Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431

www.elsevier.de/aeue

Control theoretic model for congestion pricing and impact of instability

Sebastian Zimmermann, Ulrich Killat∗

Department of Communication Networks, Hamburg University of Technology, Denickestrasse 17, 21071 Hamburg, Germany

Received 26 May 2004; received in revised form 11 November 2004

Abstract

A control theoretic model for single bit resource marking (SBRM), a TCP variant based on congestion pricing theory, ispresented in this article. We also examine the impact of instability on networks and show that instability causes decreasedutilization. From our results we conclude that SBRM is more efficient than TCP Reno with RED queues.� 2005 Elsevier GmbH. All rights reserved.

Keywords: Congestion pricing; Control theory; Instability; SBRM; RED; Adaptive RED

1. Introduction

The development of fluid-flow models for TCP and thecongestion pricing framework [1–5] allowed the analyticalcalculation of the equilibrium rates, packet marking proba-bilities, queue sizes, and delays. It became possible to viewthe rate control problem as an optimization problem and thusoptimize the goodput and the use of network resources us-ing these models. However, fluid-flow models are only validwhen the system is stable. Therefore, it is essential to ensurestability.

Although the cycle of packet marks or drops and rateadjustments can also be viewed as a closed-loop feedbacksystem, only recently control theoretic models have beenapplied to the congestion control algorithms of the Internet[6–10]. For example, in [6–8] it was shown that current TCPvariants in combination with RED active queue managementwill lead to unstable networks when the round-trip delaygrows beyond a certain bound or when the capacities of thelinks become very large.

∗ Corresponding author. Tel.: +49 40 42878 3249;fax: +49 40 42878 2941.

E-mail addresses: s.zimmermann@tu-harburg.de (S. Zimmermann),killat@tu-harburg.de (U. Killat).

1434-8411/$ - see front matter � 2005 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2005.01.014

In this paper we develop a control theoretic model for sin-gle bit resource marking (SBRM) [11], which is a TCP variantbased on congestion pricing theory [1–5]. “Congestion pri-cing”, also known as “resource marking”, is an optimizationstrategy that leads to optimal usage of network resources.Different implementations of congestion pricing theory wereexamined in [11,12]. SBRM showed very good performancewhile being compatible with current TCP sinks that supportthe explicit congestion notification (ECN) [13] option.

We also examine the behavior of SBRM when the systembecomes unstable and measure the impact on performanceby means of simulation. The results are compared to TCPReno with RED [14] and adaptive RED (a-RED) [15] queues.

The paper is organized as follows: Firstly, we will derivea control theoretic model for SBRM and present equationsfor the calculation of the steady state as well as stability re-gions. Then we will examine the impact of instability on thevalidity of the steady state predictions and on performanceparameters such as standard deviation of the instantaneousqueue size and utilization.

2. Control theoretic model for SBRM

For the development of fluid-flow models of TCP’s con-gestion control, differential equations are used that describe

422 S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431

the behavior of the source algorithm (window update), traf-fic aggregation and buffering, active queue management al-gorithm (marking probability), and congestion signal feed-back. The steady state of the network can then be derivedfrom setting all changes to zero, and solving the system. Forthe control theoretic approach, the differential equations areused directly. However, since the fluid-flow models are verycomplex, we linearize the differential equations around theequilibrium point. By transforming these differential equa-tions to Laplace domain, transfer functions can be found forevery part of the network. The Nyquist criterion can then beapplied to the transfer function model to determine linearstability.

