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Re-Examining Maxmin Protocols:A Fundamental Study on Convergence,

Complexity, Variations, and Performance

Wei K . Tsai and YuseokK imDepartment of Electrical and Computer EngineeringUniversity of California, I rvine, C A 92697, U.S.A.

Abstract-T his paper re-examines maxmin protocols for ABRtraffic in ATM networks in four aspects: convergence, complex-ity, variations, and performance. Fir st, the concept of “pseudo-saturation” is introduced. Most, if not all , protocols do not prop-erly handle pseudo-saturated links, and as a result, there is noguarantee for convergence to true maxmin solutions. Second, theconcept of “constraint precedence graph (CPG)” is introducedand is used to define the best possible time complexity of anymaxmin protocol. The existing complexity estimates are overlyconservative because they do not consider possible concurrentoperations. In contrast, the CPG analysis explicitly accounts for

parallelization. Third, the concept of “constraint” is generalizedand this generalization is used to derive an optimality conditionfor the maxmin problem with nonzero minimum cell rate (MCR)requirements. This optimality condition can be used in conjunc-tion with any maxmin protocol to handle the nonzero MCR re-quirements without adding excessive complexity. Finally, simula-tions suggest that the complexity analysis is inadequate to gaugeprotocol performance. A new analysis based on protocol dy-namics is called for to understand the performance.

K ey words: Integrated service network, ATM , ABR, flowcontrol, and maxmin fairness

I . INTRODUCTION

Rate-based flow control for AB R traffic in ATM networksstrives to maximize bandwidth utilization while maintaining

fairness among all the virtual connections (VC’s). T his leadsto the concept of maxmin fairness originally proposed byHayden and J affe [9, 111. M any maxmin protocols have beenproposed [ l , 2, 5, 6, 7, 10, 12, 13, 14, 151. This paper re-examines maxmin ABR flow control in four aspects: conver-gence, complexity, variations, and performance.

First, the concept of “pseudo-saturation’’ is introduced.Most, if not all, protocols do not properly handle pseudo-saturated links, and as a result, there is no guarantee for con-vergence to true maxmin solutions.

Second, the concept of “constraint precedence graph(CPG)” is introduced and is used to define the best possibletime complexity of any maxmin protocol. For most protocols,there is no global convergence proof and the complexity es-

timates are often very conservative as they do not considerconcurrent operations. In contrast, the CPG analysis explicitlyaccounts for parallelization. Based on the CPG analysis, analmost optimal-complexity protocol called CPG is derived.

Third, the concept of “constraint” is generalized and thisgeneralization isused to derive an optimality condition for the

maxmin problem with nonzero minimum cell rate (MCR)requirements. This optimality condition can be used in con-junction with any maxmin protocol to handle the nonzerominimum rate requirements without adding excessive com-plexity. The existing maxmin protocols that accommodateM CR requirements all require complex computation [10, 161.Using this new optimality condition, it is possible to do awaywith the complex computation.

Finally, simulations suggest that the complexity analysis isinadequate to gauge protocol’s performance. I t turns out that

speed of response and stability of the protocol play an evenmore important role than the complexity analysis in the per-formance. A new analysis based on protocol dynamics iscalled for to understand the performance.

11 PSEUDO-SATURATIONAND CONVERGENCE

For most maxmin protocols, there exist no global conver-gence proofs. To our best knowledge, only Charny et al. at-tempted to analytically prove the convergence of their proto-col from any arbitrary initial condition [5].However, it turnsout that if there existpseudo-saturated(PS) inks, all existingprotocols wil l fail to achieve true maxmin optimality.

First, the concept of maxmin optimality needs to be de-fined. Let Ai denote the rate of V C i . Let Cj denote the ABR

link capacity at l inkj . L et5denote the set of VC’s crossing

link j . Let 5 denote the total input rate from all the ABR

VC’s crossing link , hus F J =

F j I Cj for all l inkj .

Definition. A vector of rates {A, } is said to be muxmin air

if it is feasible and for each VC i , Ai cannot be increased

while maintaining feasibility without decreasing A, for some

VC k for which A, 5 A,.

A, .

Definition. A vector of rates { A , } s said to befeasible if

Now, let Rj denote the advertised rate at link j .

Definition. With respect to any V C rate vector {A , and

any advertised rate vector ( R j} ,V C i is said to be uncon-

strained at link j or link j is said to be a bottleneck link forVC i if VC i crosses link j and Ai 2R,; otherwise, VC i is

said to be constrained at link j if VC i crosses link j andA i eRj.

