Stabilized MAX-MIN Flow Control Using PID and PII 2 Controllers

Transcription

1 Stabiized MAX-MIN Fow Contro Using PID and PII Controers Jeong-woo Cho and Song Chong Department o Eectrica Engineering and Computer Science Korea Advanced Institute o Science and Technoogy Daejeon 5-71, Repubic o Korea. Abstract This paper describes an anaytica ramework or the weighted max-min ow contro o eastic ows in packet networks using PID and PII controer when ows experience heterogeneous round-trip deays. Our agorithms are scaabe in that routers do not need to store any per-ow inormation o each ow and they use simpe irst come irst serve (FCFS) discipine, stabe in that the stabiity is proven rigorousy when there are ows with heterogeneous round-trip deays. We irst suggest two cosed-oop system modes that approximate our ow contro agorithms in continuous-time domain where the purpose o the irst agorithm is to achieve the target queue ength and that o the second is to achieve the target utiization. The sow convergence o source rates traversing routers with empty buers which is inherent in many ow contro agorithms can be resoved by the second agorithm. Based on these modes, we ind the conditions or controer gains that stabiize cosedoop systems when round-trip deays are equa and extend this resut to the case o heterogeneous round-trip deays with the hep o Zero excusion theorem. We simuate our agorithms with optima gain sets or various conigurations incuding a mutipe botteneck network to veriy the useuness and extensibiity o our agorithms. I. INTRODUCTION Recenty many eorts have been devoted to provide a ramework or designing best-eort service networks that can oer ow-oss, ow-deay data services where ow contro pays a major roe in controing congestion as we as aocating bandwidth among users by enorcing users to adjust their transmission rate in a certain way in response to congestion in their path. The potentia advantages o such networks woud be the abiity to oer even rea-time services without the need or compicated admission contro, resource reservation or packet scheduing mechanisms. Fow contro is a distributed agorithm to airy share network bandwidth among competing data sources whie maximizing the overa throughput without incurring congestion. The most common understanding o airness or a best-eort service network is max-min airness as deined in [1]. The intuition behind the max-min bandwidth sharing is that any ow is entited to as much as bandwidth use as is or any other ow with the assumption that a ows have equa priority. This intuition naturay eads to the idea o maximizing the This work was supported in part by Korea Science and Engineering Foundation under Grant R and in part by university IT research center program o the government o Korea. bandwidth use o ows with minimum bandwidth aocation, thus giving rise to the term max-min ow contro. The appropriateness o max-min airness as a bandwidth sharing objective has recenty been questioned by Key [] who has introduced the aternative notion o proportiona airness. The objective o proportiona airness may be interpreted as being to maximize the overa utiity o rate aocations assuming each ow has a ogarithmic utiity unction. The advantage o proportiona airness is that it can achieve greater overa throughput than max-min airness by penaizing ows proportiona to number o hops in the path in rate aocation []. However, this advantage is probematic regarding airness in that the path o a ow is chosen by the routing protoco in the network, not by the user, so that it is obviousy unair to penaize ows because o their ong path ength. Thus, one can sti argue that max-min airness might be the better choice than proportiona airness rom user s viewpoint since max-min airness treats ows equay irrespective o their path ength. This paper concerns the design o minimum pus weighted max-min