A Duality Model of TCP and Queue Management Algorithms

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1 A Duality Model of TCP and Queue Management Algorithm Steven H. Low CS and EE Department California Intitute of Technology Paadena, CA 95 May 4, Abtract We propoe a duality model of end-to-end congetion control and apply it to undertand the equilibrium propertie of TCP and active queue management cheme. The baic idea i to regard ource rate a primal variable and congetion meaure a dual variable, and congetion control a a ditributed primaldual algorithm over the Internet to maximize aggregate utility ubject to capacity contraint. The primal iteration i carried out by TCP algorithm uch a Reno or Vega, and the dual iteration i carried out by queue management algorithm uch a DropTail, RED or REM. We preent thee algorithm and their generalization, derive their utility function, and tudy their interaction. I. Introduction Congetion control i a ditributed algorithm to hare network reource (called link in thi paper) among competing ource. It conit of two component: a ource algorithm that dynamically adjut rate (or window ize) in repone to congetion in it path, and a link algorithm that update, implicitly or explicitly, a congetion meaure and end it back, implicitly or explicitly, to ource that ue that link. On the current Internet, the ource algorithm i carried out by TCP, and the link algorithm i carried out by (active) queue management (AQM) cheme uch a DropTail or RED 6]. Different protocol ue different metric to meaure congetion, e.g., TCP Reno ], 5] and it variant, ue lo probability a congetion meaure, and TCP Vega 4], it turn out, ue queueing delay a congetion meaure 8]. Both are implicitly updated at the link and implicitly fed back to ource through end-to-end lo or delay, repectively. In thi paper, we preent a general model of end-to-end congetion control and apply it to undertand the equilibrium propertie of the cloed-loop ytem pecified by variou TCP/AQM protocol. The baic idea i to regard the proce of congetion control a carrying out a ditributed computation by ource and link over a network in real time to olve a global optimization problem formulated in ]. The objective i to maximize aggregate ource utility ubject to capacity contraint. We will interpret ource rate a primal variable, congetion meaure a dual variable, and TCP/AQM protocol a ditributed primal-dual algorithm to olve thi optimization problem and it aociated dual problem (Section II). Different protocol, uch a Reno, Vega, RED, and REM ], all olve the ame prototypical problem with different utility function, and we derive thee function explicitly (Section III and IV). Moreover all thee protocol generate congetion meaure (Lagrange multiplier) that olve the dual problem in equilibrium. The model implie that the equilibrium propertie of a large network under TCP/AQM control, uch a throughput, delay, queue length, lo probabilitie, and fairne, can be readily undertood by tudying the underlying optimization problem (ee later ection and 8]). Moreover, ince the problem i a concave program, thee propertie can be efficiently computed numerically. It i poible to go between utility maximization and TCP/AQM algorithm in both direction. We can tart with general utility function, e.g., tailored to our application, and then derive TCP/AQM algorithm to maximize aggregate utility, a done in, e.g., ], 6], 9], ], ]. Converely, and hitorically, we can deign TCP/AQM algorithm and then revereengineer the algorithm to determine the underlying utility function they implicitly optimize and the aociated dual problem, a we do here and in 8]. Thi i the conequence of end-to-end control: a long a the end-to-end congetion meaure to which the TCP algorithm react i the um of the contituent link congetion meaure, uch an interpretation i valid.. In Section V, we dicu the interaction of generalized Reno algorithm, and that of Reno and Vega. To appear in IEEE/ACM Tran. on Networking, October. Partial and preliminary reult appear in 5]. Thi work i uppported by NSF through grant ANI-45 and ANI-967, and ARO through grant DAAD Under ome mild aumption on the TCP and AQM algorithm that are typically atified (aumption C C in Section II)

2 It will become clear that fairne of TCP algorithm hould not be defined olely in term whether they receive the ame equilibrium rate, a commonly done in the literature, becaue the equilibrium bandwidth allocation generally alo depend on AQM, network topology, and routing, etc. We will conclude in Section VI with ome inight from the duality model and limitation of thi work. II. Duality model of TCP/AQM A network i modeled a a et L of link (carce reource) with finite capacitie c = (c l, l L). They are hared by a et S of ource indexed by. Each ource ue a et L L of link. The et L define an L S routing matrix R l = { if l L otherwie Aociated with each ource i it tranmiion rate x (t) at time t, in packet/ec. Aociated with each link l i a calar congetion meaure p l (t) at time t. Following the notation of ], let y l (t) = R lx (t) be the aggregate ource rate at link l and let q (t) = l R lp l (t) be the end-to-end congetion meaure for ource. In vector notation, we have ( T denote tranpoe) y(t) = Rx(t) and q(t) = R T p(t) Here, x(t) = (x (t), S) and q(t) = (q (t), S) are in R S +, and y(t) = (y l(t), l L) and p(t) = (p l (t), l L) are in R L + (R + denote non-negative real). Source can oberve it own rate x (t) and the end-to-end congetion meaure q (t) of it path, but not the vector x(t) or p(t), nor other component of q(t). Similarly, link l can oberve jut local congetion p l (t) and flow rate y l (t). The ource rate x (t) i adjuted in each period according to a function F baed only on x (t) and q (t): for all, x (t + ) = F (x (t), q (t)) () The link congetion meaure p l (t) i adjuted in each period baed only on p l (t) and y l (t), and poibly ome internal (vector) variable v l (t), uch a the queue length at link l. Thi can be modeled by ome function (G l, H l ): for all l, p l (t + ) = G l (y l (t), p l (t), v l (t)) () v l (t + ) = H l (y l (t), p l (t), v l (t)) () where G l i non-negative o that p l (t). Here, F model TCP algorithm (e.g., Reno or Vega) and (G l, H l ) model AQM (e.g., RED, REM); ee the next ection. We will often refer to AQM by G l, without explicit reference to the internal variable v l (t) or it adaptation H l. We aume that () () ha a et of equilibria (x, p). The fixed point of () define an implicit relation between equilibrium rate x and end-to-end congetion meaure q : x = F (x, q ) Aume F i continuouly differentiable and F / q in the open et A := {(x, q ) x >, q > }. Then, by the implicit function theorem, there exit a unique continuouly differentiable function f from {x > } to {q > } uch that q = f (x ) > (4) To extend the mapping between x and q to the cloure of A, define f () = inf {q F (, q ) = } (5) poibly. If (x, ) i an equilibrium point, F (x, ) = x, then define f (x ) = (6) Define the utility function of each ource a U (x ) = f (x )dx, x (7) that i unique up to a contant. Being an integral, U i a continuou function. Since f (x ) = q for all x, U i nondecreaing. It i reaonable to aume that f i a nonincreaing function the more evere the congetion, the maller the rate. Thi implie that U i concave. If f i trictly decreaing, then U i trictly concave ince U (x ) <. An increaing utility function implie a greedy ource a larger rate yield a higher utility and concavity implie diminihing return. Now conider the problem of maximizing aggregate utility formulated in ]: max x U (x ) ubject to Rx c (8) The contraint ay that, at each link l, the flow rate y l doe not exceed the capacity c l. An optimal rate vector x exit ince the objective function in (8) i continuou and the feaible olution et i compact. It i unique if U are trictly concave. A the ource are coupled through the hared link (the capacity contraint), olving for x directly, however, may require coordination among poibly all ource, and We abue notation to ue L and S to denote et and their cardinality.

