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Sizing Internet Router Buffers, ActiveQueue Management, and the Lur’e

ProblemChristopher M. Kellett, Robert N. Shorten, and Douglas J. Leith

Abstract— Recent work in sizing Internet routerbuffers has shown that for a large number of users,most buffers will be over-provisioned. This workimplies that it is possible to significantly reducequeueing delays while incurring little reduction inutilization of the egress link under stationary trafficconditions. While this work is of undoubted valuein the context of network research, it is neverthelessdifficult to see how these recent results may beused to provision network buffers. In particular, net-work conditions (traffic mix, number of TCP usersetc.) vary over short time scales. Consequently, thebounds derived under the assumption of stationaritymake little sense in real network environments. Ourprincipal contribution in this note is to propose anactive queue management (AQM) scheme that strivesto regulate link utilization by adjusting the drop-tail buffer size. This work exploits recent results onbuffer sizing in a way that takes the non-stationarityof the arrival process into account. In particular,our algorithm strives to achieve its target utilizationusing the smallest possible drop-tail buffer size.In addition, and perhaps more importantly, ourformulation reveals a direct connection between theAQM’s stability/convergence and a classical problemin stability theory; namely, the Lur’e problem. Fi-nally, packet-level simulation results are presented todemonstrate the efficacy of our proposed algorithm.

I. I NTRODUCTION

Buffers are used at Internet routers to tem-porarily store incoming packets when the arrivalof packets received exceeds the capacity of theegress link. This is done to maintain a high-level of utilization of link capacity and to ac-commodate bursty traffic. The classical heuristic

Authors are with the Hamilton Institute, National Universityof Ireland, Maynooth, Co. Kildare, Ireland, and are supportedby Science Foundation Ireland Grant 04/IN3/I460. E-mails: chris.kellett@nuim.ie, robert.shorten@nuim.ie,doug.leith@nuim.ie

for sizing router buffers in the Internet, namelythe Bandwidth-Delay Product (BDP) rule, statesthat the amount of buffering necessary to maintainfull utilization of a single bottleneck link is givenby the product of the bottleneck link capacityCand the ‘typical’ round-trip timeRTT betweensource and destination. However, the BDP ruleis becoming increasingly unviable as it impliesextremely large network and expensive buffers aslink speeds scale up to gigabit level and above.

Recent research has suggested that the assump-tions under which the BDP heuristic is usefulare overly conservative and several authors havesuggested novel sizing strategies that significantlyreduce the amount of required buffering in suchlinks. Appenzelleret al. [1], and subsequentlya number of other researchers, have observedthat, under appropriate assumptions, the necessarybuffering to maintain high utilization of the egresslink is no more thanC×RTT√

N, where N is the

number of long-lived Transport Control Protocol(TCP) flows accessing the congested router,C

is the bottleneck link capacity, andRTT is theround-trip time between source and destination.Avrachenkovet al. [2] improved this bound to(C×RTT )2

32N3 . The underlying rationale for theseresults is based on the fact that, under typicalcircumstances, at any congestion event only afraction of TCP flows reduce their sending rates.

Unfortunately, these sizing strategies are depen-dent on the number of flows and the mix of trafficin a network, which are dynamic and difficult toestimate quantities. This fact immediately callsinto question the utility of such buffer sizingstrategies in real communication networks. Forexample, Figure 1 shows the queue occupancy for

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Fig. 1. Slowly varying number of users. Simulation pa-rameters: 200 Mbps link, 500 packet queue, RTTs uniformlydistributed between 10ms-300ms.

a single bottleneck link with drop-tail queue asthe number of flows is gradually changed. It canbe seen that a large persistent queue exists, withonly small variations in queue occupancy, when alarge number of flows share the link. Hence, theability of the buffer to accommodate packet burstsis effectively that of a small buffer in the manyflow regime and this fact immediately calls intoquestion the need for large buffers in this type ofscenario. It can also be seen that when smallernumbers of flows share the link a large queueremains necessary in order to ensure high linkutilization. Since the statistics of arriving trafficat a link will typically not be stationary, a buffersize derived under one set of assumptions may bewholly inappropriate in other circumstances.

In addition to non-stationarity, another funda-mental problem is that, in practice, we expect linktraffic to include a complex mix of bursty, on-off flows, a mix of connection sizes, a mix ofTCP and UDP traffic (e.g. voice/video), a mixof congestion control action (not only the manyvariants of TCP but also TFRC (TCP FriendlyRate Control) and other proprietary streaming al-gorithms)etc. Current approaches to buffer sizingare model-based. That is, a parameterized trafficmix is considered and buffer sizing is analyzed asa function of these traffic parameters (e.g. numberof long-lived flows). Such approaches inevitablyrequire consideration of the impact of mismatchbetween model and reality.