ow contro [4], a generaization o max-min ow contro, where each ow is associated with two parameters, its weight w i and minimum rate requirement m i, such that the minimum rate o each ow is guaranteed as requested during the entire hoding time o the ow and the bandwidth unused ater aocating the minimum rates is shared by a ows in the weighted max-min sense. An increase in the weight o a ow eads to an increase in the bandwidth share o the ow with the assumption that users pay more or a higher weight. Let us deine ow i s source rate, say a i,tobea i w i i + m i where i is the max-min air share o the bandwidth unused ater aocating the minimum rates to a ows. Let us denote the set o a inks, the set o a ows and the set o ows traversing through ink by L, N and N(), respectivey. Then, the weighted max-min airness can be deines as oows. Deinition 1: A rate vector <a 1,, a N > is said to be easibe i it satisies a i, i N and i N() a i α T µ, L. Deinition : A rate vector <a 1,, a N >, where a i = w i i + m i, is said to be weighted max-min air i it is easibe, and or each i N and easibe air rate vector < 1,, N > or which i < i, there exists some i with i i > i. Gobecom /4/$. 4 IEEE

2 Here µ denotes the capacity o ink and αt is a constant deining target ink utiization ( < αt 1). Note that admission contro is necessary to ensure i N() m i <αt µ or a L so that the minimum rate o each ow is guaranteed as requested during the entire hoding time o the ow. Deinition can be restated more inormay as oows: a rate vector <a 1,, a N > is said to be weighted maxmin air i it is easibe and or each user i N, its air rate i cannot be increased whie maintaining easibiity without decreasing the air rate i or some user i or which i < i. In this paper, our goa is to provide a contro-theoretic ramework based on deterministic uid modes that reveas not ony the existence o such a distributed iterative agorithm but aso an expicit stabiity condition o the agorithm in presence o ows with heterogeneous round-trip deays. A. Our Contributions We propose two contro-theoretic max-min ow contro modes and agorithms. The irst agorithm satisies Deinition or αt = 1 such that in the steady-state, bandwidth at every botteneck ink is used to the u whie the minimum pus weighted max-min airness is maintained in bandwidth sharing. Moreover, the queue ength at every botteneck ink converges to the target vaue, say qt, thereby achieving constant queueing deay expressed by q T µ. In contrast, the second agorithm satisies Deinition or < αt < 1 such that in the steady-state, every botteneck ink achieves its target utiization (αt µ ) and hence virtuay zero queueing deay whie the minimum pus weighted max-min airness is maintained. The motivation behind the second agorithm is making the queueing deay at each ink to be virtuay zero and improving transient perormance by absorbing transient overshoots occurring beore convergence at the expense o reduced ink utiization. But the major advantage o the second agorithm is that the sow adaptation o source rates traversing routers with empty buers is overcome with this agorithm. The suggishness o PI controers based on queue ength is aso pointed out in [5]. In summary, the ormer can oer zero-oss, constant-deay data services at u utiization o botteneck inks whereas the atter can oer zero-oss, zerodeay data services and aster rate adaptation at the expense o reduced botteneck ink utiization. In the ormer, the dierence between queue ength and target queue ength, i.e., q (t) qt, is used as a congestion measure at each ink and the max-min air rate i is computed by a PID (proportiona integra derivative) controer o this queue-ength based congestion measure. In the atter, i N() a i(= w i i + m i ) αt µ is used as a congestion measure at each ink and the max-min air rate i is computed byapii (proportiona integra doube integra) controer