ABR Congestion Control in ATM NetworksOrhan Ç. IMER�, Sonia COMPANSy, Tamer BA�ARzx, R. SRIKANT{Introdu tionAsyn hronous Transfer Mode (ATM) is the underlying te hnology enabling B-ISDN (broad-band integrated servi es digital network). B-ISDN was introdu ed as the su essor to narrowbandISDN after the latter fell short of meeting the high demand for bandwidth required by emergingappli ations su h as real-time video and HDTV (High-De�nition Television). B-ISDN envisionstransmission of �xed-size pa kets ( ells) over digital virtual ir uits at rates ex eeding 150 Mbps.ATM is basi ally a pa ket-swit hing te hnology with 53 byte long ells. The small ell size makesit possible to build swit hes that an a ept and swit h a large bat h of ells. ATM is asyn hronousin the sense that it has no requirements that ells rigidly alternate among the various sour es (i.e., ells arrive randomly from di�erent sour es).An international non-pro�t organization, the ATM Forum, was established several years ago,with the primary obje tive of promoting and extending the use of ATM produ ts and servi es. TheATM standards set by the ATM Forum de�ne mainly the user-network interfa e; that is, the waya omputer owned by a private user an onne t to the network and ommuni ate through it. Forthat, �ve di�erent servi es are available and are used for di�erent types of ommuni ation [1℄:� Constant Bit Rate (CBR): This servi e ategory is used by onne tions that request a stati amount of bandwidth that is ontinuously available during the onne tion lifetime. Telephoneand television use this servi e.� Variable Bit Rate (VBR): The ell rate is variable and is mainly intended for bursty sour es.This servi e an be real-time VBR (video onferen ing) or nonreal-time VBR (multimediae-mail).� Available Bit Rate (ABR): In this servi e ategory, the ell rate depends on the availabilityof the network. It is basi ally designed for bursty tra� whose bandwidth range is knownroughly. A rate ontrol me hanism is spe i�ed by the ATM Forum. Examples of this servi e ategory in lude browsing the Web and e-mail.�Department of Ele tri al and Computer Engineering and Coordinated S ien e Laboratory, University of Illinois,1308 West Main Street, Urbana, IL 61801-2307, USA. Email: imer�uiu .eduyDepartment of Ele tri al and Computer Engineering and Coordinated S ien e Laboratory, University of Illinois,1308 West Main Street, Urbana, IL 61801-2307, USA. Currently with Al atel Business Systems, 32 Avenue Kleber,92707 Colombes Cedex, Fran e. Email: sonia. ompans�al atel.frzDepartment of Ele tri al and Computer Engineering and Coordinated S ien e Laboratory, University of Illi-nois, 1308 West Main Street, Urbana, IL 61801-2307, USA. Tel: (217) 333-3607; Fax: (217) 244-1653; Email:tbasar�de ision. sl.uiu .edu.xAuthor for orresponden e.{Department of General Engineering and Coordinated S ien e Laboratory, University of Illinois, 1308 West MainStreet, Urbana, IL 61801-2307, USA. Tel: (217) 333-2457; Fax: (217) 244-1642; Email: rsrikant�uiu .edu.

� Unspe i�ed Bit Rate (UBR): This ategory uses the leftover network apa ity. UBR servi edoes not spe ify tra� -related servi e guarantees. Ba kground �le transfer uses this servi e.When a virtual ir uit (VC) is established between a sour e and a destination, both the ustomer(sour e) and the arrier (swit h) must agree on a ontra t de�ning the Quality of Servi e (QoS).The ATM Forum de�nes several QoS parameters, su h as Peak Cell Rate (PCR), Minimum CellRate (MCR), Cell Loss Ratio (CLR), and Cell Delay Variation (CDV). In CBR and VBR servi es,a tra� ontra t spe ifying the QoS guarantees is negotiated during the virtual ir uit setup phaseand is maintained for the duration of the onne tion. For ABR servi e, only lower and upperbounds (MCR and PCR) on the bandwidth of a virtual ir uit are spe i�ed at setup. Therefore,with CBR and VBR tra� , it is generally not possible for the sour e to de rease its rate, evenwhen an intermediate node be omes ongested, be ause of the QoS guarantees made at VC setuptime. However, ABR sour es might adjust their rates to the level of available servi e at times of ongestion. Thus, ABR tra� an be used to ontrol ongestion in the network.About four years ago, the ATM Forum adopted a rate-based ongestion ontrol s heme for theABR servi e [1℄. In this s heme, expli it rate ontrol messages are sent from intermediate nodes tothe sour es using spe ial ells alled Resour e Management (RM) ells. The goal of this ongestion ontrol me hanism is to fairly share the bandwidth left over from high-priority tra� (CBR andVBR) among the ABR sour es while making sure that the links throughout the network are fullyutilized.An ATM network onsists of several nodes (swit hes) inter onne ted via bi-dire tional links.We say that a swit h is bottlene ked if the in oming ABR ell rate at any one of its output portsis larger than the available rate to serve the ABR ells. Clearly, at a given instant, there may bemultiple bottlene ks in the network, as in Fig. 1.

Bottleneck Switch

Switch

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Figure 1: A Generi ATM network with multiple bottlene ks.Although the rate-based ongestion ontrol s hemes are standardized, developing good expli it2