0-7803-5417-6/ 99/ $10. 001999 IEEE. 811

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Definition. A maxmin protocol is one that satisfies the

following condition: For every link j , f Fc j , Cj, and Ncj do

not change any more, the protocol will eventually assign theadvertised rate of

C. -F c.

R . =I N j -Nc j

to every V C which crosses link j and is unconstrained at link

j , where F cj is the input rate from constrained VC’ s at link ,

and Ncj is the number of constrained VC’s at link .

Definition. A maxmin protocol is said to be convergent ifgiven an arbitrary initial rate vector { Ai ) nd an arbitrary

initial advertised rate vector ( Rj ), the protocol will eventu-

ally assign the unique max-min rate vector {A ; } to the set of

VC’s.

In order to prove the convergence, Charny et al. introducedthe concept of marking-consistency (M -consistency) [5]. Es-

sentially, M-consistency means that once Rj is calculated atlink j based upon F cj and Nc,, Ai of constrained V C i

should be strictly less than Rj .

her dissertation [4] as follows:

Charny handled the PS links in her protocol appeared in

(2)

However, in both [4] and [5], the convergence proofs didnot take into account the PS links. In fact, in [5] Charny et al.changed the code segment handling the PS l inks as fol lows:

If Ncj equals to N, , Rj =C j - F cj +maxiEv,A ; .

If Ncj equals to N j , Rj =C j . (3)

The code segment (2) appears to be critical for the protocol

to converge. Let’s consider the following counter example asdepicted in Fig. 1 where Charny’s modified protocol withcode segment (3)does not converge.

vc 3

R,=i5 I R,=40 R,=30

R, ’=15 V C ~,’=26 R,’=34

V C , : Virtual Connection iC , : L i n k capacity of L i n k j

R ,Initial advertised rate o f L i n k j

R,’ :dax-Min advertised rate of L i n k j

Fig. 1. Counter example

Note that Link 2 is PS since both V C 1and VC 2 are con-

strained at L ink 1 and L ink 3, respectively. From the set ofinitial advertised rates (R ,=15, R, =40, R3=30),since Nc,

is 2 (=N,), R, becomes 41 (=C ,) and stays at 41 forever.

Note that the maxmin value of R2 is 26.

In the protocol ASAP presented at [15], Rj is not updated

if Ncj equals to N j In the same counter example, the proto-

Construct-CPG(N:network, L :l evel)

1. For each link k in N , calculate Rk=Ck N , .

2. For each link k in N , do the foll owing:

2-1. If Rk=min [ R I link & link k share a joint V C), add link k

to the set of level L links.

For each link k in the set of level L links, do the following:3-1. Remove link k and all the VC’ s crossing link k fromN .

3-2. Add link k as a child node in the CPG to any level L-1 link thatshares a joint V C with link k.

3-3. UpdateCjandNj for all l i nkj sharing a joint V C with link k:

3.

Cj=Cj-Rk. (number of removed vc ’ s)Nj =Nj - number of removed VC ’s)

4. If N is not empty, call Construct-CPG(N, L +1).Otherwise, stop.

Fig. 2. CPG algorithm

col does not converge. The code segment (2) (or other codesegment handling PS links correctly) should be adapted inevery maxmin protocol.

111. THECPG ANALY SISND THECPG PROTOCOL

A. The Constraint Precedence Graph

The Constraint Precedence Graph (CPG) is similar to thecomputation precedence graph commonly used in the com-

putation theory except that the precedence relationships aredefined in terms of convergence sequence of link advertisedrates to their maxmin advertised rates (MAR’ S). In this sec-tion, the advertised rate (A R) at a link i s defined as the maxi-

mum rate all ocated to the V C’ s crossing the link. The CPGanalysis is based on a simple observation. I f a link’s MA R issmaller than that of another link and both links share someVC’ s, then the AR of the link with the smaller M A R has toconverge to its MA R before the AR of the link with the largerM AR can converge to its MA R. The precedence relationshipis then expressed in terms of a graph. This graph can be con-structed using the CPG algorithm shown in Fig. 2.

The CPG algorithm is a global synchronous algorithm. Themain di fference between the CPG algori thm and the globalalgorithm cited in most maxmin protocol papers published sofar is that the CP G algorithm is a parallel algorithm, while in

every paper where a global algorithm is used as the model,the algorithm is inherently serial. I n the CPG algorithm, Steps2 and 3 can be carried out at each link in parallel. T he CP Ganalysis is illustrated using the sample network configuration

depicted in Fig. 3. The degree of parallelization can be visu-

alized in the CPG shown in Fig. 4.

