Bursty Fluid Approximation of TCP for Modeling

Internet Congestion at the Flow Level.

Daniel Genin

NIST, Gaithersburg, MD

Email: dgenin@nist.gov

Vladimir Marbukh

Senior Member, IEEE,

NIST, Gaithersburg, MD

Email: marbukh@nist.gov

Abstract—We propose an improved model for TCP flow

level congestion performance. Conventional fluid approximation

models yield incorrect results for TCP congestion at the flow

level due to throughput overestimation. We introduce a more

accurate model incorporating some of the packet level burstiness,

by approximating TCP sources as on-off fluid sources. Our

model is largely based on the work of D.Anick, D.Mitra and

M.M.Sondhi exploring buffer statistics of the superposition of

on-off sources. Incorporating some of the TCP burstiness at

the packet level while retaining the over all fluid approximation

framework allows us to substantially improve the accuracy of

the flow level congestion model at the price of a relatively small

increase in mathematical complexity. The model is extensively

validated against ns2 simulations and shown to perform better

than the M/M/1/B based model typically used in this context.

Index Terms—TCP, flow level congestion, fluid approximation,

congestion control

I. INTRODUCTION

It is a well known fact that TCP is currently responsible for

transporting the bulk of Internet traffic. This makes models of

TCP performance essential tools for network operators as well

as application designers. Significant progress has been made

in understanding the steady state behavior of multiple TCP

flows traversing a network, e.g. [6],[10],[3],[7],[1],[11] to cite

just a few of the many publications on the topic. However,

in real networks, like the Internet, the number of active

flows is constantly changing and, in recent years, particular

attention has been given to the study of TCP performance

under a steady arrival stream of document transfer flows.

This mimics what has been dubbed elastic traffic, comprising

web page, music, video, and miscellaneous file downloads.

Performance of such flows is measured by their duration and

the degree to which they accumulate in the network reducing

individual user throughput. Note that this type of models is

different from the class of models with a fixed number of

continuously transmitting flows. The latter are mainly used to

study equilibrium properties and near equilibrium dynamics

of TCP on the timescales much shorter than the duration of a

typical document download. On the other hand, in the models

with elastic traffic, often called flow level congestion models

since the quantity of interest is the number of concurrent flows

in the network, the number of flows changes as active flows

finish and new flows arrive.

Flow level congestion can be studied by running detailed

packet level simulators such as ns2. This approach while

giving highly accurate results is computationally intensive

and for realistic networks prohibitively so. Moreover, simulations yield information about specific combinations of

network parameters, requiring a large number of simulations to

map out performance over the range of operational network

conditions. At the opposite extreme are fluid approximation

based mathematical models, which assume separation of time

scales between flow duration and convergence to equilibrium

transmission rate under TCP. These models ignore the details

of packet traffic all together, instead assuming efficient, i.e.

without waste, link bandwidth partitioning between concurrent

flows [5]. The latter approach while lacking the extensive

computational requirements of the packet level simulations is

also significantly less accurate. Thus, as is often the case we

are faced with a trade off between accuracy and computational

intensiveness. Our goal is to improve this trade off by introducing a fluid approximation based, and hence computationally

undemanding, model, which at the expense of some increase

in mathematical complexity performs better than the currently

available fluid approximation models.

0.0 0.2 0.4 0.6 0.8 1.0

0

500

1000

1500

2000

Ρ

nHΡL

Fig. 1. Number of concurrent flows vs. load based on 1 (red), ns2 (blue)

and the proposed model (black).

As a reference point we take the flow level congestion

model studied in [5], which has gained wide acceptance in

the literature. In [5] the number of concurrent flows on a link

is hypothesized to behave as the number of customers in an

M/G/1 processor sharing queue. Assuming efficient bandwidth
utilization by TCP this gives

n ̄ =

ρ

1 − ρ

(1)

for the mean number of concurrent flows on the link, where

ρ is the offered load. Figure 1 compares n ̄(ρ) deduced from

ns2 simulations and the equation (1) for a 1 Gbps link loaded

with a Poisson arrival stream of exponentially distributed file

transfers of mean size 100KB. We note that we concentrate

on the case of a single link as a fundamental test case on

which a model claiming any degree of accuracy must perform

adequately and on which results about more complicated

networks may be built.

It is clear from the diagram that the analytical model (1)

significantly underestimates the number of concurrent flows.

