Controller Design for Congestion Control: Some Comparative Studies

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1 Cotroller Desig for Cogestio Cotrol: Some Comarative Studies Teresa Alvarez, Hector de las Heras, Javier eguera Abstract The cogestio cotrol roblem i data etworks is aalyzed here from the oit of view of cotrol systems, ad some solutios are comared. By aalyzig i detail their liearized trasfer fuctios, it is discussed why TCP Westwood ad TCP Neweo reset differet behavior whe dealig with the same etwork coditios, showig why TCPWestwood gives better erformace. Moreover it is show that adequate stability ad erformace ca be obtaied, eve i the resece of the wide variatios i the arameters of the etworks, as log as the cotroller is roerly desiged. Idex Terms cogestio cotrol, cotrol systems, robust erformace, TCP Westwood, TCP Neweo, trasfer fuctio. I. INTODUCTION outers are cetral comoets i moder data etworks, as they simlify the coectio of several etworks, directig data to the fial destiatio. Whe the router receives more data tha the maximum limit it ca rocess the cogestio haes, causig excessive trasmissio delays, blockig of ew coectios, ad losses of ackets (see, for examle, [], [], ad refereces therei). If this cogestio is ot roerly treated cogestive collase occurs, with the router rovidig very low throughut. To correct this roblem moder routers imlemet rotocols that iclude some cogestio cotrol techiques, which rovide feedback to the revious odes i the etwork, i order for them to adat the traffic to the maximum caacity of the cogested router (see, for examle, [4], [5]). This feedback is frequetly rovided by droig ackets: whe the loss of some ackets i a data lik is detected, the rotocols imlemeted i revious odes reduce the traffic. May cogestio cotrol algorithms have bee roosed: this aer cocetrates o those imlemeted i two of the most oular rotocols: We assume that the rotocol imlemeted at the trasort layer is the Trasmissio Cotrol Protocol (TCP), with a Active Queue Maagemet (AQM) algorithm imlemeted i the etwork scheduler withi the routers. These AQM algorithms imlemet the cogestio cotrol algorithm: a queue is imlemeted at the iterface with each lik, that holds the ackets that have beig scheduled (by TCP) to go Mauscrit received March 4. Work fuded by MiCI DPI- 589-C5-5. T. Alvarez, H. de las Heras ad J. eguera are with the Uiversity of Valladolid (Telf: ; tere@autom.uva.e. out i that lik; before the queue is full, i order for the seder to detect the cogestio roblem as soo as ossible some ackets are droed (or marked, deedig o the secific algorithm) [5,6]. Comared with other cogestio cotrol algorithms (such as the dro-tail imlemeted i may router, AQM marks the ackets robabilistically to imrove sharig the badwidth betwee differet liks, the imrovig the rocessig of bursty flows ad avoidig sychroized oscillatios betwee the liks. May AQM algorithms have bee roosed ad tested (see, for examle, [5]). Moreover, eve if the same AQM techique is imlemeted results are differet deedig o the TCP rotocol used. This aer cocetrates o discussig the advatages of usig algorithms derived from techiques that are stadard i cotrol systems ad have beig extesively tested i alicatio areas other tha data etworks (see [7,8] for some geeral referece of cotrol systems i geeral) cosiderig two differet TCP variats: Westwood ad New eo. These cotrol system techiques ormally require a aroximate model of the system to be cotrolled, exressed as liear dyamic trasfer fuctio; some dyamical models of data cogestio were already roosed i [], ad its liearizatio was discussed i []. From the may cotroller desig methodologies roosed i cotrol systems, PID cotrol is robably the most widely used to develo AQM algorithms. We ca cite [], [], [4], [5] ad