IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006 1313

A Two-Time-Scale Design for Edge-Based Detectionand Rectification of Uncooperative Flows

Xingzhe Fan, Kartikeya Chandrayana, Murat Arcak, Senior Member, IEEE,Shivkumar Kalyanaraman, Member, IEEE, and John Ting-Yung Wen, Fellow, IEEE

Abstract—Existing Internet protocols rely on cooperative be-havior of end users. We present a control-theoretic algorithm tocounteract uncooperative users which change their congestioncontrol schemes to gain larger bandwidth. This algorithm rectifiesuncooperative users; that is, forces them to comply with theirfair share, by adjusting the prices fed back to them. It is to beimplemented at the edge of the network (e.g., by ISPs), and canbe used with any congestion notification policy deployed by thenetwork. Our design achieves a separation of time-scales betweenthe network congestion feedback loop and the price-adjustmentloop, thus recovering the fair allocation of bandwidth upon a fasttransient phase.

Index Terms—Network congestion control, singular-perturba-tions, uncooperative flow control.

I. INTRODUCTION

I N A NETWORK which does not differentiate among users,the equilibrium rate for any user is primarily decided by the

congestion control being used [1]. With new software advance-ments, however, “uncooperative” users can change their con-gestion control schemes to gain more than their fair share ofbandwidth, at the cost of cooperative users. This uncooperativebehavior can lead to TCP unfriendliness, congestion collapse[3], [4] and, to a traffic-based denial-of-service to cooperativeusers [5], [6]. Detecting uncooperative users, and “rectifying”their flow rates to comply with cooperative rates, and thus, im-proving quality of service for individual flows becomes an im-portant emerging problem in network management.

Among rectification mechanisms proposed in the literature,the majority are “router-based” that is, they modify the routeralgorithm to detect and limit uncooperative flows, e.g., Active

Manuscript received November 2, 2004; revised September 10, 2005;approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor E. Knightly.This work was supported in part by the RPI Office of Research through anExploratory Seed Grant, and in part by the NSFC Two-Bases Project (No.60440420130), and the Outstanding Overseas Chinese Scholars Fund ofChinese Academy of Sciences (No. 2005-1-11), China.

X. Fan is with the Electrical and Computer Engineering Department, Univer-sity of Miami, Miami, FL 33146-2509 USA (e-mail: fan.rpi@gmail.com).

K. Chandrayana is with Cisco Systems Inc., San Jose, CA 95134 (e-mail:karchand@cisco.com).

M. Arcak is with the Electrical, Computer and Systems Engineering Depart-ment, Jonsson Engineering Center, Rensselaer Polytechnic Institute, Troy, NewYork 12180-3590 USA (e-mail: arcakm@rpi.edu).

S. Kalyanaraman is with the Rensselaer Polytechnic Institute, Troy, New York12180-3590 USA (e-mail: shivkuma@ecse.rpi.edu).

J. T.-Y. Wen is with the Department of Electrical, Computer, and SystemsEngineering and the Department of Mechanical, Aerospace, and Nuclear Engi-neering, Center for Automation Technologies and Systems (CATS), RensselaerPolytechnic Institute, Troy, NY 12180-3590 USA (e-mail: wenj@rpi.edu).

Digital Object Identifier 10.1109/TNET.2006.886351

Queue Management (AQM) schemes or scheduling disciplines.In [3] and [4], the authors study AQM schemes, and investi-gate the effect of uncooperative flows on network throughputand loss rates. Flow Random Early Drop (FRED), a modifiedRED scheme, is proposed in [7] to detect uncooperative users,and to limit their rates by increasing their packet drop probabil-ities. In [8], the authors combine the BLUE queue managementalgorithm with a Bloom filter to detect and rate-limit uncoop-erative flows. Several other rate-based schemes are surveyed in[9]. Scheduling schemes, such as ack-spacing, have been sug-gested to manage uncooperative flows in [10].

More recently, edge-based price-adjustment mechanismshave been proposed in [11] and [12], which manage uncooper-ative flows only at edge routers. A significant advantage of thisapproach is that it does not require core network upgrades andcan be implemented without performing per flow managementat routers. By estimating each flow’s incoming rate and usingit to label flow’s packet, the Core-Stateless Fair Queueing(CSFQ) algorithm in [11] computes the forwarding probabilityfrom link fair rate estimation. However, this design only appliesto network in which all nodes implement Fair Queueing. In[12], the authors manage uncooperative flows by mappingtheir utility function to a specified target network behaviorat the edge. This study, however needs to estimate the utilityfunction to achieve this edge-based price adjustment, and isthus restricted to a specific form of TCP.