rate omputation algorithms is still an open issue. As the link speeds ontinue to rise, the delay-bandwidth produ t (i.e., the produ t of the round-trip propagation delay and the link apa ity)in reases. An issue of importan e that arises in this ontext is how to deal with a tion delays,whi h is the time from the moment ontrol information is sent to a sour e, until an a tion is takenby it, and until subsequently that a tion a�e ts the state of the swit h that initiated that ommand.In this arti le, we onsider a ontrol-based mathemati al model that helps us address this prob-lem. Our initial modeling assumption is that of a single bottlene k swit h shared by several ABRsour es. Although in a real network topology, the sour es may be inter onne ted in several ways,resulting in multiple bottlene ks (Fig. 1), the single bottlene k assumption admits theoreti al aswell as experimental justi� ation [2℄ and provides a good starting point for derivation of e�e tiverate ontrollers. More pre isely, we present three di�erent ongestion ontrol algorithms for ABRrate ontrol and study their performan e in a simulated network environment. The mathemati almodel uses idealized linear queue dynami s, already introdu ed in [3℄ - [5℄, but the simulation modeltakes the saturation nonlinearities into a ount. Parti ularly, in a real network, queue length ata swit h must lie between zero and the size of the bu�er, and this is taken into a ount in thesimulation model. Two of the algorithms we present formulate the ongestion ontrol problem as asto hasti team where players are users sharing the bottlene k swit h. In this formulation, the ap-proa h involves a model for the available bandwidth as an autoregressive (AR) pro ess, driven by anarbitrary, independently distributed random sequen e, and minimization of an obje tive fun tionalthrough whi h most of the design riteria are re�e ted [6℄ - [7℄. The third algorithm we present isbased on a deterministi model of the available bandwidth. In this s heme, the emphasis is more onrobustness against the variations in round-trip delays and estimating the number of sour es sharingthe bottlene k swit h.Several other types of ABR ongestion ontrol designs have been onsidered. We brie�y sum-marize these here to ompare and ontrast them with our approa h. The simplest feedba k ontrolme hanism is alled rate mat hing. In rate mat hing, the node measures the average rate availableto ABR sour es at periodi intervals and simply divides a fra tion of this apa ity equally amongthe various users. This is the basi approa h used in [8℄, although several modi� ations are used inthe a tual implementation. The main advantage of this s heme is its simpli ity, but it is di� ult to ontrol queue length optimally to avoid bu�er over�ows. However, this s heme is stable (i.e., thequeue length remains bounded in an appropriate sto hasti sense [9℄). Queue length information isnot used in the basi algorithm, although [8℄ allows one to in orporate queue length information inan ad ho manner. Alternatively, this problem an be viewed as a feedba k ontrol problem wherequeue length is used for expli it feedba k. This approa h is used in [10℄ - [11℄ to study this problemusing lassi al ontrol te hniques or using a state-spa e approa h. As in rate mat hing, the primarygoal is not optimality, but simply queue length stability. In these approa hes, the available band-width to ABR sour es is treated as an unmodeled disturban e. Thus, the algorithms in [10℄ - [11℄ensure stability in the presen e of this disturban e, but do not address the issue of performan e. Ina re ently published work [12℄, a losed-loop proportional-derivative ontroller is proposed, whi ha hieves max-min fairness plus queue length stability, but the design falls short of addressing theissue of robustness against un ertainty in delays.Available Bit Rate Servi eFlow Control Model for ABRIn the ABR servi e, the sour e adapts its rate to hanging network onditions. As mentioned inthe introdu tion, information about the state of the network, su h as bandwidth availability, state3

of ongestion, and impending ongestion, is onveyed to the sour e through spe ial ontrol ells alled RM ells. ABR �ow ontrol o urs between a sour e and a destination, whi h are onne tedvia bi-dire tional links. The forward dire tion is the dire tion from the sour e to the destination,and the ba kward dire tion is the dire tion from the destination to the sour e.A sour e generates forward RM ells every Nrm data ells, where Nrm is generally taken to be32. These ells travel along the same path as the data ells but are treated spe ially by the swit hesalong the way (Fig. 2). The swit h may:� Dire tly insert feedba k ontrol information into RM ells by using the Expli it Rate (ER)�eld of RM ells.� Provide binary feedba k by marking the Congestion Indi ation (CI) or No In rease (NI) bitin the RM ells.� Indire tly inform the sour e about ongestion by setting the Expli it Forward CongestionIndi ation (EFCI) bit in the data ell header, and rely on the destination to onvey ongestioninformation ba k to the sour e by marking the CI bit in the ba kward RM ells it generates.� Spontaneously generate ba kward RM ells and ship them ba k to the sour e.Note that Fig. 2 is only a generi representation of the ontrol loop, as there may be more thanone swit h between the sour e and the destination. Ea h ABR sour e has a urrent ell rate, ACR(A tual Cell Rate), whi h it must modify upon re eiving feedba k from the network via RM ells.ACR always falls somewhere between MCR and PCR. When a sour e sends out a forward RM ell,it sets the ER �eld of the RM ell to the rate at whi h it would urrently like to transmit. As theRM ell passes through the various swit hes on the way to the destination and ba k to the sour e,those that are ongested may redu e the ER. When the sour e re eives the RM ell ba k, it takesone of the following a tions depending on the settings of the ER �eld and CI and NI bits.Destination Switch Source

RM Cell

Data CellFigure 2: ABR tra� management model.� When there is no ongestion (both CI and NI bits are not set), ACR an be in reased (butnot above PCR) by a quantity RIF*PCR, where RIF is the rate in rease fa tor. However,ACR annot be in reased above the expli it rate spe i�ed in the ER �eld.� When a sour e re eives a ba kward RM ell with CI bit set, it de reases its ACR (but notbelow MCR) by a quantity RDF*PCR, where RDF is the rate de rease fa tor. However, againACR of the sour e annot be larger than the expli it rate spe i�ed in the ER �eld. Both RIFand RDF are negotiated in the VC setup phase.� Finally, when the sour e re eives a ba kward RM ell with only the NI bit set, it sets its rateto the minimum of ACR and ER. 4