Initially, the AR of each link is set to its fair share, i.e.,

(link capacity)/(the number of VC’s crossing the link). Based

on this initial A R, L ink 1 is the most bottlenecked link. T hus,the AR of Li nk 1 is same as its M A R. However, for L ink 2,the AR is initially 40, but since V C 2 cannot take more than35, Link 2 increases its AR to 45 (=80 - 35) which nowequals to its M A R. Now, let’s look at L ink 3. Initially, theARof Link 3 is 50.However, since V C 3can take only 40, L ink3 increases its AR to 60 (=100- 40). Note that this happensbefore Link 2 updates its AR to 45. When Li nk 2 updates its

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AR to 45, Link 3 decreases its AR down to 55 (=100- 45)

which now equals to its MAR. L ikewise, L ink 4 updates itsAR from 60 to 80 and down to 75 which is its MAR. Notethat once theAR of L ink 2 converges to itsMAR, both L ink 3

and L ink 4can update their AR’s concurrently.

vc 3

VC,: Virtual Connection i

C,: Link capacity 0 1 Link i

R.:lnitial advertised rate of Link j

Ri’:M’ax-Min advertis ed rate of Link j

Fig. 3. Sample network co nfiguration

CGK Level 1

Link 2 Level 2Aink 3 Link 4 Level 3

Fig.4. CPG for sample network configuration

Note that the CPG depends only on the underlying maxminproblem and is independent of the protocol one might use tocompute and all ocate the rates. I t is our conjecture that theCPG defined above represents a unique convergence partialordering that cannot be violated by any maxmin protocol.With the CPG defined above, some analytical results nowfollow.

Definition. CPG(i) denotes the set of links in the nodes atthe i-th level of the CPG.

Definition. Given a set of feasible rates{Ai

1, linkj

is saidto be constrainedby link k if there exists a VC crossing both

links j and k and Rk 5Rj, where Rk and Rj are the MAR’Sof

link k and link , respectively.

Intuitively, each link in the child node of the CPG is con-strained by the links in its parent node. Note that the links ineach branch of a node in the CPG can independently con-

verge into their maxmin rates.

Proposition 1. For every maxmin problem, the associatedCPG is unique.

Proof. The proof is trivial by the CPG algorithm.

Definition.’ Let { A i * } denote the set of source rates com-puted by the CPG algorithm, i.e., {Ai *}= { A i }where A; is the

rate assigned to the link crossed by VC i when VC i is re-moved at step 3of the CPG algorithm.

Proposition 2. The rates { A*}computed by the CPG algo-rithm for any maxmin problem are maxmin fair.

Proof. According to the proposition in [3,pg.5271, { A * } s

maxmin fair iff every VC has a bottleneck link. Supposek E

CPG (1). Then, by the property of the CPG algorithm, all

VC’s crossing link k has a bottleneck at link k. By the CPGalgorithm, each V C being removed at the I-th level of theCPG has a bottleneck link at the I-th level of the CPG. By

Proposition 3. For any maxmin problem, suppose that { A , )

Proof. The proof is trivial by the CPG algorithm.

Thus, the maxmin source rate { A , * } vector is unique, whilethe link rate vector is not unique.

Definition.Let V’ denote the set of VC’s crossing the linksin the CPG at the level 1 but not crossing the links in the CPGat the levels from 1through ( 1 - 1).

The VC’s in V‘ are called the level 1 VC’ s. Note that thelevel 1 VC’s are removed at the I-th level in the CPG algo-rithm.

Definition. Let V[ denote the set of level 1 VC’ s crossing

induction the proposition is proved.

is any set of maxmin fair rates, then ( A , } = { A , * ) .

li nkj, i.e., V‘ =V‘ nV .

Definition. Let 5 denote the sum of the maxmin optimal

rates of the level 1through level 1 VC’s crossing linkj.Thus,1

Fj ’ = Z A *m=iEV,”’

Definition. Let C: denote the capacity of link j at the

beginning of the I-th level of the CPG algorithm. Therefore,

Cj =C j - C!-l. Cj will be called the remaining capacity of

link j at level 1.

Definition. Let Nj denote the number of VC’s crossing

l i nkj at the beginning of the I-th level of the CPG algorithm.

Therefore, Nj =N j

-15’+V’ +...+?!-‘I. Nj will be called

the number of remaining V C’ s at l ink j at level.