In this paper, we develop an edge-based price-adjustment al-gorithm using tools from control theory. Rather than address aspecific protocol, we develop our design within the optimizationframework of Kelly [1], [2], [13]–[15], which is applicable to di-verse types of networks, and encompasses numerous protocolssuch as TCP Reno, TCP Vegas, FAST [16], [17] etc. Our algo-rithm recovers the cooperative share of bandwidth prescribed inKelly’s framework, with a new feedback loop implemented atthe edge router, and, hence, referred to as the “edge supervisor”.It detects uncooperative users by comparing their sending rateswith “audit” rates calculated according to an ideal, cooperative,model, and increases their price feedback. Although in this de-sign edge supervisor does perform per flow management by thisprice adjustment loop, core routers, which are in general morecomplex than edge routers, do not perform per flow manage-ment, and therefore the implementation complexity is signifi-cantly reduced. Our algorithm is independent of congestion no-tification policy deployed by the network, and thus, can be usedwith any Active Queue Management scheme, as well as DropTail queueing.

We design the price adjustment loop to evolve in a fastertime-scale than the existing price feedback loop from the links,

1063-6692/$20.00 © 2006 IEEE

1314 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006

because, then, uncooperative flows are rectified during a fasttransient phase, after which stability and convergence proper-ties of the desired cooperative network model is recovered. In-deed, using tools from singular perturbations theory [18], [19,Chapter 11], we prove that the fast and slow feedback loops,when combined, ensure convergence of the sending rates totheir cooperative values. The type of convergence establishedis “semi-global” [19], which means that any desired region ofattraction can be achieved by increasing the feedback gain ofthe price-adjustment loop.

The paper is organized as follows: Section II overviewsKelly’s primal and dual flow control algorithms. Section IIIstudies the primal algorithm and presents our price adjustmentdesign for uncooperative users. Section IV extends this designto the dual algorithm. In Section V, we implement our priceadjustment algorithms in ns-2 and evaluate their performancefor various single and multi-bottleneck topologies, for bothmarking and dropping congestion notification policies, and withand without AQM schemes. In particular, we show that givena standard network behavior like TCP-Friendliness, our algo-rithm forces uncooperative users to comply with their fair-shareof the bandwidth. Conclusions are given in Section VI. In thepaper, some proofs are omitted due to space limitations.

II. OVERVIEW OF KELLY’S PRIMAL AND DUAL

FLOW CONTROL ALGORITHMS

In Kelly’s framework [1], network flows are modeled as theinterconnection of users and communication links as shown inFig. 1. Packets from each users (with sending rate ) are routedthrough the links with the aggregate link rate

(1)

where is the forward routing matrix. Each link has a fixedcapacity , and based on its congestion and queue size, a linkprice, is computed:

(2)

The link price information is then sent back to each source withthe aggregate source price

(3)

where , since the links only feed back price informa-tion to the users that utilize them.

Kelly formulated the flow control task as a static optimiza-tion, and dynamic stabilization problem. The static optimiza-tion problem computes the desired equilibrium by maximizingthe sum of the source utility functions , while complyingwith capacity constraints in the links:

(4)

The dynamic problem is to design the source rate update lawbased on the aggregate price, and the link price update law basedon the aggregate rate, to guarantee stability of the equilibrium.For this problem, Kelly introduced two dynamic algorithms:

Fig. 1. Network flow control model.

The Primal Algorithm consists of a first order source update law,and a static penalty function for the link to keep the aggregaterate below its capacity:

(5)

The penalty functions are designed to enforce the linkcapacity constraints , , i.e., to keep theaggregate rate below its capacity .

The Dual Algorithm consists of a static source update and afirst order dynamic price update:

(6)

where the positive projection for a general function isdefined as

if , or andif and .

