developing explicit rate control...

By Orhan Cagri Imer, Sonia Compans, Tamer Basar, and R. Srikant

Developing Explicit Rate Control Algorithms

TM (asynchronoustransfer mode) is the

underlying technologyenabling B-ISDN

(broadband inte-grated ser vices

digital network). B-ISDN was intro-duced as the successor to narrow-band ISDN after the latter fell short ofmeeting the high demand for band-width required by emerging applica-tions such as real-time video and highdefinition TV (HDTV). B-ISDN envi-sions the transmission of fixed-sizepackets (cells) over digital virtual cir-cuits at rates exceeding 150 Mb/s.

ATM is basically a packet-switchingtechnology with 53-byte-long cells. The small cell size makesit possible to build switches that can accept and switch alarge batch of cells. ATM is asynchronous in that it has no re-quirements that cells rigidly alternate among the varioussources (i.e., cells arrive randomly from different sources).

An international nonprofit organization, the ATM Forum,was established several years ago, with the primary objec-tive of promoting and extending the use of ATM products andservices. The ATM standards set by the Forum define theuser-network interface; that is, the way a computer owned by

a private user can connect to the network and communicatethrough it. For that, five different services are available andare used for different types of communication [1]:

• Constant Bit Rate (CBR): This service category is usedby connections that request a static amount of band-width that is continuously available during the con-nection lifetime. Telephone and television use thisservice.

• Variable Bit Rate (VBR): The cell rate is variable and ismainly intended for bursty sources. This service can

38 IEEE Control Systems Magazine February 2001

©20

00Im

age

100

Ltd.

Imer, Bas,ar ([email protected]), and Srikant are with the Department of Electrical and Computer Engineering and CoordinatedScience Laboratory, University of Illinois, 1308 West Main Street, Urbana, IL 61801-2307, U.S.A. Compans is with Alcatel Business Sys-tems, 92707 Colombes Cedex, France.

0272-1708/01/$10.00©2001IEEE

be real-time VBR (video conferencing) or nonreal-time VBR (multimedia e-mail).

• Available Bit Rate (ABR): In this service category, thecell rate depends on the availability of the network. It isbasically designed for bursty traffic whose bandwidthrange is known roughly. A rate control mechanism isspecified by the ATM Forum. Examples of this servicecategory include browsing the Web and e-mail.

• Unspecified Bit Rate (UBR): This category uses theleftover network capacity. UBR service does not spec-ify traffic-related service guarantees. Background filetransfer uses this service.

• Guaranteed Frame Rate (GFR): This service categoryis intended to support non-real-time applications re-quiring a minimum rate guarantee. It does not requireadherence to a rate control protocol. An example ap-plication is frame relay interworking.

In principle, any UBR or GFR application can take advan-tage of the ABR flow control protocol to achieve a low cellloss ratio. Thus, the network traffic generated by varioussources can be thought of as composed of CBR, VBR, andABR connections only.

When a virtual circuit (VC) is established between asource and a destination, both the customer (source) andthe carrier (switch) must agree on a contract defining thequality of service (QoS). The ATM Forum defines three QoSparameters:

• cell delay variation (CDV),• maximum cell transfer delay (maxCTD),• cell loss ratio (CLR).In CBR and VBR services, a traffic contract specifying the

QoS guarantees is negotiated during the virtual circuitsetup phase and is maintained for the duration of the con-nection. The ABR service, on the other hand, does not re-quire bounding the delay or the delay variation experiencedby a given connection. Therefore, with CBR and VBR traffic,it is generally not possible for the source to decrease itsrate, even when an intermediate node becomes congested,because of the QoS guarantees made at VC setup time. How-ever, ABR sources might adjust their rates to the level ofavailable service at times of congestion. Thus, ABR trafficcan be used to control congestion in the network.

About four years ago, the ATM Forum adopted a rate-based congestion control scheme for the ABR service [1]. Inthis scheme, explicit rate control messages are sent from in-termediate nodes to the sources using special cells calledresource management (RM) cells. The goal of this conges-tion control mechanism is to fairly share the bandwidth leftover from high-priority traffic (CBR and VBR) among theABR sources while making sure that the links throughoutthe network are fully utilized.

An ATM network consists of several nodes (switches) in-terconnected via bidirectional links. We say that a switch isbottlenecked if the incoming ABR cell rate at any one of itsoutput ports is larger than the available rate to serve the

ABR cells. Clearly, at a given instant, there may be multiplebottlenecks in the network, as in Fig. 1.

Although the rate-based congestion control schemes arestandardized, developing good explicit rate computation al-gorithms is still an open issue. As the link speeds continueto rise, the delay-bandwidth product (i.e., the product of theround-trip propagation delay and the link capacity) in-creases. An issue of importance that arises in this context ishow to deal with action delays, which is the time from themoment control information is sent to a source, until an ac-tion is taken by it, and until subsequently that action affectsthe state of the switch that initiated that command. In thisarticle, we consider a control-based mathematical modelthat helps us address this problem.

Our initial modeling assumption is that of a single bottle-neck switch shared by several ABR sources. Although in areal network topology the sources may be interconnected inseveral ways, resulting in multiple bottlenecks (Fig. 1), thesingle bottleneck assumption admits theoretical as well asexperimental justification [2] and provides a good startingpoint for the derivation of effective rate controllers. Moreprecisely, we present three different congestion control al-gorithms for ABR control and study their performance in asimulated network environment. The mathematical modeluses idealized linear queue dynamics, already introduced in[3]-[5], but the simulation model takes the saturationnonlinearities into account. Particularly, in a real network,queue length at a switch must lie between zero and the sizeof the buffer, and this is taken into account in the simulationmodel. Two of the algorithms we present formulate the con-gestion control problem as a stochastic team where playersare users sharing the bottleneck switch. In this formulation,the approach involves a model for the available bandwidthas an autoregressive (AR) process, driven by an arbitrary,independently distributed random sequence, andminimization of an objective functional through which mostof the design criteria are reflected [6]-[7]. The third algo-rithm we present is based on a deterministic model of the

February 2001 IEEE Control Systems Magazine 39

SwitchBottleneck Switch

Figure 1. A generic ATM network with multiple bottlenecks.

available bandwidth. In this scheme, the emphasis is moreon robustness against the variations in round-trip delaysand estimating the number of sources sharing the bottle-neck switch.

