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AQM Stability in Multiple Bottleneck Networks
Dario Bauso
Dip. di Ing. dell’Automazione
e dei Sistemi, DIAS
Universit
´
a di Palermo
Palermo, Italia +39-091481119
Email: Bauso@ias.unipa.it
Laura Giarr
´
e
Dip. di Ing. dell’Automazione
e dei Sistemi, DIAS
Universit
´
a di Palermo
Palermo, Italia +39-091481119-17
Email: giarre@unipa.it
Giovanni Neglia
Dipartimento di Ing. Elettrica, DIE
Universit
´
a di Palermo
Palermo, Italia +39-0916615286
Email: giovanni.neglia@tti.unipa.it
Abstract— In this paper, we highlight that multiple bottlenecks
can affect the performance of Active Queue Management con-
trollers, which are usually configured on a single bottleneck basis,
as if each controller were the only element regulating the TCP
traffic along its path. To see this, we consider a network scenario
where RED is configured at each router, according to previously
developed control theoretic techniques. These configuration rules
assure stability in a single bottleneck scenario. Yet, we show
that instability may arise when two link become congested. We
justify this result through a multiple bottleneck model and give
guidelines for new cooperative AQM controllers.
I. INTRODUCTION
AQM has been proposed to support end-to-end TCP conges-
tion control in the Internet [1]. AQM controllers operate at the
network nodes to detect incipient congestion and indicate it to
TCP sources, which reduce their transmission rate in order to
prevent worse congestion. Usually packet drops are used for
congestion indication.
Many AQM schemes have been proposed [2], [3], [4], [5],
whose algorithms usually rely on some heuristics and their
performances appear to be highly dependant on the considered
network scenario (see, e.g., [6], [7], [8], as regards the well-
known Random Early Detection -RED- algorithm).
This paper is motivated by the consideration that the dis-
tributed fashion of TCP flows control across the network has
not been explicitly considered up to now. As a matter of fact
TCP flows may turn to be controlled at the same time by
two or more nodes acting independently according to their
AQM settings. According to our opinion, this can hardly
affect AQM algorithms performance. In particular, we propose
a counterexample to show that RED controllers, configured
according to [9], do not prevent from instability if two nodes
face congestion at the same time (this is referred to as multiple
bottleneck scenario).
This paper is organized as follows. Section II recollects
some results from [9], which will be referred to in the
following sections. In Section III we present a multiple bottle-
neck network scenario, that exhibits instability. The presence
of instability is derived from performance metrics obtained
through simulations. In Section IV, we provide an analytical
insight to better understand the experimental results. Finally,
conclusive remarks and further research issues are given in
Section V. In particular we discuss the development of new
cooperative congestion local controllers under the assumption
that a congested node may communicate its state to the
neighbors.
II. S
INGLE BOTTLENECK MODEL
The starting point in [9] is the model described by the
following coupled, nonlinear differential equations:
˙
W (t)=
1
R(t)
−
W (t)W (t − R(t))
2R(t − R(t))
p(t − R(t)) (1)
˙q(t)=
W (t)
R(t)
N(t) − 1
q(t)
C (2)
where 1
q
=1if q>0, 1
q
=0otherwise. Symbols used in
the equations above are summarized in the following table.
W expected TCP window size (packets);
q expected queue length (packets);
R round-trip time;
C link capacity (packets/sec);
T
p
propagation delay (secs);
N load factor (number of TCP sessions);
p probability of packet drop;
The first equation represents the TCP window, that increases
by one every round trip time, and halves when a packet
loss occurs. Packet loss rate is computed as the dropping
probability times the number of packets sent per time unit.
The round trip time is related to the propagation delay and the
queue occupancy by the following relation: R = T
p
+
q
C
.The
second equation represents the variation of queue occupancy
as the difference between the input traffic and the link capacity.
AQM schemes determine the relation between the dropping
probability and the nodes congestion status.
Here we considered RED as AQM scheme. RED configu-
ration is specified through four parameters: the minimum and
the maximum threshold (THR
min
, THR
max
), the maximum
dropping probability in the region of random discard P
max
,
and the weight coefficient w
q
. RED can be modelled by the
following equations (refer to [2] for RED operation):
˙x(t)=−Kx(t)+Kq(t) (3)
p(x)=
0, 0 ≤ x<THR
min
(x−THR
min
)P
max
THR
max
−THR
min
,THR
min
≤ x<THR
max
1,THR
max
≤ x,
(4)
where K = − ln(1 − α)/δ and δ is the time between two
queue samples, and can be assumed to be equal to 1/C for a
congested node.
