New Framework of Back Diffusion-based
Autonomous Decentralized Control and Its
Application to Clustering Scheme
Chisa Takano

Masaki Aida

Masayuki Murata

Graduate School of Information Sciences, Graduate School of System Design,
and Makoto Imase
Hiroshima City University,
Tokyo Metropolitan University,
Graduate School of Information Science
Hiroshima-shi,Hino-shi, Tokyo 191-0065, Japanand Technology, Osaka University,
Hiroshima 731-3194, Japan
Email: maida@sd.tmu.ac.jp
Suita-shi, Osaka 565-0871, Japan
Email: takano@hiroshima-cu.ac.jp
Email: {murata, imase}@ist.osaka-u.ac.jp

Abstract—We previously proposed a framework for autonomous decentralized control by introducing a partial differential equation based on local-action theory. Although each nodein the network acts autonomously on the basis of only the local
information directly available to it, our control proposal leads tooptimal performance for the whole network. In addition, we haveproposed flow control based on a diffusion equation to achieveautonomous load-balancing in the network and have confirmedthat realizes congestion avoidance. This paper introduces a newautonomous decentralized control scheme that creates the spatial
structures of finite size by using renormalization transformation
and back diffusion drift. Moreover, we use the proposed control
scheme to realize autonomous decentralized clustering in ad hoc
networks.

I. I NTRODUCTION
Mobile ad hoc network (MANET) [1] has attracted a great
deal as a likely network configuration technology with which
to realize the long-desired ubiquitous networking. MANET
has no communication infrastructure and is a self-configuring
network of mobile systems connected by wireless links. It can,therefore, offer a flexible communication environment that isrobust against disasters and changes in event sites. For ad
hoc network control, many techniques have been introduced,
especially with regard to route selection. The representative
flat-type routing technologies in MANET are classified aseither proactive schemes [3], [4] or reactive schemes [5],[6]. Both are effective if the MANET does not have many
mobile systems. As the network size grows, however, these
protocols have the problem that the size of the routing table
stored at each individual node becomes large. As a result, theflooding overhead of route requests increases network load.
Against this problem, a large number of studies have looked
at clustering [7], [8], [9]. In clustering, mobile nodes are
divided into different virtual groups (clusters) and routing isbased on intra-cluster and inter-cluster calculations. Clusterhead candidate selection is one of the important problems inclustering schemes, and [10], [11] have proposed algorithms to
select cluster-heads appropriately in accordance with network
conditions.

The common basic framework demanded in those algorithms is adaptability to network environments, and it is hoped
to have the following flexibility: when there is no clusterhead candidate in the neighborhood, the mobile node itself
stands as a candidate and becomes a cluster head. On the other
hand, when there are two or more cluster-head candidates,
we need a highly effective and efficient metric that canrealize optimal cluster-head selection. Examples of the metrics
possible include node degree, cluster size, mobility speed, andbattery energy [12]. In general, the cluster-head must have full
information about the state of other nodes or of the overall
network to optimize the cluster structure. Unfortunately, it isdifficult to collect global information about the network given
the network structure and the need to minimize the informationexchange frequency over the MANET. Therefore, a MANET
needs an autonomous decentralized clustering mechanism inwhich each node acts autonomously on the basis of just local
information they are aware of, while the decisions made byeach node optimize the state of the whole network.
In this paper, we introduce an autonomous decentralized
clustering method that allows the nodes to act based only on
the information they are aware of while realizing environmental adaptability and overall network optimization.
II. AUTONOMOUS D ECENTRALIZED C ONTROLM ECHANISM BASED ON L OCAL I NTERACTION
This section describes the framework of the autonomous decentralized control mechanism based on local interaction [13].
When thinking about the interaction of two objects that are
separated in space, there are two theories depending on theaction method; “action at a distance (non-local interaction)”and “action through a medium (local interaction)”. “Non-local
interaction” is the direct interaction of two objects that are notlinked by any field or medium. For example, Newton’s law of
universal gravitation is an example of “non-local interaction”;
the force of gravity created between object A and object B, F ,is directly proportional to mass m1 and m2 of the interacting
objects, and inversely proportional to the square of the distanceFig. 1.

