Usage of Pythagorean Triple Sequence in OSPF

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1 Commuicatios ad Network,,, Published Olie February ( Usage of Pythagorea Triple Sequece i OSPF Simo Tembo, Ke-ichi Yukimatsu, Shohei Kamamura, Takashi Miyamura, Kohei Shiomoto, Atsushi Hiramatsu Departmet of Computer Sciece ad Egieerig, Akita Uiversity, Akita City, Japa NTT Network Service Systems Laboratories, NTT Corporatio, Tokyo, Japa {tembo,yukimatu}@ie.akita-u.ac.jp, {kamamura.shohei, miyamura.takashi, shiomoto.kohei, hiramatsu.atsushi}@lab.tt.co.jp Received November, ; revised December 6, ; accepted December 9, ABSTRACT Shuttig dow a lik for the purposes of a scheduled routie maiteace does cause the forwardig path to chage. If these chages are ot doe i a required order will cause ot oly trasiet micro loops but also a overload i some liks. Curretly, some ISP operators use a graceful lik shutdow procedure by first settig up the Iterior Gateway Protocol (IGP) lik metric to MAX_METRIC ad the shutdow the lik. I this paper, we preset a Pythagorea Triple Metric Sequece as a method to use to shutdow a lik durig such etwork operatios. Coductig a lik shutdow of ay desired lik for maiteace purpose is a very delicate duty that requires extreme care to prevet trasiet loops durig such topological chages. We thus wish to demostrate that there exists a Pythagorea Triple Metric Sequece for ay give lik that ca be used to shutdow a lik durig the routie maiteace by ISPs. Keywords: MAX_METRIC; Lik Shutdow; Pythagorea Triple Metric Sequece; Target Metric; Forwardig Path; Plaed Failures. Itroductio Iteret Service Providers (ISPs) have a obligatio to observe the coditios set i the Service Level Agreemets (SLA) with their customers. Oe of the coditios is to esure that ISPs miimize the duratio of the loss of coectivity. Losses of coectivity are commo occurreces most especially i etworks with larger topologies, although most of the times such losses of coectivity are uavoidable as the ISPs have to shut dow the equipmet for them to be able to coduct the scheduled routie maiteace. The lik metric of a lik ca always be icreased to a larger metric by progressively icreasig the metric of the lik by oe, util the target metric is reached. Therefore the lik ca be shut dow, by icreasig its lik metric util it becomes large that it caot carry packets aymore. Whe the target metric has bee reached, the lik ca the be safely shutdow []. Shuttig dow a lik for the purposes of a scheduled routie maiteace aloe accouts for about % of the total failures []. The shuttig dow of a lik due to these operatios is cosidered as plaed failures. I this paper, we make referece maily to these plaed failures. Some ISP operators do eve use some procedures to coduct a graceful lik shutdow []. The procedure ivolves by first settig up the Iterior Gateway Protocol (IGP) lik metric to MAX_METRIC ad the shutdow the lik that is scheduled for routie maiteace. Shuttig dow a lik for the purposes of a scheduled routie maiteace does cause the forwardig path to chage. For istace, usig a Simple Network model (i Figure ) the forwardig matrices i Figures ad helps to illustrate the effect of shuttig dow a lik o the forwardig path. The forwardig matrix i Figure is the forwardig iformatio bases (FIB) for all the 5 odes whe we have o lik failure. However, i Figure Figure. Simple etwork model all lik metrics equal to. Copyright SciRes.

