Journal of Communications Vol. 9, No. 10, October 2014

Hierarchical Performance Evaluation Method for Describing theInteractions between Networks and Users
Sota Hatakeyama1, Chisa Takano2, and Masaki Aida1
1

2

Graduate School of System Design, Tokyo Metropolitan University, Tokyo 191-0065, Japan
Graduate School of Information Sciences, Hiroshima City University, Hiroshima 731-3194, Japan
Email: sota-hatakeyama@sd.tmu.ac.jp; takano@hiroshima-cu.ac.jp; aida@tmu.ac.jp

reservation. If a request is rejected, the correspondinguser might generate retry traffic.
 Retry traffic generated due to a shortage of callprocessing resources.
A shortage of processing resources causes the responsetime of the call-processing system to increase.
Psychological factors will induce impatient users to retry
their requests. Because prior requests are not cancelled,
one user can create duplicate requests.
A variety of queuing models with retry traffic havebeen well studied [1], [2]. Conventionally, therelationship between the link bandwidth and the input
rate of call setup requests has been investigated using the
M/G/s/s model. Let us consider the M/G/s/s based retry
model called the multi-server model in [1]. The
expression M/G/s/s represents a model in which service(call-setup) requests arise in a Poisson manner, enter thesystem, receive service from one of s servers, and thenleave the system, and in which service requests are
discarded if all s servers are busy. An M/G/s/s retrial
queue model represents an M/G/s/s model in which
discarded service requests are stored in a retry queue andre-enter the system after a certain elapsed time
determined by an exponential distribution. This model isstable if the length of the retry queue does not diverge[2]; its stability condition is known to be

Abstract—Information networks are an important social
infrastructure, and we should ensure their stable and sustainable
operation. Since retry traffic greatly impacts network stability,
we should, in particular, consider retry traffic when designingand controlling communication network systems. We
previously proposed a hierarchical method that uses the quasistatic approach to evaluate system performance in the presenceof retry traffic. This approach is based on the fact that system
response times are much shorter than the users’ perceivable
time-scales. For the sake of simplicity, previous work
considered a specific model that assumed that all users’
behaviors were synchronized with respect to a certain timeinterval even though users behave independently. This paper
introduces a performance evaluation method, based on theframework of the quasi-static approach, for networks in which
retry traffic is generated by users autonomously.
Index Terms—Retry traffic, IP telephony, system stability,
quasi-static approachI.

INTRODUCTION

Signaling systems play a crucial role in supporting all
virtual channel (VC) communication services, including
IP telephony and conventional telephone networks. The
congestion in the current Internet is caused byoverloading of not only the communications links but
also the processing resources in the signaling systems.
Recently, problems with commercial IP telephonysystems have been reported in Japan. One of the keycauses of such problems is overloading of the signaling
functions provided by the call-processing system. In
particular, the retry traffic generated by users has aserious negative impact on system stability. Here, retrymeans multiple user attempts to set up a connection.
In general, retries are generated by two different
factors as described below.
 Retry traffic generated due to a shortage of link
bandwidth.
This shortage (e.g., resource for VCs) triggers rejectionof requests for connection setup or for bandwidth(1)
where λ0 is the arrival rate of primary service requests,
excluding retry requests, per unit time, and 1/η is theaverage service time. The above model incorporates retry
traffic that arises due to a resource shortage in the link
bandwidth, but it does not consider retry traffic that arises
due to resource shortages in the call-processing system.
However, such traffic is expected to serious impact
system stability. This is because that the prior requests
are not cancelled when impatient users generate retry
traffic, duplicate requests will exist in the system. In thispaper, we focus on a call-processing system that isconfigured using a VC-based communication service
model (e.g., IP telephony), and consider the stability ofthe system under the impact of retries. We consider how
to evaluate the probability that the system will become
unstable.
Our previous research focused on an IP telephonysystem, and discussed the properties of retry traffic

