pg are qualified to join setSA. At the same time, the vertices fc, g, f , l, b, ag arequalified to join SB. Finally, the RESULT VERTEXset SA D fi, j, m, n, k, e, o, pg, and the number ofvertices in SA equals the threshold. Then thePARTITION TWO algorithm stops and outputsthe SA.

PARTITION TWOfwhile �E�G� � VE A 6D ��f

e1 D max edge �E�G� � VE A�;let e1 be h vi , vji; S A D S A [ vi , vj ; iniS A D S A;e2 D max edge �E�G� � VE A � VE B � NE�S A��;let e2 be h vm, vni; S B D S B [ fvm, vng; iniS B D S B;if (DECOMPOSITION2(S A, S B� D SUCCESS)

return S A // DONE SUCCESSFUL!else if �E�G� � VE A � fh vi , vjig � fh vm, vnig 6D �)f // FAILED this time

VE B D VE B [ fh vm, vnig;S A D iniS A; S B D iniS B

gelsef

VE A D VE A [ fh vi , vjig; VE B D �;S A D iniS A; S B D iniS B

gg // end of while// FAILED ! find no solution

g

Table 4. Pseudo-code of the algorithm PARTITION TWO

In some cases, starting with an SA, SB vertex pairwill not find the answer. Then we must find thenext possible SB and perform the procedure again.If all the possible SB are used, then we change theother SA and try again. If we still cannot find thesolution with all possible pairs in graph G, thenwe partition graph G with the PARTITION ONEmethod. In other cases, we find the candidatevertex LA and LB are the same. We must checkthe total number of vertices in SA, SB, and LA. If thevalue is less than the threshold, we combine the SA,SB, and LA into S1, and perform the PARTITIOIN-ONE algorithm with S1 for starting. If the valueis more than the threshold, we find another LB

and continue the algorithm. If no other availablecandidate vertex can be used, we must start withother vertices sets SA and SB. The pseudo-code ofthe algorithm PARTITION TWO is illustrated inTable 4.

Analytical ModelIn order to evaluate a network structure, we

propose an analytical model to measure the costsof a given network structure based on the PNNIprotocol. This analytical model is divided into twoparts: tracking cost model and locating cost.



int DECOMPOSITION2(S A, S B)fwhile (!CTFS)f

if((CE A D NE�S A�� 6D �)f // find next candidate vertex of set S A

L A D fv j v /2 S A [ S B & u 2 S A & h u, vi D max edge�CE A�gif (L A D �) Check (S A, S B) // all neighbors are vertices in the set S B

else Check Connected (S A [ L A, S B)gelse Check (S A, S B) // no candidate vertex is foundif((CE B D NE�S B�� 6D �)f // find next candidate vertex of set S B

L B D fy j y /2 S B [ S A & x 2 S B & h x, yi D max edge�CE B�gif (L B D �) Check(S A, S B) // all neighbors are vertices in the set S A

else Check Connected (S A [ L A, S B [ L B�while (L B D L A & CE B 6D ��f // candidate of set S A S B are the same

if (j S Aj C j S Bj C j L Aj < r Ł j V �G�j�fS1 D S A [ S B [ L A;VE D fh m, ni j h m, ni 2 E�G� & m 2 S A & n /2 S Bg;Do Partition I starting with S1 and VE

gelsef // check next candidate vertex of set S B

L B D fy j y /2 S B [ S A [ L A & x 2 S B & h x, yi D max edge�CE B�gif (L B D ��f

CFTS D true; breakgelse Check Connected(S A [ L A, S B [ L B�

gg // end of while

g // end of ifelse Check(S A [ L A, S B) // no candidate node is foundS A D S A [ L A;S B D S B [ L B;if (j S Aj < R Ł j V �G�j� // DO AGAINelse return SUCCESS

g // end of whilereturn FAILED

gCheck(S A, S B)f // procedure of Check the size of the S A and S B; Check the graph is connected;

if (j S Aj C j S Bj � R Ł j V �G�j)if�V �G� � �S A [ S B) is CONNECTED)f

S1 D S A [ S B;VE D fh m, nij h m, ni 2 E�G� & m 2 S A & n /2 S Bg;Do Partition I starting with S1 and VE

gelsefCFTS D true; return true;g

elsefCFTS D true; return true;

ggCheck Connected(S A, S B)f //procedure of Check whether the graph is connected

if�V �G� � �S A [ S B) is CONNECTED) return true;else return false;

g

Table 4. (Continued)



In the following sub-sections we will brieflydescribe the tracking and locating proceduresdefined by the PNNI standard and explain howto evaluate the tracking cost and locating costindividually.