2.1. Window evolvement under SBRM

With SBRM, a fraction of 1 − pn(t) of all incoming ac-knowledgments will increase the congestion window by

�cwnd(t)+ = �wn

RTTn

cwndn(t), (1)

where pn(t) is the end-to-end marking probability observedat source n, � is a small constant gain, wn is the willingness-to-pay, RTTn is the round-trip time, and cwndn(t) is the cur-rent congestion window [11]. The rate of congestion windowupdates is identical to the arrival rate of acknowledgmentsxn(t − RTTn(t)), where

xn(t − RTTn(t))�cwndn(t − RTTn(t))

RTTn(t − RTTn(t)). (2)

xn(t) is the rate of source n at time t in packets per second.Thus, the window is increased by (1) at a rate of

xn(t − RTTn(t)) (1 − pn(t)).

Similarly, a fraction of pn(t) of all incoming acknowl-edgments will decrease the congestion window by

�cwnd(t)− = �

(wn

RTTn

cwndn(t)− 1

)

at a rate of

xn(t − RTTn(t)) pn(t).

Put together, the congestion window under SBRM evolvesaccording to

cwnd′n(t) = �

(wn

RTTn(t)

cwndn(t)− pn(t)

)cwndn(t − RTTn(t))

RTTn(t − RTTn(t)). (3)

While TCP Reno uses a fixed gain of 1, with SBRMthe gain can be set using the willingness-to-pay parameterwn and gain parameter �. The willingness-to-pay is usedto distinguish priorities of individual sources, � influencesconvergence speed and stability.

2.2. SBRM queue

A SBRM queue marks packets at each link l ∈ L(n) witha probability ml(t), where L(n) is the set of all links thatare used by source n. The end-to-end marking probability isthen

pn(t) = 1 − �l∈L(n)(1 − ml(t − �bln(t))),

where �bln(t) is the backward delay from link l to source n.

Assuming that ml(t) is small, we approximate

pn(t)�∑

l∈L(n)

ml(t − �bln(t)). (4)

All flows aggregate at link l to a load of

yl(t) =∑

n∈N(l)

xn(t − �fln(t)), (5)

where �fln(t) is the forward delay from link l to source n,

and N(l) is the set of all sources that use link l. ThusRTTn(t)=�f

ln(t)+�bln(t) for any l ∈ L(n). It can be modeled

as propagation delay dn plus queuing delay of all queues onthe path

RTTn(t) = dn +∑

l∈L(n)

bl(t)

cl

, (6)

where cl is the capacity of link l, and bl(t) denotes theinstantaneous queue size at link l. For bl(t) > 0 the queuelength evolves according to

b′l (t) = yl(t) − cl . (7)

Each packet leaving the SBRM queue is marked with theprobability

ml(t) = 1 − exp(−�[bl(t) − b0]+), (8)

where � > 0 is a scaling factor, and b0 is the target queuesize [11].

2.3. Steady state

The steady state is determined from Eqs. (3)–(8) by set-ting cwnd′(t)=b′

l (t)=0. Table 1 shows the resulting formu-las simplified for a single bottleneck scenario. For compar-ison, the corresponding formulas for TCP Reno+RED1 aregiven in Table 2. They were taken from [16]. Using (2)and (6) the average congestion window can be calculated

1 It is assumed that the mean queue length is between minimum andmaximum thresholds.

S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431 423

Table 1. Steady state of SBRM in a single bottleneck link topology

Average rate x∗n = wn∑N

i=1wi

c

Mean marking probability p∗ =∑N

n=1wn

c

Mean queue length b∗ = b0 − 1� ln

(1 − p∗)

Table 2. Steady state of TCP Reno with RED queues in a singlebottleneck link topology [16]

Average rate x∗n = 1

RTTn

1∑Ni=1

1RTT

i

c

Mean marking probability p∗ = 2

2+c2

(1∑N

n=11

RTTn

)2

Mean queue length b∗ = thmin + 1�p∗

as follows:

cwnd∗�x∗ RTT∗, (9)

where

RTT∗ = d + b∗

c.

2.4. Linearization

To simplify the system of differential equations describingSBRM (3)–(8), and to study stability of SBRM around theequilibrium (steady state), the differential equations are lin-earized. For further simplification, the time-varying round-trip delay is approximated by its equilibrium value exceptfor the window-rate-relation (2), where the time-varying de-lay (6) must be used [7].

