Understanding CHOKe. Ao Tang Jiantao Wang Steven H. Low California Institute of Technology, Pasadena, USA

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1 Understanding CHOKe Ao Tang Jiantao Wang Steven H. Low California Institute of Technolog, Pasadena, USA Astract A recentl proposed active queue management, CHOKe, is stateless, simple to implement, et surprisingl effective in protecting TCP from UDP flows. As UDP rate increases, even though the numer of UDP packets in the queue rises, its andwidth share eventuall drops to zero, in stark contrast to the ehavior of a regular FIFO uffer. We derive a detailed model of CHOKe that accuratel predicts this, and other ehaviors of CHOKe, and validate the model with simulations. Its ke features are the incorporation of the feedack equilirium of TCP with dropping proailit and the spatial characteristics of the queueing process. CHOKe produces a leak uffer where packets can e dropped as the move toward the head of the queue. This lea to a spatiall non-uniform distriution of packets and their velocit, and makes it possile for a flow to simultaneousl maintain a large numer of packets in the queue and receive a vanishingl small andwidth share. This is the main mechanism through which CHOKe protects TCP from UDP flows. I. INTRODUCTION TCP is elieved to e largel responsile for preventing congestion collapse while the Internet has undergone dramatic growth in the last decade. Indeed, numerous measurements have consistentl shown that more than 9% of traffic on the current Internet are still TCP packets, which, fortunatel, are congestion controlled. Without a proper incentive structure, however, this state of affair is fragile and can e disrupted the growing numer of non-rate-adaptive e.g., UDP-ased) applications that can monopolize network andwidth to the detriment of rate-adaptive applications. This has motivated several active queue management schemes, e.g., [6], [3], [4], [3], [8], [], [], that aim at penalizing aggressive flows and ensuring fairness. The scheme, CHOKe, of [] is particularl interesting in that it does not require an state information and et can provide a minimum andwidth share to TCP flows. In this paper, we provide an analtical model of CHOKe that explains its spatial characteristics. The asic idea of CHOKe is explained in the following quote from []: When a packet arrives at a congested router, CHOKe draws a packet at random from the FIFO first-in-first-out) uffer and compares it with the arriving packet. If the oth elong to the same flow, then the are oth dropped; else the randoml chosen packet is left intact and the arriving packet is admitted into the uffer with a proailit that depen on the level of congestion this proailit is computed exactl as in RED). Appeared in Proceedings of IEEE Infocom, San Francisco, CA, April 3. The surprising feature of this extremel simple scheme is that it can ound the andwidth share of UDP flows regardless of their arrival rate. In fact, as the arrival rate of UDP packets increases without ound, their andwidth share approaches zero! An intuitive explanation is provided in []: the FIFO uffer is more likel to have packets elonging to a misehaving flow and hence these packets are more likel to e chosen for comparison. Further, packets elonging to a misehaving flow arrive more numerousl and are more likel to trigger comparisons. As a result, aggressive flows are penalized. This however does not explain wh a flow that maintains a much larger numer of packets in the queue does not receive a larger share of andwidth, as in the case of a regular FIFO uffer. It turns out that a precise understanding of this phenomenon requires a detailed analsis of the queue dnamics, the ke feature of our model. A simple model of CHOKe is also presented in [] that assumes a Poisson packet arrival process and exponential service time. The Poisson assumption is critical in order to use the PASTA Poisson Arrival Sees Time Averages) propert to compute drop proailities. This model however ignores oth the feedack equilirium of the TCP/CHOKe sstem and the spatial characteristics of the queue. Here, we adopt a deterministic fluid model that explicitl models oth features Section II). Our model predicts, and simulations confirm, that as UDP rate ecomes large, not onl does the total numer of UDP packets in the queue increase, more importantl, the spatial distriution of UDP packets ecomes more and more concentrated near the tail of the queue, and drops rapidl to zero toward the head of the queue. Hence even though the total numer of UDP packets in the queue is large, all of them will e dropped efore the advance to the head. As a result the UDP andwidth share drops to zero, in stark contrast to a non-leak FIFO uffer where UDP andwidth shares would approach as its input rate increases without ound. Our current model is too complex to e solved analticall. We outline a numerical solution Section II-C), and compare our numerical results with detailed ns- simulations. The match accuratel not onl with average ehavior of CHOKe as reported in [], ut also with much finer spatial characteristics of the queueing process. The technique presented here should e applicale to analzing the queueing process in other tpes of leak uffer. II. MODEL In general, one can choose more than one packet from the queue, compare all of them with the incoming packet, and drop those from the same flow. This will improve CHOKe s performance, especiall when there are multiple unresponsive sources

