On the Stability of Autonomous Decentralized Flow Control
in High-Speed Networks with Asymmetric Conﬁgurations
Chisa Takano† , Masaki Aida‡ and Shin-ichi Kuribayashi‡† Trafﬁc Engineering Division, NTT Advanced Technology Corporation
2-4-15, Naka-cho, Musashino-shi 180-0006, Japanchisa@m.ieice.org‡ NTT Information Sharing Platform Laboratories, NTT Corporation
Musashino-shi 180-8585, Japan{aida.masaki,kuribayashi.shinichi}@lab.ntt.co.jp
Abstract
In high-speed network, the principle adopted for a timesensitive ﬂow control mechanism should be autonomous decentralized control. In this mechanism, each node in a network handles its local trafﬁc ﬂow only on the basis of theinformation it is aware of, although it is desirable that thedecision-making at each node leads to high performance ofthe network as a whole. To implement this mechanism, weneed ﬁrst to investigate the relationship between the ﬂow
control mechanism of each node and network performance.
In particular, the stability of the network performance isimportant. Our previous studies have proposed a simple
mechanism for autonomous decentralized ﬂow control andhave shown that it has desirable stability. However, since
we used a simple, symmetric, and homogeneous networkmodel having a single bottleneck in the evaluation, the results cannot be generalized. In this paper, by using more
complex network conﬁgurations in evaluations, we showthat the principle of autonomous decentralized ﬂow control
can be applied to high-speed network with asymmetric conﬁgurations.

1 Introduction
In a high-speed network, propagation delay becomes thedominant factor in the transmission delay because the speedof light is an absolute constraint. Therefore, at any given
time, a large amount of data is being propagated on links
in the network (Fig. 1). The amount of such data is characterized by the bandwidth-delay product, i.e., the propagation distance multiplied by the transmission rate. Therefore, in high-speed and/or long-distance transmission, there
is more data in transit on the links than there is in the nodes.

low-speed
network

high-speed
network

packetlink
nodepacket

node

link

node

node

Figure 1. Effect of large bandwidth-delay
product.
Since the state of the nodes changes rapidly in accordancewith their clock speed, but the speed of light is constant, it
is impossible to exert time-sensitive control based on collecting global information about the network. If we allow
sufﬁcient time to collect global information, the data sogathered is too old to apply to time-sensitive control. So,
in a high-speed network, the principles adopted for timesensitive control are inevitably autonomous decentralized
systems [1, 2, 3].
This paper focuses on ﬂow control realized as an autonomous decentralized system. In our model, nodes in networks handle their local trafﬁc ﬂow themselves based only
on the information they are aware of. Since time-sensitive
control in high-speed networks cannot collect global information about the network, nodes can use only restricted local information. We assume that each node can be aware of
the following information: the distance between the node
and adjacent nodes, the number of packets stored in thenode at the present moment, and the feedback informationthat is received from the adjacent nodes. It is, of course,
desirable that the decision making of each node should leadto high performance of the whole network. In ﬂow control,
we use the total throughput of a network as a global perfor-

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mance measure.
In our previous studies, we have investigated the behaviors of local packet ﬂow and the global performance measure when a node is congested, and demonstrated an appropriate ﬂow control model through simulation results [3]. In
addition, we investigated the stability and adaptability of the
network performance when the capacity of a link is changed[4].
However, since we used a simple, symmetric, and homogeneous network model having a single bottleneck in theevaluation, the results cannot be generalized. In this paper,by using more realistic network models in evaluations, weshow that the framework of autonomous decentralized ﬂow
control can be applied to high-speed network with asymmetric conﬁgurations.
This paper is organized as follows. In Section 2, we discuss related works, comparing them with our work by categorizing ﬂow control mechanisms. In Section 3, we describe a performance measure for the whole network andour ﬂow control model. In Section 4, we show the way to set
the values of the parameters used in our ﬂow control model.
This is required for applying to asymmetric network models. In Section 5, we describe two simulation models andthe corresponding results. One simulation model has twobottlenecks and the other model includes links which differ in length. From the evaluations, our ﬂow control modelis shown also to be effective in high-speed networks withcomplex and asymmetric conﬁgurations. Finally, we conclude this paper in Section 6.

