IEICE TRANS. COMMUN., VOL.E99–B, NO.2 FEBRUARY 2016

402

PAPER

Proof Test of Chaos-Based Hierarchical Network Control Using
Packet-Level Network Simulation
Yusuke SAKUMOTO†a) , Chisa TAKANO††b) , Masaki AIDA†c) , Members, and Masayuki MURATA†††d) , Fellow

SUMMARY Computer networks require sophisticated control mechanisms to realize fair resource allocation among users in conjunction with
eﬃcient resource usage. To successfully realize fair resource allocation in
a network, someone should control the behavior of each user by considering fairness. To provide eﬃcient resource utilization, someone should
control the behavior of all users by considering eﬃciency. To realize bothcontrol goals with diﬀerent granularities at the same time, a hierarchicalnetwork control mechanism that combines microscopic control (i.e., fairness control) and macroscopic control (i.e., eﬃciency control) is required.
In previous works, Aida proposed the concept of chaos-based hierarchical network control. Next, as an application of the chaos-based concept,
Aida designed a fundamental framework of hierarchical transmission rate
control based on the chaos of coupled relaxation oscillators. To clarify therealization of the chaos-based concept, one should specify the chaos-basedhierarchical transmission rate control in enough detail to work in an actual
network, and confirm that it works as intended. In this study, we implement
the chaos-based hierarchical transmission rate control in a popular network
simulator, ns-2, and confirm its operation through our experimentation. Results verify that the chaos-based concept can be successfully realized in
TCP/IP networks.
key words: chaos, hierarchical network architecture, congestion control, TCP global synchronization, TCP (transmission control protocol),RED (random early detection)

1. Introduction
In computer networks (hereinafter simply referred to as“networks”), many users have to share the limited resources (e.g., link bandwidth). Such networks need control
mechanism to realize fair resource allocation among users
while providing eﬃcient resource utilization. To realize fairresource allocation in a network, someone should control
the behavior of each user by considering fairness; to provide eﬃcient resource utilization, someone should control
the behavior of all users by considering eﬃciency. These
controls are characterized by the granularity of their control
targets. Since each user is essentially a microscopic control
target, the fairness control corresponds to microscopic control, whereas eﬃciency control for all users corresponds toManuscript received June 15, 2015.
Manuscript revised September 28, 2015.
†
The authors are with Tokyo Metropolitan University, Hinoshi, 191-0065 Japan.
††
The author is with Hiroshima City University, Hiroshima-shi,
731-3166 Japan.
†††
The author is with Osaka University, Suita-shi, 565-0871
Japan.
a) E-mail: sakumoto@tmu.ac.jpb) E-mail: takano@hiroshima-cu.ac.jpc) E-mail: aida@tmu.ac.jpd) E-mail: murata@osaka-u.ac.jp
DOI: 10.1587/transcom.2015EBP3256

macroscopic control. Therefore, as shown in Fig. 1(a), torealize fairness and eﬃciency simultaneously, we must realize a hierarchical network control mechanism that combines
both microscopic and macroscopic controls.
In [1], Aida proposed the concept of chaos-based hierarchical network control. Chaos is the state of disorderobserved at a microscopic scale in a deterministic system,
but has a stable structure at a macroscopic scale. Aida’s
concept utilizes such a hierarchical structure of chaos, as
shown in Fig. 1(b)), to design hierarchical network control.
The hierarchical structure of chaos is briefly described asfollows. We denote the network state by a point in a multidimensional space, and represent the time evolution of the
network state by the trajectory of this point. In a deterministic system, the trajectory for an initial point is determined
without randomness; however, if a deterministic system ischaotic, the future point is unpredictable unless its initialpoint is known with perfect accuracy. This is caused by sensitivity to initial conditions of chaos, which means that thediﬀerence between points that originate from slightly different points increases exponentially with time. Due to thesensitivity to initial conditions, any trajectory in a chaotic
system appears to be random at a microscopic scale as illustrated in Fig. 1(b); however, regardless of the initial point,
all trajectories are constrained by a common stable attractor (i.e., a chaos attractor) at a macroscopic scale. This property is called the structural stability of chaos. In [1], Aida
explained that the hierarchical structure of chaos is usefulfor designing hierarchical network control schemes.
As an application of the above chaos-based concept,
Aida also proposed a fundamental framework for hierarchical transmission rate control based on the chaos of coupled
relaxation oscillators [2] (hereinafter simply referred to as“coupled oscillators”). He first designed the control rule forthe transmission rate of each flow on the basis of the deterministic rule that underlies coupled oscillators. Due tothe sensitivity to the initial conditions of chaos, transmission
rates of flows appear to be random; thus, this control avoids
severe network congestion. Such randomization would be
eﬀective for various network problems (e.g., the TCP global
synchronization problem [3] and buﬀerbloat [4]). Morespecifically, this control rule deterministically restricts theflow with the largest transmission rate among all flows sharing a bottleneck link; thus, realizing fairness among flows.
Note that the control rule involves a parameter that impacts
the eﬃcient usage of bottleneck links. Aida then designed
the parameter setting necessary to achieve eﬃcient usagec 2016 The Institute of Electronics, Information and Communication Engineers
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Fig. 1