Several other types of ABR congestion control designshave been considered. We briefly summarize these here tocompare and contrast them with our approach. The sim-plest feedback control mechanism is called rate matching. Inrate matching, the node measures the average rate availableto ABR sources at periodic intervals and simply divides afraction of this capacity equally among the various users.

This is the basic approach used in [8], although severalmodifications are used in the actual implementation. Themain advantage of this scheme is its simplicity, but it is diffi-cult to control queue length optimally to avoid buffer over-flows. This scheme, however, is stable (i.e., the queue lengthremains bounded in an appropriate stochastic sense [9]).Queue length information is not used in the basic algorithm,although [8] allows one to incorporate queue length infor-mation in an ad hoc manner. Alternatively, this problem canbe viewed as a feedback control problem where queuelength is used for explicit feedback. This approach is used in[10]-[11] to study this problem using classical control tech-niques or using a state-space approach. As in rate matching,the primary goal is not optimality, but simply queue lengthstability. In these approaches, the available bandwidth toABR sources is treated as an unmodeled disturbance. Thus,the algorithms in [10]-[11] ensure stability in the presenceof this disturbance, but do not address the issue of perfor-mance. In a recently published work [12], a closed-loop pro-portional-derivative controller is proposed, which achievesmax-min fairness plus queue length stability, but the designfalls short of addressing the issue of robustness against un-certainty in delays.

ABR ServiceFlow Control Model for ABRIn the ABR service, the source adapts its rate to changingnetwork conditions. As previously mentioned, information

about the state of the network, such as bandwidth availabil-ity, state of congestion, and impending congestion, is con-veyed to the source through special control cells called RMcells. ABR flow control occurs between a source and a desti-nation, which are connected via bidirectional links. The for-ward direction is the direction from the source to thedestination, and the backward direction is the directionfrom the destination to the source.

A source generates forward RM cells every Nrm data cells,where Nrm is generally taken to be 32. These cells travelalong the same path as the data cells but are treated specially

by the switches along the way (Fig. 2). The switch may:• Directly insert feedback control information

into RM cells by using the explicit rate (ER) fieldof RM cells.

• Provide binary feedback by marking the conges-tion indication (CI) or no increase (NI) bit in theRM cells.

• Indirectly inform the source about congestion bysetting the explicit forward congestion indication (EFCI)bit in the data cell header, and rely on the destination toconvey congestion information back to the source bymarking the CI bit in the backward RM cells it generates.

• Spontaneously generate backward RM cells and shipthem back to the source.

Note that Fig. 2 is only a generic representation of thecontrol loop, as there may be more than one switch betweenthe source and the destination. On the establishment of anABR connection, the source specifies to the network both amaximum required bandwidth and a minimum usable band-width. These are designated as peak cell rate (PCR) and theminimum cell rate (MCR), respectively. The MCR may bespecified as zero. The bandwidth available from the net-work may vary, but should not become less than MCR. EachABR source has a current cell rate, allowed cell rate (ACR),which it must modify upon receiving feedback from the net-work via RM cells. The ACR always falls somewhere betweenMCR and PCR. When a source sends out a forward RM cell, itsets the ER field of the RM cell to the rate at which it wouldcurrently like to transmit. As the RM cell passes through thevarious switches on the way to the destination and back tothe source, those that are congested may reduce the ER.When the source receives the RM cell back, it takes one ofthe following actions depending on the settings of the ERfield and CI and NI bits.

• When there is no congestion (both CI and NI bits arenot set), ACR can be increased (but not above PCR) bya quantity RIF×PCR, where RIF is the rate increase fac-tor. ACR cannot be increased, however, above the ex-plicit rate specified in the ER field.

• When a source receives a backward RM cell with CIbit set, it decreases its ACR (but not below MCR) bya quantity RDF×PCR, where RDF is the rate decreasefactor. However, again the ACR of the source cannotbe larger than the explicit rate specified in the ER


Source Switch Destination

RM CellData Cell

Figure 2. ABR traffic management model.

In the ABR service, the sourceadapts its rate to changingnetwork conditions.

field. Both RIF and RDF are negotiated in the VCsetup phase.

• Finally, when the source receives a backward RM cellwith only the NI bit set, it sets its rate to the minimumof ACR and ER.

Performance CriteriaNumerous algorithms have been andare being developed for ATM ABRcongestion control. To be able tostudy and compare these algorithmson equal grounds, we need to setsome performance measures inde-pendently of the particular algorithmunder investigation. The main goal ofABR control is to provide fairnessamong all VCs with a minimal cell lossratio and maximal utilization of network resources. The lat-ter two of these objectives can be achieved by regulating thequeue length at bottleneck nodes around a desirable level.Tracking such a nominal queue length (whose exact value isdetermined based on QoS requirements) is desirable toavoid losses due to overflow and waste of the buffer capac-ity due to underflow.

Fairness is an issue that requires more discussion, as itmay be hard to visualize what is meant by a fair allocationin a large network with multiple bottlenecks. The mostwidely accepted notion of fairness is the max-min fairnesscriterion [15]. Under this criterion, the fair share of eachconnection contending for a given link bandwidth should

be equal to ( ) / ( )µ − −r N Mu u . Here µ is the available ser-vice rate for ABR sources at a particular switch, N is thenumber of active sessions at that particular switch, r u isthe total rate of connections that are bottlenecked else-where or are limited by their PCRs, and M u is the number ofsuch connections.

Basic Model of an ATM SwitchGenerally, an ATM switch has several input and outputlines. The number of input lines is almost always thesame as the number of output lines, because the links arebidirect ional . Cel ls arr ive on the input l inesasynchronously, but the switching is done synchronouslywith the help of a master clock. Each input line is con-nected to a common bus, through which the incoming cellsare directed to their corresponding output ports. Most ofthe commercial ATM switches use output queueing to pre-vent high cell loss rates. In output queueing each outputline has a finite buffer, where the incoming cells are servedon a FIFO (first in, first out) basis. Associated with each


SV1 DV1

SV2 DV2

SV3 DV3

SV4 DV4

SV5

DV5SW1 SW2 SW3 SW4

SV1:-Asterix

-Terminator

-Two On/Off Sources

SV2:-Soccer SV3:-Bond SV4:-Lambs

-Simpsons

SV5:-Starwars

-ATP

-Two On/Off Sources

Figure 3. Four-node configuration: VBR sources.

The goal of ABR control: providefairness among all virtual circuits with aminimal cell loss ratio and maximalutilization of network resources.

output line is an ER controller that suggests a data rate foreach ABR VC.

