Computer Networks 51 (2007) 114–133

www.elsevier.com/locate/comnet

Beyond fluid models: Modelling TCP mice in IP networksunder non-stationary random traffic

G. Carofiglio, M. Garetto, E. Leonardi *, A. Tarello, M. Ajmone Marsan

Dipartimento di Elettronica, Politecnico di Torino, Italy

Received 7 September 2005; accepted 10 April 2006Available online 24 May 2006

Responsible Editor: I. Matta

Abstract

Fluid models of IP networks are based on a set of ordinary differential equations, that provide an abstract deterministic

description of the average network dynamics. When IP networks operate close to saturation, fluid models were proved toprovide reliable performance estimates. Instead, when the network load is well below saturation, standard fluid modelslead to wrong performance predictions, since all buffers are forecasted to be always empty, so that the packet discard prob-ability is predicted to be zero. These incorrect predictions are due to the fact that fluid models, being deterministic in nat-ure, do not account for the random traffic variations that may induce temporary congestion of some network elements. Inthis paper we discuss three different approaches to describe random traffic variations in fluid models, considering random-ness at both the flow and packet levels. With these approaches, fluid models allow reliable results to be obtained also in thecase of IP networks that operate well below their saturation load. Numerical results are presented to prove the accuracyand the versatility of the proposed approaches, considering both stationary and non-stationary traffic regimes.� 2006 Elsevier B.V. All rights reserved.

Keywords: Stochastic processes; Queueing theory; Simulation; IP networks; TCP mice; Fluid models

1. Introduction

Fluid models have been recently proposed as ascalable and accurate tool for performance analysisof IP networks loaded by TCP Traffic (see the nextsection for an overview of previous work on fluid

1389-1286/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.comnet.2006.04.013

* Corresponding author. Tel.: +39 011 5644133.E-mail addresses: carofiglio@mail.tlc.polito.it (G. Carofiglio),

garetto@mail.tlc.polito.it (M. Garetto), leonardi@mail.tlc.polito.it (E. Leonardi), tarello@mail.tlc.polito.it (A. Tarello), ajmone@mail.tlc.polito.it (M. Ajmone Marsan).

models). Fluid models adopt an operational view ofthe network; that is, the network is described througha set of ordinary differential equations that providean abstract deterministic description of the averagenetwork dynamics and neglect the detailed short-term, packet-by-packet behavior, whose descriptionprevents the scaling of traditional approaches tothe study of large IP networks. Differential equationsare then solved numerically, obtaining estimates ofthe time-dependent network behavior.

Fluid models of IP networks were originallydeveloped for long-lived TCP flows (often called

.

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 115

elephants). In this case, the model complexity, i.e.,the number of differential equations to be solved,is independent of the number of elephants and oflink capacities; in addition, the differential equationswere proved to capture the limiting network behav-ior, when the number of flows grows very large [2,3].The characteristics of TCP elephants, and in partic-ular the dynamics of their congestion window, aresuch that the load of network elements is broughtclose to saturation, so that a deterministic descrip-tion of the network produces accurate results. Inthe case of short-lived TCP flows (called mice), themost numerous today in the Internet, the load ofnetwork elements is determined by the flow genera-tion rate and by the average flow length, and is nor-mally (much) less than 1. This situation is such thata purely deterministic description of the networkwith standard fluid models leads to wrong perfor-mance predictions, since all buffers are forecastedto be always empty, and the packet discard proba-bility is predicted to be zero (except for the possibil-ity of oscillating behaviors close to saturation in thelimiting case of an infinite number of mice [13]).

Fluid models were recently extended to networkscenarios comprising mice, by accounting for thestochastic aspects that characterize IP traffic, andthat may produce short-term congestion in thenetwork elements [5,6]. With this approach, fluidmodels allow meaningful results to be obtainedalso in the case of TCP mice. Randomness affectsthe IP network dynamics at two different levels: (i)at the flow level, since both the arrival times andthe lengths of TCP mice are random; (ii) at thepacket level, since the packet emission process ofTCP sources is driven by the window dynamics,which are influenced by losses, hence random; inaddition, also the packet lengths are random. In thispaper we provide a broader extension of fluidmodels to tackle traffic randomness, exploring threedifferent approaches to integrate stochastic aspectsinto standard fluid models. Indeed, we add tothe Montecarlo approach adopted in [5,6], thepossibility of describing randomness with Brownianmotions.

With the first approach, the differential equationsdescribing the dynamics of both the TCP flows andthe IP network buffers are transformed into a set ofcoupled stochastic differential equations which arenumerically solved using a combined Montecarlo-fluid method.

With the second approach, while the stochasticdifferential equations describing the TCP flows

dynamics are still solved using a Montecarlo method,a second-order approximation is applied to the bufferprocesses, and the buffer length evolution is repre-sented as a Brownian motion, so that the bufferlength distribution dynamics is driven by a Fok-ker–Planck equation.

With the third approach, a second-order approx-imation is used to describe both the TCP flows andthe buffer dynamics. In this case, the model com-prises a set of coupled deterministic differential equa-tions describing the evolution of the distribution ofthe number of active TCP flows, of the window sizeof active TCP flows, and of buffer lengths.

2. Previous work on fluid models

Fluid models were first proposed in [1] to studythe interaction between TCP elephants and a REDbuffer in a packet network consisting of just one bot-tleneck link, either ignoring TCP mice [1], or model-ling them as unresponsive flows [7] which introduce astochastic disturbance. In this case, fluid modelsoffer a very attractive alternative to packet-basedsimulators, since their complexity (i.e., the numberof differential equations to be solved) is independentof both the number of TCP flows and the linkcapacity.

Structural properties of the fluid model solutionwere analyzed in [8], while important asymptoticproperties were proved in [2,3]. In the latter works,it was shown that fluid models correctly describethe limiting behavior of the network when both thenumber of TCP elephants and the bottleneck linkcapacity jointly tend to infinity.

The single bottleneck model was then extended toconsider general multi-bottleneck topologies com-prising RED routers in [9,10]. In all cases, the setof ordinary differential equations of the fluid modelmust be solved numerically, using standard discreti-zation techniques. In Section 2.1 we briefly summa-rize the fluid models proposed in [1,9,10], whichconstitute the starting point for our work.

An alternative fluid model was proposed in[11,12] to describe the dynamics of the average win-dow for TCP elephants traversing a network ofdrop-tail routers. The behavior of such a networkis pulsing: congestion epochs in which some buffersare overloaded (and overflow) are interleaved toperiods of time in which no buffer is overloaded,and no loss is experienced, due to the fact that previ-ous losses forced TCP sources to reduce their send-ing rate. In such a setup, a careful analysis of the

116 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

average TCP window dynamics at congestion epochsis necessary, whereas sources can be simply assumedto increase their rate at constant speed between con-gestion epochs.

An extension that allows TCP mice to be consid-ered is also outlined in [11,12]. On the basis of theproposed model, since the dynamics of TCP micewith different size and/or different start times followdifferent evolutions, each mouse must be describedwith two differential equations; the first represent-ing the average window evolution, and the seconddescribing the workload evolution. As a conse-quence, one of the nicest properties of fluid models,the insensitivity of complexity with respect to thenumber of TCP flows, is lost.

In [4] a different description of the dynamics oftraffic sources is proposed, that exploits partial dif-ferential equations to analyze the asymptotic behav-ior of a large number of TCP elephants through asingle bottleneck link fed by a RED buffer.

In [5] we built on the approach in [4], showingthat the partial differential equation description ofthe source dynamics allows the natural representa-tion of mice as well as elephants, with no sacrificein terms of scalability of the model.

The limiting case of an infinite number of TCPmice is considered in [13], where it is proved that,even in the case of loads less than 1, deterministicsynchronization effects may lead to congestion andto packet losses.

When the network workload is composed ofa finite number of TCP mice, it normally happensthat link loads, that result from the product ofthe mice arrival rate times the average mice length,are well below link capacities. In this case, thedeterministic nature of fluid models leads to pre-dicting that buffers are always empty and thusnever overflow, which contradicts the observationsmade on real packet networks. This discrepancyis due to the fact that, in underload conditions,the stochastic nature of the input traffic plays afundamental role in the network dynamics, whichcannot be captured by the determinism of fluidmodels.

In [5,6] we first discussed the possibility of inte-grating stochastic aspects within fluid models,adopting a Montecarlo approach.

In this paper, we further investigate the possibil-ities for the integration of randomness in fluid mod-els, proposing two new approaches, which rely onsecond-order Gaussian approximations of the sto-chastic processes driving the network behavior.