These observations – non-stationarity of traf-fic arrivals and the need to support complex,poorly characterized arrival processes – lead usnaturally to consider an adaptive, measurement-based framework for drop-tail queues. Rather thanbuilding ever more elaborate traffic models in anattempt to better capture realistic behavior, wepropose a different approach. The quantities ofinterest: utilization, loss, delay,etc, are readilymeasured for a current choice of buffer size. Weargue that this information can be used directlyto adapt the buffer size with the aim of regulatingutilization. That is, we consider dynamically vary-ing maximum queue length1 based on feedbackof the measured level of utilization, see Figure2. We refer to this paradigm for active queuemanagement (AQM) as Active Drop-Tail (ADT).

ObserverController/

(e.g., Utilization)

Traffic

Max QueueSize

Measurements

Queue

Fig. 2. Active Drop-Tail Architecture.

The ADT paradigm evidently allows the queueto adapt to changing traffic characteristics. Imple-mentation is based on readily available queue mea-surements (utilization,etc) rather than estimationof arrival process parameters. A key issue in suchan adaptive approach is, however, that the trafficarrival process in internet links is typicallyelastic.That is, there exist flow-level feedback loops thatseek to roughly match offered load to networkcapacity. The flow-level feedback loops are cou-pled together at links where queueing occurs.Changes in the queue parameters can thereforechange the arrival process itself. When we adjustthe queue parameters in response to changes in thearrival process, this immediately raises concernsas to adverse interactions leading, for example,to oscillations or other instabilities. This stabilityquestion is made particularly challenging by the

1In other words, we regularly compute the maximum allow-able queue size and, when a packet arrives, if it causes thebuffer to exceed the maximum allowable size, the packet isdiscarded. Otherwise, it is enqueued.

3

Queue size

Utilisation

u*

q*

100%

Qmax

Fig. 3. Utilization curves for a number of traffic scenarios

facts that (i) the arrival process is time-varying,highly complex and poorly characterized and (ii)there is, in fact, a network of queues with queueingpossible at multiple bottlenecks simultaneously.

Our contribution in this paper is twofold. Firstly,we demonstrate that the foregoing stability ques-tion can be formulated as a Lur’e problem. Con-sequently we are able account for uncertaintyand non-stationarity in network traffic patterns inthe form of a sector bounded time-varying non-linearity. Networks of coupled queues can then betreated inside a passivity or other interconnectionframework. Secondly, we illustrate the use of thisobservation to design an ADT algorithm whichstrives to minimize queueing delay subject to aminimum desired average target utilization.

II. BUFFER SIZING AND THELUR’ E PROBLEM

We take as our starting point the proposed ADTarchitecture illustrated in Figure 2. In the sequelwe confine consideration to linear control strate-gies although the approach can be readily extendedto encompass nonlinear strategies. Our basic as-sumption in formulating our stability problem isthat average utilizationu(k + 1) is a function ofthe buffer sizeq(k) and the traffic arrival process.That is, u(k + 1) = f(q(k), θk) where θk is anexogenous input determined by the traffic arrivalprocess, see Figure 3. We assume thatf(q(k), θk)is one-to-one and continuous for fixedθk.

The ADT paradigm requires thatu(k) be esti-mated at each sample step. This essentially implies

K ) ( ˆ k q ) ( ˆ u

z

+

-

k+1

-1

g( , ) 0

n(k)

Fig. 4. ADT and the Lur’e problem

a slow sampling rate and the existence of anobserver to estimate the average utilization overeach sampling period.

Let u∗ be the desired reference average utiliza-tion of the feedback system, and letq∗ be thecorresponding equilibrium queue size on one ofthe possible utilization curves. Define:

u(k) = u(k) − u∗, (1)

q(k) = q(k) − q∗. (2)

Then, we can model the relationship betweenutilization and queue size as

u(k + 1) = g(q(k), θk) + n(k) (3)

where g(·, ·) is a first and third quadrant sectorbounded nonlinearity, andn(k) is a bounded ex-ogenous disturbance term due to non-stationarityof the network traffic (i.e. moving between thecurves in Figure 3).

Then, the closed loop ADT system can bedepicted as in Figure 4 where:

k1 ≤g(q(k), θk)

q(k)≤ k2. (4)

Furthermore, having restricted attention to a linearcontroller/observer pair, we see that the ADTparadigm can be seen as a linear system in feed-back with a static, memoryless, sector-boundednonlinearity.