Hierarchical network control vs. the hierarchical structure of chaos.

on the basis of the chaos attractor established by the control rule. Since all flows use this parameter setting, it corresponds to controlling all flows. To confirm the feasibility of this chaos-based concept, we intend to specify thechaos-based hierarchical transmission rate control schemesuch that it can work in an actual network, and confirm thatit operates as intended.
In this paper, we describe our implementation of chaosbased hierarchical transmission rate control in the popular
network simulator ns-2 [5]; furthermore, we experimentally confirm its operation. In our experiments, we clarify that (a) fairness among users is realized when applying
the chaos-based control rule to each flow, and (b) bottleneck links are eﬃciently utilized when using the parametersetting proposed in [1]. Our results verify the possibilityof successfully realizing the chaos-based concept in TCP/IP
networks.
The rest of the paper is organized as follows. In Sect. 2,
we introduce another hierarchical structure used in network
control design as well as other network controls that areaware of chaos. In Sect. 3, we explain the chaos-based hierarchical transmission rate control. In Sect. 4, we provide
details regarding our implementation of chaos-based hierarchical transmission rate control in ns-2. In Sect. 5, we show
experimental results obtained from our implementation. In
Sect. 6, we conclude this paper and describe scope for futurework.
2. Related Worknovel hierarchical network architecture inspired by systems
in nature [10]. This network architecture has multiple layers to reflect diﬀerent time/space scale dependencies, and ishierarchized by combining the behaviors of the microscopic
and macroscopic layers quite similar to nature system models. As an example, to combine these behaviors, Aida introduced a renormalization group for networks in [10]. Hierarchical network control based on chaos [1] is another example of hierarchical network control based on Aida’s hierarchical network architecture.
Some hierarchical network control mechanisms basedon the hierarchical network architecture of [10] have been
proposed [11]–[13]. In [11], [12], the authors use a partial diﬀerential equation (PDE) to combine the behaviors ofmultiple layers. A PDE and its solution respectively correspond to a microscopic rule and its macroscopic property.
They designed the microscopic control based on a PDE, andthe macroscopic control based on initial and boundary conditions that determine PDE solutions. In [13], the authors
designed a microscopic control (i.e., autonomous decentralized control) based on Markov Chain Monte Carlo, which
is a design method to control the probability distribution ofa system macroscopic variable. The hierarchical network
controls introduced in [11]–[13] were built by designing a
microscopic rule through which macroscopic behavior canbe indirectly controlled. By contrast, hierarchical network
control based on chaos [1] is designed to be able to simultaneously control diﬀerent behaviors (e.g., fairness and eﬃciency) at both the microscopic and macroscopic layers.

2.1 Hierarchical Structure of Network Control

2.2 Chaos in Network Controls

The standard network architecture — the Open Systems Interconnection model — has a hierarchical structure consisting of multiple abstraction layers to hide implementation details. Thanks to this hierarchical structure, developers caneasily implement network controls that work within a specific abstraction layer; however, problems due to the hierarchical structure have been reported, and ad hoc solutions
have been proposed in [6]–[9].
To solve the problems caused by the standard networkarchitecture using a fundamental approach, Aida proposed a

The behaviors of a particular network controls show chaotic
behaviors [14]–[16]. In [14], Chen et al. reported that thedeterministic model of TCP congestion control is chaotic;
as such, they designed a mechanism to stabilize the TCP
congestion control by removing the chaotic behavior. While
in existing research, it is common to remove the behaviorof chaos from unstable network controls, Aida’s concept attempts to deliberately utilize chaos in network control.