Variation around the equilibrium is given by

�cwnd′n(t) = �n

⎛⎝ wn

RTT∗n

·⎛⎝ ∑

l∈L(n)

�bl(t − �fln)

cl

⎞⎠

− wn

cwnd∗n

�cwndn(t)

−cwnd∗n

RTT∗n

�pn(t)

⎞⎠ . (10)

Note that the variation of the congestion window dependson the queue length �bl(t − �f

ln) and on the marking proba-bility �pn(t). In the transfer function model, these two de-pendencies will be reflected by two separate source transferfunctions P

sourcedelayn

and Psourcemarkn

(cf. Fig. 1).

Aggregated link rates are linearized using (5) and (6)

�yl(t) =∑

n∈N(l)

�xn(t − �fln(t))

=∑

n∈N(l)

⎛⎝�cwndn(t − �f

ln)

RTT∗n

−cwnd∗n

RTT∗2

n

·∑

k∈L(n)

�bk(t − �fkn)

ck

⎞⎠ , (11)

where the round-trip time around the equilibrium is derivedfrom (6)

�RTTn(t) =∑

l∈L(n)

�bl(t)

cl

. (12)

Similarly, the buffer process is linearized using (7)

�b′l (t) = �yl(t). (13)

Using (4) and (8), the linearized SBRM marking algo-rithm is finally described by

�ml(t) = �l e−�l ([b∗l −b0]+) �bl(t), (14)

�pn(t) =∑

l∈L(n)

�ml(t − �bln). (15)

2.5. Single bottleneck link model (Laplace domain)

After linearization, Eqs. (10)–(15) can be transformed toLaplace domain to yield the transfer function model describ-ing each component of the control system. For example, (10)is written in Laplace domain as follows:

�cwndn(s) =� wn

RTT∗n

�b(s)ck

e−�fns

s + � wn

cwnd∗n

−� cwnd∗

n

RTT∗n

�m(s) e−�bns

s + � wn

cwnd∗n

:= �cwnddelayn (s) + �cwndmark

n (s).

The transfer function model then becomes simplified fora single bottleneck link

Psourcemarkn

(s) = �cwndmarkn (s)

�mn(s) e−�bns

= −� cwnd∗

n

RTT∗n

s + � wn

cwnd∗n

Psourcedelay

n(s) = �cwnddelay

n (s)

�bn(s) e−�fns

=� wn

RTT∗n

1c

s + � wn

cwnd∗n

Pbuffer(s) = �b(s)

�yrate(s)

= 1

s + 1c

∑Ni=1

cwnd∗i

RTT∗2i

e−�fi s

Paqm(s) = �m(s)

�b(s)= � e−�([b∗−b0]+).

424 S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431

buffer

sourcemark

sourcedelay

sourcemark

sourcedelay

aqmn

f1

fN

b1

f1

bN

fN

1

N

Fig. 1. Single bottleneck link model with heterogeneous sources.

Note that all of these transfer functions have poles in theleft-hand plane, i.e. the real parts of the poles are negative.Thus, each component of the open-loop is stable. The com-plete control system is depicted in Fig. 1.

We now calculate the overall transfer function of the con-trol system

F(s) =N∑

n=1

(e−�b

ns Psourcemarkn

(s)e−�f

ns

RTT∗n

)

Pbuffer(s)

(1 −

N∑n=1

(Paqm(s) e−�b

ns

Psourcemarkn

(s)e−�f

ns

RTT∗n

+ e−�fns

Psourcedelay

n(s)

e−�fns

RTT∗n

)Pbuffer(s)

)−1

.

