Multiple time-delays system modeling and control for router management

Multiple time-del ays system modeling and control f or r outer man agement Y assine Ariba ∗ † , Fr ´ ed ´ eric Gouaisbaut ∗ † and Y ann Labit ∗ ∗ † October 2008 Abstract This paper in vestigates the ov erload problem of a single congested router in TCP ( T ran smission Contr ol Proto col ) networks. T o cope with the congestion phe- nomenon , we design a feedback control based on a multiple time-delays model of the set TCP/A QM ( Active Queue Mana gement ). Indeed, using robu st control tools, especially in the quadratic separation frame work, the TCP/A QM model is rewrit- ten as an i nterconnected system and a structured state feedback i s constructed to stabilize t he network variab les. Finally , we i llustrate the proposed methodology with a numerical ex ample and simulations using NS-2 [11] simulator . 1 INTR ODUCTION In I P networks, active queue m anagem ent (A QM), embedd ed in router, reports to TCP sources its p rocessing load. The objective is to man age th e buffer utilization as well as the queuein g delay . This has motiv ated a hu ge a mount o f work aiming at un- derstandin g the congestion phen omeno n and achieving better p erfor mances in terms of Quality of Service (QoS). As a matter of fact, there has been a growing r ecognition that the n etwork itself must p articipate in congestion control and ressource man agement [8], [26]. The A QM principle consists in dropping (or marking when ECN [34] option is en- abled) some packets b efore buffer saturates. Hence, fo llowing the Ad ditive-Inc r ease Multiplicative-Decrease (AIMD) be havior o f TCP , sources r educe their conge stion window size a voiding then the full saturatio n of th e router . Basically , A QM suppo rt TCP f or con gestion av oida nce and fee dback to the la tter wh en traffic is to o heavy . In- deed, an A QM drops/mar ks incoming packet with a gi ven prob ability related to a con- gestion index (such as qu eue le ngth or delay s) allowing then a k ind o f co ntrol on the buf fer occu pation at route rs. V arious m echanisms have been pro posed in the ne twork commun ity f or th e d evelopment of A QM such as R an dom E arly Detec tion (RED) [14], Random Early Marking (REM) [2], Adaptive V irtual Qu eue (A VQ) [24] and many oth- ers [35]. Their perfo rmances ha ve been e valuated [ 13], [35] and empirical studies [26] have sh own th e ef fe ctiv eness of these algorithms. As it has been highligh ted in the litterature (see for examp le [27], [ 17] an d refer- ence therein), A QM acts as a controller supporting TCP for congestion con trol and can ∗ Uni versit ´ e de T oulouse; UPS, 11 8 Route de Narbonne, F-31062 T oulouse, Franc e. † LAAS; CNRS; 7, av enue du Colonel Roche, F-31077 T oulouse, France . { yariba,fgouaisb ,ylabit } @laas.f r be refor mulated a s a f eedback con trol p roblem . Then, a significant resear ch has b een dev oted to the use of control th eory to develop more efficient A QM. Using dyn amical model developed by [29], some P ( Pr oportion al ), PI ( Pr oportiona l Inte gral ) [17 ] h ave been designed . In th e same fram ew ork , other tools have been used to extend this pre- liminary work such as a PID co ntroller [3 5], [12] or robust contro l [33]. Howe ver, most of these papers d o no t take into account the delay an d ensu re the stability in closed loo p for all delays which could be very conservati ve in p ractice. The study of congestion p roblem in time d elay systems fra mew or k is not new and has been successfu lly exploited. Nevertheless, most of these works have be en dedi- cated to the stability an alysis of network s composed o f homo geneo us sources (see fo r example [25], [31], [9], [28] and [37]). In this paper, networks with heterogen eous sources are consid ered introdu cing then several delays and inc reasing the model co m- plexity . Regardin g th e A QM design prob lem, some works have already been d one in [21] an d [1 6]. In [16], the construction of th e A QM r equired to inv oke the Ge neral- ized Ny quist Th eorem and [21] provides a delay depend ent state feed back inv olvin g delay compen sations with a memo ry feedb ack con trol. Or ev en in [3], delays are re- placed by a Pad ´ e appro ximation which is kn own to be not so acc urate. All th ese latter methodo logies are interesting in th eory but sorely suitable in practice. While these latter studies have co nsidered the sim plified mod el of TCP/A QM from [2 9], we use in this contribution a more accurate mo del presented in [27]. In deed, contrary to [2 1], [3] and [16], bo th f orward and bac kward delays are taken