2 Related Works
In general, the mechanism used for ﬂow control for highspeed networks should satisfy the following requirements:

node
...

packet stream
Ji-1
Ji

i-1

i

feedback...

i+1

Figure 2. Interaction between nodes.
in a time-scale shorter than the round-trip delay, it is insufﬁcient to apply to decision-making in very short time-scale.
In our previous studies, we have investigated the characteristics of autonomous decentralized ﬂow control in ahigh-speed network, considering the two above requirements [1, 2, 3]. We proposed a simple and effective methodof ﬂow control in [3] and investigated its stability and adaptability in [4].

3 Flow Control Model
3.1 Performance Measure
Each packet in a network is either in a node or on a link.
Since the packets currently stored in nodes are not beingtransmitted over the network, it is natural to deﬁne the total
throughput of the network as a global performance measure
as follows. We deﬁne the total throughput of a network attime t as the amount of data being propagated on the network [1, 2, 3, 10]. In other words, it is the number of packets being propagated on all links in the network at time t.
On the other hand, the only packets we can control
are those stored in nodes, and not those being propagated.
Thus, higher performance of the whole network involves
many uncontrollable packets being propagated on links.
Therefore, inappropriate ﬂow control cannot produce a statethat has high performance and stability.

1. With regard to the collection of information: it must be
possible to collect the information used in the control.

3.2 Node Model

2. With regard to the delay in applying control: the control should take effect immediately.

Figure 2 shows the interaction of our ﬂow control between nodes using the network model with a simple 1dimensional conﬁguration. All nodes have two incoming links and two outgoing ones for a one-way packet
stream and for feedback information, that is, node i (i =0, 1, 2, . . . ) transfers packets to node i + 1 and node i + 1
sends feedback information (node information) to node i.
For simplicity, we assume that packets have a ﬁxed length
in bits.
All nodes are capable of receiving and sending node
information from/to adjacent downstream and upstream
nodes, respectively. Each node i can receive node information sent from the downstream node i + 1, and can send thenode information about node i itself to the upstream nodei − 1. When node i receives node information from downstream node i + 1, it determines the transmission rate for

There are many other papers that study optimization of
ﬂow control problems in a framework of solving linear programs [5, 6, 7, 8, 9]. Their works assume the collection of
global information about the network, but it is impossible torealize such a centralized control mechanism in high-speed
networks. In addition, solving these optimization problems
requires enough time to be available for calculation, and so
it is difﬁcult to apply them to decision-making in a very
short time-scale.
Decentralized ﬂow control by end hosts including TCP
is widely used in the current networks, and there is much research in this area [8, 9]. However, since the end-to-end or
the end-to-node control does not apply to decision-making

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packets to the downstream node i + 1 using the received
node information and adjusts its transmission rate towards
the downstream node i + 1. The framework of node behavior and ﬂow control is summarized as follows:• Each node i autonomously determines the transmission rate Ji based only on information it is aware of,that is, the node information obtained from the downstream node i + 1 and its own node information.
• The rule for determining the transmission rate is thesame for all nodes.
• Each node i adjusts its transmission rate towards thedownstream node i + 1 to Ji .
(If there are no packets in node i, the packet transmission rate is 0.)• Each node i autonomously creates node informationaccording to a predeﬁned rule and sends it to the upstream node i − 1.
• The rule for creating the node information is the samefor all nodes.
• Packets and node information both experience thesame propagation delay.
As mentioned above, the framework of our ﬂow control model involves both autonomous decision-making by
each node and interaction between adjacent nodes. Thereis no centralized control mechanism in the network. Moreprecisely, it is impossible to realize centralized control in ahigh-speed network environment.