From (6), the unique equilibrium for the dual control law is ob-tained from the equations

(7)ifif

(8)

which, as shown in [1], correspond to the solution of the opti-mization problem (4), in which ’s play the role of Lagrangemultipliers for the capacity constraints. For the primal controllaw (5), the equilibrium obtained from

(9)

(10)

approximates the optimality condition (7)–(8) with the helpof the penalty functions . The stability of these twoalgorithms and their extensions has been established in [2],[13]–[15], [17], [20]–[22], and [24].

III. UNCOOPERATIVE USERS IN KELLY’S PRIMAL ALGORITHM

We now assume that some users, which we call “uncooper-ative”, use more aggressive utility functions to increase their

FAN et al.: A TWO-TIME-SCALE DESIGN FOR EDGE-BASED DETECTION AND RECTIFICATION OF UNCOOPERATIVE FLOWS 1315

share of bandwidth; that is, instead of in (5), they im-plement :

(11)

To rectify these uncooperative users, we propose that the super-visor at the edge of the network (e.g., Internet service providers)adjusts the price feedback from its nominal value to . Anideal design of would be

(12)

which replaces in (11) with the cooperative .However, this design is not implementable because isnot known to the supervisor. Instead, in our design, we obtainan estimate of with the help of the cooperative referencemodel:

(13)

The thus calculated differs from by , which,from (11)–(13), is governed by

(14)

This means that, if we design the price adjustment to be

(15)

with a sufficiently high gain , then the variable evolvesin a faster time scale than , and reaches the quasi-steady state

. Thus, after a fast transient, our design(13), (15) approximates the non-implementable scheme (12).For cooperative users, where , (13) and (15)yield , which means that no price adjustment is applied.Note that in (13) is not necessarily the same for each user.This means that the supervisor can intentionally set up differentutility functions, and use this flexibility to only adjust high band-width flows while leaving low bandwidth flows without rectifi-cation.

The algorithm (13), (15) is depicted with a block diagram inFig. 2. In implementation, the edge router performs the priceadjustment (15) every round-trip time. The phrase “fast timescale” is used to indicate that when is large in (15), the su-pervisor subsystem approaches its quasi-steady-state faster thanthe dynamics of the network. In Theorem 1 below, we use toolsfrom singular-perturbations theory [18], [19] to prove that (13),(15) achieves asymptotic stability of the cooperative value in(9)–(10).

Theorem 1: Consider the network (1)–(3), where some usersimplement the uncooperative algorithm (11), rather than (5).Suppose are increasing and sufficientlysmooth functions, , andand as for . Then, theprice adjustment algorithm (13), (15) ensures that, for any com-pact set of initial conditions , there existssuch that, if , then and remain bounded, and

converges to the cooperative value in (9)–(10).The assumptions of Theorem 1 on the utility functions

are standard in the literature [1], [14], [25]. In particular, the

Fig. 2. Price adjustment for uncooperative users in Kelly’s primal algorithm.

assumption as ensures that ispositively-invariant, i.e., if is initially in , it will remainin for all . It is satisfied by commonly used utilityfunctions such as (variant of TCP Reno)and (TCP Vegas) [14]. For others, suchas (TCP Reno), we canmodify Theorem 1 and prove stability by using positive projec-tion functions as in [15]. It is reasonable to make the same as-sumptions for as for , because cheating users wouldtypically change the parameters of the nominal utility functions,such as in TCP Vegas above. However, this assumption ex-cludes some traditional unresponsive flows referred to as UDPor CBR, in which, users send data at a constant rate without ac-knowledging any feedback from the network.

Proof: To represent the algorithm (11), (13), and (15) inthe standard singularly perturbed form [18], [19], we let

(16)

(17)

and obtain(18)

(19)

An inspection of (18) and (19) shows that the equilibrium foris same as the cooperative in (9)–(10), and the equilibriumfor is

(20)

To shift this equilibrium to 0, we define

(21)

and rewrite (18)–(19) as

(22)

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where we use the vector notation ,. and are diag-

onal matrixes of the source controller gains and ,, and is a vector whose th com-

ponent is the derivative of the utility function .Likewise, and consist of the penaltyfunctions and uncooperative utility functions .