In what follows, we focus on a particular output line of anATM switch. Other output lines can be treated in a similarmanner. The model we adopt here is a discrete-time model,where a time unit corresponds to the interval over whichthe rate available to ABR sources is determined (that is, theinterval over which measurements are made). Further, thismeasurement interval is assumed to be long enough for theswitch to be able to process several cells—a reasonable as -

sumption if the link speeds are high and packet sizes aresmall. This allows us to ignore the cell-level dynamics andmodel the ABR traffic as a fluid.

Let rn denote the total number of cells that arrive in one ofthe output buffers of an ATM switch in the interval [ , )n n +1 ,and let µ n denote the number of cells that depart from thisbuffer in the same time interval. Note thatµ n represents theavailable bandwidth (unused by higher priority traffic, par-ticularly CBR and VBR). Denoting the queue length at time nby qn and ignoring the boundary effects on the queue dy-namics as in [16], we have the evolution

q q rn n n n+ = + −1 µ . (1)

Let there be a total number of N connections (sources)switched through the output line under study, and the num-ber of cells that arrive from source m during time-slot[ , )n n +1 be denoted by rmn . Clearly

r rn mnm

N

==

∑1

.

In general, rn has two components: 1) The number of cells thatarrive from uncontrolled sources, i.e., those sources which arebottlenecked elsewhere in the network or are limited by their

PCR constraints. The ER controller at this switchhas no control over these sources. In otherwords, the ER field of an uncontrolled source iseither overwritten at some other switch or is re-placed by its PCR value at the source. We denotethis component of rn by rn

u . 2) The number of cellsthat arrive from controlled sources, which arebottlenecked at this switch. The ER fields of RMcells of this type of sources are modified toachieve several traffic-related service guaran-tees. We assume that each source, either con-trolled or uncontrolled, has an MCR = 0. If the

MCRs are positive, we can reserve the minimum cell rate foreach user and make the assumption that a source will attemptto send at a rate no smaller than its reserved MCR.

Let there be a total number of M controlled sources andthe number of cells that arrive from controlled source m inthe interval[ , )n n +1 be denoted by rmn

c . Then, we have the fol-lowing relation between rn , rmm

c , and rnu :

r r rn mnc

m

M

nu= +

=∑

1

.

In the analysis to follow, we assume that the switch exer-cises ER ABR congestion control, which amounts to settingRIF =1, RDF = 0. Note that for a controlled connection, the PCR


SW1 SW2 SW3 SW4

SA2

DA2

SA3 DA3

DA1

SA1

DAi: Destination ABRiSAi: Source ABRi

Bottleneck Switch

Figure 4. Fairness configuration: ABR sources.

For a controlled connection, thePCR constraint never becomesactive, as this would invalidate theassumption that the source iscontrolled.

constraint never becomes active, as this would invalidate theassumption that the source is controlled. Also note that bothrn andµ n represent cell rates with the unit of time taken as thesampling interval over which the measurements are made.

As indicated earlier, the decision regarding the rate atwhich each source should transmit is made by the switch,and the type of information accessible to the switch affectsthis decision. Since ATM technology relies on thehigh-speed switching of packets, the amount of overheadcaused by the ER controller should be kept at a minimallevel. At time n +1, when the decision on the rate of eachcontrolled source has to be made, three pieces of informa-tion are available to the switch without any delay: queuelength,qn , available bandwidth,µ n , and the total number ofcells arrived in the interval [ , )n n +1 . Extracting these num-bers does not require any elaborate measurements, as qn

can be measured by looking at how many cells in the bufferare waiting to be served, whileµ n and rn can be determined

by counting the number of cells left and arrived in theinterval[ , )n n +1 , respectively. In general, to be able to fairlydivide the bandwidth among all sources, the switch needsto know how many controlled sources are being served at agiven time. Calculating the number of VCs requires theswitch to look at the header of every single RM cell and tellwhat source it originated from and what destination it isrouted to. Even though theoretically possible, this taskbrings in a computational overhead slowing down the op-eration of the ATM switch. Moreover, the number of VCs, N,calculated this way may not be equal to the actual numberof controlled sessions, M, as some of the sources might belimited by their PCRs, bottlenecked at some other switchin the network, or just too bursty to be exercised any con-trol over. Therefore, it is desirable to develop an ER controlalgorithm that relies only on the knowledge of the threeeasily accessible quantities described above.

As mentioned earlier, it takes time from the moment theER decision is made by the switch until an action is taken bya source, and until subsequently that action affects the stateof the node that initiated the action. Thus, the cell rate ofsource m at time n, rmn , is actually an outcome of an actiontaken dm time units earlier, where dm represents the actiondelay for source m and is taken to be independent of time n.Without any loss of generality, we assume that the dms areordered such that

0 1≤ ≤ ≤ ≤d d dM�


SW1 SW2 SW3 SW4

SA4SA1

DA1SA2

SA3 DA3

DA4

DA2

SAi: Source ABRi

DAi: Destination ABRi

Bottleneck Switch

Figure 5. Max-min fairness configuration: ABR sources.

Table 1. Action delays for the fairness model.

SW1 SW2 SW3 SW4

ABR1 0 3 6 10

ABR2 0 3 6

ABR3 0 3

where d corresponds to the maximum round-trip net-work delay.

Congestion Control Algorithms

Controller 1:A Certainty Equivalent Controller

Basic AssumptionsThe available bandwidth for ABR, µ n , may change in an un-predictable way since the transmission rate of VBR traffic istime varying. The bandwidth available to the controlledsources, on the other hand, is subject to further uncertaintydue to the variations in the uncontrolled traffic, rn

u . Theseobservations motivate us to consider the CBR, the VBR, andthe uncontrolled ABR traffic collectively as interference,modeled by an AR process, which is stable:

γ µ µ ξn n nu u

nr r:= − = − + (2)

ξ α ξni

p

i n i n= + φ=

− −∑1

1.(3)

Here µ is the constant nominal service rate, r u is the nomi-nal rate of uncontrolled sessions, p is the order of the ARprocess, α i i p, , ,=1 � are the parameters characterizing theprocess, and the driving term { }φ ≥n n 1 is a zero-mean i.i.d.Gaussian sequence with variance k2.

Given the dynamics of the AR process, the source rates,rmn

c , are determined by minimizing a cost function that quan-tifies the tradeoff between two partially conflicting goals:steady queue length around a nominal value and fair alloca-tion and actual sharing of the available bandwidth. The costfunction adopted for this purpose is


Table 2. Action delays for the max-min fairnessmodel.