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3. Chaos-Based Hierarchical Transmission Rate Control
In [1], as an application of the aforementioned chaos-based
concept, Aida proposed a fundamental framework for hierarchical transmission rate control based on the chaos of
coupled oscillators [2], which is a chaotic deterministic system. Each oscillator has displacement as its state, and theinteraction among oscillators generates chaos with oscillator displacements appearing to be random at a microscopic
scale.
3.1 Chaos-Based Control Rule for the Transmission Rateof Each Flow

Fig. 2 Time series of transmission rates of flows A and B when using thechaos-based control rule with α = 0.0, β = 0.5, ρ = 1.0 Mbit/s2 , rmax = 1.0
Mbit/s, ∂A = {B}, and ∂B = {A}.

In [1], Aida designed a control rule for the transmission rateof each flow according to the deterministic rule that models
the displacement of each oscillator. The designed control
rule adjusts transmission rate ri (t) of flow i by using the following steps.
1. At the start time for flow i, ri (t) is set to 0, and increasing rate si (t) of ri (t) is set to ρ (ρ > 0).
2. Flow i increases ri (i) by si (t) per unit time until the condition of either step 3 or step 4 is satisfied.
3. If ri (t) = rmax , flow i decreases from ri (t) to α ri (t)where 0 ≤ α < 1, and si (t) is set to ρ. Return to step 2.
4. If r j (t) = rmax for flow j that has an adjacent relationship to flow i, flow i sets si (t) to β ρ where 0 < β < 1.
Return to step 2.
Note that threshold rmax is a positive parameter of the chaosbased control rule, and strongly aﬀects the eﬃcient utilization of bottleneck links. The dynamics of transmission rateri (t) and the dynamics of increasing rate si (t) are describedbyd ri (t)= si (t) − si (t) (1 − α) rmax δ(rmax − ri (t)),
dtd si (t) =s j (t) (β ρ − si (t)) δ(r j (t) − rmax )dtj∈∂i+si (t) (ρ − si (t)) δ(ri (t) − rmax ),

(1)

(2)

where δ(x) is the delta function, and ∂i is the set of flowsthat have an adjacent relationship to flow i. Flows in ∂i are
selected from the set of flows sharing a link according to acertain rule.
Next, we provide an execution example of the abovechaos-based control rule. Figure 2 shows time series dataof the transmission rates of flows A and B following thechaos-based control rule. In the execution example, flow A
starts its transmission slightly earlier than flow B. Immediately after starting the transmission of flow B, transmission
rates of flows A and B are approximately the same; however, they appear to be random over time. This property
helps to prevent severe network congestion. The fairness

Fig. 3Fig. 4

Lorenz attractor.

State space described by two vector fields.

between flows A and B seems to be realized since a flow
with a higher transmission rate is always decreased according to step 3. In the execution example, flows appear to
be random, but the trajectory of (rA (t), rB (t)) is constrained
within the Lorenz attractor, as shown in Fig. 3 [2]. This is
explained by the following reason. There are two combinations of (sA (t), sB (t)); thus, a trajectory of (rA (t), rB (t)) traverses the two vector fields shown in Fig. 4. In the figure,when a point of (rA (t), rB (t)) proceeds to the upper side or
right side of a vector field, the point appears at the corresponding lower or left side. By pasting each pair of the corresponding sides shown in Fig. 4, we obtain a 3D structure
similar to two jointed tori. The obtained structure is topologically equivalent to the Lorentz attractor [2]. In [1], Aida
designed a setting method of threshold rmax on the basis of
the chaos attractor, the details of which are described in thenext subsection.