For comparison we also include the corresponding plot for the

proposed model. The main reason for this discrepancy is the

assumption of efficient capacity utilization under TCP. As our

results show TCP throughput can be as low as 60% of the raw

link capacity depending on the propagation delay and router

buffer size.

We propose to correct the above shortcoming by bringing

some of the packet level dynamics back into the model.

The proposed modification is based on the Anick-MitraSondhi(AMS) model introduced in [2] to study queuing statistics of multiplexed on-off sources. Certain aspects of TCP

packet dynamics, described in Section III, make the model

particularly well suited for representing multiplexing of a large

number of TCP flows on a link. In spite of the complexity of

the AMS model a relatively simple and accurate approximation

can be easily deduced based on the work presented in [2].

The rest of the paper is arranged as follows. We begin with

a discussion of existing TCP throughput models and a brief

literature review in Section II. In Section III we present the

case for using the AMS model followed by an adaptation of

the AMS model to the present setting in Section IV. In Section

V we derive a simple approximation for packet loss probability

based on the dominant eigenmode approximation introduced

in [2]. The model is validated in Section VI against ns2

simulations. We then show that the mean-field approximation

for the flow level congestion model based on AMS performs

substantially better than (1 and tolerably well when compared

with ns2 simulations. Section VII summarizes our findings and

discusses directions for future work.

II. THE PROBLEM

The main shortcoming of the model in (1) is that it

completely disregards packet level dynamics of TCP assuming

instead that the all available bandwidth is completely divided

between concurrent flows. This would be an adequate approximation if TCP was at least approximately efficient in taking

advantage of the link capacity. However, the actual throughput

of a link multiplexing even a very large number of TCP flows

can be far below its raw capacity depending on the round-trip

propagation delay and buffer size. The main problem, thus,

is to find an accurate approximation for the TCP throughput

in terms of the intrinsic link parameters and the number of

concurrent flows, where by throughput we mean the number

of packets successfully delivered per second on a link with the

given parameters and a given number of concurrent flows, and

by TCP we mean TCP-Reno. (Note, however, that the packet

loss probability model derived below may be applicable to

other versions of TCP using the “self-clocking” mechanism

of sending a new packet only when an acknowledgment is

received.) Since what can actually be computed based on

the mechanics of the congestion avoidance algorithm is the

transmission rate, throughput is computed as

throughput = transmission rate × (1−packet loss probability)

(2)

TCP transmission rate is closely linked with packet loss

probability on the link since TCP relies on packet loss for

detecting network congestion and controlling its transmission

rate. The square root relationship between packet loss probability and transmission rate

x =

1

T

s

8/3

p

, (3)

where x stands for throughput and p for packet loss probability

is well known and has been substantially validated by live Internet measurements [4], [6]. However, packet loss probability

is not an intrinsic property of the link since it itself depends

on the transmission rate. Thus, (3) gives only the general

form of the fixed point equation determining throughput. To

obtain an equation for throughput in terms of the intrinsic link

parameters p must in turn be expressed in terms of these same

parameters and transmission rate.

If the aggregate packet arrival process were a Poisson

process the packet loss probability could be approximated by

the buffer overflow probability from an M/M/1/B or M/M/1/∞

(if the buffer is very large) queuing model. However, it has

been demonstrated that TCP packet arrival process is not a

Poisson process even when the number of multiplexed flows

is large [8]. In any case, it is easy to see that M/M/1/* models

yield incorrect throughput by simply comparing the graphs of

the solution to (3) with

p(x) = (x/c)

B(1 − x/c)/(1 − (x/c)B+1) (4)

or

p(x) = (1 − c/x)

+, (5)

where c is router capacity, corresponding to buffer overflow

probabilities for M/M/1/B and M/M/1/∞ respectively, and

results of ns2 simulations. As Figure 6 shows M/M/1/* model

predict that throughput is essentially equal to raw link capacity

while ns2 simulations show that depending on the propagation

delay and buffer size the throughput can be as low as 60%

of the raw link capacity. In spite of this apparently stark

inaccuracy M/M/1/* packet loss models continue to be used

in fluid flow approximation models [9].
114.70 114.71 114.72 114.73 114.74 114.75

0

200

400

600

800

1000

Time HsL

Flow number

Fig. 2. Packet arrival times separated by flow number

III. STRUCTURE OF TCP PACKET ARRIVAL PROCESS

Poor performance in M/M/1/* based packet loss models is

due mainly to the non-Poisson character of the packet arrival

process. Under TCP packets tend to arrive in tight batches (or

bursts) which greatly increases the number of dropped packets

relative to a Poisson stream with the same average rate.