2 7 S. TEMBO ET AL. NODE TO FROM Figure. Forwardig matrix for simple etwork with o lik failure. NODE TO FROM Figure. Forwardig matrix for simple etwork with a lik betwee odes ad shutdow. whe we shutdow the lik betwee odes ad does cause chages i the FIB, such that ode becomes the ext hop for traffic from ode destied for odes ad. The chages i the forwardig path whe the lik is shutdow, if are ot doe i a required order, will cause ot oly trasiet micro loops but also a overload i some liks. Shuttig dow the lik due to scheduled routie maiteace is cosidered as a predictable failure ad that the effect of such topological chages ca be reduced by correctly choosig updatable order that does ot oly avoids trasiet routig loops but also avoids cogestio ad lik overflow [-8]. I this paper we show that by use of the Pythagorea Triple Properties we ca compute the Pythagorea Triple Metric Sequece that we ca use to cofigure the etwork topology such that the desired lik ca be shutdow. I our Pythagorea Triple Metric Sequece method, we use the Pythagorea Triple Sequece {,, 5} to determie a sequece of metrics as target metrics to use to shutdow a lik. I this research we performed experimets usig the etwork model i Figure i which we assumed that all liks have same IGP metrics of value cofigured to each lik. We the performed experimets usig the etwork topology i Figure i which we assumed that the lik metric of Figure are ot uiform. To evaluate ad validate our Pythagorea Triple Metric Sequece method we directly varied ad cofigured the IGP lik metric of each ad every lik i the etwork topology to {,,,, 5} sequece by cosiderig each lik as a lik desired to be shutdow. For performace evaluatio we have used a Simple Network, COST9, ad HLDA as example of etwork topology models for evaluatio purposes ad whilst for validatio purposes we have used K 5 COMPLETE, as our litmus test etwork topology. I all the cases we use a gravity model type of traffic geeratio accordig to the populatio distributio. I our performace evaluatio we demostrate at what stage of the {,,,, 5} sequece is the shuttig dow possible. We performed simulatios for each ad every lik i each of the etwork topologies to verify whe each lik stops carryig traffic, ad thus validatig the lik beig shutdow. The rest of the paper is orgaized as follows. I Sectio, we itroduce the Pythagorea Triple Sequece Method. We explai the overview of our idea ad itroduce our algorithm. I Sectio, we discuss the experimets ad evaluatio of the results. Fially, i Sectio, we provide the coclusio.. Algorithm to Shutdow a Lik Usig the Pythagorea Triple Sequece.. Overview It has bee proved, that the IGP lik metric of a lik ca always be icreased to a larger metric by progressively icreasig the metric of the lik by oe (see Equatio ()), util the target metric is reached without causig trasiet forwardig loops. m m () m m where is the iitial metric ad is the ew metric. It has bee proved further that the lik ca be shut dow by icreasig its IGP lik metric util it becomes large that it caot carry traffic aymore. It was show that whe the target metric has bee reached, the lik ca the be safely shutdow []. It was illustrated i [], that we ca have a sequece of three elemets to sufficietly provide a loop free covergece: m, m, mt We also kow that curretly, ISP operators must first set the IGP lik metric to MAX_METRIC to gracefully reroute traffic before shuttig dow the lik. A expressio that ca be expaded to a equatio expressed as follows: 6 MAX_METRIC 56 () () I Equatio (), we have a differece of two squares, which is a property of the Pythagorea Triple. It is from this fact that we believe if the property of Equatio () holds for ay give etwork topology the there must exist a Pythagorea Triple that ca be used to shut dow the lik cosiderig the fact that to perform the graceful shut dow, ISPs are curretly usig a Pythagorea property as illustrated i Equatio (). It was illustrated i [5] that there exist ifiitely may right-agled triagles with itegral sides i which the Copyright SciRes.