Definition. The ratio Cj/Nj is said to be the remaining

fair shareof link j at level 1 of the CPG and the ratio C IN;

is said to be the fair share of l inkj .

Proposition 4. (a) For every link k E CPG(1) (k is in anode at the I-th level the CPG), the CPG algorithm will termi-

ninate with Rk =

N ’ ’

(b) For every link k E CPG(l), the remaining fair share oflink k at level 1is the maxmin optimal rate for all VC’ s in theset v,L.

(c) Let L be the number of levels in the CPG, then for every

link k E CPG(l),

Proof. Obvious by the CPG algorithm and Propostion 3.

With the above results, now it is possible to prove a lowerbound complexity result for the class of maxmin protocols

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which isdefined below.

Definition. Given a VC rate vector ( A i }and an advertisedrate vector (Rj},a switch is said to have inconsistent con-straint status if one of the followings is true:

(a) The value of F c, computed based on the informationstored at the switch is not equal to the the sum of rates forconstrained VC’ s at link j with respect to the rate vectors

(b) The value of Nc, computed based on the information

stored at the switch is not equal to the number of constrainedVC’s at link j with respect to the rate vectors ( ( A i )and (R,}).

Definition. Assuming that V C i crosses li nkj , let0: denotethe feedback latency from li nk j to source i and from source i

back to l inkj . Then the maximum feedback latency D is de-fined to be:

( { A i }and IRjI).

D =max ( D/ I all V C i crosses li nkj, for all l inkj ).

Proposition 5. For any convergent max-min protocol, the

best possible “worst-case” time complexity of convergence isL .D +TRM where L is the number of levels in the associated

CPG.Proof. Suppose the CPG has only one level. Since the ini-

tial rate vectors ( ( A i }and ( R j ) )are arbitrary, there exists aninitial state of the protocol such that there is a link with aninconsistent constraint status. In the worst case, a convergentmax-min protocol will need at least D +TRM units of time to

allow the switch to gather i nformation to correct its constraintstatus. Note that TRM is needed for the switch to receive at

least one FRM cell. Every protocol cannot start updating i tsadvertised rate without an RM cell.

Now suppose that the proposition is true for max-minproblems with a CPG having 1 levels. Consider a max-min

problem with (1+1) levels in the CPG. By the induction hy-pothesis, a convergent max-min protocol will need at least ID

units of time to compute (R,) for all the links j E CPG(1).Since the initial rate vectors ( ( A ; } and (Rj})are arbitrary,there exists a state of the protocol such that there exists a link

k E CPG(I+I) with an inconsistent constraint status. In theworst case, a convergent proper max-min protocol will needadditional D units of time for link k to gather information tocorrect its constraint status. Thus any convergent proper max-

min protocol will need at least ( I+1)D+ TRMunits of time to

compute (R,) for all the links j E CPG(l+1).The proposition

B. The CPC Protocol

is proved by induction.

The CP G algorithm can be developed into a distributedprotocol (called the CPG protocol) as follows. The main dif-ference between the CP G protocol and other maxmin proto-cols based on (1) is that the CP G protocol uses a new conceptof “settle status”. A V C is said to be settled if its rate hasreached the maxmin rate. The formula used to compute theadvertised rate at link k is given below:

(4)- FS,

Rk =Nk -NSk

where F s, is the total flow from settled vc’ s and Ns, is the

number of settled VC’s at link k . This formula may looksimilar to (l),but the difference is quite signif icant. In (4),

one can say that once a V C becomes settled, the VC hasreached its maxmin rate. O n the other hand, in formula (l),

the constrained VC may or may not have reached its maxminrate. Because of the conservative nature of the formula, theresponse speed of the CPG protocol is slower than that of themaxmin protocols we have simulated. However, in terms ofconvergence complexity, the CPG protocol can be shown toachieve a complexity of 0.75L.D+TRMon the average and

1.5L.D+TRM in the worse case. The pseudo code of the CP G

protocol is presented in the Appendix.

C. The Complexity Estimates of Maxmin Protocols

So far, the only maxmin protocol that accompanies a cor-

rect convergence proof and a correct complexity proof is theclass of the Marking-Consistency protocol (M CP) by Charnyet al. However, as pointed out in Section 11 Charny et al. didnot realize the importance of pseudo-saturated links, and thustheir proof appeared in [6] is incomplete. Charny et al. alsotook an overly conservative approach in estimating the com-plexity of the M CP . With the help of the CPG analysis, thecomplexity of M CP can be shown to be upper bounded by2L .D +TRM Without CPG, a good estimate of M CP would

be 2 N L .D + TRM, where N , is the number of the distinct

maxmin advertised rates. The proof of these rates can be eas-ily obtained by studying [Cha95, Cha961, carefully account-ing all the steps, and making sure that pseudo-saturated linksdo not change the complexity in any essential way. W e shall

not provide a detailed proof here.