SW1 SW2 SW3 SW4

ABR1 0 3 6 10

ABR2 0 3

ABR3 0

ABR4 0 3

0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400 1600 1800

Cel

ls p

erT

ime

Uni

t

Time Unit

0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400 1600 1800

Cel

ls p

erT

ime

Uni

t

Time Unit

0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400 1600 1800

Cel

ls p

erT

ime

Uni

t

Time Unit

ABR1 Mean 1494.2

ABR2 Mean 1495.4

ABR3 Mean 1495.9

Figure 6. Fairness model: Source rates under Controller 1.

( )JN

E q Qk

r aN

nmm

M

mnc

m nn

= − + −

→∞ ==

∑lim sup1 12

21

2

1

( ) γN

∑

(4)

where Q is the target queue length, kms are positive constants,amm

M

=∑ =1

1, and E{}⋅ denotes expectation over the statistics ofthe AR process. Exact fair sharing implies a Mm =1 / , whichwill be the initial value for each am . Note that this choice of am

requires the knowledge of the number of controlled VCs, M.The first term in (4) represents the penalty for deviating

from the desired queue length, Q, while the second term repre-sents a penalty (for each source) for deviating from the autho-rized transmission rate, which we have taken to be a fractionof the available bandwidth for each source to achieve max-minfairness. The kms quantify the relative priority given to eachsource—the larger km is, the lower the priority.

Derivation of Controller 1To arrive at a somewhat simpler configuration, we first in-troduce the shifted variables

x q Q

u r a rn n

mn mnc

mu

:

: ( )

= −= − −µ

where xn stands for the state and umn for the control. Thenthe state dynamics (1)-(3) become

x x un n mnm

M

n

n ii

p

n i n

+=

+=

+ −

= + −

= + φ

∑

∑

11

11

1

ξ

ξ α ξ

and the cost function is

JN

E xk

u aN

nmm

M

mn m nn

N

= + −

→∞ ==

∑lim sup1 12

21

2

1

( ) ( )ξ∑

.

Let

~ : , ~ : ( ~ , , ~ )u u a u u umn mn m n n n Mn= − = ′ξ 1 � .

Then the cost function can be written as

JN

E xk

uN

nmm

M

mnn

N

= +

→∞ ==

∑∑lim sup1 12

21

2

1

( ) ( ~ ).

Also, in terms of ~u, the dynamics for x become


0 500 1000 1500Time Unit

0 500 1000 1500Time Unit

0 500 1000 1500Time Unit

0

0.5

1

1.5

2N

umbe

r of

Cel

ls

0

500

1000

1500

Num

ber

of C

ells

0

0.5

1

1.5

2

2.5

3

Num

ber

of C

ells

q1

q3 Mean 695.6

q4

0 500 1000 1500Time Unit

0

0.5

1

1.5

2

2.5

3

Num

ber

of C

ells

q2

Figure 7. Fairness model: Queue lengths under Controller 1.

x x un n mnm

M

+=

= + ∑11

~ .

If there was no delay (i.e., d d dM1 2 0= = = =� ), then thestandard discrete-time linear regulator theory [17] wouldapply, yielding the unique solution

~ [ ]u R bb s bs xn n= − + ′ −1

where

b

Rk k

M

M

′ =

=

×: ( )

: , ,

1 1 1

1 1

1

12 2

�

�diag

and s is the unique positive root of

[ ]s s b R bb s bs= + − + ′ −1 1 1'( ) .

This can be solved explicitly to yield

s km

M

m= + + =

=

−

∑1 1 42

22

1

2

1ζ ζ; : .

The original controller would then be (with zero delay)

u u an n m n= +

~ .

1

1

1

�ξ

(5)

Since the impact ofumn on the network would be felt onlyafter dm time units, here we invoke certainty equivalence,where, in the solution (5) for each controller umn ,m M=1 2, , , ,� we replace the queue length and bandwidth bytheir estimates dm time units later. These considerationslead to the following certainty equivalent controller, whichwe will refer to as Controller 1:

u p x a m Mm n m n d n m n d nm m, | |� � , , ,∗

+ += − + =ξ 1 � . (6)

Here pm is the mth component of [ ]R bb s bs+ ′ −1 and �|xn d nm+ ,

�|ξn d nm+ are the predicted values of xn dm+ and ξn dm+ , respec-

tively, based on the what is known at time n and given that allother controllers are also in the form (6). These predictorsare generated by


0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400 1600 1800

Cel

ls p

erT

ime

Uni

t

Time Unit

0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400 1600 1800

Cel

ls p

erT

ime

Uni

t

Time Unit

0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400 1600 1800

Cel

ls p

erT

ime

Uni

t

Time Unit

ABR1 Mean 1492.8

ABR2 Mean 1496.3

ABR3 Mean 1496.5


� � � � ,| | , | |x x u jn j n n j n m n j d nm

M

n j nm+ + − + − −=

+ −= + − ≥∑1 11

1ξ 1

11

;

� ;

� � , ; �

|

| | |

x x

j

n n n

n j n ii

p

n j i n n k n n

=

= ≥ =+=

+ − −∑ξ α ξ ξ ξ − ≥k k, ;0

and

� :� �

, || |u

p x a j d

um n j d nm n j n m n j n m

mm+ − −

+ − + −=− + ≥ +

11 1 1ξ if

, .n j d mmj d+ − − < +

1 1if

These are the recursive equations generating the predictorsfor the queue length and rate information at a future time,where the future time is the current time plus the action de-lay for the corresponding source. For example, �

|ξn j n+ de-notes the predicted value at time n of the value of ξ at somefuture time n j+ , based on the information available at timen. A similar interpretation holds for �

|xn j n+ .The above algorithm is relatively easy to implement.