2.1. Fluid models of IP networks

In this section we briefly summarize the originalfluid model presented in [1,9,10] and the partialdifferential equations approach described in [5].Finally, we propose an extension of the latterapproach for TCP mice, which accounts for time-outs and fast-recovery.

Consider a network comprising K router outputinterfaces, equipped with FIFO buffers, feeding linksat rate C (the extension to non-homogeneous datarates is straightforward). The network is fed by Iclasses of TCP elephants; all the elephants withinthe same class follow the same route through the net-work, thus experiencing the same round-trip time(RTT), and the same average loss probability. Attime t = 0 all buffers are assumed to be empty. Buf-fers drop packets according to their average occu-pancy, as dictated by a RED (Random EarlyDetection [14]) active queue management (AQM)scheme.

2.1.1. TCP source evolution equations

In [1,9,10,18], simple differential equations wereused to describe the behavior of TCP elephants overnetworks of IP routers adopting a RED AQMscheme. We refer to this original model with thename MGT.

The temporal evolution of the average windowsize of TCP sources in the ith class of elephants,Wi(t), is described by the following differentialequation:

dW iðtÞdt

¼ 1

RiðtÞ� W iðtÞ

2kiðtÞ; ð1Þ

where Ri(t) is the average RTT for class i, and ki(t) isthe loss indicator rate experienced by TCP flows ofclass i. The differential equation is obtained byassuming that elephants are always in congestionavoidance (CA) mode, so that the window dynamicsevolve according to the additive increase multiplica-tive decrease (AIMD) scheme. The window increasein CA mode is linear, and corresponds to one packetper RTT. The window decrease rate is proportionalto the rate at which congestion indications are re-ceived by the source, and each congestion indicationimplies a reduction of the window by a factor 2.

In [5] we extended the fluid model presented in[1,9,10,18]. With our approach, that will be namedI04, rather than just describing the average TCPwindow evolution, we probabilistically model thedynamics of the entire population of TCP flows

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 117

sharing the same path. This approach leads to sys-tems of partial derivatives differential equations,and produces more flexible and accurate models,while preserving the scalability property with respectto the number of flows.

To begin, consider a fixed number of TCPelephants. We use Pi(w, t) to indicate the numberof elephants of class i whose window is 6w at timet. For the sake of simplicity, we consider just oneclass of flows, and omit the index i from all the vari-ables. The sources dynamics are described by the fol-lowing equation, for w P 1:

oP ðw; tÞot

¼Z 2w

wkða; tÞ oP ða; tÞ

oada� 1

RðtÞoP ðw; tÞ

ow;

ð2Þwhere k(w, t) is the loss indication rate. The intuitiveexplanation of the formula is the following. Thetime evolution of the population described byP(w, t) is governed by two terms: (i) the integral ac-counts for the growth rate of P(w, t) due to thesources with window between w and 2w that experi-ence losses; (ii) the second term describes the de-crease rate of P(w, t) due to sources increasingtheir window with rate 1/R(t).

2.1.2. Network evolution equationsIn both MGT and I04 models, Qk(t) denotes the

(fluid) level of the queue in the kth buffer at time t;the temporal evolution of the queue level is describedby

dQkðtÞdt

¼ AkðtÞ½1� pkðtÞ� � DkðtÞ; ð3Þ

where Ak(t) represents the fluid arrival rate at thebuffer, Dk(t) the departure rate from the buffer(which equals the link capacity Ck, provided thatQk(t) > 0), and the function pk(t) represents theinstantaneous loss probability at the buffer, whichdepends on the packet discard policy. An explicitexpression for pk(t) is given in [9] for RED buffers,while for drop-tail buffers

pkðtÞ ¼maxð0;AkðtÞ � CÞ

AkðtÞIfQkðtÞ¼Bkg ð4Þ

which is greater than zero when the queue length isequal to Bk, the buffer size of queue k.

Let Tk(t) denote the instantaneous delay experi-enced at buffer k at time t, given by

T kðtÞ ¼ QkðtÞ=Ck:

If Fk indicates the set of elephants traversing bufferk, and Ai

kðtÞ and DikðtÞ are the arrival and departure

rates at buffer k referred to class i elephants, respec-tively, it follows that:

AkðtÞ ¼Xi2Fk

AikðtÞ;

Z tþT kðtÞ

0

DikðaÞda ¼

Z t

0

AikðaÞ½1� pkðtÞ�da

which means that the total amount of fluid arrivedup to time t leaves the buffer by time t + Tk(t), sincethe buffer is FIFO.

2.1.3. Source–network interactions

Let k(h, i) be the hth buffer traversed by elephantsin class i along their path Pi comprising Li links. TheRTT Ri(t) perceived by such elephants satisfies thefollowing expression:

Ri t þ gi þXLi

h¼1

T kðh;iÞðtkðh;iÞÞ !

¼ gi þXLi

h¼1

T kðh;iÞðtkðh;iÞÞ;

ð5Þwhere gi is the total propagation delay, and tk(h, i) isthe time when the fluid injected at time t by the TCPsources reaches the hth buffer along its path Pi. Notethat (5) comprises the propagation delay gi in a sin-gle term, as if it were concentrated only at the lasthop; the inclusion of the propagation delay at eachhop just requires a formal modification in the recur-sive equation of tk(h, i), that we omit for the sake ofsimplicity. We have:

tkðh;iÞ ¼ tkðh�1;iÞ þ T kðh�1;iÞðtkðh�1;iÞÞ: ð6Þ

The loss rate indicator in the MGT model isgiven by

kiðtÞ ¼ aW iðtÞRiðtÞ

pFi ðtÞ; ð7Þ

where Wi(t)/Ri(t) is the instantaneous emission rateof TCP sources, a is a calibration parameter, andpF

i ðtÞ is the instantaneous loss probability experi-enced by elephants in class i:

pFi ðtÞ ¼ 1�

YLi

h¼1

½1� pkðh;iÞðtkðh;iÞÞ�:

In our I04 extension the quantity k(w, t) can becomputed as

k w; t þ RðtÞð Þ ¼ wpF ðtÞRðtÞ ; ð8Þ

where the source window at time t + R(t) is used toapproximate the window at time t. Intuitively, thisloss model distributes the lost fluid over the entire

118 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

population, proportionally to the window size. Notethat by so doing, no calibration parameter isnecessary.

Finally, in both models:

AkðtÞ ¼X

i

Xq

riqkDi

qðtÞ þX

i

eik

W iðtÞRiðtÞ

N i; ð9Þ

where eik ¼ 1 if buffer k is the first buffer traversed

by elephants of class i, and 0 otherwise; riqk is derived

by the routing matrix, being riqk ¼ 1 if buffer k

immediately follows buffer q along Pi; finally, Ni isthe number of class i elephants.

2.1.4. Extensions to mice

The extension of fluid models to a finite popula-tion of mice was provided in [5] for the I04 model.

The model of TCP mice described in this sectionfurther extends the one presented in [5], by jointlytaking into account the effects of the source maxi-mum window size Wmax, of the TCP fast recoverymechanism, which prevents the TCP transmissionwindow to be halved more than once in eachround-trip time, of the initial slow-start phase upto the first loss, and of timeouts.

Assuming flow lengths to be exponentially dis-tributed, with average L, we can write:

oP nlðw; tÞot

¼� 1

RðtÞoP nlðw; tÞ

ow� 1

RðtÞL

Z w

1

aoP nlða; tÞ

oada

�Z w

1

kða; tÞoP nlða; tÞoa

þ 1

RðtÞP lðw; tÞ; ð10Þ

oP lðw; tÞot

¼Z minð2w;W maxÞ

1

kða; tÞpNTOðaÞoP nlða; tÞ

oada

þZ minð2w;W maxÞ

1

kða; tÞpNTOðaÞoP sða; tÞ

oada

þZ W max

1

kða; tÞpTOðaÞoP nlða; tÞ

oa

þZ W max

1

kða; tÞpTOðaÞoP sða; tÞ

oa� 1

RðtÞP lðw; tÞ;

ð11ÞoP sðw; tÞ

ot¼� w

RðtÞoP sðw; tÞ

ow� 1

RðtÞL

Z w

1

aoP sða; tÞ

oada

�Z w

1

kða; tÞoP sða; tÞoa

daþcðtÞ; ð12Þ

where Pnl(w, t) represents the window size distribu-tion of TCP sources in congestion avoidance modewhich have not experienced losses in the last

round-trip time; Pl(w, t) represents the window sizedistribution of TCP sources in congestion avoidancemode which have experienced losses in the lastround-trip time; Ps(w, t) represents the window sizedistribution of TCP sources in slow-start mode;and Wmax represents the maximum window size.Finally, pTO(w) = 1 � pNTO(w) = min(1,3/w) repre-sents the probability that packet losses induce atimeout of TCP transmitters having window size w,as proposed in [15].