To obtain a more accurate picture of the packet arrival

process we studied the ns2 simulation traces (for configuration

details see Section VI). Figure 2 diagrams packet arrival times

split vertically by flow number on a timescale of a round

trip propagation time. As can be seen from the figure packets

from a given flow arrive in tight batches. For the purposes of

the discussion we will define a batch as a group of packets

belonging to the same flow and such that the inter-arrival time

between successive packets is less than 1% of the round-trip

time.

We make the following observations about the structure of

the arrival process.

I) The aggregate batch arrival process is well approximated by a Poisson process (Fig. 3). This is a consequence of it being a superposition of a large number

of periodic processes with nearly identical periods and

random phases. Periodicity in this case refers only to the

time of arrival and not to the batch size. Furthermore, the

batch arrival rate is very close to RTT/N, where RTT

is the round-trip propagation delay and N the number of

concurrent flows, suggesting that each batch corresponds

to a congestion window’s worth of packets sent in reply

to the acknowledgments for the previous congestion

window. We make this our working hypothesis (to be

validated by comparison with ns2 simulations in Section

VI)

II) The batch size distribution is well approximated by a

Gaussian tail distribution for most parameter values

(Fig. 4). Assuming that our hypothesis about the correspondence between batch and congestion window sizes

is correct, the parameters of the batch size distribution

are determined by the operation of the TCP congestion

control. Bacelli et al. [1] have deduced the congestion

window size distribution based on the assumption that

0.2 0.4 0.6 0.8

10-5

10-4

0.001

Batch inter-arrrival time HmsL

LogHProbabilityL Fig. 3. Sample log probability plot of batch inter-arrival time distribution

based on ns2 packet traces.

0 20 40 60 80

-12

-10

-8

-6

-4

Batch size HpktsL

Log Probability

(b)

Fig. 4. Sample log plot of ns2 empirical batch size distribution (red) and

the approximating Gaussian tail distribution (blue).

the packet loss process is a variable rate Poisson process.

It is interesting to note that the stationary congestion

window size distribution obtained in [1] is an infinite

sum of functions of the form exp(−aw2), where a > 0

is a constant coefficient and w is the congestion window

size, which are exactly Gaussian tail distributions.

III) Sizes of consecutive batches in the aggregate stream

are very nearly uncorrelated. Although, the sizes of

consecutive batches for a given flow are correlated since

if the batch size is equal to the congestion window size

then the size of the succeeding batch is either 1 greater

or is half of the size of the preceding one. On the other

hand, when batch arrivals from all flows are aggregated

in a single stream then the sizes of consecutive batches

are very nearly uncorrelated.

IV. ANIK-MITRA-SONDHI MODEL ADAPTATION

A simple fluid approximation model capable of reproducing

the main features of the arrival process described above is the
Annik-Mitra-Sondhi model [2], originally developed to model

queuing statistics of ATM multiplexers.

We briefly describe an adaptation of the AMS model to

the present setting. The arrival process is modeled as a

superposition of N on-off sources. The “on” periods arrive

according to a Poisson process with rate λ = 1/RTT.

The duration of “on” periods is taken to be exponentially

distributed with parameter μ, μ

−1 = w/C, where w is

the mean batch (or congestion window) size and C is the

router capacity. According to the observations of the previous

section a more accurate model would probably result if the

“on” periods were distributed according to a Gaussian tail

distribution but as results of Section VI show, exponential

distribution may already provide an adequate approximation

and is much easier to deal with mathematically. During an

“on” period data arrives at a constant rate C (same as the router

capacity). Arriving data is processed and discharged from the

queue at the same rate C. The buffer can hold up to B units

(packets) of data. Once the buffer is full any subsequently

arriving data is discarded.

The mean number of concurrently active sources is easily

computed to be Nλ/μ since if “on” periods are treated

as customers then the system is isomorphic to an M/M/∞

queuing system. Given that source transmission rate is C, we

have the necessary stability condition in terms of the offered

load

ρ =

Nw

CT

< 1. (6)

In TCP terminology this translates into the natural requirement

that for the queue to remain finite the aggregate congestion

window must be smaller than the bandwidth-delay product.