Manuscript received July 24, 2014; revised October 16, 2014.
This work was supported by a Grant-in-Aid for Scientific Research(B) No. 26280032 (2014-2016) from the Japan Society for the
Promotion of Science.
Corresponding author email: aida@tmu.ac.jp
doi:10.12720/jcm.9.10.737-744©2014 Engineering and Technology Publishing737

Journal of Communications Vol. 9, No. 10, October 2014

caused by resource shortages in the link bandwidth and/or
the call-processing system [3]–[6]. To understand the
dynamics of retries, we should understand theinteractions of the users and the system. A fundamental
approach for describing interactions is the decompositionof timescales. However, the decomposition models inearlier papers were primitive. In [3], we introduced thequasi-static approach to describe fluctuations in traffic.
This approach assumes that state transitions of the systemoccur on a timescale significantly (infinitesimally) shorter
than the timescale of human perception. This means thatthe system might as well be working at infinite speed.
This assumption cannot take account of fluctuations intraffic. In [4]–[6], it is assumed that all users’ behaviors
are synchronized with respect to a certain time interval,
but this assumption is unrealistic.
This paper focuses on retry traffic caused by resource
shortages in the call-processing system and introduces aretry traffic model in which users’ behaviors are mutuallyindependent. That is, we develop a new realistic model ofretry traffic, and clarify that system stability can beevaluated by using the conventional quasi-static approach.
First, we briefly explain time scale decomposition for
describing the retry traffic in Sec. II. Next, we show theconventional quasi-static approach as the conventional
retry traffic model and its issues, in Sec. III. Then, we
introduce a new retry traffic model that can describe
users’ behaviors properly as an extension of the quasistatic approach in Sec. IV. The validity of the proposedmodel is verified through an evaluation of systemstability in Sec. V. Finally, we conclude our discussion in
Sec. VI.

a new call setup. Such retry traffic will be generated
without canceling the existing call setup requests.
Let us consider how retry traffic arises. It is natural toassume that the intensity of retry traffic depends on thelength of the queue in the M/M/1 model. It is logical toconsider that retry traffic dependent on the number ofqueued contents in the call-processing system. If weassume that each call setup request waiting in the callprocessing system generates retry traffic at a certain rate ε,then a diagram depicting the state transition rate withrespect to the number of requests in the M/M/1 model
will be like that shown in Fig. 2. In this figure, λ0 is theprimary arrival rate of call setup requests, excluding retry
traffic, per unit time, while 1/μ is the average service timeof the call-processing system. For this system to have asteady-state probability, the infinite sum on the righthand side of the following equation must exist.

Therefore, if ε > 0, the system is unstable. Since anincrease in retry traffic does not result in the divergenceof the waiting time of an actual call-processing system
under normal operating conditions, we can conclude thata model in which retry traffic is generated in proportion
to the number of call setup requests in the call-processing
system is not realistic.

Fig. 2. State transition diagram for the case where retry traffic is
generated in proportion to the current number of call setup requests inthe call-processing system

Let us identify what is inappropriate in the above
model. In general, state transitions of the call-processing
system occur on a timescale much shorter than humans
can perceive. It is natural to assume that the intensity of
the retry traffic is not proportional to the number of call
setup requests in M/M/1 at the present time, but
proportional to the average number of call setup requests
in M/M/1 on a longer timescale. The timescale should belong enough for humans to perceive. Let T be theminimum timescale perceptible to humans. We assume
that the call-processing system is in a steady state on atimescale smaller than T, and that retry traffic affects thesystem on a timescale larger than T. In the following, wesummarize the assumptions of such a quasi-static state forthe generation of retry traffic.
 The system can be assumed to be in a steady state ona timescale smaller than T.
 Changes in the system are observed at discrete times
occurring at an interval of T.
 Retry traffic generated from users at a certain discretetime, t = k, is determined by the steady-state
probability of the system at t = k − 1.