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R
0
C
2
2N
2
s+
2N
R
2
0
C
e
−sR
0
N
R
0
s+
1
R
0
L
1+
s
K
- - - -
∂p∂W ∂q
TCP Dynamic Queue Dynamic RED Control Law
Fig. 1. Block diagram of linearized RED control system
2 3 4
1
9
7
5
8
6
Fig. 2. Network topology
TABLE I
N
ETWORK PARAMETERS
Link Capacity (Mbps) Propagation Delay (ms)
1-4 20 15
2-3 10 5
3-4 20 10
4-7 10 10
5-1 20 15
6-2 20 5
8-2 20 15
3-9 20 10
The linearized system (TCP sources, congested node queue
and AQM controller) can be represented by the block diagram
of Figure 1, where L = P
max
/(THR
max
− THR
min
).
The open-loop transfer function of the system in Figure 1
is:
F (s)=
L
(RC)
3
(2N)
2
e
−sR
1+
s
K
1+
s
2N
R
2
C
1+
s
1
R
(5)
In [9] the authors present RED configuration rules, that
guarantee the stability of the linear feedback control system
in Figure 1 for N ≥ N
−
and R
0
≤ R
+
.
III. A
N INSTABILITY EXAMPLE
We consider a parking lot network whose topology is
depicted in Figure 2. The capacity and the propagation delay
of each link are reported in Table I. Packet size is 1500 bytes.
Links between nodes 4 and 7 and between nodes 2 and 3 will
play the role of bottlenecks.
The RED algorithm is deployed at nodes 4 and 2, respec-
tively to manage the output queues for the link 4 − 7 and
−5 −1 0 5
−20
−15
−10
−5
0
5
N=8
N=4
N=12
Fig. 3. Nyquist plots for the considered RED configuration and N =4, 8, 12
flows
2 − 3. In what follows we refer to these buffers simply as
node 4 buffer and node 2 buffer, without specifying the link,
or as queue 4 (q
4
) and queue 2 (q
2
).
Our RED configuration relies on the control theoretic anal-
ysis of RED presented in [9]. Nevertheless, we do not adopt
exactly the configuration rules proposed there, since their
high stability margins do not allow simple counter-example,
but we use common thumb rules and then we verify RED-
configuration stability through the Nyquist plot of the open
loop transfer function.
We recall that the Nyquist criterion allows one to study the
stability of the closed loop system through the polar plot of
the open loop transfer function F (jω). For the functions we
are interested in, the closed loop system is stable if and only
if the plot does not encircle the point (−1, 0).
We choose THR
min
=2, THR
max
=20, P
max
=5%,
and w
q
=0.002. This configuration guarantees stability if the
number of flows is greater than or equal to N
−
=7and the
Round Trip Time is lower than or equal to 110ms. Figure 3
shows the Nyquist plot of the open loop transfer function (5)
for R = 110ms and different number of flows N.
Simulations were conducted through ns v2.1b9a [12]. We
used TCP Reno implementation.
A. Single Bottleneck
A primary question is which metric is particularly suitable
to catch instability phenomena. In this sense, though instability
is by many authors addressed looking at the amplitude of
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0
5
10
15
20
10 12 14 16 18 20
queue occupancy
time (s)
THRmin
THRmax
Fig. 4. Instantaneous buffer occupancy with number of flows N =8
queue size oscillations, we will better refer to the normalized
standard deviation as a more suitable metric to analyze in-
stability phenomena. For example, when the number of flows
decreases, stability margins decrease according to the linear
model developed in [9], and one could expect larger queue
oscillations. Yet, at the same time the queue average value
decreases and the physical constraint of positive queue values
can determine smaller oscillations. Ultimately, the cause is the
RED coupling of queue length and loss probability, which lets
the operating point depend from the network conditions, like
the load level. From a control theoretic point of view one says
that the RED controller has steady state regulation errors.
Now, in order to analytically show how instability of the
linear model concretely affects the network performance, we
first present some results regarding the single bottleneck
scenario.