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3.2 Setting Threshold rmax
Threshold rmax is the key to realizing the eﬃcient utilizationof bottleneck links. If rmax is set to too large a value, thesource of each flow can inject many packets into the network. As a result, the total traﬃc rate of flows arriving at
a bottleneck link can exceed the link bandwidth of the bottleneck link, and packet losses occur due to buﬀer overflow.
Conversely, if rmax is set to too small a value, bottlenecklinks are not well utilized. Therefore, care should be taken
when setting rmax .
Threshold rmax is one of parameters that determines theshape of the chaos attractor established by the chaos-based
control rule. In [1], Aida derives a setting for rmax on thebasis of the chaos attractor as
2 1 + (N − 1) βC,
(3)rmax = N 2 + (N − 1) β + (N − 1) α β
where N is the number of flows sharing a link, and C is thelink bandwidth.

Fig. 5

An example of unfairness caused by RED.

those of low-rate TCP flows. In this sense, RED imposes
unfairness between high- and low-rate flows.
Chaos-based hierarchical transmission rate control canrandomize transmission rates of flows by the property ofchaos. Moreover, since chaos-based hierarchical transmission rate control deterministically restricts the flow withthe largest transmission rate, it avoids the unfair situation
caused by RED. Moreover, this restriction on the largest
flow would be also eﬀective to alleviate buﬀerbloat [4],which is a phenomenon in which the router at a bottleneck
link persistently buﬀers too many packets.

3.3 Characteristics
In this subsection, we explain the characteristics of chaosbased hierarchical transmission rate control by comparing itwith other transmission rate controls.
TCP transmits most of its traﬃc on the Internet, and thewell-known TCP transmission rate control mechanism [17]attempts to avoid severe network congestion by monitoring
packet losses in the network. In TCP control, the source ofeach TCP flow increases its transmission rate until packet
loss is detected, whereupon the rate is then drastically decreased; however, such TCP control fails to avoid severe
network congestion in certain situations [3]. One such situation, called TCP global synchronization, occurs when thetime evolution of transmission rates of TCP flows sharing
a bottleneck link are synchronized by TCP control. Owing to TCP global synchronization, severe network congestion and low utilization of the bottleneck link are alternately
triggered. To solve TCP global synchronization problem,
the time evolution of TCP transmission rates should be randomized.
Random early detection (RED) [18] is the most famous
solution to the TCP global synchronization problem. RED
randomly drops a packet at a bottleneck link, thus randomizing the transmission rates of TCP flows. Consequently,
TCP control supported by RED is successful in avoiding severe network congestion and low utilization of a bottleneck
link; however, RED causes a new undesirable situation. As
an example, we consider the situation in which a bottleneck
link is shared by one TCP flow with a transmission rate of
100 Mbit/s and 100 TCP flows with a transmission rate of
1 Mbit/s. In this situation, RED should drop a packet fromthe high-rate TCP flow with respect to fairness, but instead
selects a packet from a low-rate TCP flow with a probability of 50% because 50% of packets arriving at the router are

4. Implementation in a Packet-Level Network Simulator
Here, we describe the fundamental framework of hierarchical transmission rate control based on chaos and implement
it in the popular network simulator, ns-2 [5]. Network simulator ns-2 is designed on the basis of TCP/IP, and is often
used to verify the operation of network control mechanisms
in TCP/IP networks. Through experiments on ns-2, we clarify the feasibility of the concept of the chaos-based hierarchical network control proposed in [1].
4.1 Fundamentals
The steps of the chaos-based control rule explained in
Sect. 3.1 should be executed at a router. If these steps are
executed at the source host of a flow, the source host has
to gather information regarding flows sharing a bottleneck
link. In particular, to execute step 4 of the chaos-based control rule, the source host of flow i has to gather transmission
rates r j (t) of flow j in ∂i; however, in general, it is diﬃcult to
exchange such information regarding flows between source
hosts only using standard TCP functions. Even if source
hosts can exchange flow information, the chaos-based control rule may fail to generate chaos because of the delay in
executing step 4. To avoid this problem, we steps should be
performed at a router. As shown in Fig. 6, each router calculates the transmission rates of flows according to the proposed steps, and notifies source hosts of calculated transmission rates using packet headers. To realize such notification
via packet headers, we refer to the procedures of explicit
congestion control (XCP) [19], [20], which passes information regarding transmission rates calculated by a router to asource host.

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Fig. 6
host.

Notification of transmission rate calculated by a router to source

Fig. 7

Dumbbell topology used in our experiments.