Fig. 1. System model

II. TIME SCALE DECOMPOSITION FOR DESCRIBINGRETRY TRAFFIC
This section describes time scale decomposition thatcan describe a retry traffic model that can take account of
resource shortages in call-processing systems. The
decomposition gives a foundation to the quasi-static
approach shown in the next section. The significant features of the decomposition include macroscopic systembehavior on a human perceptible timescale.
The call-processing system model should describe thesystem behavior related to call setup at a server, and forthis we use the M/M/1-based model. The M/M/1-basedmodel is a combination of a simple M/M/1 model and a
retry traffic model that includes resource shortages in thecall-processing system, with retry traffic generated by
impatient users (Fig. 1). That is, some of the users who
have been kept waiting for execution of call setup request©2014 Engineering and Technology Publishing738

Journal of Communications Vol. 9, No. 10, October 2014

where 1/μ is the average service time of the callprocessing server, and ε is a positive constant indicating
the intensity of the retry traffic generated due to a callprocessing resource shortage.
Next, let us consider the stability of the system. We
assume that the requirement for the system to be stable isthat traffic, including retry traffic, does not diverge aftersufficient elapsed time. Namely,

Note that the condition that the steady state is achieved
on a finite timescale T implies that transitions of thesystem occur on a timescale much (infinitesimally)
shorter than the human perceptible timescale.
In other words, the system works at essentially infinite
speed.
The value of T must satisfy the following
requirements:
 T must be a timescale on which humans can actuallyperceive an increase in the waiting time for their callrequests.
 T must be sufficiently longer than the timescale forwhich it is possible to assume that the system is in a
steady state on a timescale smaller than T.
It is well known that the tolerable waiting time todisplay a website is about 8 seconds; a longer wait
exceeds the typical user’s patience Many webpages aredesigned so as to satisfy this so-called 8-second rule [7].
Measurements of waiting time have verified that a user’s
interest fades after 10 seconds or so [8]. In our case,however, the tolerable waiting time must be smaller thanthat for webpages. Reference [9] classifies the key timeintervals according to human perception as follows: Delay of 0.1 second is perceived as instantaneousaccess.
 Delay of 1.0 second is the limit for a user’s thought
flow to remain uninterrupted.
 Delay of 10 seconds is the limit for keeping a user’s
attention/focus on the dialogue.
The last category coincides fairly well with the 8second rule for websites in terms of both the duration andthe explanation. Since the generation of retry traffic
seems to correspond to the second category, one second
is a reasonable value for T.
Changes in the state of the system are examined at
discrete intervals T. The retry traffic at time t = k isdetermined by the steady-state probability of the systemat time t = k −1 (see Fig. 3).

(3)This can be verified by defining the functions of λ, f(λ)
and g(λ) as follows, and determining if they intersect.
(4)(5)
Since λ0 > 0, f(0) > g(0). The relationship between f(λ)
and g(λ) in some typical cases is shown in Fig. 4. If the
input traffic at a certain time is λ, the input traffic at thenext time instant becomes {g-1 ◦ f }(λ), followed by { g-1 ◦f }2 (λ), etc. In general, the input traffic after the passage
of n unit times becomes { g-1 ◦ f }k (λ). As shown in theleft chart in Fig. 4, if f(λ) and g(λ) do not intersect, {g-1 ◦f}k(λ) → ∞ (k→ ∞ ). If, on the other hand, the twointersect in the manner shown in the right chart of Fig. 4,
then the system is stable for all λ to the left of the rightmost intersection.

Fig. 4. System stability possible if f(λ) and g(λ) intersect.

III. QUASI-STATIC APPROACH: CONCEPT AND ISSUESIn the previous sections, we assumed that a steady state
was achieved on finite timescale T. This implied that thesystem works at an infinitely high speed. However, sincereal systems have finite speeds, the steady state is neverachieved in a finite time interval. In addition,synchronization of users’ behaviors with respect tointerval T, shown in Fig. 3, is unrealistic. The quasi-static
approach tries to avoid these problems.
A. Concept of Quasi-Static Approach
In this subsection, we briefly explain the concept of thequasi-static approach. Let us consider how to examine
system stability to input traffic under the condition thatthe system works at a finite speed. First, we clarify
certain problems in the traditional approaches to
examining stability. Then, we clarify the concept of thequasi-static approach can avoid the problems.