2 []. Here, we focus on the modeling of a single drop candidate packet. The analsis can e extended to the case of multiple drop candidates. The general setup for our model and our simulations is shown in Figure. We focus on the single ottleneck FIFO uffer at Fig.. TCP sources S S S3 SN UDP Network topolog R capacit=c dela=d R TCP destinations D D D3 DN UDP ink router R where packets are queued and drained at a rate of c packets per second. The uffer is shared N identical TCP flows and a single UDP flow. All TCP flows have a common round trip propagation dela of d secon. We assume the sstem is stale and model its equilirium ehavior. A. Notations Quantities rate, acklog, dropping proailit, etc) associated with the UDP flow are indexed. Those associated with TCP flows are indexed i =,..., N. Since the TCP sources are identical, these quantities all have the same value, and hence we will refer to flow as the generic TCP flow. These are equilirium quantities which we assume exist. We collect here the definitions of all the variales and some of their ovious properties: i : packet acklog from flow i, i =,. : total acklog; = + N. r: Congestion ased dropping proailit. The spatial properties of CHOKe are insensitive to the specific algorithm, such as RED, to compute this proailit, as long as it is the same for all flows. In general, r = g, τ) for some function g as a function of aggregate acklog and queueing dela τ. h i : The proailit that an incoming packet of flow i, i =,, is dropped CHOKe: p i : h i = i overall proailit that a packet of flow i, i =,, is dropped efore it gets through, either CHOKe or RED: p i = h i + r rh i ) function f as a function of overall loss proailit p and queueing dela τ at equilirium. τ: common round-trip queueing dela. Round-trip time is d+ τ. It is important to keep in mind that x is the onl independent variale; all other variales listed aove are functions of x, though this is not made explicit in the notations. B. TCP/CHOKe model A packet ma e dropped, either on arrival due to CHOKe or congestion e.g., according to RED), or after it has een admitted into the queue when a future arrival from the same flow triggers a comparison. Let p i e the proailit that a packet from flow i is eventuall dropped. To see wh p i is related to CHOKe and RED dropping proailities according to ), note that ever arrival from flow i can trigger either packet loss from the uffer, packet loss due to RED, or packet losses due to CHOKe. These events happen with respective proailities of h i ) r), h i )r, and h i. Hence, each arrival to the uffer is accompanied an average packet loss of h i + h i )r + h i ) r) Hence we take the overall loss proailit p i to e the packet loss rate h i + h i )r. We now derive this proailit from another perspective. Consider a packet of flow i that eventuall goes through the queue without eing dropped. The proailit that it is not dropped on arrival is r) h i ). Once it enters the queue, it takes τ time to go through it. In this time period, there are on average τx i packets from flow i that arrive at the queue. The proailit that this packet is not chosen for comparison is ) τxi Hence, the overall proailit that a packet of flow i survives the queue is p i = r) h i ) ) τxi ) A simple interpretation of a leak uffer is as follows: x i is the source rate of flow i and x i r) h i ) is the rate at which flow i enters the queue after CHOKe and congestionased dropping. This flow splits into two flows: one eventuall exits the queue and the other is dropped inside the queue CHOKe. The rate of the former flow is flow i s throughput x i p i ) and the rate of the latter flow is its leak rate x i h i, so that the sum to the input rate x i r) h i ). Since the link is full utilized, the flow throughputs sum to link capacit: x p ) + Nx p ) = c x i : The explanation of ) is provided elow. source rate of flow i, i =,. The spatial properties of CHOKe are insensitive to the specific TCP algorithm, such as Reno or Vegas. In general, x = fp, τ) for some This completes the description of the model. In summar, the independent variale is UDP rate x. The ten dependent variales of the model are: acklogs i of flow i, i =, ; total acklog = +N.