refereces therei. I order to make these studies the methodologies described will be alied to a secific commuicatio roblem: usig simulatio would make ossible to make a fair comariso, by reroducig the same traffic coditios whe testig each algorithm. This aer will reset a comariso betwee Westwood TCP ad Neweo TCP. The o liear fluid models are liearized ad the trasfer fuctio models are the obtaied. These trasfer fuctios will be comared usig the ste resose, the frequecy resose, the locatio of oles ad zeros uder differet etwork cofiguratios. These studies will hel to uderstad why TCP Westwood behaves better whe there are chages i the traffic coditios, umber of users or delay i the etwork, regardless of the tye of commuicatio (wired or wireles. The orgaizatio of the aer is the followig: Sectios ad describe the TCP Westwood ad TCP Neweo rotocol ad their fluid models. Sectio 4 resets a comariso betwee TCP Westwood ad TCP Neweo. Sectio 5 briefly describes the cotroller to be imlemeted

2 ad fially Sectio 6 discusses some exerimets. A discussio cocludes the aer. II. TCP WESTWOOD TCP Westwood (TCPW) [6] is a oular cogestio cotrol algorithm that has show to rovide robust selectios for the cogestio cotrol arameters (i fact, it is imlemeted with mior modificatios as art of the Liux kerel). It is articularly relevat whe acket losses are ot oly geerated by cogestio cotrol algorithms (for examle i wireless etwork. TCPW relies o studyig the stream of returig ackowledgmet ackets (ACK to adat some arameters of the cogestio cotrol algorithm [6], i articular the size of the so-called cogestio widow (cwd) that fixes the maximum umber of bytes that ca be acceted at each lik: this widow is slowly icreased util some ackowledgemets are ot received or a timeout exires. A. Fluid-flow Model The oliear dyamical model of TCPW ([5], [6], [7], [8]) is ow reseted, as it will be the base of the models used for cotroller desig ad comariso. If W( deotes the size of the TCPW widow at a give lik i ackets, the umber of ackets at the corresodig queue (:queue legth) ca be derived from the followig equatios as follows: dq N( W ( () dt T dw W ( W t ( t ( dt q t ( T T t ( ) W t ( T t ( q t ( T t ( ) Where T is the roagatio delay (i secod, is the lik caacity (i ackets/, N( is the umber of TCPW sessios (load factor) ad ( is the roud-tri time (equal to /+T) ad ( is the variable that is modified by the AQM algorithm: the robability of discardig a acket (or markig it i some AQM imlemetatio. B. Model for cotroller desig As it has bee metioed, a liear model with costat arameters is frequetly used to desig cotrol systems, as it simlifies the aalysis ad desig of cotrollers. This kid of models ca be easily derived by liearizatio: i our case, temorarily assumig that the umber of active TCPW sessios, the lik caacity ad the roud-tri time do ot chage, ad discardig high-frequecy comoets, a Taylor aroximatio of () ad () at a give set of oeratig arameters gives the followig trasfer-fuctio descritio of our system (see []or [5] for detail: () s C T N e C N N s C C s T T ( Or equivaletly: CQ s e N ( (4) N N C s s C C Q Where C, N, ad Q are the values of, N(, ( ad at the curret workig oit. q ref () Figure : Block diagram of the AQM feedback cotrol system The static gai of this liearized system would be: C T Q C N T C Q N With the system resodig as a stable secod-order system with delay, uderdamed if T>N. III. TCP NEWENO TCP Neweo (TCPN) is still robably the most widely used cogestio cotrol techique (with several variat, although TCPW rovides better erformace whe the lik icludes large aths or might geerate uexected acket losses (like wireless etwork. This simler algorithm ca be modelled as follows: dw dt - Cotroller (5) dq N( W ( (6) dt T t W ( W qt T T t ) AQM Queue t The corresodig trasfer fuctio model is the C s e N ( N s s C Or equivaletly: C s e N ( N N s s C C q (7) (8) (9)