Performan e CriteriaNumerous algorithms have been and are being developed for ATM ABR ongestion ontrol. Tobe able to study and ompare these algorithms on equal grounds, we need to set some performan emeasures independent of the parti ular algorithm under investigation. The main goal of ABR rate ontrol is to provide fairness among all VCs with a minimal ell loss rate and maximal utilizationof network resour es. The latter two of these obje tives an be a hieved by regulating the queuelength at bottlene k nodes around a desirable level. Tra king su h a nominal queue length (whoseexa t value is determined based on QoS requirements) is desirable in order to avoid losses due toover�ow and waste of the bu�er apa ity due to under�ow.Fairness is an issue that requires more dis ussion, as it may be hard to visualize what is meant bya fair allo ation in a large network with multiple bottlene ks. The most widely a epted notion offairness is the max-min fairness riterion [15℄. Under this riterion, the fair share of ea h onne tion ontending for a given link bandwidth should be equal to (��ru)=(N�Mu). Here � is the availableservi e rate for ABR sour es at a parti ular swit h, N is the number of a tive sessions at thatparti ular swit h, ru is the total rate of onne tions that are bottlene ked elsewhere or are limitedby their PCRs, and Mu is the number of su h onne tions.Basi Model of an ATM Swit hGenerally, an ATM swit h has several input and output lines. The number of input lines isalmost always the same as the number of output lines, be ause the links are bi-dire tional. Cellsarrive on the input lines asyn hronously, but the swit hing is done syn hronously with the help ofa master lo k. Ea h input line is onne ted to a ommon bus, through whi h the in oming ellsare dire ted to their orresponding output ports. Most of the ommer ial ATM swit hes use outputqueueing to prevent high ell loss rates. In output queueing ea h output line has a �nite bu�er,where the in oming ells are served on a FIFO (�rst in, �rst out) basis. Asso iated with ea h outputline, there is an ER ontroller that suggests a data rate for ea h ABR VC.In what follows we fo us on a parti ular output line of an ATM swit h. Other output lines anbe treated in a similar manner. The model we adopt here is a dis rete-time model, where a timeunit orresponds to the interval over whi h the rate available to ABR sour es is determined (thatis, the interval over whi h measurements are made). Further, this measurement interval is assumedto be long enough for the swit h to be able to pro ess several ells �a reasonable assumption if thelink speeds are high and pa ket sizes are small. This allows us to ignore the ell-level dynami s andmodel the ABR tra� as a �uid.Let rn denote the total number of ells that arrive in one of the output bu�ers of an ATM swit hin the interval [n; n+ 1), and let �n denote the number of ells that depart from this bu�er in thesame time interval. Note that �n represents the available bandwidth (unused by higher prioritytra� , parti ularly, CBR and VBR). Denoting the queue length at time n by qn, and ignoring theboundary e�e ts on the queue dynami s as in [16℄, we have the evolutionqn+1 = qn + rn � �n: (1)Let there be a total number of N onne tions (sour es) swit hed through the output line understudy, and the number of ells that arrive from sour e m during time-slot [n; n+ 1) be denoted by5

rmn. Clearly, rn = NXm=1 rmn:In general, rn has two omponents: 1) The number of ells that arrive from un ontrolled sour es,i.e., those sour es whi h are bottlene ked elsewhere in the network or are limited by their PCR onstraints. The ER ontroller at this swit h has no ontrol over these sour es. In other words, theER �eld of an un ontrolled sour e is either overwritten at some other swit h, or is repla ed by itsPCR value at the sour e. We denote this omponent of rn by run. 2) The number of ells that arrivefrom ontrolled sour es, whi h are bottlene ked at this swit h. The ER �elds of RM ells of thistype of sour es are modi�ed to a hieve several tra� related servi e guarantees. We assume thatea h sour e, either ontrolled or un ontrolled, has an MCR = 0. If the MCRs are positive, we anreserve the minimum ell rate for ea h user and make the assumption that a sour e will attempt tosend at a rate no smaller than its reserved MCR.Let there be a total number of M ontrolled sour es, and the number of ells that arrive from ontrolled sour e m in the interval [n; n + 1) be denoted by r mn. Then, we have the followingrelation between rn, r mm, and run: rn = MXm=1 r mn + run:In the analysis to follow, we assume that the swit h exer ises ER ABR ongestion ontrol, whi hamounts to setting RIF = 1, RDF = 0. Note that for a ontrolled onne tion, the PCR onstraintnever be omes a tive, as this would invalidate the assumption that the sour e is ontrolled. Alsonote that both rn and �n represent ell rates with the unit of time taken as the sampling intervalover whi h the measurements are made.As indi ated earlier, the de ision on the rate at whi h ea h sour e should transmit is made bythe swit h, and the type of information a essible to the swit h a�e ts this de ision. Sin e ATMte hnology relies on high speed swit hing of pa kets, the amount of overhead aused by the ER ontroller should be kept at a minimal level. At time n+ 1, when the de ision on the rate of ea h ontrolled sour e has to be made, three pie es of information are available to the swit h withoutany delay: queue length, qn, available bandwidth, �n, and the total number of ells arrived in theinterval [n; n+1). Extra ting these numbers does not require any elaborate measurements, as qn anbe measured by looking at how many ells in the bu�er are waiting to be served, while �n and rn anbe determined by ounting the number of ells left and arrived in the interval [n; n+1), respe tively.In general, to be able to fairly divide the bandwidth among all sour es, the swit h needs to knowhow many ontrolled sour es are being served at a given time. Cal ulating the number of VCsrequires the swit h to look at the header of every single RM ell and tell what sour e it originatedfrom and what destination it is routed to. Even though theoreti ally possible, this task brings ina omputational overhead slowing down the operation of the ATM swit h. Moreover, the numberof VCs, N , al ulated this way may not be equal to the a tual number of ontrolled sessions, M ,as some of the sour es might be limited by their PCRs, bottlene ked at some other swit h in thenetwork, or just too bursty to be exer ised any ontrol over. Therefore, it is desirable to developan ER ontrol algorithm that relies only on the knowledge of the three easily a essible quantitiesdes ribed above.As mentioned in the introdu tion, it takes time from the moment the ER de ision is made bythe swit h until an a tion is taken by a sour e, and until subsequently that a tion a�e ts the state6