Two aggregates, each one of four TCP flows (N =8), enter
the network through node 5 and node 6 with destination node
7 (solid lines in figure 2). The link between nodes 4 and 7 is
congested.
Figure 4 shows the instantaneous queue occupancy time-
plot for the buffer at node 4. RED should be able to keep the
queue occupancy within the two thresholds (dotted lines).
Let us progressively reduce the number of flows through
the network and see if instability occurs as claimed in [9].
In Figure 5 the buffer occupancy is shown to revisit with a
higher frequency the regions associated to buffer overload and
underload (out of RED thresholds).
Numerical results for the throughput and the normalized
standard deviation are shown in Table II. As the total flow
number decrease from 8 to 6 we note that i) the throughput
over the link 4 − 3 reduces from 9.80 Mbps to 9.70 Mbps,
ii) both the average queue occupancy and the oscillation
amplitude decrease, respectively from 10.0 to 8.19 and from
5.26 to 4.64, and iii) the normalized standard deviation, i.e.
the ratio between standard deviation and mean, increases from
0.52 to 0.56.
Experimental results show that instability predicted by the
model in [9] leads to reduced link utilization and higher
normalized oscillations (higher jitter in percentage).
Conversely, if we increase the number of flows, higher
0
5
10
15
20
10 12 14 16 18 20
queue occupancy
time (s)
THRmin
THRmax
Fig. 5. Instantaneous buffer occupancy with N =6
0
5
10
15
20
10 12 14 16 18 20
queue occupancy
time (s)
THRmin
THRmax
Fig. 6. Instantaneous node 4 buffer occupancy in a two bottleneck scenario
throughput and lower jitter can be achieved.
Node 2 buffer has the same RED configuration. Table II
shows similar results when only the link 2 − 3 is congested,
due to flows coming from nodes 6 and 8.
B. Two Bottlenecks
We now draw the attention to the fact that buffer occupancy
instability, may arise when flows through node 4 are in part
already controlled by some other congested upstream node, for
instance, node 2 when link 2 − 3 is congested (see Figure 2).
To recreate artificially such a scenario, let us introduce an
additional aggregate entering the network from node 8, with
destination node 9 (dotted line in figure 2). Node 4 buffer
occupancy for a 4-flows aggregate exhibits a high oscillatory
behavior in figure 6.
The numerical values stored in the last three rows of Table II
support quantitatively our claims rising from Figures 6. In par-
ticular the normalized oscillation values of node 4 buffer are
comparable to the value stored in the fifth row, corresponding
to a single bottleneck instability scenario due to a low number
of flows (N
5
+N
6
=4<N
−
). From Table II instability arises
also at node 2.
Note that, though the number of flows at each node and the
flow round trip time should assure stable operation, instability
arises due to the traffic aggregate from 6 to 7, which traverses
both the congested links.
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TAB LE II
NUMERICAL RESULTS
N
6
N
5
N
8
Thr
6
Thr
5
Thr
8
queue
4
queue
4
queue
2
queue
2
occupancy oscillation occupancy oscillation
6 6 0 5.36 4.57 - 13.6 0.41 0.94 0.26
4 4 0 5.39 4.41 - 10.0 0.52 0.95 0.25
3 3 0 5.29 4.41 - 8.19 0.56 0.96 0.28
2 2 0 5.32 4.17 - 6.31 0.64 0.97 0.43
0 4 0 - 9.49 - 5.51 0.72 0 0
4 0 4 4.92 - 4.92 0 0 10.48 0.48
4 4 4 3.60 6.06 6.12 8.05 0.73 9.36 0.62
4 4 6 3.03 6.59 6.82 7.51 0.75 11.60 0.53
4 4 8 2.59 7.03 7.33 7.16 0.74 11.60 0.45
This example shows the limits of local AQM configuration
ignoring the distributed nature of TCP flows control in a
multiple bottleneck scenario. If we consider the configuration
rules given in [9], instability probably does not arise in such a
simple example, but there is a reduction of stability margins.
This modifies the system dynamic response and reduces the
system robustness to the flows number and the round trip time
variation.
IV. T
HE ANALYTICAL INSIGHT
In this section, we provide an insight into the physical
causes of instability in our counter-example. We start from
a nonlinear two bottleneck model of the network with some
simplifying assumptions, and prove that the system is instable.