3 Thus, it ca be see that the TCPNeweo static gai is ad the dyamic resose corresods to a secod order overdamed system with delay, ad real oles laced at N, C () Where the domiat ole is give by -N/(C ). Please, take ito accout that oly if N/C is bigger tha / the domiat ole would be -/. IV. COMPAISON BETWEEN TCPW AND TCPN esearch has show ([6]) that TCPW is a good aroach whe there are wireless liks i the etwork. This sectio resets a comariso usig the trasfer fuctio models of TCPW ad TCPN i order to achieve some coclusios regardig the static gai, the oe loos ad the frequecy resose of each system. This aalysis will allow desigig adequate cotrollers. The umber of users (N) ad the TT () should have coheret values. I [] ad [] some guidelies o how to choose these values were give. The best sceario is to have may users ad a small delay. The worst situatio is to have few users ad a big delay. This coclusio is true for both TCP aroaches uder study. First, let s comare the static gai of both systems: Q C C Q N - TCPW static gai (5): - TCPN static gai (): C 4N Which oe is bigger? It deeds o the arameters values, but (usually) (5) will be bigger tha (), i.e., the iut will be amlified: Q C C C QN 4N Practical examles will show this. This is a very imortat iferece for the desigig cotrollers: whe the umber of users or the delay chage i the etwork, TCPW will be more robust tha TCPN. Both systems have o zeros ad both have two oles. However, the TCPN trasfer fuctio always has two real oles ad the TCPW system s oles ca be real or comlex cojugate. The deomiators of both trasfer fuctios are: - TCPW: s N N C s C C Q N N - TCPN: s s C C The secod ad first order terms are equal, the differece is o the ideedet, term ad they have very similar: N C N versus is the same tha C Q C C versus Q C Whe is greater tha? It deeds o C, ad Q Q, but i real etworks the iitial queue legth will be smaller tha the times C. As a result, for a give N, C, ad Q, the two oles of TCPW are always laced betwee the two oles of TCPN. Aother coclusio is that TCPW will be a bit faster tha TCPN. This also ca be justified lookig at the aalytical solutio of the oles: If we look at the solutio of the secod order equatio of TCPN: oles, N C N C N N 4 C C 4N 4N 4 C C O the other had, the oles of TCPW will be give by: oles, N C N C N C 4 C Q N 4N 4N 4 C C Q 4N 4N Whe, the the oles will be comlex 4 Q C cojugate. A secod order system is characterized by this equatio: Kw s w s w ad w the atural frequecy. Cosiderig TCPW: Kw s w CQ N w s w s C N Q C (, where is the damig ratio N C C N s Q C

4 N C C N Q C C N Q CN C Q Whe the oles are comlex cojugate: VI. EXAMPLES This sectio comares TCPW ad TCPN uder differet traffic coditios. The basic toology is deicted i Figure. Differet scearios will be cosidered. oles, w jw V. CONTOLLE DESIGN This sectio briefly describes how a cotroller ca be imlemeted ito a AQM aroach. Figure shows the basic structure. As it has bee metioed the cotrol actio i AQM algorithm is robability of droig (or markig) ackets (, with is adated by comarig the curret queue legth with a desired value q ref (a ercetage of the real caacity of the queue): see Figure. We have chose the PID cotroller [] as the AQM cogestio cotrol algorithm, as this cotroller is widely used ad gives good results. The structure of the cotroller is: u t t K e t e K e t t de d Td T i dt t de Ki e d K d dt t () It must be oited out that this is the ideal equatio: Whe the cotroller is imlemeted, the derivative term should be treated as a first order filter. Figure : Dumbbell toology A. First exerimet: Fixed N, chagig The first batch of exerimets cosiders the arameters show i Table I. We cosider a fixed N ad a chagig. The greater the delay, the smaller the iitial markig robability. TABLE I NETWOK CONFIGUATIONS N C P Q Case Case Case Case Case TABLE II. Fixed N, chagig Caso Caso Caso Caso 4 Caso 5 TCPW tf TCPN tf s 7.85s 5.6 s.6s. 7 s.74s. 756 s.9s. 467 s.9s s 7.85s 7.9 s.6s. 987 s.74s. 5 s.9s. 58 s.9s. 4 TCPW s ga TCPN s ga TCPW bad TCPN bad TCPW Poles TCPN Poles TCPW Set tim.69 sec..55 sec. 7. sec sec..6 sec TCPN Se Ti.6 sec.8 sec 54 sec 95.5 sec 8 sec