To determine stability of the system, the open-loop transferfunction must be calculated. For homogeneous sources in asingle bottleneck link model, the loop gain is given by

L(s) = N� cwnd∗ e−RTTs − �w 1

ce−2�f s

RTT∗2 (s + � w

cwnd∗)

· � e−�(b∗−b0)1

s + Nc

cwnd∗RTT∗2 e−�f s

. (16)

Stability criteria can now be applied to this transfer function.As validation will prove, the Nyquist criterion is applicable.

2.6. Stability regions

Using the transfer function (16), stability can be deter-mined for arbitrary sets of parameters. Keeping wn = 20,� = 0.2, and � = 0.006 constant, we varied the number ofsources N from 20 to 60, the link capacity c from 8 to15 packets/ms, and the minimum round-trip delay d from50 to 102 ms. Selecting a parameter set, we plotted Nyquis

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

TCP/SBRM Nyquist plot: N=30 c=10 pkts/ms

Re(L)

Img(

L)

50 ms54 ms58 ms62 ms66 ms70 ms74 ms78 ms82 ms86 ms90 ms94 ms98 ms102 ms

Fig. 2. SBRM Nyquist plots.

diagrams for several minimum round-trip times and thendetermined the critical delay at which the system changesfrom stable to unstable (cf. Fig. 2).

Fig. 3 shows the stability regions that can be derivedfrom the analysis. The stable region is in the lower left cor-ner. For comparison, we also calculated stability regions forTCP Reno with RED queues. The RED model was takenfrom [7].

As the capacity of the link increases, the round-trip de-lay must decrease to maintain stability. This is a significantfinding for actual networks: While in the past the backbonenetwork was mostly the limiting factor, nowadays the accesslink is often the bottleneck. The readily available capacityof the access link has increased in the last 15 years from2400 bps to more than 8 Mbps. In future, even higher accesslink capacities will be available. The round-trip delay, onthe other hand, cannot be decreased arbitrarily. Of its com-ponents transmission delay, packetization delay and queu-

S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431 425

8 9 10 11 12 13 14 1550

55

60

65

70

75

80

85

90SBRM: stability region

capacity [pkts/ms]

dela

y [m

s]

N=20N=30N=40N=50N=60

(a)

8 9 10 11 12 13 14 1550

55

60

65

70

75

80

85

90RED: stability region

capacity [pkts/ms]

dela

y [m

s]

N=20N=30N=40N=50N=60

(b)

Fig. 3. Stability regions: (a) SBRM; (b) TCP Reno+RED.

ing delay, only the queuing delay will decrease as capac-ity increases. Thus, instability might become a significantproblem. Assuming a bottleneck capacity of 100 Mbps, REDqueues, packet sizes of 1500 bytes, and 50 active sources,the system will be unstable for round-trip delays roughlylarger than 100 ms, which can easily be exceeded in today’sInternet. Also, for TCP Reno with RED, the critical delaystrongly depends on the number of active sources. If only20 sources are active, instability will already occur when theround-trip time exceeds 60 ms. This dependency is signifi-cantly reduced for SBRM.

2.7. Validation

To validate the results obtained in the stability analysis,we simulated the original differential equations (3)–(8) inSimulink. The development of the queue length over time

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

time [s]

queu

e si

ze [p

acke

ts]

SBRM queue: N=30, c=10 pkts/ms, d=50 ms

(a)

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

time [s]

queu

e si

ze [p

acke

ts]

SBRM queue: N=30, c=10 pkts/ms, d=90 ms

(b)

Fig. 4. SBRM queue trajectory (Simulink simulation): (a) RTT:50 ms (stable); (b) RTT: 98 ms (unstable).

was recorded for the same sets of parameters used in theprevious analysis. Fig. 4 shows the queue size trajectory ofthe Simulink model for SBRM.

