Recently it has been shown that the active queue management schemes implemented in the routers of communication networks supporting transmission control protocol (TCP) flows can be modeled as a feedback control system. Based on a delay differential equations model of TCPs congestion-avoidance mode different control schemes have been proposed. Here a robust controller is designed based on the known techniques for H∞ control of systems with time delays

Quet P.-F.

Özbay H.

Quet, P-F.

Özbay, Hitay

Bilkent University Institutional Repository

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004 1031On the Design of AQM Supporting TCP Flows UsingRobust Control TheoryPierre-François Quet and Hitay ÖzbayAbstract—Recently it has been shown that the active queue manage-ment schemes implemented in the routers of communication networks sup-porting transmission control protocol (TCP) flows can be modeled as afeedback control system. Based on a delay differential equations model ofTCPs congestion-avoidance mode different control schemes have been pro-posed. Here a robust controller is designed based on the known techniquesfor control of systems with time delays.Index Terms—Active queue management (AQM), communication net-works, control of uncertain systems with time delays, control.I. INTRODUCTIONRecently several mathematical models of active queue management(AQM) schemes supporting transmission control protocol (TCP) flowsin communication networks have been proposed [1]–[3]. From thesemodels a control theory-based approach can be used to analyze or to de-sign AQM schemes. The authors of [2] have derived a delay differentialequations model of TCPs congestion avoidance mode and further sim-plified this model focusing the design of a proportional–integral con-troller on the low-frequency dynamics, considering the high-frequencydynamics as parasitic. Their controller could guarantee some robust-ness with respect to the network parameters uncertainties. However,if the uncertainties to be tolerated for stability are “relatively” large,the system’s response becomes sluggish. Motivated by their work, wedesign in this note an H1 controller for their original linear system,without neglecting high-frequency dynamics, that ensures robust sta-bility and good performance for a wider range of network parametersuncertainties. As in [2], we assume that the AQM mechanism bringsthe system to the neighborhood of an equilibrium (operating point), sothat we take the same linear model. Large deviations from this oper-ating point (e.g., in the form of TCP time-out and slow-start phases,buffer overflow, empty queue) are ignored.Other control theoretic based design of AQM include [4]–[6] whilethe importance of considering time delays is pointed out in [7]–[9] and ageneral overview of Internet congestion control literature can be foundin [10].II. MATHEMATICAL MODEL OF AN AQM SCHEME SUPPORTING TCPFLOWSWe consider in this note the network configuration consisting of asingle router receivingN TCP flows, we assume that the AQM schemeimplemented at the router marks packets using explicit congestion no-tification (ECN) [11] to inform the TCP sources of impending con-gestion. In the following, we ignore the TCP slow start and time outManuscript received October 23, 2002; revised August 23, 2003. Recom-mended by Associate Editor C. D. Charalambous. This work was supported inpart by the National Science Foundation under Grant ANI-0073725.P.