5 Pole-Zero Ma Westwood 4 Oe loo BODE Westw ood eal Axis.5 Pole-Zero Ma Westwood Magitude (db) eal Axis Figure : Case ad 5, oe loo oles Table II shows a comariso betwee TCPW ad TCPN i terms of oe loo trasfer fuctio, static gai, badwidth ad oles ad zeros Frequecy (rad/sec) Figure 5: Oe loo TCPW Magitude Bode lot Oe loo BODE New reo 4 As it was hited i revious sectio, the sa of variatio of TCPW s static gai is smaller tha the oe of TCPN. I fact, the stadard deviatio of TCPW static gai is. 4 ad TCPN stadard deviatio is.9 6. If a cotroller is tued for a certai etwork arameters ad the there are chages, TCPW will erform much better. Magitude (db) Lookig at the last two rows of Table II ad Figure, we ca see that the oe loo oles of TCPW lie betwee the oles of TCPN. Moreover, the greater the delay, the closer the oles to the origi, i.e., the slower the system. - - Oe loo ste Westwood, N=5, x 5 Oe loo ste Neweo, N=5, , Figure 4: Oe loo ste resoses ad settlig times Figure 4 shows the oe loo ste resose of both TCP variats cosiderig the delay. Settlig times are smaller for TCPW (as show i Table II). The greater the delay is, the slower the system ad greater the gai are. Clearly, TCPN is slower tha TCPW. Figures 5 ad 6 deict the oe loo magitude Bode diagram of each system. Agai, TCPN has a worst behavior tha TCPW. TCPW s badwidth is bigger tha TCPN s oe: Westwood is more robust whe there are erturbatios or oise i the system Frequecy (rad/sec) Figure 6: Oe loo TCPN Bode diagram A. Secod exerimet: Chagig N, fixed The secod exerimet cosiders a fixed ad a chagig umber of users: TABLE III NETWOK CONFIGUATIONS N C P Q Case Case Case Case Case Observig Table IV, it ca be cocluded that the bigger N give smaller the static gai. Agai the stadard deviatio of the static gai is smaller for TCPW (984) tha for TCPN (86497). Moreover, the badwidth icreases with the umber of users. As i exerimet, all the oles are stable (Table IV ad Figure 7). I fact, TCPW s oles are always laced betwee TCPN s oles. As is costat, the farthest ole is located at -/.

6 TCPW tf TCPN tf TABLE IV. CHANGING N, VAYING Caso Caso Caso Caso 4 Caso 5 65 s. 57s s. 57s s.6s.7 s. 748s s. 96s s.6s.9877 s.74s. 5 s. 96s s 4. 59s s 4. 59s. 95 TCPW s ga TCPN s ga TCPW bad TCPN bad TCPW Poles ±.5i -.96 ±.64i -.59 ±.85i TCPN Poles TCPW Set tim 4.6 sec.55 sec.8 sec.46 sec.95 sec TCPN Set tim 7. sec.8 sec. sec 7. sec.97 sec Oe loo BODE Westw ood 5 Pole-Zero Ma N=4 Imagiary Axis Newreo eal Axis Westwood Magitude (db) N= 5 Pole-Zero Ma Imagiary Axis eal Axis Westwood Figure 7: Exerimet. Case ad 5, oe loo oles - - Frequecy (rad/sec) Figure 9: TCPW magitude Bode lot, ex Oe loo BODE New reo Oe loo ste Westwood, N, =. N= N= N= Magitude (db) 6 4 N= Oe loo ste Neweo, N, =. -5 N= N= Figure 8: Exerimet : Oe loo ste resoses ad settlig times The settlig times (Figure 8, Table IV) are ot so differet as i exerimet. Agai TCPW is faster tha TCPN ad the settlig times are more similar. The magitude Bode diagrams (Figures 9 ad ) are more similar tha i exerimet. - - Frequecy (rad/sec) Figure : TCPN magitude Bode lot, ex A. Third exerimet: Mixed situatios The fial exerimet will cosider mixed situatios (Table V). The objective is to desig a PID cotroller for the sceario cosidered as a ormal situatio ad to test the behavior of this cotroller i other scearios.