The left hand side of (10) represents the variationof the number of TCP sources in congestion avoid-ance which at time t have window less than w andhave not experienced losses in the last round-triptime. The negative terms on the right hand sideof the same equation indicate that this numberdecreases because of the window growth (first term),of connections’ completion (second term), and oflosses (third term). The fourth positive term indicatesan increase due to the elapsing of a round trip timeafter the last loss.

Similarly, the left hand side of (11) represents thevariation of the number of TCP sources in conges-tion avoidance which at time t have window less thanw and have experienced losses in the last round-triptime. The positive terms on the right hand side indi-cate that this number increases because of losses thateither do not induce or do induce a timeout (respec-tively, first and second, and third and fourth terms)for connections that either were in slow-start mode(second and fourth terms) or had not experiencedlosses in the last round-trip time (first and thirdterms). The fifth negative term indicates a decreasedue to the elapsing of a round trip time after the lastloss.

Finally, the left hand side of (12) represents thevariation of the number of TCP sources which attime t have window less than w and are in slow-startmode. The negative terms on the right hand sideindicate that this number decreases because of thewindow growth (first term), of connections’ comple-tion (second term), and of losses (third term). Thefourth positive term indicates an increase due tothe generation of new connections, that alwaysopen in slow-start mode with window size equalto 1.

We wish to stress the fact that (10)–(12) providequite a powerful tool for an efficient representationof TCP mice, since a wide range of distributions(including long-tail distributions) can be approxi-mated with an arbitrary degree of accuracy by amixture of exponential distributions [16].

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 119

3. On the accuracy of fluid models

The fluid models that we presented so far providea deterministic approximate description of the net-work dynamics, thus departing from the traditionalapproach of attempting a probabilistic descriptionof the network by means of stochastic models, suchas continuous-time or discrete-time Markov chainsand queueing models.

First we observe that fluid models exhibit a niceinvariance property, as proven in [8,17], which pro-vides the theoretical foundation to their scalability.In particular, the whole set of equations describingour I04 model, along with its extension proposedhere, is invariant under the linear transformation ofparameters and variables summarized in Table 1.The top row of Table 1 reports the transformationswhich map the original network parameters intothose of the transformed network, being g 2 Rþ themultiplicative factor applied to the model parame-ters. Basically, the transformed network is obtainedfrom the original network multiplying by g the num-ber of elephants N and the arrival rate of mice c, aswell as all transmission capacities and buffer sizes.Table 1 in the bottom row reports the transforma-tions which relate the modeled behavior of the origi-nal network to that of the transformed network.

This invariance property is extremely interesting,since it suggests that the behavior of very large sys-tems can be predicted by scaling down networkparameters, and studying smaller-scale systems.This result was confirmed by simulation experi-ments reported in [8,17], in case of congested linksloaded by TCP elephants.

However, it is known that in some cases the net-works dynamics do not exhibit such an invarianceproperty [8]. When this is the case, the invarianceproperty of fluid models actually suggests that theyare not suitable to analyze all systems.

In this section, we critically discuss the effective-ness of fluid models to accurately predict the

Table 1Mapping of input parameters and dynamic variables from theoriginal fluid model to the scaled model

Unchanged Multiplied by g

Inputparameters

g, L N, B, C

pmax c(t), thmin, thmax

Dynamicvariables

R(t), k(w, t),p(t), W(t)

Q(t), D(t), A(t),P(w, t), Pmax(t)

pF(t), T(t) Pl(w, t), Pnl(w, t), Ps(w, t)

dynamics of real networks. To this end, in orderto understand where the invariance property for fluidmodels comes from, we have to step back to thetheoretical foundations of fluid models, recalling therelations between the fluid models dynamics andthe dynamics of the underlying stochastic networks.

Fluid models have been proved (at least in simpleconfigurations) to describe the asymptotic behaviorof the underlying stochastic networks, after the so-called fluidification process.

Consider a single-link network where buffers aremanaged according to an AQM (Active QueueManagement) scheme, and are fed by long-livedTCP connections. We build a sequence of modelsby scaling up parameters according to Table 1, insuch a way that a scaling parameter g = n corre-sponds to the nth element of the sequence.

Consider the average TCP window size in the nthelement of the sequence,

wnðtÞ ¼PnN

i¼1W iðtÞnN

and the ‘‘normalized’’ queue size

qnðtÞ ¼QnðtÞ

n:

It was proved in [2,3] that the sequence of trajec-tories (wn(t), qn(t)) admits w.p.1. the solution of theassociated fluid model as unique limit for n!1.

Now, consider a Æ/G/1/B queue, under the mildassumption that the arrival process at the queue sat-isfies the strong law of large numbers (SLLN). Webuild a sequence of models by scaling up the arrivalrate, the service rate and the buffer size. The evolu-tion of each element of the sequence is described byLindley’s equation:

Qnðt þ dÞ ¼ ½QnðtÞ þ Anðt; t þ dÞ � nCd�nB0 ;

where ½x�B0 ¼ minðmaxðx; 0Þ;BÞ and An(t, t + d) rep-resents the amount of bits entering the queue in[t, t + d). Due to the SLLN:

limn!1

Anðt; t þ dÞn

! kd w:p:1:

It is immediate to verify that q(t) = limn!1 qn(t) sat-isfies the following deterministic version of Lindley’sequation:

qðt þ dÞ ¼ ½qðtÞ þ kd� Cd�B0whose solutions are identically null for all values ofq ¼ k

C < 1 and t > B/C, and identically equal to B

for all values of q ¼ kC > 1 and t > B/C.

120 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

As a consequence, any non-overloaded buffer ina packet network subject to an arrival process satis-fying the SLLN is replaced in the fluid model (limit)by an empty queue.

Thus, it is not surprising that, in situations inwhich queue size and packet loss dynamics areessentially driven by random fluctuations of thearrival processes, the predictions of fluid modelscannot be accurate.

In scenarios comprising essentially TCP ele-phants, the TCP closed-loop control mechanism isable to drive the system toward a permanent near-congestion state, in which bottleneck links are fullyutilized. In these conditions, short-term randomfluctuations of the arrival processes play a minorrole on the queueing dynamics, and deterministicfluid models are quite accurate in predicting the net-work dynamics [10,5].

However, scenarios dominated by long-lived(infinite) TCP connections (elephants) are not fullyrepresentative of what happens over the Internet,where most of the TCP flows are very short (mice).In more realistic scenarios, where a significant com-ponent of the traffic is associated with TCP mice,and the average link utilization is sufficiently farfrom saturation, the TCP closed-loop congestionmechanism is not able to drive the system towarda permanent near-congestion state. In this case, onlysporadic congestion arises at queues, due to stochas-tic fluctuations of the packet arrival process. Thesephenomena are completely neglected in fluid mod-els, which, as explained above, predict that everynetwork queue is permanently empty if q < 1.

For these reasons, we believe that any modelused to assess the performance of TCP/IP networksin scenarios in which the link average utilization issufficiently far from saturation cannot ignore ran-dom fluctuations, intrinsic in packet network traffic.The fluidification process cannot thus be taken to itsextreme, rather, we have to stop short of a purelydeterministic model in order to define an efficienttool, whose predictions have not just an asymptoti-cal significance, but can accurately describe thebehavior of real (finite) networks.

Randomness impacts the system behavior essen-tially at two levels.

• Randomness at flow level is due to the stochasticnature of the arrival and completion processesof TCP mice. The number of active TCP micein the network varies over time, and the offeredload at network queues changes accordingly.

The slowly varying modulation effect on traffic(i.e., on packet arrival processes at queues) pro-vided by flow dynamics may have a great impact,since it is an important source of LRD effects innetwork traffic.

• Randomness at packet level is due to the burstynature of the arrival and departure processes ofpackets at buffers on time scales shorter thanthe round trip time. These effects are completelyignored by pure fluid models, however theymay have significant impact on the network per-formance. In particular, the ‘‘short-term’’ bursti-ness of TCP traffic has been recently proved to beresponsible for high queueing delays and spo-radic buffer overloads, even when the averagelink utilization is much smaller than 1 [19–21].

In the next section we describe three differentapproaches to take into account the effects of ran-dom fluctuations on the network performance. Thefirst approach is the one that we presented in [5],and that we refer here as I04. The two otherapproaches are novel.