3 3 congestion ased dropping proailit r, CHOKe dropping proailities h i, and overall dropping proailities p i, i =,. TCP rate x and queueing dela τ. The relations among these variales define our model. For ease of reference, we reproduce these ten equations here: p i = h i + r rh i, i =, 3) p i = r) h i ) ) τxi, i =, 4) h i = i, i =, 5) = + N 6) c = x p ) + Nx p ) full utilization) 7) x = fp, τ) TCP) 8) r = g, τ) e.g. RED) 9) Let z = p, p, h, h,,,, x, r, τ) denote the ten dependent variales. Then the aove equations 3) 9) can e expressed as F z, x ) = ) This can e regarded as implicitl defining z as a function of x. We assume: A: Given an x, there is a unique solution zx ) that satisfies ). C. Numerical solution of TCP/CHOKe model The set of nonlinear equations 3) 9) that models the TCP/CHOKe sstem can e solved numericall. The solution can then e used in the differential equation model descried elow to solve for spatial properties of the leak uffer under CHOKe; see Section III. The nonlinear equation ) can e solved minimizing the quadratic cost: min z Jz) := F z, x ) T W F z, x ) with an appropriate choice of positive diagonal weighting matrix W. A solution z of TCP/CHOKe satisfies Jz ) = min z Jz) =. Matla is used to implement the aove procedure. The weighting matrix is chosen such that each component in the vector W z is in the range [ ] near the fixed point. A direct search method [7] for multidimensional unconstrained nonlinear minimization implemented in matla is used for this optimization prolem. The search algorithm is stopped when Jz) is smaller than.5. The solution is accuratel validated with ns- simulations in section IV. III. SPATIAL CHARACTERISTICS A. Spatial distriution and packet velocit If a packet cannot e dropped once it has een admitted into a FIFO queue, then, clearl, the queueing dela τ and andwidth share µ i are τ = c and µ i = i ) For a leak uffer where a packet can e dropped while it advances toward the head of the queue, ) no longer hol, and the queueing dela and andwidth share depend criticall on the spatial characteristics of the queue. The ke to their understanding is the spatial distriution of packets in the queue and the flow rate velocit at which packets move through the queue) at different positions in the queue. We now define these two quantities and relate them to the variales previousl defined. Let [, ] denote a position in the queue, with = eing the tail and = the head of the queue. In a leak uffer, the queueing dela of a packet that eventuall exits the queue is no longer the acklog it sees on arrival divided the link capacit. This is ecause it advances toward the head oth when packets in front of it exit the queue and when the are dropped CHOKe. To model this dnamics, define as the velocit at which the packet at position moves toward the head of the queue: = d dt For instance, the velocit at the head of the queue equals the link capacit, v) = c. Then, the queueing dela τ is given in terms of as τ = τ dt = More generall, define, for x, τ) τ) = d ) 3) vs) which can e interpreted as the time for a packet to reach position from position. Clearl, τ) = τ. Let ρ i ) e the proailit that the packet at position elongs to flow i, i =,. As usual, we have ρ ) + Nρ ) =, for all [, ] 4) The average numer of flow i packets in the entire acklog satisfies i = ρ i )d 5) More importantl, the andwidth share µ i is the proailit that the head of the queue is occupied a packet from flow i: µ i = ρ i ) 6) Note that if the queue is not leak, then the spatial distriution of packets will e uniform, ρ i ) eing independent of position : ρ i ) = ρ i ) for all [, ] This, together with 5), implies the andwidth share in ), i.e., the andwidth share depen onl on the total numer of flow i packets in the queue. When the queue is leak, however,