3.3 Diffusion-Type Flow Control Mechanism
This subsection brieﬂy reviews our ﬂow control model
described in [3, 4]. In this paper, we focus on the stability of ﬂow control in the congested state, and we consider
packet ﬂow in a heavy-trafﬁc environment. The packet ﬂow
is deﬁned as the number of sent packets per unit of time,and it is the same as the transmission rate toward the downstream node in a heavy-trafﬁc environment. That is, we let
the packet ﬂow be Ji (t) if the transmission rate speciﬁed bynode i is Ji (t). This is because node i has sufﬁcient packets
to transfer. Hereafter, we identify the packet ﬂow with thetransmission rate speciﬁed by the node.
The packet ﬂow Ji (t) should be controlled by the behavior of node i in the framework described in Sec. 3.2. Thismeans the packet ﬂow can be expressed using the node information obtained from the downstream node i + 1 and its
own node information. We deﬁne the packet ﬂow as
Ji (t) := α ri (t − di ) − D (ni+1 (t − di ) − ni (t)),(1)

where ni (t) denotes the number of packets in node i attime t, ri (t − di ) is the target transmission rate speciﬁedby the downstream node i + 1 as node information, α (> 0)and D (> 0) are constants, and di denotes the propagation delay between node i and node i + 1. In addition,(ri (t − di ), ni+1 (t − di )) is notiﬁed from the downstream
node i + 1 with the propagation delay di . The ﬁrst term on
the right hand side of Eq. (1) reﬂects the target rate speciﬁed by the downstream node, and the second term is proportional to the gradient of the packet density. We call α andD the ﬂow intensity multiplier and the diffusion coefﬁcient,
respectively.
If there is no packet loss in the network, the temporalvariation of ni (t) is expressed asni (t + ) − ni (t) =  [Ji−1 (t − di−1 ) − Ji (t)] ,(2)

where  > 0 is a small number.
To estimate the temporal variation roughly, we replace iwith x and apply continuous approximation. Then the propagation delay becomes di → 0 for all i and the packet ﬂow
is expressed as
J(x, t) = α r(x, t) − D∂n(x, t),∂x

(3)and the temporal variation of the number of packets at x is
expressed as a diffusion type equation,∂r(x, t)∂ 2 n(x, t)∂n(x, t)
= −α
+D
,
∂t
∂x
∂x2

(4)

by using the continuous equation∂n(x, t) ∂J(x, t)
+
= 0.
∂t
∂x

(5)

That is, our method aims to perform ﬂow control using theanalogy of a diffusion phenomenon. We can expect thatpackets in the congested node to be distributed to the wholenetwork and normal network conditions to be restored aftersome time.
In our diffusion-type ﬂow control, node i’s packet transmission rate to the downstream node i + 1 is determinedasJi (t) = α ri (t − di ) − Di (ni+1 (t − di ) − ni (t)),(6)

where the diffusion coefﬁcient D in Eq. (1) has been replaced with Di , which depends on i. The meaning of thisreplacement is revealed in the next section.
In addition, the node information of node i sent to the
upstream node i − 1 is determined asri−1 (t) = Ji (t).

(7)

In the framework of Eqs. (6) and (7), the node informationof i speciﬁed to the upstream node i − 1 is a pair of values(ri−1 (t), ni (t)).

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4 Parameter Tuning for the Diffusion-Type
Flow Control

linkallocatedbandwidth

linkallocatedbandwidthnode

The diffusion-type ﬂow control includes two parameters:
the ﬂow intensity multiplier α and the diffusion coefﬁcient
Di . These parameters should be appropriately determinedbased on only the local information that node i is aware of.
In [3, 4], the values of α and Di are chosen to be independent of i. Diffusion-type ﬂow control with these parameters exhibits appropriate performance for simple andsymmetric network. However, since all the links in suchnetworks are the same lengths, it is an unrealistic situation.
Therefore, we consider an appropriate way to determine thevalues of the parameters for networks with links of different
lengths.
Let us consider again the packet ﬂow by continuous approximation at a certain point x = 0,
J(0, t) = α r(0, t) − D∂n(0, t).
∂x