4. Models of randomness

The three approaches that we propose to accountfor random traffic fluctuations are named:

1. Flow-Montecarlo packet-Montecarlo (FMPM).In this case, the deterministic differential equa-tions of the fluid model are transformed into sto-chastic differential equations, which are thensolved using a Montecarlo method. This approachwas originally presented in [5].

2. Flow-Montecarlo packet-Brownian (FMPB). Inthis case, the randomness at the flow level is stillmodelled using a Montecarlo technique, whilethe randomness at the packet level is modelledby analytically describing the queues distributionevolution through a set of deterministic func-tional equations, adopting a second-order Gauss-ian approximation of the queue dynamics, i.e., adiffusion approximation. We also present a sim-plified version of this model, where the queuedynamics are decoupled from the source dynam-ics, taking advantage of the fact that the formerare much faster than the latter.

3. Flow-Brownian packet-Brownian (FBPB). Inthis case, a diffusion approximation is adoptedto analytically model the randomness at boththe flow and the packet level.

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 121

4.1. Flow-Montecarlo packet-Montecarlo (FMPM)

approach

A first approach to account for randomness con-sists in transforming the deterministic differentialequations of the fluid model into stochastic differen-tial equations, which are then solved using a Monte-carlo method.

The reader may have the impression that thisapproach contradicts the main idea on which fluidmodels are based. Indeed, fluid models are, in princi-ple, obtained by describing the evolution of every sin-gle agent in the system (i.e., TCP sources and queues)in terms of stochastic differential equations and thenasymptotically scaling up the system, so as to obtaina set of deterministic differential equations whichrepresent the asymptotical queue and source behav-ior. Transforming again the obtained deterministicfluid equations into stochastic differential equationsat a first look may seem coming back to the originalpoint. However, this is not true, since fluidificationhas permitted averaging over TCP sources individualbehavior, obtaining a single deterministic equationwhich describes, statistically, the evolution of anentire population of TCP sources. Making stochasticthe ‘‘average’’ TCP sources fluid equations, we pre-serve one the most important property of fluid mod-els, which is the fact that the model structure(number of equations) is insensitive with respect tothe number of active TCP flows. Essentially, we eval-uate the impact of randomness on the aggregatedynamics of the TCP population, rather than oneach single source.

More in detail:

• Randomness at flow level. The deterministic micearrival rate c(t) in (11) can be replaced by a Pois-son counter with average c(t). The deterministicmice completion process can be replaced by aninhomogeneous Poisson process whose averageat time t is represented by the sum of the secondterms on the right hand side of (10) and (12). Asopposed to the mice arrival process, which isassumed to be exogenous, the mice completionprocess depends on the network congestion: whenthe packet loss probability increases, the rate atwhich mice leave the system decreases.

• Randomness at packet level. The workload emit-ted by TCP sources, rather than being a continu-ous deterministic fluid process with rate Wi(t)Ni/Ri(t), can be taken to be a Poisson process withbulk arrivals, in which the batch size distribution

is matched to the actual window size distributionof the TCP sources. This approach was originallyproposed in [21,22].

An important observation is that the addition ofrandomness in the model destroys the invarianceproperties described in Section 4. This can beexplained very simply by considering that the lossprobability predicted by any analytical model of afinite queue with random arrivals (e.g., anM/M/1/B queue) depends on the buffer size, usuallyin a non-linear way. Instead, according to the trans-formations of Table 1, the loss probability shouldremain the same after scaling the buffer size. Inunderload conditions, it is clearly wrong to assumethat the packet loss probability does not changewith variations of the buffer size.

4.2. Flow-Montecarlo packet-Brownian (FMPB)

approach

The second approach to account for randomnesstreats differently the randomness at the flow leveland the randomness at the packet level. While atthe flow level randomness is still modelled using aMontecarlo technique, at the packet level random-ness can be modelled by analytically describing thequeues distribution evolution through a set of deter-ministic functional equations. For this purpose, weadopt a second-order Gaussian approximation ofthe queue dynamics, i.e., a diffusion approximation.

Diffusion models were widely applied in the mid’70s to obtain product-form approximations of thesteady-state behavior of open non-Markovian FIFOqueueing networks [23]. The evolution of each queuek is represented by independent Reflected BrownianMotion processes, with assigned average infinitesi-mal drift mk and variance r2

k . The steady-state queuedistribution qk(x) is governed by a stationary Fok-ker–Planck equation:

0 ¼ �mkdqkðxÞ

dxþ 1

2r2

k

d2qkðxÞdx2

under the conditionsZ 1

0

qkðxÞdx ¼ 1� q; mkqkð0Þ ¼1

2r2

k

dqkðxÞdx

����0

in which the parameters mk and r2k depend, respec-

tively, on the first and second moment of the inter-arrival and service times at the queue, according tothe following rules for single server systems:

122 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

mk ¼ E½Ak� � lk; r2k ¼ E½Ak�a2

k þ lks2k :

E[Ak] and lk are, respectively, the average arrivalrate and service rate at queue k, and a2

k and s2k the

squares of the variation coefficients of inter-arrivaland service times. We notice that coefficient a2

k ,which depends on the second moment of the depar-ture processes at the queues preceding queue k

along the customers’ paths, can be obtained foropen networks with a single class of customers bysolving a system of linear equations [24].

More recently, multidimensional (non-productform) diffusion approximations of queueing net-works were proved to generate an asymptoticallyexact description of the queue distribution evolutionfor several families of open multiclass queueing net-works in the heavy traffic regime.

The numerical solutions of multidimensional dif-fusion models become computationally intractablewhen the number of queues in the network exceedsvery few units. For these reasons, we propose someextensions of the classical product-form diffusionmodels for the representation of queueing dynam-ics, which allow multiclass queueing systems subjectto non-stationary arrival processes to be considered.

Let qk(x, t), with 0 < x < B, represent the proba-bility density function associated with the length ofqueue k at time t, and B the buffer size. let PðkÞ0 ðtÞrepresent the probability for the queue to be empty,and PðkÞB ðtÞ the probability for the queue to be full.According to our diffusion approximation, thedynamics of q(x, t) are driven by the following Fok-ker–Planck equation with two absorbing barriers at0 and B:

oqkðx; tÞot

¼�mkðtÞoqkðx; tÞ

oxþ1

2r2

kðtÞo2qkðx; tÞ

ox2

þAkðtÞPðkÞ0 ðtÞdðx�1ÞþlkPðkÞB ðtÞdðx�Bþ1Þ;

ð13ÞdPðkÞ0 ðtÞ

dt¼�mkðtÞqkð0; tÞþ

1

2r2

kðtÞoqkðx; tÞ

ox

����0

�AkðtÞPðkÞ0 ðtÞ;

ð14ÞdPðkÞB ðtÞ

dt¼þmkðtÞqkðB; tÞ�

1

2r2

kðtÞoqkðx; tÞ

ox

����B

�lkPðkÞB ðtÞ

ð15Þ

with parameters for the drop-tail case1: lk = Ck/E[Sk],mk(t) = Ak(t) � lk and r2

k ¼ AkðtÞa2kðtÞ þ lks2

k , being

1 If an AQM packet discard policy is employed at buffer k,mk(t) = Ak(t)[1 � pk(t)] � lk.

E[Sk] the average service time, and a2kðtÞ and s2

k thesquares of the variation coefficients of the inter-arrival and service times at queue k, respectively.In addition, to fully characterize the queueingdynamics, we assume an initial condition qk(x,0)and fix as boundary constraints qk(B, t) = 0 andqk(0,t) = 0.

The diffusion approximation provides a descrip-tion of the time-variant queue dynamics in termsof mean and variance of the packet arrival and ser-vice processes. As a consequence, in order to cor-rectly represent the queue dynamics, it is necessaryto estimate the burstiness of the packet arrival pro-cess at queue k, measured by a2

kðtÞ, which is a func-tion of the burstiness of the departure processes atqueues preceding queue k along the customers’paths. A deterministic path is associated with eachclass of TCP flows. As a consequence, the applica-tion of classical methodologies defined for single-class networks of queues, like the one proposed in[25], may lead to significant errors in the estimationof the coefficient of variation of inter-arrivals atqueues. We have therefore extended the classicalmethodologies to obtain the coefficients of variationof inter-arrivals at queues for multiclass queuingnetworks in which each class of customers followsa deterministic routing.

The most important step of our procedure con-sists in relating the burstiness of the packet outputprocess from a queue to the burstiness of the packetarrival process at the next queue, for each TCP flow.First, we suppose the flow departure process fromeach queue to be a renewal process; then, weapproximate a generic packet arrival process witha batch Poisson process where geometrically distrib-uted batches are used to fit the first two moments ofthe inter-arrival times. This approach is justified bythe fact that, under the diffusion approximation,two renewal processes are indistinguishable if theirfirst and second moments are equal.