4 4 the spatial distriution can e highl non-uniform. The andwidth share ρ i ) of flow i depen on the spatial distriution of packets onl at the head of the queue and does not depend directl on the distriution at other positions or the total numer of flow i packets, in stark contrast to the case of non-leak uffer. This is the underling reason wh UDP packets can occup almost half of the queue, et receiving ver small andwidth share: when UDP rate is high, ρ ) decreases rapidl from = to = with ρ ) ; see Section III-C. We have completed the definition of spatial distriution ρ i ) and velocit of packets in the queue. We now derive an ordinar differential ODE) equation model of these quantities. B. ODE model of ρ i ) and We will derive an ODE model for ρ ) and ; ρ ) can e otained from 4). Consider a small volume dt of the one-dimensional fluid) queue at position. The amount of fluid packets) in this volume that elongs to flow i is ρ i )dt, i =,. For instance, ρ i )v)dt, i =,, is the amount of fluid that arrives at the tail of the queue, packets that are not dropped CHOKe or congestion ased dropping on arrival and admitted into the uffer. Hence ρ i )v) = x i r) h i ), i =, 7) Another oundar condition is the packet velocit at the head of the queue mentioned aove: v) = c 8) Suppose the small volume ρ i )v)dt of fluid our tagged packet ) arrives at the uffer at time, and reaches position at time τ). During this period [, τ)], there are x i τ) packet arrivals from flow i, and each of these arrivals triggers a comparison. The tagged packet is selected for comparison with proailit / each time. We model this saing that the fluid is thinned a factor /) xiτ) when it reaches position at time τ). Thus ρ i )v) ) xiτ) = ρ i ) 9) Note that this is the same argument that lea to ). Taking logarithm on oth sides and using 3) to eliminate τ), we have where lnρ i )) = lnρ i )v)) + βx i β := ln ) Differentiating oth sides with respect to, we get β x β x = ρ ) ρ ) + v ) = ρ ) ρ ) + v ) vs) ) ) Now ) ρ ) + ) N ρ ) iel v ) = β ρ )x + ρ ))x ) ) where we have used 4). Sustituting ) into ), we get ρ ) = βx x ) ρ ) ρ )) 3) Hence the spatial distriution ρ i ) and packet velocit is given the two-dimensional sstem of nonlinear differential equations ) 3) with oundar conditions 7) 8). Since the right-hand sides of ) 3) are continuousl differentiale in v, ρ ), there exists a unique solution in its interval of existence []. We make an important remark. Given x, quantities such as τ,, i are uniquel determined ) assumption A. At the same time and ρ ) are uniquel determined the differential equations ) 3) with oundar conditions 7) 8). The relations ) and 5) etween these two sets of quantities are not necessaril true a priori. Even though the seem reasonale ased on their phsical interpretation, the nonetheless remain a postulation: A: Relations ) and 5) hold. Note that 4) hol without assumption and defines ρ ). C. Structural properties In this susection, we prove some structural properties of the velocit and spatial distriution ρ ). The are illustrated in Figure. Theorem. For all x, packet velocit is a convex and strictl decreasing function with v) = x h ) + Nx h )) r) and v) = c. It is linear if and onl if x = x. Proof. Using 4), ) can e rewritten as v ) = βρ )x + Nρ )x ) < where β = ln /) <. Hence is strictl decreasing. Differentiating again and using ρ ) + Nρ ) = from 4), we have From 3), we have v ) = βρ )x + x ρ )N) = βx x )ρ ) v ) = β x x ) ρ )ρ ) N with equalit if and onl if x = x. Hence is convex and is linear if and onl if x = x. The oundar values of follows from 7) sum over i) and 8). Given x, define ρ := 3 x ) x x

5 5 Differentiating ), we have after some algera x = x x x x ρ ) = β x x )ρ ) ρ )) v ) 3ρ ))x x ) x ) c c c r r There are three possile cases: ρ ) ρ : 3ρ ))x x ) x and ρ ). In this case ρ) is convex decreasing. ρ ) ρ : 3ρ ))x x ) x and ρ ). In this case ρ) is concave decreasing. ρ ) > > ρ ): Then ρ ) for and ρ ) for. Hence ρ) is strictl concave decreasing efore the inflexion point and strictl convex decreasing after. a) Velocit x = x x > x x ) ) ) ) Spatial distriution ρ ) Fig.. Illustration of Theorems,, 4, 6 and 7 and the inflexion point ρ ) = ρ r r Theorem. Suppose x > x. Then ρ ) is a strictl decreasing function, with Moreover, ρ ) = x h ) x h ) + Nx h ) if ρ ) ρ, then ρ ) is convex. if ρ ) ρ, then ρ ) is concave. if ρ ) < ρ < ρ ), then ρ ) is concave for and convex for. Proof. From 3), we have ρ ) = βx x )ρ ) ρ )) Since β < and x x, ρ ), i.e., ρ ) is a decreasing function. The value of ρ ) follows from 7) and Theorem. Theorems and are illustrated in