—Tracking Cost Model—

Based on the PNNI protocol, we define a topologyupdate process that occurs when a mobile switchmoves from one place to another or powers on/off.The topology update process is invoked to informthe related switches whose topology database mustbe updated to reflect a topology change. We definethe following term used in our tracking algorithm.The ancestors-are-siblings level aij of nodes i and jis the level at which the ancestors of the two nodesbelong to the same peer group.

In the power-on case, the topology updates mustbe propagated from the lowest level up to level 1.After the topology updates are propagated up tothe highest level and then down to the lowestlevel, all switches in the network will updatetheir topology databases and have informationabout this new mobile switch. If mobile switchpowers off, the mobile tracking procedure is thesame as that for the power-on case. If a mobileswitch moves, instead of updating all the topologydatabases in the network, we adopt a schemeupdating only a limited number of topologydatabases. Suppose a mobile switch moves from anold switch o to a new switch k. For node x, whereaxk < aok, only those topology databases of nodesm, where amk ½ aok, need to be updated.

We define the topology update cost uij as the costof a processing topology update when a mobileswitch moves from node i to node j. Based on thedescription in previous paragraphs, mobile switchmoving causes updating of a topology databaseof a subtree in the hierarchical network structure.Let the ancestors of the two nodes, i and j, besiblings in peer group PG(A) at level aij. If ��X�denotes the number of nodes (switches) in peergroup PG(X), ��A� nodes should be informed toupdate their topology databases at level aij. Onthe other hand, these ��A� LGNs (logical groupnodes) at level aij represent ��A� peer groups, calledPG(A, 1), PG(A,2), . . ., PG(A, ��A�), at level aij C 1.Hence, the number of nodes at level aij C 1 tobe informed to update their topology databases

is ��A, 1� C ��A, 2� C ��A, 3� C . . . C ��A, ��A�� D∑��A�d1D1 ��A, d1�. We can apply this analysis from

level aij down to lower levels. In general, there are

NN�aij C x� D��A�∑

d1D1

��A,d1�∑d2D1

��A,d1,d2�∑d3D1

Ð Ð Ð

��A,d1,d2,d3,ÐÐÐ,dx�1�∑dx D1

��A, d1, d2, d3, . . . , dx �

nodes must be notified at level aij C x. Assume thata topology update request is flooded through aminimal spanning tree within each peer group. Ifthere are ��X� nodes in a peer group PG(X), therewill be ��X� � 1 edges in such a tree. Therefore,a local topology update for peer group PG(X)exchanges 2[��X� � 1] messages and updates ��X�topology databases. The total numbers of messagesexchanged and total number of topology databasesupdated when a mobile switch moves from nodei to node j are given in equations (4) and (5)respectively. From equations (4) and (5), uij isderived in equation (6).

Total number of messages

D [NN�aij� � 1]

C [NN�aij C 1� � NN�aij�]

C [NN�aij C 2� � NN�aij C 1�]

...

C [NN�L� � NN�L � 1�]

D NN�L� � 1 �4�

Total number of database

D NN�aij� C NN�aij C 1� C Ð Ð Ð C NN�L� �5�

uij D cŁmessage2[NN�L� � 1]

C cŁdb[NN�aij� C NN�aij C 1� C Ð Ð Ð C NN�L�] �6�

Based on the hypothesis we made earlier, amobile switch always moves to the immediateneighboring switches of the switch to which itcurrently connects. If prob�i ! j� denotes theprobability that a mobile switch moves from nodei to node j, for all immediate neighboring switchesj of a switch i, the summation of probabilitiesthat a mobile switch moves from switch i toits immediate neighboring switches equals 1, i.e.∑

8j prob�i ! j� D 1.



When ‘one’ mobile switch moves from switch ito its immediate neighboring switches, the updatecost Ui of switch i is calculated by equation (7).As we mention above, the average probabilitythat a mobile switch connects to switch i isprob�stay i�. So the average update cost for switchi is prob�stay i�ŁUi. Finally, the average trackingcost (CTracking�H�) of the network hierarchy H is thesummation of average update costs of each switchin the network. Mathematically, it is expressed as

Ui D ∑8j

prob�i ! j�Łuij �7�

CTracking�H� Dn∑

iD1

prob�stay i�ŁUi �8�

—Locating Cost Model—

The PNNI 1.0 standard does not explicitlyconsider locating procedure for mobile switchesor mobile endpoints before connection setup.Therefore, we propose a routing algorithm. Thesource switch selects a path to the destinationbased on the network structure known to it. Whena call setup arrives at the entry switch of a peergroup, it selects a complete path within the currentpeer group. The locating cost is defined as the costof setting up a connection from sender to receiver.The locating cost consists of (1) the shortest-pathdecision cost by the source switch and entry nodesin the PNNI hierarchy and (2) the setup cost by theswitches along the path.

