BOTTLENECK BRANCH MARKING FOR NOISE CONSOLIDATION

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1 BOTTLENECK BRANCH MARKING FOR NOISE CONSOLIDATION IN MULTICAST NETWORKS Jordi Ros, Wei K. Tsai ad Mahadeve Iyer Departmet of Electrical ad Computer Egieerig Uiversity of Califoria, Irvie, CA {jros, wtsai, Abstract The oisy feedback cosolidatio problem i poitto-multipoit ATM multicast etworks is studied. A ew algorithm, which keeps track of the M smallest available rates (AR) from the braches at each brachig poit, is proposed. This algorithm has zero respose delay, oise stability (defied i this paper), ad small probability of oise. The probability model assumes o kowledge of the distributio of the available rate from the braches. Both aalytical ad simulatio results demostrate the superiority of the ew algorithm. 1 Itroductio Multipoit commuicatio has rapidly become the mode of choice for sharig data delivery across a etwork. A importat problem here is the oisy feedback cosolidatio problem i poit-to-multipoit commuicatio across coectio orieted etworks such as ATM or MPLS etworks. I poit to multipoit coectios, a source has to trasmit at the available rate (AR) of the most bottleecked lik i the whole multipoit tree. This meas that the source has to collect the state of all the braches ad feedback cosolidatio at the brachig poits becomes ecessary. I all sigalig protocols, backward resource maagemet (BRM) cells set from leaf odes periodically arrivig at the brachig poit brig (feedback) the state of the braches i the form of available rates, stored i the explicit rate (ER) field i the BRM cell. BRM cells are periodically set back to the root brigig the ewest bottleeck rate to the source. The bottleeck rate is obtaied by miimizig over all leafs the available rates. If a costly per-brach accoutig is to be avoided, this miimizatio is carried out at a brachig poit by keepig track of oly oe variable called MER (miimum explicit rate). Sice this miimizatio has to be carried out i a distributed asychroous way, a sigalig protocol has to decide whe to re-start the miimizatio. The re-start ca be doe i two ways: oe is to reset MER to a default large value from time to time, the other is to reset it after at least oe feedback value has bee received from each brach. I the first approach, the MER values set back to the source might ot be the true bottleeck rate i the multipoit tree, i which case a oise is said to be geerated. I the secod approach, oe has to wait for a log time, especially whe there are o-resposive leafs, defeatig the very purpose of timely feedback. Such are the key issues i the commoly kow cosolidatio oise problem [fah97]. This paper presets a study of the cosolidatio feedback issue i multipoit coectios for ATM etworks. We first preset a review of the curret solutios of the problem. The, we itroduce a ew approach: the Bottleeck Brach Markig (BBM) algorithm. By itroducig the idea of storig the ID of the brach, we prove that some of the weakesses of the previous approaches ca be overcome. The trade-off betwee performace ad complexity is reflected i the parameter M, the umber of braches the switch keeps track of. This techique happes to have two ice properties. First, we prove that eve if the umber of braches N teds to ifiity, the probability of havig oise does ot ted to 1. I this paper, a "worst-case" probability model of the cosolidatio oise is costructed. Secod, the oise probability decreases expoetially with the umber of braches M the switch keeps track of. For example, we prove that for M=2 (oly the 2 most bottleeck braches are stored) ad N = (there are ifiite umber of braches), the oise probability is about less tha 1 8. This result says that we do t eed ifiite storage to cosolidate ifiite 1 This research is supported by the Natioal Sciece Foudatio, award ANI , uder the CISE ANIR program. This work is also supported i part by the Uiversity of Califoria, Irvie, ad the Geeralitat de Cataluya through a Balsells Fellowship to Jordi Ros. 1

2 umber of brach ER values but just a small storage. Eve though the BBM scheme has bee preseted to solve the particular oise cosolidatio problem for multicast ATM coectios, it should be cosidered as a geeral tool that ca solve ay geeral cosolidatio feedback problem. I these problems, a brachig poit receives feedback from may differet sources ad has to make a decisio about which feedback is passed back to the root. We prove that the BBM approach ca dramatically reduce the complexity while still achieve excellet performace. This paper is orgaized as follows. Sectio 2 itroduces the feedback cosolidatio issues ad sectio 3 reviews the existig cosolidatio algorithms. The BBM algorithm is itroduced i sectio 4 with its performace aalysis i sectios 5 ad 6. Simulatios ad cocludig remarks follow i sectios 7 ad 8. 