of the node that initiated the a tion. Thus, the ell rate of sour e m at time n, rmn, is a tually anout ome of an a tion taken dm time units earlier, where dm represents the a tion delay for sour em and is taken to be independent of time n. Without any loss of generality, we assume that thedm's are ordered su h that 0 � d1 � : : : � dM � �dwhere �d orresponds to the maximum round-trip network delay.Congestion Control AlgorithmsController 1: A Certainty Equivalent ControllerBasi AssumptionsThe available bandwidth for ABR, �n, may hange in an unpredi table way sin e the trans-mission rate of VBR tra� is time varying. The bandwidth available to the ontrolled sour es, onthe other hand, is subje t to further un ertainty due to the variations in the un ontrolled tra� ,run. These observations motivate us to onsider the CBR, VBR and the un ontrolled ABR tra� olle tively as interferen e, modeled by an AR pro ess, whi h is stable: n := �n + run = �+ ru + �n; (2)�n = pXi=1 �i �n�i + �n�1 : (3)Here � is the onstant nominal servi e rate, ru is the nominal rate of un ontrolled sessions, p isthe order of the AR pro ess, �i; i = 1; : : : ; p are the parameters hara terizing the pro ess, and thedriving term f�ngn�1 is a zero-mean i:i:d: Gaussian sequen e with varian e k2.Given the dynami s of the AR pro ess, the sour e rates, r mn, are determined by minimizinga ost fun tion that quanti�es the tradeo� between two partially on�i ting goals: steady queuelength around a nominal value and fair allo ation and a tual sharing of the available bandwidth.The ost fun tion adopted for this purpose isJ = lim supN!1 1NE( NXn=1"(qn �Q)2 + MXm=1 1k2m (r mn � am n)2#) (4)where Q is the target queue length, km's are positive onstants, PMm=1 am = 1, and Ef�g denotesexpe tation over the statisti s of the AR pro ess. Exa t fair sharing implies am = 1=M , whi h willbe the initial value for ea h am. Note that this hoi e of am requires the knowledge of the numberof ontrolled VCs, M , whi h an be determined at the expense of a slower swit hing rate.The �rst term in (4) represents the penalty for deviating from the desired queue length, Q, whilethe se ond term represents a penalty (for ea h sour e) for deviating from the authorized transmissionrate, whi h we have taken to be a fra tion of the available bandwidth for ea h sour e to a hievemax-min fairness. The km's quantify the relative priority given to ea h sour e�the larger km is,the lower the priority.Derivation of Controller 1To arrive at a somewhat simpler on�guration, we �rst introdu e the shifted variablesxn := qn �Qumn := r mn � am�7

where xn stands for the state and umn for the ontrol. Then the state dynami s (1)-(3) be omexn+1 = xn + MXm=1umn � �n�n+1 = pXi=1 �i �n+1�i + �n;and the ost fun tion isJ = lim supN!1 1NE( NXn=1 "(xn)2 + MXm=1 1k2m (umn � am�n)2#) :Let ~umn := umn � am�n; ~un := (~u1n; : : : ; ~uMn)0:Then the ost fun tion an be written asJ = lim supN!1 1NE( NXn=1" (xn)2 + MXm=1 1k2m (~umn)2 #) :Also, in terms of ~u, the dynami s for x be omexn+1 = xn + MXm=1 ~umn:If there was no delay (i.e., d1 = d2 = : : : = dM = 0), then the standard dis rete-time linearregulator theory [17℄ would apply, yielding the unique solution~un = �[R+ bb0s℄�1bs xnwhere b0 := � 1 1 : : : 1 �1�M ;R := diag� 1k21 ; : : : ; 1k2M � ;and s is the unique positive root ofs = 1 + s[1� b0(R+ bb0s)�1bs℄:This an be solved expli itly to yields = 1 +p1 + 4�22 ; �2 := � MXm=1 k2m��1:8

The original ontroller would then be (with zero delay)un = ~un + am 26664 11...1 37775 �n: (5)However, sin e the impa t of umn on the network would be felt only after dm time units, here weinvoke ertainty equivalen e, where, in the solution (5) for ea h ontroller umn; m = 1; 2; : : : ;M; werepla e the queue length and bandwidth by their estimates dm time units later. These onsiderationslead to the following ertainty equivalent ontroller, whi h we will refer to as Controller 1:u�m;n = �pm xn+dmjn + am �n+dmjn ; m = 1; : : : ;M : (6)Here pm is the mth omponent of [R+ bb0s℄�1 bs and xn+dmjn, �n+dmjn are the predi ted valuesof xn+dm and �n+dm , respe tively, based on the what is known at time n and given that all other ontrollers are also in the form (6). These predi tors are generated byxn+jjn = xn+j�1jn + MXm=1 um;n+j�1�dmjn � �n+j�1jn ; j � 1 ;xnjn = xn;�n+jjn = pXi=1 �i �n+j�ijn ; j � 1 ; �n�kjn = �n�k ; k � 0 ;and um;n+j�1�dmjn := 8<: �pmxn+j�1jn + am�n+j�1jn if j � dm + 1um;n+j�1�dm if j < dm + 1 :These are the re ursive equations generating the predi tors for the queue length and rate informationat a future time, where the future time is the urrent time plus the a tion delay for the orrespondingsour e. For example, �n+jjn denotes the predi ted value at time n of the value of � at some futuretime n+ j, based on the information available at time n. A similar interpretation holds for xn+jjn.The above algorithm is relatively easy to implement. The estimator algorithms are simple s alaroperations, and the s alar solution of the Ri ati equation s has already been obtained expli itly. Insummary, an easily implementable version of Controller 1 is given below in the form of pseudo ode:Pseudo ode for the node's omputation at time n using Controller 1for j = 1 to dM dofor m = 1 to M doif (n+ j � 1� dm � n)um;n+j�1�dmjn = �pmxn+j�1 + am�n+j�1endifendforxn+j = xn+j�1 + MXm=1 + um;n+j�1+dmjn � �n+j�1endfor 9