into acco unt (tha t is, we do not neglect forward d elays). Then, cong estion co ntrol of n etworks consisting in het- erogen eous TCP source s is transfor med into a stabilization prob lem for multip le time delays s y stems. Using a rob u st analy sis frame work and especially quadra tic separation approa ch developed for time-d elay systems b y [15], a stabilizing A QM is designed. The proposed control mec hanism enables QoS in terms of R TT ( Round T rip T ime ) and delay jitter which are re lev ant fe atures for strea ming and real- time applications over I P networks. No te that the approach employed in this paper allo ws the formulatio n of the problem into matrix inequalities [ 7] that provide systematic stability condition, easy to test. The p aper is o rganized as follows. The second part p resents the m athematical model of a network co mposed of a single router and se veral heterogeneo us sour ces supportin g TCP . Section I II is dedicated to the design of the A QM ensur ing the stab i- lization of TCP . Section IV p resents a nu merical example and simulation results using NS-2. 2 NETWORK D YNAMICS In this paper, we consider a network consisting of a single router and N hetero geneou s TCP sources. By heteroge neous, we mean that eac h source is linked to the router with different prop agation times ( see Figure 1). Sin ce the bottlen eck is shared by N flo ws, TCP ap plies the co ngestion av oidan ce algorith m to cope with the n etwork saturation [18]. Deterministic fluid -flow mode ls have been widely used ( see [2 7], [29], [20] and [36] and re ferences therein ) to describe cong estion co ntrol and A QM sche mes in IP networks. These m odels cap ture the m ean beh avior o f the TCP d ynamic. While many studies dealing with network con trol in the automatic c ontrol th eory f ramework consider the mo del pr oposed by [29], we use in th is paper the model introduced in [27] and d escribed by (1). Contrary to the f ormer, the m odel (1) takes into account Figure 1: Network topolo gy Figure 2: A single conn ection the forward an d b ackward delays and do es no t make the simplifyin g assumption that ( W ( t − τ ) / W ( t ))( 1 − p ( t − τ b )) = 1 [21]. The mode l and notations are as follo w:              ˙ W i ( t ) = W i ( t − τ i ) τ i ( t − τ i ) ( 1 − p i ( t − τ b i )) 1 W i ( t ) − W i ( t − τ i ) τ i ( t − τ i ) W i 2 p i ( t − τ b i ) ˙ b ( t ) = − C + ∑ N η i W i ( t − τ f i ) τ i ( t − τ f i ) τ i = b ( t ) C + T p i = τ f i + τ b i (1) where W i ( t ) is the co ngestion window size of the sou rce i , b ( t ) is the queu e leng th o f the buffer at the r outer, τ i is the round trip time ( R TT) perceived by the sourc e i . Th is latter q uantity can b e decom posed as the sum of th e forward and backward delays ( τ f i and τ b i ), standing for, respectively , the trip time from t h e source i to the router (the one way) and fro m the router to the sou rce via the receiver (the re turn) (see Fig ure 2). C , T p i and N are param eters related to the network co nfiguratio n an d represent, resp ec- ti vely , the link capacity , th e propaga tion time of the path taken by the conn ection i and the n umber of TCP so urces. η i is the numb er of session s e stablished by source i . The signal p i ( t ) co rrespon ds to th e drop probability of a packet. In this pap er , the objec ti ve is to d evelop a m ethod which com putes the app ropriate dropp ing proba bility ap plied at the route r in orde r to r egulate the q ueue len gth o f th e buf fer b ( t ) to a desired level ( Figure 3). Since control depends on the system state, it is required to ha ve access to them. Howe ver, c ongestion windows W i are not measurable. So that, we pro pose to re formu late the model (1) such that state vector can b e m easured. T o this end, r ates of eac h flo w x i , e xp ressed as x i ( t ) = W i ( t ) τ i ( t ) , will b e considered. Hence, Figure 3: Network contro l the dynamic o f this new quantity is o f the form ˙ x i ( t ) = d dt  W i ( t ) τ i ( t )  = ˙ W i ( t ) − x i ( t ) ˙ τ i ( t ) τ i ( t ) . Based on the expressions o f ˙ W ( t ) , ˙ b ( t ) , τ i ( t ) (see equ ation (1)) and ˙ τ ( t ) = ˙ b ( t ) C , a n ew model of the TCP behavior is deri ved      ˙ x ( t ) = x ( t − τ ) x ( t ) τ ( t ) 2 ( 1 − p ( t − τ b )) − x ( t − τ ) x ( t ) 2 p ( t − τ b ) + x ( t ) τ ( t ) − x ( t ) τ ( t ) C ∑ N η i x i ( t − τ f i ) ˙ b ( t ) = − C + ∑ N η i x i ( t − τ f i ) . (2) Remark 1 Th is m odel transformation allows us to u se x i instead of W i which is mor e suitable to h andle. I ndeed, n umer ous works ha ve developed tools that en able flow rates measur emen