Let 1/b represent the mean batch size; the prob-ability for a batch to have size x is P(batch = x) =b(1 � b)x�1. It is easy to relate b to the desired burs-tiness a2

k for the arrival process:

b ¼ 2

a2k þ 1

: ð16Þ

We can now relate the parameters of the departureprocess to the arrival process parameters by observ-ing that the packet inter-departure time for each flowis supposed to be a generalized hyper-exponential.Indeed, the inter-departure time between the nth

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 123

packet and the n + 1st packet departure equals theservice time of the nth packet, if packets n andn + 1 belong to the same batch, i.e., with probability1 � b. Note that we assume that packets belonging tothe same batch cannot be interleaved in the queuewith packets of other batches, since a batch arrivalis instantaneous. More complex is, instead, the eval-uation of the distribution of the inter-departure timewhen the nth packet and the n + 1st packet belong todifferent batches. In this case, their departures can beinterleaved either by packets belonging to otherflows, or by periods in which the queue remainsempty. In this case, we assume the inter-departuretime distribution to be exponential, and we fix itsaverage so as to obtain the desired average flow load.

Once we have obtained the first two moments forflows departing from queues, (9) can be used tocompute the average packet arrival rate at queues.The equation can be easily generalized to computethe burstiness of the arrival processes at queues,by assuming independent flows coming from differ-ent queues and merging at the considered queue.

4.2.1. SFMPB: simplified FMPB approach

In high-speed networks, the queue dynamics areoften much faster than the source dynamics, whichevolve over time scales comparable with the roundtrip time. This is true in particular when link capac-ities and buffer sizes are selected so as to make themaximum queueing delay negligible with respectto the propagation delay. When this is the case,the interaction between sources and queues can berather accurately represented assuming a quasi-sta-tionary queueing behavior. This implies that thequeue distribution can be expressed analytically interms of the instantaneous arrival process parame-ters at the queue, by solving the stationary versionof (13). The stationary queue distribution can beeasily computed; it results:

PðkÞ0 ðtÞ¼1

1þqe�cðtÞðB�1Þ þ qðtÞ1�qðtÞ ½1� e�cðtÞðB�1Þ�

;

qkðxÞ¼

qðtÞ1�qðtÞP

ðkÞ0 ðtÞð1� e�cðtÞxÞ for x6 1;

qðtÞ1�qðtÞP

ðkÞ0 ðtÞðecðtÞ �1Þe�cðtÞx for 16 x6B�1;

qðtÞ1�qðtÞP

ðkÞ0 ½e�cðtÞðx�1Þ � e�cðtÞðB�1Þ� for x>B�1;

8>>><>>>:

PðkÞB ðtÞ¼PðkÞ0 ðtÞqðtÞe�cðtÞðB�1Þ

being c ¼ �2mkðtÞr2

k ðtÞand qðtÞ ¼ AkðtÞ

lkðtÞ.

This entails a much simpler way of accountingfor the packet-level randomness. The flow-level

randomness can be accounted for through a Monte-carlo method, thus obtaining quite an efficient solu-tion procedure.

4.3. Flow-Brownian packet-Brownian (FBPB)

approach

A fully analytical model of the system dynamicscan be obtained by incorporating in the equationsthe description of the stochastic evolution of thenumber of active flows in the network.

As pointed out in [26], at least in simple configu-rations, the evolution of the number of flows in apacket network can be rather accurately describedby different queuing models in different workingregimes: an M/G/1 queueing model is accuratewhen the packet network is lightly loaded and littleor no congestion arises; an M/G/1-PS model is moresuited to the estimation of the number of flows whenthe network is heavily congested. In the moderatecongestion regime, modelling the system is morecomplex, since short congestion phases alternate tonon-congestion phases. This, however, is the mostinteresting regime, in which the packet-level andflow-level dynamics closely interact, requiring acarefully developed model to accurately predictperformance.

In order to cope with different queuing behavior,we use a time-inhomogeneous Processor-Sharingqueue to represent the evolution of the number ofactive flows for each class i of TCP connections.Each job arriving at the queue represents a newTCP connection of class i. Upon arrival, each jobplaces a demand to the server for an amount of workwhich represents the transmission resources neededalong the route through the network to accomplishthe transfer of the data carried by the TCP connec-tion (proportional to the flow length); finally, thequeue service capacity at time t equals the instanta-neous aggregate throughput obtained by TCP con-nections belonging to class i.

Notice that we expect the instantaneous through-put of TCP connections belonging to class i to beroughly proportional to the number of active flowswhen the system is lightly loaded, since the through-put of each individual flow is only limited by theintrinsic TCP flow-control dynamics (such as themaximum window) and it is not significantly influ-enced by the presence of other flows; in this caseour time-inhomogeneous Processor-sharing queuereproduces the dynamics of an M/G/1. When thesystem is severely congested, instead, the bandwidth

124 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

obtained by class i TCP flows is limited only by theavailable capacity along the network path. In thiscase the aggregate throughput of class i connectionsdoes not change with a variation in the number ofactive competing flows. In this case our modelbehaves similarly to a M/G/1-PS queue.

Unfortunately, the exact transient analysis ofsuch a queueing system appears to be prohibitive,so that we are forced to approximately representthe inhomogeneous Processor Sharing behaviorusing a second-order Brownian motion approxima-tion of the queueing system evolution.

Let n(y, t) be the probability density function ofthe number of active sources, and let s(w, tjy) andS(w, tjy), respectively, be the conditional pdf andPDF of the sources with window w, given that y

flows are active.The evolution of n(y, t) is given by the following

Fokker–Plank equation:

onðy; tÞot

¼ � o

oy½ðcðtÞ � fðtjyÞÞnðy; tÞ�

þ 1

2

o2

oy2½ðcðtÞ þ fðtjyÞÞnðy; tÞ�;

where f(tjy) represents the instantaneous endingrate of active flows, which is directly proportionalto the instantaneous throughput achieved by classi connections

fðtjyÞ ¼Z 1

0

wð1� pF ðtjyÞÞ=LRðtjyÞ sðw; tjyÞdw:

Note that we have approximated the TCP flow com-pletion process with a time-variant Poisson Processwith equal rate.

Let us now consider the evolution of S(w, tjy),which is coupled to the evolution of n(y, t) throughthe following equation:

oSðw; tjyÞnðy; tÞot

¼ 1

RðtjyÞ sðw; tjyÞ þZ 2w

wkða; tjyÞsða; tjyÞda

� �nðy; tÞ

þ cðtÞ �Z w

0

að1� pF ðtjyÞÞ=LRðtjyÞ sða; tjyÞda

� �nðy; tÞ

� o

oy½ðcðtÞ � fðtjyÞÞSðw; tjyÞnðy; tÞ�

in which the first term in the right-hand side repre-sents the effects of the TCP window evolutionmechanism (note that this term is directly relatedto the right-hand side of Eq. (2)); the second term

represents the direct effect on the conditional win-dow distribution of the flows which start and end;the last term, finally, represents the indirect effecton the conditional window size distribution of a var-iation on the number of active TCP flows.

The evolution of the queue length distribution,which is coupled to the dynamics of the numberof active flows, is represented by the law that drivesthe evolution of the conditional queue length distri-bution qk(x, tjy), given the number y of active flows:

oqkðx; tjyÞnðy; tÞot

¼þ �mkðtjyÞoqkðx; tjyÞ

oxþ 1

2r2

kðtjyÞo

2qkðx; tjyÞox2

� �nðy; tÞ

þ o

oy½ðfðtjyÞ � cðtÞÞqkðx; tjyÞnðy; tÞ�

in which the first term on the right-hand side repre-sents, similarly to the right-hand side of (13), the ef-fects on the conditional queue distribution relatedto queue dynamics; the second term expresses theindirect effect on the conditional queue distributionof a variation on the number of active TCP flows.

4.4. Some comments on the three approaches

The three approaches mainly differ in the method-ology used to describe the randomness present inthe system dynamics. The first approach relies on abrute-force Montecarlo technique, thus studyingthe system dynamics through a sample-path analy-sis. The second approach attempts a probabilisticsecond-order approximate description of the ran-dom queue dynamics over their state space, througha partial differential equation which governs thequeue distribution evolution. Finally, the thirdapproach permits a fully probabilistic second-orderapproximation description of the random systemdynamics through differential equations which rep-resent the evolution of all the involved distributions.