W e would, however, like to highlight the key steps in theconvergence and complexity proof.

Definition. Thefirst round-trip update after t at link k is

defined to be the first possible update at link k after every VCcrossing link k has at least one BR M cell leaving link k aftert , and this B RM cell arrives at the source and changes thesource state, and then the source of thi s VC sends an FRMcell to arrive at link k , at the time of the link update.

Definition. Let tf be the time at which the first link update

is performed at link k after the disturbance. Let

t o =max{t::k is a link}; thus, t o is the time at which the first

link updates have been performed at all links in the network.

Furthermore, let ti be the time at which the first round-trip

update after to is performed at link k . Let

ti =max{ti:k isa link}; thus, ti is the time at which the first

round-trip update after t o is performed at all links in the net-

work. Using induction, let t‘ be recursively defined forI = :..L.

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The following lemma is the key to the convergence proof.

L emma 1. For all links k, assuming that all the denomi-

nators in the following expressions are positive,

Proof.The proof is trivial and is omitted.

Definition. Let CS, denote the constraint status of V C i .

CS, can take only one of the two values: constrained or un-

constrained

Definition. An advertised rate Rk at link k is said to be

marking consistent with the set of source rates {A,;CS,}lcV,

associated with all the VC’s crossing link k if A, 2 Rk implies

that CS, =unconstrained.

Assuming marking consistency, using Lemma 1, one caneasily prove the following Lemma:

L emma 2. Assume that R,,,(t)=R!,, , or any link m whichis at level I or lower, and R, (t)2Rj for any link j which is

level 1+1or higher, for all t2t’ .Let ti,l be the time at which

the fi rst round-trip update after t‘ is computed at link k . Let

tf =max{tl,,:k is a link}. Let t;,* be the time at which the first

round-trip update after t: is computed at link k . Let

ti =max{tlS2: is a link]. Then, for all links j at level 1+1 or

higher and for all r 2 ti ,

R,(t)2 R;+’ (5)

The key here is the condition (5). This turns out to be avery nice stopping (terminating) criterion for the optimization

algorithm (the Marking Consistency Protocol). Once (5) issatisfied, L emma 1 implies that any link at level 1+1 willhave a smaller advertised rate than all the links at levelshigher than 1+1, which share a common VC with the level1+1 link, within one round-trip update time. For all maxminprotocols other than MCP and the CPG protocol, this condi-tion isvery hard to be proven.

For the CPG protocol, on the other hand, there is no needto rely on M arking Consistency to show the convergence andcomplexity of the protocol. The CPG protocol mimics theglobal CPG algorithm, thus, i ts convergence is triviallyproven. The only issue is the complexity estimate. In the CPGprotocol presented in Appendix, bi -directional feedback isused: both FRM and BRM are used to inform both upstream

links and downstream links of settle status and rates of settledVC’s. In the following analysis, it is assumed that TRM <<D .

In the worst case, the complexity is 1.5L .D+ TRM. his can

be seen as follows. In the CPG protocol, a link has to wait forall its unsettled VC’s to confirm that those unsettled VC’s cantake the advertised rate before the link can declare that it hasconverged (settled) or not. The confirmation would take one

full round trip delay ( Dunits of time) in the worst case. Oncea link converges, all its unsettled VC’s will change their settlestatus to SET TLED. Then all other l inks need to be informedof the changes in the settle status and rates of these newly

settled VC’s. Thi s propagation of information wi ll take 1/2the round trip delay (OSD units of time). We have thusproved the following proposition.

Proposition6. The CPG protocol converges to a maxminoptimal advertised rate vector from any arbitrary initial con-dition within 1.5L.D + TRM units of time.

Assume a very simplistic probabil ity model where the av-erage time required for the protocol to converge is equal tothe time needed as if every link behaves exactly as the aver-age link. This approach is known as the certainty equivalenceprinciple in stochastic control. I n this simplistic stochasticmodel, the time for an average link to confirm that it has con-verged to its maxmin rate will take OSD. The time for an av-erage link to relay the changes in settle status and rates of

newly settled VC’ s to another average link will take 0.250.Thus the average complexity reduces to 0.75L.D +TRM. t

should be noted that using the simplistic stochastic model, theaverage complexity of the Marking Consistency Protocol re-ducesto L.D+T,,.