3 hence i infeaible in a large network. The key to undertanding the equilibrium of () () i to regard x(t) a primal variable, p(t) a dual variable, and (F, G) = (F, G l, S, l L) a a ditributed primaldual algorithm to olve the primal problem (8) and it Lagrangian dual (ee 6]): min max (U (x ) x q ) + p l c l (9) p x l Hence, the dual variable i a precie meaure of congetion in the network. The dual problem ha an optimal olution ince the primal problem i feaible. We will interpret the equilibria (x, p ) of () () a olution of the primal and dual problem, and that (F, G) iterate on both the primal and dual variable together in an attempt to olve both problem. We ummarize the aumption on (F, G, H): C: For all S and l L, F and G l are nonnegative function. Moreover, equilibrium point of () () exit. C: For all S, F are continuouly differentiable and F / q in {(x, q ) x >, q > }; moreover, f in (4) are nonincreaing. C: If p l = G l (y l, p l, v l ) and v l = H l (y l, p l, v l ), then y l c l with equality if p l >. C4: For all S, f are trictly decreaing. Condition C guarantee that (x(t), p(t)) and (x, p ). C guarantee the exitence and concavity of utility function U. C guarantee the primal feaibility and complementary lackne of (x, p ). Finally condition C4 guarantee the uniquene of optimal x. Theorem : Suppoe aumption C and C hold. Let (x, p ) be an equilibrium of () (). Then (x, p ) olve the primal problem (8) and the dual problem (9) with utility function given by (7) if and only if C hold. Moreover, if aumption C4 hold a well, then U are trictly concave and the optimal rate vector x i unique. Proof. The dicuion after the definition (7) of U prove the econd claim when C4 hold, o we only prove the firt claim. By duality theory (e.g.,, Propoition 5..5]), (x, p ) i primal-dual optimal if and only if x i primal feaible, p i dual feaible, complementary lackne hold, and x = arg max x L(x, p ) () where L i the Lagrangian of (8) defined a: L(x, p) = U (x ) + p l (c l R l x ) l Hence, to prove the firt claim, we only need to etablih (). Now max L(x, x p ) = max x = U (x ) + ( p l c l ) R l x l ( ) U (x ) x R l p l + p l c l l max x l By contruction of U, we have from (7) and (4) that, for any equilibrium at which x >, (x, p ), U (x ) = f (x ) = q = l R l p l () Note that if q =, then () hold by (6). If x =, we have from (5) But, () () implie that U () = f () q () L x (x, p ) with equality if x >. Since L(x, p ) i concave in x, thi i the neceary and ufficient Karuh-Kuhn- Tucker condition for x to maximize L(x, p ) over x. Hence the proof i complete. Hence, variou TCP/AQM protocol can be modeled a different ditributed primal-dual algorithm (F, G, H) to olve the global optimization problem (8) and it dual (9), with different utility function U. Thi computation i carried out by ource and link over the Internet in real time in the form of congetion control. Theorem characterize a large cla of protocol (F, G, H) that admit uch an interpretation. Thi interpretation i the conequence of end-to-end control: it hold a long a the end-to-end congetion meaure to which the TCP algorithm react i the um of the contituent link congetion meaure, under ome mild aumption on the TCP and AQM algorithm that are typically atified (aumption C C in Section II). Note that the definition of utility function U depend only on TCP algorithm F. The role of AQM (G, H) i to enure that the complementary lackne condition of problem () () i atified (condition C). The complementary lackne ha a imple interpretation: AQM hould match input rate to capacity to maximize utilization at every bottleneck link. Any AQM that tabilize queue poee thi property (ee (6) below) and generate a Lagrange multiplier p that olve the dual problem. In the following ection, we apply Theorem to interpret TCP Reno with RED and with REM, and TCP Vega with DropTail. We firt derive an algorithm model (F, G, H) from protocol decription, and then ue (7) to derive the utility function U which the protocol implicitly optimize. The reult are ummarized in Table I.