To prove asymptotic stability of we use theLyapunov function

(23)

in which, the first and the second terms, are identical to the Lya-punov function used in [1], [15] for the proof of the stabilityof Kelly’s Primal algorithm, while the third term is a quadraticLyapunov function for the dynamics of subsystem. This Lya-punov function is positive definite and radially unbounded in

, and yields the derivative

(24)

where

(25)

(26)

We show in Lemma 1 below that, on any compact set ofthat includes , we can choose small enough to

ensure is negative definite. The conclusion of Theorem 1 fol-lows from this lemma because, from , we have

and, thus , whichmeans that for any set as in the statement of the theorem, wecan find a corresponding region of attraction in coordi-nates, which does not depend on . Since is also independentof , we can select a level set of that encompasses this regionof attraction, and design from Lemma 1 to render negativedefinite in this level set.

Lemma 1: Let the assumptions of Theorem 1 hold, and letand be defined as in (25)–(26). Then, for any com-

pact set of that includes , there existssuch that if for all , then givenin (24) is negative definite on .

In Theorem 1, we require that the edge supervisor setequal to . However, it is not difficult to show that the proofholds true for small errors between and . Time delaysin the network are not considered in Theorem 1. We wish toemphasize, however, that the high-gain component of our feed-back design is limited to the local price adjustment loop, and notthe price feedback loop, which is subject to the network delays.

Fig. 3. Price adjustment for uncooperative users in Kelly’s dual algorithm.

Thus, our correction algorithm can be combined with conges-tion control algorithms that are robust to time delays, such thosein [23]–[25].

In implementation, it may also be necessary to know howlarge the gain must be selected. While, in principle, such avalue can be obtained from the calculation of in the proof,this value may be conservative, and depends on the class ofutility functions employed by uncooperative users. A morepractical value can be obtained by monitoring whether the un-cooperative rates persists and by increasing the gain accord-ingly. A further discussion on the choice of this gain is given inSection VI-C.

IV. PRICE ADJUSTMENT FOR KELLY’S DUAL ALGORITHM

We next study Kelly’s dual algorithm where uncooperativeusers implement, instead of (6)

(27)

We assume , , which means that theuncooperative sending rate is larger than the cooperative rate.To counteract such uncooperative users, the supervisor must re-place the nominal price feedback with

(28)

which, when substituted in (27), results in the cooperative rate(6). Because a direct solution of (28) would require the knowl-edge of , which is not available to the supervisor, we pro-pose the dynamic algorithm

(29)

(30)

depicted in Fig. 3. The equilibrium of (30) is achieved when

(31)

which indeed coincides with the cooperative rate (6). Weachieve asymptotic stability of this equilibrium, again, bydesigning the adaptation gain to be sufficiently high:

Theorem 2: Consider the network (1)–(3), (6) and(27), where and are as in Theorem 1, and

FAN et al.: A TWO-TIME-SCALE DESIGN FOR EDGE-BASED DETECTION AND RECTIFICATION OF UNCOOPERATIVE FLOWS 1317

Fig. 4. Topologies used in simulations. (a) Single-bottleneck. (b) Multi-bottleneck.

, . Then, the price adjustmentalgorithm (27), (30), ensures that, for any compact setof initial conditions , there exists such that, if

, then , and remain bounded, and andconverge to the cooperative values and in (7)–(8).

V. IMPLEMENTATION AND SIMULATION SETUP

We have implemented the uncooperative framework pre-sented in this paper in the Network Simulator (ns-2). While wehave studied both dynamic (Section III) and static (Section IV)users, in simulations we implement the method of Section IIIbecause of the prevalence of TCP, which is dynamic and canbe modeled as in (11) (see [1]). We added an edge-based su-pervisor, which adjusts the price feedback to the uncooperativeusers with the following algorithms.

1) Let be the rate of uncooperative user as in (11).2) Define an “audit” rate, , which represents the cooperative

rate computed as in (13).3) Calculate the marking or dropping probability at the edge

as , where , and isthe gain in (15).

4) For each incoming packet mark/drop a packet with a prob-ability .

The implementation of this feedback adjustment dependsupon the congestion notification policy deployed in the net-work. In our simulations we present the results with scenarioswhere marking (ECN) and dropping are used as congestionnotification policies. The framework presented in this paper isindependent of the buffer management policy deployed in thenetwork; that is, it works with any Active Queue Managementscheme as well as with simple Drop Tail queueing.

We note that, unlike the static link assumption in Section III,AQM and Drop-Tail in simulations make use of queue lengthand, hence, are dynamic algorithms. An extension of the proofof dynamic-source dynamic-link algorithms would be possible,but lengthy. The stability properties observed in simulations inthe next section are indeed consistent with those predicted byTheorem 1.