First of all, it is worth emphasizing that the threeapproaches rely on quite similar assumptions. As aconsequence, we may expect their results to be veryclose. Of course, the need of introducing second-order approximations may reduce the accuracy ofthe last two approaches with respect to the firstone. However, the progressive disappearance of sto-chastic elements in the equations defining the secondand third approaches can have several beneficialnumerical effects, making their results numericallymore stable. In addition, the approaches 2 and 3

0.0001

0.001

0.01

0.1

50 100 150 200 250

pdf

Queue distribution

FMPBFMPM

ns-2FDPB

FMPB-Poisson

Fig. 1. Queue size distributions obtained with the ns simulator,the FMPM approach, and the FMPB approach.

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 125

are potentially more tractable from an analyticalviewpoint, thus permitting a theoretical investigationof the properties of their solutions, and hopefullyleaving open the possibility of deriving analyticalsolutions at least in simple setups.

From the computational complexity viewpoint,the three approaches entail an increasing degree ofcomplexity. While the first approach requires onlya marginal adaptation of the solver developed fordeterministic fluid models, the second approachrequires the efficient solution for each queue of theassociated Fokker–Planck equation. However, inthe cases in which the quasi-static approximationfor the queue dynamics can be successful applied,the second method typically becomes more effi-cient than the first. Finally, the third approachrequires the efficient solution of 3-dimensionalpartial differential equations. Unfortunately, sophis-ticated numerical techniques are required to effi-ciently solve large-dimensional systems of partialdifferential equations, and our approach, based ona simple finite-difference method (described in [17]),is not powerful enough to efficiently solve the fully-analytical model. For this reason, in this paper wejust report numerical results for the first two models,leaving for future work the development of anefficient finite-elements tool for the solution of thefully analytical model.

As we already noted, the introduction of random-ness in fluid models breaks the invariance property ofdeterministic fluid models which constitutes the maintheoretical support to the insensitivity of the numer-ical complexity of deterministic fluid models withrespect to link and buffer capacities. As a conse-quence, when considering randomness, the insensi-tivity of the numerical solution complexity is lost.Indeed, since the speed at which queue processesevolve is related to the capacity of the associatedqueue, the discretization step in time associated withthe equation driving the queue dynamics cannotbe chosen independently from the queue capacity(this consideration does not apply to the SFMPBapproach, in the case in which the quasi-staticapproximation can be successfully applied).

5. Numerical results

In this Section we present numerical results refer-ring to several different network scenarios. First, inSection 5.1 we compare the proposed modellingapproaches under stationary traffic conditions, dis-cussing the impact of randomness at the flow and

packet levels. Then we move on to non-stationarytraffic scenarios, to demonstrate one of the mainstrengths of our methodology, which is the abilityto predict the behavior of large but finite popula-tions of TCP mice under transient network condi-tions. In Section 5.2 we study a dynamic scenarioin which a link is temporarily overloaded by a flashcrowd of new TCP connection requests. In Section5.3 we examine the case of a link failure in the net-work backbone, and the effect of the rerouting of alarge number of TCP flows. We wish to stress thefact that we were forced to select simple network set-ups in which ns simulations are feasible, in order tobe able to compare the results of our models againstthose obtained by detailed packet-level simulations.

5.1. Stationary traffic scenarios

5.1.1. A first simple scenario

We consider one bottleneck link fed by a drop-tail buffer. The buffer size is equal to 256 packets;the packet size, here and in the rest of the paper,is assumed constant, equal to 1250 bytes; the linkcapacity is C = 100 Mb/s; the propagation delaybetween TCP sources and buffer is 30 ms. Two typesof TCP mice coexist, short mice and long mice. Foreach of the two classes, the mice length is geometri-cally distributed, but the mean is 20 packets forshort mice, and 2000 packets for long mice. Bothtypes of mice are opened according to a stationaryPoisson Process, with rates 200 and 2 mice/s, respec-tively. The resulting link load is q = 0.8.

Fig. 1 presents a comparison of the queue lengthdistributions obtained with the ns simulator and the

Table 2Average queue lengths (AQL), average loss probabilities (ALP) and average completion times (ACL) of short and long flows at loadq = 0.8, for different values of link capacity

C (Mb/s) FMPB ns

AQL ALP ACT (20) ACT (2000) AQL ALP ACT (20) ACT (2000)

10 59.15 3.0 · 10�3 0.71 9.36 68 2.0 · 10�3 0.72 8.9100 41.31 8.6 · 10�4 0.32 3.18 34 7.7 · 10�4 0.32 2.8

1000 28.69 1.2 · 10�4 0.29 2.10 25 7.5 · 10�5 0.30 2.3

Table 3Parameters of the hyper-exponential distribution used to approx-imate the Pareto distribution

Prob. Mean length

7.88 · 10�1 6.481.65 · 10�1 23.263.70 · 10�2 80.658.34 · 10�3 279.71.87 · 10�3 970.24.22 · 10�4 33769.46 · 10�5 118622.10 · 10�5 430864.52 · 10�6 176198

126 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

model, in this latter case using both the FMPM andthe FMPB approaches. A good match can beobserved between the results obtained with the simu-lator and both models in this simple scenario. Thefigure also shows the queue length distributionobtained with the packet Brownian approach whenrandomness at the flow level is neglected (FDPB,flow deterministic packet Brownian): the compari-son against the curves which consider randomnessat both packet and flow level indicates that flow-levelrandomness plays a significant role on the tail of thequeue size distribution, and helps quantify theimpact of the flow-level dynamics on the overall sys-tem performance. Finally, in order to assess theimportance of the packet level (short term) trafficburstiness, we report also the he curve (labelledFMPB-Poisson) where the arrival process at thequeue is an inhomogeneous Poisson process (i.e.,the impact of batch arrivals is neglected). The differ-ence in behavior is striking, and demonstrates theimportance of incorporating in the model an accu-rate description of the ‘‘short term’’ traffic burstinessinduced by the TCP protocol.

We next consider the same scenario with trafficload q = 0.8, for three different values of link capac-ity: 10 Mb/s, 100 Mb/s, 1 Gb/s. Table 2 reports theaverage queue lengths (AQL), the average loss prob-abilities (ALP), and the average completion times(ACT) of short and long mice, for the three valuesof link capacity, as predicted by both the FMPBmodel and ns simulations.2 The results obtainedwith ns can be observed to match reasonably wellthe predictions of the FMPB approach.

5.1.2. Realistic flow size distribution

In order to generate a TCP traffic load similar towhat is observed on the Internet, we consider flowsizes distributed according to a Pareto distribution

2 ns Simulations were run for a sufficiently long time to meet a10% confidence interval maximum width with confidence level0.95.

with shape parameter equal to 1.2 and scale param-eter equal to 4.

Using the algorithm proposed in [16], we approx-imated the Pareto distribution with a hyper-exponen-tial distribution of the 9th order, whose parametersare reported in Table 3. The resulting average flowlength is 20.32 packets. Correspondingly, 9 classesof TCP ‘‘mice’’ are considered in our model. Themaximum window size is set to 64 packets for allTCP sources.

First, we consider a 1 Gb/s single bottleneck linkfed by a drop-tail buffer, with capacity equal to 256packets.

The top two rows of Table 4 report the resultsobtained with the SFMPB model and ns forq = 0.9. A good agreement between model predic-tions and simulation results can be observed; mostof the model predictions fall within the confidenceinterval of the ns results. We notice, however, thatresults obtained with ns are affected by a large uncer-tainty, as shown by the large width of confidenceintervals reported in the table. This is mainly dueto the fact that we were unable to run simulationslonger then 5000 s due to intrinsic limitations ofthe simulator (the maximum number of events thatcan be scheduled is bounded).

Table 4 also reports model predictions for q =0.95 and q = 0.97, both considering and ignoringthe flow-level randomness (respectively, labelled with

Table 4Average loss probability (ALP), average queue length (AQL) andaverage completion times (ACT) in seconds of the nine classes ofmice; the drop tail case

ALP AQL ACT [s]

SFMPB,q = 0.9

2.22 · 10�3 49.92 0.137,0.189,0.2610.418,1.06 3.2710.6,41.1,119

ns,q = 0.9

[1.65–2.64] · 10�3 [51.32–61.54] [0.125,0.133][0.185,0.194][0.240,0.263][0.371,0.401][0.780,0.863][2.30,2.69][7.38,9.67][25.3,36.9][70.3,157]

SFMPB,q = 0.95

4.94 · 10�3 71.29 0.138,0.192,0.2730.497,1.30 4.1513.2,51.4,156

SFDPB,q = 0.95

3.79 · 10�3 69.04 0.137,0.191,0.2580.415,1.00,3.2111.1,40.1,140

SFMPB,q = 0.97

1.5 · 10�2 145.01 0.145,0.214,0.3530.871,2.71,9.0732.8,111.6,476

SFDPB,q = 0.97

6.52 · 10�3 103.52 0.14,0.208,0.2980.608,1.70,5.5319.1,68.6,276

0

100

200

300

400

500

600

700

800

0 2000 4000 6000 8000 10000 12000 14000

Num

ber

of f

low

s

Time (s)

SFDPB All flowsSFMPB All flowsSFDPB Long miceSFMPB Long mice

Fig. 2. A comparison between active flow number evolution forSFMPB and SFDPB at q = 0.97.