umn = �pmxn+dm + am�n+dmController 2: Optimal ControllerRe all that in the derivation of Controller 1, we �rst assumed zero delays, solved the simpli�edLQR (linear quadrati regulator) problem, and then in orporated the delays into ontrollers throughestimators using the ertainty-equivalen e prin iple. An alternative approa h, whi h leads to theoptimal solution of the problem, is to augment the state spa e in an appropriate way. Basi ally, onemust introdu e new state variables for those sour es that have nonzero delays. Sin e this is a tediouspro ess with many algebrai details, we do not in lude the derivation of the optimal ontroller here,but instead refer the interested reader to [18℄. The optimal solution is hara terized in terms of thesolution of a dis rete-time algebrai Ri ati equation (DARE), whose dimension is determined bythe magnitude of the largest delay and the order of the AR pro ess des ribing the available apa ity.Controller 3: A Robust Adaptive Controller Under Un ertaintyBasi AssumptionsIn our earlier paper [6℄, by onsidering a one-node example with perfe t delay information andinstantaneous feedba k, we showed that delay is an important fa tor in any design of rate �ow ontrollers and hen e must be expli itly taken into a ount in any realisti model of high-speednetworks, as we have done above. However, in a realisti network, there are several deviations fromthis ideal model.Both Controller 1 and Controller 2 assume omplete knowledge of delays in the network. Al-though the end-to-end round-trip delay may be known to the swit h as part of the FRTT (�xedround-trip time) omputation performed at VC setup time, we an still have small errors in delayestimates. One sour e of error is the assumption that the delay is an integer multiple of the timeunit (measurement interval), whi h may not be true. Further, there is variability in the delays dueto queues in the virtual ir uits. Finally, RM ells are generated only every 32 data ells and hen efeedba k is not instantaneous.In the derivations of Controller 1 and 2, we also assumed that the swit h has a ess to the infor-mation about the number of ontrolled sour es, M . Re all that we need this number to determinethe fair share of ea h sour e. As mentioned earlier, in a realisti environment with bursty sour es,determination of M may not be an easy task. Hen e the swit h must have a method of estimatingthis parameter.Motivated by these observations, we want to develop a ontroller that is robust to un ertaintyin delays, and at the same time adaptive to the number of sour es. Thus, in deriving what we allController 3, we only require the knowledge of �d and �M , whi h denote the upper bounds on theround-trip delay and the number of ontrolled sessions, respe tively.Although the available bandwidth for ABR, �n, and the rate of un ontrolled sessions, run, may hange with time, as in the previous se tion, we assume here that both �n and run are onstantsdenoted by � and ru, respe tively. These assumptions are justi�ed if �n and run vary slowly omparedto the time onstant of the losed-loop system. With these simplifying assumptions, the stateequation (1) be omes qn+1 = qn + MXm=1 r mn + ru � �:Sin e the rate of the ontrolled sour e m at time n is a tually an out ome of an a tion taken by the10

swit h dm time units earlier, we have r mn = n�dmwhere n denotes the ommand issued by the swit h at time n.Derivation of Controller 3As the a tual number of ontrolled sour es, M , is not known to the swit h, we start our analysisby proposing a method to estimate this �gure. Let us �rst onsider the ase when ru = 0 (i.e.there are no un ontrolled sessions). Our method relies on a simple observation: If the swit h sendsout the same ommand, say , for ( �d + 1) time units, at the end of the ( �d + 1)st step, all of the ontrolled sour es in the system will be transmitting at rate . Hen e, at the end of the ( �d + 1)sttime step, if we divide the in oming ell rate, rn, by the assigned rate , we obtain the a tualnumber of ontrolled sour es, M , at the swit h. In order to have a running estimate of this �gure,one an onstru t an estimator, Mn, and update it every ( �d+ 1) time units using this s heme. Wenote that this algorithm onverges to the exa t value of M in only a �nite number of steps. Havingdetermined M in this manner, one an set the ommand at the next time slot to be equal to �Mand a hieve fairness, whi h in this parti ular ase also orresponds to max-min fairness, as we tookru = 0.Now if the rate of the un ontrolled onne tions, ru, is not zero, the above s heme fails to onverge to the a tual number of bottlene ked onne tions, M . However, it still does onverge, butto a di�erent value. In fa t, as we will show shortly, in al ulating the share of ea h ontrolled onne tion, if one uses the number to whi h the algorithm onverges, then the resulting bandwidthallo ation is max-min fair.Before pro eeding further, we note that the hoi e of the rate above is ompletely arbitrary,be ause it does not play any role in the estimation pro ess. Thus, we are free to pi k any rate largerthan zero for the algorithm to work. Mathemati ally speaking, letting Mn denote the estimate ofM at time n, we propose the following estimator: n( �d+1) = n( �d+1)+1 = � � � = (n+1)( �d+1)�1;Mn( �d+1) = Mn( �d+1)+1 = � � � = M(n+1)( �d+1)�1 = rn( �d+1) (n�1)( �d+1)where the �rst equation sets n to the same value for �d+1 steps, while the se ond equation is used toestimate M every �d+1 time units. For ease of notation, let us introdu e the folowing subsequen es:qsn := qn( �d+1); sn := n( �d+1); rsn := rn( �d+1); M sn := rsn sn�1 = M + ru sn�1 (7)with M s0 = �M . To omplete the design of Controller 3, we need to spe ify how sn should be sele tedto a hieve the dual goal of max-min fairness and queue length stability. We propose the followingdesign for sn: sn = max( �M sn � �(qsn �Q); 0) (8)where � > 0 is the gain to be sele ted to ensure stability, and the max fun tion is introdu ed toensure that the swit h asks the sour e to transmit at a positive rate in ex ess of MCR, as requiredby the QoS spe i� ations. In (8), the term ��(qsn �Q) is introdu ed to drive the queue length, qn,to the desired set point, Q, by providing negative feedba k in the losed-loop system dynami s.11

Note that if qn onverges to Q, the ommand of the node for ontrolled sour es, sn, onvergesto the solution of the following equation: s1 = max( �M + ru s1 ; 0) ;whi h is given by s1 = max��� ruM ; 0� = �� ruMwhere the se ond equality follows from the obvious onstraint ��ru � 0. Hen e, if we an show thatthe losed-loop queue dynami s, qn, onverge to Q, then the rate of ontrolled sessions onverges tothe max-min fair share of the available bandwidth. In fa t, it an be shown that if the ontrollergain, �, is pi ked su h that 0 < � < 1( �d+ 1) �M �1� ru� � ;then the robust ontrol poli y, given by (7)-(8), a hieves queue length ontrol plus fair share ofthe available bandwidth [20℄. The proof of this fa t is rather tedious and is hen e omitted in thisarti le. Note that, as with Controllers 1 and 2, the algorithm we propose here does not su�er from omputational omplexity, be ause there is a single design parameter, namely �, to be tuned, andto determine the ER, the swit h has to perform only two divisions, one multipli ation, and twoadditions per output line, every ( �d + 1) time units. Moreover, the information the swit h needsto perform these al ulations, frsn; qsn; �g, is lo ally available, and just a few memory elements peroutput port are required, as there are only four numbers, f �d; �M;Q; �g, to be stored.SimulationsSimulation ModelIn simulations, we onsider a four-node high-speed network where the link speed is the speed oflight, the servi e rate o�ered by every link is 1 Gbps, and the distan e between adja ent nodes is onstant and equal to 1000 Km. We take the time unit to be the time required to serve 5,000 ells(5; 000 � 53 � 8 bits). Then, the propagation delay for a ell (i.e., the time required for a ell totraverse a link) is t = 1� 1063� 108 � 10953� 8 � 15; 000 � 1:6 time units;where we have again taken the speed of propagation to equal the speed of light. A ordingly, we onsider here the four-node network depi ted in Fig. 3. In the �gure, SV i represents a set ofVBR sour es as detailed under the �gure and DV i is the destination of SV i. Moreover, the VBRsour es onsidered are of two types: video tra es obtained from [24℄ - [22℄ and simulated ON-OFFsour es. The latter are bursty sour es simulated by us, whi h alternate between ON and OFF statesa ording to a Markov hain, and when in the ON state, ells arrive at a onstant rate.Superimposed on the model in Fig. 3, whi h depi ts only VBR sour es, we onsider two on�g-urations for the ABR sour es:� Parking lot model: This model will allow us to evaluate the fairness of our algorithms whenthere is only one bottlene k node. This network has a �parking lot� on�guration [25℄ where12