Example of local interaction.

|r1 − r2 | between the objects as follows:
F (r1 , r2 ) = G

m1 m2,|r1 − r2 |2

where G is the gravitational constant. Function F (r1 , r2 ) is
decided by the distance, that is to say that it is necessary toknow the value of the distance, in other word, the positionsof objects A and B. From the viewpoint of network control,
this is the same as the requirement established by centralized
control that all global information about the network state must
be collected.
In the case of “local interaction,” on the other hand, anyvariation in the physical value at a point of space is transmitted
to the adjacent point via the medium of the gravitational field
at a limited speed. A temporal variation in such a gravitational
field can be described by using a partial differential equation.
We use the following example to elucidate “local interaction.”: When we let a few drops of black ink fall into a glass tube
filled with water, the ink density distribution follows a normal
distribution and the ink spreads through the whole tube bydiffusion (Fig. 1). In this process, the action within a minuteregion of water in the glass tube is very simple: the ink diffuses
from the higher density side towards the lower density side.
The rate of ink diffusion is proportional to the density gradient.
Even though each segment acts autonomously and only local
information is available, the ink density distribution throughout
the glass tube exhibits orderly behavior.
Let the density function (density distribution) of “a certain
quantity” at time t in space be p(x, t). In the diffusion
phenomenon, flow J(x, t) of “a certain quantity” in space isexpressed as,
J(x, t) = −κ∂p(x, t),∂x(1)where κ, which is a positive constant, (> 0). This equationmeans that “a certain quantity” flows at a rate which is
proportional to the gradient of the density. When the total
amount of “a certain quantity” in the system is unchanged,
the continuous equation,∂p(x, t)∂J(x, t)
=−
∂t
∂x

(2)

holds.
We substitute Eq. (2) for Eq. (1) and get the temporal
evolution equation of the density distribution,∂p(x, t)∂ 2 p(x, t)
=κ
,
∂t
∂x2

(3)

where this is the ordinary diffusion equation. In this equation, when the density distribution is a delta function as
the initial condition, the density becomes uniform over time.
Since Eq. (3) is a linear differential, we can obtain generalsolutions by superposing the normal distribution under evenmore complex initial conditions. As seen in an example ofthe diffusion phenomenon, we can consider a system that isbased on local interaction as the framework of the autonomous
decentralized control from the viewpoint of engineering. In
other words, while the sub-systems autonomously run using
only local information, the state of the whole system is optimized indirectly through the corresponding temporal evolution
equation. Variants of the autonomous decentralized control
based on such local interaction can be summarized as follows:• Think about the macro characteristic that the state of thewhole system should have, and find the partial differential
equation (corresponding to Eq. (3)) whose solution has sucha characteristic.
• Think about local interaction (corresponding to Eq. (1)) thatthe partial differential equation describes, and design subsystems so that they act according to the rule that is obtained
from the local interaction.
• As a result, while sub-systems act only based on local
information autonomously, they push the whole system into
the preferable state as a solution of the partial differential
equation.

In general, in autonomous decentralized systems whose subsystems independently perform local interaction, the wholesystem might fall into system-deadlock if the sub-systems
are not designed properly. The framework of autonomous
decentralized control based on local interaction avoids thisproblem and gives us a hint as to how to properly design local
actions of sub-systems so that the whole system is pushed into
the desired state.
As an example of autonomous decentralized control basedon local interaction, we previously proposed a diffusion-typeflow control (DFC) mechanism [14], [15] that overcomes thedifficulty of controlling high-speed networks. In this control
mechanism, the state of the whole network is controlled indirectly through the autonomous action of each node; each nodemanages its local traffic flow on the basis of only the local
information directly available to it, by using predetermined
rules. By applying DFC, the distribution of the total numberof packets in each node in the network becomes uniform overtime, and it exhibits orderly behavior. This property is suitablefor fast recovery from congestion. In Fig. 2, we show the
temporal evolution of the number of packets stored in eachnode. The target flow is between node 1 and node 30, while
the background traffic flows between node 15 and node 30.
The target flow and the background flow start at simulation
time t = 0 s and t = 0.1 s, respectively. The horizontal axes
denote node ID (1–29) and the vertical axes denote the numberof packets stored at the node. In the case with TCP without
DFC, after the start of the background traffic flow (0.1 s),
all the stored packets were at node 15, resulting in packet

Fig. 2.