Each router sets threshold rmax by Eq. (3); here, therouter uses the number of flows sharing its managed link,
N, as well as its own link bandwidth C.
This study assumes that the chaos-based control mechanism is activated in all routers at links that may possibly
become bottlenecks. We detail here the chaos-based hierarchical transmission rate control required to confirm its operation on the dumbbell topology shown in Fig 7. Eﬃcient
implementation and support of general topologies are beyond the scope of this paper.
4.2 Design Details

time period, the router removes the flow from its list. If any
updates to its list occur, the router resets rmax by Eq. (3).
To set ∂i of flow i, adjacency relationships among flows are
determined by the entry order in the list. Namely, ∂i if N ≥ 2
is given by⎧ −1⎪
if o(i) = 1{o (2)}
⎪
⎪
⎨ −1
{o
(N
−
1)}
if
o(i) = N ,(4)∂i = ⎪⎪⎪⎩ {o−1 (o(i) ± 1)} otherwise
where o(i) is the function that returns the index number offlow i in the list, and o−1 (k) is the function that returns theidentifier of the k-th flow in the list. In general, o−1 (o(i)) = i.
If N = 1, ∂i is ∅ (i.e., empty set). Consider the example
when a router holds list (A, B, C). In this example, o(A) =1, o(B) = 2, and o(C) = 3. According to Eq. (4), ∂A ={o−1 (2)} = {B}, ∂B = {o−1 (1), o−1 (3)} = {A, C} and ∂C ={o−1 (3 − 1)} = {B}.
Each router calculates the transmission rates of flows
in its list according to the steps explained in Sect. 3.1. Whenthe router receives a packet of a flow, it writes the transmission rate of the flow to the header of the received packet justas in an XCP router.
4.2.3 Procedures for Destination Hosts
Regarding destination hosts, each destination host handles
received packets according to XCP. Similar to XCP destination hosts, each destination host reads the header information of a received packet of a flow, and then writes the transmission rate of the flow to the header of the corresponding
ACK packet.

4.2.1 Procedure for Source Hosts
5. Experimentation
Regarding the procedure for source hosts, each source hosttransmits packets according to XCP [20]. Each XCP source
host restricts the number of packets available for simultaneous release by its congestion window size, and updates
its congestion window size using the change amount in theheader of ACK packets. Similar to XCP source hosts, eachsource host updates its congestion window size using header
information (e.g., the transmission rate calculated by therouter at a bottleneck link). In this update, each source
host calculates its new congestion window size as the product of the notified transmission rate and the base round trip
time (RTT), which is the RTT without queuing delay time.
Some congestion control mechanisms [21], [22] also use thebase RTT to control transmission rates.
4.2.2 Procedure for Routers
Regarding routers, each router forwards packets according
to XCP. We therefore explain the procedures that diﬀer from
XCP below.
Each router has a list of flows passing through its link.
If a router receives a packet of a flow that is absent fromits list, the router adds the flow to the bottom of its list. If arouter receives no packets of a flow in its list within a certain

We confirm the validity of the chaos-based hierarchicaltransmission rate control through experimentation. Morespecifically, we confirm that (a) fairness among flows is realized when using the chaos-based control rule, and (b) eﬃcient usage of the bottleneck link is realized when using theparameter setting of threshold rmax proposed in [1]. Throughour experiments, we will clarify the feasibility of realizing
the chaos-based concept in TCP/IP networks.
Our experiments compare the performance of chaosbased control with that of RED. Note that RED randomizes
flow behavior using a pseudo-random number generator,
whereas chaos-based control randomizes them by the property of chaos. We investigate the diﬀerences between RED
and chaos-based control in terms of fairness and eﬃciency.
Through our experiments, we clarify the performance differences produced by both mechanisms to randomize flow
behavior. Chaos-based control uses more information (i.e.,
the number of flows, N, and the link capacity C) than RED,so we expect that chaos-based control will outperform RED.
5.1 Experimental Setting
For our experimental setup, we use the dumbbell topology

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Table 1

Parameter settings.

chaos-based hierarchical transmission rate control
relaxation ratio α
1/2
deceleration ratio β
1/2√
2
increasing rate ρREDminimum threshold minth
100
maximum threshold maxth
500
packet loss probability at maxth
0.1
weight parameter wq
0.002
commonnumber of sender, N
10bandwidth of non-bottleneck link, Cnon
1,000
bandwidth of bottleneck link, C
100
propagation delay of a link, τ
5 ms
buﬀer size of a link, Q
5,000
packet size (i.e., MSS) P
512

Mbit/s2
packetpacketMbit/sMbit/sFig. 8Time series of transmission rates of flows 1–5.