3 S. TEMBO ET AL. 75 legths of two o-hypoteuse sides differ by. That is, the triple sides as a sequece will be; x, x, y Comparig Equatios ()-(), we ca see some similarities i these expressios, i.e. the first term i Expressio () m x, the secod term m x ad the third term mt y. This aalogy i the Expressios () ad () also provide us with the motivatio to believe that the use of the Pythagorea Triple Sequece is worth explorig. Further, our idea is very much ecouraged by the fact that i the graceful shuttig dow of a lik method a Pythagorea property as illustrated i Equatio () is valid. It was further stated i [5], that there exist positive itegers x ad y such that the legths of the sides of the triagle are x, x ad y respectively fulfils the Pythagoras Theorem, that is: x x y Examples of the Pythagorea triples are {,, 5} ad {,, 9}. The {,, 5} triple ad its multiple {,, 5} are the oly Pythagorea triple that are i arithmetic progressio ad cosecutively icremetig. It is for reaso we have decided to use this particular type of Pythagorea Triples Sequece. We kow that i Arithmetic Sequece the differece betwee oe term ad the ext is a costat as give Equatio (6), a fact that we see i the triple sequece of {,, 5}. aa, da, d, (6) where: a is the first term, ad d is the commo differece betwee the terms... Pythagorea Triples Usig Euclid Formula I mathematics it has bee show that if {a, b, c} is a Pythagorea triple, the so is {ka, kb, kc} for ay positive iteger k, ad that the smallest Pythagorea Triple is {,, 5} whe k. As illustrated i [6], Euclid provided a formula to fid Pythagorea triples from ay two positive itegers m ad, where m. For istace usig Euclid formula i terms of a sequece (see Figure ) we have {a, b, c}: a m b m c m As a Example, Table illustrates the possible Pythagorea triples usig the Euclid formula give i Equa- () (5) (7) (8) (9) m m m Figure. Euclid s parameterizatio. Table. Example of pythagorea triples. m TRIPLE,, 5 5,, 7,, 5 5 9,, 6 5, 6, 6 tios (7)-(9). Further, as a Example usig the Euclid formulas i (7-9), assumig we have triples give as a se- quece m, m, m, ad where m for a to be a positive umber. Thus Equatio () ca be computed as follows: 6 MAX_METRIC 56 Assumig m 56 ad a m 65,55; MAX_METRIC b m 5 c m 65,57 The above example shows that Equatio () fulfils the Euclid formula, agai qualifyig our idea of usig the Pythagorea Triple Sequece Method whe shuttig dow the lik for routie maiteace purpose... The {,, 5} Triple Algorithm As stated earlier i this paper, the {,, 5} triple ad its multiples {,, 5} are the oly triples that are i arithmetic progressio ad cosecutively icremetig. It is these two reasos that we have chose to use this particular type of Pythagorea Triples Sequece i this paper. I our method we show that usig the Pythagorea Triple Metric Sequece {,, 5}, ad give that is the iitial elemet of the sequece {,,,, 5} ad the hece the commo differece of the arithmetic progressio, we ca determie the target lik metric ad directly recofigure to the lik desired to be shutdow. I Table below we illustrate the idea of our algorithm i determiig the target metric. Whe we wish to shutdow ay give lik i ay give topology we cofigur- Copyright SciRes.

4 76 S. TEMBO ET AL. Table. The {,, 5} triple algorithm. PRIMITIVE TRIPLES,, 5,, 5,, 5 6, 8,,, 5 9,, 5,, 5, 6,,, 5,, 5 ig all other liks i the etwork topology with a uiform IGP lik metric which we deote as, except for the lik we wish to shutdow to which we vary its lik metric as per sequece {,,,, 5}.. Results ad Discussio To validate our idea of the Pythagorea Triple Metric Sequece, we have used several differet topologies as etwork models for evaluatio purposes such as ) a Simple Network Figures ad 5; ) COST9 i Figure 6; ) HLDA i Figure 7; ad ) K 5 COMPLETE i Figures 8. We have used gravity model as the traffic coditio accordig to the populatio distributio. The routig algorithm assumed is a miimum cost routig. We have assumed that the lik metric is iversely proportioal to lik capacity... Use of Pythagorea Triple to Shutdow a Lik The value of was varied as show Table to avoid etwork over load. The results show below i Figures 9 ad, illustrate the validity of our method. We validate that by varyig the lik metric of liks as per the values i the sequece {,,,, 5}, it is oly whe the lik metric reaches the values of all or some of the elemets of the {,, 5} sequece that the liks stops to carry traffic. Thus at this poit the lik is cosidered to be safe for shutdow as the lik utilizatio goes dow to zero. The graph i Figure shows that 5% of liks i the Simple Network (i Figure ), ca be shutdow whe the lik metric reaches {} of the {,, 5} sequece. We further observe that ALL liks ca be shut dow after the lik metric reaches {, 5} of the {,, 5} sequece. Therefore we ca say that whe the lik metric of the Simple Network is uiform, 5% of the liks ca be shutdow as per {,, 5} sequece thus makig our algorithm to be valid. Whereas whe we varied the values of the lik metric (see Figure 5) so that we achieved o uiform lik metric sceario, our simula- Figure 5. Simple etwork model all lik metrics NOT equal to. Figure 6. COST9 model ( odes, 5 liks). Figure 7. HLDA model ( odes, 6 liks). Copyright SciRes.