Figure. The figure also shows the asmptotic properties of and ρ ), to which we now turn. D. Asmptotic properties We will prove that and ρ ) take the form shown in the right-hand column of Figure asmptoticall as x. We make more assumptions: A3: Given an x, the solution zx ) of ) is continuous in x and that lim x zx ) exists. Denote lim x zx ) z = p, p,...). A4: The pointwise limits of and ρ ) as x, denoted v ) and ρ ), exist. Moreover, relations ), 3) and 5) are satisfied in the limit with zx ), and ρ ) replaced z x ), v ) and ρ ) respectivel. A5: The TCP equilirium) algorithm fp, τ) is continuous in its arguments. Moreover, x = fp, τ) < when p >. A6: The congestion ased dropping g, τ) is continuous in its arguments. Moreover, g, τ) as. Conditions A5 and A6 are nonrestrictive: A5 sas that the TCP rate is finite if there is an loss, and A6 sas that if acklog grows without ound then eventuall all incoming packets will e dropped. Note that we are not assuming that the limit functions v ) and ρ ) satisf the ODEs of Section III-B. We start with a result that sas that regardless of the UDP rate x, ever flow, including UDP flow, occupies less than half of the queue. This implies that asmptoticall as x, congestion ased dropping proailit r < and queueing dela τ >. These properties are used later to prove the asmptotic UDP throughput and the asmptotic spatial properties of the leak uffer of CHOKe. Theorem 3. ) For all x, i /, i =,. ) As x a) h = r)/ r) and h = /N r). ) r <. c) x < and τ >. d) i <, i =,

6 6 Proof. ) From 3), h i + h i )r = p i and hence using 5), we have i = h i r r The right-hand side is a decreasing function for r with a maximum value of / at r =. ) Since x ρ ) c for all x, p must tend to as x. Hence h + h )r = p. Then h r)/ r), or r)/ r). Since = + N, we have h /N r). Suppose r =. From 3), h i + h i )r = p i and hence, for i =,, r h i h i with equalities if and onl if h i = for i =,. But this contradicts that h + Nh =. Hence r <. Consider TCP flow i =. Since h = /N r )) > from part a), 4) implies that p >. B assumption A5, x = fp, τ ) <. Now suppose for the sake of contradiction that τ =. Then 4) implies ut 3) implies p = r ) h ) p = r ) h ) h ielding h =, contradicting that h = /N r )) >. Hence τ >. Finall, if i =, then assumption A6, we have r = lim x g i, τ) =, contradicting ). We next show that the UDP throughput vanishes as x grows without ound. This result is also otained independentl in [9] and in [4], using different metho. Recall 3) that for x τ) = vs). Theorem 4. As x, x p ) = ρ )c. Proof. From 3), we have ρ ) ρ ) ρ )) = βx x ) where β = ln /). Integrating oth sides from to, we get ln Hence ρ ) ρ ) ln ρ ) ρ ) ρ ) = = βx x )τ) ae βx x)τ) + ae βx x)τ) 4) where a = ρ ) ρ ) 5) UDP throughput share x p )/c is ρ ) evaluated at = where τ) = τ. From Theorem 3c), lim x τ) = τ >, and hence Lemma 5 elow, lim x ρ ) =. The following lemma implies that, asmptoticall as x, not onl does the throughput of UDP x p ), moreover, all UDP packets are dropped efore the first point where τ) is nonzero. Lemma 5. If lim x τ) >, with possil a function of x, then lim ρ ae βx x)τ) ) = lim = x x + ae βx x)τ) Proof. We will show that the numerator of 4) ten to as x. From Theorem, we have ρ ) = and hence from 5), we have The numerator is then x h ) x h ) + Nx h ) a = x h ) Nx h ) x h ) Nx h ) eβx x)τ) 6) From Theorem 3, as x, h i <, i =,, and x <. Moreover, from Theorem 3d), < and hence β = ln / ) <. Then, if lim x τ) >, it can e shown that a grows at most linearl to, while e βx x)τ) ten at least exponentiall to. Hence the numerator ae βx x)τ. ρ The next result sas that the asmptotic spatial distriution ) of UDP takes the form shown in Figure. Theorem 6. Let := r )/ r ). Then {, < ρ ) =, < Proof. From Theorem 4, we know ρ ) =. Hence we can define ŷ := inf{ ρ ) = } Let e an point with ρ ) =. One exists ecause, from proof of Lemma 5, we know ρ ) = see 6)). Consider the midpoint etween and ŷ, = + ŷ)/. It suffices to prove that i) either ρ ) = or = ŷ, and ii) ŷ =. i) Suppose ρ ) <. We need to show that = ŷ. Since ρ ) is decreasing for an finite x x Theorem ), its limit ρ ) is nonincreasing in. Hence, ŷ. Suppose for the sake of contradiction that δ := ŷ >.