We present simulation results for both single and multi-bot-tleneck topologies, depicted in Fig. 4(a) and (b). All the accesslinks are configured to have a capacity equal to four times thatof bottleneck links. The bottleneck links capacity and delay isfixed at 0.8 Mb/s and 20 ms, respectively, unless specificallystated. (However, for some results presented in the paper, wewill relax these configuration settings.) For all simulations re-ported in this paper, the simulation time is 150 seconds, and

rate (or throughput) measurements are taken every 0.5 seconds.Each router has a buffer equal to one bandwidth delay product.In setups where the bottleneck routers have Random Early Drop(RED) buffer management policy deployed, the correspondingmaximum and minimum threshold are set at andwhere is the total buffer length; the queue weight was set to0.002 and the maximum dropping probability to 0.1.

The multi-bottleneck topology in Fig. 4(b) shows one flowbetween source S1 and destination D1. This flow traverses boththe bottleneck links and henceforth in this paper we will callthis a long flow. The two flows between the source destinationpairs [S2-D2] and [S3-D3] go over only one bottleneck link and,therefore, in the rest of paper we refer to them as short flows.

We refer to flows, which under same operating conditions, getmore rate than TCP as selfish flows. This definition is also oftencommonly referred to as TCP-Friendliness. Since almost 90%of the traffic carried on the Internet uses TCP, we chose TCP-Friendliness as our definition of conformant flows. In this paperall transport protocols are rate based. Thus, all TCP-Friendlyschemes use equation based rate control scheme (TCP FriendlyRate Control—TFRC) presented in [27] and all selfish schemesare variants of TFRC which have conservative decrease algo-rithms, i.e., upon congestion they decrease more slowly thanTCP. We would like to refer the reader to [12] for ways togenerate selfish flows. In this paper we have also assumed thatall the flows are persistent flows, i.e., they have infinite data totransfer. However, we will also present the results for the sce-narios where we have both persistent and short web traffic com-peting for bandwidth.

VI. SIMULATION RESULTS

In this section we evaluate our algorithm for both single andmulti-bottleneck topologies, with various degree of flow multi-plexing. We also test its robustness in the presence of mice likeweb traffic and reverse path congestion.

A. AQM-Based Network

In this section we assume that Active Queue Management(AQM) policies are deployed on the bottleneck routers, and allrouters have RED queues installed on them. Fig. 5 shows theresults where two flows compete for bandwidth in a single bot-tleneck scenario. Among these flows, the first is TCP-Friendlywith , while the other is uncooperative withthe utility function . If both competing flowswere TCP-Friendly, they would have shared the bandwidth eq-uitably. However, Fig. 5(a) shows that the uncooperative flow

1318 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006

Fig. 5. Single-bottleneck topology: equitable sharing of bandwidth enforced by the edge supervisor. (a) Without rectification. (b) With rectification.

Fig. 6. Multi-bottleneck scenario where (a) shows the ideal bandwidth sharing, (b) shows the aggravated unfair sharing in the presence of uncooperative flows,and (c) shows the rectification of uncooperative flows with our edge supervisor.

Fig. 7. Background web traffic: (a) ideal bandwidth sharing scenario; (b) aggravated unfair sharing in the presence of uncooperative flows; and (c) rectificationof uncooperative flows with our edge-based supervisor.

grabs a larger share of the bandwidth. With our edge-based al-gorithm, in Fig. 5(b) the two flows share the bottleneck band-width equitably. Simulations with other uncooperative utilityfunctions in the single-bottleneck scenario, not presented here,yield similar results. The multi-bottleneck topology is shownin Fig. 4(b) and has two bottleneck links. The TCP-Friendlylong-flow, which goes over both bottleneck links, competes forbandwidth against the two uncooperative short-flows, which goover only one bottleneck link. Fig. 6(a) shows the result corre-sponding to the cooperative setup where both the long and shortflows use TCP-Friendly rate control scheme. Fig. 6(b) showsthe result for the setup where we replace the TCP-Friendly shortflows with uncooperative rate control schemes with utility func-tion . Once again, we see that, the uncoopera-tive flows get an unfair share of the bandwidth and almost forcea traffic volume based denial of service attack. When we em-ploy our edge-based supervisor, with , we re-cover the ideal bandwidth sharing of bottleneck links, as shownin Fig. 6(a).