4 4 Reno- Reno- F (x (t), q (t)) Utility F (x (t), q (t)) TCP ] + x (t) + q(t) D q (t)x ( ) (t) / D tan x D ] + x (t) + x(t)dq(t) q (t)x (t) Utility D log xd x D + x (t) + D if x (t) < x (t) Vega F (x (t), q (t)) x (t) D if x (t) > x (t) x (t) otherwie Utility α d log x AQM r l (t + ) b l ρ RED G l (y l (t), p l (t), v l (t)) (r l (t + ) b l ) b l r l (t + ) b l ρ (r l (t + ) b l ) + m l b l r l (t + ) b l r l (t + ) b l H l (y l (t), p l (t), v l (t)) b l (t + ) = b l (t) + y l (t) c l ] + r l (t + ) = ( α l )r l (t) + α l b l (t) REM G l (y l (t), p l (t), v l (t)) φ r l(t+) H l (y l (t), p l (t), v l (t)) b l (t + ) = b l (t) + y l (t) c l ] + r l (t + ) = r l (t) + γ(α l b l (t) + y l (t) c l )] + Delay G l (y l (t), p l (t), v l (t)) p l (t + ) = p l (t) + y l(t) c l ] + TABLE I Summary: duality model of TCP/AQM algorithm. Notation are explained in Section III and IV. D

5 5 III. Reno/AQM For TCP, we only model the congetion avoidance phae and ignore other (important) apect uch a low-tart and fat retranmit/fat recovery. For AQM, it i ueful to ditinguih between meaure of congetion and feedback of congetion meaure. TCP Reno, for intance, ue lo probability a a meaure of congetion. The value of thi congetion meaure can be fed back to ource either by dropping packet or etting an ECN bit with thi probability. In thi paper, we are concerned with the deign of congetion meaure and it equilibrium propertie, and our AQM model do not capture the feedback mechanim. We will henceforth ue marking to refer to either dropping a packet or etting an ECN bit. A. (F, G, H) model In thi ubection, we preent model of TCP Reno, RED and REM. The implication of thee model will be given in the following ubection and in the Concluion ection. We only model the average behavior of AIMD and doe not differentiate between TCP Reno 5] and it variant uch a NewReno, SACK, etc. All thee protocol (henceforth referred to a Reno ) increae the window by one every round trip time if there i no mark in the round trip time, and halve the window otherwie. There are two verion of multiplicative decreae. Older variant of Reno halve the window every time a mark i detected, wherea new verion of Reno halve the window only once if there i one or more mark in the round trip time. We will call the former verion Reno- and the latter Reno-; a we will ee below, they have lightly different utility function and fairne property. For both verion, we interpret packet marking probability a a meaure of congetion. Under DropTail, a packet that arrive to a full buffer i dropped. We do not know a convenient expreion for the dynamic of marking probability. A model of lo rate that ha been ued, e.g., in 7], ], i that for a bufferle queue, p(t + ) = c/ x (t)] +. Thi model i uitable for the penalty function approach to olving (8), but not the duality approach becaue of the feaibility contraint. Hence, we only preent model for RED and REM. Let w (t) be the window ize. Let D be the equilibrium round trip time (propagation plu equilibrium queueing delay), which we aume i contant, a cutomary in the literature, e.g., ], ]. Let x (t) defined by x (t) = w (t)/d be the ource rate at time t. The time unit i on the order of everal round trip time and ource rate x (t) hould be interpreted a the average rate over thi timecale. Dynamic maller than the timecale of a round trip time i not captured by the fluid model. A. Reno- Let p l (t) be the marking probability at link l at time t. We make the key aumption that the end-to-end marking probability q (t) to which ource algorithm react i the um of link marking probabilitie: q (t) = l R l p l (t) () Thi i reaonable when p l (t) are mall, in which cae q (t) = l L ( p l (t)) l L p l (t). In period t, it tranmit at rate x (t) packet per unit time, and receive (poitive and negative) acknowledgment at approximately the ame rate, auming every packet i acknowledged. On average, ource receive x (t)( q (t)) number of poitive acknowledgment per unit time and each poitive acknowledgment increae the window w (t) by /w (t). It receive, on average, x (t)q (t) negative acknowledgment (mark) per unit time and each halve the window. Hence, in period t, the net change to the window i roughly x (t)( q (t)) w (t) x (t)q (t) 4w (t) Then the ource algorithm F (x (t), q (t)) of Reno- i given by: x (t + ) = x (t) + q (t) D ] + q (t)x (t) (4) The quadratic term ignifie the property that, if rate double, the multiplicative decreae occur at twice the frequency with twice the amplitude. A. Reno- Reno- increment the window by per round trip time D if there i no mark, and halve the window once in each round trip time if there i one or more mark. We model thi a follow: in each period t (which i on the order of a few round trip time), the window increae by /D with probability ˆq (t) and decreae by w (t)/d with probability ˆq (t), where ˆq (t) i the end-to-end probability that at leat one packet i marked in period t in the path of. Again, let p l (t) denote the probability that a packet i marked at link l in period t, and q (t) be the endto-end packet marking probability given by (). We model ˆq (t) a ˆq (t) = w (t)q (t) The factor 4 i motivated by conidering a ingle Reno flow, where in maller timecale than that of the fluid model, the window ocillate between 4 w(t) and w(t) with an average of w (t). It i more cutomary to replace the factor 4 by in the literature, a we will do in numerical example in Section V.