7 7 We first prove that v ) <. Consider for an finite x the quantit ρ ), the TCP flow rate at position in the queue. Using 9) and 7), we have ρ )v ) ρ )v) = x r) h ) Taking limit as x, we have ρ )v ) x r ) h ) which is ounded Theorem 3. Note that ρ ) < definition of. Since ρ )+Nρ ) =, we have taking limit of the corresponding finite-x expression as x ) ρ ) = ρ )v ) ρ )v ) + Nρ )v ) < Hence, ρ )v ) < implies ρ )v ) <. This in turn implies that v ) < since ρ ) > definition of otherwise, = ŷ and we are done). Now define e the midpoint of and ŷ, = + ŷ)/. Then ρ ) > definition of ŷ. We now show that v ) < implies ρ ) =, a contradiction. From Theorem, is strictl decreasing in and hence τ) defined 3) satisfies τ ) = > = Taking limit as x, we have τ ) δ v ) vs) vs) v ) δ v ) > where the last inequalit follows from v ) <. Hence ρ ) = Lemma 5. But this contradicts the definition of ŷ since < ŷ. Hence we must have = ŷ. ii) The aove shows that ρ ) takes the form shown in Figure. Then, 5) and Theorem 3, we have using assumption A4) r r = This completes the proof. ŷ d = ŷ The next result proves the shape of v ). Theorem 7. Let = r )/ r ). Then {, < v ) = c β x ), < Proof. We first prove for < and then for >. Assume there exists < such that v ) <. Consider = + )/. Since v ) is nonincreasing in, we have v ) v ) < for an. Hence, using assumption A4, we have τ ) = v ) v s) v s) > Then Lemma 5 implies that ρ ) =. Since <, this contradicts theorem 6. So for <, v ) =. For >, we prove the theorem in 4 steps. Step : v ) < for all >. Using 9) and 7), and taking limit as x, we have ρ )v ) ρ )v ) = x r ) h ) But Theorem 6, ρ ) = ρ ))/N = /N for >, and Theorem 3, the right-hand side is finite. Hence v ) < for all >. Step : τ ) > for >. Fix >. Since is strictl decreasing Theorem ) we have, for each finite x, τ) = > > + vs) vs) v ) + Taking limit as x, we have for > τ ) v ) + > Step 3: There exists an integrale function H x ) such that, for all x and >, H x ) := x ρ ) + x ρ )) H x ) From 4) we have x ρ ) = x ae βx x)τ) + ae βx x)τ) Since τ ) > for all >, the proof of Lemma 5 shows that, as x, a grows at most logarithmicall to while e βx x)τ) grows at least exponentiall to. Hence the numerator x ae βx x)τ) ten to as x. Moreover, m) := max x x ae βx x)τ) is finite. Hence, for x sufficientl large so that a > ), we have x ρ ) m)

8 8 Note that m) is independent of x. Hence if we define H x ) = m) + x, then H x ) H x ). Since x is finite for all x Theorem 3), H x ) is integrale over, ). Step 4: v ) = c β x ) for >. Fixed >. From ), we have, for each x, = c β = c β ρ s)x + ρ s))x ) H x s) Taking limit as x, we have v ) = c β lim x H x s) From Step 3, H x s) is upper ounded an integrale function and converges pointwise to H ) = x as x. Hence Legesgue convergence theorem applies see [, p.9], [, pp. 5]): lim x H x s) = H s) = x ) Hence, v ) = c β x ) for >. This completes the proof. We summarize these structural properties. First, when x is large, the spatial distriution ρ ) decreases rapidl toward the head of the queue. This means that most of the UDP packets are dropped efore the reach the head. It is therefore possile to simultaneousl maintain a large numer of packets concentrating near the tail) and receive a small andwidth share, in stark contrast to the ehavior of a non-leak FIFO uffer. Indeed, as x grows without ound, UDP share drops to. Second, the packet velocit is infinite efore the position ecause UDP packets are eing dropped at an infinite rate until. IV. SIMULATION RESULTS We implemented a CHOKe module in ns- version.9 and have conducted extensive simulations using the network shown in Figure to stud the equilirium ehavior of CHOKe. There is a single ottleneck link from router R to router R shared N TCP sources and one UDP source. The UDP source sen data at constant rate CBR). For all our simulations, the link capacit is fixed at c = Mps and the round trip propagation dela is d = ms. Packet size is KB. The simulation time is 3 secon. We use RED+CHOKe as the queue management with RED parameters: min th = packets, max th = 5 packets, p max =.5. This defines the function g in our model 9). For the TCP function in 8), we use [5]): p = + x d + τ) 7) To e precise, = x is a function of x. To make the domain of integration independent of x, extend H x ) and H x ) to, max x x ) defining them to e zero for not in, x ). We var UDP sending rate x from.5pkts/s to 5 pkts/s and the numer N of TCP flows from to 64 to oserve their effect on the equilirium ehavior of CHOKe. We measure, and compare with our numerical solutions, the following quantities ) aggregate queue size ) UDP andwidth share µ = ρ ) 3) TCP throughput x p ) = µ c = ρ )c 4) spatial distriution ρ ) of UDP packets The results illustrate oth the equilirium ehavior of CHOKe and the accurac of our analtical model. The show the ailit of CHOKe to protect TCP flows and agree