1) Background Traffic: In this section we will evaluate ourframework in the presence of short web flows on a multi-bot-tleneck topology. In this setup, there is one long TCP-Friendlyflow, which competes for bandwidth against one short uncoop-erative flow, on each bottleneck. HTTP sources are now addedto these persistent flows. Each http page sends a single packetrequest to the destination, which then replies with a file of sizewhich is exponentially distributed with 12 1-kb packets. After asource completes this transfer it waits for a random time, whichis exponentially distributed with a mean of 1 second and thenrepeats the process.

Our results in Fig. 7 show that these web traffic or mice flowstake some fraction of the bottleneck bandwidth while the per-sistent TCP and uncooperative flows compete for the remainingbandwidth. Fig. 7(a) shows the results where, in the presenceof mice traffic, all the persistent flows used a TCP-Friendlyrate control scheme. This corresponds to the ideal bandwidth-sharing scenario. The results shown in Fig. 7(b) correspondsto the setup where the persistent short flows, i.e., flows which

FAN et al.: A TWO-TIME-SCALE DESIGN FOR EDGE-BASED DETECTION AND RECTIFICATION OF UNCOOPERATIVE FLOWS 1319

Fig. 8. Reverse path congestion: (a) ideal bandwidth sharing scenario; (b) aggravated unfair sharing in the presence of uncooperative flows; and (c) rectificationof uncooperative flows with our edge-based supervisor.

Fig. 9. Higher flow multiplexing with background traffic and reverse path congestion in a multi-bottleneck setup. (a) Ideal sharing. (b) Without rectification.(c) With rectification.

go over only one bottleneck, use a selfish rate control scheme.We see that the selfish flows take an unfair share of the bottle-neck bandwidth. Thereupon, we introduce our edge-based su-pervisor, with , to rectify the misbehaving persistentshort selfish flows, and recover the equitable rates as in Fig. 7(c).

2) Cross Traffic: We next evaluate our framework in thepresence of reverse path congestion. For this case we used amulti-bottleneck topology with one persistent TCP-Friendlyflow traversing two bottleneck links competing against selfishflows going over just one bottleneck. For creating reverse pathcongestion, we added short TCP-Friendly flows on the reversepaths. Fig. 8(a) shows the results corresponding to the idealcase, i.e., all the flows used TCP-Friendly rate control schemeswhile Fig. 8(b) shows the results when the short flows on theforward path used an uncooperative rate control scheme. Wethen added our edge based rectification agent and the corre-sponding results are plotted in Fig. 8(c) (with ).We see that in the presence of edge-based supervisor, the selfishflows are effectively managed.

3) Higher Flow Multiplexing With Background Traffic andReverse Path Congestion: In the previous sections we have de-tailed the performance of our edge-based rectification algorithmfor various single and multi-bottleneck scenarios. However, topresent the efficiency and the robustness of our scheme we lim-ited the number of competing flows to 3. In this section we willincrease the number of competing flows.

For the results presented in this section, the TFRC flows com-peted for bandwidth against selfish flows on a multi-bottlenecktopology [shown in Fig. 4(b)]. However, we increase the ca-pacity of the bottleneck links to 8 Mb/s and that of access linksto 80 Mb/s. The links delays were not changed and the bottle-

neck buffer was set to one bandwidth delay product. Finally, allthe routers have RED queue management schemes deployed onthem. The RED queue configurations are detailed in Section V.For the edge-based rectification agent we set the as .

Fig. 9 shows the results for the scenario where five TFRCflows compete for bandwidth against selfish flows. Further, oneach bottleneck there were five selfish short flows. To these per-sistent flows, we also added short web transfers which occu-pied 10% of the bottleneck bandwidth. (The details about howwe generated these flows is provided in Section VI-A-1. On av-erage, at any given time, there were 10 short flows on each bot-tleneck.) For this simulation we also setup flows on the reversepath. Specifically, on each bottleneck in the reverse path therewere five flows competing for bandwidth and thus creating con-gestion on the reverse path. For the results presented in this sec-tion we plot the throughput of one flow from each group: TFRCLong flows or flows which go over two bottleneck links, selfishshort flows from Group 1 or flows which go over the first bot-tleneck only; and, finally, the selfish short flows from Group 2or flows which go over the last bottleneck only.