o this aggregate-ow based congestion measure. We show that the cosed-oop characteristics o the network under these two dierent agorithms are actuay identica, yieding the identica stabiity condition. By appeaing to the Nyquist stabiity criterion [] and the Zero excusion theorem in robust contro theory [7], we derive the suicient and necessary condition + m Q Fows bottenecked at ink b(i) Q a () t Q Fows bottenecked at ink 1 r() t 1 t r () Q at () 1 w Q Sources Fig m 1 w 1 b Q b 1 () t - () t - Link min[ ] - L () t + () t Link L Link + L The network architecture or weighted max-min airness. or the asymptotic stabiity o the network as an expicit and usabe unction o the upper bound τ o a round-trip deays ( τ τ i or a i N where τ i is the round-trip deay o ow i). Moreover, we ind optima controer gains or both PID and PII controers to maximize the asymptotic decay rate o the cosed-oop dynamics, thereby achieving aster convergence. Finay, both PID and PII controers are highy scaabe in that the computationa compexity o the ink agorithm is O(1) with respect to number o ows passing through a ink and no per-ow queueing impementation is necessary at any ink. Some reated works are introduced in [8]. II. NETWORK MODEL AND CONTROLLERS In this section, we propose network modes and controers which achieve weighted max-min airness. The network architecture with mutipe sources and inks is depicted in Fig. 1. Let us consider a botteneck ink L. Then, the dynamics o the buer o the ink can be written by q (t) = i N() a i(t τ, i ) µ, q (t) > [ i N() a i(t τ, i ) µ ] +, q (t) = where a i (t) is the sending rate o source i, τ, i is the orwardpath deay rom source i to ink, µ is the ink capacity o the ink and the saturation unction [ ] + max[, ] represents that the q (t) cannot be negative. A source i sends packets according to air rate vaue assigned by the network. To achieve weighted max-min airness, et us assume that the source sends packets according to the minimum vaue among the air rate vaues assigned by the inks aong the path o its ow. Thus we assume the oowing source agorithm. + (1) a i (t) =m i + w i min [ (t τ,b i )], () L(i) }{{} i(t) where L(i) is the set o inks through which ow i traverses, (t) is the rate vaue assigned by the ink on the path o ow i and τ,b i is the backward-path deay rom ink to source i. Because min[ ] operation is taken over a inite number o inks, Gobecom /4/$. 4 IEEE

3 there shoud exist at east one ink such that = min[ ]. Thereore, each ow i has at east one botteneck L(i). There are severa assumptions empoyed or the anaysis o the network mode. A.1. We assume that the sources are persistent unti the cosed-oop system reaches steady state. By persistent, we mean that the source aways has enough data to transmit at the aocated rate. A.. There are two deays, say, the orward-path deay τ, i and the backward-path deay τ,b i, which incude propagation, queueing, and transmission and processing deays. We denote the sum o two deays by τ i and assume that this is constant. A. The PID Contro Mode To contro ows and to achieve weighted max-min airness, we use a PID ink controer at each ink. In PID ink controer mode, there is a speciied target queue ength qt to avoid underutiization o the ink capacity. Because we have a nonzero target queue ength qt, PID mode impies that αt = 1 in Deinition 1. Each ink cacuates the common eedback rate vaue (t) or a ows traversing through the ink according to PID contro mechanism. In genera, a proportiona term increases the convergence speed o transient responses and reduces errors caused by disturbances. An integra term is necessary to eiminate steady state error and it decreases the size o stabiity region. A derivative term adds some damping and extends the area o stabiity region and it aso improves the perormance o transient periods. Let us denote the set o ows bottenecked