Diffusion effect of DFC

loss. After that, the TCP window size was reduced and thenumber of stored packets decreased. In the case of TCP with
DFC, the latter prevents the stored packets from building up
at a particular node. Since packet loss was avoided, the TCP
window size did not reduce and high network efficiency wasachieved. Through the introduction of DFC, each node acts
cooperatively to avoid packet loss even though the decisionmaking of each node is based only on local information.
III. R ENORMALIZATION TRANSFORMATION OF THE
DIFFUSION PHENOMENON FOR AUTONOMOUS
DECENTRALIZED CONTROL

So far, we have referred to the diffusion effect in describing
autonomous decentralized control based on local interaction.
We turn now to a structure of practical size that offers
autonomous decentralized control for patterns other than the
diffusion effect.
When we calculate the temporal evolution of density distribution p(x, t) which is the solution of Eq. (3),√while dithering
the x axis simultaneously within the ratio of 2κt/σ , we getdensity distribution q(x, t) as follows:
√√
!
√
2κt
2κt
q(x, t) :=px, t ,(4)σ
σwhere σ is a positive constant. Carrying out both the temporal
evolution (diffusion) and the scale-variation (dithering) at thesame time can be regarded as a kind of renormalizationtransformation of the density distribution p(x, t). The density
distribution q(x, t) for which we execute the renormalizationtransformation follows
µ
∂
∂
1
∂
∂2
q(x, t) =x + σ 2 2 q(x, t).
(5)
∂t
2t ∂x
∂x
This differential equation satisfies the following equation:
lim q(x, t) = √

t→∞

1
2πσ 2

x2

e− 2σ2 .

(6)

For the normal diffusion equation (3), the above equationmeans that the solution approaches a normal distribution
asymptotically with time regardless of the initial condition.
By use of this mechanism, we can realize a new autonomous
decentralized control approach that operates within a limited
structure. However, using Eq. (5) instead of expression Eq. (3)raises two problems with regard to autonomous decentralized
control.
• The existence of 1/(2t) on the right hand side of Eq. (5)makes, over time, the change in the distribution small.
• It is necessary to set the coordinate systems in the network
because the drift term for Eq. (5) directly depends on x.
We consider the latter problem in a following section.
Let us then discuss the transformation of the time-scale.
The distribution q(x, t) in Eq. (5) should have a fast temporal
evolution so that 1/(2t) on the right hand side of Eq. (5) canbe eliminated.
If a constant, c > 0, is used instead of 1/(2t) in Eq. (5),we get:µ∂∂∂∂2q(x, t) = c
x + σ 2 2 q(x, t),
(7)
∂t
∂x
∂x
where we construct the renormalization transformation√√!
√2κe2ct2κe2ct2ctq(x, t) :=
p
x, e
.
σ
σ

(8)

Here, for the reason that time merely passes fast in Eq. (8),the limiting distribution (6) is invariable under the redefinitionof renormalization transformation.
It is necessary to specify the flow corresponding to Jq (x, t)of the continuous equation,∂∂q(x, t) = −Jq (x, t),∂t∂x

if we are to associate Eq. (7) with autonomous decentralized
control. Jq (x, t), which satisfies this equation, is expressed as
Jq (x, t) = −c x q(x, t) − c σ 2∂q(x, t),∂x(9)where c denotes the rate of temporal evolution of the density distribution and σ 2 denotes the variance of the normal
distribution that is converged on. By the way, the temporal
evolution equation Eq.(8) obtained by the renormalizationtransformation is an example of the Ornstein-Uhlenbeck process [16]. For the reason given above, if we make the local
action of each node follow Eq. (9), the limiting distribution
is expressed as Eq. (6), regardless of the configuration of theinitial distribution.
The next section describes a method for designing autonomous decentralized control schemes that do not dependon the coordinate system.
IV. L OCAL ACTION THEORY BASED ON BACKDIFFUSION - BASED POTENTIAL
It is necessary to decide the coordinate system in the
network, because the drift term −c x q(x, t) of Eq. (9) depends
on x. Here, we compare the flow of original drift term withthe flow for the drift term based on the absolute value of
coordinate x. Figure 3 shows that the direction and the size ofthe drift are invariable in terms of absolute coordinate values.