Fig. 3. Temporal evolution of the input rate in discrete interval model.

Let λk be the input rate, including retry traffic, at timek. We assume that the input rate at time k + 1 is
(2)©2014 Engineering and Technology Publishing739

Journal of Communications Vol. 9, No. 10, October 2014

Let n be the number of call setup requests arising ininterval T. If retry traffic from the call-processing system
is proportional to the average number of call requests
waiting in M/M/1 at time k (in interval T), the second
term on the right-hand side of (2) is replaced withbehavior at infinite speed, and considers the gap between
that behavior and the system behavior at finite speed as
fluctuations (see Fig. 5). In other words, the change factor
in the state of the system is decomposed into two parts,
one being the deterministic change factor which is
described by system behavior at infinite system speed andthe other factors being described as stochastic
fluctuations. This approach is effective for a model with ascale that cannot be realistically handled by both theMarkov model and the simulation technique.

(6)kwhere Qi denotes the number of call requests waiting in
the M/M/1 system immediately before the i-th request
arrives. If the system works at infinite speed, that is, n →∞, we have

B. Conventional Quasi-Static Approach
We define input rate Λ(t;T) at time t as
(8)

(7)

where <QT>t is a kind of the average number of callrequests in the call-processing M/M/1 system during

Fig. 5 shows the appropriate approaches with respect
to the system speed. The horizontal axis plots the system
speed from slow to infinitely high. Let us consider thecase where the system works at a very low speed, forexample n = 1. This case corresponds to the situation inwhich a human can detect the systems present state,
meaning that the human’s perception is exceptionallysensitive or the system speed is very slow. In this case,the next value of input traffic depends only on the latest
kvalue of Q1 . That is, the system can be described by theMarkov model of Fig. 2, which corresponds to slow in
Fig. 5.

interval [t − T, t] and ε is a positive constant. This is anextension of (2) written using continuous time t. The
input rate Λ(t;T) is given by the sum of the rate λ0 forprimary traffic and the rate for retry traffic, which is
proportional to the average number of call requests.
In the conventional quasi-static approach, <QT>t isdefined as the average during interval T immediately
before the present time t, that is,(9)
where Q(t) is the number of call requests in the system attime t. This model makes the rate of retry traffic rate attime t proportional to the average number of call-setup
requests in [t − T , t]. If users were able to react to thevariation of the system state immediately, T → 0 then(10)
This corresponds to the system model described by thestate transition diagram shown in Fig. 2.
As a preliminary for describing the quasi-static
approach, we show the Langevin equation and the
Fokker- Planck equation [10]. Let X(t) be a randomvariable. Let us investigate the temporal evolution of X(t)as follows. We assume that the temporal evolution of X(t)is given by

Fig. 5. Concept of quasi-static approach.

In general, if n > 1, we can use a Markov model with
an n-dimensional state space that is constructed by thekkkpast n states { Q1 , Q2 , ..., Qn }. That is, it corresponds
to medium in Fig. 5. However, if the system works athigh speed, n ≫ 1, the state space explodes and becomes
intractable.
Next, we consider the applicability of simulation
techniques. If we require the probability of the system be-6
coming unstable, for example, to be less than 10 in thelong-term operation of an IP telephony system, more than
68910 (usually 10 to 10 ) runs of the corresponding longterm simulation are required. Such large-scale
simulations are not realistic.
The quasi-static approach proceeds follows. In order toanalyze the stability of the high (but finite) speed system,
the quasi-static approach first examines the sys- tem©2014 Engineering and Technology Publishing(11)
where g1(X) and g2(X) are functions of X(t), and W(t) is
the Wiener process. The first and second terms on theright-hand side of (11) are deterministic and stochastic
parts of the temporal evolution, respectively.
Equivalently, this equation can be written in thefollowing form
(12)
where ξ(t) is Gaussian white noise such that E[ξ(t)] = 0740