The estimator algorithms are simple scalar operations,and the scalar solution of the Riccati equation, s, has al-

ready been obtained explicitly. In summary, an easilyimplementable version of Controller 1 is given below in theform of pseudocode:

Pseudocode for the node’s computation at time n usingController 1for j dM=1 to do

for m M=1 to doif (n j d nm+ − − ≥1 )

� � �, |u p x am n j d n m n j m n jm+ − − + − + −= − +1 1 1ξ

endifendfor� � � �

, |x x un j n j m n j d nm

M

n jm+ + − + − +=

+ −= + −∑1 11

1ξendforu p x amn m n d m n dm m

= − ++ +� �ξ

Controller 2: Optimal ControllerRecall that in the derivation of Controller 1, we first assumedzero delays, solved the simplified linear quadratic regulator(LQR) problem, and then incorporated the delays into con-trollers through estimators using the certainty-equivalenceprinciple. An alternative approach, which leads to the opti-


0 500 1000 1500Time Unit

0 500 1000 1500Time Unit

0 500 1000 1500Time Unit

0

0.5

1

1.5

2N

umbe

r of

Cel

ls

0

500

1000

1500

Num

ber

of C

ells

Num

ber

of C

ells

q1

q3 Mean 695.9

0 500 1000 1500Time Unit

Num

ber

of C

ells

q2

q4

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

Figure 9. Fairness model: Queue lengths under Controller 2.


mal solution of the problem, is to augment the state space inan appropriate way. Basically, one must introduce new statevariables for those sources that have nonzero delays. Sincethis is a tedious process with many algebraic details, we donot include the derivation of the optimal controller here, butinstead refer the interested reader to [18]. The optimal solu-tion is characterized in terms of the solution of a discrete-time algebraic Riccati equation (DARE), whose dimension isdetermined by the magnitude of the largest delay and the or-der of the AR process describing the available capacity.

Controller 3: A Robust AdaptiveController Under UncertaintyBasic AssumptionsIn our earlier paper [6], by considering a one-node examplewith perfect delay information and instantaneous feedback,we showed that delay is an important factor in any design ofrate flow controllers and hence must be explicitly taken intoaccount in any realistic model of high-speed networks, aswe have done above. In a realistic network, however, thereare several deviations from this ideal model.

Both Controllers 1 and 2 assume complete knowledge ofdelays in the network. Although the end-to-end round-tripdelay may be known to the switch as part of the fixedround-trip time (FRTT) computation performed at VCsetup time, we can still have small errors in delay esti-mates. One source of error is the assumption that the delayis an integer multiple of the time unit (measurement inter-val), which may not be true. Further, there is variability inthe delays due to queues in the virtual circuits. Finally, RMcells are generated only every 32 data cells and hence feed-back is not instantaneous.

In the derivations of Controllers 1 and 2, we also assumedthat the switch has access to the information about the num-ber of controlled sources, M. Recall that we need this numberto determine the fair share of each source. As mentioned ear-lier, in a realistic environment with bursty sources, determi-nation of M may not be an easy task. Hence, the switch musthave a method of estimating this parameter.

Motivated by these observations, we want to develop acontroller that is robust to uncertainty in delays, and at thesame time adaptive to the number of sources. Thus, in deriv-ing what we call Controller 3, we only require the knowledge

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of d and M , which denote the upper bounds on the round-tripdelay and the number of controlled sessions, respectively.

Although the available bandwidth for ABR, µ n , and therate of uncontrolled sessions, rn

u , may change with time, asin the previous section, we assume here that bothµ n and rn

u

are constants denoted by µ and r u , respectively. These as-sumptions are justified ifµ n and rn

u vary slowly compared tothe time constant of the closed-loop system. With these sim-plifying assumptions, the state equation (1) becomes

q q r rn n mnc

m

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+=

= + + −∑11

µ .

Since the rate of the controlled source m at time n is actuallyan outcome of an action taken by the switch dm time unitsearlier, we have

r cmnc

n dm= −

where cn denotes the command issued by the switch at time n.

Derivation of Controller 3As the actual number of controlled sources, M, is not knownto the switch, we start our analysis by proposing a methodto estimate this figure. Let us first consider the case whenr u = 0 (i.e., there are no uncontrolled sessions). Our methodrelies on a simple observation: If the switch sends out thesame command, say c, for( )d +1 time units, at the end of the( )d +1 st step, all of the controlled sources in the system willbe transmitting at rate c. Hence, at the end of the ( )d +1 sttime step, if we divide the incoming cell rate, rn , by the as-signed rate c, we obtain the actual number of controlledsources, M, at the switch. To have a running estimate of thisfigure, one can construct an estimator, �M n , and update it ev-ery( )d +1 time units using this scheme. We note that this al-gorithm converges to the exact value of M in only a finitenumber of steps. Having determined M in this manner, onecan set the command c at the next time slot to be equal toµ / M and achieve fairness, which in this particular case alsocorresponds to max-min fairness, as we took r u = 0.

Now if the rate of the uncontrolled connections, r u , is notzero, the above scheme fails to converge to the actual number


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ABR1 Mean 1392.9

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Figure 11. Max-min fairness model: Source rates under Controller 1.

of bottlenecked connections, M. It still does converge, how-ever, but to a different value. In fact, as we will show shortly, incalculating the share of each controlled connection, if oneuses the number to which the algorithm converges, then theresulting bandwidth allocation is max-min fair.

Before proceeding further, we note that the choice of therate c above is completely arbitrary, because it does notplay any role in the estimation process. Thus, we are free topick any rate larger than zero for the algorithm to work.Mathematically speaking, letting �M n denote the estimate ofM at time n, we propose the following estimator:

c c c

M M

n d n d n d

n d n d

( ) ( ) ( )( )

( ) (

,

� �

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+

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=

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− +

= ⋅⋅ ⋅ = =1 1 1 1 1

1

1 1) ( )( )

( )

( )( )

�Mr

cn d

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n d

where the first equation sets cn to the same value for d +1steps, while the second equation is used to estimate M everyd +1time units. For ease of notation, let us introduce the fol-lowing subsequences:

q q c c r r Mr

cns

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u

ns

Mr

c

(7)with �M Ms

0 = . To complete the design of Controller 3, weneed to specify how cn

s should be selected to achieve thedual goal of max-min fairness and queue length stability. Wepropose the following design for cn

s :

cM

q Qns

ns n

s= − −

max�

( ),µ β 0

(8)

whereβ > 0 is the gain to be selected to ensure stability, andthe max function is introduced to ensure that the switchasks the source to transmit at a positive rate in excess ofMCR =0, as required by the QoS specifications. In (8), theterm − −β( )q Qn

s is introduced to drive the queue length, qn ,to the desired set point, Q, by providing negative feedbackin the closed-loop system dynamics.