IV . SOURCEONSTRAINEDAXMIN ROTOCOLS

There exist two cases where a source can be constrained onthe entire VC: the source could be limited by the maximumrate (or PCR) it can transmit or associated with the minimumcell rate (MCR) requirement imposed by the network-usercontract. For the case of non-zero M CR, the A TM forum hasadopted a few options in the T M 4.0 specif ication. Amongthose options, the most difficult one to handle is the option of

M ax(M axmin rate, M CR): the source must be allocated theM CR if its maxmin rate is smaller than its MCR, otherwise,the source must be allocated the maxmin rate.

The key idea to solve this problem is to recognize that theM CR constraint i s a constraint by itself. I n the original max-min formulation, a constrained VC at a link is one that is un-

ableto increase its rate to reach the advertised rate at the link.A straightforward generalization is to define a constrained VCat a link as one that is unable to either increase or decrease itsrate to reach the advertised rate. This simple observation,trivial it may appear to be, is a key to design distributedmaxmin protocol s without per-VC computation. There appearto be two maxmin protocols from the literature that handle theM CR constraint [IO, 161. The GM M protocol presented in

[IO] requires per-VC-sorting, which is quite undesirable. Thepurpose of this section is to introduce a new optimality con-dition that can be used to design a non-per-VC-sorting proto-col for handling MCR constraints.

We shall adopt the definition of generalized maxmin opti-mality by Hou et al. For each VC i , let PCRi and MCRi de-

note the PCR and MCR, respectively. In addition, i t is as-

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sumed that the sum of the MCR’s of all the VC’s crossingeach link is no larger than the capacity of the link.

Definition. A vector of source rates {A,} is said to bef ea-

sible if 5I j for all link j , and for each VC i ,

MCR, 2 A, 2 PCR, .

Definition.A vector of source rates {A,] is said to be Gen-

erali zed Maxmin (GMM)optimal if it is feasible and for eachVC i, A, cannot be increased while maintaining feasibility

without decreasing A, for some V c k for which A, I ,.

Now, every VC i will be modeled as a virtual link i con-necting to the source i , the virtual link having an upper f lowlimit (capacity) PCR, and a lower flow limit MCR,, the

source i being-allowed to be able to send at arbitrary rates.There will be only one VC, which is i crossing the virtuallink i , and the rate A, is constrained to satisfy

MCR, 2 A, 5 PCR, .Using this model, one can show that there

are really two kinds of bottleneck links in a GM M optimalrate assignment: capacity-bottleneck (same as the originalbottleneck) links and MCR-bottleneck links. Both the capac-ity-bottleneck links and MCR-bottleneck may include virtuallinks.

Definition. Given a feasible rate vector {A,}, for each link

j , the advertised rate Rj is defined to be

R, =max{Ai:VCiEVj&A, M C R , }

Definition. Let 6 denote the set of links crossed by VC i.

Given a feasible rate vector {A,], for each source i , the pre-

maxmin rate A Y is defined by

A,? =min{Rj : j E 4 ) .

Definition (TheGM M Optimality C ondition). A feasiblerate vector {A,} is said to satisfy the GM M optimality condi-

tion if for all source i , A, =MCR, implies that A,!“ 5MCR,,

and A, #MCR, implies that AY >MCR, and A, =A Y .

Proposition 7. A feasible source rate vector {Ai] is GMM

optimal iff it satisfies the GM M optimali ty condition.

Proof. (If part): Suppose that the rate vector {Ai) is GM M

optimal. Consider any source i with A, =MCR,. Suppose

further that Am >MCR,. Let link j be a link with the adver-

tised rate equal’ to AY , then there exists a source k crossing

link j with the rate AY . Now A,(= Am)>A, can be reduced if

we increase the rate of source i without violating the feasibil-

ity constraint at link j . This violates the assumption that {A,}

is GM M optimal.Similarly, consider any source i with Ai #MCR,. There are

two sub-cases to consider. First suppose that Am 5MCR,.

Since {A,} is feasible, we must have A, >MCR, 2 Am. Let

link j be a link with the advertised rate equal to AY , then

Const r uct - GMM CPG( N: network, tlevel)

1. For each link k in N, et Rk=C, N, (if link k is a virtual link then

set R, to PCR, of VC i crossing link k).