7 TABLE V NETWOK CONFIGUATIONS N C P Q Case Case Case Case The oe loo ste resoses are show i Figure. The treds ad commets regardig static gai, badwidth ad oles follow those give for exerimets ad. It ca be see that the bigger the delay, the bigger the gai is. I fact, whe the delay is big, TCPN is clearly slower tha TCPW (Figure ). Amlitude Closed loo resose. Westw ood N=8,=. N=6,=.5 N=5,=. N=8,=.. Oe loo ste Westwood N=6,=.5 - N=4,=.9 N=8,=. - N=5,= x 5 Oe loo ste Neweo Figure : TCPW closed loo resose, queue evolutio. x Closed loo resose. POBABILITY. Westw ood N=4,=.9 N=8,6,5 Amlitude - N=6,=.5 N=5,=. N=4,=.9 - N=8,= Figure : TCPW ad TCPN oe loo ste resose Case 4 is the workig sceario: 5 users ad a delay of. secods. Usig the classical method of Ziegler- Nichols [8] followed by some fie tuig, two cotrollers were tued, oe for TCPW ad the other for TCPN.: TCPW: K= -.5, Ti=5, Td=. TCPN: K= -., Ti=, Td=.5 The TCPW cotroller is a PI, the derivative term has bee set to zero because there was o imrovemet whe icludig this term. Figure shows the closed loo resose of the TCPW system for the 4 scearios uder study. It reresets the evolutio of the router s queue (cotrolled variable) whe a ste is alied. Figure shows the cotrol variable (robability): how it chages to drive the queue to the desired value. Figures ad 4 deict the closed loo behavior of the TCPN system for the four scearios uder study. The cotrolled variable (queue size at the router) corresods to Figure (best settlig time 8. sec., case ad the worst time is 867 sec. i Case) ad the robability of markig a acket is showed i Figure Figure : TCPW closed loo resose, robability Observig Figures ad, we ca coclude that the ste resose is stable i all cases. Whe N ad are closer to the omial sceario, the results are better. But what it is really imortat is that the system is robust to chages i the arameters. Lookig at Figures 4 ad 5 some coclusios are similar: the most similar the sceario to the omial oe (fourth grah) the better the results. But, ow there is a stability roblem: whe N=4 ad =.9, the closed loo system (lat with the cotroller) is ustable. It is true that the cotroller has ot bee tued for this situatio. Nevertheless as TCPN has show to be less robust (see revious sectios ad exerimets ad ) to chages i the arameters, this is a situatio that ca arise i reality.

8 Ste esose.5 N=8,=. System: N=8,=. Settlig : x 6 Ste esose Ste Time esose (sec).5 N=6,=.5 System: N=6,=.5 Settlig : Ste Time esose (sec).5.5 N=5,=. System: N=5,=. Settlig : N=4,=.9 Figure 4: TCPN closed loo resose, queue evolutio. x Ste esose Ste Time esose (sec) x Ste esose x Closed loo resose. POBABILITY. New eo Figure 5: TCPN closed loo resose, robability VI. CONCLUSION The aer has reseted a detailed aalysis of TCP Westwood ad TCP Neweo i terms of oles, static gai ad frequecy resose of its liearized trasfer fuctios, arametrized i terms of the etwork arameters. This has made ossible to comared their exected resoses for the same traffic coditios: The mai coclusio is that as variatios i the static gai ad badwidth are smaller for TCP Westwood ad its badwidth is bigger, the cotrolled system will be more robust i the iheret resece of chages i the etwork arameters: it has bee show that it ca work roerly i a wide rage of situatios as it has bee show