at ink and its cardinaity by Q and Q.Theink agorithm with The PID controer that uses the dierence between q (t) and qt as its input is given by [ (t) = 1 t )] + (g P e 1 Q (t) w +gi e 1 (t)dt + gdė 1 (t) () where e 1(t) q (t) qt is the error signa between contro target and current output signa and, g P > and g I,g D. Here, Q w denotes the sum o ocay bottenecked ows weights, i.e., Q w i Q w i. For convenience in deriving our resuts, we use the deinition ρ i w i / Q w. Then it is satisied that i Q ρ i =1and ρ i > because we require w i >. Suppose that the cosed-oop system has an equiibrium point at which the derivatives o the system variabes are zero, i.e., im t q (t) =, im t q (t) =qt, im t a i (t) = a is and im t (t) =s.tobemoreorma,theseto ows bottenecked at ink is given by Q = {i i N() and a is = w i s + mi} (4) and the set o a ows not bottenecked at ink but traversing through ink, N() Q, is given by N() Q = {i i N() and a is = w i b(i) s + m i and b(i) s < s }. (5) where b(i) L(i) (b i ) is some botteneck or ow i N() Q. Then the equation (1) impies that the ink capacity µ in PID ink controer mode is uy utiized as oows. a is = µ. () i N() Using (), and the deinitions (4) and (5), we obtain (w i s + m i )+ (w i s b(i) + m i )=µ i Q i N() Q which estabishes that the PID contro mode achieves the oowing weighted max-min airness property r is w i s = wi Q w µ w i b(i) s m i. (7) i N() Q i N() B. The PII Contro Mode Instead o using e 1(t), one can use e (t) i N() a i(t τ, i ) αt µ as an input o ink controers where αt is the target utiization o ink and shoud be a positive vaue smaer than 1. In this case, one can use PII contro mode as oows because we now use rate error signa instead o queue error signa. [ (t) = 1 t t t )] + (h Q w P e (t) +hi e (t)dt + h I e (t)dtdt (8) where h P,h I and h I >. It shoud be remarked that PID and PII mode are not identica because ė 1(t) = q (t) = i N() a i(t τ, i ) µ e (t) or q (t) >. In this mode, the purpose o contro is to achieve the target utiization, α T. In PII mode, note that q (t) = (1 αt )µ < when q (t) > and e (t) =. Thereore, this mode contros ows so that the queue ength at steady state becomes zero at the cost o some degree o underutiization. In PID mode, note that e 1(t) cannot be smaer than qt because q (t) cannot be negative. Thus, one axiomatic advantage o PII mode is that the contro dynamics are not saturated at q (t) =because the controer uses the rate error signa as its input instead o the queue error signa. Thus the main physica saturation noninearity o the PID contro mode can be overcome by this mode. For steady state anaysis, oowing a simiar way given in Section II-A except that µ αt µ and im t q (t) =, r is is given as oow. r is w i s = wi Q w α T µ w i b(i) s m i. (9) i N() Q i N() This shows that the PII contro mode aso achieves weighted max-min airness property. III. STABILITY ANALYSIS Athough we presented a mutipe botteneck network architecture in Section II, rigorous stabiity anaysis o these kinds o modes has been shown to be very diicut in [9] due to the dynamics couping among inks empoying FCFS (irst come irst serve) discipine. In [9], though such dynamics Gobecom /4/$. 4 IEEE

4 -1+j Fig.. P Imag. Axis g D Rea Axis Nyquist pot o G (jω). couping exists in theory, the eect o couping was shown to be negigibe through simuations. Recenty, Wydrowski et a. [1] aso showed that the dynamics couping is o a very weak orm. Thus, in this section, we drop the superscript and the anaysis is ocused on a singe botteneck mode. We conjecture that our anaytica resuts can be extended to mutipe botteneck modes without signiicant modiication. We describe the stabiity conditions or controer gains or two network modes when the saturation unctions empoyed in (1), () and (8) are reaxed. The main contribution o our anaysis is that we ind the equivaent stabiity condition