First, let us consider that a switch s wantsto set up a connection to a mobile switch thatlinks directly to a switch r. Due to the featuresof the distributed routing algorithm, each entryswitch (i.e. the border switch that receives thesetup request from another peer group) of a peergroup makes its decision to choose a local shortestpath inside the peer group. The number of nodes(LGNs) on the selected path at level i determineshow many peer groups participate in the pathdecision procedure at level i C 1. Let the ancestorsof s and r be siblings in peer group PG(B) at levelasr. If ��X� denotes the number of nodes on theshortest path found within peer group PG(X), therouting algorithm should find a local shortest pathin one peer group at level asr and ��B� local shortestpaths in ��B� peer groups, labeled by PG(B,1),PG(B,2), . . ., PG(B, ��B�) respectively, at level asrC1.

Recursively repeating this extension, we can derivethat NPG�asr C x� peer groups should participatein the path decision procedure at level asr C x, aspresented in equations (9), (10) and (11). Therefore,there are [NPG�asr� C NPG�asr C 1� C Ð Ð Ð C NPG�L�]participating peer groups and the length of selectedpath is

(∑��B�e1D1

∑��B,e1�e2D1

∑��B,e1,e2�e3D1 Ð Ð Ð∑��B,e1,e2,ÐÐÐ,eL�asr �1�

eL�asr D1

ð ��B, e1, e2, Ð Ð Ð , eL�asr�)

� 1.

NPG�asr C x� D��B�∑e1D1

��B,e1�∑e2D1

��B,e1,e2�∑e3D1

Ð Ð Ð

��B,e1,e2,ÐÐÐ,ex�2�∑ex�1D1

��B, e1, e2, Ð Ð Ð , ex�1� �9�

NPG�asr � D 1 �10�

NPG�asr C 1� D ��B� �11�

We define rij as the routing cost for selectinga route and setting up links along the path fromswitch i to switch j. Depending on this definition,rij is shown in equation (12). We assume thedistribution of calls to a mobile switch follows auniform distribution, i.e., it is equally probable fora mobile switch to receive a call originating fromany switch. The average routing cost of makinga call from any switch to a mobile switch thatconnects to switch i is given in equation (13).As we mention above, the average probabilitythat a mobile switch connects to switch i isprob�stay i�. So the average routing cost for settingup a call to mobile switch through switch i isprob�stay i�ŁRsi . Consequently, the average locatingcost (CLocating�H�) of the network hierarchy H is thesummation of the average routing cost of eachswitch in the network. CLocating�H� is given below inequation (14).

rij D cshortŁ [NPG�aij� C NPG�aij C 1� C Ð Ð Ð

C NPG�L�]

C clinkŁ

��B�∑

e1D1

��B,e1�∑e2D1

Ð Ð Ð��B,e1,e2,ÐÐÐ,eL�aij

�1�∑eL�aij

D1

ð ��B, e1, e2, Ð Ð Ð , eL�aij �

� 1

�12�

Ri D 1n � 1

∑j 6Di

rij �13�



CLocating�H� Dn∑

iD1

prob�stay i�ŁRi �14�

—Average Total Cost—

Based on the tracking and locating procedure wementioned above, the total cost of a PNNI hierarchyconsists of the tracking cost and locating cost. Toestimate the average total cost, the call arrival rateof a mobile switch ��c� and the movement rate ofa mobile switch ��m� will be given. The averagetracking cost per unit time, the average locatingcost per unit time, and the average total cost perunit time are given below respectively.

Average tracking cost per unit time D �mCTracking�H�

Average locating cost per unit time D �cCLocating�H�

Average total cost per unit time D �mCTracking�H�

C �cCLocating�H�

We use the CMR (call-to-mobility ratio), denotedas � (defined as the number of calls arrivals permove) to quantify the dependence of the averagetotal costs on the CMR. The average total cost permove is

Ctotal �H� D CTracking�H� C � Ð CLocating�H� �15�

Experiments and ResultsIn this section we present some experiments

for our new partition algorithms. We discuss theimpact of the PNNI hierarchy cost after perform-ing the partitions and compare the performance ofthese two algorithms with the case of no partition-ing. In addition, we demonstrate the gain ratio ofthe two algorithms.

—Experiment 1—

The PNNI hierarchy architecture of Experiment1 is shown in Figure 5. Experiment parameters aregiven in Table 5. We have 6 peer groups in thegraph. First four peer groups are grouped togetherto form a logical group called ‘A’, the other twopeer groups are grouped into logical group ‘B’. The

Number of switches 225

Number of peer group 6

Number of PNNI hierarchylevels

3

Number of peaks Randomnumberfrom 1 to 5

Partitioned peer group size 100

Clen, Cnln, Cnpg, Cntn 1, 1, 1, 1

R 0.5

Number of experimentalcases

100

Table 5. Experiment parameters

peer group in the architecture to be partitioned isPG (B.2).