through examles. Moreover, the TCP Westwood trasfer fuctio model has damig greater tha.8. esults are ecouragig ad more work is beig doe to use this iformatio, such as usig techiques for tuig the cotroller that takes ito accout the exected arameter variatios. EFEENCES [] Azuma, T., T. Fujita, M. Fujita (6). Cogestio cotrol for TCP/AQM etworks usig State Predictive Cotrol. Electrical Egieerig i Jaa, 56, [] Aström K.J. ad T. Hägglud (6), Advaced PID cotrol. ISA. NC. [] Deg, X., S. Yi, G. Kesidis. C.. Das (). A cotrol theoretic aroach for desigig adative AQM schemes. GLOBECOM, 5, [4] Jacobso, V. (988). Cogestio avoidace ad cotrol. ACM SIGCOMM 88. [5] yu, S., C. um, C. Qiao (4). Advaces i Active Queue Maagemet (AQM) based TCP cogestio cotrol. Telecommuicatio Systems, 5, 7-5. [6] Floyd, S., & Jacobso, V. (99). adom early detectio gateways for cogestio avoidace. Networkig, IEEE/ACM Trasactios o, (4), 97. [7] Mathis, M., Semke, J., Mahdavi, J., & Ott, T. (997). The macroscoic behavior of the TCP cogestio avoidace algorithm. ACM SIGCOMM Comuter Commuicatio eview, 7(), [8] Dorf,. C. (995). Moder cotrol systems. Addiso-Wesley Logma Publishig Co., Ic. [9] Frakli, G. F., Powell, J. D., & Emami-Naeii, A. (994). Feedback cotrol of dyamics systems. Addiso-Wesley, eadig, MA. [] Hollot, C.V., V. Misra, D. Towsley, W. Gog (). Aalysis ad Desig of Cotrollers for AQM outers Suortig TCP flows. IEEE Trasactios o Automatic Cotrol, 47, [] Bolajraf, M., Tadeo, F., Alvarez, T., & ami, M. A. (). Statefeedback with memory for cotrolled ositivity with alicatio to cogestio cotrol. IET cotrol theory & alicatios, 4(), 48. [] Hohebichler, N. (9). All stabilizig PID cotrollers for time delay systems. Automatica, 45, [] Silva, G., A. Datta ad S. Bhattacharyya. PID cotrollers for timedelay systems. Birkhäuser, 5. [4] Log, G.E., B. Fag, J.S. Su ad Z.Q. Wag (). Novel Grahical Aroach to Aalyze the Stability of TCP/AQM Networks. Acta Automatica Siica, 6, 4-. [5] Teresa Alvarez, Diego Martíez (), Hadlig the Cogestio Cotrol Problem of TCP/AQM Wireless Networks with PID Cotrollers, i IAENG Trasactios o Egieerig Techologies, ages [6] M. di Berardo, L. A. Grieco, S. Mafredi, ad S. Mascolo," Desig of robust AQM cotrollers for imroved TCP Westwood cogestio cotrol", Proc. of the 6th Iteratioal Symosium o Mathematical Theory of Networks ad Systems (MTNS 4), Katholieke Uiversiteit Leuve, Belgium, July, 4. [7] Che, J., F. Pagaii, M.Y. Saadidi,. Wag ad M. Gerla (6). Fluid-flow aalysis of TCP Westwood with ED. Comuter Networks, 5, -6. [8] Mascolo, S., C. Casetti, M. Gerla, M.Y. Saadidi,. Wag. TCP Westwood: badwidth estimatio for ehaced trasort over wireless liks, i: Proceedigs of Mobicom,. [9] Hollot, C.V. ad Y. Chait (). Noliear stability aalysis for a class of TCP/AQM etworks. I Proceedigs of the 4th IEEE Coferece o Decisio ad Cotrol, Orlado, USA. [] Al-Hammouri, A.T., V. Liberatore, M.S. Braicky, ad S.M. Phillis (6b). Parameterizig PI cogestio cotrollers. Proc. First Iteratioal Worksho o Feedback Cotrol Imlemetatio ad Desig i Comutig Systems ad Networks, Vacouver, CANADA. [] Al-Hammouri, A.T., V. Liberatore, M.S. Braicky, ad S.M. Phillis (6a). Comlete stability regio characterizatio for PI- AQM. SIGBED eview, (). ACKNOWLEDGMENT The authors would like to thak Prof. F. Tadeo for so may fruitful discussios.