Journal of Communications Vol. 9, No. 10, October 2014and E [ξ(t)ξ(t’)] = δ(t – t’). This equation is called theLangevin equation. Let the probability density function of
X (t) be p(x, t). Then, the temporal evolution function ofp(x, t) is given by the following Fokker-Planck equation,

This is composed of the difference between increment
X(t+dt, dt) and decrement X(t− T, dt) (see Fig. 7).
Here, since the input traffic follows a Poisson process,the variance of the input traffic is equal to the input rate.
In addition, since we consider large-scale systems manycall requests, the Poisson distribution can be approximated by a Gauss distribution with no correlation.
Therefore, by the quasi-static approach, the number ofarriving customers can be expressed as(13)
The first and second terms of (13) are called the drift
term and the diffusion term, respectively. The drift term
describes the deterministic motion of p(x,t) and the
diffusion term describes the fluctuation around the drift
motion. The values of g1(X) and g2(X) govern thestrength of drift and diffusion motions, and they denotes
variation of the mean value and its standard deviation,
respectively. These mechanisms are easily recognizedthrough the following explanation. Let us introduce
potential function U(x) of the drift motion as

(16)

(17)

(14)
In the form of Langevin equation, we have
Fig. 6 shows an example of the potential function. The
drift motion occurs toward the direction that the value ofthe potential decreases. The minimum point and the wallof the potential correspond to the left and rightintersections in the right panel of Fig. 4, respectively. In
addition to the drift motion described as Fig. 6, the quasistatic approach can describe some fluctuations by the
diffusion term.

(18)

Fig. 6. Example of potential function and the corresponding drift motion.

Next, we consider the temporal evolution of retry
traffic by the quasi-static approach. We define the actual
number of customers arriving during [t − T, t] as X(t,T).
Under the limit of higher system speed, lim T→∞ X(t,T)/T =Λ(t;T) a.s. In general, X(t,T) is not stationary but thevariation occurs very slowly. Here, we define theinfinitesimal variation of X(t,T) as

IV. EXTENSION OF QUASI-STATIC APPROACH
A. Randomness of Retry Users
In order to avoid the fact that an individual user
depends on a specific periodic interval of T, we assume
that the call-setup requests (customers) waiting forservice by a M/M/1 call-processing system generate retry
traffic after randomized intervals. Since we assume theinput traffic obeys a Poisson process, the randomized
interval follows an exponential distribution.
Each customer in the system at an arbitrary point oftime retries the request at a constant rate ε per unit time.
The randomized time interval, which is from the arbitrary(15)

Fig. 7. Variation of input rate

©2014 Engineering and Technology PublishingThis equation has been verified by comparison against
simulation results in a certain situation that all the users
are synchronized as described in Fig. 3 [6]. The temporal
evolution equation (18) has been improved from thedeterministic model (2) in following two ways.
 (18) is stochastic model and it takes fluctuations into
consideration.
 The time parameter is improved from discretevariable k to continuous variable t.
However, synchronization of users’ behaviors is notcompletely avoided. The users’ behaviors in the timeinterval of T affect only the next time interval. It isnatural that the behavior of individual users should be
independent of the specific periodic timing of T
introduced by us. To avoid this problem, we introduce anextension of the conventional quasi-static approach in thenext section.