Up to a certain minimum round-trip delay the system isstable. If the delay becomes larger, the system will be un-stable. We determined the critical delay at which the sys-tem changes from stable to unstable from the queue plotsand compared them with the critical delays previously deter-mined from the Nyquist model. For each parameter set, weplotted the critical delays from the model (x–axis) vs. thesimulation (y–axis). Good matches are scattered around theidentity line (cf. Fig. 5a). We also determined the frequencyof the oscillations (critical frequency) for both model andsimulation and plotted them (cf. Fig. 5b).

Considering a resolution of 4 ms, the figures show a nearlyperfect match for critical delay and critical frequency. Thus,

426 S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431

50 60 70 80 90 100 11050

60

70

80

90

100

110

Sim

ulin

k si

mul

atio

n

model

SBRM: critical round-trip propagation delay [ms]

(a)

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 31.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Sim

ulin

k si

mul

atio

n

model

SBRM: critical frequency at critical delay [Hz]

(b)

Fig. 5. SBRM: Critical delay and critical frequency of model andSimulink simulation: (a) Critical delays; (b) Critical frequencies.

the simulation corroborates the transfer function model (16)with respect to characteristic instability parameters.

We also validated our results using a packet based simula-tion in the ns-2 network simulator [17]. Since packet arrivalsalso introduce oscillations, stability is difficult to determinefrom the queue trajectories in the region near the critical de-lay. We therefore only simulated at critical delay and calcu-lated the fast Fourier transform from the instantaneous queuesize. Then we identified the frequency of the first peak (cf.Fig. 6). This is the critical frequency, which was again com-pared against the model (cf. Fig. 7). Again, there is a goodmatch indicating validity of our stability analysis.

Fig. 6 also shows that the amplitude of the oscillationsis much higher for RED than for SBRM. This property canalso be calculated for SBRM from the loop-gain model (16)and be seen in the Bode plots for both SBRM and RED (cf.Fig. 8).

0 1 2 3 4 5 6 7 8 9 10-0.05

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Mag

nitu

de

FFT

0 10 20 30 40 50 600

20

40

60

80

100

120

time [s] (sampled at 20 Hz)

queu

e si

ze [p

kts]

Queue trajectory (N=30, c=10 pkts/ms, d=90 ms)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Mag

nitu

de

FFT

0 10 20 30 40 50 600

20

40

60

80

100

120

time [s] (sampled at 20 Hz)

queu

e si

ze [p

kts]

Queue trajectory (N=30, c=10 pkts/ms, d=90 ms)

(b)

(a)

Fig. 6. Spectral analysis of instantaneous queue size (ns-2 simu-lations): (a) SBRM; (b) TCP Reno+RED.

At critical frequency – where the phase becomes −180◦–for SBRM (cf. Fig. 8a) the gain is small and becomes onlyslightly larger as the round-trip delay is increased from 50to 102 ms. For RED (Fig. 8b), on the other hand, the gainis quickly becoming larger for increasing round-trip times.Thus, the effect of instability will be visible in actual net-works sooner with RED than with SBRM. This will bedemonstrated in the following section.

3. Impact of instability

In the previous section we showed that instability willlead to low-frequency oscillations of the queue size. In thissection, we analyze the impact of instability for actual net-works. Since the instantaneous queue size will change atevery packet arrival, high frequency fluctuations are intro-duced. Further, each source will react at a frequency inverse

S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431 427

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 31.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

ns-2

sim

ulat

ion

model

SBRM: critical frequency at critical delay [Hz]

Fig. 7. SBRM: Critical frequency of model and ns-2 simulation.

10-1 100 101 102

10-1 100 101 102

-60

-40

-20

0

20

40TCP/SBRM Bode plot: N=30 c=10 pkts/ms

w

20*l

og10

(|L|

)

-800

-600

-400

-200

0

w

phas

e [°

]

50 ms102 ms

(a)

10-2 10-1 100 101 102-80

-60

-40

-20

0

20

40TCP/RED Bode plot: N=30 c=10 pkts/ms

w

10-2 10-1 100 101 102

w

20*l

og10

(|L|

)

-1000

-800

-600

-400

-200

0

phas

e [°

]

50 ms102 ms

(b)

Fig. 8. Bode plots: (a) SBRM; (b) TCP Reno+RED.

to the round-trip delay, introducing another type of fluctua-tion to the queue size. For these reasons, the low-frequencyoscillation due to instability might not have direct impact onperformance parameters until its amplitude becomes verylarge. We therefore examined the impact of instability onfluid-flow model predictions of the steady state and on per-formance parameters.