5 S. TEMBO ET AL. 77 Table. Topologies for our performace evaluatio. No. TOPOLOGY Nodes Liks. SIMPLE NETWORK 5 6. K 5 COMPLETE 5. COST9 5. HLDA 6 UTILIZATION Figure 8. K 5 COMPLETE (5 odes, liks) lk - lk LINK METRIC 5 lk - lk - Figure 9. The graph validates that we are able to shutdow some liks i the Simple Network (i Figure ) ad COST9; ALL liks i HLDA ad K 5 COMPLETE after the lik metric reaches {,, 5} of {,,,, 5} sequece. UTILIZATION lk - lk LINK METRIC 5 Figure. The graph validates that other liks i the Simple Network (i Figure ); COST9 ad ALL liks i the Simple Network (i Figure 5) ca be shutdow ONLY after the lik metric reaches {, 5} of {,,,, 5} sequece. tio results show i Figure show that we ca ONLY shutdow the lik after the lik metric reaches the values of {, 5} of the our algorithm {,, 5}. Thus we ca say that NO liks i the Simple Network (i Figure 5), ca be shutdow whe the lik metric reaches {} of the {,, 5} sequece. However, ALL liks ca be shutdow whe the lik metric reaches {, 5} of the {,, 5} sequece. I Figure, below we show the simulatio results whe each ad every lik was each i tur cofigured ad simulated for a lik shutdow i COST9, HLDA ad K 5 COMPLETE topologies usig our algorithm. The validity resposiveess of these etwork topologies to our lik shutdow method is that COST9 respoded favorable to the {,, 5} algorithm is 96%, HLDA is % ad K 5 COMPLETE is also %. These results demostrate that our method is valid to perform a lik shutdow... Use Pythagorea Triple to Shutdow Two Liks To further validate our method we performed experimets o the Simple Network topology (Figure ), usig our algorithm to simultaeously shutdow two liks i the etwork topology. The graphs show i Figures - 5 demostrate that we are ONLY able to simultaeously shutdow TWO liks whe we have a lik metric combiatio of {, 5} or {5, 5}. We have further validated our method by performig similar experimets of simultaeously shuttig dow two liks for COST9, HLDA ad K 5 COMPLETE topologies usig our algorithm. The graphs show i Figures 6 demostrate the resposiveess of the TWO liks shutdow of each topology. The illustratio is that, for COST9, we are able to simultaeously shutdow TWO liks oly whe we have a lik metric combiatio of {, }, {, 5} ad {5, 5}. Whereas i the case of the HLDA model, the resposiveess of our algorithm is % as we are able to simultaeously shutdow TWO liks for all lik metric combiatio. That is, we are able to simultaeously shutdow TWO liks for all lik metric combiatio of {, }, {, }, {, 5} {, }, {, 5} ad {5, 5}. This similar results, is obtaied for the K 5 COM- PLETE, which we have used as our Litmus Test for our algorithm. The results show that the K 5 COMPLETE s performace is % validatio as we are able to simultaeously shutdow TWO liks all the lik metric combiatio of {, }, {, }, {, 5} {, }, {, 5} ad {5, 5}. But as for the case of the Simple Network topology, we are oly able to simultaeously shutdow TWO liks whe we have a Copyright SciRes.