2 Feedback Cosolidatio Issues The followig are the most importat issues that eed to be addressed whe implemetig a cosolidatio algorithm. Cosolidatio oise. The available rate received from all the braches has to be miimized ad set back to the source. This ca be doe by usig a per-brach accoutig of this available rate. However, because this may be too expesive, curret algorithms use oly oe field that is updated every time a BRM cell is received ad is reset to ifiity every time a BRM cell is set. Because of this simplificatio, some algorithms ted to sed feedback to the source with values that do ot correspod to the most bottleecked brach but to some other brach. We call this effect cosolidatio oise. Noise stability. We will show that some of the curret algorithms ted to be ustable i terms of oise. I other words, oisy feedback (feedback which does ot brig a correct value) teds to produce more oisy feedback. Trasiet respose ad level sesitivity. I order to cosolidate the bottleeck rate, some algorithms wait for the feedback from all the braches to be received before sedig a BRM cell. This icurs i a higher trasiet respose delay. Because this effect is produced at each brachig poit, the cosolidatio algorithm may tur out to suffer level sesitivity: the trasiet respose is sesitive to the umber of levels i the multipoit tree. Ratio BRM/FRM. Because i multipoit coectios FRM (forward RM cell) cells are duplicated, the umber of BRM cells may icrease with the umber of leafs of the tree. We defie the ratio BRM/FRM as the umber of BRM cells received at the source for each cell set by the same source. I order to avoid implosio of BRM cells at the source, the cosolidatio algorithm has to cotrol the ratio BRM/FRM so that it coverges to oe. Brach resposiveess. Some algorithms eed to implemet additioal code i order to overcome the case i which oe of the braches is ot resposive. This additioal codig icurs additioal complexity i the complete algorithm. Complexity. The complexity of the algorithm eeds also to be studied. Based o the previous papers ([Re98], [Fah98]), the followig will be assumed i this paper to judge the algorithm complexity: -Turig aroud the cells is more complex tha passig through the cells. -Per-brach accoutig is a expesive solutio ad should be avoided. 3 Cosolidatio Algorithms I this sectio, we preset some of the more relevat cosolidatio algorithms curretly defied. Before this, we defie the cocept of cosolidatio algorithm. Defiitio 1. Cosolidatio Algorithm. We defie a cosolidatio algorithm at a brachig switch as a algorithm that fulfills the followig actios: 1) Periodically receives the available badwidth from each brach. 2) Processes the set of available badwidth i each brach to obtai the most bottleecked brach ad stores it i a field called Miimal Explicit Rate (MER). 3) Periodically seds the value of MER to the root. For the rest of this sectio, we show a quick review based o [Fah98], which presets 7 cosolidatio algorithms. Because of space limitatio, here we oly examie algorithms 1, 3, 4 ad 6 (usig the same otatio i [Fah98]). For a further review, refer to [Fah98]. 3.1 Algorithm 1 The mai idea for this algorithm is that BRM cells are retured from the brachig poit whe FRM cells are received, ad cotai the miimum of the values idicated by the BRM cells received from the braches after the last BRM cell was set. FRM cells are duplicated ad multicast to all braches at the brachig poit. 3.2 Algorithm 3 I this approach, the brachig poit does ot tur aroud the FRM cells, but the BRM cell that is received from a brach immediately after a FRM cell has bee received by the brachig poit is passed back to the source, carryig the miimum ER value sice the last BRM cell was set. Agai, FRM cells are duplicated ad multicast to all braches at the brachig poit. 3.3 Algorithm 4 The mai idea i this algorithm is that a BRM cell is passed to the source oly whe BRM cells have bee received from all braches after the last time a BRM was 2

3 set. FRM cells are duplicated ad multicast to all braches at the brachig poit. 3.4 Algorithm 6 This approach reflects the idea that there is o eed to wait for feedback from all the braches whe a overload situatio has bee detected. I this case, the overload is immediately idicated to the source by sedig a BRM cell ad, hece, the respose time is reduced. Because this may cause extra BRM cells to be set, the ratio BRM over FRM may tur out to be bigger tha oe. To avoid this problem, algorithm 6 keeps track of the umber of extra cells set to the source. Whe feedback from all leaves idicates uder-load ad the value of extra BRM cells is more tha zero, this particular feedback is igored. 