ts, especially in anoma ly detection fr amework (see for e xample [4], [22]). Besides, The measur e o f the aggr egate flow has already b een pr opo sed a nd successfully exploited in [24] and [ 2 1] for the r ealization of th e A VQ (Adap tive V irtua l Queue) and a PID type A QM r espectively . Our work focuses on the c ongestion contr ol o f a single ro uter with a static to polog y ( N and η i are con stant). Moreover , fo r the mathem atical tractability , we make the usual assumption [27], [17], [ 21] that all d elays ( τ i , τ f i and τ b i ) are time in v aria nt when they appear as argument o f a v ar iable (for example x i ( t − τ i ( t )) ≡ x i ( t − τ i ) ). This latter assump tion is valid as lon g as the que ue length rem ains close to its equilibriu m value and wh en the que ueing delay is smaller than p ropag ation delay s. Defining an equilibriu m point      τ i 0 = T p + b 0 / C ˙ b ( t ) = 0 ⇒ ∑ N η i x i 0 = C ˙ x i ( t ) = 0 ⇒ p i 0 = 2 2 +( x i 0 τ i 0 ) 2 , (3) model (2) can be linearized:      ˙ x 1 ( t ) . . . ˙ x N ( t ) ˙ b ( t )      = A      δ x 1 ( t ) . . . δ x N ( t ) δ b ( t )      + A d       δ x 1 ( t − τ f 1 ) . . . δ x N ( t − τ f N ) δ b ( t )       + B    δ p 1 ( t − τ b 1 ) . . . δ p N ( t − τ b N )    (4) Figure 4: Implem entation of an A QM where δ x i . = x i − x i 0 , δ b . = b − b 0 and δ p i . = p i − p i 0 are the state variations aro und the equilibriu m point (3). Matrices of the equation (4) are define d by A =      a 1 0 0 h 1 0 . . . 0 . . . 0 0 a N h N 0 0 0 0      , B =      e 1 0 0 0 . . . 0 0 0 e N 0 0 0      A d =      f 1 η 1 . . . f 1 η N 0 . . . . . . . . . 0 f N η 1 . . . f N η N 0 η 1 . . . η N 0      , with a i = − 1 − p i 0 x i 0 τ 2 i 0 − x i 0 p i 0 2 , h i = − 2 ( 1 − p i 0 ) C τ 3 i 0 , f i = − x i 0 τ i 0 C and e i = − 1 τ 2 i 0 − x 2 i 0 2 . Remark that a multiple time delays sy stem (4) is obtained with a particular f orm since each compo- nent of the state vector is delaye d by a different quantity related to the comm unication path. Thus, in Section 3 an appropriate m odeling of (4) with a structured f orm (in quadra tic separation fram ew ork ) is prop osed in o rder to formu late the stability co ndi- tion. The p roblem of regulation is tack led in Sectio n 3 with th e design o f a stabilizing state fe edback f or m ultiple time delays systems in o rder to gua rantee a Q oS. Hen ce, the dropp ing probability p i associated to source i will be computed at the router by p i ( t ) = p i 0 + k 1 δ x i ( t − τ f i ) + k 2 δ b ( t ) (5) with k 1 , k 2 are compon ents wh ich have to be d esigned (see Figure 4). Note that the dropp ing p robab ility p i perceived by the source i will be delayed p i ( t − τ b i ) because of the backward delay . Thus, a structured state feedb ack of the form (5 ) is p ropo sed: • to av oid unnatural signals with d ifferent d elay com bination s in the state f eed- back: δ x i ( t − τ f i − τ b j ) for i , j ∈ { 1 , ..., N } introducing then many additional delays, • it p rovides ligh t computation s (with less operation s) reducing then the processing time at router, Figure 5: An intercon nected system • it provides a dece ntralized contro l if the dropp ing strategy is perfo rmed at end hosts when emulating A QM [5]. Applying the structured state feedback type control law p i ( t ) to eac h source i , ∀ i ∈ { 1 , . . . , N } , th e following i n terconne cted system is o btained      δ ˙ x 1 ( t ) . . . δ ˙ x N ( t ) ˙ b ( t )      = A      δ x 1 ( t ) . . . δ x N ( t ) δ b ( t )      + A d       δ x 1 ( t − τ f 1 ) . . . δ x N ( t − τ f N ) δ b ( t )       + BK 1    δ x 1 ( t − τ 1 ) . . . δ x N ( t − τ N )    + BK 2    δ b ( t − τ b 1 ) . . . δ b ( t − τ b N )    (6) where matrices gains K 1 and K 2 are stru ctured as K 1 = d iag { k 11 , . . . , k 1 N } and K 2 = d iag { k 21 , . . . , k 2 N } . Equation (6) represents thus the mean behavior of TCP regulated by a stru ctured state feedback type A QM around an equilibrium point. K 1 and K 2 can be derived fr om the stability analysis of the interco nnection (6). T o this end , we pro pose to design these latter gains by a suitab le modelin g of (6) ap plying then the q uadratic sep aration principle [32], [15] for the stability condition . 