in continuous-time domain or the case ows experience heterogeneous round-trip deays, and the stabiity condition depends ony on a given upper bound o round-trip deays. Due to space imitation, we concentrate on the PID contro mode and simiar arguments or PII contro mode are given in [8]. A. Singe-Deay Case To anayze the singe-deay case o the PID contro mode, we simpy set Q to be 1, then there is ony one ow which is bottenecked at the ink. We aso regard this case as the situation where a round-trip deays o ows are equa and ρ i are chosen to be ρ i = w i / Q w, i Q. This case aows us to drop the subscript o τ 1, so that the round-trip deay o ow 1 be τ. By the singe-deay assumption, ρ i can be set as oows. ρ 1 = w 1 Q w =1 and ρ i =, i >1. (1) By Eq. () and pugging Eq. () into Eq. (1), we can get the oowing equations. ë 1 (t) =w 1 (t τ), (t) = 1 Q w [g P ė 1 (t)+g I e 1 (t)+g D ë 1 (t)]. Then the corresponding Lapace transorm o the open-oop system is given by G(s) =. ( g D + g P s + g ) I }{{ s exp( τs). (11) } G (s) By s = jω, the oowing equations, to which we now appy Nyquist stabiity criterion [], are obtained. G(jω)=G (jω)exp( jτω), G (jω)=g D j gp ω gi ω. (1) Note that the Nyquist pot o G (jω), which is depicted in Fig. starts at j in the third quadrant where ω = + and ends at g D where ω =+. Inerring rom Fig., we can see that the condition g D < 1 is necessary because the Nyquist pot o G(jω) wi encirce or touch 1+j uness the condition is satisied. Let us denote by P and ω the point at which Nyquist pot o G (jω) intersects with the unit circe and the vaue o ω at P, respectivey. As shown in Fig., φ is the ange between P and 1+j. More precisey, φ = arccos( Re[G (j ω)]). (1) Since the Nyquist pot o G(jω) is the Nyquist pot o G (jω) rotated by τω in the cockwise direction, it is required by Nyquist stabiity criterion that τω < φ. Beore proving the theorem or singe-deay case, we need the oowing proposition. (Its proo is in [8].) Proposition 1: I there exists a unique vaue ω in < ω < and ωτ < π such that G(j ω) =1,Im[G(j ω)] <, and G(jω) > 1 or a ω< ω, then Im[G(jω)] < is satisied or a ω in <ω ω. With the hep o Proposition 1, the equivaent stabiity condition or the singe-deay case now can be stated as oows. (Its proo is in [8].) Theorem 1 (Singe-Deay Case, PID Mode): The cosed-oop system o the PID mode with a singe deay τ is asymptoticay stabe i and ony i g D < 1 and the deay is bounded by τ< arccos ( g I ) ω g D. (14) ω B. Expicit Stabiity Conditions Athough we acquired the equivaent condition or the stabiity o our cosed-oop system, the conditions are impicit and do not aow easy choice o controer gains, g P, g I and g D. To obtain more expicit stabiity conditions, we proceed in the oowing way. We assume that τ is ixed to a vaue and that g D, g P > and g I. We wi ind expicit conditions or controer gains. Now, there are three variabes, i.e., g P, g I and g D, concerned with the stabiity conditions. For mathematica tractabiity, we wi ignore the case τ = and use the oowing deinitions o variabes. ω 1. = ωτ, GD. = gd, G P. = gp τ, G I. = gi τ. I we rewrite Eq. (14) and the condition or ω in terms o new variabes assuming τ>, it oows that ( ) GI <ω 1 < arccos (15) and ( GI ω 1 ω 1 ) ( GP + ω 1 ) =1. (1) Coroary 1 (Expicit Stabiity Region): The stabiity condition given in Theorem 1 is equivaent to the oowing equations. Gobecom /4/$. 4 IEEE