2 M ID AR Root Brachig switch Figure 1. Brach switch model 4 The BBM Algorithm Brach 1 Brach 2 Brach N There are two fudametal reasos for which the previous algorithms suffer from several cosolidatio issues (1) The MER value has to be reset to ifiity (or to the peak cell rate) every time a BRM cell is set back to the root. Because of this, if a BRM cell is set before all the feedback has bee cosolidated, oise may be geerated. (2) The oly way to icrease MER is to wait for it to be reset. I other words, suppose that the most bottleecked brach icreases its available rate ad seds its feedback to the brachig switch. Because there is o way for the switch to kow that this particular feedback comes from the most bottleecked brach, it caot icrease MER. Istead, it has to wait for a reset of MER ad for a ew miimizatio iteratio to determie the ew bottleeck available rate. We suggest that both problems ca be solved if the switch stores the idetifier of the brach for which it has stored the available rate. Ideed, ote that (2) ca be solved sice ow the switch ca idetify that the most bottleecked brach has icreased its available rate. Moreover, this provides the meas to icrease MER without havig to reset its value ad, hece, (1) is also solved. Upo the receipt of a FRM cell Multicast FRM cell to all braches; Upo the receipt of a BRM cell NumberOfBRMsReceived ++; IF BRM.ER is oe of the Mth most bottleeck Store BRM.ER ad the brach ID i BBM; IF BRM.ER is the most bottleeck Sed to the root this BRM cell; / so that respose time = / IF NumberOfBRMsReceived == N NumberOfBRMsReceived ; ELSE ExtraBRM ++; ELSE IF NumberOfBRMsReceived ==N NumberOfBRMsReceived ; IF ExtraBRM == ; BRM.ER Smallest AR i BBM cell; Sed to the root this BRM cell; ELSE ExtraBRM --; Discard this BRM cell; / to esure BRM/FRM 1 / ELSE Discard this BRM cell; Figure 2. BBM Algorithm The previous reasoig is the motivatio for the Bottleeck Brach Markig (BBM) algorithm. The BBM algorithm keeps track of the M most bottleecked braches. For this, the switch stores a matrix with M etries icludig both the idetifier of the brach (ID) ad the last available rate received at the brachig ode, as show i fig. 1. We call this matrix the BBM matrix ad its size M ca be adjusted depedig o the scalability ad performace requiremets. For example, if N is the umber of braches, the the case M=N is equivalet to a per-brach approach. Fig. 2 presets the pseudocode of the BBM algorithm. The BBM matrix has M etries, ad each etry cosist of both the ID ad the available rate. A ew feedback is stored i this structure oly if its available rate is smaller tha ay of the available rates i the BBM matrix. There are two ways for which a brachig switch may sed a BRM cell back to the root: (1) Wheever a ew bottleeck appears: Note that this way the algorithm has zero respose delay. However, that may cause the ratio BRM/FRM be bigger tha oe. To cotrol this, the algorithm stores the umber of extra BRM cells set because of a ew bottleeck i the ExtraBRM field. (2) Wheever the umber of feedback values received sice the last time a BRM cell was set is equal to the umber of braches: This oe is implemeted with the NumberOfBRMsReceived field. However, if ExtraBRM > 1 the o BRM cell is set. This way the 3

4 algorithm esures covergece of the ratio BRM/FRM to 1. 5 Performace aalysis I this sectio, the issues defied i sectio 2 are examied oe by oe for each of the cosolidatio algorithms preseted so far. Noise Cosolidatio. Algorithms 1 ad 3 suffer from the oise cosolidatio problem, sice the MER is reset every time a BRM cell is set ad a BRM cell may be set before the feedback from the most bottleecked brach has arrived. Algorithms 4 ad 6 do ot suffer from this problem sice a BRM cell waits for all the feedback before beig set. We ow show that the BBM scheme may also iduce some erroeous feedback. Suppose M=1 ad that the two most bottleecked braches have a available rate of AR 1 ad AR 2, respectively, so that MER = AR1. Assume ow that a ew feedback rate bigger tha AR 2 arrives from the most bottleecked brach. The ew MER should be ow AR 2. However, because the switch does ot keep track of the secod most bottleecked rate, it caot tell the correct MER. Note that this situatio ca be solved by makig M = 2, but still we ca fid some other more complex situatios i which this case also happes to fail. The previous property proves that if M is smaller tha the total umber of braches, the some situatios may iduce oise. However, i the followig sectios we will prove that these situatios happe to be very ulikely. Moreover, eve if some oise is produced, we will prove that the BBM scheme is stable ad that it quickly coverges to the correct value without oscillatios. Noise Stability. It ca be prove that if the etwork is i steady state (we assume a etwork is i steady state if o parameter chages i time) the algorithms 1 ad 3 do ot have oise. Ituitively, this is because i this particular case the frequecy i which BRM cells arrive at the brachig switch is equal to the frequecy at which they are set to the root. The they ca sychroize i such a way that by the time a BRM cell is set, all the feedback has bee cosolidated. The BBM algorithm also does ot suffer from oise i steady state. I this case, however, the reaso is that as log as the available rates i the etwork do ot chage, the rates i the BBM matrix ed up covergig to the M most bottleecked rates. Suppose ow that while beig i steady state the source makes a mistake ad trasmits at a higher rate tha the allowed for a short period of time. The the iterarrival time of FRM cells at the brachig switch will decrease. I algorithms 1 ad 3 this implies a decrease of the iter-departure time of a BRM cell at the same switch ad, the, the chaces of sedig this BRM cell before all the feedback has bee cosolidated are bigger. This may produce a ew oisy BRM cell that whe