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-two on/off sourcesFigure 3: Four-node on�guration: VBR sour es.three ABR sour es with di�erent delays (see Fig. 4) are bottlene ked at the third node. Theone-way propagation delay from one swit h to another is 1.6 time units. Sin e the delaysused in the ontrol algorithms are integers, we will take the nearest integers to the a tualdelays. For example, the a tual a tion delay between sour e ABR1 and the se ond swit h is2 � 1:6 = 3:2 time units. The orresponding delay used in the mathemati al model is then3. Table 1 spe i�es the a tion delays (the ones used by the ontrol algorithms) between thesour es and the swit hes. Re all that the a tion delay hara terizing a sour e with respe t toa node is de�ned as the time taken from the moment the node sends ontrol information to asour e (via a ba kward RM ell) until that sour e takes the orresponding a tion, and untilsubsequently that a tion a�e ts the state of the node. For the adaptive ontrol algorithm,we take �d = 10, whi h is the largest possible network delay in this on�guration. Note thatto implement the adaptive robust ontroller, we do not need the delay information given inTable 1. SW1 SW2 SW3 SW4

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Bottleneck SwitchFigure 4: Fairness on�guration: ABR sour es.13

Table 1: A tion delays for the fairness model.SW1 SW2 SW3 SW4ABR1 0 3 6 10ABR2 0 3 6ABR3 0 3� Max-min model: The parking lot model has only one bottlene k node. This does not allowus to fully evaluate the max-min fairness properties of our algorithms. Hen e we onsiderthe model depi ted in Fig. 5. Three of the ABR sour es are bottlene ked at the third node.The remaining ABR sour e (SA2) will then use the remaining apa ity at the se ond swit h(a ording to the max-min riterion [15℄), whi h then be omes bottlene ked. The networkeventually ends up with two bottlene k nodes. Table 2 spe i�es the a tion delays used in themathemati al model. SW1 SW2 SW3 SW4

SA4 SA1

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Bottleneck SwitchFigure 5: Max-min fairness on�guration: ABR sour es.Table 2: A tion delays for the max-min fairness model.SW1 SW2 SW3 SW4ABR1 0 3 6 10ABR2 0 3ABR3 0ABR4 0 3For the simulation examples, we used the following parameter values:� Time unit: Time required to serve 5,000 ells� Target queue length: Q = 700� ICR = 300, MCR = 0, PCR = 4500 14

� RIF = 1, RDF = 0� Weights: km = 1 8m (Controllers 1 and 2)� Maximum number of sessions: �M = 5 (Controller 3)� Maximum network delay: �d = 10 (Controller 3)� Controller gain: � = 1=(1:2 � �M � ( �d+ 1)) ' 0:0152 (Controller 3)For Controllers 1 and 2, the nominal servi e rate � and the AR pro ess parameters �i are estimatedonline using the Yule-Walker algorithm [26℄, assuming that the order of the AR pro ess is 8: Thisis dis ussed in the next subse tion. For Controller 3, we take � to be equal to 4500 ells per timeunit. The propagation delay from one node to the next is 1.6 time units. Note that the a tual delayis variable sin e the ells go through node bu�ers, whi h leads to additional queueing delays.Following the urrent ATM standards, the feedba k me hanism has been implemented using RM ells that are generated by the sour es every 32 data ells. Even though the mathemati al modelfor the ontrollers an only use integer values for delay, the a tual one-way propagation delay forthe RM ells is 1:6 in the simulations. ATM standards allow for measurement of propagation delaysin the signaling proto ol. However, queueing delay remains variable and unknown. Thus, one ofthe goals of our simulation is to show that the impa t of these two types of approximations (to thea tual delays) on the performan e of Controllers 1 and 2 is negligible.The AR Pro essTwo of our algorithms are based on modeling the available apa ity (i.e., total apa ity minus the apa ity used by the VBR and un ontrolled ABR sour es) as an AR pro ess (see (1)-(2)). The orderp, the parameters �i, and the varian e of the noise have to be determined. We wish to emphasizethat the varian e of the noise is determined as part of the estimation of the AR parameters but isnever a tually used in the ontrol algorithms, as it is not needed.Tuning the parameters of the AR pro ess and �nding the best possible value of p is a hallengingtask in general. Several methods exist to al ulate the parameters, �i, on e p is �xed. We use theYule-Walker algorithm [26℄ to determine p and �i's from the data that is observed online.Let T be the time interval over whi h we attempt to �t an AR model to the available apa ity.Then one riterion to determine the best order for an AR pro ess is the �nal predi tion error (FPE),de�ned by FPE = �2T + pT � pwhere � is the estimate of the varian e of the noise aused by the ross tra� . The order p of thepro ess that minimizes the FPE is the best order [26℄. The time interval in our simulations wasT = 200, and using this, we determined that p = 8 gives good results. For the sake of brevity, wedo not in lude the details here.Simulation ResultsIn simulations, all rates are expressed in ells per time unit.FairnessFirst, by onsidering the �parking lot� on�guration (Fig. 4), we study the fairness of our designswhen there is only one bottlene k node (SW3). In a single-bottlene k-node s enario, fairness is the15

apability of the algorithm to fairly distribute its available apa ity among the various sour esdespite the presen e of di�erent delays.� Controllers 1 and 2: When VBR sour es are present, the mean available bandwidth at SW3is around 4500 ells per time unit. In addition, re all that sour es ABR1, ABR2, and ABR3have delays of 6, 3, and 0, respe tively. The simulations for both ontrollers show that the apa ity is fairly shared among the three sour es (Figs. 6 and 8) with a mean rate of around1495. Although the sour es have di�erent delays, the design manages to provide fairness. Thequeue at the third node is regulated around 700, as expe ted (Figs. 7 and 9).0 200 400 600 800 1000 1200 1400 1600 1800