6 6 where w (t) i the window ize. Thi would be jutified if packet in the ame window are marked independently of each other and the packet marking probability q (t) i mall, in which cae ˆq (t) = ( q (t)) w(t) w (t)q (t). Then, the average change in window ize in period t i = ( ˆq (t)) w (t) ˆq (t) D D w (t)q (t) D q (t)x (t)w (t) Hence the ource algorithm F (x (t), q (t)) for Reno- i given by: = A. RED x (t + ) x (t) + x (t)q (t)d D ] + q (t)x (t) (5) RED 6] maintain two internal variable, the intantaneou queue length b l (t) and average queue length r l (t). They are updated according to b l (t + ) = b l (t) + y l (t) c l ] + (6) r l (t + ) = ( α l )r l (t) + α l b l (t) (7) where α l (, ). Then, (the gentle verion of) RED mark a packet with a probability p l (t) that i a piecewie linear increaing function of r l (t): r l (t) b l ρ p l (t) = (r l (t) b l ) b l r l (t) b l (8) ρ (r l (t) b l ) + m l b l r l (t) b l r l (t) b l where ρ = m l b l b l and ρ = m l b l The equation (6) (8) define the model (G, H) for RED. A.4 REM REM ] alo maintain two internal variable, intantaneou queue length b l (t) and a quantity called price r l (t). A in RED, b l (t) i modeled by (6); the price r l (t) i updated according to r l (t + ) = r l (t) + γ(α l b l (t) + y l (t) c l )] + (9) where γ > and α l > are contant. It mark packet with a probability that i exponential in price r l (t): p l (t) = φ r l(t) () where φ > i an REM parameter. In practice, (9) can be replaced by r l (t + ) = r l (t) + γ(α l (b l (t) ˆb ] + l ) + y l (t) c l ) where ˆb l i a target equilibrium backlog. A larger ˆbl generally yield a higher utilization epecially when the queue ocillate widely ]. With thi verion, the equilibrium queue length in Theorem below i b l = ˆbl. (9) correpond to etting ˆb l =. Exponential marking probability () i ueful for etimating end-to-end price l L r l (t) at the ource. Since thi i not ued by Reno, other increaing function can alo be ued, a explained in ]. For intance, the marking probability can be linear in price r l (t): p l (t) = min{ρr l (t), } () for ome contant ρ >. The verion with nonzero target queue length ˆb l and linear marking probability i equivalent to the PI controller of 9]. Other propoed AQM uch a Adaptive Virtual Queue of ] can alo be modeled in the form of () (). The equation (6), (9), and () or () define the model (G, H) for REM. B. Utility function of Reno In thi ubection, we derive the utility function of Reno- and Reno- and how that, with RED or REM, they olve both the primal and dual problem. Note that all reult of thi ubection apply to a network that contain both Reno- and Reno- ource and both RED and REM link. Lemma : The function (F, G, H) that model Reno-, Reno-, RED and REM (equation (4) ()) atify condition C, C, C4. Proof. Clearly, condition C i atified. For both Reno- and Reno-, when x >, F i continuouly differentiable and F / q. For Reno-, For Reno-, q = q = x D + =: f (x ) () x D (x D + ) =: f (x ) () Hence f (x ) i trictly decreaing for both Reno- and Reno-, implying trict concavity of their utility function. Hence condition C and C4 are both atified. Combining () and (7), the utility function of Reno- (4) i ( ) / U (x ) = tan x D (4) D

7 7 Similarly, from (), the utility function of Reno- (5) i: U (x ) = D log x D x D + (5) Note that the utility function of Reno- and Reno- imply that, unlike Vega, it i poible for a ource that travere many bottleneck link to receive zero bandwidth (when it end-to-end price i unit). The following reult applie Theorem to Reno with RED or with REM. It implie that the equilibrium queue length with RED depend on the problem intance (network topology, routing, number of ource, etc.) and RED parameter, and hence inevitably grow a load increae. RED parameter can be tuned, tatically or dynamically, to reduce equilibrium queue length, but only at the expene of potential intability; ee Example below. In contrat, the equilibrium queue length with REM i zero regardle of load. Theorem :. Let (x, p ) be an equilibrium of a network that contain both Reno- and Reno- ource and both RED and REM link. Then (x, p ) olve the primal (8) and the dual problem (9), with utility function given by (4) for Reno- and (5) for Reno- ource. Moreover, the equilibrium rate vector x i unique.. If link l implement RED, then the equilibrium queue length b l atifie b l > b l at link with p l >. If link l implement REM, then b l =. Proof. By Lemma, C, C, C4 are atified by combination of (4) (). Given an equilibrium (x, p ), to how that it i primal-dual optimal, we need to check that C i alo atified. From (6), y l c l with both RED and REM and hence primal feaibility i atified. Suppoe p l >. If link l implement RED, then from (7) and (8), b l = r l > b l (6) but b l > implie that y l = c l. If link l implement REM, then p l > implie r l > and hence, from (9), Conider a ingle link with capacity c hared by a et of Reno- ource with round trip delay D. From () and y l = c l, the equilibrium rate are x = D D c, and the equilibrium marking probability i p = c D + > where D = ( ). D If the ource are Reno- intead, then the equilibrium rate are the ame (ue ()), but the marking probability p = c D + cd > i typically lower ince cd i uually greater than packet. If RED i ued, the equilibrium probability p determine the equilibrium queue length through the marking probability function. Inverting (8), we have (ince b l = r l and p > ) b l = { bl + ρ p l if < p l m l b l + ρ (p l m l) if m l < p l In particular, a the number of ource increae, D decreae, and hence both p l and b l increae. Indeed, b l under RED grow toward twice the maximum threhold a load increae: lim b l = lim D p b l = b l To reduce equilibrium queue length b l, a large m l (max p) and a mall b l (max th) hould be ued. But thi increae the lope ρ and compromie tability; ee 4]. Hence, RED parameter can be tuned either to maintain tability or reduce equilibrium queueing delay. α l b l + y l c l = (7) We know yl c l. If yl < c l, then (6) implie that b l = ; but thi contradict (7). Hence yl = c l (and b l = ). We have thu hown that complementary lackne i atified with both RED and REM, and hence C i atified and (x, p ) i primal-dual optimal. Moreover, (6) alo how that b l > b l when p l > with RED. With REM, the preceding argument how that b l =. Thi complete the proof. Example : Reno/RED at a ingle link Remark:. The relation () and () imply that Reno- and Reno- dicriminate againt ource with large D, a well known in many previou tudie, e.g., 5], 6], ], ]. Moreover, () for Reno- can be rewritten a / q / x = D q D q in packet per unit time, when probability q i mall, a relation widely oberved previouly. Some author, e.g., 9], ], aume that Reno increae it window by every round trip time determinitically. Thi correpond to replacing ( q (t)) by in (4), which hold