arrivig at the source will make the source trasmit at a higher rate tha the allowed. Now we are uder the same coditio tha we were at the begiig ad oscillatio is produced. This shows that algorithms 1 ad 3 are ustable protocols. I other words, a sigle oisy BRM cell may iduce the etwork to oscillate forever. Note that this is ot the case of algorithms 4, 6 ad the BBM algorithm. This is because for these cases, the iter-departure time of the BRM cells does ot deped o the iter-arrival time of the FRM cells. I the particular case of the BBM algorithm, eve though a casual oisy BRM cell may be geerated (we proved this i the previous subsectio), this oe will ot iduce more oisy BRM cells ad stability is immediately reached agai. Trasiet Respose ad Level Sesitivity. Algorithms 1, 3, 4, ad 6 suffer trasiet delay sice i ay of them a BRM cell arrived at the switch with the most bottleeck value is ot esured to be set immediately back to the root. Oly algorithm 6 ca sed this BRM cell immediately i a overload situatio. However, it caot do the same for a uder-load situatio, this is whe the most bottleeck brach icreases its rate. For this case, i algorithm 6 the BRM cell will wait for all feedback to arrive icurrig uderutilizatio of the etwork. The BBM scheme achieves zero trasiet delay, i both overload ad uder-load situatio. This is because it keeps track of the brach ID ad hece ca tell whe the brach icreases or decreases its available rate. BRM/FRM. All preseted algorithms solve the BRM/FRM issue so that it teds to oe. Resposiveess. Algorithms 4 ad 6 wait for all the braches to respod before sedig a BRM cell. The, if ay of the braches happes to become o-resposive, BRM cells will be ever set back. To solve this, the algorithm has to add more complexity so that this kid of situatios ca be detected. Note that the rest of the algorithms do ot suffer from o-resposive situatios, sice i these other approaches BRM cells ever wait for all the feedback. Complexity. Algorithm 1 turs aroud cells so it ca be cosidered more complex, sice most studies argue that turig aroud RM cells has a high implemetatio cost. The rest of the algorithms do t tur aroud cells ad i this sese, they are simpler. The cost of BBM ca be cotrolled by icreasig or decreasig the umber of etries (M) i the BBM matrix. Big values for M meas high performace with more complexity, ad vice versa. Oe may woder what is the miimal umber of etries i the BBM matrix ecessary to achieve a good eough performace for a fixed umber of braches N. The followig sectio provides the aswer to this questio. It will be show that the performace icreases expoetially as a fuctio of M. Moreover, we will show 4

5 that eve if the umber of braches is very high (eve ifiity), small values of M ca achieve very good performace results. 6 Noise Probability Aalysis I this sectio, we preset a theoretical aalysis of the oise probability for the BBM algorithm. This aalysis turs out to be the key to uderstad the beefits of this approach. We will prove that a BBM scheme ca dramatically reduce the complexity while achievig still good performace. 6.1 The Model Fig. 2 showed the brach switch model used for the mathematical aalysis of the oise probability. We focus o a sigle brachig poit coectio with N braches ad 1 root. The brachig switch implemets BBM ad oly keeps track of the M most bottleecked braches, where 1 M N. For the purpose of our mathematical aalysis we will redefie BBM as a N 2 matrix such that for i= 1,..., N, BBM ( i,1) ad BBM ( i,2) store the ID of the ith most bottleecked brach ad its last available badwidth received, respectively. Note that eve though we have ow exteded this matrix to iclude all N braches available i the switch, still the actual implemetatio of the BBM scheme cosists of the submatrix icludig oly the M most bottleecked braches. We defie the time dimesio i terms of iteratios. A iteratio is a period of time i which a complete set of cosecutive feedbacks from all the braches has arrived. For the purpose of our mathematical aalysis, we will assume that withi a iteratio, oe ad oly oe feedback from each brach arrives. Eve though this may seem urealistic, i sectio 7.3 we will prove through simulatios that this assumptio does ot affect the results i this paper. We use Ν to deote a particular iteratio i such a way that iteratio +1 occurs just after iteratio. Usig this otatio, f i deotes the available badwidth feedback received from brach i at iteratio ad BBM (, i j) is the fial state of the BBM matrix at the ed of iteratio. We also defie the subsets of idetifiers BBM h ad BBM t as BBMh = { BBM ( 11, ), BBM ( 21, ),..., BBM ( M, 1 )} BBMt = { BBM ( M + 11, ), BBM ( M + 21, ),..., BBM ( N, 1)}, respectively, where the subidex h ad t stad for the head ad the tail of the BBM matrix. We will assume that withi a iteratio, ay order of arrivals of the feedback values is equally likely ad that the available rates i the braches are idepedet ad idetically distributed (iid). Eve though urealistic, this model proves to be a worst-case assumptio. Note that i a realistic sceario, braches that belog to a cogested area i the etwork