3.1. Impact of instability on validity of fluid-flowmodel predictions

Fluid-flow models are only applicable when the system isstable. To evaluate how fluid-flow model predictions differfrom the actual values when the system becomes unstable,we calculated the mean queue size and the average conges-tion window size for both steady-state model and simulation.For the model, the formulas shown in Tables 1 and 2, andEq. (9) were used. Simulations were done of the continuousdifferential equation model (3)–(8) in Simulink and of ac-tual implementations of SBRM and TCP Reno+RED in thens-2 network simulator [17]. The same set of parameters asin the previous section was used. For each parameter set wethen plotted the value for the model (x–axis) vs. the corre-sponding value for the simulation (y–axis). Good matchesare scattered around the identity line.

Figs. 9 and 10 show the average congestion windowand average queue size of the steady state model and theSimulink, respectively, ns-2 simulations for TCP Reno withRED queues.

As long as the system is stable, the match is nearly per-fect (black dots). However, as soon as the system becomesinstable, differences between model and simulation are in-creasing (gray dots). While the mismatch between the modeland simulation is only small for the average congestion win-dow, it becomes very large for the average queue size. Inboth cases, the values of the simulations are smaller thanthe predictions by the model. This is problematic, as lowermean queue sizes further increase the chance that the queueruns empty, causing underutilization of the link. Even worse,too low average congestion windows result in a lower send-ing rate than possible, also indicating underutilization. Anetwork operator not knowing about the stability proper-ties of RED or another adaptive queue management algo-rithm might try to increase the target queue size in orderto maintain good utilization, which will, however, not solvethe problem of oscillations and related jitter.

Fig. 10b analogously shows the mean queue sizes ofmodel vs. ns-2 simulation. For the unstable region, thisplot confirms the findings of Fig. 9b. However, for thestable region the mean queue sizes achieved by the ns-2simulation tend to be smaller than the ones predicted bythe model. This discrepancy is most likely a result of thequeue sometimes running empty which is not consideredin the fluid-flow model and thus not visible in the previousplots.

428 S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Model [pkts]

Sim

ulin

k si

mul

atio

n [p

kts]

RED: average congestion window (stable and unstable)

stableunstable

(a)

(b)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Model [pkts]

Sim

ulin

k si

mul

atio

n [p

kts]

RED: average queue size (stable and unstable)

stableunstable

Fig. 9. TCP Reno/RED: Average queue and congestion windowsizes of model and Simulink simulation: (a) Average congestionwindow size (model vs. Simulink); (b) Average queue size (modelvs. Simulink).

The corresponding plots for SBRM are shown in Figs.11 and 12. Here, the average size of the congestion windoweven then matches the prediction by the model when the sys-tem is unstable. Thus, the average transmission rate roughlystays the same as the theoretical value – maintaining highutilization as we will show later. Like RED, however, themean queue size becomes smaller than the prediction.

3.2. Impact of instability on performance

As said before, oscillations will degrade utilization of thelink: As the amplitude of the oscillation grows, the queueis more likely to run empty. Additionally, oscillations willintroduce jitter which is especially bad for interactive mul-

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

ns-2

sim

ulat

ion

[pkt

s]

Model [pkts]

RED: average congestion window (stable and unstable)

stableunstable

(a)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

ns-2

sim

ulat

ion

[pkt

s]

Model [pkts]

RED: average queue size (stable and unstable)

stableunstable

(b)

Fig. 10. TCP Reno/RED: Average queue and congestion windowsizes of model and ns-2 simulation: (a) Average congestion win-dow size (model vs. ns-2). (b) Average queue size (model vs. ns-2).

timedia traffic. These effects are visible in Fig. 13, wheremean queue size, standard deviation and link utilization areshown for RED, adaptive RED (a-RED), and SBRM. Pa-rameters of the simulations are given in Table 3.