8 8 when the marking probability i mall. Thi model give x = //D q, with a correponding utility function U (x ) = / x D (8) a ued in 9], ] (ignoring a contant term). For Reno-, () can be approximated by q = x D (x D + ) x D when x D, or when q i mall. Then Reno- ha the ame utility function a Reno- given by (8).. By duality theory, given a dual optimal p, the rate vector x given by x = U (q ) (9) i the (unique) optimal rate vector, where q = l R lp l. The rate adjutment proce of Reno, (4) or (5), can be regarded a a moothed verion of thi trategy, in the following ene. Let x (t) = U (q (t)) be the target rate determined by (9), given p(t), uing the utility function of Reno- or Reno-. Then uing (4) for Reno-, we have x (t) = U (q (t)) = D q (t) q (t) We can then rewrite the rate adjutment (4) in term of the target rate x (t) a: x (t + ) = x (t) + q + (t) ( x (t) x (t))] Hence, intead of etting the rate x (t + ) directly to the target rate x (t) in one tep, Reno- move the current rate x (t) toward the target rate x (t) by adding an amount proportional to the difference of their quare, q (t)(x (t) x (t))/. For Reno-, from (), the target rate x (t) mut atify q (t) = x (t)d (x (t)d + ) Hence F in (5) can be rewritten in term of the target rate x (t) a = x (t + ) x (t) + ( D x )] + (t)(x (t)d + ) x (t)(x (t)d + ) i.e., increae rate if x (t) < x (t) and decreae otherwie.. The approach taken here follow that in 7] where queue management mechanim are modeled entirely by (G l, H l ). In contrat, the model in 5] include the marking probability function a a part of F, making utility function dependent on AQM a well a TCP algorithm. IV. Vega/DropTail A duality model of Vega ha been developed and validated in 8]. In thi ection, we ummarize the main reult. We conider the ituation where the buffer ize i large enough to accommodate the equilibrium queue length o that Vega ource can converge to the unique equilibrium. In thi cae, there i no packet lo in equilibrium. It i hown in 8] that Vega ue queueing delay a congetion meaure, p l (t) = b l (t)/c l, where b l (t) i the queue length in period t. The update rule i therefore G l (y l (t), p l (t)) given by (dividing both ide of (6) by c l ): p l (t + ) = p l (t) + y ] + l(t) () c l Hence, AQM for Vega doe not involve any internal variable. Given p(t), or q (t), let x (t) given by: x (t) = α d q (t) () be the target rate, where α i a parameter of Vega, and d i the round trip propagation delay of ource, aumed known by. The update rule for ource rate i then F (x (t), q (t)) given by: x (t + ) = x (t) + D (x (t) x (t)) () where (z) = if z >, if z <, and if z =. In equilibrium, we have x = x = α d /q. Hence U (x ) = α d /x or U (x ) = α d log x. The following reult i proved in 8]. It implie, in particular, that we can compute the queue length at each link by olving a imple concave program. Theorem 4 (8]). An equilibrium (x, p ) of Vega/DropTail a modeled by () () olve the primal (8) and the dual problem (9), with utility function U given by U (x ) = α d log x Moreover, x i unique and weighted proportionally fair.. The equilibrium queue length at link l are c l p l. Again the rate adjutment of Vega () can be interpreted a a moothed verion of (9) with the utility function given in the theorem. Intead of etting the rate x (t + ) in one tep to the target rate x (t) determined by (9), Vega move the current rate x (t) cloer to the target rate x (t) by /D in each tep. V. Generalization and TCP-friendline In thi ection we derive the utility function of Renolike algorithm, and conider the interaction of different TCP algorithm.

9 9 A. Reno-like algorithm Conider algorithm that increae the rate x (t) by α (x (t)) on each poitive acknowledgment, and decreae it by β (x (t)) on each mark. Then F in (4) and (5) are generalized to x (t + ) = x (t) + ( q (t))x (t)α (x (t)) q (t)x (t)β (x (t))] + () Reno- i a pecial cae with α (x ) = /x D and β (x ) = x /, and Reno- with α (x ) = /x D and β (x ) = /D. We will index thee algorithm by their increae-decreae function (α, β ). From () we have in equilibrium q = α (x ) α (x ) + β (x ) := f (x ) (4) and hence the utility function i α (x ) U (x ) = α (x ) + β (x ) dx (5) A ource algorithm (α, β ) i aid to be TCP-friendly if it equilibrium rate coincide with Reno. In the following, we will ue Reno- in the definition of TCPfriendline; however an analogou analyi applie to Reno-. Equating (4) for Reno- and (4), we ee that an algorithm (α, β ) i TCP-friendly if and only if or α (x ) α (x ) + β (x ) α (x ) β (x ) = = + x D x D (6) Hence, TCP-friendline of a Reno-like algorithm depend on the increae-decreae function only through their ratio. A an illutration, we conider a cla of Reno-like algorithm called binomial algorithm in ]. Thee algorithm are indexed by a pair (k, l) and correpond to α (x ) = α/x k+ D k+ (7) β (x ) = βx l D l (8) for ome contant α, β > (Reno correpond to (k, l) = (, )). Subtituting (7) (8) into the condition (6) for TCP-friendline yield α (x ) β (x ) = α β (x D ) k+l+ which implie the k +l rule of ]: a binomial algorithm i TCP-friendly if and only if k + l = and α/β =. The utility function of binomial algorithm can be derived from (5) and (7) (8) to be U (x ) = ( ) α n dy D β + y n y=xd(β/α) n where n = k + l +. The cla of n = include the AIAD algorithm and ha a utility function U (x ) = ( ) α D β log β + x D α The cla of n = i TCP-friendly when α/β = and ha a utility function U (x ) = ( ) α β D β tan x D α For n =, the utility function i 4, pp. 6] ( ) α n U (x ) = D β + x D (β/α) n log x D (β/α) n + x D(β/α) n + tan x ) D (β/α) n x D (β/α) n B. Interaction: binomial algorithm The duality model provide a convenient framework in which to tudy the interaction of different TCP/AQM cheme, provided all TCP algorithm ue the ame congetion meaure. Once the cheme under tudy are characterized by (F, G, H) and their utility function, the equilibrium rate and performance uch a lo, delay and queue length can be obtained by olving the concave program (8). Cloe-form olution are uually unavailable for general network topology, but numerical olution can be efficiently computed to provide inight on the equilibrium propertie, uch a throughput and fairne. For ingle-link network, cloed-form olution can be eaily obtained, a we now illutrate. We firt conider the interaction of binomial algorithm (k, l), which ue the ame congetion meaure, marking probability. In the next ubection, we conider the interaction of Reno and Vega, which ue different congetion meaure. Conider a ingle link hared by N n type-n ource with equilibrium round trip delay D n, where k + l + = n, n =,,. Let x n be the common equilibrium rate of all type-n ource. Let p be the common equilibrium marking probability. Since, in equilibrium, p = U (x ) for all ource, we have from (4), p = ( + λx n ndn) n and x n = D n ( p λp ) n (9)