Fig. 4.
Fig. 3.

back diffusion

Coordinate values and the direction of the drift

Therefore, we replace x, that appears to the drift term forEq. (9), by some function f (x, t), as follows:
Jq (x, t) = −c f (x, t) q(x, t) − c σ 2∂q(x, t).
∂x(10)

The temporal evolution of distribution q(x, t) that corresponds
to this is given by
µ
∂
2
∂
∂
2 ∂
q(x, t) = cf (x, t) + σq(x, t).
(11)
∂t
∂x
∂x2
The introduction of f (x, t) eliminates the need to set spacecoordinates in the network.
As a more intuitive explanation, we consider the following
potential function Φ(x, t) instead of function f (x, t):f (x, t) = −∂Φ(x, t)
.
∂x

(12)

Choosing Φ(x, t) appropriately yields autonomous decentralized control that does not depend on the coordinate system.
In general, in order to decide beforehand the potentialfunction Φ(x, 0) that satisfies some global condition, we must
know the macro-scale information related to the entire system,
not the micro-scale information related to system elements.
This renders problematic autonomous decentralized systems
that operate only with local information. To resolve thisconundrum, we consider how to decide the drift term from
distribution q(x, t). Because the potential function Φ(x, t)
should have the effect to maintain the distribution within therange of the limited space, in opposition to the effect ofdiffusion, Φ(x, t) is decided as the equation,µ∂∂ 2 q(x, t)Φ(x, t + dt) = − q(x, t) − κ0
dt
,
(13)
∂x2
where κ0 > 0. This equation can be expressed as follows:• We let the time progress of the diffusion phenomenon withthe diffusion coefficient κ0 be reversed (back diffusion).
• Next, we reverse the distribution (up and down), and regard
the completed distribution as the potential.
The temporal evolution of the distribution becomes a flat statein the foward direction of time for general diffusion (Fig. 4).
For back diffusion, on the other hand, the temporal evolutionof the distribution goes against the direction of time and theconfiguration of the distribution becomes sharp with time.
Inherent in back diffusion, the phenomenon of a decrease inentropy does not happen in the natural world, but it can berealized easily in the framework of autonomy decentralized
control.
Due to the effect of the drift term, including the potential,
the peak of distribution q(x, t) is emphasized and the distribution shape is sharpened. The effect of the diffusion term,on the other hand, is to flatten the distribution. A structure offinite size can be formed by balancing one effect against theother.
V. AUTONOMOUS DECENTRALIZED CONTROL
CLUSTERING IN AD HOC NETWORKS

This paper applies the above autonomous decentralized
control technologies to realize clustering (of finite size configuration) to ad hoc networks. To simplify the discussion, we
adopt a one-dimensional network model in which the spacecoordinate x corresponds to node ID. Moreover, each node
has a numerical value, q(x, t), that indicates its fitness to be
the cluster head.
R
We assume that the total number q(x, t) dx for numericalvalue q(x, t) of each node x in the network is constant. Eachnode interacts with the adjacent nodes and the numericalvalue is exchanged. As a result, distribution q(x, t) changes
over time. Concrete interaction follows the local action rules(10) described in the previous section, and q(x, t) follows the
temporal evolution Eq. (11). At this time, the peak in the shapeof distribution q(x, t) corresponds to a cluster head, and thebottom corresponds to the boundary of the cluster.
We now explain the concrete process by which q(x, t)changes and how a cluster structure is generated.
The peak of distribution q(x, t) is emphasized and thecluster head candidate emerges, for the effect of the potentialfunction Φ(x, t) which is obtained by using the back-diffusionof the current distribution q(x, t) for the drift term in Eq. (11).
On the other hand, the effect of the diffusion term of temporal
evolution Eq. (11) softens the peak of the distribution q(x, t).
A structure appears by balancing the effect of the drift termagainst that of the diffusion term.
We explain the characteristics of the proposed method by
Fig. 5. Let us consider the initial condition in which there
are two or more cluster-head candidates on the left, andthere is one small-scale cluster-head candidate in the right. In
areas that are over-endowed with cluster-head candidates, thecandidates are integrated autonomously to form one cluster.
When there is no candidate in the surrounding oppositely, it
becomes a cluster head even if it is not a powerful cluster
head candidate. We can invent the structure that adjusts tolocal circumstances by the autonomous decentralized control.

Fig. 5.

Integrating and forming clusters according to local circumstances

The control information that each node sends is only q(x,t).
Each node sends the information to only adjacent nodes, andflooding other than adjacent nodes is unnecessary. Therefore,
there are not a lot of amounts of the control information
forwarded.