Fig. 9(a) shows the ideal sharing of the bottleneck when theshort flows are also TCP-Friendly. Fig. 9(b) shows that in theabsence of any policing the uncooperative flows get more shareof the bandwidth at the expense of TFRC flows. With our recti-fication algorithm the fair share of the TFRC flows is restored;see Fig. 9(c).

4) ECN Enabled Network: We conclude this section withthe results from a multi-bottleneck test case where the REDscheme is configured to mark the packets (instead of droppingthem). Once again, we used a setup where one persistent TCP-Friendly flow competed for bandwidth against one short unco-

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Fig. 10. ECN Enabled Network: (a) ideal bandwidth sharing scenario; (b) aggravated unfair sharing in the presence of uncooperative flows; and (c) rectificationof uncooperative flows with our edge supervisor.

Fig. 11. Drop tail queues and single bottleneck topology: rectification of uncooperative flows. (a) Without rectification. (b) With rectification.

Fig. 12. Drop tail queues: (a) ideal bandwidth sharing scenario; (b) aggravated unfair sharing in the presence of uncooperative flows; and (c) rectification ofuncooperative flows with our edge supervisor.

operative flow. Fig. 10(a) shows the ideal bandwidth sharingwhile Fig. 10(b) shows that in the presence of uncooperativeflows the resulting bandwidth sharing is unfair. When we intro-duced the edge-based supervisor, with , the bandwidthis shared fairly.

B. Drop-Tail Queues

In this section, we look at the alternative scenario, where weassume that the network operates with Drop-Tail queues only.We extensively tested with various single and multi-bottlenecktopologies and with different kind of uncooperative schemes.Fig. 11(a) shows the results with a single bottleneck topologywhere one uncooperative and TCP-Friendly flows compete forbandwidth. As the figure shows, the uncooperative flows gainan unfair share of the bottleneck bandwidth, which are thencorrected with our design. For this simulation the was set to

.

Fig. 12 shows the results with a multi-bottleneck setup whereone long TCP-Friendly flow is sharing bandwidth with short un-cooperative flows. Fig. 12(a) shows the ideal bandwidth sharingand Fig. 12(b) shows the unfair sharing of the bottleneck inthe presence of uncooperative flows. Finally, in Fig. 12(c) wepresent results with our rectification agent when .

C. Effect of Gain on Rectification of Selfish Users

The performance of our edge-based rectification algorithmdepends on the gain in (15). As detailed below, simulationstudies indicate that too small or too large values of this maydeteriorate the performance. Indeed, Theorem 1 disallows smallvalues of because, otherwise, the desired two-time-scale be-havior is not achieved. Although Theorem 1 allows arbitrarilylarge values for , in practice, such high-gain leads to satura-tion of dropping or marking schemes, which violate the “small

FAN et al.: A TWO-TIME-SCALE DESIGN FOR EDGE-BASED DETECTION AND RECTIFICATION OF UNCOOPERATIVE FLOWS 1321

Fig. 13. Effect of gain � on the steady rates of uncooperative and TFRC flows. (a) � = 10 . (b) � = 2:5� 10 . (c) � = 10 .

marking probability” approximation in (3). We see in the fol-lowing simulations that large might result in “over-penaliza-tion”, which means that uncooperative flows receive even lessthan their fair share.

Consider the multi-bottleneck setup shown in Fig. 4(b)with one long TFRC flow and one short uncooperative flowon each bottleneck. In Fig. 6 we presented simulations with

. In Fig. 13 we compare this result with[Fig. 13(a)] and with [Fig. 13(c)].

We note that a high value of may result in over-penalization,and on the other hand, with a very small value of the selfishusers are not sufficiently penalized and they continue to get moreshare of the bottleneck link(s) at the expense of cooperativeusers. However, for intermediate values, such asin Fig. 13(b), we recover the ideal shares for the uncooperativeand the cooperative users.

For all the results reported in this paper we have found that theideal range of lies between the interval to . We alsoextensively evaluated the edge-based rectification model for dif-ferent value of selfishness, i.e., users chose different values of

, and found observation on consistent with those reportedabove. A judicious choice of this gain, however, deserves furtherinvestigation.