Note that if qn converges to Q, the command of the nodefor controlled sources, cn

s , converges to the solution of


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cM

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which is given by

cr

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Ms

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= −max ,

µ µ0

where the second equality follows from the obvious con-straintµ − ≥r u 0. Hence, if we can show that the closed-loopqueue dynamics, qn , converge to Q, then the rate of con-trolled sessions converges to the max-min fair share of theavailable bandwidth. In fact, it can be shown that if the con-troller gain, β, is picked such that

( )01

11< <

+−

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µd M

r u

,

then the robust control policy, given by (7)-(8), achievesqueue length control plus fair share of the available band-width [20]. The proof of this fact is rather tedious and ishence omitted in this article. Note that, as with Controllers 1and 2, the algorithm we propose here does not suffer fromcomputational complexity, because there is a single designparameter, namely β, to be tuned, and to determine the ERthe switch has to perform only two divisions, one multipli-cation, and two additions per output line, every ( )d +1 timeunits. Moreover, the information the switch needs to per-form these calculations, { , , }r qn

sns µ , is locally available, and

just a few memory elements per output port are required, asthere are only four numbers, { , , , }d M Q β , to be stored.

SimulationsSimulation ModelIn simulations, we consider a four-node high-speed networkwhere the link speed is the speed of light, the service rate of-fered by every link is 1 Gb/s, and the distance between adja-cent nodes is constant and equal to 1000 Km. We take thetime unit to be the time required to serve 5000 cells (5000 ×


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ABR1 Mean 1321.3

ABR3 Mean 1479.3 ABR4 Mean 1478.2

ABR2 Mean 3212.1


53 ×8 bits). Then, the propagation delay for a cell (i.e., thetime required for a cell to traverse a link) is

t = ××

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. time units,

where we have again taken the speed of propagation toequal the speed of light. Accordingly, we consider here thefour-node network depicted in Fig. 3. In the figure, SVi rep-resents a set of VBR sources as detailed under the figureand DVi is the destination of SVi. Moreover, the VBRsources considered are of two types: video traces obtainedfrom [22]-[24] and simulated ON-OFF sources. The latterare bursty sources simulated by us, which alternate be-tween ON and OFF states according to a Markov chain, andwhen in the ON state, cells arrive at a constant rate.

Superimposed on the model in Fig. 3, which depictsonly VBR sources, we consider two configurations for theABR sources:

• Parking lot model: This model will allow us to evaluatethe fairness of our algorithms when there is only onebottleneck node. This network has a “parking lot” con -figuration [25] where three ABR sources with differentdelays (see Fig. 4) are bottlenecked at the third node.The one-way propagation delay from one switch to an-other is 1.6 time units. Since the delays used in thecontrol algorithms are integers, we will take the near-est integers to the actual delays. For example, the ac-tual action delay between source ABR1 and thesecond switch is 2 16 3 2× =. . time units. The corre-sponding delay used in the mathematical model isthen three. Table 1 specifies the action delays (theones used by the control algorithms) between thesources and the switches. Recall that the action delaycharacterizing a source with respect to a node is de-fined as the time taken from the moment the nodesends control information to a source (via a backwardRM cell) until that source takes the corresponding ac-tion, and until subsequently that action affects thestate of the node. For the adaptive control algorithm,


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we take d =10, which is the largest possible networkdelay in this configuration. Note that to implement theadaptive robust controller, we do not need the delayinformation given in Table 1.

• Max-min model: The parking lot model has only onebottleneck node. This does not allow us to fully eval-uate the max-min fairness properties of our algo-rithms. Hence we consider the model depicted inFig. 5. Three of the ABR sources are bottlenecked atthe third node. The remaining ABR source(SA2) will then use the remaining capacityat the second switch (according to themax-min criterion [15]), which then be-comes bottlenecked. The network eventu-ally ends up with two bottleneck nodes.Table 2 specifies the action delays used inthe mathematical model.

For the simulation examples, we used the follow-ing parameter values:

• Time unit: Time required to serve 5000 cells• Target queue length: Q = 700• ICR = 300, MCR = 0, PCR = 4500• RIF =1, RDF = 0• Weights: k mm = ∀1 (Controllers 1 and 2)• Maximum number of sessions: M = 5 (Controller 3)• Maximum network delay: d =10 (Controller 3)• Controller gain: β = × × + −1 12 1 0 0152/ ( . ( ))~ .M d (Con-

troller 3).For Controllers 1 and 2, the nominal service rate µ and

the AR process parameters α i are estimated online usingthe Yule-Walker algorithm [26], assuming that the order ofthe AR process is eight. This is discussed in the next subsec-tion. For Controller 3, we takeµ to be equal to 4500 cells pertime unit. The propagation delay from one node to the nextis 1.6 time units. Note that the actual delay is variable sincethe cells go through node buffers, which leads to additionalqueueing delays.

Following the current ATM standards, the feedback mecha-nism has been implemented using RM cells that are generatedby the sources every 32 data cells. Even though the mathemat-ical model for the controllers can only use integer values fordelay, the actual one-way propagation delay for the RM cells is1.6 in the simulations. ATM standards allow for measurementof propagation delays in the signaling protocol. Queueing de-lay, however, remains variable and unknown. Thus, one of thegoals of our simulation is to show that the impact of these twotypes of approximations (to the actual delays) on the perfor-mance of Controllers 1 and 2 is negligible.

The AR ProcessTwo of our algorithms are based on modeling the availablecapacity (i.e., total capacity minus the capacity used by theVBR and uncontrolled ABR sources) as an AR process (see(1)-(2)). The order p, the parameters α i, and the variance ofthe noise have to be determined. We wish to emphasize that

the variance of the noise is determined as part of the estima-tion of the AR parameters but is never actually used in thecontrol algorithms, as it is not needed.

Tuning the parameters of the AR process and finding thebest possible value of p is a challenging task in general. Sev-eral methods exist to calculate the parameters, α i, once p isfixed. We use the Yule-Walker algorithm [26] to determine pand α is from the data that is observed online.

Let T be the time interval over which we attempt to fit an

AR model to the available capacity. Then one criterion to de-termine the best order for an AR process is the final predic-tion error (FPE), defined by

FPE = +−

�σ2 T pT p

where �σ is the estimate of the variance of the noise causedby the cross traffic. The order p of the process that mini-mizes the FPE is the best order [26]. The time interval in oursimulations wasT = 200, and using this, we determined thatp = 8 gives good results. For the sake of brevity, we do not in-clude the details here.

Simulation ResultsIn simulations, all rates are expressed in cells per time unit.

FairnessFirst, by considering the “parking lot” configuration (Fig. 4),we study the fairness of our designs when there is only onebottleneck node (SW3). In a single-bottleneck node sce-nario, fairness is the capability of the algorithm to fairly dis-tribute its available capacity among the various sourcesdespite the presence of different delays.