Fig. 6. Evaluation results of clustering method according to local circumstances

VI. E VALUATION
A. Evaluation of clustering method according to local circumstancesFirst, we evaluate whether to generate a cluster while
dynamically changing each node role according to local circumstances. The network model used in the simulation is the
one-dimensional model (the number of nodes is 200). The
initial state of distribution q(x, t), which represents cluster
head fitness, is shown in the upper left of Fig. 6. The networkmodel has the feature of torus type and both sides in onedimensional model are connected (the right side of node 200 is
node 0). The initial state exhibits three cluster head candidates
on the left side, and one cluster head candidate on the rightside. The horizontal axis denotes node ID and the vertical axis
denotes the value of distribution q(x, t).
If two cluster head candidates close to each other, theyshould be merged to form one cluster according to local
circumstances and parameter σ 2 . When there is no clusterhead candidate in a neighborhood, a cluster head candidate
should become a cluster head even if it is not necessarily
prime candidate. The results of evaluating the cluster head
fitness distribution from this viewpoint are shown in the upper
right of Fig. 6 and in the lower of this figure. Parameter σ 2
is set to values of 0.1, 0.2 or 0.4 in the graphs, and c = 1,
κ0 = 0.1. In this evaluation, the distribution is prone to form
smooth clusters by setting the operation interval of the drift
term more than the operation interval of the diffusion term.
We set the drift interval τdrift = 10 while the diffusion intervalτdiﬀ is 1. The plots show the results after sufficient time has
past (t = 100). We can see from the results that when σ 2
is small(large), the cluster head candidates in the left side arenot(are) integrated. Note that the cluster head on the right sideis present in all cases.
B. Evaluation of clustering method for the state without
structure
In Fig. 7, similarly, we show the impact of the cluster
structure in the condition where the initial fitness of eachcluster head is given at random. Distribution q(x, t), which
denotes cluster head fitness, is set to a uniform random decimal
number between [0, 1], the upper left of Fig. 7. The evaluation

Fig. 7. Evaluation results of clustering method for the state without structure

results of the cluster head fitness distribution are shown in theupper right of and in the the lower of Fig. 7. Parameter σ 2
is set to values of 0.1, 0.2 or 0.4 in the graphs, and c = 1,
κ0 = 0.1. The drift interval τdrift is 5 and the diffusion intervalτdiﬀ is 1. From these results, we can see that if parameter σ 2 issmall(large), many(few) small(large) clusters are created. Thisis reasonable because increasing parameter σ 2 strengthens
the diffusion effect which promotes cluster integration. The
proposed method forms clusters autonomously where their
number and size are determined by parameter σ 2 .
C. Evaluation of temporal evolution for the number of clusters
Next, we investigate the variation of the number of clusters
for parameter σ 2 that denotes strength of the diffusion effect.
As for the network model, the initial fitness of each cluster isassumed to be at random as well as the previous evaluation.
Parameter σ 2 is set to values of 0.1, 0.2 or 0.4, and c = 1,
κ0 = 0.1 and the drift interval τ = 5. For calculating thenumber of clusters N we add one to N when cluster head
fitness q(x, t) of each node is larger than those of adjacentnodes (right and left nodes). We evaluate the number ofclusters N ten times by changing the random seed. In Fig.8,
the horizontal axes denote the simulation time and the verticalaxes denote the mean number of clusters Nmean for ten results.
The three lines in this figure show the results for the cases
where parameter σ 2 = 0.1, 0.2, and 0.4. We can see in Fig.8

ｎFig. 8.

Temporal variation of the mean number of clusters for σ 2

back-diffusion potential; it forms the basis of our new clustering method for ad hoc networks. Though each node acts
autonomously based only on local information, the decisionmaking of each node leads to meaningful cluster formation
appropriate for overall network conditions. Evaluations clarified that the proposed method can realize a cluster structure
autonomously in response to local circumstances in ad hoc
networks. We can see that when there is no cluster-headcandidate in the neighborhood, the mobile node itself stands
as a candidate and becomes a cluster head, and on the other
hand, when there are two or more cluster-head candidates,
the node corresponding to metric is selected as cluster-headamong them.
ACKNOWLEDGMENT
This work was partly supported by Grant-in-Aid for Young
Scientists (B) (21700086) from the Japan Society for the Promotion of Science, and Early-concept Grants for Exploratory
Research on New-generation Network from NICT.
R EFERENCES

Fig. 9.