(8)

If we change the scale of x as x → ξ = a x, (a > 0), thenwe haveJ(0, t) = α r(0, t) − D∂n(0, t).
∂ξ

(9)This operation corresponds to changing the distance between nodes by using a scale parameter a. Since the rulefor determining the transmission rate should be the same for
all nodes, the form of Eq. (9) is unchanged under x → ξ.
Therefore, by choosing D = D/a, we obtain
J(0, t) = α r(0, t) − D∂n(0, t)
.
∂x

(10)

This implies that, for the propagation delay di between
node i and node i + 1, we should choose the diffusion coefﬁcient as
Di ∝

1
,
di

(11)

and the ﬂow intensity multiplier α is independent of di .
For the ﬂow intensity multiplier α, [4] shows that thevalue of α ≥ 1.0 enables stability of the performance.

5 Simulation
In our simulation study, we consider the case where thecapacity of a link in the network is suddenly reduced to anarrow bandwidth. This situation occurs when there is anaccident to links, an occupancy of bandwidth caused bybackground trafﬁc, and so on. For example, Fig. 3 illustrates a change of available bandwidth inﬂuenced by cross
trafﬁc. In high-speed networks, no node is aware of the

….

node
….

Figure 3. Example of a bottleneck link.
change of the link state and the new capacity of the link. We
investigate the performance and stability of the diffusiontype ﬂow control model through observations of the total
throughput of the network.
Although the diffusion-type ﬂow control mechanism described in the previous sections achieves both desirable performance and stability, those results are obtained from simulations using the simple, symmetric, and homogeneous
network model, as shown in [3, 4]. The network models
used in [3, 4] have just one bottleneck link and the lengthsof all links are the same. In this section, we investigate
the effectiveness of the diffusion-type ﬂow control mechanism through simulation using two different types of network model, which are more complicated than the model
used in [3, 4]. The ﬁrst set of models include two bottleneck links and the second set of models include links which
differ in length (in other words, different propagation delays).

5.1 Evaluation for Network with Multiple Bottlenecks
In this subsection, we consider network models containing two narrow bandwidth bottleneck links, and investigate
the performance of the diffusion-type ﬂow control mechanism.
5.1.1 Network Model and Simulation Conditions
Figures 4 and 5 show our network models, which are closed
networks with 1-dimensional conﬁgurations and toroidal
boundaries. Both models have two congested nodes andcorresponding two bottleneck links in the network. In network model 1, the congested nodes are located at opposite
sides from each other as shown in Fig. 4, and in networkmodel 2, they are conﬁgured at adjacent positions as shownin Fig. 5. All the other nodes and links are in the same condition as each other. These models simulate the situations
in which congestion occurs at the congested nodes.
Detailed conditions of our network models are listed below:

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wherecongested node

0
1

59
2

...

33

32

...

28

29

31
30

narrow bandwidth

Figure 4. Network model (model 1) with twobottleneck links (i = 0 and 29).

0
1

59
2

...

33

32

...

28

29

congested node31
30

α = 1.0.

(14)
(15)
(16)

5.1.2 Simulation Results: Total Throughput and Its
Stability

narrow bandwidth

Figure 5. Network model (model 2) with twobottleneck links (i = 29 and 30).