• Controllers 1 and 2: When VBR sources are present, themean available bandwidth at SW3 is around 4500 cellsper time unit. In addition, recall that sources ABR1,ABR2, and ABR3 have delays of six, three, and zero, re-spectively. The simulations for both controllers showthat the capacity is fairly shared among the threesources (Figs. 6 and 8) with a mean rate of around1495. Although the sources have different delays, thedesign manages to provide fairness. The queue at thethird node is regulated around 700, as expected (Figs.7 and 9).

• Controller 3: When there are no VBR sources, the avail-able service rate at SW3 is exactly 4500 cells per timeunit. As can be seen from Fig. 10, our algorithm


The ABR service plays a centralrole in regulating the networktraffic.

achieves fair share. Also the queue at SW3 convergesto the desired value of 700. In addition, the algorithmfinds the actual number of sources at SW3 in d + =1 11time units, as expected. The rate of convergence ofsource rates as well as the queue length depend on thecontroller gain β. A smaller value of β results in asmaller overshoot but a larger settling time.

Max-Min FairnessThe configuration under consideration is depicted inFig. 5. The main bottleneck node is the third one, wherethe capacity should be equally distributed among thesources ABR1, ABR3, and ABR4. Then, since ABR1 doesnot use its fair share at node 2, ABR2 should use the re-maining capacity and increase its cell rate to make up forthe difference.

• Controllers 1 and 2: With VBR sources present, themean available capacities are around 4870, 4720, 4480,and 4860 cells per time unit for switches 1, 2, 3, and 4,respectively. If the capacity at SW3 is equally distrib-uted among the sources ABR1, ABR3, and ABR4, eachone of these sources should transmit at around 1500

cells per time unit. Then, ABR2 should use the unusedcapacity at SW2 and increase its cell rate to around 3200.Nevertheless, our basic algorithm sets the weights tothe exact fair share (1 / M for M sources), and the dropobserved in the queue length at switch 2 is not substan-tial enough to increase the rate of ABR2. Therefore,max-min fairness is not achieved with the original con-trollers. An adaptive weight algorithm needs to be im-plemented to adapt the weights to a max-min fairnessconfiguration. This problem has been addressed in[27], where the authors determine the actual activity ofeach source and deduce from that a max-min fair share.We adopt a similar idea to choose the weights am , as de-scribed below. Consider a switch with M sources goingthrough it. We first evaluate the mean cell rate of eachsource m, say meanCCR m( ), by using the CCR (currentcell rate) field in the RM cells. If a source uses less than1 /M times the capacity of the switch, then its am is re-duced to the fraction that it actually uses. Such sourcesare called underloading sources. As a result, the remain-ing capacity is fairly allocated to the rest of the sources.


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To account for variability, we actually compare asource’s bandwidth utilization to a number slightlysmaller than1 / ,M say 0 85. /M .

The switch then computes the source optimalrates using these new weights. For example, considerthe three sources ABR1, ABR2, and ABR3 at theswitch. The total available capacity is 3000 cells pertime unit. Source ABR1 is bottlenecked somewhereelse at 500 cells per time unit. Then the new distribu-tion of the weights is aABR1 =1 6/ , aABR2 = 5 12/ , andaABR3 = 5 12/ . Therefore, the optimal rates at the pres-ent switch will be 500 cells per time unit for ABR1 and1250 cells per time unit for ABR2 and ABR3. Theserates will be used as feedback information.

The simulations done with the adaptive weight al-gorithm indicate that max-min fairness is indeedachieved (Figs. 11, 12, 13, and 14). Initially (with all amsset to the fair share), the bottleneck node is the thirdone and sources ABR1, ABR3, and ABR4 share fairlythe capacity at this switch (around 1500 cells per timeunit). Meanwhile, source ABR2 only uses half of thecapacity of switch 2. Actually its cell rate is slightlyhigher than the fair share. Because of the small queuelength, the design appears to increase the allowed cellrate slightly. Once ams at each switch are adapted,source ABR2 should increase its rate to 3220 cells pertime unit while the other sources still transmit at 1500cells per time unit. The simulations show that sourceABR2 effectively catches up (Figs. 11 and 13).

• Controller 3: One advantage of using Controller 3 isthat it achieves max-min fair bandwidth allocationwithout necessitating any sort of adjustment of pa-rameters, assuming of course that the available ser-vice rate remains constant. We simulate the configura-tion depicted in Fig. 5 taking the constant service rateto be 4500 cells per time unit for each switch. As thesimulation results indicate (Fig. 15), the algorithmconverges to the max-min fair allocation rather fastwhile stabilizing the queue around the desired setpoint, Q = 700 at SW3.

ConclusionsIn this article, we have presented a control-theoretic ap-proach to designing ABR congestion control algorithms. ATMnetworks deal with different types of traffic. Among the sev-eral services offered by ATM, the ABR service plays a centralrole in regulating the network traffic, as it is the only servicecategory that uses explicit feedback from the network. Wehave presented several algorithms for ABR congestion con-trol and have shown that they perform well under various cri-teria such as basic fairness and max-min fairness whileachieving high bandwidth utilization. In addition, excessivecell loss is avoided by controlling the queue length. In our de-signs, the network delay is explicitly taken into account,which we believe is useful in modeling network behavior over

links with high delays, such as satellite ATM networks, or IPover ATM. Furthermore, we have shown that one of our algo-rithms is robust to uncertainty in round-trip delays, and atthe same time it can adapt to the variations in the number ofsources sharing a switch. These properties make this algo-rithm easier to apply in volatile networks, where both the net-work delay and the number of active sessions are not knownor cannot be predicted accurately beforehand. In summary,we strongly believe that control theoretic design tools can beeffectively used in designing high-performance algorithms inthe context of ATM ABR congestion control.

AcknowledgmentThe research reported here was supported by NSF GrantNSF ANI 98-13710, NSF CAREER Award NCR 9701525, AFOSRMURI Grant AF DC 5-36128, and a grant from Rome Labs (AirForce Contract F30602-96C0156) through Scientific SystemsCompany, Inc.

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[5] S. Keshav, “A control-theoretic approach to flow control,” in Proc.SIGCOMM’91, Zurich, Switzerland, Sept. 1991, pp. 3-15.

[6] E. Altman, T. Bas,ar, and R. Srikant, “Robust rate control for ABR sources,”in Proc. IEEE INFOCOM, San Francisco, CA, Mar. 1998.