5 G I = =.1 =. =. =.4 =.5 =. =.7 =.8 =.9 Stabiity Region G P Fig.. Expicit stabiity region in terms o, G P and G I. <G P < { < 1, (17) arccos( ) 1 G D i < cos(ω ), ω sin(ω ) i cos(ω ) < 1, (18) G I <ω 1 ( + cos(ω 1 )) i arccos( ) ω, ω ( + cos(ω )) < G I <ω 1 ( + cos(ω 1 )) i ω < arccos( ), (19) where ω. is the vaue maximizing the unction ωsin(ω) over the interva <ω<π, ω 1 is the unique soution o G P = ωsin(ω) over the interva <ω ω, and ω is the unique soution o G P = ωsin(ω) over the interva and ω <ω<arccos( ) which exists ony when the condition ω < arccos( ) is satisied. The proo o this coroary is in [8]. This coroary aows us to draw an exact stabiity region, provided that we are given a vaue o. With the hep o Coroary 1, an expicit stabiity region is depicted in Fig. or various vaues o. Notaby, stabiity region corresponding to =is exacty the same to the stabiity region ound in [11] where PI controer was used or ow contro. C. Heterogeneous-Deay Case In this section, we prove a theorem that aows us to contro ows with heterogeneous round-trip deays ony with the knowedge o a given upper bound o round-trip deays. This point is important because a router may not store round-trip deay vaues o ows because doing so inevitaby compes a router to store per-ow inormation. With canceation o Eq. (1), we now consider more genera situation where a round-trip deays o ows can be dierent and the sum o ρ i is ess than or equa to 1. The reason or aowing i Q ρ i < 1 wi be cear soon. Simiar to Section III-A, we can get the oowing open-oop transer unction. G(s) =. ( g D + g P s + g ) I }{{ s ρ i exp( τ i s). () } i Q G (s) Beore proving the theorem, we need a proposition. TABLE I PARAMETERS USED FOR SIMULATION µ β T 1 α T T δ λ CPS DPS NCP W 1.1µ bytes 5bytes Proposition (-1+j Excusion Theorem): Given a ixed vaue ω, et us deine the vaue set V (ω) = Z = G(jω,ρ i,τ i ) ρ i 1, τ i τ. i Q The system is asymptoticay stabe i and ony i the oowing two conditions are satisied: There exists a Q -tupe vector (ρ 1,, ρ Q, τ 1,, τ Q ) such that the system with the open-oop transer unction o G(s, ρ i,τ i ) is asymptoticay stabe. For a ω, the vaue set V (ω) does not touch the point 1+j, i.e., 1+j / V (ω). Basicay, this proposition is a direct appication o the Zero excusion theorem which is one o main resuts in robust contro theory. Expanation and proo o the theorem can be ound in [7]. Denoting by τ an upper bound o τ i, i.e., max i Q τ i τ, we are ready to state our main resut. (Its proo is in [8].) Theorem (Heterogeneous-Deay Case, PID Mode): The cosed-oop system o the PID mode with heterogeneous deays is asymptoticay stabe or a τ i τ and or a ρ i satisying i Q ρ i 1 i and ony i the cosed-oop system o the singe-deay case with deay τ is asymptoticay stabe. This theorem guarantees that a network is stabiized or a combinations o τ i τ i routers know ony one upper bound o round-trip deays, i.e., τ, by choosing a controer gain set (,G P,G I )=(g D,g P τ,g I τ ) contained in stabiity region depicted in Fig.. Observe that the cosedoop dynamics shoud be better when the τ is more tighty chosen. A method or the estimation o Q w is expored in [8] because a router without per-ow inormation cannot know the exact sum o w i. By appeaing to Theorem, we can see that it is competey sae to overestimate Q w, i.e., ˆQ w Q w where ˆQ w is the estimate o Q w, because i Q ρ i = i Q w i/ Q w is aowed to be smaer than 1. IV. SIMULATION RESULTS Due to space imitation, readers are encouraged to reer to [8] or inding optima gain sets o PID, PII, PI and II modes. For determination o main parameters incuding β, τ and q T, and estimation o Q w, readers are aso encouraged to reer to [8]. Here we give simuation resuts in two scenarios to demonstrate the perormance o our agorithms and to compare the perormance o two modes. The simuations are done using the ns- simuator [1]. The argest round-trip propagation deay on paths traversing through a ink is set to ms. We assume that a packets are enqueued in the same buer Gobecom /4/$. 4 IEEE