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Figure 6: Fairness model: Sour e rates under Controller 1.� Controller 3: When there are no VBR sour es, the available servi e rate at SW3 is exa tly4500 ells per time unit. As an be seen from Fig. 10, our algorithm a hieves fair share. Alsothe queue at SW3 onverges to the desired value of 700. In addition, the algorithm �ndsthe a tual number of sour es at SW3 in �d + 1 = 11 time units, as expe ted. The rate of onvergen e of sour e rates as well as the queue length depend on the ontroller gain �. Asmaller value of � results in a smaller overshoot but a larger settling time.Max-Min FairnessThe on�guration under onsideration is depi ted in Fig. 5. The main bottlene k node is thethird one, where the apa ity should be equally distributed among the sour es ABR1, ABR3, andABR4. Then, sin e ABR1 does not use its fair share at node 2, ABR2 should use the remaining apa ity and in rease its ell rate to make up for the di�eren e.� Controllers 1 and 2: With VBR sour es present, the mean available apa ities are around 4870,4720, 4480, and 4860 ells per time unit for swit hes 1, 2, 3, and 4, respe tively. If the apa ityat SW3 is equally distributed among the sour es ABR1, ABR3, and ABR4, ea h one of thesesour es should transmit at around 1500 ells per time unit. Then, ABR2 should use the unused16

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Figure 8: Fairness model: Sour e rates under Controller 2.17

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apa ity at SW2 and in rease its ell rate to around 3200. Nevertheless, our basi algorithmsets the weights to the exa t fair share (1=M for M sour es), and the drop observed in thequeue length at swit h 2 is not substantial enough to in rease the rate of ABR2. Therefore,max-min fairness is not a hieved with the original ontrollers. An adaptive weight algorithmneeds to be implemented in order to adapt the weights to a max-min fairness on�guration.This problem has been addressed in [27℄, where the authors determine the a tual a tivity ofea h sour e and dedu e from that a max-min fair share. We adopt a similar idea to hoosethe weights am, as des ribed below.Consider a swit h with M sour es going through it. We �rst evaluate the mean ell rate ofea h sour e m, say meanCCR(m), by using the CCR ( urrent ell rate) �eld in the RM ells.If a sour e uses less than 1=M times the apa ity of the swit h, then its am is redu ed to thefra tion that it a tually uses. Su h sour es are alled underloading sour es. As a result, theremaining apa ity is fairly allo ated to the rest of the sour es. To a ount for variability, wea tually ompare a sour e's bandwidth utilization to a number slightly smaller than 1=M; say0:85=M:The swit h then omputes the sour e optimal rates using these new weights. For example, onsider the three sour es ABR1, ABR2, and ABR3 at the swit h. The total available apa ityis 3000 ells per time unit. Sour e ABR1 is bottlene ked somewhere else at 500 ells pertime unit. Then the new distribution of the weights is: aABR1 = 1=6, aABR2 = 5=12, andaABR3 = 5=12. Therefore, the optimal rates at the present swit h will be 500 ells per timeunit for ABR1 and 1250 ells per time unit for ABR2 and ABR3. These rates will be used asfeedba k information.The simulations done with the adaptive weight algorithm indi ate that max-min fairness isindeed a hieved (Figs. 11, 12, 13, and 14). Initially (with all am's set to the fair share), thebottlene k node is the third one and sour es ABR1, ABR3, and ABR4 share fairly the apa ityat this swit h (around 1500 ells per time unit). Meanwhile, sour e ABR2 only uses half ofthe apa ity of swit h 2. A tually its ell rate is slightly higher than the fair share. Be auseof the small queue length, the design appears to in rease the allowed ell rate slightly. On eam's at ea h swit h are adapted, sour e ABR2 should in rease its rate to 3220 ells per timeunit while the other sour es still transmit at 1500 ells per time unit. The simulations showthat sour e ABR2 e�e tively at hes up (Figs. 11, 13).� Controller 3: One advantage of using Controller 3 is that it a hieves max-min fair bandwidthallo ation without ne essitating any sort of adjustment of parameters, assuming of oursethat the available servi e rate remains onstant. We simulate the on�guration depi ted inFig. 5 taking the onstant servi e rate to be 4500 ells per time unit for ea h swit h. As thesimulation results indi ate (Fig. 15), the algorithm onverges to the max-min fair allo ationrather fast while stabilizing the queue around the desired set point, Q = 700 at SW3.Con lusionsIn this arti le, we have presented a ontrol-theoreti approa h to designing ABR ongestion ontrol algorithms. ATM networks deal with di�erent types of tra� . Among the several servi eso�ered by ATM, the ABR servi e plays a entral role in regulating the network tra� , as it isthe only servi e ategory that uses expli it feedba k from the network. We have presented severalalgorithms for ABR ongestion ontrol and have shown that they perform well under various riteria19

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Figure 15: Max-min fairness model: Sour e rates under Controller 3.su h as basi fairness and max-min fairness while a hieving high bandwidth utilization. In addition,ex essive ell loss is avoided by ontrolling the queue length. In our designs, the network delay isexpli itly taken into a ount, whi h we believe is useful in modeling network behavior over links withhigh delays, su h as satellite ATM networks, or IP over ATM. Furthermore, we have shown that oneof our algorithms is robust to un ertainty in round-trip delays, and at the same time it an adaptto the variations in the number of sour es sharing a swit h. These properties make this algorithmeasier to apply in volatile networks, where both the network delay and the number of a tive sessionsare not known or annot be predi ted a urately beforehand. In summary, we strongly believe that ontrol theoreti design tools an be e�e tively used in designing high performan e algorithms inthe ontext of ATM ABR ongestion ontrol.A knowledgmentsResear h reported here was supported by NSF Grant NSF ANI 98-13710, an NSF CAREERAward NCR 9701525, AFOSR MURI Grant AF DC 5-36128, and a Grant from Rome Labs (AirFor e Contra t F30602-96C0156) through S ienti� Systems Company, In .Referen es[1℄ The ATM Forum, Te hni al Committee, �Tra� management, Version 4.1,� pp. 43-55, af-tm-0121.000, Mar h 1999.[2℄ J-C. Bolot, �End-to-end pa ket delay and loss behavior in the Internet,� in Pro eedings of ACMSig omm '93, San Fran is o, CA, September 1993, pp. 289-298.22