3 ST ABILIZA TION AND QoS GU ARANTEE: DESIGN OF AN A QM The stability of the intercon nected system (6) d epends on m atrices K 1 and K 2 . This section aims to de velop a m ethod that provid es such stabilizing matrices. Th us, th e quadra tic separatio n fr amework is consid ered and the fo llowing theorem will be em- ployed [32]. Theorem 1 Given two po ssibly non- squared matrices E , A an d a n unce rtain ma trix ∇ b elongin g to a set Ξ . The un certain system r epresented on Figur e 5 is stab le for all matrices ∇ ∈ Ξ if a nd only if it exis ts a matrix Θ = Θ ∗ satisfying condition s  E − A  ⊥∗ Θ  E − A  ⊥ > 0 (7)  1 ∇ ∗  Θ  1 ∇  ≤ 0 . (8) It is then required to tran sform the initial system ( 6) into a feedbac k system of the form of Figure 5 wh ere a linear eq uation is conn ected to a line ar uncertain ty ∇ . T he key idea (p ropo sed by [ 15] fo r the single time-delay systems case) co nsists in associating the delay o perator as an un certainty wh ich must be bound ed. Hence, Theorem 1 may be ap plied to th e m ultiple tim e delays system ( 6) b y r ewritting it as an interconn ected system (see Figure 5) with E = 1 , w z }| {     X ( t ) ˆ x τ f ( t ) ˆ b τ b ( t ) ˆ x τ ( t )     = ∇ z }| {     s − 1 1 4 D τ f D τ b D τ     z z }| {     ˙ X ( t ) ˆ x ( t ) ˆ b ( t ) ˆ x ( t )     (9) and E z z }| {     ˙ X ( t ) ˆ x ( t ) ˆ b ( t ) ˆ x ( t )     = A z }| {     A ¯ A d BK 2 BK 1 E 1 0 0 0 E 2 0 0 0 E 1 0 0 0     w z }| {     X ( t ) ˆ x τ f ( t ) ˆ b τ b ( t ) ˆ x τ ( t )     (10) where ˆ b ( t ) =    b ( t ) . . . b ( t )    , ˆ b τ b ( t ) =    b ( t − τ b 1 ) . . . b ( t − τ b N )    , ˆ x τ f ( t ) =    x 1 ( t − τ f 1 ) . . . x N ( t − τ f N )    , ˆ x τ ( t ) =    x 1 ( t − τ 1 ) . . . x N ( t − τ N )    , ˆ x ( t ) =    x 1 ( t ) . . . x N ( t )    , D ♦ =    e − ♦ 1 s 0 . . . 0 e − ♦ N s    , (11) X ( t ) = [ ˆ x ′ b ( t )] ′ . T he delay matrix o perator s D ♦ ( ♦ repr esents τ f , τ b or τ ) mu st b e defined to create the delayed signals ˆ x τ f , ˆ x τ and ˆ b τ b (11). W e aim at proving the stab ility ( i.e. n o poles in the righ t hand side of the comp lex plane fo r all values of the delay and fo r all values of the uncer tainty ∇ ∈ Ξ ) which problem can be recast in the presen t framew or k as the well- posedness of th e fee dback system for all s ∈ C + , for all v alues of the delays ( τ i , τ f i and τ b i , i = { 1 , . . . , N } ) an d all admissible uncertain ties ∇ ∈ Ξ . Then a con servati ve ch oice of quad ratic separator that fullfils (8) is of the form Θ =             0 0 0 0 − P 0 0 0 0 − Q f 0 0 0 0 0 0 0 0 − Q b 0 0 0 0 0 0 0 0 − Q 0 0 0 0 − P 0 0 0 0 0 0 0 0 0 0 0 0 Q f 0 0 0 0 0 0 0 0 Q b 0 0 0 0 0 0 0 0 Q             , (12) with P ∈ R N + 1 × N + 1 > 0 and Q ♦ = d iag ( q ♦ 1 , . . . , q ♦ N ) wh ere q ♦ i are p ositiv e scalars fo r all i = { 1 , . . . , N } an d for ♦ = { f , b , / 0 } (see [15] for a simpler case). Then, it re mains to test the first cond ition (7). Sin ce th is ine quality d oes no t d epend on delay s, th e derived cond ition is said to b e I ndepen dent Of Delays (IOD). Conse- quently , th is latter criter ion provides state feedb ack ga ins that stabilize th e system for all p ossible values of delay s. It thus appears th at this meth od is very co nservati ve and it would be interesting to ha ve a condition depend ing o n delays. W e aim n ow at d eriving a Dela y Depend ent r esult ( i . e . the well known DD ap- proach wh ich ensures the stability for all values of the de lay betwe en zero and an upper b ound ) with the same meth odolog y . T o do so note that the results were delay indepen dent becau se op erators e − ♦ i s , when s ∈ C + , can only be characterize d as un- certainties norm bounded by 1. T o get delay dependen t r esults it is therefore needed to have ch aracteristics that d epend o n u pper bou nds of delays. This can be don e no ting that for all s ∈ C + and a gi ven delay h ∈ [ 0 ¯ h ] one h as | s − 1 ( 1 − e − hs ) | ≤ ¯ h and th is operator is such that V ( s ) = s − 1 ( 1 − e − hs ) ˙ X ( s ) where V ( s ) a nd ˙ X ( s ) are the L aplace transform s respectiv ely o f v ( t ) = x ( t ) − x ( t − h ) and ˙ x ( t ) . Intro ducing this n ew o perator for each delay τ i , τ f i and τ b i , i = { 1 , . . . , N } leads to write the delay dep endent stability problem of (6) as a well-posedness problem of the system in Figure 5 with E =                 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 E 1 0 0 0 − 1 0 0 E 2 0 0 0 0 − 1 0 E 1 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0                 , w =             X ( t ) ˆ x τ f ( t ) ˆ b τ b ( t ) ˆ x τ ( t ) w 1 ( t ) w 2 ( t ) w 3 ( t )             , A =                 A ¯ A d BK 2 BK 1 0 0 0 E 1 0 0 0 0 0 0 E 2 0 0 0 0 0 0 E 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 E 1 − 1 0 0 − 1 0 0 E 2 0 − 1 0 0 − 1 0 E 1 0 0 − 1 0 0 − 1                 , z =             ˙ X ( t ) ˆ x ( t ) ˆ b ( t ) ˆ x ( t ) ˙ ˆ x ( t ) ˙ ˆ b ( t ) ˙ ˆ x ( t )             , (13) where ¯ A d = A d  1 N 0 1 × N  , E 1 =  1 N 0 N × 1  , E 2 =    0 N 1 . . . 