We generate 100 cases by assigning peaks in peergroup PG(B.2) randomly. The peaks mean high-density regions of mobile switches. By calculating,we obtain move probabilities and stay probabilitiesof each switch. By the equation mentioned abovemobility patterns are easy to construct. We appliedtwo partition algorithms to the mobility patterns.The average tracking cost (CTracking) and the averagelocating cost (CLocating) of the original hierarchy (Ho)and partitioned hierarchy (Hp) for 100 cases arecalculated. The degradation ratio of CTracking and theincrease ratio of CLocating between Ho and Hp in eachcase can be calculated by equations (16) and (17).Figures 6 and 7 illustrate the degradation ratio ofCTracking, the increase ratio of CLocating, and the ratiodifferences of the 100 cases for PARTITION ONEand PARTITION TWO. We sort the results by ratiodifference of each case and then average the resultsper five cases.

CTracking�Ho� � CTracking�Hp�

CTracking�Ho��16�

CLocating�Hp� � CLocating�Ho�

CTracking�Ho��17�

�[

Ctotal�Hp� � Ctotal �Ho�

Ctotal�Ho�

]�18�

We also study the relation between call-tomobility ratio (CMR) and total cost degradationratio of the original and partitioned topologies forour two partition algorithms, as given in Figure 8.



A B

PG(A) PG(B)

A.1

A.2

A.3

A.4

B.1

B.2

PG(A.1)

PG(A.2)

PG(A.3)

PG(A.4)

PG(B.1)

PG(B.2)

Figure 5. The PNNI hierarchy architecture of Experiment 1

Total cost degradation ratio of each (original,partitioned) topology pair can be obtained byequation (18). Each curve in Figure 8 is theaverage of the total cost degradation ratios of the100 (original, partitioned) topology pairs underdifferent CMRs.

I n order to construct a PNNI networkhierarchy or dynamically change the peer

group size, a partition algorithm is essential.

—Discussion—

As shown in Figures 6 and 7, in 75% of casesof Experiment 1, the degradation ratio of trackingcost is higher than the increase ratio of locating

cost. Two switches, which belong to the samepeer group before being partitioned, are dividedinto different peer groups. When a mobile switchmoves between these two switches, the numberof switches that should be notified by a trackingprocedure increases after partition. That meansthe number of switches in those peer groupswhose parents are siblings of the parents of thepartitioned peer groups will also influences thetracking cost. We also observe that the effectsof PARTITION ONE and PARTITION TWO arealmost equivalent on average to the results shownin Figure 8. When the CMR is very small,partitioning a large peer group with congregatedmobile switches will reduce by 30% the totalcost of the PNNI network. Even though theCMR increases to about 15, it is beneficial topartition a large peer group by using our partitionalgorithms.



T1/P1

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Cases sorted by ratio difference

Rat

io

L_cost increase ratio

T_cost degradation ratio

Ratio difference

Figure 6. Experimental results of the CTracking degradation ratios, CLocating increase ratios and the ratiodifferences of the 100 cases for PARTITION ONE

T1/P2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Cases sorted by Ratio difference

Rat

io

L_cost increase ratio

T_cost degradation ratio

Ratio difference

Figure 7. Experimental results of the CTracking degradation ratios, CLocating increase ratios and the ratiodifferences of the 100 cases for PARTITION TWO

ConclusionsTwo partition algorithms for large peer groups

in the PNNI hierarchy are proposed in this

article to support peer group reconfiguration andmaintain optimal network structure. The premiseof this article is that there exists an optimal peergroup size with the lowest network cost. In order



−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Gai

n R

atio

Partition 1 vs Partition 2

P1 meanP2 mean

0 5 10 15 20 25 30CMR

Figure 8. The total cost degradation ratio under different CMRs

to reconstruct a PNNI networks hierarchy ordynamically change the peer group size, a partitionalgorithm is essential.

The principle of our partition algorithm isbased on the congregation of mobile switches.When the mobile switches move around, theymight sometimes aggregate in a peer group.This phenomenon shows that mobile switchesfrequently move in a small region within a peergroup. The proposed partition algorithms intendto look for the congregation area on the topologyand set the area apart to form a new peer group.PARTITION ONE proceeds from one link whilePARTITION TWO, which proceeds from twolinks, is a variant of Partition ONE. Pseudo-codesfor these two partition algorithms are provided inthis paper.

An analytical model used to analyze the trackingand locating costs of a PNNI topology is alsoproposed. This can be used to measure thegain ratio of network costs between the originaltopology and the partitioned one under differentcall-to-mobility ratios. Experimental results showthat it is beneficial to partition a large peergroup by using our partition algorithms even

though the CMR increases to 15 and proves ourhypothesis.

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Network topology management in a mobile-switch ATM network: dynamic partition algorithms