V. MANY FLOW ASYMPTOTIC LIMIT

In spite of certain mathematical complexity of the AMS

model it turns out, nevertheless, to be explicitly solvable in

the case when the buffer is infinite. The reduction to a linear

system of ordinary differential equations (o.d.e.’s) and an

elegant solution utilizing a generating function of eigenvectors

is lucidly laid out in [2]. The end result is a closed formula

for stationary probability of buffer overflow beyond x solely

in terms of the parameters of the model — λ, μ, C, and

N. We note that there are some minor differences in model

parametrization between [2] and the present exposition so not

all formulas translate directly.

Serious mathematical difficulties arise when the buffer size

is taken to be finite, which, of course, is the case we are

most interested in. It is quite likely that explicit formula for

buffer overflow probability in this case cannot be found due

inherent non-linearity of the problem. In view of these mathematical difficulties we make the crude assumption (justified

by numerical simulations) that the buffer overflow probability

for finite buffer AMS has the same exponential form as the

infinite buffer AMS but with a scaled exponent. Note that this

adds only one extra degree of freedom to the model.

In the infinite buffer case the stationary buffer content

probability distribution is described by a linear system of

o.d.e.’s. The solution of such a system is a linear combination

of eigenmodes corresponding to eigenvalues of the coefficient

matrix. However, as is often the case the dominant eigenmode carries most of the weight with higher eigenmodes

contributing relatively little so that the dominant eigenmode

alone is already a good approximation to the exact solution.

The dominant eigenmode approximation for buffer overflow

beyond x for the infinite buffer case is [2]

G(x) ≈ ρ

N





N−b

Y

Cc−1

i=1

zi

zi + r



 e

−rx

, (7)

where zi are the eigenvalues of the linear system, r = −z0

and ρ is the offered load. The eigenvalues ziturn out to be

solutions of a family of quadratic equations parametrized by i

(see [2] for details). Since we are interested in the statistics for

large N we approximate zi and ρ by their values in the limit

of large N with C = cN, where c is the per flow capacity.

Passing to the limit we get

ρ = λ/μ

r = −

μ − λ

c

for very large N. Furthermore, straightforward computations

show that products of conjugate eigenvalues (eigenvalues come

in pairs corresponding to quadratics) converge to ρ

−2

as N

tends to infinity, so that

ρ

N

N−b

Y

Cc−1

i=1

zi

zi + r

≈ ρ (8)

for very large N. Putting this all together and substituting

for λ and μ from Section IV we obtain an approximation

for probability of buffer overflow as a function of congestion

window size w

p(w) =

w

cT

e

−α(c/w−1/T)B/c

. (9)

where α is the unknown factor compensating for the finite

buffer size.

Observe that p(w) has the correct behavior at the extremes:

limw→0p(w) = 0

lim

w→cT

p(w) = 1,

and is monotonically increasing in w.

VI. VALIDATION

We validate the model derived in Section V by comparing

its predictions with ns2 simulations. The model network we

use for validation consists of 1000 TCP sources aggregated at

a bottleneck router with 1Gbps capacity and a buffer of size B

pkts ranging from 50 pkts to 300 pkts in increments of 50 pkts

(Fig. 5). The sources are limited only by the capacity of the

router. The main component of propagation delay is the delay

on the link connecting the router and the sink node, is denoted

by T0 ms and ranges from 50 ms to 300 ms in increments of 50

ms. The propagation delay T on the access links, connecting
SinkRouter

Sink

Sink

Sink

Sink

Sink

Sink

Source

Router

Source

Source

Source

Source

Source

Fig. 5. Simulated network configuration.

individual sources to the router, is uniformly distributed in

the interval [0, 10%T0]. This variability in the propagation

delay is introduced to avoid phase-locking between flows.

All sources are controlled by TCP-Reno with fast retransmit

disabled. The sources begin transmitting after a small random

delay uniformly distributed in [0, .1] seconds and transmit

continuously throughout the simulation. The delay in the start

of transmission is also meant to destroy any correlation that

may arise had the flows been started simultaneously.

The finite buffer correction factor, α, was calibrated against

numerical simulations of the finite buffer AMS model executed

over a three dimensional grid spanned by T, B and w. The

least squares fit over the resulting data produced α = 1.4 so

that the AMS model packet loss probability is given by

p(w) =

w

cT

e

−1.4(c/w−1/T)B/c

. (10)

Finally, combining (10) with (3) and using x = w/T we have

an equation for the approximate mean congestion window size

in terms of the basic link parameters

w =

s

8/3

(w/cT)e−1.4(c/w−1/T)B/c . (11)

Figure 6 compares throughput computed from ns2 simulations (blue), AMS model (black) and M/M/1/B model (red).