10 where λ = β/α. Hence x D = (x n D n ) n = ( p)/λp for all n. Since ( p)/λp i greater than if and only if (4) below hold, we have Theorem 5: Conider clae of binomial algorithm (k, l) indexed by n with n = k + l + that hare the ame link. The window w n = x n D n of type-n ource i related to that of type- ource by Figure how the equilibrium rate x, x, x when N = N = and the number N of aggreive ource varie from to. Figure (a) how the rate of individual ource wherea Figure (b) how the aggregate rate, ummed over all ource of the ame type. A oberved in ], type- ource are more w = wn n, n =,,....5 Equilibrium rate If then p < α α + β (4) Rate (pkt/m).5 type Otherwie w > w >....5 type N type w w... with equalitie if and only if equality hold in (4). Reno ource are of type n = ource with α =, β = / and λ = /. 4 Then (4) become p < /, which uually hold in practice. The theorem then implie that the window ize of a type-n binomial ource i no larger than that of a Reno ource if and only if n. We cloe by preenting a numerical example. Example : Binomial algorithm Conider a link of capacity c hared by N type- ource, N type- ource, and N type- ource, all with the ame round trip delay of D = m. From (9) we have Rate (pkt/m) (a) Individual rate Aggretate equilibrium rate type type type N where x = η6 D, x = η D, x = η D (4) (b) Aggregate rate Fig.. Equilibrium rate a type ource varie. N =,..., N = N =, D = m, c = pkt/m. η := ( p λp ) 6 (4) Since N x + N x + N x = c the link capacity, we have N η 6 + N η + N η = cd (4) Hence we can olve the polynomial in (4), and then obtain marking probability p from (4) and equilibrium rate from (4). We compute the cae for α =, β = / and λ = /, under which type n = ource are Reno. We fix the number of Reno ource, N =, and vary the number N or N to oberve the effect of unfriendly ource on equilibrium rate. The link capacity i c = packet/m. aggreive than Reno, while type- ource are le aggreive. Moreover, the preence of type ource can eize a diproportionally large amount of bandwidth: when there i jut one type- ource, x =.88 pkt/m while x =. pkt/m and x =.7 pkt/m (when there are no type- ource, x =. pkt/m and x =.9 pkt/m). A N increae, while individual rate x drop, the aggregate rate of all type- ource rie harply. Figure how the individual and aggregate rate when N varie from to, while keeping N = N =. The effect of polite ource i much le dramatic than that of aggreive ource. The aggregate hare of all type- ource range from 8% to 75% a N varie from to. 4 We ignore the factor 4 in thi ection; ee footnote.

11 .4.. Equilibrium rate type equilibrium rate if and only if ( ) αv d v = ( p n) q v p n Rate (pkt/m) Rate (pkt/m) type N (a) Individual rate Aggretate equilibrium rate type type N type type (b) Aggregate rate Fig.. Equilibrium rate a type- ource varie. N = N =, N =,..., D = m, c = pkt/m. C. Interaction: Reno and Vega Suppoe Reno and Vega ource hare the ame network. Under what condition will they receive the ame equilibrium rate? Thi i not a traightforward a for binomial algorithm becaue Reno and Vega ue different congetion meaure, marking probability for Reno and queueing delay for Vega. A Reno-like ource i TCP-friendly a long a it increae-decreae ratio atifie (6). Note that thi mean that if uch a ource i friendly under any condition (network topology, routing, etc.), then it i friendly under all condition. In contrat, Vega ource can receive more, equal, or le bandwidth than Reno ource depending on the network condition. Specifically, let q v be the end-toend queueing delay of a Vega ource v, in equilibrium, and let p n be the end-to-end marking probability of a Reno ource n haring the ame network, among other ource. Then, the equilibrium rate of the Vega ource v i x v = α v d v /q v, where α v > i a protocol parameter and d v i the round trip propagation delay of v. The equilibrium rate of the Reno- ource n i x n = ( p n )/p n. Hence they receive the ame Hence, whether v i TCP-friendly or not depend on the network condition through equilibrium queueing delay q v and marking probability p n. But thee equilibrium propertie depend not only on TCP Reno and Vega algorithm F, but alo on AQM algorithm (G l, H l ) and it parameter etting, a well a network topology, routing, and link capacity. Hence, TCP-friendline of a cheme that ue a different congetion meaure hould not be defined imply in term of it equilibrium bandwidth hare, becaue one can generally find cenario where the cheme receive higher bandwidth hare than TCP Reno and cenario where the revere i true. To be concrete, conider Reno (again, we conider only Reno- ource though the analyi applie to Reno- ource a well) and Vega ource haring a ingle link employing RED or REM. Reno ource react to RED or REM mark by halving it rate. If Vega react to mark in the ame way, then it behavior would be imilar to Reno. Hence we tudy the cae where Vega ource ignore RED mark and only react to delay in it path a it doe under DropTail. Under REM, we ue the Vega/REM algorithm in 8] in which a Vega ource etimate the price and ue it to replace queueing delay in etting it rate. Notice that RED ue queue length b(t) a an internal variable that determine both the marking probability for Reno and queueing delay for Vega. Hence we can regard b(t) a a common congetion meaure to which Reno and Vega react under RED. 5 For REM, the common congetion meaure can be taken to be the price variable. The following example how that AQM can have a big effect on the equilibrium rate allocation when ource react to different congetion ignal. Example : Reno and Vega under RED Suppoe there are N Reno ource and N Vega ource. The round trip propagation delay for type-i ource i d i, i =,, o that the round trip time i D i = d i +b/c in equilibrium, where b i the queue length, c i the link capacity, and b/c i the queueing delay. Vega ource all have parameter α o that each keep α d packet in the buffer in equilibrium. Conider the cae where the link ue RED with marking probability that depend on queue length b: p(b) = b b b b, b b b (44) 5 The utility function of Reno i different under thi formulation; ee 5].