Foreach Link k in N, dothe following:

2-1. If Rk =min [R,I l inkj & link k shareajoint VC}, add link kto the set of level L links.

2-2. Else if link k is a virtual link and MCR,>min [R, I l inkj & link

k share ajo int VC 1, add link k to the set of level L links, and setR, =MCR,

3. For each link k in the set of level L links, do the following:3-1. Remove link k and all the VC’ s crossing link k fromN.3-2. Add link k asa child node in the CPG to any level L-1 link thatshares a joi nt VC with link k.3-3. Update C and N for all link sharing ajoint VC with link k:

2.

C =C -Rk. number of removed VC ’s)N =N - number of removed VC’s)

4. If N is not empty, call Consf r ucf - GMM- CPG( N, +1).Otherwise, stop.

Fig. 5. GM M CPG algorithm

there exists a source k crossing link j with the rate AY . Butnow the condition Ai >A k(= AY ) violates the definition of

R,. Second, suppose that Ai f AY . Now A k(= AY )>Ai can

be reduced if we increase the rate of source i without violat-ing the feasibility constraint at link j. This violates the as-sumption that {A,} is GM M optimal. This completes the proof

of (If part).(Only if part): Suppose that the rate vector {A,} satisfies the

GM M optimality condition. Consider any source i withA, =MCR,. The GM M optimali ty condition implies

AY I CR,. L et li nkj be a link with the advertised rate equal

to AY . If one wants to increase Ai , then in order to keep fea-

sibility at l i nk j one must decrease a VC’srate which is less

than Ai (one cannot decrease the rate of any VC at its M CRvalue).

Similarly consider any source with A, f MCR,. The GM M

optimality condition implies Am >MCR, and Ai =A Y . Let

link j be a link with the advertised rate equal to AY . If one

wants to increase A ,, then in order to keep feasibil ity at l i nk j

one must decrease a VC’s rate that is no larger than Ai (one

cannot decrease the rate of any V C at its MCR value)..

With the above theory, it is quite easy to see that the GM Moptimality can be formulated in a way that a constrained VCdoes not have to be defined as one that cannot raise its rate toreach the.advertised rate at its bottleneck link. Once a con-strained V C is defined to one that is unable to raise its rate,

sorting seems to be necessary, see [lo]. The above proposi-tion gets rid of this limitation. As a result, it is now possibleto design a maxmin protocol wi thout using per-V C sorting. Infact, all maxmin protocols can be adapted using the newmodel and the optimality condition derived (Proposition 7).

Due to space limitation, we shall not present a full protocol;instead, we shall il lustrate the idea using the CPG algorithm.The CPG algorithm has been developed into a distributed

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maxmin protocol (the pseudo-code is provided in A ppendix).The original C PG algorithm can be modified as shown in Fig. 5. We call this new algorithm GM M CPG algorithm.

Definition. Let {Ai * } enote the set of rates computed by

the GM M CPG algorithm, i.e., {Ai *}= ( Ai } here Ai is therate assigned to the link crossed by V C i when VC i is re-moved at Step 3 of the GM M CPG algorithm.

Proposition 8. The rates {A*} computed by the GM MCPG algorithm for any GM M maxmin problem are GMMoptimal.

Proof. The proof is straightforward and is omitted.

Finally, if one uses the CP G protocol , the above adaptationwill give a time complexity of 1.5Ls.D +TRM where Ls is

the number of levels in the CPG generated by the GM M CPGalgorithm. Since virtual links are included, the CPG graphwill contain N , +N, number of nodes, where N , is the num-

ber of links and N, is the number of sources (virtual links). In

this case, we expect the number of levels Ls to be signifi-cantly smaller than N , +N, since massive parallelization is

possible in the presence of the virtual links. This complexityresult represents a significant improvement.

v. PERFORMANCEOFMAXMINROTOCOIS

In this section, three maxmin protocols (A SA P [15], M CP[l o] , and ERI CA from "unpublished" [131) are simulatedusing NI ST's ATM Network Simulator [SI. Note that ASA Pis the only protocol that does not require per-V C-accounting.Both ERI CA and MCP use per-VC-accounting, while M CPuses, in addition, per-V C-sorting.

Fig. 6 shows the network configuration used in the simula-tion. SE S and DES stand for source end-system and destina-

tion end-system, respectively. L inks 1and 2 are 20 km longand all other links are 0.2kmlong.