For the analytical models the throughput was computed as

throughput = (1 − p(x))x, (12)

where x is the equilibrium transmission rate computed from

the square root law (3). The 36 link parameter pairs of T0 and

B are sorted into six groups of six, first, by increasing T0 and,

then by increasing B. In Figure 6 each pane corresponds to

a group of six combinations with the same T0. This format

permits us to display all 36 parameter combinations on an

easily readable two dimensional graph. As the figure indicates

the AMS model is generally closer to the ns2 throughput curve,

although, for low propagation delay and for large buffer sizes

AMS error is similar to or larger than the M/M/1/B error.

As we will see shortly, however, where flow level congestion

is concerned, overestimating throughput causes much larger

errors than underestimating. Note, also, that it is the non-trivial

lower bound on TCP throughput that is important in practice

since the upper bound is obvious.

Figure 7 compares relative error in predicted throughput

for AMS and M/M/1/B models, computed relative to the

throughput measured in ns2 simulations. Although, AMS performs better than M/M/1/B the relative error is still substantial

reaching as high as 35% in the worst case. While the AMS

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T0=50ms T0=100ms T0=150ms T0=200ms T0=250ms T0=300ms

40

60

80

100

120

Link parameters

Throughput HpktsmsL

Fig. 6. Throughput computed from ns2 simulations (blue) and square root

law with p given by AMS model(black), M/M/1/B (red). The horizontal line

at 122 pkts/ms is the router capacity and also the throughput predicted by the

M/M/1/∞ model.

T0=50ms T0=100ms T0=150ms T0=200ms T0=250ms T0=300ms

0.0

0.2

0.4

0.6

0.8

Link parameters

Relative errror

Fig. 7. Relative error for the AMS model with α=1.4 (black), α=4 (dashed

black) and M/M/1/B (red).

packet loss probability formula is a product of a large number

of approximations and simplifying assumptions our numerical

experiments suggest that the bulk of the error is caused by the

passage from the discrete packet model to continuous fluid

approximation. We note in passing that increasing α to 4

significantly improves the performance of the model (Figure

7 dashed line),but this parameter choice can only be justified

a posteriori by comparison with ns2 data and we have no

theoretical justification for selecting this particular value.

We return now to the question of flow level congestion

model. Instead of continuously transmitting flows we now have

a steady stream of arriving document transfer flows. Let the

flow arrival rate be l flows per second and the mean document

size be b bytes then the aggregate load is lb B/s. Let c ̃(n) be

the throughput of the link when the number of active flows

is n. If the aggregate load does not exceed the capacity of

the link the number of concurrent flows will converge to an

equilibrium value n ̄ which must satisfy

c ̃(n ̄) = lb (13)

This is a mean-field approximation which is asymptotically

exact as the flow arrival rate and capacity tend to infinity,
0.0 0.2 0.4 0.6 0.8 1.0

0

500

1000

1500

2000

2500

Ρ

nHΡL

(a)

0.0 0.2 0.4 0.6 0.8 1.0

0

500

1000

1500

2000

Ρ

nHΡL

(b)

Fig. 8. Mean number of active flows as a function of utilization for 1Gbps

link with B = 100 pkts and T0 = 100 ms (a) and T0 = 300 ms (b). Flow

size is exponentially distributed with mean 100 KB.

diminishing the relative size of fluctuations about the equilibrium. Thus, given the throughput in terms of the link

parameters it is possible to estimate the equilibrium number

of flows in the system. Figure 8 shows mean number of flows

on a 1Gbps link as a function of utilization, lb/C, under a

load of exponentially distributed elastic flows with mean size

100 KB, for ns2, the flow congestion model (1) and the AMS

based mean-field model. We note that to compute c ̃ for the

AMS mean-field model we used a modification of the square

root law (3) for large p

w =

s

8(1 − p)

3p

(14)

found in [6]. While the AMS based mean-field model is far

from perfect it can be seen to predict the equilibrium number

of flows in the system much more accurately than (1).