12 i.e., the marking probability rie from to over the interval b, b]: From (), Reno equilibrium rate atifie p(b) = + x (d + b/c).5. Equilibrium rate Combining with (44), we have.5 x = ( ) (b b) c (45) b b b + cd Rate (pkt/m)..5. Vega From (), the equilibrium rate of Vega ource are.5 Reno x = α d b/c Since N x + N x = c, we have (46) N (a) Individual rate ( ) (b b) N + α d N b b b + cd b = (47) 4 5 Aggretate equilibrium rate Hence we can obtain equilibrium queue length b by olving (47), and then equilibrium rate uing (45) (46). All ource have a round trip propagation delay of d = d = m. We fix the number of Reno ource, N =, and vary the number N of Vega ource from to. Each Vega ource ha α =. pkt/m o that it keep α d = pkt in it path in equilibrium. RED parameter are b = 5 pkt and b = 4 pkt. The link capacity i c = packet/m. Figure how the individual and aggregate rate of Reno and Vega in equilibrium a the number of Vega ource increae from to. The behavior i qualitatively imilar to the interaction of Reno with aggreive binomial ource hown in Figure, with individual Vega ource eizing a larger proportion of bandwidth than individual Reno ource. The aggregate hare of all Vega ource rie a the number of Vega ource increae. Each Vega ource keep α d = packet in the link. The equilibrium backlog determine the marking probability p(b) which then determine the ource rate of Reno through (45). The rate of Vega ource are proportional to their hare of the buffer occupancy (ee (46)). Figure 4 how the number of Reno and Vega packet in the queue. A the number of Vega ource increae, Vega packet in the queue exceed Reno packet and they receive a larger aggregate bandwidth. Aggregate rate (pkt/m) Reno Vega N (b) Aggregate rate Fig.. Equilibrium rate under RED a Vega ource varie from to. N =, N =,...,, D = m, c = pkt/m. Queue length (pkt) Reno Equilibrium queue length under RED Vega Example 4: Reno and Vega under REM We repeat Example with REM. In thi cae, the price r can be regarded a the common congetion meaure to which Reno and Vega react. The marking probability i given by (). We ue φ =. (other REM Fig. 4. Equilibrium queue and Vega hare under RED. N =, N =,...,, D = m, c = pkt/m. N

13 parameter do not affect equilibrium). Combining () with () and noting that D = d ince backlog i zero (Theorem ), we obtain Reno ource rate a: x = d (φr )/ (48) With the Vega/REM algorithm of 8], Vega ource rate are given by (46) with queueing delay b/c replaced by price r. Since N x + N x = c, we have N d (φr )/ + α d N r = c (49) Hence we can obtain the equilibrium price r by olving (49) and then rate from (48) and modified (46). The reult are hown in Figure 5. For thi example, Rate (pkt/m) Aggregate rate (pkt/m) Equilibrium rate N4 (a) Individual rate Aggretate equilibrium rate Vega Reno Vega N4 (b) Aggregate rate Reno real time to maximize aggregate utility ubject to capacity contraint. Different TCP algorithm have different utility function and we have derived the utility of Reno and Vega; ee Table I. Thi model can be ued to analyze equilibrium propertie, uch a throughput, lo, delay and fairne, of a network that contain different TCP ource and different AQM link, a long a they ue a common meaure of congetion. The duality model ha everal intereting implication. Firt, it i well-known that a bottleneck queue can fluctuate about the buffer capacity under Reno or DropTail, generating packet loe. What i more intriguing i that increaing the buffer ize doe not reduce lo rate ignificantly, but only grow the queueing delay. According to the duality model, lo probability under Reno i the Lagrange multiplier, and hence it equilibrium value i determined olely by the network topology and the utility function of the ource, independent of link algorithm and buffer ize. Increaing the buffer ize with everything ele unchanged doe not change the equilibrium lo probability, and hence a larger backlog mut be maintained to generate the ame lo probability. Thi mean that with DropTail, the buffer at a bottleneck link i alway cloe to full regardle of buffer ize. With RED, ince lo probability (Lagrange multiplier) i increaing in average queue length, the average queue length mut increae teadily a the number of ource grow. Second, it i well-known that TCP Reno dicriminate againt connection with large propagation delay. Thi i borne out by the duality model, a dicued in Remark of Section III. TCP Vega achieve proportional fairne a it ha a log utility function (o doe Reno- approximately). Third, when Reno and Vega ource hare a common network, Vega ource may receive more, equal, or le bandwidth than Reno ource, depending on the network topology and AQM algorithm at the link. In general, the friendline of TCP algorithm that adopt different congetion meaure depend not only on themelve, but alo on their environment uch a AQM algorithm and network parameter. Thi ugget that TCP-friendline that i defined olely in term of ource algorithm i too retrictive. We have only tudied the equilibrium propertie and have ignored the tability and dynamic of thee protocol. The global tability of REM in the abence Fig. 5. Equilibrium rate under REM a Vega ource varie of delay i etablihed in ] uing a Lyapunov argument. Local tability of Reno/RED ha been tudied from to. N =, D = m, c = pkt/m. in 8], 4]. It would be intereting to invetigate delayed global tability of variou TCP/AQM protocol. Vega receive much le bandwidth than Reno, both individual rate and the aggregate. Here we derive the utility function U from rate adjutment algorithm F. One can turn the quetion VI. Concluion around and tailor utility function U to application, We have preented a duality model of everal TCP/AQMand then deign a TCP algorithm F to optimize it. protocol. It interpret thee protocol a ditributed primal-dual algorithm carried out over the Internet in