-F. Quet was with the Department of Electrical Engineering, The Ohio StateUniversity, Columbus OH 43210 USA. He is now with the GM R&D Center,Warren, MI 48090-9055 USA (e-mail: quet.1@osu.edu).H. Özbay is with the Department Electrical and Electronics Engineering,Bilkent University, Bilkent, Ankara TR-06800, Turkey, on leave from the De-partment of Electrical and Computer Engineering, The Ohio State University,Columbus OH 43210 USA (e-mail: ozbay.1@osu.edu).Digital Object Identifier 10.1109/TAC.2004.829643Fig. 1. AQM implementation.mechanisms, thus providing a model and analysis during the conges-tion avoidance mode only. In TCP, the congestion window size (W (t))is increased by one every round trip time if no congestion is detected,and is halved upon a congestion detection. This additive-increase mul-tiplicative-decrease behavior of TCP has been modeled in [1] by thefollowing difference equation (case of one TCP flow interacting witha single router):dW (t) =dtR(t) W (t)2dN(t) (1)withR(t) = q(t)=C+Tp where Tp is the propagation delay, q(t) is thequeue length at the router,C is the router’s transmission capacity, thus,q(t)=C is the queuing delay and R(t) is the round trip time delay, anddN(t) is the number of marks the flow suffers. In a network topologyof N homogeneous TCP sources and one router, a model relating theaverage value of these variables and the router’s queue dynamics be-comes [2]_W (t) =1R(t) W (t)2W (t R(t))R(t R(t))p(t R(t)) (2)_q(t) =N(t)R(t)W (t)  C+(3)where p(t) is the probability of packet mark due to the AQM mecha-nism at the router. Here we use the notation [x]+ = x if x  0, and[x]+ = 0 if x < 0.The linearization of (2) and (3) about the operating point is carriedout in [2] and the perturbed variables about the operating point satisfy_W (t) =  NR20C(W (t) + W (t R0)) 1R20C(q(t)  q(t R0)) R0C22N2p(t R0) (4)_q(t) =NR0W (t) 1R0q(t) (5)where the operating point is defined by the solution (R0;W0; p0) ofthe following set of equations:R0 =q0C+ Tp (6)W0 =R0CN(7)p0 =2W 20(8)for a desired equilibrium queue level q0. Then,W (t) = W0+ W (t),and similarly for R(t); p(t); q(t). Clearly, implementation of the con-troller depends on q0, see Fig. 1. In the random early detection (RED)algorithm, q0 is adjusted by setting appropriate parameters to satisfyp0 = LRED(q0  minth) (9)where LRED and minth are the AQM-RED parameters (LRED is theratio of a small change in packet mark probability to a small change inqueue length, and minth is the minimum queue length beyond which0018-9286/04$20.00 © 2004 IEEE1032 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004packet marking is applied linearly), see [1], [2]. Thus, (6)–(9) define theoperating point that is adjusted by the RED parameters. Then, aroundthe operating point, RED can be seen as a linear proportional controllerwith gain LRED.Here, we consider the same linear plant derived in [2]. Note that forthe linearization the time-varying nature of the roundtrip time delay inthe terms “t   R(t)” is ignored and these terms are approximated by“t   R0.” However, the queue length still depends on the round-triptime in the dynamical equation (3).From (4) and (5) we derive the transfer function from p to qq(s)p(s)=  NW 302A(s)e R s1 + A(s)R0se R s(10)whereA(s) =1W0(R0s)2 + (W0 + 1)R0s+ 2:Considering a negative feedback control systemwith the AQMbeingthe controller, the system to be controlled is given byP (s) =NW 302A(s)e R s1 +A(s)R0se R s(11)=NW 302e R sR CNR20s2 + R CN+ 1 R0s+ 2 +R0se R s:(12)Lemma 2.1: The plant P defined in (11) is stable for all positivevalues of R0; C , and N .Proof: The poles of the transfer function A are in the left-halfpart of the complex plane for all values of the parameters W0 and R0positives, thus A is always stable. We also haveA(s)R0se R ss=j!=R0!(2 W0R20!2)2+ (W0 + 1)2(R0!)2< 1for all positive values of the parameters W0 and R0. So according tothe Nyquist stability test the transfer function11 +A(s)R0se R sis stable for all positiveW0 andR0. We can thus conclude that the plantP defined in (11) is stable for all positive values of R0; C , and N .In the following, we design an AQM scheme based on H1 controltechniques that improves the system’s transient while stabilizing theplant, and ensures robustness with respect to uncertainties in the valuesof the system’s parameters.III. MODELING OF THE PARAMETRIC UNCERTAINTIES OF THE PLANTNote that P in (11) can be written asP (s) = P2(s)P1(s)1 + P1(s)(13)withP1(s) = A(s)R0se R s (14)andP2(s) =NW 302R0s: (15)We assume that the parameters have known nominal valuesNn; R0n; Cn;W0n = (R0nCn=Nn) and that we have the followingbounds for their uncertainty:jN  Nnj  N+jR0  R0nj  R+0jC   Cnj  C+:Assuming that (R0n  R+0 ); (Cn C+); (Nn  N+) are pos-itives, we also havejW0  W0nj  W+0whereW+0 = max(R +R )(C +C )(N  N ) W0nW0n  (R  R )(C  C )(N +N ):The plant P can be written asP (s) = Pn(s)(1 + P (s)) (16)where Pn(s) is the nominal plant (P (s) defined from the nominalvalues of the parameters N;R0; C) and P (s) is the multiplicativeplant uncertainty. We would like to design a controller for the nominalplant Pn(s) that would also stabilize the actual plant P (s) that wouldbe obtained by the same linearization process around the actual equi-librium point. In the following our goal is to find a bound W2(s) forthe multiplicative plant uncertaintyP (s). For clarity of presentationwe omit in the following the argument of the transfer functions (i.e.,we just write P instead of P (s)). Note thatP1n +P11 + P1n +P1(P2n +P2)=P1nP2n1 + P1n11 + P1+P+PP1 + P1+P+P (P +P )P P1 + P1+P=P1nP2n1 + P1n1 +  P1+P+ PP+ P (P +P )P P1 + P1+PsoP =  P1+P+ PP+ P (P +P )P P1 + P1+P=11 + P1n +P1P1P1n+P2P2n(1 + P1n)+P1P1nP2P2n(1 + P1n) : (17)It can be shown that a boundW2 of the multiplicative plant uncertainty,i.e., W2 that satisfiesjP (s)js=j!  jW2(s)js=j! 8! 2+ (18)is (see details of the derivation in the Appendix)W2(s) = a+ bs+ cs2 (19)with a; b, and c defined in (38)–(40).IV. H1 OPTIMIZATION PROBLEMWe design the controller to minimize the following H1 cost func-tion:infW1(1 + Pn(s)C(s)) 1W2Pn(s)C(s)(1+ Pn(s)C(s)) 11=: opt (20)IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004 1033where the infimum is taken over allC stabilizingPn, andW1(s) = 1=sfor good tracking of step-like reference inputs (desired queue size).Applying the formulas given in [12] and [13], the optimal solution to(20) is found as (a similar derivation can be found in [14])Copt(s) =2cNnW 30n W0nR20ns2 + (W0n + 1)R0ns+ 2 +R0nse R ss2 11 + a c s+ F (s)(21)where F is a finite impulse response filter defined asf(t) =(b2 + a2   a2c22) cos t+ c2 + a2b2   1 sin t ; for t < R0n0; otherwise(22)witha2 =1c2   a2x(23)b2 =(b2   2ac)2   c2c22+ 2px  2   a2c22x(24)c2 =px (25)where x is the unique positive root ofx3 +b2   2ac  a22c22x2   (2   a2)[(2ac  b2)2 + c2]c44x  (2   a2)2c44= 0 (26)and  is determined as the largest root of1  ce R ss(s+ a2)(s2 + b2s+ c2) s== 0: (27)V. IMPLEMENTATION ISSUESThe packets are marked with probability p(t) computed by theH1controller (21) according to the block diagram shown in Fig. 1 whereq0 is the desired steady-state queue length andp0 =2N2nqC+ Tpn2C2n(28)is calculated from (6)–(8).VI. SIMULATIONSThe simulations are carried out using simulink, the nonlinearmodel defined by (2) and (3) representing the dynamics of N TCPflows loading a router. The router implements the AQM schemedefined by (21). The following scenario is considered.• Nominal values known to the controller:Nn = 50 TCP sessions,Cn = 300 packets/s, Tp = 0:2 s, so R0n = 0:533 s andW0n =3:2 packets (we assume here a fluid model and, thus, we do notconsider packetization issues).• Real values of the plant: N = 40 TCP sessions, C = 250packets/s, Tp = 0:3 s, so R0 = 0:7 s and W0 = 4:375 packets.• The following controller design parameters are considered:N+ = 10;R+0 = 0:1;C+ = 50, which impliesW+0 = 2:3417.Fig. 2. Comparison between and PI controllers.Fig. 3. Comparison between , PI controllers and RED.For comparison we also simulate the PI AQM scheme proposed in [2].The parameters of the PI controller are computed as suggested by theauthors of [2] using the aforementioned known nominal values. We cansee in Fig. 2 that the H1 controller performs significantly better thanthe PI controller for this set of parameters.In Figs. 3 and 4, we analyze the robustness of the two schemeswith respect to variations in the network parameters. The outgoing linkcapacity C is a normally distributed random signal with mean 