Fig. 9Time series of transmission rates of flows 6–10.

packetbyte

shown in Fig. 7. The bottleneck link is shared by N flows.
When we use chaos-based control, queue management at
the bottleneck link is set to Droptail. When we use RED for
the queue management, transmission rates of flows are controlled by TCP Reno [17], and we use TCP-Reno+RED fornotational distinction. Overall, we use the default parameters shown in Table 1. Note that we use 512 bytes as the
TCP packet size (i.e., the maximum segment size (MSS)),similar to other studies [23]–[25]. Below, we show resultsfor diﬀerent number of flows, N, while keeping the same
packet size. Results for diﬀerent packet sizes can be estimated from those for diﬀerent N’s because of the following reason. Chaos-based control is a transmission rate control mechanism using congestion window, and its behavior
for a parameter configuration would be identical to that forother parameter configurations that have the same numberof transmitted packets per unit time. The doubling of thepacket size halves the number of transmitted packets perunit time as same as the doubling of N. Hence, we can estimate results when the packet size doubles from those when
N doubles.
In our simulations, each source host starts transmitting
randomly within the time range [0, 1] s. After starting atransmission, each source host continues transmitting until
the end of the simulation. We repeat the simulation experiments with diﬀerent random sequences of start times until
the 95% confidence interval of measurements becomes sufficiently small. Destination hosts have suﬃcient processing
power to handle received packets, so transmission rates offlows are not restricted by the processing performance ofdestination hosts. As a simple setting for the base RTT usedin the calculation of congestion window size, we set 0.03 s
for source hosts.
5.2 Confirmation of Operation of the Chaos-Based Transmission Rate Control Rule
To confirm the operation of the chaos-based control rule,
we investigate the behavior of flows and the fairness amongflows.
Figures 8 and 9 plot time series data of transmission

rates of flows. In the figures, the transmission rates of flows
are approximately the same immediately after transmission
begins; however, they appear to be random over time. Fromthese results, we can confirm the randomization eﬀect ofchaos-based control.
Next, we investigate the fairness among flows. For ourfairness metric, we use the standard deviation (SD) of thetransmission rates of flows. The SD at time t is defined as
SD(t) =where1
ri (t) =
Δ

⎛
⎞2
N
N
⎜⎜⎜ 1 ⎟⎟1  2
ri (t) − ⎜⎜⎝ri (t)⎟⎟⎟⎠ ,
N i=1
N i=1

(5)

 t
ri (x) d x.

(6)

t−Δ

In the above equation, Δ is the calculation interval time ofSD, representing the time scale of interest in the fairness determination. The integration on the right hand side of Eq. (6)is obtained by counting the number of packets transmitted
by the source host of flow i from t − Δ to t.
Figures 10(a)–10(c) show time series data for the SDof transmission rates of flows for Δ = 0.1, 0.5, and 1.0 s,
respectively. According to the figures, the SD for chaosbased control is almost always smaller than that for TCPReno+RED, regardless of Δ. Then, from Fig. 11, chaosbased control has a smaller SD than TCP-Reno+RED, regardless of N. Therefore, chaos-based control has higher
fairness than RED. This is because chaos-based control can
avoid the unfair situation of RED explained in Sect. 3.3.
Chaos-based control uses N and C when calculating
threshold rmax by Eq. (3). As another control using the same

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Fig. 12

Time series of the transmission rate at the bottleneck link.

Fig. 13 Number of flows vs. the time average of the transmission rate at
the bottleneck link.

5.3 Eﬀectiveness of Setting Threshold rmax by Eq. (3)Fig. 10flows.

Time series of standard deviation (SD) of transmission rates ofFig. 11 Number of flows vs. standard deviation (SD) of transmission
rates of flows.

information, we can design a simple control that sets transmission rates of flows to C/N. Note that the simple control is equivalent to chaos-based control if relaxation ratio αof chaos-based control approaches to 1. When we use thissimple control, the transmission rates of all flows are same.
Hence, the SD of transmission rates of flows under the simple control is zero. According to Figs. 11, the fairness ofchaos-based control is worse than that of the simple control.