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Journal of Communications Vol. 9, No. 10, October 2014

chosen time point to the time of the corresponding retry
traffic arrival, obeys an exponential distribution withmean of T. Fig. 8 illustrates the above retry traffic model.
Let Q(s) be the number of customers in the system at times. The contribution to the input rate of retry traffic atcurrent time t (s < t) from Q(s) at the point of the past s isduring a certain time interval and derive the number of
input traffic per unit time. That is, it is impossible toknow the input traffic rate of (19) by measurement at any
one moment. Let X(t,T) be the number of actual input
traffic generated in time period [t − T, t] whose length isT. If T → ∞ , the observed input traffic rate X(t)/T
approaches the theoretical value,(εQ(s)/T)exp((t− s)/T)dt. Then input rate at time t isexpressed as

(21)
(19)

However, as seen in Fig. 5, (21) is not correct for a
finite T and we should additionally consider fluctuations.
The infinitesimal variation of X(t,T) is expressed as

This is an example of (8) and an extension of (9) [11].
The second term on the right hand side of (19) denotes
retry traffic rate. This is the exponentially weighted
moving average of the number of customers of the past.
Figure 9 shows examples of the temporal evolution ofcustomers and the weight. The quantity of this weights
decays exponentially by time constant T. Similar to (10),if users were able to react to variation of the system state
immediately, T → 0 then(22)
After the infinitesimal time dt has elapsed, X isincreased by A(t;dt) and is decreased by D(t;dt). Because
tran- sitions in system state are attributed to user behavior,
stochastic process X varies slowly at rate T. In addition,
since we consider large-scale system (λ0≫1) and there
are a lot of call requests, the input Poisson process can beapproximated by a Gauss distribution with no correlation.
Therefore, the infinitesimal variations A and D are
expressed as(20)This means the new model (19) is also an extension of
the Markov model described in Fig. 2.

(23)

(24)
Fig. 8. Relationship between the number of requests Q(s) at s and theretry at t caused by Q(s)Thus, we can obtain(25)

In the form of Langevin equation, (25) is expressed as

Fig. 9. Weighted average.

(26)
B. Temporal Evolution Equation for the Extended
Quasi-Static Approach
The extended input traffic model (19) derived in theprevious subsection fundamentally differs from theconventional model (9). In the extended model of (19),individual users autonomously generate retry traffic
where generation is independent of any specific interval,
T. In this subsection, we consider the temporal evolution
equation of the input traffic based on the extended model(19).
In order to measure the rate of input traffic actually
generated, we need to count the input traffic generated©2014 Engineering and Technology Publishingand the result, in the form of the Fokker-Planck equation,
is

(27)

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Journal of Communications Vol. 9, No. 10, October 2014

parameters are as follows: λ0 = 500, μ = 1000, ε = 1.0, T= 1.0. Fig. 10 shows the potential function corresponding to the above condition. The minimal point of

The above results are the same as that of the conventional
quasi-static approach (18).
Note that these input models are very different
although their forms of temporal evolution are the same.
The reason why we get the same equations is that theprocedure of measuring the input rate is the same as theprocedure of moving average of the input rate.
Fortunately, we can use the same temporal evolution
equation even though the input traffic model becomescomplex.

the potential function exists at X ≃ 500. We set the initial
distribution p(x,0) as the δ-function at X = 600. Figures
14 and 15 show the cumulative distribution function ofX(t,T) at t = 1 and t = 5, respectively. We can recognizethat the quasi-static approach can properly describe thechange of the distribution. Through the above
experiments, we can confirm that the quasi-static
approach can describe the behavior of the systemdeviated from the stable point by perturbation.

V. SIMULATION RESULTS
This subsection verifies the validity of (26) and (27) bycomparison with Monte Carlo simulation. In thissimulation model, each user generates retry traffic
independently of the specific periodic interval of T. Thisis a major difference from the evaluation of theconventional quasi- static approach shown in [6].
Fig. 10 shows the potential function with parameters of
λ0 = 800, μ = 1000, ε = 0.5, T= 1.0. The horizontal axis
denotes X(t,T) and vertical axis denotes the potentialfunction U(X). As the initial condition, the p(x,0) is a δfunction at the minimal point of the potential function.
The wall of the potential function is an absorption state(i.e., once it reaches there, it can never re- turn). Fig. 11
and Fig. 12 show the cumulative distribution function ofX(t, T) at t = 1 and t = 50, respectively. We can recognizethat the quasi-static approach can properly describe thebehavior of input traffic.