with those aggregate measurements of []. Moreover, the exhiit the fine structure of queue dnamics, and confirm the spatial character of our CHOKe model. We next discuss these results in detail. A. Effect of UDP rate x First we stud the effect of UDP sending rate on queue size and andwidth allocation. The numer of TCP sources is fixed at N = 3. We var the UDP sending rate x from. link capacit to link capacit. The results are in Figure 3. The aggregate queue length steadil increases as UDP rate x rises. UDP andwidth share µ = ρ ) rises, peaks, and then drops to less than 5% as x increases from.c to c, while the total TCP throughput follows an opposite trend eventuall exceeding 95% of the capacit. We have shown oth TCP throughput the total numer of TCP packets, from all N flows, transmitted the link during each simulation and TCP goodput sum of largest sequence numers in each simulation. TCP goodput does not count retransmitted packets and is hence lower than TCP throughput. These results match closel those otained in [], and with our analtical model. Figure 4 shows the spatial distriution ρ ) of UDP packets compare with Figure )). To get the packet distriution from each simulation x value), we took J = 3 snapshots of the queue ever ms for 3 secon. From the J sample queue sizes j, we first calculated the average avg := j j /J. The distriution was estimated over this range [, avg ], as follows. For each [, avg ], the sample distriution is calculated as ρ) = j ) J where j ) is if the packet in position j / avg of the jth snapshot is UDP, and otherwise. This distriution is plotted in Figure 4, together with the numerical solution of our CHOKe model. When x =.c Figure 4a)), UDP packets are distriuted roughl uniforml in the queue, with proailit close to.8 at each position. As a result, its andwidth share is roughl %. As x increase, ρ ) concentrates more and more near the tail of the queue and drops rapidl toward the head, as predicted Theorems and 6. Also marked in Figure 3), are the UDP andwidth shares corresponding to UDP rates in Figure 4. As expected the UDP andwidth shares in 3) are equal to ρ ) in Figure 4. When j

9 queue size pkts) 5 UDP Packets Spatial Distriution Queue Position 3 a) UDP rate x =.c UDP sending rate pkts/s) a) Queue size.9.8 UDP andwidth share x =c,share=.474 UDP Packets Spatial Distriution Queue Position..5 x =.c,share=.777 x =c,share=.95 ) UDP rate x = c 3 UDP sending rate pkts/s).9 ) UDP andwidth share µ x 4 UDP Packets Spatial Distriution TCP Total Queue Position.5 c) UDP rate x = c.5 Througput Goodput Throughput 3 UDP Sending Rate pkts/s).9.8 c) Total TCP throughput µ c Fig. 3. Effect of UDP rate x on queue size and andwidth allocation. N = 3, x =.c to c, c = Mps, simulation duration = 3sec. UDP Packets Spatial Distriution Queue Position

10 x > c, even though roughl half of the queue is occupied UDP packets, almost all of them are dropped efore the reach the head of the queue! B. Effect of numer N of TCP flows Figure 5 shows the effect of N on aggregate queue size and on per-flow TCP throughput µ c = ρ )c. 4 Queue Size pkts) C=5pkts/s, N=4,UDP rate=5pkts/s Queue Size pkts) Numer of TCP Sources N) a) Queue size Fig. 6. Queue Size at N= Time s) small N. With smaller N, each TCP source gets a larger andwidth share, and a larger TCP sending rate requires a lower dropping proailit. However, when CHOKe is active, it imposes a lower ound on the dropping proailit: from ) p i = h i + h i )r h i N 8) TCP Thoughput per Source pkts) Numer of TCP Sources N) where the last inequalit follows from 5) and the fact that UDP packets occup at most half of the queue Theorem 3)). We can estimate the minimum N with which CHOKe is alwas active. Approximate the TCP function in 7) p = Comining with 8) to get x d + τ) N x d + τ) When UDP sending rate is large, TCP flows take almost all the andwidth, so ) TCP throughput per flow µ c x c N Fig. 5. Effect of numer N of TCP flows on aggregate queue and per-flow TCP throughput µ c. N = 64, x = 5 pkts/s, c = 5pkts/s, simulation duration = 3s Not surprisingl, the queue size increases and per-flow TCP throughput decreases with N as the queue ecomes more congested. Again, the simulation and analtical results match ver well, further validating our model. V. DISCUSSION Our model captures well the equilirium ehavior of CHOKe under the assumption that the queue size is etween and. This hol if c is sufficientl small or N is sufficientl large. A sample queue size is shown in Figure 6. It ma not hold for Around, queueing dela is roughl τ c Putting all these together, the minimum numer of TCP flows required for CHOKe to remain active is roughl N 3 cd + ) Using the same parameters as in the last section, we can estimate the minimal N to e When N = 8, queue size oscillates around = packets, constantl turning CHOKe on and off, as shown in Figure 7 compare with Figure 6).