Through further simulation, we confirm whether the rmax
setting proposed in [1] realizes eﬃcient utilization of thebottleneck link.
Figure 12 plots time series data of the transmission rateof all flows at the bottleneck link. As a reference, we also
show results for TCP-Reno+RED in Fig. 12. From the figure, chaos-based control almost always fully uses the link
bandwidth. Figure 13 shows the time average of the transmission rate at the bottleneck link for diﬀerent numbers offlows, N. According to this result, chaos-based control realizes complete utilization of the bottleneck link, regardlessof N.
If link bandwidth is fully utilized but packet losses occur owing to buﬀer overflow, we cannot say that chaos-based
control eﬃciently utilizes the bottleneck link. Hence, we
confirm the queue length of the bottleneck link. Figure 14
shows time series data for the queue length of the bottleneck
link. As a reference, we show results for TCP-Reno+RED
in Fig. 14. From the figure, we observe that chaos-based
control always yields smaller queue lengths than the buﬀer
size (i.e., 5,000 packets), so there is no possibility of packet
loss owing to buﬀer overflow. The same result is obtained,
regardless of N, as shown in Fig. 15. The simple control
introduced at the end of Sect. 5.2 can realize zero queue
length and complete link utilization at the bottleneck link.
From Fig. 15, we observe that the queue length of chaosbased control is larger than that of the simple control that

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Fig. 14

Time series of the queue length of the bottleneck link.

Fig. 16 kmax vs. the time average of the transmission rate at the bottleneck link

Fig. 15 Number of flows vs. time average of the queue length of thebottleneck link.

Fig. 17link

kmax vs. the time average of the queue length of the bottleneck

sets transmission rates of flows to C/N.
Finally, we compare the the results for diﬀerent rmax
settings to clarify the eﬀectiveness of the settings proposed, which is given by
in [1]. In this experiment, we use rmax
rmax
= kmax rmax ,(7)

where kmax is a positive constant. In the right side of theabove equation, rmax is given by Eq. (3). Hence, if kmax = 1,
the setting by Eq. (3) is same as that by Eq. (7).
Figures 16 and 17 show time average values of thetransmission rate at the bottleneck link and the time averagevalues of the queue length of the bottleneck link for diﬀerent
values of kmax , respectively. According to Fig. 16, the transmission rate of the bottleneck link is equal to the bottleneck
link bandwidth if kmax ≥ 1. Conversely, from Fig. 17, we observe that the queue length of the bottleneck link increases
as kmax increases. Therefore, we conclude that kmax = 1 isthe best parameter setting, regardless of N.
As evident from above, chaos-based control can realize eﬃcient utilization of the bottleneck link with the rmax
setting proposed in [1].
5.4 Experimentation Considering Actual Networks
It is unrealistic to expect all flows to support chaos-based
control in actual networks. There are usually some flows notunder chaos-based control that would share a bottleneck linkwith flows under chaos-based control. Hence, we perform
experiments using the network topology shown in Fig. 18.
This topology contains background traﬃc not under chaosbased control (i.e., UDP traﬃc with 20 Mbit/s).

Fig. 18 Dumbbell topology used in the experiment when generatingbackground traﬃc.

Fig. 19 Dumbbell topology used in the experiment using settings with
diﬀerent RTTs.

In general, flows with diﬀerent RTTs share a bottleneck link. Hence, we then perform experiments using the
network topology shown in Fig. 19. In this network topology, link delay li between source of flow i and the router atthe bottleneck link are set to a random value from 0 to 10
ms. Furthermore, the base RTT of flow i is set to 2 li + 20
ms.
Figures 20 and 21 show transmission rates and queue
lengths of the bottleneck link when generating background
traﬃc, respectively. Note that we showed the transmission
rate without background traﬃc to investigate the eﬃciencyof flows under chaos-based control. From these figures, weIEICE TRANS. COMMUN., VOL.E99–B, NO.2 FEBRUARY 2016

410

More specifically, we confirmed that (a) the chaos-based
control rule realizes fairness among users, and (b) the settingof the parameter used in the control rule proposed in [1] realizes eﬃcient utilization of bottleneck links. These results
clarified the feasibility of realizing the chaos-based concept
in TCP/IP networks.
As future work, we plan to apply the concept of chaosbased hierarchical network control to other network control
designs.
Fig. 20 Number of flows vs. the time average of the transmission rate at
the bottleneck link in the experiments with considering actual networks.