6 78 S. TEMBO ET AL. Liks ca be shutdow at lik metric of {} of the {,, 5} sequece Liks ca be shutdow ONLY whe lik metric reaches {, 5} of the {,, 5} sequece % % Performace % 8% 6% % % % % Simple Network All liks do NOT have SAME Lik Metric Simple Network Model 5% Simple Network - All liks SAME Lik Metric have Figure. Performace compariso of the Simple Network Models, that is whe the value of the lik metric is uiform for all liks ad whe it is ot as show i Figures ad 5. Liks that respods favourably to {,, 5} algorithm Liks that ca be shutdow ONLY whe the lik metric reaches {, 5} of the {,,,, 5} sequece Performace % 9% 8% 7% 6% 5% % % % % % 96% % % 5% 5% % % % SIMPLE NETWORK COST9 HLDA K5 COMPLETE Topology Figure. Performace of shuttig dow a lik.,,, 5,.5. UTILIZATION UITILIZATION lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - Lik ID lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - Lik ID Figure. The graphs illustrate that i the Simple Network (i Figure ); we are ot able to shut dow a lik whe we have lik metric combiatio of {, } ad {, }. lik metric combiatio of {, 5} or {5, 5}. I Figure 7, we show the performace of the TWO liks shutdow usig our algorithm for the Simple Network, COST9, HLDA, ad K 5 COMPLETE topologies. The validity resposiveess of these etwork Figure. The graphs illustrate that i the Simple Network (i Figure ); we are able to shutdow oly ONE lik whe we have lik metric combiatio of {, 5} ad {, }. topologies to the TWO lik shutdow is that the Simple Network respods favorable to the {,, 5} algorithm by %, COST9 respose is 5%, whereas HLDA ad K 5 COMPLETE is % for both of them. The results demostrate that our method is valid to Copyright SciRes.

7 S. TEMBO ET AL. 79 UTILIZATION ,, 5 5, 5 lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - lk - Lik ID Figure 5. The graphs illustrate that i the Simple Network (i Figure ); we are able to shutdow TWO liks whe we have a lik metric combiatio of {, 5} ad {5, 5}. MAXIMUM LOAD No Lik Failure SIMPLE NETWORK HLDA COST9,,, 5, Lik Metric Combiatio K5 COMPLETE, 5 5, 5 Figure 6. Effect of shuttig dow TWO liks o Maximum Load. Performace Performace of {,, 5} algorithm whe perfomig the Two Liks Shutdow % 8% 6% % % % % SIMPLE NETWORK 5% COST9 % HLDA Topology % K5 COMPLETE the fact that the {, 5} is a subset of the Pythagorea Triple Metric {,, 5} Sequece. The similar lik shutdow experimets reveal that were performed for other topologies, as show i figure, reveals that usig our Pythagorea Triple Metric {,, 5} Sequece, the validity is 96% for COST9 model, % for HLDA model ad % for K 5 COM- PLETE. To further validate our method we performed experimets by shuttig dow TWO liks simutaeously for all the four topologies. The results show i Figure 7, summaries our dicoveries that our method still remais valid eve whe we coduct ad shutdow TWO liks simultaeously. To further explai the effect of the wrog combiatio of the lik metric as what effect it has o the performace of the topology, we use Figures 8- of the Simple Network, to illustrate this fact. A example of the compariso of performace is give i Figures 8- of the Simple Network, whe just by a mere swappig of the lik metric of the topology what effect that it has o the Maximum Load. Lets us for istace say we call the model SN whe the lik metric for Lik -/- is equal to, ad for Lik -/- is equal to 5 as show Figure 9. We the call the model SN whe the lik MAXIMUM LOAD No Lik Failure SN- SN-,,, 5,, 5 5, 5 Lik Metric Combiatio Figure 8. Effect o maximum load (for SN ad SN) by mere swappig the lik metric combiatio of the Simple Network. Figure 7. Performace of shuttig dow TWO Liks. perform a TWO liks shutdow... Performace Evaluatio Whe each ad every lik of the Simple Network model, i Figure, was each i tur cofigured ad simulated for a lik shutdow usig our Pythagorea Triple Metric {,, 5} Sequece, 5% of the liks validates our algorithm whilst the other 5% the sequece could oly be shutdow whe their lik metric reached values of {, 5}, the results which we ca still cosider as valid 5 Figure 9. Simple Network (SN) with two Forwardig Trees for trasportig traffic whose destiatio is odes ad (Lik -/- =, Lik -/- = 5). Copyright SciRes.