Fig. 12. Distribution of the input traffic at t = 50.0

Fig. 13. Potential function

Fig. 10. Potential function.

Fig. 14. Distribution of the input traffic at t =1.0

Fig. 11. Distribution of the input traffic at t = 1.0

Next, we investigate the situation in which the initialcondition is not at the minimum point. As the initialcondition, we set the delta-function to a position shiftedfrom the minimum point of the potential function. The©2014 Engineering and Technology Publishing

Fig. 15. Distribution of the input traffic at t =5.0

VI. CONCLUSIONS
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Journal of Communications Vol. 9, No. 10, October 2014[7] Z. Re-search, “The need for speed II,” Zona Market Bulletin, no. 5,

In this paper, we have shown how to properly accountfor retry traffic generated by interaction between users
and networks. To model the interaction properly, ourapproach describes the retry traffic randomly generated
by individual users. The model can characterize thenatural behavior of users unlike the conventional model
which uses a specific periodic interval.
Fortunately, although our proposed model is more
complex than the conventional model, the temporal
evolution equation of the input traffic can be written in
the same form. Therefore, our proposed model does notneed to introduce additional complex procedures.
Experiments support the fact that our proposed model
properly de- scribes the behavior of retry traffic.
This paper focused on retry traffic generated by inadequate resources in the call-processing system. Since
retry probably significantly impacts various communication services, we need to investigate retry traffic models
for the various services. In these processes, we expectthat the concept of time scale decomposition and thequasi- static approach will prove to be useful tools.

pp. 1–9, April 2001.

[8] Fiona Fui-Hoon Nah, “A study on tolerable waiting time: Howlong are Web users willing to wait?” Behaviour & Information
Technology, vol. 23, no. 3, pp. 153–163, 2004.
[9] J. Nielsen, “Response times: The three important limits,” Usability
Engineering, J. Nielsen, Ed., ch.5, Academic Press, 1993.
[10] N. G. van Kampen, Stochastic Processes in Physics andChemistry, Elsevier, 1992.
[11] M. Aida, “Using a renormalization group to create ideal
hierarchical network architecture with time scale depen- dency,”
IEICE Transactions on Communications, vol. E95-B, no. 5, pp.
1488–1500, 2012.
Sota Hatakeyama received the B.E. degreefrom Tokyo Metropolitan University, Japan,
in 2011. He is currently a graduate student
working toward the M.E. degree at theGraduate School of System Design, Tokyo
Metropolitan University. His primary interests
include ad hoc networks.

Chisa Takano received the B.E. degree in
Telecommunication Engineering from Osaka
University, Japan, in 2000, and received the
Ph.D. in Telecommuni-cations Engineeringfrom Tokyo Metropolitan University, Japan,
in 2008. In 2000, she joined the Traffic
Research Center, NTT Advanced Technology
Corporation (NTT-AT). 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 computer networks anddistributed systems. She received the IEICE’s Young Researchers’Award in 2003.

ACKNOWLEDGMENT
This work was supported by a Grant-in-Aid for Scientific Research (B) No. 26280032 (2014–2016) from theJapan Society for the Promotion of Science.
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©2014 Engineering and Technology Publishing744

Masaki Aida received his B.S. degree inPhysics and M.S. degree in Atomic Physics
from St. Paul’s University, Tokyo, Japan, in
1987 and 1989, respectively, and the Ph.D. in
Telecommunications Engineering from the
University of Tokyo, Japan, in 1999. In April
1989, he joined NTT Laboratories. 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 School of System Design, Tokyo Metropolitan University
since April 2007. His current interests include traffic issues in computercommunication networks. He received the Best Tutorial Paper Award of
IEICE Communications Society in 2013. He is a member of the IEEE,the IEICE, and the Operations Research Society of Japan.