L Zvc"3

'Cl.C2 = 150 Mbps

Ai : Allowed Ce l l Rale (ACR) of VC iCj : Link Capacity of Link j " _ I"x- Vi rtual Connection ( V C )

Fig.6. Simulation configuration

In order to simulate dynamic changes in available band-

width for ABR traffic, a VBR VC is added at Link 1and an-other VBR V C is added at Link 2, both of which transmitcell s at 48-72 M bps with an on-interval of 5 msec and an off-interval of 5 msec. Two VBR VC's are 180degrees out of

phase. We use 155M bps as the actual capacities of L inks 1and 2. We ran all simulations for 150msec. Table I lists theparameter values used in the simulation.

We ran all three protocols at 5 different target utilizationlevels (SO%, 85%, 90%, 95%, and 99%).For each protocol,we generate a curve relating the maximum queue size and thetotal throughput as shown in Fig. 7. The maximum queue size

vs. total throughput curve represents a tradeoff curve that canbe used to identify the optimal operating point for each proto-col . The use of these tradeoff curves removes a bias in com-paring protocols: total throughput and maximum queue sizedepend on one another and one can potentially vary one ofthem to make the other variable to appear nice.

TABLE ISIMULA TION PARAMETERS

PCR =155 Mbps

MCR =0.155M bps

I TRM.100p sec1 NRM=31

Averaging interval forERICA =1 msecI ICR=7.75Mbps 1 S(ER1CA)=0.1 I

From Fig. 7, ASA P appears to perform best in the presenceof dynamic V BR traffic, even though ASA P has the leastcomputation requirement. One reason that ASA P performsbest might be that A SA P significantly reduces global incon-

sistency of constraint status (by using the bottleneck ID con-cept), while M CP and ERICA do not. In order to reduceglobal inconsistency, A SA P'S speed of response is slightlyslower, but global consistency appears to be even more im-portant than the speed of response for performance. For moredetail, the reader is referred to our paper [15]. The irony isthat there is no convergence proof, no complexity estimatesavailable for ASA P. On the other hand, the CPG protocolperforms (its performance is not shown in this paper) consis-tently worse than ASA P and MCP even though its complexityis the best. One needs to be cautious in interpreting the resultfor ERICA as we did not include the queue control optionsand other options in this simulation.

- >Throughput vs 0 Su e at different arget utilization evels (60% 85%, 90%. 95%. 99%)

13.5 4 4.5 5 5.5 6 . 6 5 7 7.5

Total Throughput (cel ls) at 150 mx 10'

Fig. 7.M aximum queuesize vs . total throughput

VI. CONCLUDINGEMARKS

The CPG analysis not only identifies parallelism in theexecution of maxmin protocols but also provides a framework

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to analyze the maxmin problem in a fundamental way. A s a

result, the CP G analysis opens up an entirely new avenue tostudy and develop maxmin protocols. M any research issuesremain to be tackled. The CPG protocol could be improvedusing the bottleneck ID scheme (used in ASA P). The current

version of our CPG protocol has not been optimized. It hasthe best time complexity but the performance does not live upto our expectation. T he GM M optimality condition should beadapted into other maxmin protocols to efficiently handle theM CR requirements.

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[5] A . Charny, D. Clark, and R. J ain, “Congestion Controlwith Explicit Rate Indication,” Proc. ICC’95, J une 1995,

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[101Y . Hou, H. Tzeng, and S. Panwar, “A Generali zed M ax-M in Rate Al location Policy and Its Distributed Imple-mentation Using the ABR F low Control M echanism,”

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Conf. High Perf. Networking, Sep. 1995, pp.143-154.[13]S. Kalyanaraman, R . Jain, S. Fahmy, R. Goyal, and B.

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CN

NsFsR

Link-StateACR[i]

VC-State[i]

151W. Tsai, Y . K im, and L . Hu, “ASAP: A Non-Per-VCAccounting Max-M in Protocol for A BR Flow Controlwith Optimal Convergence Speed,” IEEE SICON ’98,Singapore, July 1998, pp.139-153. Available from

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161L . K alampoukas, ‘Congestion M anagement in H ighSpeed Networks,’ Ph.D. Dissertation, University of Cali-fornia at Santa Cruz, A ugust 1997.

Total ABR CapacityNumber of active VC’ sNumber of settled VC’sTotal input rate from the settled VC’sAdvertised cell rate calculated by the switchSettle statusof linkACR of VC i

Settle statusof VC i

APPENDIX: HECPG PROTOCOL

(a) RM ceil fieldsI ACR I Current cell rate (CCR) of the source

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