8 8 S. TEMBO ET AL. 5 Figure. Simple Network (SN) with two Forwardig Trees for trasportig traffic whose destiatio is odes ad SN(Lik -/- = 5, Lik -/- = ). metric for Lik -/- is, ad for Lik -/-is 5 as show i Figure. The chages ca eve be see i the chages of the forwardig trees for traffic destied for odes ad. I Figure 9, the forwardig tree idicates that the lik betwee odes ad is still i use such that for traffic destied for ode from ode ad vice versa cotiues to use lik -/-. Whilst Figure illustrates that the lik betwee ad is shutdow such that the forwardig tree for traffic destied for odes from ode use ode as the ext hop, whereas for the case of traffic destied for ode from ode use ode as the ext hop. Figures ad provide the forwardig matrices for both SN ad SN. Further the overall effect wrog lik combiatio has o performace as show i Figure 8 whe we compare SN graph to that of graph SN. The lik combiatio betwee poits {, } to {, 5}, shows that despite ot achievig the TWO lik shutdow simultaeously which is the objective for this lik combiatio we are already experiecig the icrease i the maximum load as early as the poit {, }. This icrease i maximum load problem proves worst still whe we cosider comparig the performace of the SN, as show i Figure with other topologies this ca be said SN lik combiatio is ideed poor. The K 5 COMPLETE was used as a Litmus Test of our idea ad which from the results we have see that i both experimets, that is a lik shutdow experimets ad a TWO lik shutdow experimets, the Pythagorea Triple Sequece performed % validatio. The additioal explaiatio that we ca give, that ca hopefully provide as some scietific explaatio of the experimetal results our method is by usig properties such as Node Degree, Clusterig Coefficiet ad Expected Path Legth. Whe comparig performace of these topologies usig properties such as Node Degree, Clusterig Co- NODE FROM TO Figure. Forwardig Matrix for a SN Model (i Figure 9). NODE TO FROM Figure. Forwardig Matrix for a SN Model (i Figure ). Performace % 8% 6% % % % Performace of {,, 5} algorithm whe perfomig the Two Liks Shutdow 7% % 5% % % SN SN COST9 HLDA K5 COMPLETE Topology Figure. Compariso of performace of shuttig dow two liks. efficiet (also called the fried of my fried is also my fried ), ad Expected Path Legth (also called Close Cetrality ) [9-], we use Figures -6 to explai these pheomea i compariso to the performace our algorithm. It ca be explaied that the Clusterig Coefficiet (which is ofte discribed as the tedecy for triagles of coectios to appear frequetly i etworks) ad the Average Node Degree for Simple Network are lowest as compared to both COST9 ad HLDA. However whe we compare COST9 ( odes, 5 liks) ad HLDA ( odes, 6 liks) both have the same umber of odes, ad their oly differece betwee them is the extra lik i HLDA topology. This extra lik helps HLDA model to have some improvemets i the Average Node Degree, Clusterig Coefficiet ad Expected Path Legth as we ca see i Figures -6. It could be said that, possibly it is same extra lik that helps HLDA model to respod more favourable eve to our Pythagorea Triple Sequece as compared to COST9. However the major determiat i our algorithm could be said to be the Expected Path Legth, sice it relates more to our assumptio of the Miimum Cost Routig. This fact ca be explaied usig Figure 6. We see Copyright SciRes.