Acknowledgements
The authors thank Ms.Yuri Takahashi of KDDI Corporation
for her help in implementing chaos-based transmission rate
control in the network simulator, ns-2. The authors also
thank Mr. Liu Yongchao of Tokyo Metropolitan University
for his help to in gathering simulation results. This work was
supported by JSPS KAKENHI Grant Number 25540032
and 15K15985.
References

Fig. 21 Number of flows vs. time average of the queue length of thebottleneck link in the experiments with considering actual networks.

Fig. 22 Number of flows vs. standard deviation (SD) of transmission
rates of flows in the experiments with considering actual networks.

can confirm that chaos-based control realizes eﬃciency similar to the results of Sect. 5.3. Figure 22 shows the SD oftransmission rates of flows under chaos-based control. According to these results, chaos-based control has similar effectiveness in terms of fairness to those of results presented
in Sect. 5.2. Figures 20–22 also show results from experiments using diﬀerent RTTs. For such experiments, we obtain the same conclusions regarding fairness and eﬃciency
when using scenarios with background traﬃc and those presented in Sects. 5.2 and 5.3.
6. Conclusions and Future WorkWe implemented chaos-based hierarchical transmission rate
control in the popular network simulator, ns-2, and confirmed its successful operation through various experiments.

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Yusuke Sakumotoreceived M.E. and Ph.D.
degrees in the Information and Computer Sciences from Osaka University in 2008 and 2010,
respectively. He is currently an assistant professor at Graduate School of System Design,
Tokyo Metropolitan University, Japan. His research work is in the area of analysis and designof network controls. He is a member of IEEE,IPSJ and IEICE.

Chisa Takano
received the B.E. degree
in Telecommunication Engineering from Osaka
University, Japan, in 2000, and received the
Ph.D. in Telecommunications Engineering from
Tokyo Metropolitan University, Japan, in 2008.
In 2000 she joined the Traﬃc Research Center,
NTT Advanced Technology Corporation (NTTAT). Since April 2008, she has been an Associate Professor of the Graduate School of Information Sciences, Hiroshima City University.
Her research interests are in the area of compute
networks and distributed systems. She received the IEICE’s Young Researchers’ Award in 2003.

Masaki Aida
received the B.S. degreein Physics and M.S. degree in Atomic Physics
from St. Paul’s University, Tokyo, Japan, in
1987 and 1989, respectively, and received the
Ph.D. in Telecommunications Engineering fromthe University of Tokyo, Japan, in 1999. After joining NTT Laboratories in April 1989, hehas been engaged in research on traﬃc issues incomputer communication networks. From April
2005 to March 2007, he was an Associate Professor at the Faculty of System Design, Tokyo
Metropolitan University. He has been a Professor of the Graduate Schoolof System Design, Tokyo Metropolitan University since April 2007. Hereceived the Best Tutorial Paper Award of IEICE Communications Society
in 2013. Prof. Aida is a member of the IEEE, IEICE, and the Operations
Research Society of Japan.

Masayuki Murata
received the M.E. and
D.E. degrees in Information and Computer Sciences from Osaka University, Japan, in 1984 and1988, respectively. In April 1984, he joined
IBM Japan’s Tokyo Research Laboratory, asa Researcher. From September 1987 to January 1989, he was an Assistant Professor with
the Computation Center, Osaka University. In
February 1989, he moved to the Department ofInformation and Computer Sciences, Faculty ofEngineering Science, Osaka University. In April
1999, he became a Professor of Osaka University, and since April 2004,
he has been with the Graduate School of Information Science and Technology, Osaka University. He has contributed more than four hundredand fifty papers to international and domestic journals and conferences.
His research interests include computer communication networks and performance modeling and evaluation. He is a member of the IEEE, ACM,
IEICE, the Internet Society, and IPSJ.