250packets/s and variance 50 added to a pulse of period 60 s, amplitude60 packets/s. The number of TCP flows N is a normally distributedrandom signal with mean 45 and variance 30 added to a pulse of pe-riod 20 s and amplitude 10. The propagation delay Tp is a normallydistributed random signal with mean 0.8 s and variance 0.05 s addedto a pulse of period 20 s and amplitude 0.2 s. The controllers have thefollowing value known to them: C = 300 packets/s, N = 50; Tp =0:7 s and the desired queue length is q0 = 100 packets. In additionthe H1 controller uses the following design parameters: N+ =10;R+0 = 0:1, and C+ = 50. RED [15] has the following pa-rameters: pmax = 0:1;minth = 80;maxth = 150, (these determinethe AQM-RED controller gain asLRED = (pmax)=(maxth minth))and queue averaging weight = 0:0001. It can be seen that both the1034 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004Fig. 4. Values of , and corresponding to Fig. 3.PI and theH1 controllers perform better than RED as the initial over-shoot is reduced, and the system oscillates less under parameter vari-ations. The wide oscillations of RED can cause the queue to becomeempty, thus decreasing the utilization of the outgoing link.VII. CONCLUDING REMARKSWe developed in this note a robust AQM control scheme supportingECN and TCP flows. The simulation experiments based on a non-linear fluid flow model, that includes delays, show that the proposedAQM scheme performs better than RED and the proportional-integralscheme by obtaining faster transients and less oscillatory responses,which translates into higher link utilization, low packet loss rate andsmall queue fluctuations. A challenging extension of this work wouldbe to consider networks having multiple links and sources, but suchan extension is not trivial due to the interaction between the controlschemes of each router.APPENDIXFrom (17), we see that we can bound the multiplicative plant uncer-tainty in the following way:jP (s)js=j!<11  upper bound onjP1n +P1js=j! P1P1n s=j!+P2P2n(1 + P1n)s=j!+P1P1n s=j! P2P2n(1 + P1n)s=j!: (29)Note that1W0nR20ns2 + (W0n + 1)R0ns+ 2=1W Rs2 + 2!0 + !20(30)with!0 =1R0n2W0n =W0n + 12p2W0n:We haveW0n + 1  2pW0n 8W0nthus  1p28W0nand, consequently, the transfer function (30) do not exhibit any reso-nance phenomenon and1W0nR20ns2 + (W0n + 1)R0ns+ 2 s=j! 1W0nR20ns2 + (W0n + 1)R0ns+ 2 s=0 12:P2(s) = P2n(s) + P2(s)  P2n(s)=NnW30n +Nn 3W20nW0 + 3W0nW20 +W30 +N(W0n +W0)32R0ns 1 +RR  NnW30n2R0ns=Nn 3W20nW0 + 3W0nW20 +W30 +N(W0n +W0)3  NnW 30nRR2R0ns 1 +RRP1P1n s=j!= 1  P1n +P1P1n s=j!= 1  R0n +R0R0n e R s W0nR20ns2 + (W0n + 1)R0ns+ 2(W0n +W0)(R0n +R0)2s2 + (W0n +W0 + 1)(R0n +R0)s+ 2 s=j! 1 + 1 + R+0R0n12W0nR20ns2 + (W0n + 1)R0ns+ 2 s=j! : (32)IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004 1035Fig. 5. Case 0.Fig. 6. Case 0.Similarly, we havej(W0n +W0)(R0n +R0)2s2+(W0n +W0 + 1)(R0n +R0)s+ 2js=j!  2:The discriminant of the second order polynomial in s formed by thedenominator of the transfer function P1(s) is = (W0 + 1)2R20   8W0R20= R20(W20   6W0 + 1):A sketch of the bode plot of P1 is shown in Figs. 5, and 6. For the caseof  < 0 corresponding to Fig. 5 an upper bound for the magnitudeof P1(s) evaluated on the imaginary axis is jP1(!m)j where !m =(1=R0) (2=W0). For the case of  > 0 corresponding to Fig. 6 anupper bound for the magnitude of P1(s) evaluated on the imaginaryaxis is jP1(!m)j where !m is the geometric mean between the rootsof the second-order polynomial in s formed by the denominator of thetransfer function P1(s), which is also !m = (1=R0) (2=W0). Thus,we havejP1n +P1js=j! < maxW ;RR0n +R02 !m< maxW122W0n +W0:Assuming W0n + W0 > 1, we havejP1n +P1js=j! <p22: (31)We also have the first equation shown at the bottom of the previouspage, as well as (32), also shown at the bottom of the previous page,and a bound of the magnitude ofP2(s) on the imaginary axis isjP2(s)js=j! < Z2ss=j!withZ2 =Nn 3W20nW+0 + 3W0n(W+0 )2 + (W+0 )32R0n 1  RR+N+(W0n +W+0 )3 +NnW30nRR2R0n 1  RRso we haveP2P2n s=j!