9 S. TEMBO ET AL. 8 Average Node Degree Average Node Degree = [*L/].55.7 SIMPLE NETWORK COST9 HLDA Topology. K5 COMPLETE Figure. Average ode degree of Simple Network, COST- 9, HLDA ad K 5 COMPLETE. Clusterig Coefficiet Clusterig Coefficiet =[(K-)/(K-)].5.55 SIMPLE NETWORK COST9 HLDA Topology.5 K5 COMPLETE Figure 5. Clusterig coefficiet of Simple Network, COST- 9, HLDA ad K 5 COMPLETE. Expected Path Legth SIMPLE NETWORK Expected Path Legth =[log N/log K].58.5 COST9 Topolgy HLDA.6 K5 COMPLETE Figure 6. Expected Path Legth of Simple Network, COST- 9, HLDA ad K 5 COMPLETE. that topologies with miimum Expected Path Legth, that the closer the Expexted Path Legth is to UNITY the better the result is see i respose to the Pythagorea Triple Sequece. The K 5 COMPLETE helps to explai this fact. Whe we compare the all the three properties Expected Path Legth, Clusterig Coefficiet ad Average Node Degree of K 5 COMPLETE to COST9, HLDA ad Simple Network, it is oly the Expeted Path Legth that the K 5 COMPLETE model shows some superiority. This property easily helps to provide the explaatio that the Pythagorea Triple Sequece, is valid to use as a lik shutdow method ad that the determiat pheomea i our method is miimum cost routig resposiveess.. Coclusios We have preseted a ew lik shutdow method for ay give lik desired to be shut dow for routie maiteace. The key idea of our method is the itroductio of the Pythagorea Triple Metric Sequece {,, 5} to use to cofigure a lik that is desired to be shut dow for routie maiteace. Whe a lik is scheduled for routie maiteace the lik ca be cofigured to oe of the metric i the sequece as the target metric before shut dow. I our experimets, each lik was cofigured to {,,,, 5} sequece ad we performed some simulatios, ad it was discovered that it is oly whe the lik metric reached values of {,, 5} were able to perform a lik shutdow. The simulatios result validated our idea of usig the Pythagorea Triple Metric Sequece {,, 5} as a lik shut dow method. I our experimets to performig a lik shutdow, we discovered that usig our Pythagorea Triple Metric {,, 5} Sequece, for the Simple Network model 5% of the liks validates our algorithm whilst the other 5% the sequece could oly be shutdow whe their lik metric reached values of {, 5}, the results which we cosider as valid the fact that the {, 5} is a subset of the Pythagorea Triple Metric {,, 5} Sequece. The similar lik shutdow experimets revealed that as for other topologies, the validity respose is 96% for COST9, % for HLDA ad % for K 5 COM- PLETE. To further validate our algorithm we performed experimets by shuttig dow TWO liks simutaeously for all the four topologies. The validity respose is % for the Simple Network model, COST9 respose is 5%, whereas HLDA ad K 5 COMPLETE is % for both of them. The simulatio results have validated our idea to use the Pythagorea Triple Metric Sequece {,, 5} as a lik shutdow method. We further established that our method has positive resposiveess to miimum cost routig, as we have illustrated that topologies with Expected Path Legth ear UNITY value ted to have good performace. As for future work, we pla to ivestigate further the cosequeces of usig other Pythagorea Triple Sequeces other tha the {,, 5}. REFERENCES [] P. Fracois, M. Shad ad O. Boaveture, Disruptio Free Topology Recofiguratio i OSPF Networks, Proceedigs of IEEE INFOCOM, Achorage, 6- May 7. [] A. Markopoulou, G. Iaaccoe, S. Bhattacharyya, C.-N. Chuah ad C. Doit, Characterizatio of Failures i a IP Backboe, IEEE INFOCOM, Vol.,, pp. 7- Copyright SciRes.

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