1    and w 1 ( t ) = ˆ x ( t ) − ˆ x τ f ( t ) , w 2 ( t ) = ˆ b ( t ) − ˆ b τ b ( t ) , w 3 ( t ) = ˆ x ( t ) − ˆ x τ ( t ) and with the augm ented un certain operator ∇ = Diag  s − 1 1 4 , D τ f , D τ b , D τ , I τ f , I τ b , I τ  (14) where I ♦ = d iag ( 1 − e − ♦ 1 s s , . . . , 1 − e − ♦ N s s ) , ♦ = { f , b , / 0 } . Delay opera tors D ♦ are thus isolated as w ell as op erators I ♦ ensuring a b etter bou nd on each delays ( delay depen - dent c ase, see [15]). Expressing the covering set on every block o f ∇ , a c onservati ve choice of separator satisfying the second inequa lity ( 8) of Theor em 1 is Θ =  Θ 11 Θ 12 Θ ′ 12 Θ 22  (15) with Θ 11 = d iag  0 N + 1 , − Q f 0 , − Q b 0 , − Q 0 , − Q f 1 T f 2 , − Q b 1 T b 2 , − Q 1 T 2  , Θ 12 = d iag ( − P , 0 6N ) , Θ 22 = d iag  0 N + 1 , Q f 0 , Q b 0 , Q 0 , Q f 1 , Q b 1 , Q 1  , Q ♦ 0 = d iag  q ♦ 01 , . . . , q ♦ 0 N  , Q ♦ 1 = d iag  q ♦ 11 , . . . , q ♦ 1 N  , T ♦ = d iag  τ ♦ 1 , . . . , τ ♦ N  , (16) where P ∈ R N + 1 × N + 1 > 0 and q ♦ i are p ositiv e scalars for all i = { 1 , . . . , N } and for ♦ = { f , b , / 0 } . Since, the inequality (8) is satisfied b y definition of the oper ator bounds of th e uncertain matrix ∇ , it r emains to verify the first on e (7). Af ter som e algebraic manipulatio ns, i t can b e shown that th is latter inequality can be expressed as   Ξ 1 Ξ 2 Q 1 T Ξ 2 Q f 1 T f ∗ Q 1 T 0 N ∗ ∗ Q f 1 T f   > 0 (17) where Ξ 1 = N T 1 Θ N 1 + N T 2 Θ N 1 + N T 1 Θ N 2 , Ξ 2 =  0 N × 2N + 1 E 1 BK 2 E 1 BK 1  T , N 1 =                   A ¯ A d 0 N + 1 × 2N E 1 E 2 E 1 0 3N × 3N E 1 A E 1 ¯ A d 0 N × 2N 0 N × N + 1 E 2 ¯ A d 0 N × 2N E 1 A E 1 ¯ A d 0 N × 2N 1 4N + 1 E 1 E 2 E 1 − 1 3N                   , N 2 =         0 N + 1 × 2N + 1 BK 2 BK 1 0 3N × 4N + 1 0 N × 2N + 1 E 1 BK 2 E 1 BK 1 0 N × 4N + 1 0 N × 2N + 1 E 1 BK 2 E 1 BK 1 0 7N + 1 × 4N + 1         . (18) This latter conditio n giv es the following theorem: Theorem 2 F o r given scalars τ i , τ b i and τ f i for i = { 1 , ..., N } , if there e xists a N + 1 × N + 1 positive definite matrix P and N × N d iagonal positive matrices Q f k , Q b k , Q k with k = { 0 , 1 } an d K 1 , K 2 such that the inequ ality ( 17) is satisfie d, then the system (6) is stable. The state feedbac k g ains K 1 and K 2 are thus derived solvin g the inequality (17) of Theorem 2. Regarding th e synth esis p roblem , since K 1 and K 2 are decision variables, condition (17) is bilinear and a g lobal optimal solution cannot be foun d. Nevertheless, the feasibility problem can still be tested to provide a suboptimal solution: ⊲ using a BMI solver as penbmi [23]. ⊲ using a relaxation algorith m [30]: • s t e p 0. Initializatio n: K 1 = K 1 0 , K 2 = K 2 0 . • s t e p 1. Run LMI com putation (in equality (17) with fixed K 1 , K 2 ) ⇒ backu p P 0 = P , Q 1 0 = Q 1 , Q f 1 0 = Q f 1 . • s t e p 2. Set P = P 0 , Q 1 = Q 1 0 , Q f 1 = Q f 1 0 and K 1 , K 2 are f ree, run LMI computatio n. • s t e p 3. If con dition feasible: stop algo rithm. Other wise: retur n to s t e p 0 It is worthy to n ote that the state feedback gains K 1 and K 2 is easily and routinely derived solvin g the matrix inequality (17 ) while m any A QM suc h RED are well known to be difficult to tune as it has been stated in [10], [6]. 4 NS-2 SIMULA TI ONS In this section, we p erform simulatio ns with the network simulator NS-2 [11] (release 2.30) to validate the exposed theo ry . Throug hout this par t, the e fficiency of the pro- posed mechanism is evaluated and com pared to few existing A QM. Consider the n u- merical example of the Figur e 6. So, the objective is to regulate the queu e leng th of the ro uter to a d esired level b 0 = 100 packets while the max imal buffer size is set to 400 packets. Propag ation times are as illustrated o n Figure 6. The lin k bandw idth is fixed to 1 0 M b ps , that is 25 00 packet/s co nsidering packet size of 500 by tes. Hence, at the equilib rium the qu eueing delay is equ al to 40 ms . Each of the thre e sources u ses TCP/Reno and establishes 10 co nnection s gene rating long lived T CP flows (like FTP connectio ns). Upon these latter specifications, the equilibriu m point (3) is deriv ed: b 0 = 1 00 p kt , τ 1 0 = 1 50 ms , τ 2 0 = 2 50 ms , τ 3 0 = 3 50 ms , x 1 0 = x 2 0 = x 3 0 = 8 3 . 33 pkt / s (rate for each connection s of the three sou rces) and p 0 = 10 − 3 [ 9 . 508 3 . 444 1 . 