Figure 8 also raises several questions. First, it appears to

indicate that for the same average number of flows on the

link the TCP throughput is lower when flows come and go

than when flows transmit continuously. This can be seen from

the AMS model underestimating the number of flows for a

given utilization compared with the ns2 simulations in Figure

8 (a), which corresponds to link parameters for which AMS

underestimates TCP throughput for continuously transmitting

flows by about 15%. If the TCP throughput was the same

for elastic traffic as for continuously transmitting flows the

order of curves in Figure 8 (a) would have been opposite. The

reason for decrease in throughput under elastic traffic could be

the cost of setting up and tearing down of TCP connections.

Figure 8(a), also, shows that at high loads the AMS model

departs from ns2 simulations in another critical way. The

reversal in the AMS curve shows that increasing the number of

active flows beyond a certain threshold causes the throughput

to decrease contrary to what ns2 indicates. The reason for

this behavior is that the transmission rate under AMS model

cannot exceed router capacity C, while packet loss probability approaches 1 as the bandwidth per flow tends to zero,

leading to a corresponding drop in throughput. This picture

might indeed have been accurate if TCP always remained in

congestion avoidance mode. In congestion avoidance mode at

most one packet per round-trip time can be sent without a

corresponding acknowledgment from the sink and the rate of

acknowledgment packets clearly cannot exceed the router capacity. At very high loads, however, time-outs and subsequent

slow start phases begin to account for a significant portion

of transmitted data. This appears to be born out by ns2 data

showing that the mean transmission rate exceeds the router

capacity when the number of flows is very large (even when

flows are transmitting continuously).

VII. CONCLUSION

We presented a new model for TCP throughput based

on an old model of Anick, Mitra and Sondhi for modeling

buffer statistics of ATM multiplexers. Additional accuracy is

achieved by accounting for some of the packet level burstiness,

while retaining fluid approximation approach yields an explicit

formula in terms of the basic link parameters. The presented

model provides a lower bound on the TCP throughput and is

typically within 20% of the ns2 simulation throughput, which

under most circumstances is better than the commonly used

M/M/1/B based models. Using the AMS-based throughput

formula in the simple mean-field model of flow level congestion also gives substantially better results than the commonly

used formula based on the assumption of efficient bandwidth

partitioning.

On the other hand, the AMS model clearly still has a lot

of room for improvement. The fluid approximation itself is

a major cause of error in the model eliminating which, even

partially, is likely to substantially boost the accuracy of the

model. At high loads the accuracy of the model will also be

improved by accounting for the TCP slow start phase. Finally,

more data is necessary to address validity of the AMS meanfield approximation for flow level congestion performance.

REFERENCES

[1] Franc ̧ois Baccelli, David R. McDonald, and Julien Reynier. A meanfield model for multiple tcp connections through a buffer implementing

red. Perform. Eval., 49(1/4):77–97, 2002.

[2] M.M. Sondhi D. Anick, D. Mitra. Stochastic theory of a data handling

system with multiple sources. ICC’80; International Conference on

Communications, Seattle, 1, 1980.
[3] D. Tan F. Kelly, A. Maulloo. Rate control for communication networks:

shadow prices, proportional fairness and stability. J. Oper. Res. Soc.,

49:237–252, 1998.

[4] Sally Floyd. Connections with multiple congested gateways in packetswitched networks. part 1: One-way traffic. ACM Computer Communication Review, 21:30–47, 1991.

[5] S. Ben Fredj, T. Bonald, A. Proutiere, G. Regni  ́ e, ́ and J. W. Roberts.

Statistical bandwidth sharing: a study of congestion at flow level.

SIGCOMM Comput. Commun. Rev., 31(4):111–122, 2001.

[6] D. Towsley J. Padhye, V. Firoiu and J. Kurose. Modeling tcp reno

performance: a simple model and its empirical validation. ACM,

Transactions on Networking, 8(2):133–145, 2000.

[7] R. Srikant L. Ying, G. Dullerud. Global stability of internet congestion

controllers with heterogeneous delays. Transactions on Networking,

14(3):579–591, 2006.

[8] Vern Paxson and Sally Floyd. Wide-area traffic: The failure of poisson

modeling. IEEE/ACM Transactions on Networking, 3:226–244, 1995.

[9] R. Srikant. Mathematics of Internet congestion control. Willey, 2001.

[10] D. Towsley V. Misra, W. Gong. Fluid-based analysis of a network of

aqm routers supporting tcp flows with an application to red. SIGCOMM,

2000.

[11] G. Vinnicombe. On the stability of end-to-end congestion control for the

internet. Univ. Cambridge Tech. Rep. CUED/F-INFENG/TR.398, 2001.