6 S1 S S 1 R1 S4 S5 S km Sink 4 1Mbps Link 1 Link km 1Mbps R km Fig. 4. Mutipe botteneck network used or Scenario 1 TABLE II Sink1 FLOW MODELS USED FOR SCENARIO 1 AND FAIR RATE. (THE UNITS OF m i AND FAIR RATE ARE IN Mbps AND THE UNITS OF D i ARE IN km. THE UNITS OF ARRIVAL AND DEPARTURE TIME ARE IN SECONDS.) Src. m i w i D i Arr. Dept. Sink S1 5(4.75) 1 8 Sink1 S 15(14.5) 1 Sink1 S 4 5 Sink S4 (19) 8 Sink S5 1 Sink S Sink Src. Fair Rate in Time Interva S1 5(.75) (19) 1(19.95) (19) 18.(17.4) S 75(71.5) (57) (59.85) (57) 55(5.5) S - (19) 1(15.) (19).7(5.) S4 (57) 5(47.5) 44(41.8) 5(47.5) 7.(9.7) S5 4(8) (8.5) 4(.8) (8.5) - S - - 1(15.) - - and served simpy with FCFS discipine and w max is set to. Other parameters used or simuation are given in Tabe I where is one data packet transmission time, i.e., DPS/µ. Note that the target queue engths are qt 5kbytes with µ = 1Mbps and the queueing deay o a ink with q (t) = qt is approximatey 4ms or the PID mode. Simuation resuts or PID, PII, PI and II modes are respectivey denoted by G PID, G PII, G PID and G PII. A. Mutipe Botteneck Network In the irst scenario, we investigate various properties o our agorithm. In [9] and [1] authors showed through simuations and anaysis that the oca stabiity condition derived in the neighborhood works we or the FCFS discipine. By appeaing to this resut, we here consider a scenario where two bottenecks exist. The network coniguration is shown in Fig. 4 where the botteneck inks 1 and have the ink capacity o 1Mbps. The ow modes used in this scenario and theoretica rates o ows satisying the weighted max-min airness property or the PID mode are summarized in Tabe II. For the PII mode, changes are indicated in parentheses when needed. At t =s, the queue engths at ink 1 and are aready stabiized with 4 ows, S1, S, S4 and S5. For our gain sets, vaues o q (t) and ˆQ w at ink 1 and, and source transmission rates a i (t) are shown in Fig. 5. For G PID and G PID, the queue engths are controed to the target queue ength except transient periods. For G PII and G PII,the Queue Length(kbytes) at ink Queue Length(kbytes) at ink Est. Sum o Weights at ink Est. Sum o Weights at ink Source Tx. Rate(Mbps) G PII G PII quick adaptation quick adaptation S quick adaptation S 4 4 S 5 S S S Time(sec) Fig. 5. Resuts o Scenario 1: From top to bottom - Queue ength at ink 1 (q 1 (t)), Estimated sum o ocay bottenecked ows weights at ink 1 ( ˆQ 1 w ), Queue ength at ink (q (t)), Estimated sum o ocay bottenecked ows weights at ink ( ˆQ w ) and Source Transmission Rates (a i (t)). queue ength are neary zero except transient periods, at the cost o 5% underutiization. It can be observed that the queue ength overshoot is smaer and the rate adaptation is aster when a three-term controer (G PID or G PII ) is used instead o a two-term controer (G PID or G PII ). The overshoots o queue ength at t = 5s and t = 1s are mainy due to the smaness o Q w in this scenario. At t =5s, the queue ength with the gain set G PID is sti being stabiized ater the departure o S5 at t =s because the error signa e 1(t) is saturated to qt due to the saturation noninearity at q (t) =. G PII and G PII achieve theoretica air rates in advance o two gain sets used or the PID mode. S is bottenecked at ink rom t =5sto t =sand bottenecked at ink 1 as S5 stops its transmission at t =s. The actua source transmission rates approach to the theoretica air rates given in Tabe II except transient period. We can see that our agorithms work we even i mutipe bottenecks exist. B. Simpe Network With Short-ived Fows In the second scenario, we investigate the eect o shortived ows to our agorithm. We use a simpe network shown in Fig. where 1 persistent sources with D i = i 18km, i {1,,, 1} and 15 on-o sources with D i = (i 1) 1km, i {11, 1,, 5} exist. A on-o source is modeed by a two-state birth-death mode where the dwe time periods in on and o state are exponentiay distributed Gobecom /4/$. 4 IEEE