760 ] ′ . Remark 2 I t is worthy to note that th e second equ ation in (3) enab les to choose arbi- trarily the rate assigned to ea ch sou r ces. Th is latter o ption a llows us to make a service differ entiation between senders. In the pr evious e xample, w e choose to assign the same sending rate for all sour ces (and for each connections) ensuring then fairness. Figure 6: Network topolo gy: an example The interco nnected sy stem of the form (6) mod eling a such network c onfigur ation is then written as     δ ˙ x 1 ( t ) δ ˙ x 2 ( t ) δ ˙ x 3 ( t ) ˙ b ( t )     =     − 2 . 02 0 0 − 0 . 22 0 − 0 . 75 0 − 0 . 05 0 0 − 3 . 88 − 0 . 01 0 0 0 0         δ x 1 ( t ) δ x 2 ( t ) δ x 3 ( t ) δ b ( t )     +     − 2 . 22 − 2 . 22 − 2 . 22 0 − 1 . 33 − 1 . 33 − 1 . 33 0 − 0 . 95 − 0 . 95 − 0 . 95 0 20 20 20 0         δ x 1 ( t − 0 . 025 ) δ x 2 ( t − 0 . 05 ) δ x 3 ( t − 0 . 075 ) δ b ( t )     +     − 912 0 0 0 − 884 0 0 0 − 876 0 0 0       K 1   δ x 1 ( t − 0 . 15 ) δ x 2 ( t − 0 . 25 ) δ x 3 ( t − 0 . 35 )   + K 2   δ b ( t − 0 . 125 ) δ b ( t − 0 . 2 ) δ b ( t − 0 . 275 )     (19) Then, we have to find matrices K 1 and K 2 such th at the feedba ck system (19) is sta- ble and the regulation ar ound th e equ ilibrium p oint is ensu red. Applying T heorem 2 exposed in section 3, tw o stabilizing ma trix gains K 1 = 10 − 3   − 0 . 09 0 0 0 0 . 61 0 0 0 0 . 76   , K 2 = 10 − 3   0 . 27 0 0 0 0 . 13 0 0 0 0 . 08   (20) can b e fou nd. Con sidering the ne twork of Fig ure 6, w e h ave simulated the conges- tion phenom enon at a router applying the drop tail mech anism and d ifferent A QM. Effect of these latter d ropp ing strategies on the qu eue le ngth have been ev alu ated and are illustrated on Figure 7. T able 2 presents additional character istics of the results. As expected, the d rop t ail mechanism maintains the qu eue size c lose to the buffer overflow (maximal buffer size: 400 p kt ) in volving m any un contro lled dropp ed packets. Futher- more, a such large queue size implies a lar ge queueing delay and also large oscillations providing an important delay jitter (see T able 2). In order to maintain a controlled (and desired) queueing delay with a low jitter , the queue length at the router should be regulated. This issue is tackled with the use of A QM (adjustments o f th e dif fer ent setting parameters are shown in table 1 ). On Figu re T able 1: Adjustme nt of parameter setting of each A QM RED min t h =50, max t h =300, w Q =5.99e- 06, max p =0.1, f s =160Hz REM γ =0.003 , Φ =1.001, q re f =100p kt PI a=1.483 e-05, b=1.47 9e-05, q re f =100p kt, f s =160Hz SF gains K 1 and K 2 (20), equilibriu m point (3) 7, contro l per forman ces of A QM can b e me asured by two observations: the transient perfor mance (in p articular the speed of response) a nd the stead y state error contro l. It can be observed that the state feed back ( SF ) (2 0) ensures an efficient control since the queue len gth ma tches its equ ilibrium faster than o ther cases. Secondly , SF provides a better regula tion causing less o scillations and good precision. Th ese characteristics ca n be verified in T able 2 which analyses e ach respon se fr om Figur e 7 . In th e simulation, A QM was designed to regulate qu eue size at 100 pkt (equilibr ium) av oiding severe con- gestion and en suring then a stable que ueing delay of 40 ms . First, in T able 2 , the mea n (and average qu eueing d el a y since the queue ing delay is expressed b y b ( t ) / C ) shows the accu racy of each A QM according the desired pr escribed queue len gth. Next, the s t a nd ard d eviat ion (and average d el ay jit t er ) sho ws th e ab ility of each A QM to main- tain the q ueue length (and queue ing delay) close to the desired e quilibrium pr oviding then an efficient regu lation. Note that REM has the lowest que ueing delay , howe ver the o bjective is to gua rantee the qu eue leng th stability inducin g then a queuein g delay equals to 4 0 ms . If a lower queueing delay is req uired, one ju st has to c hange the eq ui- librium point (redu cing the desired q ueue length). Besides, the stability o f the conge stion phenomeno n keepin g a stable queue length (and thus a stab le queueing delay), allows to control the R TT ( τ i for i = { 1 , ..., N } ) for all sources to a desired value with low v ariations (see Figu re 8). Than ks to the ef ficient regulation applied by S F , a QoS in terms of R TT and d elay jitter is h ence g uaranteed . It provides thu s an interesting feature which i s relev ant for streaming and real-time ap- plications. Altho ugh RED performs a go od regulation on t h e q ueue leng th and th e R TT at the steady state, its respo nse time is very slow wh ich makes RED inefficient against short-lived traffic perturbation s [1]. T able 3 presents statistics related to the packet ar riv al rates of each user wh en dif- ferent A QM are implemented at the r outer and b ears out the a ccuracy and the co ntrol efficiency o f the pro posed S F . The prescribed eq uilibrium rate fo r each connection (ac- cording to (3)) that establishes fairne ss is x 1 0 = x 2 0 = x 3 0 = 8 3 . 