[3℄ J-C. Bolot, �A self-tuning regulator for overload ontrol in ommuni ation networks,� in Pro- eedings of the 31st IEEE Conferen e on De ision and Control, Tu son, AZ, De ember 1992,pp. 1022-1023.[4℄ J-C. Bolot and A.U. Shankar, �Analysis of a �uid approximation to �ow ontrol dynami s,� inIEEE INFOCOM '92, Floren e, Italy, May 1992, pp. 2398-2407.[5℄ S. Keshav, �A ontrol-theoreti approa h to �ow ontrol,� in ACM SIGCOMM '91, Zuri h,Switzerland, September 1991.[6℄ E. Altman, T. Ba³ar, and R. Srikant, �Robust rate ontrol for ABR sour es,� in Pro eedings ofIEEE INFOCOM, San Fran is o, CA, Mar h 1998.[7℄ E. Altman, T. Ba³ar, and R. Srikant, �Congestion ontrol as a sto hasti ontrol problem witha tion delays,� Automati a, vol. 35, pp. 1937-1950, De ember 1999.[8℄ S. Kalyanaraman, R. Jain, S. Fahmy, R. Goyal, and B. Vandalore, �The ERICA swit h algo-rithm for ABR tra� management in ATM networks,� IEEE/ACM Trans. on Networking, vol.8, no. 1, pp. 87-98, February 2000.[9℄ O. Ait-Hellal, E. Altman, and T. Ba³ar, �Rate-based �ow ontrol with bandwidth information,�European Transa tions on Tele ommuni ations, vol. 8, no. 1, pp. 55-65, 1997.[10℄ L. Benmohamed and S. M. Meerkov, �Feedba k ontrol of ongestion in pa ket swit hing net-works: The ase of a single ongested node,� IEEE/ACM Transa tions on Networking, vol. 1,no. 6, pp. 693-707, 1993.[11℄ L. Benmohamed and Y. T. Wang, �A ontrol-theoreti ABR expli it rate algorithm for ATMswit hes with per-VC queueing,� in Pro eedings of IEEE INFOCOM, San Fran is o, CA, Mar h1998.[12℄ A. Kolarov and G. Ramamurthy, �A ontrol-theoreti approa h to the design of an expli itrate ontroller for ABR servi e,� IEEE Transa tions on Networking, vol. 7, no. 5, pp. 741-753,O tober 1999.[13℄ A. Kolarov and G. Ramamurthy, �A ontrol theoreti approa h to the design of losed loop ratebased �ow ontrol for high speed ATM networks,� in Pro eedings of IEEE INFOCOM, 1997.[14℄ S. Mas olo, D. Cavendish, and M. Gerla, �ATM rate based ongestion ontrol using a Smithpredi tor: An EPRCA implementation,� in Pro eedings of IEEE INFOCOM, San Fran is o,CA, Mar h 1996.[15℄ D.P. Bertsekas and R. Gallager, Data Networks. Englewood Cli�s, NJ: Prenti e-Hall, 1987.[16℄ E. Altman and T. Ba³ar, �Optimal rate ontrol for high speed tele ommuni ation networks,�in Pro eedings of the 34th IEEE Conferen e on De ision and Control, New Orleans, Louisiana,De ember 1995, pp. 1389-1394.[17℄ B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadrati Methods. EnglewoodCli�s, NJ: Prenti e-Hall, 1990.[18℄ O.Ç. Imer and T. Ba³ar, �Optimal solution to a team problem with information delays: Anappli ation in �ow ontrol for ommuni ation networks,� in Pro eedings of the 38th IEEEConferen e on De ision and Control, Phoenix, AZ, De ember 1999.23

[19℄ C. Fulton, S.-Q. Li, and C. S. Lim, �UT: ABR feedba k ontrol with tra king,� in Pro eedingsof IEEE INFOCOM, 1997, pp. 806-815.[20℄ O.Ç. Imer, T. Ba³ar, and R. Srikant, �A robust adaptive ontrol algorithm for ABR servi e inATM networks,� in Pro eedings of IEEE ICCCN, Las Vegas, AZ, O tober 2000 (to appear).[21℄ M.W. Garrett. Contributions Toward Real-Time Servi es on Pa ket Swit hed Networks. PhDthesis, Columbia University, May 1993.[22℄ O. Rose, �Statisti al properties of MPEG video tra� and their impa t on tra� modeling inATM systems,� in Pro eedings of the 20th Annual Conferen e on Lo al Computer Networks,Minneapolis, MN, 1995, pp. 397-406.[23℄ �MPEG-1 frame size tra es,� available over the Internet via anonymous ftp from ftp-info3.informatik.uni-wuerzburg.de, in dire tory /pub/MPEG/.[24℄ �Video tra es,� opyright ( ) 1992, Bell Communi ations Resear h, In (Bell ore), available overthe Internet via anonymous ftp from thumper.bell ore. om, in dire tory pub/vbr.video.tra e.[25℄ R. Jain, �Congestion ontrol and tra� management in ATM networks: Re ent advan es anda survey,� Computer Networks and ISDN Systems (Netherlands), vol. 28, no.13, pp. 1723-38,O tober 1996.[26℄ P.J. Bro kwell and R.A. Davis, Times Series: Theory and Methods. New York: Springer Verlag,1991.[27℄ S. Fahmy, R. Jain, S. Kalyanaraman, R. Goyal, and B. Vandalore, �On determining the fairbandwidth share for ABR onne tions in ATM networks,� in Pro eedings of the IEEE Interna-tional Conferen e on Communi ations (ICC) 1998, June 1998.

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