will likely remai cogested, whereas braches that are ot cogested, e.g. because they have larger capacities, will likely remai ot cogested. This realistic sceario would make the BBM matrix more static reducig the probability of havig oise. 6.2 Theory of the Noise Probability Defiitio 2. Most Bottleecked Brach. Let f i be the last available rate feedback that has arrived from brach i at a brachig switch with N braches. We defie the most bottleecked brach at this switch as f = mi{ fi, i = 1... N}. (6.1) Defiitio 3. Noise Probability. We defie the oise probability of a cosolidatio algorithm as the probability that MER is ot equal to f. Note that the above defiitio does ot deped o the source, which meas that the oise probability is ot ecessary the same as the probability that the source receives a wrog feedback value. I fact, this last probability is lower tha the oise probability, because a switch may have MER f but ever sed this value back to the root. The, the oise probability i defiitio 3 is a upper boud of the probability that the source receives a wrog feedback value. Property 1. Cosistecy. We say that the BBM matrix is cosistet if BBM (,) 12 = f. The the followig is true, (1.a) If BBM 1 is cosistet ad there exists at least oe etry i the BBM matrix which has ot received the feedback from its correspodig brach after iteratio - 1, the the BBM matrix is still cosistet. (1.b) If a ew feedback arrives while i iteratio 1 with a available rate smaller tha BBM ( M, 2), the the BBM matrix is cosistet for the rest of the th iteratio. The proofs of the previous properties are straightforward ad we omit them. Lemma 1. Set Partitio. Let A i be the set of evets that satisfy the coditio i ad oly i feedback values from braches with idetifier i BBM 1 t arrive at iteratio earlier tha oe or more feedback values from braches with idetifiers i BBM 1 h, with i=,1,..., N M. The the followig is true, 1) A N M i is a partitio. I other words, A i = U, i= where U is the set that icludes all possible evets A i A j =, for i j. at iteratio, ad { } 5

6 2) M( N M)!( M + i 1)! A i =. i! Proof. Let s begi provig (1). N M A i = U is i= obvious if we cosider that i =,1,..., N M covers all cases. A i A j { } =, for i j, is also true because of the i ad oly i feedback values statemet i the defiitio of evet A i. To prove (2), we cosider a permutatio i the set of feedback values { f i, i = 1,..., N} as a chroological arrival orderig of these feedback values. To cout the umber of elemets i A i, we ow cosider the steps to be doe i order to build this set. At every step, we cout the umber of added evets. Step 1. Get i ad oly i feedback values from braches with idetifier i BBM 1 t. These are a total umber of evets of, N M (6.2) i Step 2. Add the brach ID s of these feedback values to the set BBM 1 h ad remove them from the set BBM 1 t. Build all possible permutatios cosiderig this two-set partitio of the whole set of brach ID s. This makes a total umber of evets of, ( M + i)!( N M i)! (6.3) I additio, cosiderig step 1 we have a total umber of evets of, N M ( M + i )!( N M i )! (6.4) i Note that the previous expressio icludes all evets i i the set A j, sice whe cosiderig all j= A j permutatios we are also icludig those evets i for j = 1,..., i 1. This fact is used i the last step. Step 3. We subtract the overlappig evets i the total umber of evets obtaied i step 2. Mathematically, i i 1 i j j j= j= A = A A = N M ( M + i )!( N M i )! (6.5) i N M ( M + i 1!( ) N M i + 1)! i 1 Fially, simplifyig the previous expressio we obtai the total umber of evets i A i, M( N M)!( M + i 1)! A i = (6.6) i! Lemma 2. Noise ecessary coditio. Let j be the idetifier of a brach such that either 1) j belogs to the set BBM 1 h, or 2) j belogs to the set BBM 1 t ad the feedback from j at iteratio arrives earlier tha oe or more feedback values from braches with idetifiers i BBM 1 h. If such a brach exists ad the available rate brought 1 by it is smaller tha BBM ( M, 2), the for the duratio of the iteratio we have that BBM (,) 12 = f. I other words, there is o oise at iteratio. Proof. If a feedback from a brach j brigs a 1 available rate smaller tha BBM ( M, 2), the by secod part of lemma 1 the BBM matrix will be cosistet after the arrival of j s feedback ad for the rest of the iteratio. Now suppose that j satisfies (1). The, by the first part of lemma 1 we have that the BBM matrix was cosistet before the arrival of j s feedback. Suppose that j satisfies (2). The by the time that j s feedback arrives, there exists at least oe etry i the BBM matrix that has still ot received feedback. Because of this, we ca apply first part of lemma 1 ad this implies that the BBM matrix was cosistet before the arrival of j s feedback. Now we have proved that for the etire iteratio (after ad before j s arrival), the BBM matrix is cosistet ad, hece, oise is ot produced withi this iteratio. Note that the previous lemma gives a sufficiet (ot ecessary) coditio of ot havig oise. This is equivalet to a ecessary coditio for havig oise. We will use this lemma to provide ad upper boud to the oise probability. Lemma 3. Assume that the set of available rates arrived at the brachig switch are idepedet ad idetically distributed. The, the probability that a ew available rate that has arrived from a arbitrary brach j is bigger tha BBM ( M, 2 ) is equal to, N!H([-N+M-1, M],[1+M],1) P( ARj BBM ( M,2)) = (6.7) (M-1)!