33 pkt / s . As it can b e seen, only S F is able to keep arriv al ra tes comparativ ely close to the eq uilibrium v alue. Futhermo re, based on th e Jain’ s fairness ind ex = ( ∑ x i ) 2 ( n ∑ x 2 i ) [19] (the mo re the index is close to 1, th e more th e distribution of the resou rces is f air), we ob serve that SF app lies a fair strategy . 5 CONCLUSION In th is p aper the design of an A QM for con geston contro l of a sing le ro uter has bee n presented. The consider ed topology consists in se veral TCP sou rces sending long - li ved flows th rough a router t o their respective rec eiv ers.T o supp ly the TCP congestion control mech anism, an A QM mu st be im plemented to the rou ter . Based on a m odified mathematical mo del o f the protoc ol behavior , a such A QM has been developed with control theory tools. I ndeed, in a tim e delay system frame work, a control law has been 0 10 20 30 40 50 60 70 80 0 100 200 300 400 DropTail 0 10 20 30 40 50 60 70 80 0 100 200 300 400 RED 0 10 20 30 40 50 60 70 80 0 100 200 300 400 REM 0 10 20 30 40 50 60 70 80 0 100 200 300 400 PI 0 10 20 30 40 50 60 70 80 0 100 200 300 400 time (s) State feedback Figure 7: Evolution of the queue length b ( t ) (p kt): th e desired size is 100 pkt s T able 2: Some statistics on the qu eue length for different A QM DT RED REM PI SF M ean ( p kt ) 317.6 1 03.8 94.8 99.4 10 2.3 St and . d ev . ( pkt ) 84 21.7 70.7 35 .1 23. 3 A verage queuein g d el ay ( ms ) 127 41.5 37.9 3 9.8 40 .9 A verage d el ay jit t er ( ms ) 33.6 8.7 28.3 14 9.3 T able 3: So me statistics on arriv al rates for dif fer ent A QM RED REM PI SF users Mean (pkt/s) Stand. dev . (pk t/s) 1 2 3 147 83 1 48 61 3 6 372 1 2 3 147 81 60 56 34 23 1 2 3 146 88 109 66 35 335 1 2 3 104 91 88 41 30 40 Jain’ s fairness index 0 .9450 0.870 3 0.957 9 0.994 6 0 10 20 30 40 50 60 70 80 250 300 350 400 450 SF 0 10 20 30 40 50 60 70 80 250 300 350 400 450 PI 0 10 20 30 40 50 60 70 80 250 300 350 400 450 RED 0 10 20 30 40 50 60 70 80 250 300 350 400 450 Time (s) REM Figure 8: Evolution of the R TT of con nections from source 3 (ms): the expected value is 350 ms propo sed and then th e stability analy sis of the f eedback sy stem has b een p erform ed. Consequently , the regulation of flows and the queue s ize o f at the router is ensured. At last, a numerical example and simulations have shown the effecti veness of the prop osed methodo logy . Refer ences [1] Y . Ariba, Y . L abit, and F . Gouaisbau t. Design an d perf ormance ev alu ation o f a state-space based aqm. In IARIA Interna tional Confer ence on Communication Theory , Reliability , and Quality of Service , pages 89–94, July 2008. [2] S. Athu raliya, D. Lapsle y , and S. Low . An enhanced random early marking algo- rithm for inter net flow control. I n IEE E INFOCOM , pages 1425– 1434 , December 2000. [3] J. A weya, M . Ouellette, and D. Y . M ontuno . A simple mech anism for stabilizing network qu eues in tcp/ip networks. Int. Journal of Network Man agement , 17:27 5– 286, 2007. [4] P . Barford and D. Plonka. Cha racteristics of network traffic flow anom alies. In In Pr oceedings of the ACM SI GCOMM Internet Mea sur ement W orkshop , November 2001. [5] S Bhan darkar, A. L. Narasimh a Redd y , Y . Zhang , and D. Logu inov . Em ulating aqm from end h osts. In A CM SIGCOMM ’07: Pr oceeding s of th e 2007 con- fer ence on Applica tions, technologies, a r chitectur es, and p r otocols for computer communica tions , pages 3 49–3 60, 2 007. [6] T . Bon ald, M . May , and J.-C. Bolot. Analytic ev aluation of red per forman ce. In IEEE INFOCOM , volume 3, pages 1415 –142 4, March 200 0. [7] S. Boyd, L. El Gh aoui, E. Feron , and V . Balakrishnan. Linea r Matrix Inequa lities in System an d Contr ol Theory . SIAM , Philade lphia, USA, 1 994. in Stud ies in Applied Mathematics, v o l.15. [8] B. Braden , D. Clark, and J. Crowcroft. Recom mendatio ns on queue management and congestion av oidance in the interne t. RFC 2309, April 1998 . [9] C.K. Chen , Y .C. Hun g, T .L. Liao, and J.J. Y a n. D esign of robust active queue managem ent controllers for a class of tcp co mmunica tion netw or ks. Information Sciences. , 177(1 9):405 9–4071, 2007. [10] M. Christiansen , K. Jeffay , D. Ott, and F . Smith. T u ning red fo r web traffic. In AC M/SI GCOM , pages 139–15 0, 2 000. [11] K. Fall and K. V aradh an. Th e n s manual. notes and d ocumen tation on the software ns2-simulator, 20 02. URL: www .isi.edu/nsnam /ns/. [12] Y . Fan, F Ren, and C. Lin. Design a pid controller for activ e queu e management. In IE EE Interna tional Symposium o n Compu ters and Commun ication (ISCC) , volume 2, pages 985–990, 2003. [13] V . Firoiu and M. Bo rden. A stu dy o f activ e queue ma nagemen t for c ongestion control. In IEEE INFOCOM , v olum e 3, pages 1435 – 1444, March 2000. [14] S. Floyd and V . Jaco bson. Random early detection gateways for conge stion av oid - ance. IEEE/ACM T r an sactions on Networking , 1:397– 413, August 1993. [15] F . Gouaisba ut and D. Peaucelle. A note on stability of time delay systems. In 5 t h IF AC Symp osium on Robust Con tr ol Design (R OCOND’06) , T oulou se, France, July 2006. [16] H. Han , C. V . Hollot, Y . Chait, an d V . Misra. Tcp networks stabilized by buf fer- based aqms. In IEEE INFOCOM , pages 964–974, March 2004. [17] C. V . Hollot, V . Misra, D T owsley , an d W . Gon g. Analysis an d de sign of co n- trollers for aqm r outers suppo rting tcp flows. IEEE T rans. o n Automat. Con tr ol , 47:945 –959 , June 2002. [18] V . Jacobson. Con gestion avoidance and control. In A CM SI GCOMM , pa ges 3 14– 329, Stanford, CA, August 1988. [19] R. Jain , D. Chiu , and W . Hawe. A qu antitative measure of fairness a nd discrim- ination for resou rce allocation in shared computer systems. T ech nical Report TR301, Digital Equipmen t C o rp, 1984. [20] F . P . Kelly , A. Maulloo , and D. T an. Rate contro l for commu nication n etworks: shadow p rices, propo rtional fairness and stability . Journal of the Operational Resear ch Society , 49:237–2 52, M arch 1998. [21] K. B. Kim. Design of feed back co ntrols suppor ting tcp b ased on the state sp ace approa ch. In IEEE T r ans. o n Automat. C o ntr ol , volume 51 (7), July 2006. [22] Seong Soo Kim and A. L . Narasimha Reddy . Netviewer: a n etwork traffic visu- alization and an alysis tool. I n LISA’05: Pr oceeding s o f the 19th con fer ence on Lar ge In stallation Sy stem Administration Con fer ence , pages 185– 196. USENIX Association, 2005. [23] M. K ocvara and M. Stingl. bilinear matrix ineq ualities: Penbm i. PENOPT GbR, http://www .penopt.c om/. [24] S. Kunniy ur an d R. Srikant. Analy sis and design of an adaptiv e virtu al queu e (avq) algo rithm for active queue man agement. I n ACM SIGCOMM , pag es 123– 134, San Diego, C A, U SA, aug 2001. [25] Y . Labit, Y Arib a, an d F . Go uaisbaut. On desig ning lyapun ov-krasovskii based controller s fo r aqm ro uters sup porting tcp flows. In 46th IEEE Con fer ence on Decision and Contr ol , pages 3818– 3823 , New Or leans, USA, Decembe r 2007. [26] L. Le, J. Aikat, K. Jeffay , an d F . Don elson Smith. The effects o f activ e queu e managem ent on web p erfor mance. In ACM SIGCOMM , pag es 265– 276, August 2003. [27] H. S. Low , F . Pagan ini, and J.C. Doyle. I nternet Congestion Contr ol , volume 22, pages 28–4 3. IEEE Control Systems Magazine, Feb 2002. [28] S. Manfredi, M. d i Bernardo, and F . Ga rofalo. Robust output feed back active queue management contr ol in tcp networks. In IEEE Co nfer ence on Decision and Contr ol , pages 1004– 1009 , December 2004. [29] V . Misra, W . Gon g, and D T owsley . Fluid- based analy sis of a network of aqm routers supp orting tcp flows with an ap plication to red. In ACM SIGCOMM , pages 151– 160, August 2 000. [30] S.I. Niculescu . Delay Effects on Stability . A Robust Contr ol Ap pr oach , volume 269 of Lectu r e Notes in Con tr ol an d Information Scie nces . Springer-V erlag , Heildelberg, 2001. [31] A. Papachristodo ulou. G lobal stability of a tcp /aqm proto col for arbitr ary n et- works with delay . I n IEEE CDC 2004 , pages 1029– 1034, December 2 004. [32] D. Peaucelle, D. Arzelier, D. Henrion , an d F . Go uaisbaut. Qu adratic separ ation for feedback connection of an uncertain matrix an d an implicit linear transforma- tion. A utoma tica , 43(5):795– 804, 2007 . [33] P . F . Quet and H. ¨ Ozbay . On the design of aqm suppor ting tcp fl ows using robust control theory . IEEE T rans. on A utomat. Contr ol , 49:10 31–10 36, June 2004 . [34] K. K. Ramak rishnan a nd S. Floyd. A prop osal to a dd explicit con gestion no tifi- cation (ecn) to ip. RFC 2 481, January 1999 . [35] S. Ryu, C. Rump , an d C. Qiao. Ad vances in acti ve queue man agemen t (aqm ) based tcp congestion control. T eleco mmunicatio n Systems , 4:317 –351 , 2004 . [36] R. Srikant. Th e Mathematics of Internet Congestion Contr ol . Birk hauser, 2004 . [37] D. W ang and C. V . Hollot. Robust a nalysis and design o f contro llers for a sin- gle tcp flow . In IE EE Interna tional Confer ence on Commu nication T echnology (ICCT) , volume 1, pages 276– 280, April 2003.