Temporal variation of the mean number of clusters for τdriftthat Nmean falls sharply at the start (from time t = 0 to t = 10)
for every σ 2 , and the decrease rate of Nmean becomes small
with time. Finally, it settles down in an almost constant value
and becomes stable. In the case of σ 2 = 0.4, larger clusters
are created and the number of clusters decreases, because thelarger σ 2 is, the higher the diffusion effect is.
Next, we evaluate the mean number of clusters Nmean forthe drift interval. The small drift interval (the high frequency ofthe drift operation) enlarges the effect of “the peak emphasis”
and “the sharpness of the distribution.” Three graphs in fig.9
show the results when σ 2 = 0.1, 0.2, and 0.4. The lines in thisfigure show the results for the cases where the drift intervalτdrift = 5, 10, 15, and 20.
In the case where σ 2 = 0.1，Nmean is large when the drift
interval is small (τdrift = 5). When the drift interval is larger,
on the other hand, Nmean is small and it doesn’t make muchdifference among three results (τdrift = 10, 15, and 20) in
the lower of this figure. Then, the drift interval τdrift doesn’t
influence much the mean number of clusters in the case whereσ 2 = 0.2 and 0.4, and the values of Nmean are almost the same
regardless of τdrift . To create a lot of clusters with the small
cluster size, we have to set the parameter σ 2 and τdrift to besmall. On the other hand, larger clusters are created by usinglarger σ 2 .
VII. C ONCLUSION
In this paper, we introduce an autonomous decentralized
control approach that uses renormalization transformation and[1] http://www.ietf.org/html.charters/manet-charter.html[2] E. Royer and C.-K. Toh, “A review of current routing protocols for ad
hoc wireless networks,” IEEE Personal Communications, pp. 46-55, April
1999.
[3] R. Ogier, F. Templin and M. Lewis, “Topology dissemination basedon reverse-path forwarding (TBRPF),” Request for Comments 3684,
February 2004.
[4] T. Clausen and P. Jacquet, “Optimized link state routing protocol(OLSR),” Request for Comments 3626, October 2003.
[5] C.E. Perkins, E.M. Royer and S. Das, “Ad hoc on-demand distance vector(AODV) routing,” Request for Comments 3561, July 2003.
[6] D. Johnson, Y. Hu and D. Maltz, “The dynamic source routing protocol(DSR) for mobile ad hoc networks for IPv4,” Request for Comments
4728, February 2007.
[7] S. Basagni, “Distributed clustering for ad hoc networks,” Proceedingof International Symposium on Parallel Architectures, Algorithms andNetworks, pp. 310–315, 1999.
[8] A.D. Amis, R. Prakash，T.H.P. Vuong and D. Huynh, “Max-min Dcluster formation in wireless ad hoc networks,” Proceeding IEEE INFOCOM 2000.
[9] T. Ohta，et al., “An adaptive multihop clustering scheme for ad hoc
networks with high mobility,” IEICE Transactions on Fundamentals vol.
E86-A no. 7, pp. 1689-1697, 2003.
[10] N. Malpani, J.L. Welch and N. Vaidya, “Leader election algorithms formobile ad hoc networks,” Proceedings of the 4th international workshop
on Discrete algorithms and methods for mobile computing and communications, pp. 96–103, August 2000.
[11] C. Johnen and L. Nguyen, “Self-stabilizing weight-based clustering
algorithm for ad hoc sensor networks,” 2nd International Workshop on
Algorithmic Aspects of Wireless Sensor Network AlgoSensors 2006,
Springer LNCS, vol. 4240, pp. 83–94, 2006.
[12] J.Y. Yu and P.H.J. Chong, “A survey of clustering schemes for mobile
ad hoc networks,” IEEE Communication Survey and Tutorial, vol. 7, no.
1, pp. 32–48, 2005.
[13] M. Aida and C. Takano, “Principle of autonomous decentralized flow
control and layered structure of network control with respect to timescales,” Supplement of the ISADS 2003 Conference Fast Abstracts, pp.
3–4, April 2003.
[14] C. Takano and M. Aida, “Stability and adaptability of autonomous
decentralized flow control in high-speed networks,” IEICE Transactions
on Communications, vol. E86-B, no. 10, pp. 2882–2890, 2003.
[15] C. Takano and M. Aida, “Diffusion-type autonomous decentralized flow
control for multiple flows,” IEICE Transactions on Communications, vol.
E90-B, no. 1, pp. 21–30, January 2007.
[16] G.E. Uhlenbeck and L.S. Ornstein, “On the theory of Brownian Motion,”
Physical Review, vol. 36, pp. 823–841, September 1930.