Table 5Average loss probability (ALP), average queue length (AQL) andaverage completion times (ACT) in seconds of the nine classes ofmice; the AQM case

ALP AQL ACT [s]

SFMPB,q = 0.9

4.15 · 10�3 35.23 0.137,0.195,0.3010.639,1.61,5.7920.3,78.2,310

ns, q = 0.9 [3.53,4.39] · 10�3 [33.7,37.6] [0.135,0.145][0.198,0.21][0.277,0.295][0.494,0.545][1.37,1.58][4.79,5.69][15.8,20.1][44.561,71.77][156.28,340.87]

SFMPB,q = 0.95

9.93 · 10�3 60.81 0.169,0.232,0.3590.69,2.28,7.7226.4,97.1,409

ns, q = 0.95 [8.64,11.0] · 10�3 [55.93,65.51] [0.181,0.202][0.256,0.289][0.369,0.425][0.75,0.91][2.33,2.85][8.31,10.3][28.1,39.1][80,124][211,434]

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 127

SFMPB abd SFDPB). Unfortunately, we wereunable to obtain ns results with a reasonable degreeof accuracy for such high loads. Comparing theSFMPB and SFDPB results, it can be observed thatflow-level fluctuations have significant effects on thesystem dynamics, especially when the system loadapproaches 1, even when the link capacity is 1 Gb/s.A better view of the effects that flow-level random-ness has on the system dynamics is provided byFig. 2, which shows the evolution of the total numberof active flows, along with the number of long mice(i.e., flows belonging to the last three branches ofthe hyper-exponential distribution) in the system,according to the SFDPB and SFMPB models atq = 0.97. The SFMPB curve shows a very slow mod-ulation of the total number of active flows induced bylong mice dynamics which play a fundamental role inthe long-range dependency observed in real Internettraffic [27]. Note the strong correlation between thedynamics of the total number of mice and of longmice only.

Table 5 presents results for the case in which thebottleneck link is fed by a buffer implementing a

gentle-RED policy, with thmin = 10, thmax = 160,pmax = 0.05, w = 0.0001. The SFMPB model predic-tions and the ns results are compared for q = 0.9and 0.95. Also in this case we observe a good agree-ment between SFMPB predictions and ns results.

Time (s)

Loa

d of

cla

ss 2

65

Mb/s

145806550 95

Fig. 4. Temporal evolution of the load produced by the flashcrowd.

0

50

100

150

200

250

300

350

0 50 100 150 200 250

Que

ue le

ngth

Time (s)

Queue 1Queue 2

Fig. 5. Queue size evolution obtained with the FMPM approach.

128 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

By comparing the model completion time resultsfor the drop-tail case at q = 0.95, against the resultsfor RED at q = 0.9 (the average loss probability isvery similar in the two cases), it is possible to seethe effects of the burstiness in the loss process onthe TCP sources dynamics. Even if the average lossprobability is almost the same, the loss processinduced by the drop-tail buffer management schemeis more bursty than the loss process under the REDpolicy. As already observed in the literature (see forexample [28]), TCP sources experience higherthroughput (thus smaller completion time) if theburstiness in the loss process increases, under thesame average loss probability. These effects are cor-rectly represented in the model.

Finally, we want to emphasize that, in this sce-nario, reproducing 5000 s of operational behaviorof the system with the SFMPB model required about30 min of CPU time; it required about 120 h with thens simulator running on the same machine. Evenafter such long simulations, the confidence intervalsobtained are quite large, mainly because of the hugevariance of the considered flow length distribution.The SFMPB model is able to produce accurateresults also in the case of LRD traffic.

5.2. Flash crowd scenario

We now study a non-stationary traffic scenario,in which a network section is temporarily over-loaded by a flash crowd of new TCP connectionrequests.

Consider a series of two links at 100 Mb/s, asshown in Fig. 3. The first link is fed by a drop-tailbuffer, while the second link is fed by a RED buffer.Both buffers can store up to 320 packets. As shownin Fig. 3, three classes of TCP mice traverse this net-work section. Mice of class 1 traverse only the firstlink, mice of class 2 traverse only the second link,and mice of class 3 traverse both links. The lengthof mice in all three classes is geometrically distrib-uted with mean 50 packets. Mice of classes 1 and 3

Q1

class 1 class 2

class 32Q

Fig. 3. Network topology of the flash crowd scenario.

offer a stationary load equal to 45 and 50 Mb/s,respectively. Mice of class 2 offer a time-varying loadthat follows the profile shown in Fig. 4, and deter-mine a temporary overload of the second queuebetween t1 = 50 s and t2 = 95 s.

Figs. 5 and 7 report, respectively, the queuelengths and the source window sizes versus time,averaging values obtained during intervals of dura-tion 1 s for a better representation. Results are com-puted with the FMPM approach. For the sake ofcomparison, Figs. 6 and 8 report the same perfor-mance indices obtained with ns. First of all, it is nec-essary to emphasize the fact that the reported curvesrefer to sample paths of the stochastic processes cor-responding to the network dynamics, not to aver-ages or moments of higher order; thus, the curvesare the result of particular values taken by the ran-dom variables at play, and the comparison betweenthe ns and the model results requires a great amountof care. In spite of this fact, we can observe a fairlygood agreement between the ns results and the

0

50

100

150

200

250

300

350

0 50 100 150 200 250

Que

ue le

ngth

Time (s)

Queue 1Queue 2

Fig. 6. Queue size evolution obtained with ns.

0

5

10

15

20

0 50 100 150 200 250

Ave

rage

Win

dow

Siz

e

Time (s)

Class 1Class 2Class 3

Fig. 7. Window size evolution obtained with the FMPMapproach.

0

5

10

15

20

0 50 100 150 200 250

Ave

rage

Win

dow

Siz

e

Time (s)

Class 1Class 2Class 3

Fig. 8. Window size evolution obtained with ns.

0

50

100

150

200

250

300

350

0 50 100 150 200 250

Que

ue le

ngth

Time (s)

Queue 1Queue 2

Fig. 9. Queue size evolution obtained with the FDPD model.

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 129

model predictions. For example, in both cases it ispossible to see that congestion arises at the secondqueue as soon as the offered load approaches 1,

i.e., around t = 50 s. After a while, congestion prop-agates back to the first queue, due to retransmis-sions of class 3 mice, which cross both queues. Attime t = 100 s, when the offered load at the secondbuffer decreases below 1, congestion rapidly endsat the second queue, but persists until t = 160 s atthe first queue, due to the large number of activemice of class 3.

For the sake of comparison, we also show inFig. 9 the predictions generated by the fully deter-ministic fluid model (flow deterministic packetdeterministic, FDPD). The fact that in this casethe network becomes overloaded allows the FDPDmodel to correctly predict congestion. However,the FDPD model predicts a behavior at the firstqueue after the end of the overload period whichis different from ns observations. Indeed, the FDPDmodel predicts a persistent congestion for the firstqueue, which gets trapped into a ‘‘spurious’’ equilib-rium point. This phenomenon is not completely sur-prising, and confirms the results presented in [13],where it was recently proved that asymptotic meanfield models of networks loaded by a large numberof short-lived TCP flows may exhibit permanentcongestion behaviors also when the load is less than1. The existence of two equilibrium points at loadsclose to 1, one stable and the other unstable, wasalso described in [29]. This pseudo-chaotic dynamicsof the FDPD model reflects a behavior that can bereally observed in ns simulations. See for examplethe ns sample path reported in Fig. 10, for the samescenario. We observe that in this case congestion atthe first queue persists for a very long time, untilrandom fluctuations bring down the system to thestable operating point.

0

50

100

150

200

250

300

350

0 50 100 150 200 250

Que

ue le

ngth

Time (s)

Queue 1Queue 2

Fig. 10. ns Sample path showing persistent congestion at queue1.

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Que

ue le

ngth

Time [s]

Queue at R2Queue at R3

Queue at R2-FDQueue at R3-FD

Fig. 12. Average queue size evolution predicted by the FMPBmodel.

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Que

ue le

ngth

Time (s)

Queue at R2Queue at R3

Fig. 13. Queue size evolution predicted by ns.