14 4 Acknowledgment: We gratefully acknowledge inightful dicuion with John Doyle (Caltech), Fernando Paganini (UCLA), and Li Zhu (NJIT) on an earlier verion of the paper. Reference ] Sanjeewa Athuraliya, Victor H. Li, Steven H. Low, and Qinghe Yin. REM: active queue management. IEEE Network, 5():48 5, May/June. Extended verion in Proceeding of ITC7, Salvador, Brazil, September. ] Deepak Banal and Hari Balakrihnan. Binomial congetion control algorithm. In Proceeding of IEEE Infocom, April. ] D. Berteka. Nonlinear Programming. Athena Scientific, ] Lawrence S. Brakmo and Larry L. Peteron. TCP Vega: end-to-end congetion avoidance on a global Internet. IEEE Journal on Selected Area in Communication, (8):465 8, October jac-vega.p. 5] S. Floyd. Connection with multiple congeted gateway in packet witched network, Part I: one way traffic. Computer Communication Review, (5), October 99. 6] S. Floyd and V. Jacobon. Random early detection gateway for congetion avoidance. IEEE/ACM Tran. on Networking, (4):97 4, Augut 99. ftp://ftp.ee.lbl. gov/paper/early.p.gz. 7] R. J. Gibben and F. P. Kelly. Reource pricing and the evolution of congetion control. Automatica, 5: , ] Chri Hollot, Vihal Mira, Don Towley, and Wei-Bo Gong. A control theoretic analyi of RED. In Proceeding of IEEE Infocom, April. edu/paper/paper.html. 9] Chri Hollot, Vihal Mira, Don Towley, and Wei-Bo Gong. On deigning improved controller for AQM router upporting TCP flow. In Proceeding of IEEE Infocom, April. paper.html. ] V. Jacobon. Congetion avoidance and control. Proceeding of SIGCOMM 88, ACM, Augut 988. An updated verion i available via ftp://ftp.ee.lbl.gov/paper/ congavoid.p.z. ] Frank P. Kelly, Aman Maulloo, and David Tan. Rate control for communication network: Shadow price, proportional fairne and tability. Journal of Operation Reearch Society, 49():7 5, March 998. ] Sriankar Kunniyur and R. Srikant. End to end congetion control cheme: utility function, random loe and ECN mark. In Proceeding of IEEE Infocom, March. http: // ] T. V. Lakhman and Upamanyu Madhow. The performance of TCP/IP for network with high bandwidth delay product and random lo. IEEE/ACM Tranaction on Networking, 5():6 5, June edu/faculty/madhow/publication/ton97.p. 4] S. H. Low, F. Paganini, J. Wang and J. C. Doyle. Linear tability of TCP/RED and a calable control. Computer Network Journal, to appear. caltech.edu. 5] Steven H. Low. A duality model of TCP flow control. In Proceeding of ITC Specialit Seminar on IP Traffic Meaurement, Modeling and Management, September 8-. 6] Steven H. Low and David E. Lapley. Optimization flow control, I: baic algorithm and convergence. IEEE/ACM Tranaction on Networking, 7(6):86 874, December ] Steven H. Low, Fernando Paganini, and John C. Doyle. Internet congetion control. IEEE Control Sytem Magazine, ():8 4, February. 8] Steven H. Low, Larry Peteron, and Limin Wang. Undertanding Vega: a duality model. J. of ACM, 49():7 5, March. 9] L. Maoulie and J. Robert. Bandwidth haring: objective and algorithm. In Infocom 99, March dmi.en.fr/\%7emitral/tcpworkhop.html. ] Matthew Mathi, Jeffrey Semke, Jamhid Mahdavi, and Teuni Ott. The macrocopic behavior of the TCP congetion avoidance algorithm. ACM Computer Communication Review, 7(), July paper/model_ccr97.p. ] Jeonghoon Mo and Jean Walrand. Fair end-to-end windowbaed congetion control. IEEE/ACM Tranaction on Networking, 8(5): , October. ] Fernando Paganini. A global tability reult in network flow control. Sytem & Control Letter, 46():5 6,. ] Fernando Paganini, John C. Doyle, and Steven H. Low. Scalable law for table network congetion control. In Proceeding of Conference on Deciion and Control, December. 4] I. M. Ryhik and I. S. Gradtein. Table of erie, product, and integral. VEB Deutcher Verlag der Wienchaften, Berlin, 96. 5] W. Steven. TCP/IP illutrated: the protocol, volume. Addion Weley, th printing. Steven. H. Low (M 9, SM 99) received hi B.S. degree from Cornell Univerity and PhD from the Univerity of California Berkeley, both in electrical engineering. He wa with AT&T Bell Laboratorie, Murray Hill, from 99 to 996 and with the Univerity of Melbourne, Autralia, from 996 to. He i now an Aociate Profeor at the California Intitute of Technology, Paadena. He ha held viiting academic poition in the US and Hong Kong, and ha conulted with companie and government in the US and Autralia. He wa a co-recipient of the IEEE Bennett Prize Paper Award in 997 and the 996 R&D Award. He i on the editorial board of IEEE/ACM Tranaction on Networking and Computer Network Journal. Hi reearch interet are in the control and optimization of network and protocol. Hi home i netlab.caltech.edu and i low@caltech.edu.