Then, we come back to single bottleneck systems, by consid-
ering only one TCP aggregate at a time, the other ones acting
as non reactive flows. Despite such system decoupling is not
correct from an analytical point of view, it allows us to get
again the linear system described in Section II, but with some
different parameters. Hence, the effect of multiple bottlenecks
can be helpfully seen as a parameter variation in the same
single bottleneck model we considered to configure the RED.
It allows us to understand why instability arises and to simply
predict the effect of some network scenario changes, such as
the number of flows and the propagation delays. The limits of
such an approximation are detailed in the following subsection.
A. Two Bottleneck Model
We extend the single bottleneck congestion model described
in Section II to the case of two congested nodes. With
reference to the network topology depicted in Figure 2 we
obtain
˙
W
5
=
1
R
5
−
W
5
W
5
(t−R
5
)
2R
5
(t−R
5
)
p
4
(t − R
5
)
˙
W
6
=
1
R
6
−
W
6
W
6
(t−R
6
)
2R
6
(t−R
6
)
(p
2
(t − R
6
)+
+ p
4
(t − R
6
) − p
2
(t − R
6
)p
4
(t − R
6
))
˙
W
8
=
1
R
8
−
W
8
W
8
(t−R
8
)
2R
8
(t−R
8
)
p
2
(t − R
8
)
˙q
4
=
W
5
R
5
N
5
+
W
6
R
6
N
6
− 1
q
4
C
4
˙q
2
=
W
6
R
6
N
6
+
W
8
R
8
N
8
− 1
q
2
C
2
(6)
where R
5
= T
p2
+
q
4
C
4
, R
8
= T
p1
+
q
2
C
2
, R
6
= T
p1
+
q
2
C
2
+
q
4
C
4
.
For sake of simplicity in (6), the time dependance is indicated
only for delayed functions.
The above model relies essentially on the assumptions of the
original single bottleneck model. One further limit is the way
node 6 traffic has been considered in queue 4 equation: this
equation ignores i) the delay from queue 2 to queue 4, and
ii) that this traffic comes from another congested node, and
therefore has been shaped by queue 2 (the outgoing traffic
cannot overcome C
2
).
B. Decoupling into three single bottleneck models
Now, we consider individually each of the three aggregates
and assume the other flows are non reactive ones, i.e., we
focus on W
i
, and assume W
j
/R
j
= W
j0
/R
j0
= cost, for
j = i . Due to congestion at nodes 2 and 4, N
5
W
50
/R
50
+
N
6
W
60
/R
60
C
4
= C
2
N
8
W
80
/R
80
+ N
6
W
60
/R
60
.We
can derive the following models for the aggregates 5 and 8
((i, j)=(5, 4) and (i, j)=(8, 2) respectively):
˙
W
i
=
1
R
i
−
W
i
W
i
(t−R
i
)
2R
i
(t−R
i
)
p
j
(t − R
i
)
˙q
j
=
W
i
R
i
N
i
+
W
60
R
60
N
6
− 1
q
j
C
j
,
(7)
and the following model for aggregate 6:
˙
W
6
=
1
R
6
−
W
6
W
6
(t−R
6
)
2R
6
(t−R
6
)
(p
2
(t − R
6
)+
+ p
4
(t − R
6
) − p
2
(t − R
6
)p
4
(t − R
6
))
˙q
4
=
W
50
R
50
N
5
+
W
6
R
6
N
6
− 1
q
4
C
4
˙q
2
=
W
6
R
6
N
6
+
W
80
R
80
N
8
− 1
q
2
C
2
.
(8)
The equation system (7) is the same as in the previous single
bottleneck system: bottleneck capacities are respectively equal
to C
eq
5
= C
4
− N
6
W
60
/R
60
= N
5
W
50
/R
50
for aggregate 5
and C
eq
8
= C
2
− N
6
W
60
/R
60
= N
8
W
80
/R
80
for aggregate 8.
Neglecting the product p
2
p
4
in comparison to the terms
p
2
and p
4
, the equation system (8) reduces to the single
bottleneck model too, where the bottleneck capacity is C
eq
6
=
C
4
− N
5
W
50
/R
50
= C
2
− N
8
W
80
/R
80
= N
6
W
60
/R
60
and
we can consider a single RED queue where P
eq
max
=2P
max
.