VII. CONCLUSION

We have presented a price adjustment algorithm for bothKelly’s primal and dual network flow control models, andtested it on the Network Simulator. This algorithm is to beimplemented at the edge of the network, and thus, does notrequire costly hardware upgrades in the entire network. It isindependent of congestion notification policy deployed by thenetwork, and thus, can be used with any Active Queue Man-agement scheme, as well as Drop Tail queueing, which makethe algorithm not only of theoretical interest but also practicalimportance. Although a suitable range for the gain in ouralgorithm was determined by simulations, a judicious choice ofthis gain deserves further investigation. An on-line adaptationfor may be possible, and is currently being pursued by theauthors.

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Xingzhe Fan received the B.E. and M.E. degreesfrom Tsinghua University, Beijing, China, andthe Ph.D. degree from the Electrical, Computer,and Systems Engineering Department, RensselaerPolytechnic Institute, Troy, NY, in 1998, 2000, and2004, respectively.

He is currently a Visiting Assistant Professor at theUniversity of Miami, Miami, FL. His research inter-ests are in nonlinear control and distributed optimiza-tion.

Kartikeya Chandrayana received the B.Tech. de-gree in electronics and communication engineeringfrom the Indian Institute of Technology, Roorkee,and the M.S. and Ph.D. degrees in computer andsystems engineering from Rensselaer PolytechnicInstitute, Troy, NY, in 1998, 2001, and 2004, respec-tively.

He is currently a Senior Software Engineer atCisco Systems, San Jose, CA. His research inter-ests include design and control of communicationnetworks including congestion control, traffic engi-

neering, quality of service, managing selfish behavior in the Internet, modelingand performance analysis of network protocols and storage networks.

Dr. Chandrayana is a member of the ACM and Sigma Xi.

Murat Arcak (S’97–M’00–SM’05) was born inIstanbul, Turkey, in 1973. He received the B.S.degree in electrical and electronics engineeringfrom Bogazici University, Istanbul, in 1996, and theM.S. and Ph.D. degrees in electrical and computerengineering from the University of California, SantaBarbara, in 1997 and 2000, under the direction ofPetar Kokotovic.

In 2001, he joined the Rensselaer Polytechnic In-stitute, Troy, NY, as an Assistant Professor of Elec-trical, Computer, and Systems Engineering. His re-

search interests are in nonlinear control theory and applications.Dr. Arcak is a member of SIAM and an Associate Editor on the Conference

Editorial Board of the IEEE Control Systems Society. He was a finalist for theStudent Best Paper Award at the 2000 American Control Conference, and re-ceived a National Science Foundation CAREER Award in 2003.

Shivkumar Kalyanaraman (S’95–A’97–M’03) re-ceived the B.Tech. degree from the Indian Instituteof Technology, Madras, in 1993, and the M.S. andPh.D. degrees in computer and information sciencesfrom The Ohio State University, Columbus, in 1994and 1997, respectively.

He is an Associate Professor at the Departmentof Electrical, Computer and Systems Engineering,Rensselaer Polytechnic Institute, Troy, NY. Hisresearch is in topics such as congestion control ar-chitectures, quality of service, last-mile community

wireless and free-space optical networks, network management, multimedianetworking, and performance analysis. His special interest lies in developingthe interdisciplinary areas overlapping with networking.

Dr. Kalyanaraman has been a member of the ACM since 1993.

John Ting-Yung Wen (S’79–M’81–SM’93–F’01)received the B.Eng. degree from McGill University,Montreal, QC, Canada, in 1979, the M.S. degreefrom the University of Illinois, Urbana, in 1981, andthe Ph.D. degree from Rensselaer Polytechnic Insti-tute, Troy, NY, in 1985, all in electrical engineering.

From 1981 to 1983, he was a System Engineer withFisher Control Company, Marshalltown, IA, wherehe worked on the coordination control of a pulp andpaper plant. From 1985 to 1988, he was a Memberof the Technical Staff at the Jet Propulsion Labora-

tory, Pasadena, CA, where he worked on the modeling and control of large flex-ible structures and multiple-robot coordination. Since 1988, he has been withthe Department of Electrical, Computer, and Systems Engineering with a jointappointment in the Department of Mechanical, Aerospace, and Nuclear Engi-neering at Rensselaer Polytechnic Institute, Troy, NY, where he is currently aProfessor. His current research interests are in the area of precision motion con-trol, distributed control, and modeling and control of mechanical and materialsystems.