7 Queue Length(kbytes) at ink Est. Sum o Weights at ink Feedback Rate(Mbps) at ink 4 1 S1 S S5 1 D 4 km R Link km 1Mbps Sink Fig.. Simpe network used or Scenario Idea or G PID, Idea or G PII, G PII Time(sec) Saturation at q (t)= G PII G PII Idea or G PID, G PID, G PII, GPII Fig. 7. Resuts o Scenario : From top to bottom - Queue ength at ink (q (t)), Estimated sum o ocay bottenecked ows weights at ink ( ˆQ w ) and Feedback rate at ink (u (t)). with the mean o 5s and 1s respectivey. w i =1and m i = are used. A portion o the simuation resuts is shown in Fig. 7. Athough the resuts are osciatory due to short-ived ows, the eedback rate and estimated sum o ows weights at ink are kept in the neighborhood o.7mbps(or.mpbs) and 15 respectivey. Note that the PII mode is good at tracking idea air rates and G PII has the smaest transient queue ength. The utiization o ink is ound to be.94,.91,.947 and.945 respectivey with the gain set G PID,, G PII and G PII. The ow utiization o G PID is caused by its sow rate adaptation when the queue ength is zero. The utiization o G PII and G PII is very cose to α T =.95 due to their ast rate adaptation. V. CONCLUSION In this paper, we provided two network modes which satisy weighted max-min airness and dispense with any kind o per-ow operation in routers. We ound equivaent stabiity conditions in two network modes with heterogeneous roundtrip deays. The theorem states that a stabiizing gain ound with singe-deay τ aso stabiizes a the networks with heterogeneous deays ess than or equa to τ and overestimation o the sum o ows weights is competey sae. We aso derived the equivaent condition or the asymptotic stabiity o the network as an expicit and usabe unction o the upper bound τ o a round-trip deays. We showed in [8] that the gain sets maximizing the asymptotic decay rates do not cause the serious perormance degradation even though they are obtained using ony an upper bound o round-trip deays. The PID mode achieves not ony u utiization but aso the target queue ength at its equiibrium point. The PII mode achieves zero queueing deays and absorbs transient overshoots in inks sacriicing some degree o utiization, and αt can be owered to absorb the transient queues when many short-ive ows exist. It aso quicky achieves air rates because the saturation noninearity at empty buers is now eiminated. We beieve that our anaytica and experimenta resuts wi pay an important roe in encouraging the usage o more sophisticated ow contro agorithms in packet networks. REFERENCES [1] D. Bertsekas and R. Gaager, Data Networks. New Jersey: Prentice Ha, 199. [] F. Keey, Charging and rate contro or eastic traic, Euro. Trans. Teecommun., vo. 8, pp. 7, [] L. Massouié and J. Roberts, Bandwidth sharing: Objectives and agorithms, in Proc. o IEEE INFOCOM 99, 1999, pp [4] B. Vandaore, S. Fahmy, R. Jain, R. Goya, and M. Goya, Genera weighted airness and its support in expicit rate switch agorithms, Comp. Commun., vo., no., pp , Jan.. [5] F. Ren, C. Lin, X. Ying, X. Shan, and F. Wang, A robust active queue management agorithm based on siding mode variabe structure contro, in Proceedings o IEEE INFOCOM,, pp. 1. [] R. C. Dor and R. H. Bishop, Modern Contro Systems. Addison- Wesey, [7] J. Ackermann, Robust Contro: Systems with Uncertain Physica Parameters. New York: Springer-Verag, 199. [8] J.-w. Cho and S. Chong, Stabiized max-min ow contro using PID and PII controers, Drat Version,. [Onine]. Avaiabe: ggumdo/pidpii.pd [9] L. Benmohamed and S. Meerkov, Feedback contro o congestion in packet switching networks: The case o mutipe congested node, Int. J. Commun. Syst., vo. 1, no. 5, pp. 7 4, [1] B. Wydrowski, L. L. H. Andrew, and M. Zukerman, MaxNet: A congestion contro architecture or scaabe networks, IEEE Communications Letters, vo. 7, no. 1, pp , Oct.. [11] S. Chong, S. H. Lee, and S. H. Kang, A simpe, scaabe, and stabe expicit rate aocation agorithm or max-min ow contro with minimum rate guarantee, IEEE/ACM Trans. Networking, vo. 9, no., pp. 55, June 1. [1] UCB/LBNL/VINT Network Simuator ns (Version.1b9a). [Onine]. Avaiabe: Gobecom /4/$. 4 IEEE