The previous results are quite intuitive. Nevertheless, we
can obtain them via linearization of the equation systems 7
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and 8 (similarly to Appendix I of [9]). Thus, we obtain the
following open-loop transfer function:
F
i
(s)=
L
eq
i
(R
i0
C
eq
i
)
3
(2N
i
)
2
e
−sR
i0
1+
s
K
1+
s
2N
i
R
2
i0
C
eq
i
1+
s
1
R
i0
(9)
where i =5, 6, 8. L
eq
i
=2L for i =6, L
eq
i
= L for i =5, 8.
These transfer function differs from transfer function in (5),
only for the parameter values.
C. Stability considerations
In this section, we justify instability results shown in Sec-
tion III, by applying the Nyquist criterion to the open-loop
transfer function in (9).
We remember that our RED configuration assure stability
for the system whose transfer loop function is (5) with N =8,
R = 110ms and C = C
4
= C
2
.
From the new open-loop transfer functions, we see that
the decrease of the number of effective flows for all the
three aggregates and the increase of the RED slope for the
aggregate 6 contribute to system instability. Yet, the decrease
of the equivalent capacity makes the system more stable. In
order to evaluate the dominating effect we have to consider
numerical values for the parameters, but we can state that as
the number of flows N
8
increases, W
5
exhibits instability. As
the number of flows N
8
increase, the aggregate 6 is going to
be harder choked, hence C
5eq
approaches C
4
and the Nyquist
plot corresponding to the transfer function (9) approaches the
dashed curve in Figure 3, which corresponds to N =4;the
plot encircles the point (−1, 0) and the corresponding closed
loop system is unstable.
With the numerical values from Table II, the single bottle-
neck models predict that W
5
is unstable, whereas W
6
and W
8
are stable: W
8
is stable due to smaller RTT in comparison to
aggregate 5 (T
p1
≤ T
p2
); as regards the window size W
6
a
smaller C
6eq
compensates the N reduction and L increase.
As regards the instability of the multiple bottleneck system,
all the variables show instability. As a matter of fact, W
5
insta-
bility implies the q
4
oscillations and hence the p
4
oscillations.
The last affect the throughput of the aggregate 6. Aggregate
6 couples the two queues and hence it yields instability to q
2
,
and so on.
Single bottleneck models allows us to simply predict for
example the effect of increasing N
8
: W
5
instability increases,
W
8
becomes more stable and the coupling between the two
queues by the aggregate 6 reduces. Hence, we expect that
instability increases at the downstream node and it decreases
at the upstream one. Such prediction is confirmed from per-
formance metrics in Table II for N
8
=6and N
8
=8.
As regards the validity of our simple analysis, let us consider
for example W
5
. Results from System 7 are more accurate as
long as i) aggregate 6 is small (W
6
(t) << W
5
(t)), or ii) it is
not small, but it is not markedly affected by the dynamics of
the aggregate 5 and of the queue 4.
V. C
ONCLUSIONS AND FUTURE WORK
In this paper we showed that RED configuration based on
a single-bottleneck assumption may not prevent from traffic
instability when congestion occurs, at the same time, in two
different locations of the network.
This suggests that the effect of multiple bottlenecks could be
counteracted by robust configuration of AQM controllers. The
network administrator should evaluate not only the minimum
number of flows at each node and their round trip time, but
he should also get more sophisticated information about traffic
matrix across the network and contemporaneously congested
nodes.
Another approach would be to implement new cooperative
AQM controllers, that base their control action on information
about the congestion status of the other nodes. Simplicity is an
obvious requirement, particularly for signalling among nodes.
We think that the Explicit Congestion Notification (ECN)
field [13] in IP packets could be usefully employed for inter-
nodes signalling. ECN has been proposed as a light in-band
signalling form between nodes and client, but it appears to be
a simple way for nodes to transmit downstream information
about their congestion status. The advantages of ECN em-
ployment are: no further network transmission resources are
required, information travels along the data path, and it can
be used by all the nodes controlling the flow.
AQM controller should monitor the ingoing traffic, evaluate
the share of traffic controlled elsewhere, by the percentage of
packets with the Congestion Experienced codepoint set (CE
packets) and set some tunable parameters (like the dropping
curve slope L) according to the controlled traffic share.
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0-7803-8533-0/04/$20.00 (c) 2004 IEEE
IEEE Communications Society
2271