• Number of nodes: m = 60. Each node is speciﬁed byi (mod 60).
• Propagation delay between adjacent nodes for all links:
1.0 (unit time).
• Index of the congested nodes:
model 1 : i = 0 and 29.
model 2 : i = 29 and 30.
• Total number of packets in the network: N = 6000.
• Maximum number of packets on a link (except the bottleneck link): Lc = 100.
• Maximum number of packets on the bottleneck link
between adjacent nodes ( i = 0 and 29 for model 1 ori = 29 and 30 for model 2 ): Lb = 25, 50, or 75 (thatis, 1/4, 1/2, or 3/4 of the bandwidth of other links, andhaving the same length).
To investigate the stability under congestion, in addition
to the above conditions, we set the initial condition for congested nodes as follows:• Number of packets in each of two congested nodes attime t = 0: 400.
• The other 5200 packets are randomly distributed
amongst the other nodes and the other links.
As a model for the diffusion-type ﬂow control, Eqs. (6)and (7), we set Di = 0.1 and use the following ﬂow control
model. Since the packet ﬂow is restricted by the link capacity, the diffusion-type ﬂow control is expressed as follows:
Ji (t) = min(max(J˜i (t), 0), Li (t)),
ri−1 = Ji ,J˜i = α ri − Di (ni+1 − ni )Lb ,(i = 0 and 29, 29 and 30),
Li =
Lc ,
(otherwise),

(12)
(13)

From the simulation results, we discuss the performanceand stability of the diffusion-type ﬂow control model,
through observations of the total throughput.
Figure 6 shows the total throughput for model 1 andmodel 2 where the capacities of the two bottleneck links
are the same, and take three different values, Lb = 25,
50, or 75. The horizontal axis denotes the simulation time
and the vertical axis denotes the total throughput (i.e., thetotal number of packets being propagated on links). The
results show that, diffusion-type ﬂow control achieves stable total throughput of the network, irrespective of the locations of the two bottleneck links and of the value of Lb .
From the quantitative point of view, for the case where thelink capacity of the bottleneck links Lb = 50, the maximum value of the sustainable total throughput (the maximum number of packets propagated stably on the links) is
3000, i.e., 50 packets/link × 60 links. Thus, the diffusiontype ﬂow control achieves 77% of the maximum value ofthe total throughput and its value is stable for both model 1
and model 2.
Similarly, Fig. 7 shows the total throughput for model 1
and model 2 where the capacities of the two bottleneck links
are different. For both model 1 and model 2, we choose
the pair of Lb s as (25,50), (25,75) or (50,75). Both graphs
in this ﬁgure show that the diffusion-type ﬂow control alsoachieves high performance and stability, in cases where thebottleneck links have different capacities. The value of total
throughput is seen to be determined by the lower capacity inthe pair of Lb s and it is almost the same as the value shownin Fig. 6
In our ﬂow control model, although no node is aware of
the bandwidth of the bottleneck link, high performance andstability are achieved without dependence on the locationsof the bottleneck links.

5.2 Evaluation for Network with Different Link
Lengths
In this subsection, we consider network models with different propagation delays between adjacent nodes and investigate the performance of the ﬂow control using thesemodels.

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# of packets on links

1000# of packets on links# of packets on links

6000

6000

model 1bottleneck link # : 0 and 29

Lb =75,75# of packets on links

2000

Lb =25,25

6000

200

400

Lb =50,50

600

time

800

Lb =50,75

2000

Lb =25,75
Lb =25,50

06000

model 2bottleneck link # : 29 and 30

200

400

600

800

time

1000

model 2bottleneck link # : 29 and 30

4000

4000

Lb =75,75
Lb =50,50
Lb =25,25
200

400

600

800

Lb =50,75

2000

2000

0

bottleneck link # : 0 and 29

4000

4000

0

model 1time
Figure 6. Total throughput of the network inmodel 1 and model 2 (in the case where the twovalues of Lb are equal).

5.2.1 Network Models and Simulation Conditions
The simulation conditions are almost the same as those in
the previous Sec. 5.1.1 except for the length of each link
and the number of bottleneck links. The differences are as
follows:• Propagation delays between adjacent nodes:

Lb =25,75
Lb =25,50

0

200

400

600

800

1000

58

...

31
30

time
Figure 7. Total throughput of the network inmodel 1 and model 2 (in the case where the twovalues of Lb are different).

0

59

1

...