11 Queue Size pkts) Fig. 7. Queue Size at N= C=5pkts/s, N=8,UDP rate=5pkts/s Times) The same phenomenon is oserved when c increases with fixed N). The lower ound on dropping proailit when CHOKe is active, p /N, eventuall prevents TCP flows from making full use of the availale capacit. An interesting positive effect is that the queue length is controlled to sta around. When there are more than one unresponsive UDP flows, CHOKe ma not e ver efficient in protecting TCP traffic. The UDP flows ma take a larger andwidth share at high sending rate. [3] S. Flod and V. Jacoson. Random earl detection gatewas for congestion avoidance. IEEE/ACM Trans. on Networking, 4):397 43, August 993. ftp://ftp.ee.ll.gov/papers/earl.ps.gz. [4] Dong Lin and Roert Morris. Dnamics of random earl detection. In Proceedings of SIGCOMM 97, pages 7 37, Septemer 997. http: // [5] Steven H. Low. A dualit model of TCP and queue management algorithms. IEEE/ACM Trans. on Networking, to appear, Octoer 3. [6] P. McKenn. Stochastic fairness queueing. In Proceedings of Infocom, pages , 99. [7] J. A. Nelder and R. Mead. A simplex method for function minimization. Comput. J., pages 38 33, 965. [8] T. J. Ott, T. V. Lakshman, and L. Wong. SRED: Stailized RED. In Proceedings of IEEE Infocom 99, March 999. ftp://ftp.ellcore. com/pu/tjo/sred.ps. [9] Rong Pan, Chandra Nair, Brian Yang, and Balaji Prahakar. Packet dropping mechanisms: some examples and analsis. In Proc. of 38th Annual Allerton Conference on Communication, Control, and Computing, Octoer. [] Rong Pan, Balaji Prahakar, and Konstantinos Psounis. CHOKe: a stateless AQM scheme for approximating fair andwidth allocation. In Proceedings of IEEE Infocom, March. [] Lawrence Perko. Differential equations and dnamical sstems. Springer, 3 edition,. [] H. L. Roden. Real analsis. MacMillan, 968. [3] I. Stoica, S. Shenker, and H. Zhang. Core-stateless fair queueing: achieving approximatel fair andwidth allocations in high speed networks. In Proceedings of ACM Sigcomm, 998. [4] Jiantao Wang, Ao Tang, and Steven H. Low. Maximum and asmptotic UDP throughput under CHOKe. Sumitted to ACM Sigmetrics, http: //netla.caltech.edu, Novemer. VI. CONCLUSIONS CHOKe is completel stateless and extremel simple to implement, et surprisingl effective in ounding the andwidth share of UDP flows. As shown in the simulations of [], as UDP source rate increases, its andwidth share eventuall drops to zero, exactl opposite to how a regular FIFO uffer ehaves. To understand this precisel requires a careful stud of the queueing process under CHOKe. In this paper, we have developed a model of CHOKe. Its ke features are the incorporation of the feedack equilirium of TCP and a detailed modeling of the queue dnamics. We have introduced the concepts of spatial distriution and velocit of packets in the queue. We prove structural and asmptotic properties of these quantities that make it possile for UDP to simultaneousl maintain a large numer of packets in the queue and receive a vanishingl small andwidth share, the mechanism through which CHOKe protects TCP flows. Finall, we remark that CHOKe algorithm ma e constantl turned on and off when the link capacit is high or the numer of TCP sources is small. This can prevent TCP flows from making full use of availale capacit ut regulate the queue size to around. REFERENCES [] Patrick Billingsle. Proailit and Measure. John Wile & Sons, edition, 986. [] W. Feng, K G. Shin, D. Kandlur, and D. Saha. Stochastic Fair Blue: A queue management algorithm for enforcing fairness. In Proceedings of INFOCOM, April.