(N-M)!M where the fuctio Hdz (,, ) is the geeralized hypergeometric fuctio, also kow as Bares's exteded hypergeometric fuctio. Proof. To simplify the otatio, we deote BBM ( M,2) by AR M. Note that AR j refers to the available rate at a arbitrary brach with idetifier j ad that AR M refers to the Mth smallest available rate from all the braches. We begi by expressig PAR ( j AR M ) i terms of coditioed probability by usig the cotiuous versio of the Theorem o Total Probability [PAP79, page 177]. Therefore, we have that, 6

7 1 P( ARj AR M) = P( ARj AR M / AR M = x) f ( x) dx (6.8) AR M I where f AR M is the probability desity fuctio (pdf) of AR M. Note that we have ormalized the available rate from a brach without a lost of geerality so that the limits of itegratio are from to 1. I the previous expressio, we have that PAR ( AR M / AR M = x) = 1 F ( x) = 1 F ( x), j ARj AR where FAR ( x ) is the cumulative distributio fuctio (cdf) of the available rate at ay brach. We ow cocer with the term f AR M. This term is the margial pdf of the Mth smallest value amog a set of N idepedet ad idetically distributed radom variables. To solve this, we apply Order Statistics Theory [Kar81]. Suppose AR M is the Mth smallest value amog a set of N idepedet ad uiformly distributed radom variables. The [Kar81, pages 1-137] proves that the pdf of AR M is the followig: M 1 N M N! x (1 x), for x 1 (6.9) f ( x) = ( M 1)!( N M)! AR M, elsewhere I order to fid a expressio for ay arbitrary distributio fuctio (recall that (6.9) correspods to the uiformly distributed case), we use the trasformatio method i [GAR94, page 155]. This method proves that a uiform distributed radom variable is obtaied whe the cdf of a radom variable is applied to this same radom variable. Mathematically, we have that AR M = F ( AR M ) (6.1) AR Now (6.9) shows the pdf for the uiform distributio case ad (6.1) shows the relatio betwee the geeral distributio, AR M, ad the uiform distributio, AR M. With both results ad kowig that F AR is mootoically icreasig, we ca use the Fudametal Theorem i [PAP79, page 126] to derive f AR M : dfar ( x) f ( x) = f ( F ( )) AR M AR AR x (6.11) M dx Substitutig f ( x) we have, AR M![ M 1 N M AR()] (1 [ AR()]) AR(), for x 1 N F x F x df x f () x = ( 1)!( )! AR M N M dx M, elsewhere Now we kow both PAR ( AR M / AR M = x) ad f AR M. Substitutig both i (6.8) ad itegratig, we obtai, j N!H([-N+M-1, M],[1+M],1) PAR ( j AR M ) = (M-1)!(N-M)!M which proves the lemma. The surprisig result i lemma 3 is the fact that PAR ( AR M ) does ot deped o the distributio j fuctio of the available rate at a brach. Though it may seem cotroversial, there is a ituitive reasoig to uderstad this property, which is ot preseted i this paper because of space costraits. Thaks to this property, we will show that there exists a upper boud for the oise probability idepedetly of the distributio fuctio of the available rate. Theorem 1. Noise Probability. The oise probability of the BBM algorithm ca be upper bouded by, N M M+ im( N M)!( M + i 1)! Poise ( ) ( ρm, N) (6.12) i= in!! where ρ M, N is equal to N!H([-N+M-1, M],[1+M],1) ρ M, N= (6.13) (M-1)!(N-M)!M Proof. Usig the Theorem o Total Probability we have, N M P( oise) = P( oise / A i) P( A i) (6.14) i= Note that we ca use this theorem because A i is a partitio of U, as proved i first part of lemma 1. Let s first calculate PA ( i ). Recall that i our model we assume that withi a iteratio, ay order of arrivals is equally likely. The, all evets i A i are equally likely ad we ca write PA ( i ) as, umber of evets i A i PA ( i ) = (6.15) umber of evets i U Sice the umber of evets i U correspods to the total umber of permutatios, this is N!, usig the secod part of lemma 1 we have that, M( N M)!( M + i 1)! PA ( i ) = (6.16) in!! We ow provide a upper boud for P( oise / A i ). From lemma 2 we ca boud the probability of ot havig oise coditioed to A i, P( oise / A i) P(coditios i lemma 3 are met/ A i) Note that the above is true sice lemma 2 is a sufficiet but ot ecessary coditio for ot havig oise. Now the probability that the coditios i this lemma are met is equal to oe mius the probability that these coditios are ot met. The later ca be obtaied by esurig that for ay brach satisfyig coditios (1) or (2) i lemma 2, its available rate brought at iteratio is bigger tha 7