2

28

congested node29

narrow bandwidth

Figure 8. Network models (model 3 and model4) with different propagation delays between
adjacent nodes.

model 3: Each distance between the node and the adjacent node is short/long (that is, small/large
propagation delay), where the ratio of the lengthof the short links to that of the long links is 1:50.
Each link state appears with the probability 1/2.
So, we choose 0.04 or 1.96 for the propagation
delays of the 60 links so that the mean propagation delay is 1.0 and the variance is 0.94.

As a model for the diffusion-type ﬂow control, we use
the ﬂow control model Eqs. (12) and (13), wheremodel 4: The length of each link follows a lognormal
distribution, where the mean delay in the networkmodel is 1.0 and the variance is 5.2.

The value of the diffusion coefﬁcient Di is explained in thefollowing subsection.

• Index of the (single) congested node: i = 29.
• Number of packets in two congested nodes at time t =0: 400.
• The other 5600 packets are randomly distributed
amongst the other nodes and the other links.
This model is shown in Fig. 8.

J˜i = α ri − Di (ni+1 − ni )Lb ,(i = 29),
Li =Lc ,(otherwise),(17)

α = 1.0.

(19)

(18)

5.2.2 Simulation Results: Total Throughput and Stability
From the simulation results, we discuss the performanceand stability of the diffusion-type ﬂow control model formodel 3 and model 4, through observations of the total
throughput. In this discussion, the value of the diffusion
coefﬁcient Di shown in Sec. 4 is justiﬁed.
Figure 9 shows the total throughput for model 3 andmodel 4, where the horizontal axis denotes the simulation
time and the vertical axis denotes the total throughput. The

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# of packets on links

6000 model 3
inverse proportion to delay

D isconstant(D=0.1)
proportion to delay

4000

bottleneck link: Lb=50# of packets on links

2000

0

400

200

600

time

800

1000

6000 model 4
inverse proportion to delay

D isconstant(D=0.1)
proportion to delay

4000

bottleneck link: Lb=50

2000

0

200

400

600

800

1000

time
Figure 9. Total throughput of the network(model 3 and model 4) (in the case where thedelays between the adjacent nodes are different).

three lines in this ﬁgure show the results from the diffusiontype ﬂow control models with1,
di
Di = 0.1, andDi = 0.1 × di .

Di = 0.1 ×

(20)
(21)
(22)

We can see from this ﬁgure that the total throughput becomes stable on a higher level of performance in the casewhen Di is inversely proportional to the propagation delay.
The other cases, Di = const. and Di is proportional to the
propagation delay, fail in ﬂow control. This result implies
that Eq. (11) is appropriate for realizing high performanceand stability in networks with asymmetric conﬁgurations.
The values of total throughput realized by the model withDi ∝ 1/di shown in Fig. 9 are almost the same as the cases
where the smaller capacity of the bottleneck links is 50 asshown in Figs. 6 and 7. This means the setting of the diffusion coefﬁcient Eq. (11) absorbs the complexity of the
network model, and realizes high performance and stability
even if the conﬁguration of the network becomes complex.

6 Conclusions
This paper has presented the performance and stability ofthe diffusion-type ﬂow control mechanism. The frameworkof this ﬂow control is an autonomous decentralized system
in high-speed networks. In the diffusion-type ﬂow control,
nodes handle their local trafﬁc ﬂow themselves based on
only the information they are aware of.

To apply this method to networks with complex andasymmetric conﬁgurations, we investigated the appropriate
values of the ﬂow intensity multiplier and the diffusion coefﬁcient, α and Di , in our ﬂow control model, and we foundthe conditions α = constant and Di ∝ 1/di , from observations of the packet ﬂow.
We have shown simulation results for two cases: multiple bottlenecks and different propagation delays. Both results show that diffusion type control achieves high performance and is stable even if the network is congested.
In particular, if we choose the diffusion coefﬁcient as
Di ∝ 1/di , the diffusion-type ﬂow control absorbs thecomplexity of the network model, and realizes high performance and stability even if the link delays in the network
are different.
We are interested in the application of the diffusiontype ﬂow control to networks with more complex topology.
These issues will be the subject of further study.

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