We are interested in the (non-local) uniformasymptotic stability of the equilibrium state ofthe system depicted in Figure 2. Due to the non-stationary nature of arriving traffic, the disturbancen(k) is typically time-varying and the best we canachieve is BIBO stability. Nevertheless, this is aclassic Lur’e problem and can be studied using aplethora of classical tools; most notably the CircleCriterion [7]. A number of remarks are in orderin view of the above discussion.

Remark 1:Stability of networks with multiplecongested routers can be guaranteed by selecting

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controller gains in each router according to theCircle Criterion, and by recalling that networks ofconnected passive elements are themselves pas-sive. Stability in the large of the entire networkfollows [11].

Remark 2:The Lur’e/passivity based analysisadvocated here can be applied to other AQMschemes; most notably the AVQ framework pro-posed by Srikant and others [8]. Work will bereported on this topic in future publications.

A. Active Drop Tail (ADT): Convergence and sta-bility

We note that many control algorithmsK arepossible. As an illustrative example we considerin this paper a simple integrator feedback (see [9]for another control technique):

q(k + 1) = q(k) + KI(u∗ − u(k)) (5)

where u ∈ [0, 1] is the utilization estimate pro-vided by the router,q(k) is the available bufferingat timek ∈ Z≥0, andKi is the gain of a integralcontrollerK = KI

1−z−1 (see Figure 4).In view of the simple controller structure used,

BIBO stability follows from the Lur’e formulationusing straightforward arguments, providedKI ∈(

0,2

k2

)

where k2 is the sector nonlinearity

upper bound.The preceding stability argument ignores other

possible elements in the feedback system suchas dynamics of a filter for estimation ofu(k),saturation elements in the feedback system, andanti-windup elements. However, the effect of suchfilters/nonlinearities is easily accommodated in theLur’e framework (see [4] or [7]).

III. S IMULATIONS

We several performed packet-level network sim-ulations using the network simulatorns-22. Inparticular, we looked at queue occupancy andutilization in three scenarios: (i) a step change inthe number of users; (ii) slowly-varying numberof users (similar to Figure 1); and (iii) a multiplebottleneck network.

2The network simulator ns-2 is available athttp://nsnam.isi.edu/nsnam/index.php/UserInformation.Necessary files for the simulations in this paper are availableat http://www.hamilton.ie/chris/adt.

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Fig. 5. Step change in users - 20 flows to 100 flows to20 flows. Simulation parameters: 100 Mbps link, 500 packetqueue, RTTs uniformly distributed between 10ms-300ms, con-troller gain of 750.

A. Step response in number of users

In Figure 5 we show the results of a suddenstep change in the number of users. In particular,from time zero we have 20 TCP users accessinga 100Mbps link. At 150 seconds, the number ofusers increases from 20 to 100. At 350 secondsthe number of users returns to 20 flows. We notethat the utilization remains near the reference of99% outside of a short time period when the stepchange occurs.

Note the persistent, but small, oscillations in thequeue size in Figure 5. This is due to the stochasticnature of the arrival traffic as seen in the noiseterm in (3). However, we do observe the expectedBIBO stability.

B. Slowly-varying number of users

By way of comparison with Figure 1, weperform the same simulation with the Drop-Tailqueue replaced by an ADT queue, with the resultsshown in Figure 6. The simulation shows a slowlyvarying number of TCP users accessing a single100Mbps link. As the top plot shows, we haveten users up to 200 seconds, after which weadd a user every two seconds up to 500 users.From 1800 seconds, a user leaves the networkevery two seconds down to ten users. Observethat ADT removes the persistent standing queueseen in Figure 1, and thus reduces queueing delay,

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Fig. 6. Slowly varying number of users. Simulation pa-rameters: 300 Mbps link, 500 packet queue, RTTs uniformlydistributed between 10ms-300ms, controller gain of 750.

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Fig. 7. Slowly varying number of users, comparison with thebound of Appenzelleret al. Simulation parameters: 200 Mbpslink, 500 packet queue, RTTs uniformly distributed between20ms-40ms, controller gain of 750.

when more users are present in the network at theexpense of a small amount of link utilization.