As previously shown in Fig. 8, gain at critical delay andthus standard deviation of the queue size increases as theround-trip delay increases. The mean queue size, on the otherhand, decreases as the queue runs empty. This can be seenin Figs. 13a–c. The decrease of standard deviation in caseof RED for round-trip delays larger than 150 ms (Fig. 13a)is due to a too low mean queue size, causing the queue torun empty regularly. The control theoretic model for REDpresented in [7] and used for the Bode plots is only validif the mean queue size is between minimum and maximumthresholds. This is not the case here any more. Obviously,

S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431 429

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Model [pkts]

Sim

ulin

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Fig. 11. SBRM: Average congestion window and queue sizes ofmodel and Simulink simulation. (a) Average congestion windowsize (model vs. Simulink). (b) Average queue size (model vs.Simulink).

as oscillations increase and mean queue size decreases, uti-lization also decreases.

For comparison, we also included adaptive RED (a-RED)[15], which was developed to automatically adjust RED pa-rameters for improved performance. Figs. 13b and e showthe corresponding plots for adaptive RED. A significant im-provement is visible. The utilization worsens at a much lowerrate. However, again the standard deviation of the queue sizeis growing, indicating oscillations. Adaptive RED suffersfrom similar problems as RED, but the impact of instabilityis significantly lower than with conventional RED. SBRM,shown in Figs. 13c and f, also suffers from instability, butcan maintain the target queue size and perfect utilization ata much larger range of round-trip delays than the other twovariants. For the full range simulated, the amplitude of the

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Fig. 12. SBRM: Average congestion window and queue sizes ofmodel and ns-2 simulation. (a) Average congestion window size(model vs. ns-2). (b) Average queue size (model vs. ns-2).

oscillations is much smaller than with RED and adaptiveRED.

4. Conclusions

A control theoretic model was developed for SBRM,which is based on congestion pricing theory. The model re-vealed stability problems also for SBRM which have to beconsidered when choosing the correct parameters. However,the impact on performance parameters such as utilizationis much lower than with RED and even lower than withadaptive RED. We also showed that instability will lead tolow frequency oscillations of the queue size and to jitter.While the resulting underutilization of the link could partlybe compensated by the network operator by setting a larger

430 S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431

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Fig. 13. Bottleneck queue characteristics: (a); queue size (RED); (b) queue size (a-RED); (c) queue size (SBRM); (d) utilization (RED);(e) utilization (a-RED); (f) utilization (SBRM).

Table 3. Simulation parameters

TCP version Parameter Value

SBRM � 0.2SBRM w 20SBRM � 0.006SBRM b0 50Reno/RED thmin 50 pktsReno/RED thmax 550 pktsReno/RED pmax 0.1Reno/RED gentle trueReno/a-RED thmin 25 pktsReno/a-RED thmax 75 pktsReno/a-RED top 0.5Reno/a-RED bottom 0.01Reno/a-RED alpha 0.01Reno/a-RED beta 0.9

target queue size, oscillations and related jitter can only beremoved by ensuring stability. Especially in the future, fornetworks with increasing capacities and new applicationsthat rely on quality parameters such as delay and jitter, thiswill become important.

Acknowledgement

This work was supported by Deutsche Forschungsgemein-schaft (DFG).

References

[1] MacKie-Mason J, Varian H. Pricing congestible networkresources. IEEE J Selected Areas Commun 1995;13:1141–9.