8 (, ). Uder coditio A i, there are M + i 1 BBM M 2 such braches. Assumig idepedet ad equally distributed braches we have, 1 ( / ) 1 ( (,2)) M + P oise A i i P ARj BBM M where AR j is the available rate of a arbitrary brach j. Equivaletly, we have, 1 ( / ) ( (,2)) M + P oise A i i P ARj BBM M Now, usig lemma 3 we have, M+ i N!H([-N+M-1, M],[1+M],1) P( oise / A i ) (M-1)!(N-M)!M which completes the prove. Theorem 2. Noise Probability Boud. The oise probability for the particular case M = 1 (i.e. the brachig switch oly keeps track of the most bottleeck brach) ad with ifiite umber of braches is at most 1 e 1. Mathematically, 1 Lim P( oise) 1 e (6.17) N M = 1 Proof. Usig theorem 1 we have that N 1 1+ i ( N 1)! Poise ( ) ( NH([-N, 1],[2],1) ) M = 1 i= N! The limit as N ca be solved by expadig the hypergeometric fuctio ad usig Stirlig s Formula [GAR94, page 45] to obtai 1 Lim P( oise) 1 e N M = 1 This last theorem may seem a surprisig result at first sight. Ideally, to achieve a oise probability of zero we eed to keep track of all the braches. If the umber of braches is ifiity, the we eed to keep track of ifiite umber of braches. However, if we oly keep track of oe brach (the most bottleeck) the oise probability does ot ted to oe. Ideed, the theorem proves that it 1 ca be bouded by 1 e Though surprisig, this result is actually ituitive if we carefully cosider lemma 2. I this case, ote that whe N teds to ifiity, the probability that the secod coditio i this lemma 1 becomes true goes to oe, sice ow BBM t has 1 ifiite elemets ad BBM h has M fiite elemets. This implies that the coditios i the lemma are ow more likely ad, because these are sufficiet coditios to avoid oise, the chaces to avoid oise are better. Actually, it is possible to fid a boud for ay arbitrary value of M. Because Stirlig s Formula turs out to be oly applicable to the case M = 1, so far o symbolic expressio has bee derived for cases M > 1. For this cases, we preset a umeric solutio. Table 1 shows the results of this aalysis. Table 1. Boud Noise Probability for ifiite umber of braches M P(oise) Fig. 3 shows the oise probability boud for a brachig switch with up to 1 braches usig a BBM matrix with up to five etries. Note that as N becomes bigger, the oise probability boud teds to the values i table 1. M=1 M=2 M=3 M=4 M=5 Figure 3. Noise Probability boud for M=1, 2, 3, 4 ad 5. I fig. 4 the oise probability boud has bee umerically evaluated for the case 2, braches ( N = 2, ) i order to approximate the case of ifiite umber of braches. Note that the y-axis is represeted i a logarithmic scale. This meas that the oise probability drops expoetially as the umber of braches we keep track of (M) icreases. I particular, the probability that oise occurs whe we oly keep track of 2 braches of a total umber of 2, braches, is about less tha 1 8. Note that this dramatically reduces the implemetatio complexity, compared to a per-brach accoutig approach, while still maitais high degree of performace. Figure 4. Noise probability boud for N=2, braches 8

9 This example shows the utility of the BBM scheme. Actually, this approach is totally geeral ad ca be see as a tool to be used i ay protocol whe the per-brach accoutig scheme is too expesive to be implemeted. 7 Simulatios I this sectio we preset some simulatios to prove the theoretical results preseted i this paper. The first simulatio shows how the cosolidatio oise issue affects each of the examied algorithms. I the secod simulatio, the trasiet respose i the brachig switch is measured for each algorithm. Fially, the third part i this sectio proves the validity of the model assumptio preseted i sectio 6.1 for the case of real etworks. ABR we use the etwork cofiguratio i fig. 5. I order to simulate a dyamic situatio i which the bottleeck lik cotiuously chages from brach to brach, we cofigure the VBR sources to geerate ON-OFF Poisso traffics with a mea burst legth of 8 µsecs ad a mea iterval betwee burst of 2 µsecs. The trasmissio rates i the ON iterval are [4, 5, 6, 7, 8] Mbps, from the most left to the most right VBR source. ACR(Mbps) x x x x Figure 5. Network Cofiguratio 7.1 Simulatio I: Noise Cosolidatio I this simulatio we evaluate the effect of the oise cosolidatio issue. Fig. 5 shows the etwork cofiguratio that has bee used. It presets a multipoit switch with 5 braches. I order to simulate bottleecks movig from brach to brach, VBR traffic is added at each brach. All lik capacities are 155 Mbps. From left to right, the trasmissio rates for the VBR sources are [5, 35, 65, 9, 4] Mbps ad their iitial trasmissio times are [, 3, 6, 9, 12] millisecods, respectively. The results i terms of available cell rate (ACR) at the source for each algorithm are preseted i fig. 6. Note that for algorithms 1 ad 3 whe a ew bottleeck brach appears oise is produced, which makes the source oscillate amog the available rates of differet braches. Algorithms 4, 6 ad the BBM algorithm, which has bee simulated with M=1, do ot suffer from oise. 7.2 Simulatio II: Trasiet Respose This secod simulatio measures the trasiet respose for each of the cosolidatig algorithms. Agai, t(microsecs) x 1 5 Figure 6. From up to dow, ACR for algorithms1, 3, 4, 6, ad BBM, respectively Fig. 7 shows, for all the algorithms, the trasiet delay that a BRM cell with a ew bottleeck rate suffers at the brachig switch before it is passed back to the root. All cases have bee simulated with the same seed i order to make a fair evaluatio. While all the algorithms suffer trasiet delay, ote that the BBM algorithm has a zero delay. Trasiet Respose (microsecs) x x x x t(microsecs) x 1 4 Figure 7. From up to dow, trasiet delay for algorithms 1, 3, 4, 6 ad BBM, respectively 7.3 Simulatio III: Model Assumptio I this sectio we will provide some simulatios to prove the validity of the mathematical model used to obtai the oise probability upper boud. I our model, we assumed that i oe iteratio (as defied i sectio 9