We perform a similar experiment for comparingour queue limitq(k) with the following boundproposed by Appenzelleret al.:

B =C × RTT

√N

,

whereC is the link capacity,RTT is the propa-gation delay, andN is the number of TCP users.In Figure 7, the middle plot shows the differencebetween the Appenzeller bound and the maximum

1 x

3 x 2 x

1 C 2 C

1 N 2 N

Fig. 8. Multiple bottleneck topology. Simulation parameters:C1 = 60 Mbps, C2 = 100Mbps, 500 packet queues.

buffer size from ADT; i.e., we plotC×RTT√N

−q(k),where q(k) comes from (5). Particularly in theregime where the Appenzeller bound is accurate(i.e., in the large number of users regime), theADT algorithm matches the bound within a fewpackets either way. Note that, for the purposes ofa more accurate comparison with the Appenzellerbound, we have chosen a narrower range of round-trip times.

C. Multiple bottlenecks

Finally, we consider the behavior of ADT inthe case where we have two bottleneck links. Weconsider a two bottleneck aggregation-type topol-ogy as shown in Figure 8. We start 50 TCP flowsfrom each of nodesx1 and x2 at time zero. At100 seconds we start another 100 TCP flows fromnode x3. Figure 9 shows that regulation to thetarget of 99% is achieved for both links. By wayof comparison, Figure 10 shows the same scenariowith standard drop-tail, rather than ADT, queues.Note that the ADT queues result in significantlyreduced queueing delay in both queues, providinga double benefit.

IV. RELATED WORK

Stability analysis of coupled AQM and TCPdynamics has previously been studied using so-called “fluid” approaches (e.g., [5], [8]). We takea different approach and suggest that this approachis preferable to the fluid approach for a number ofreasons. Firstly, the insights regarding the buffersize - utilization relationship discussed by Appen-zeller et al. are valid not only for networks ofTCP flows, but in a general sense also for net-works carrying complicated mixes of traffic. Fluidmodels, generally speaking, are unable to capturesuch complicated network traffic in a meaningfulmanner. Secondly, the buffer size - utilizationrelationship exists in a number of situations in

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Fig. 9. Left plots: 60 Mbps link; Right plots: 100 Mbps link.RTTs uniformly distributed between 10ms-300ms, controllergains of 750.

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Fig. 10. Left plots: 60 Mbps link; Right plots: 100 Mbpslink. RTTs uniformly distributed between 10ms-300ms.

which the use of fluid models cannot be justified;for example, in networks in which low numbersof TCP flows are deployed. Thirdly, the utility offluid models in the context of drop-tail queues ispresently unclear.

As previously noted, ADT is a form of AQM.Many AQM strategies have been proposed overthe past fifteen years. The first proposed AQMscheme was RED (Random Early Detection) [3].RED drops or marks packets according to aprobabilistic mechanism based on average queueoccupancy in an effort to reduce buffer overflowsand improve network fairness. To our knowledge,

the adaptive drop-tail algorithm proposed in [10]was the first algorithm to suggest varying thelength of a drop-tail queue. However, rather thanreducing queueing delays, the authors’ goal was toreduce packet retransmissions and their algorithmwill actually increase queueing delays. AdaptiveVirtual Queue (AVQ) [6] regulates utilization,similar to ADT, however based on arrival rate,rather than egress link utilization. This means thatthe mechanism by which AVQ generates drops isdifferent from ADT.

V. CONCLUSIONS

In this paper we consider the Active Drop-Tail (ADT) paradigm for active queue manage-ment. ADT is an adaptive, measurement-basedframework for drop-tail queues that respects thenon-stationarity of traffic arrivals and the needto support complex, poorly characterized arrivalprocesses. A key issue, however, is the stabilityof the coupled dynamics arising from interactionof ADT queue adjustment and flow-level con-gestion control feedback. A particular difficultyarises from the fact that the traffic arrival processis time-varying, highly complex and only poorlycharacterized. We demonstrate that this stabilityquestion can be formulated as a Lur’e problem.Consequently we are able account for uncertaintyand non-stationarity in network traffic patterns inthe form of a sector bounded time-varying non-linearity. Networks of coupled queues can then betreated inside a passivity or other interconnectionframework. We illustrate the use of this observa-tion to design an ADT algorithm which strives tominimize queueing delay subject to a minimumdesired target utilization.

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[4] G. C. Goodwin, S. F. Graebe, and M. E. Salgado.ControlSystem Design. Prentice Hall, 2001.

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[5] C. Hollot, V. Misra, D. Towsley, and W. Gong. Analysisand design of controllers for AQM routers supportingTCP flows. IEEE Transactions on Automatic Control,47(6):945–959, June 2002.

[6] S. Kunniyur and R. Srikant. Analysis and design ofan adaptive virtual queue (AVQ) algorithm for activequeue management. InProceedings of SIGCOMM’01,San Diego, California, USA, 27-31 August 2001.

[7] S. Lefschetz. Stability of nonlinear control systems.Academic Press, 1965.

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