[2] Kelly F, Maulloo A, Tan D. Rate control for communicationnetworks: shadow prices, proportional fairness and stability.J Oper Res Soc 1998;49:237–52.

[3] Gibbens R, Kelly F. Resource pricing and the evolution ofcongestion control. Automatica 1999;35:1969–85.

[4] Key P. Resource pricing for differentiated services, In: KillatU, Lamersdorf W. editors. Proceedings of: Kommunikation inVerteilten Systemen (KiVS) 2001, Berlin: Springer; February2001.

[5] AthuraliyaS, Low S. Optimization flow control, II: implemen-tation. Technical Report, Melbourne University, May 2000.Available at: http://netlab.caltech.edu/pub/papers/rem2.ps.gz.

[6] Hollot CV, Misra V, Towsley DF, Gong W. A controltheoretic analysis of RED. INFOCOM, 2001: p. 1510–19.

[7] Low S, Paganini F, Wang J, Adlakha S, Doyle J. Dynamicsof TCP/RED and a scalable control. Proceedings of the IEEEInfocom 2002, New York, 2002: p. 239–48.

[8] Low S, Doyle J. Internet congestion control: an analyticalperspective. IEEE Control Sys Mag 2002.

[9] Wang Z, Paganini F. Global stability with time delayin network congestion control. Proceedings of the IEEEconference on decision and control, 2002.

[10] Paganini F, Low S. In: Giarre L, Bamieh B, editors. Acontrol theoretical look at Internet congestion control.

S. Zimmermann, U. Killat / Int. J. Electron. Commun. (AEÜ) 59 (2005) 421–431 431

multidisciplinary research in control: the Mohammed Dahlehsymposium 2002, Berlin: Springer; 2003.

[11] Zimmermann S, Killat U. Resource marking and fair rateallocation. Proceedings of the ICC 2002, New York: IEEE;April 2002. p. 1310–4.

[12] Zimmermann S, Hamann T, Killat U. Dynamics of differentcongestion pricing strategies. In: de Souza J, da FonsecaN, de Souza e Silva E, editors. Teletraffic Engineering inthe Internet Era. Proceedings of the international teletrafficcongress – ITC-17, Salvador da Bahia, Brazil: Elsevier;September 2001. p. 1187–200.

[13] Ramakrishnan KG, Floyd S, Black DL. The addition ofexplicit congestion notification (ECN) to IP. RFC 3168.Internet Engineering Task Force, September 2001.

[14] Floyd S, Jacobson V. Random early detection gatewaysfor congestion avoidance. IEEE/ACM Trans Networking1993;1:397–413.

[15] Floyd S, Gummadi R, Shenker S. Adaptive RED:an algorithm for increasing the robustness of RED’sactive queue management. August 2001. Available at:http://www.icir.org/floyd/adaptivered/.

[16] Low S. A duality model of TCP and queue managementalgorithms. Proceedings of ITC specialist seminar on IPtraffic measurement, modeling and management, Monterey,CA, September 2000.

[17] UCB/USC/LBNL/VINT: Network simulator ns (version 2).Avaliable at: http://www.isi.edu/nsnam/ns/.

Sebastian Zimmermann studied Gen-eral Engineering Science in Hamburgand Electrical Engineering and Com-puter Science in Chicago. After comple-tion of his Master’s Thesis he workedas a Research Assistant at the Depart-ment of Communication Networks atthe Hamburg University of Technology.Currently he is a consultant with the Se-curity Services Group of Deloitte. Hisresearch interests include congestion

control, and modelling and simulation of communication networks.

Ulrich Killat studied Physics in Ham-burg and Heidelberg. After completionof his thesis he worked in the researchlaboratories of Philips in Hamburg andAachen. Since 1991 he is Professor andHead of the Department of Communi-cation Networks at the Hamburg Uni-versity of Technology. His research in-terests include traffic theory, technicalnetwork planning, and modelling andsimulation of communication networks.