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[8] S. Kalyanaraman, R. Jain, S. Fahmy, R. Goyal, and B. Vandalore, “The ERICAswitch algorithm for ABR traffic management in ATM networks,” IEEE/ACMTrans. Networking, vol. 8, no. 1, pp. 87-98, Feb. 2000.

[9] O. Ait-Hellal, E. Altman, and T. Bas,ar, “Rate-based flow control with band -width information,” Eur. Trans. Telecommunications, vol. 8, no. 1, pp. 55-65,1997.

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[11] L. Benmohamed and Y. T. Wang, “A control-theoretic ABR explicit rate al -gorithm for ATM switches with per-VC queueing,” in Proc. IEEE INFOCOM, SanFrancisco, CA, Mar. 1998, pp. 183-190.

[12] A. Kolarov and G. Ramamurthy, “A control-theoretic approach to the de -sign of an explicit rate controller for ABR service,” IEEE Trans. Networking,vol. 7, pp. 741-753, Oct. 1999.

[13] A. Kolarov and G. Ramamurthy, “A control theoretic approach to the de -sign of closed loop rate based flow control for high speed ATM networks,” inProc. IEEE INFOCOM, 1997.

[14] S. Mascolo, D. Cavendish, and M. Gerla, “ATM rate based congestion con -trol using a Smith predictor: An EPRCA implementation,” in Proc. IEEEINFOCOM, San Francisco, CA, Mar. 1996, pp. 569-576.


[15] D.P. Bertsekas and R. Gallager, Data Networks. Englewood Cliffs, NJ:Prentice-Hall, 1987.

[16] E. Altman and T. Bas,ar, “Optimal rate control for high speed telecommu -nication networks,” in Proc. 34th IEEE Conf. Decision and Control, New Or-leans, LA, Dec. 1995, pp. 1389-1394.

[17] B.D.O. Anderson and J.B. Moore, Optimal Control: Linear QuadraticMethods. Englewood Cliffs, NJ: Prentice-Hall, 1990.

[18] O.C. Imer and T. Basar, “Optimal solution to a team problem with informa -tion delays: An application in flow control for communication networks,” in Proc.38th IEEE Conf. Decision and Control, Phoenix, AZ, Dec. 1999, pp. 2697-2702.

[19] C. Fulton, S.-Q. Li, and C.S. Lim, “UT: ABR feedback control with tracking,”in Proc. IEEE INFOCOM, 1997, pp. 806-815.

[20] O.C. Imer, T. Bas,ar, and R. Srikant, “A robust adaptive control algorithmfor ABR service in ATM networks,” in Proc. IEEE ICCCN, Las Vegas, NV, Oct.2000, pp. 48-53.

[21] M.W. Garrett. “Contributions toward real-time services on packetswitched networks,” Ph.D. dissertation, Columbia Univ., New York, May 1993.

[22] O. Rose, “Statistical properties of MPEG video traffic and their impact ontraffic modeling in ATM systems,” in Proc. 20th Annual Conf. Local ComputerNetworks, Minneapolis, MN, 1995, pp. 397-406.

[23] “MPEG-1 frame size traces, ” [Online]. Available FTP:ftp-info3.informatik.uni-wuerzburg.de, /pub/MPEG/

[24] “Video traces,”1992, Bell Communications Research, Inc. [Online]. Avail -able FTP: thumper.bellcore.com, pub/vbr.video.trace

[25] R. Jain, “Congestion control and traffic management in ATM networks:Recent advances and a survey,” Computer Networks ISDN Syst. (Netherlands),vol. 28, no. 13, pp. 1723-38, Oct. 1996.

[26] P.J. Brockwell and R.A. Davis, Times Series: Theory and Methods. NewYork: Springer-Verlag, 1991.

[27] S. Fahmy, R. Jain, S. Kalyanaraman, R. Goyal, and B. Vandalore, “On deter-mining the fair bandwidth share for ABR connections in ATM networks,” inProc. IEEE Int. Conf. Communications (ICC) 1998, June 1998, pp. 1435-1451.

Orhan Cagri Imer received his B.S. degree from MiddleEast Technical University, Ankara, Turkey, in 1997 and hisM.S. degree from the University of Illinois at Urbana-Champaign in 2000, both in electrical engineering. Cur-rently, he is a Research Assistant at the Coordinated Sci-ence Laboratory at the University of Illinois. His researchinterests include control and discrete-event simulation ofcommunication networks and market-based resource allo-cation in agent systems.

Sonia Compans received the Baccalaureate fromAcademie de Toulouse, France, in 1992, the B.S. degree from

the National Polytechnic Institute of Lorraine, EcoleNationale Superieure d'Electricitè et de Mecanique, France,in 1997, and the M.S. degree in electrical and computer engi-neering from the University of Illinois at Urbana-Champaign,in 1998. Since then she has been with Alcatel in France,working in the telecommunications field.

Tamer Bas,ar received the B.S.E.E. degree from Robert Col-lege, Istanbul, in 1969, and the M.S., M.Phil, and Ph.D. degreesin engineering and applied science from Yale University, in1970, 1971, and 1972, respectively. Since 1981, he has beenwith the University of Illinois at Urbana-Champaign, where heis currently the Fredric G. and Elizabeth H. Nearing Professorof Electrical and Computer Engineering and Research Profes-sor at the Coordinated Science Laboratory. His current re-search interests are robust nonlinear and adaptive control,control of and over communication networks, risk-sensitiveestimation and control, and robust identification. He is amember of the National Academy of Engineering, a Fellow ofthe IEEE, Past President of the IEEE Control Systems Society,Deputy Editor-in-Chief of Automatica, and a member of edito-rial and advisory boards of a number of other internationaljournals. In 1993, he received the Medal of Science of Turkeyand the Distinguished Member Award of the IEEE Control Sys-tems Society, and in 1995 the Axelby Outstanding PaperAward of the same society.

R. Srikant received the B.Tech. degree from the Indian Insti-tute of Technology, Madras, India, in 1985 and the M.S. andPh.D. degrees from the University of Illinois in 1988 and1991, respectively, all in electrical engineering. He was aMember of Technical Staff in AT&T Bell Laboratories from1991 to 1995. Since August 1995, he has been with the Uni-versity of Illinois where he is currently an Associate Profes-sor in the Department of General Engineering and theCoordinated Science Laboratory and an Affiliate in the De-partment of Electrical and Computer Engineering. His re-search interests include communication networks,queueing theory, and stochastic control. He is currently anAssociate Editor of Automatica. He received an NSF CAREERaward in 1997.


developing explicit rate control...

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