130 G. Carofiglio et al. / Computer Networks 51 (2007) 114–133

5.3. Link failure scenario

As a last example, we consider the network sce-nario depicted in Fig. 11, in which every link hascapacity 100 Mb/s, and is fed by a drop-tail bufferthat can store up to 256 packets. As shown in thefigure, two classes of mice traverse this networksection. The mice length for both classes is geometri-cally distributed with mean 50 packets. Both classesof mice offer a load equal to 0.4. The first class ofmice is routed through routers R2, R3 and R4, insequence. The second class of mice is initially routedthrough R1, R3 and R4. At time t = 15 s, a failureaffects the functionality of the link R1! R3; as aconsequence, mice of class 2 are re-routed throughR1, R2, R3 and R4.

In Fig. 12 we show the time evolution of the queuelength of the buffers at routers R2 and R3, as pre-dicted by the FMPB model, both considering andneglecting the flow level randomness. Fig. 13 showsthe same curves, as observed with ns simulations.We remark that the comparison between the predic-tions of the FMPB approach and ns requires great

R1

R2 R3 R4

class 2

class 1

Fig. 11. Network topology of the link failure scenario.

care, since the ns curves depict a sample path ofthe queue evolution, while the FMPB model curvesrefer to the average queue length dynamics, undera flow-level sample path (i.e., results are averageswith respect to the packet level, and sample pathswith respect to the flow level). With this caveat, bycomparing the curves we can identify quite similarbehaviors. The largest difference between the twopredictions can be observed in the behavior of routerR2 after rerouting. Hardly any queue is observed atrouter R3 after t = 15 s using ns, due to the shapingeffect of link R2! R3, which perfectly sequencespackets arriving at router R3. The FMPB approachis not able to completely capture this phenomenon,since it does not model in detail the dependenciesbetween different network elements. It is worthnoticing, however, that the shaping effect is not

1e-05

0.0001

0.001

0.01

0.1

1

0 50 100 150 200 250

ddp

Que

ue in

R3

Time [s]

t=15t=15.50

t=16t=18

Fig. 14. Evolution of the distribution of the queue at router R2.

1e-05

0.0001

0.001

0.01

0.1

1

0 50 100 150 200 250

ddp

Que

ue in

R4

Time [s]

t=15t=15.50

t=16t=18

Fig. 15. Evolution of the distribution of the queue at router R3.

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 131

ignored in the FMPB model, which predicts a signif-icantly shorter queue at R3 with respect to that atrouter R2 (note that the queues at routers R2 andR3 experience the same average load). In this condi-tion, the queue at router R3 is perceived by the TCPsources as ‘‘almost’’ transparent, since both the lossand delay dynamics are mainly driven by the queueat router R2.

In Figs. 14 and 15 we depict the evolution of thequeue length distributions at routers R2 and R3right after the link failure, as predicted by theFMPB model. While the queue distribution dynam-ics at R2 reflect the load increases due to the rerout-ing of TCP flows, the shaping effect on the arrivalprocess at R3 has a dramatic impact on the dynam-ics of the queue distribution at R3, even if the loadof the queue at R3 does not change.

6. Conclusions

In this paper we discussed three differentapproaches to describe random traffic variations influid models, considering randomness at both theflow and packet levels. With the first approach,the differential equations describing the dynamicsof both the TCP flows and the IP network buffersare transformed into a set of coupled stochastic dif-ferential equations which are numerically solvedusing a Montecarlo approach. With the secondapproach, while the stochastic differential equationsdescribing the TCP flows dynamics are still solvedusing a Montecarlo approach, a second-orderGaussian approximation is applied to the bufferprocesses, and the buffer length evolution is repre-sented as a Brownian motion, so that the bufferlength distribution dynamics is driven by a Fok-ker–Planck equation. With the third approach, asecond-order Gaussian approximation is used todescribe both the TCP flows and the buffer dynam-ics. In this case, the model comprises a set of cou-pled partial derivatives deterministic differentialequations describing the evolution of the distribu-tion of the number of active TCP flows, of the win-dow size of active TCP flows, and of buffer lengths.With these enhancements, fluid models allow reli-able results to be obtained also in the case of IP net-works that operate well below their saturation load,because traffic is mainly due to a finite population ofTCP mice, as it is normally the case in the Internettoday. Numerical results in stationary as well asnon-stationary traffic conditions prove the accuracyand the versatility of the proposed approaches,which are capable of coping with networking sce-narios that so far could only be studied bysimulation.

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Giovanna Carofiglio received the Dr. Ingdegree in Telecommunication engineer-ing from Politecnico di Torino, Italy, in2004, where she is currently pursuing herPh.D. degree. From September 2004 toDecember 2004 she was at Intel ResearchLab, Cambridge, UK, for an internshipunder the supervision of dott.Iannac-cone. Since March 2005, she is visitingTREC-INRIA group led by Prof. Bac-celli at Ecole Normale Superieure, Paris,

France, where she has already been between April and June 2005.Her research interest are in performance evaluation of commu-

nication networks.

Michele Garetto (S’00) received theDr. Ing. degree in TelecommunicationEngineering in 2000, and the Ph.D. inElectronic and Telecommunication Engi-neering in 2004, both from Politecnico diTorino, Italy. In 2002 he was visitingscholar with the networks group of Uni-versity of Massachusetts, Amherst, and in2004 he hold a Post-Doc position at RiceUniversity, Houston. His research inter-ests are in the field of performance eval-

uation of wired and wireless communication networks.

G. Carofiglio et al. / Computer Networks 51 (2007) 114–133 133

Emilio Leonardi received a Dr. Ing.degree in Electronics Engineering in 1991and a Ph.D. in TelecommunicationsEngineering in 1995 both from Politec-nico di Torino. He is currently anAssociate Professor at the Dipartimentodi Elettronica of Politecnico di Torino.

In 1995, he visited the Computer Sci-ence Department of the University ofCalifornia, Los Angeles (UCLA), insummer 1999 he joined the High Speed

Networks Research Group, at Bell Laboratories/Lucent Tech-nologies, Holmdel (NJ); in summer 2001, the Electrical Engi-

neering Department of the Stanford University and finally insummer 2003, the IP Group at Sprint, Advanced TechnologiesLaboratories, Burlingame CA.

His research interests are in the field of: performance evalua-tion of communication networks, switching architectures, all-optical networks.

Alessandro Tarello received his M.Sc.and Ph.D. degree in Electrical andCommunication Engineering fromPolitecnico di Torino, Torino, Italy, in2002 and 2006, respectively. He visitedthe Laboratory for Information andDecision Systems at MIT, Cambridge,MA, USA, from January to December2004 and from July to October 2005.During Summer 2005 he also visited theJet Propulsion Laboratory in Pasadena,

CA, USA. He received the best student paper award at WiOPT2005. His research is on models for performance evaluation of the

TCP protocol and optimization techniques for wireless and adhoc networks.

Marco Ajmone Marsan is a Full Profes-sor at the Electronics Department ofPolitecnico di Torino, in Italy. SinceSeptember 2002 he is the Director of theInstitute for Electronics, InformationEngineering and Telecommunications ofthe National Research Council.

He holds degrees in Electronic Engi-neering from Politecnico di Torino andUniversity of California, Los Angeles.

From November 1975 to October1987 he was at the Electronics Department of Politecnico di

Torino, first as a Researcher, then as an Associate Professor.From November 1987 to October 1990 he was a Full Professor atthe Computer Science Department of the University of Milan, inItaly.

During the summers of 1980 and 1981 he was with theResearch in Distributed Processing Group, Computer ScienceDepartment, UCLA.

During the summer of 1998 he was an Erskine Fellow at theComputer Science Department of the University of Canterbury inNew Zealand.

He has coauthored over 300 journal and conference papers inthe areas of Telecommunications and Computer Science, as wellas the two books ‘‘Performance Models of Multiprocessor Sys-tems’’ published by the MIT Press, and ‘‘Modelling with Gen-eralized Stochastic Petri Nets’’ published by John Wiley.

He received the best paper award at the Third InternationalConference on Distributed Computing Systems in Miami, Fla., in1982. In 2002 he was awarded a ‘‘Honoris Causa’’ Degree inTelecommunication Networks from the Budapest University ofTechnology and Economics.

He has been the principal investigator in national and inter-national research projects in the field of telecommunication net-works.

His current interests are in the fields of performance evaluationof communication networks and their protocols.

He is a Fellow of IEEE, and a corresponding member of theAcademy of Sciences of Torino. He participates in a number ofeditorial boards of international journals, including the IEEE/ACM Transactions on Networking and Computer Networks.