10 6.1) oe ad oly oe feedback arrives from each brach. We simulate three differet traffic patters: - Traffic patter 1: correspods to the traffic patter defied by our theoretical model, i.e. exactly oe feedback from each brach arrives at each iteratio. The distributio of the order i which each feedback arrives at each iteratio correspods to a uiform distributio. - Traffic patter 2: The iterarrival time of the feedback i each brach correspods to a uiform distributio. - Traffic patter 3: The iterarrival time of the feedback i each brach correspods to a expoetial distributio. Hece, for traffic patter 2 ad 3 o assumptio regardig the umber of feedbacks arrivig from each brach i a iteratio is doe. Note also that we do ot specify the time properties of the iterarrival distributios (mea ad variace) sice the oise probability aalysis is idepedet of them. For all simulatios, the value of the available badwidth i the feedback cells is assumed to be uiformly distributed. There is o loss of geerality sice our upper boud does ot deped o the type of distributio. The results below have bee obtaied by simulatig 2 iteratios for each plotted poit i the graphs. Fig. 8a shows the results for traffic patter 1. Comparig this graph to that of fig. 3, we show that expressio (6.12) is a actual upper boud of the oise probability uder our model assumptio. Fig. 8b ad 8c show the same results applied to traffic patters 2 ad 3. This graphs show that the asymptotic behavior of the oise probability still applies eve for the real case of ay order of arrivals. Fially, fig. 8d shows for all traffic patters the scalability property of the BBM algorithm. I other words, the graph proves that the oise probability decreases expoetially with the umber of braches we keep track of. This is cosistet with the theoretical results preseted i fig Coclusios Table 2 shows a compariso betwee the cosolidatio algorithms. The BBM scheme is the oly approach that ca achieve zero trasiet respose delay. It is also stable i the sese that a oisy BRM cell does ot geerate more oisy BRM cells. It does ot suffer from the o-resposive braches issue ad esures that the ratio BRM/FRM coverges to 1. Fially, BBM defies a trade-off betwee the complexity ad the oise cosolidatio issue that ca be cotrolled with the parameter M. However, as we have proved, this trade-off turs out to be favorable, sice very low oise ca be achieved with small complexity (small M). Refereces [Fah97a] Soia Fahmy, Raj Jai, Shivkumar Kalyaarama, Rohit Goyal, Bobby Vadalore ad Xiagrog Cai, "A Survey of Protocols ad Ope Issues i ATM Multipoit Commuicatio," OSU Techical Report, August 21, [Fah97b] Soia Fahmy, Raj Jai, Rohit Goyal, Bobby Vadalore, "A switch algorithm for ABR multipoit-to-poit coectios," ATM Forum/97-185, December [Fah98] Soia Fahmy, Raj Jai, Rohit Goyal, Bobby Vadalore, Shivkumar Kalyaarama, Sastri Kota, ad Pradeep Samudra, Feedback Cosolidatio Algorithms for ABR Poit-to- Multipoit Coectios i ATM Networks, Ifocom 98, Sa Fracisco, March 1998, vol. 3 pp [Fah99] Soia Fahmy, Raj Jai, Rohit Goyal ad Bobby Vadalore, "Fairess for ABR Multipoit-to-poit Coectios," Submitted to IEEE Network Magazie, March [Gar94] Alberto Leo-Garcia, Probability ad Radom Processes for Electrical Egieerig, 2 d Editio, Addiso Wesley, Massachusetts, [Kar81] Samuel Karli, Howard M.Taylor, A Secod Course i Stochastic Processes, Academic Press, New York, [Pap79] Athaasios Papoulis, Probability, Radom Variables, ad Stochastic Processes, McGraw-Hill, Taiwa, [Re96] W. Re, K-Y Siu, ad H. Suzuki, O the Performace of Cogestio Cotrol Algorithms for Multicast ABR Service i ATM, Proceedigs of IEEE ATM 96 Workshop, Sa Fracisco, August [Re97] W. Re, K.-Y. Siu, ad H. Suzuki, Multipoit-to-Poit ABR Service i ATM Networks, IEEE Iteratioal Coferece o Commuicatios, Motreal, Caada, Jue [Re98] W. Re, K-Y Siu, ad H. Suzuki, ad M. Shiohara, Multipoit-to-multipoit ABR Service i ATM Networks, Computer Networks ad ISDN Systems, October 1998, vol. 3, o.19, pp [Sim98] MANUAL: The NIST ATM/HFC Network Simulator, Operatio ad Programmig Guide, Versio 4., December [Va99] Bobby Vadalore, Soia Fahmy, Raj Jai, Rohit Goyal ad Mukul Goyal, "QoS ad Multipoit support for Multimedia Applicatios over ATM ABR service," IEEE Commuicatios Magazie, Jauary 1999, pp Table 2. Compariso of cosolidatio algorithms Algorithm BBM Cosolidatio oise Yes Yes No No Expoetially decreases with M Noise stability Ustable Ustable Stable Stable Stable Trasiet respose > > > > delay BRM/FRM Resposiveess OK OK KO KO OK Complexity High Low Low Low Scalable with M 1

11 figure 1.4 a) b) Noise Probability Noise Probability figure N figure 3 c).4 d) Noise Probability Noise Probability N figure 4 traffic patter 1 traffic patter 2 traffic patter N M Figure 8. Noise probabilities graphs for traffic patters 1, 2 ad 3. 11