<2R0nZ2NnW 30n: (33)We also havej1 + P1njs=j!  1+R0nsW0nR20ns2 + (W0n + 1)R0ns+ 2 s=j 1 + 1W0n + 1 32(34)if we assume W0n > 1.With (31)–(34), (29) can be written asjP (s)js=j! < ~W2(!) (35)with~W2(!) =11 p221 +121 +R+0R0n (2 W0nR20n!2)2 + (W0n + 1)2R20n!2 1 + 3R0nZ2NnW 30n+3R0nZ2NnW 30n= a3 + b3 + c3!2 + d3!4anda3 =11 p221 +6R0nZ2NnW 30ne3 =1411 p2221 +R+0R0n21 +3R0nZ2NnW 30n2b3 = 4e3c3 = W20n +R20n   2W0nR20n e3d3 =W20nR40ne3:Note that we have~W2(!) < jW2(s)js=j! (36)withW2(s) = a+ bs+ cs2 (37)1036 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004wherea = a3 +pb3 (38)c =pd3 (39)b = max 2a3pd3 + 2ac+ c3;a3c3pb3+ 2ac+ c3 : (40)REFERENCES[1] V. Misra, W. B. Gong, and D. Towsley, “Fluid-based analysis of anetwork of AQM routers supporting TCP flows with an application toRED,” in Proc. ACM/SIGCOMM, 2000, pp. 151–160.[2] C. V. Hollot, V. Misra, D. Towsley, and W. B. Gong, “Analysis and de-sign of controllers for AQM routers supporting TCP flows,” IEEE Trans.Automat. Contr., vol. 47, pp. 945–959, June 2002.[3] F. Kelly, “Mathematical modeling of the internet,” in Mathematics Un-limited—2001 and Beyond, B. Engquist and W. Schmid, Eds. Berlin,Germany: Springer-Verlag, 2001.[4] T. Ott, T. Lakshman, and L. Wong, “SRED: Stabilized RED,” in Proc.IEEE INFOCOM, 1999.[5] S. Low, F. Paganini, J. Wang, S. Adlakha, and J. Doyle, “Dynamics ofTCP/RED and a scalable control,” in Proc. IEEE INFOCOM’02, NewYork, June 2002.[6] S. Kunniyur and R. Srikant, “Analysis and design of an adaptive virtualqueue algorithm for active queue management,” in Proc. SIGCOMM2001, Aug. 2001.[7] R. Johari and D. K. H. Tan, “End-to-end congestion control for the in-ternet: Delays and stability,” IEEE/ACM Trans. Networking, vol. 9, pp.818–832, Dec. 2001, to be published.[8] L. Massoulie, “Stability of distributed congestion control with heteroge-neous feedback delays,” Microsoft Research Tech. Rep., 2000.[9] G. Vinnicombe, “On the stability of end-to-end congestion control forthe internet,” Univ. Cambridge Tech. Rep., 2001.[10] S. H. Low, F. Paganini, and J. C. Doyle, “Internet congestion control,”IEEE Contr. Syst. Mag., vol. 22, pp. 28–43, Jan. 2002.[11] K. Ramakrishnan and S. Floyd, “A Proposal to add explicit congestionnotification (ECN) to IP,” RFC 2481, Jan. 1999.[12] O. Toker and H. Özbay, “ optimal and suboptimal controllers forinfinite dimensional SISO plants,” IEEE Trans. Automat. Contr., vol. 40,pp. 751–755, June 1995.[13] C. Foias, H. Özbay, and A. Tannenbaum, Robust Control of InfiniteDimensional Systems: Frequency Domain Methods. New York:Springer-Verlag, 1996.[14] P.-F. Quet, B. Ataşlar, A. İftar, H. Özbay, T. Kang, and S. Kalyanaraman,“Rate-based flow controllers for communication networks in the pres-ence of uncertain time-varying multiple time-delays,” Automatica, vol.38, pp. 917–928, June 2002.[15] S. Floyd and V. Jacobson, “Random early detection gateways for con-gestion avoidance,” IEEE/ACM Trans. Networking, vol. 1, pp. 397–413,Aug. 1993.

IEEE Transactions on Automatic Control

On the design of AQM supporting TCP flows using robust control theory

Date of Conference: 9-12 December 2003Conference Name:  42nd IEEE International Conference on Decision and Control, IEEE 2003Recently it has been shown that the AQM (Active Queue Management) schemes implemented in the routers of communication networks supporting TCP (Transmission Control Protocol) flows can be modeled as a feedback control system. Based on a delay differential equations model of TCP's congestion-avoidance mode different control schemes have been proposed. Here a robust controller is designed based on the known techniques for ℋ ∞ control of systems with time delays

Quet, P. F.

http://repository.bilkent.edu.tr/bitstream/11693/24277/1/On%20the%20design%20of%20AQM%20supporting%20TCP%20flows%20using%20robust%20control%20theory.pdf

On the design of AQM supporting TCP flows using robust control theory

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