社団法人 電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS

信学技報
TECHNICAL REPORT OF IEICE.

トラヒックの長期変動から現在のトラヒックマトリクスを推定する手法
大下

裕一†

宮村

崇††

荒川 伸一†††

大木英司††††

塩本公平††

村田 正幸†††† 大阪大学 大学院経済学研究科††† 大阪大学 大学院情報科学研究科†† 日本電信電話株式会社 ネットワークサービスシステム研究所†††† 電気通信大学 情報通信工学科
E-mail: †y-ohsita@econ.osaka-u.ac.jp, ††{miyamura.takashi,kohei.shiomoto}@lab.ntt.co.jp,†††{arakawa,murata}@ist.osaka-u.ac.jp, ††††oki@ice.uec.ac.jp
あらまし 現在のトラヒックマトリクスを得ることは、トラヒックエンジニアリングにとって必須である。しかしながら、トラヒックマトリクスを直接観測するのは難しく、推定する手法も提案されているものの、現実のネットワー
クでは、推定にもちいらたモデルが合致しない場合もあり、正確に推定することができない。本論文では、トラヒッ
クエンジニアリングによって引き起こされた経路変更を用いて、現在のトラヒックマトリクスを正確に推定する手法を提案する。提案手法では、まず、最近 M 回の観測によって得られたリンク負荷を元に、長期変動の推定を行う。そして、現在のリンク負荷に合うように、推定された長期変動に補正を加える。さらに、推定された長期変動が、現在のトラヒックに合致しなくなった場合には、その変化を検出し、合致しなくなった原因となるトラヒックに関する過去の情報を除去した上で、長期変動の推定を行いなおす。本稿では、シミュレーションにより、提案手法の評価を行い、トラヒックが変化した場合であっても、正確にトラヒックマトリクスを推定可能であることを示す。
キーワード

トラヒックエンジニアリング, GMPLS, トラヒックマトリクス推定

Estimating current traffic matrices accurately
by using long-term variations information
Yuichi OHSITA† , Takashi MIYAMURA†† , Shin’ichi ARAKAWA††† , Eiji OKI†††† , Kohei SHIOMOTO†† ,and Masayuki MURATA†††† Graduate School of Economics, Osaka University††† Graduate School of Information Science and Technology, Osaka University†† NTT Network Service Systems Laboratories†††† Department of Information and Communication Engineering, The University of Electro-Communications
E-mail: †y-ohsita@econ.osaka-u.ac.jp, ††{miyamura.takashi,kohei.shiomoto}@lab.ntt.co.jp,†††{arakawa,murata}@ist.osaka-u.ac.jp, ††††oki@ice.uec.ac.jp
Abstract Obtaining current traffic matrices is essential to traffic engineering (TE) methods. However, it is difficult to monitor traffic matrices directly. The existing estimation methods also cannot estimate them accurately. In this paper, we propose
a method for estimating current traffic matrices by using route changes introduced by a TE method. In this method, we firstestimate the long-term variations of traffic by using the link loads monitored the last M times. Then, we adjust the estimatedlong-term variations so as to fit the current link loads. In addition, when the traffic variation trends change and the estimatedlong-term variations cannot match the current traffic, our method detects mismatches. Then, so as to capture the current traffic
variations, the method re-estimates the long-term variations after removing information about the end-to-end traffic causing
the mismatches. For this paper, we evaluated our method through simulation. The results show that our method can estimate
current traffic matrices accurately even when some end-to-end traffic changes suddenly.
Key words Traffic engineering, GMPLS, Traffic matrix estimation—1—

Traffic Volume between OD pair

1. Introduction
Obtaining current traffic matrices accurately is essential to trafficengineering (TE) methods [1, 2]. By using the current traffic matrices, TE methods configure routes on a network so as to fit thecurrent traffic.
Because it is difficult to monitor traffic matrices directly, several(1) Estimate long-term
variations of traffic(2) Adjust estimated longterm variations so as to fit
current link loads(3) Detect mismatch between
estimated long-term variations
and current trafficTimemethods for estimating traffic matrices from limited information

Fig. 1 Overview of proposed method

have been proposed [3–5]. In such methods, an entire traffic matrix is estimated using link loads that can be collected much morematrices, by monitoring link loads while some routes are changed.

easily than by directly monitoring end-to-end traffic. Because the

However, the current traffic can differ from the initial traffic mon-link load is the sum of the traffic using a link, we haveX(n) = A(n)T (n),itored before the first route change. Therefore, we need to consider(1)long-term variations. By using the link loads monitored the last M
times, our method estimates the long-term variations of traffic in-where X(n) is a matrix indicating the amount of traffic on each linkstead of estimating the current traffic matrices directly. Then, we

at time n, T (n) is the traffic matrix at time n, and A(n) is the rout-obtain the current traffic matrices by adjusting the estimated long-

ing matrix. However, because the number of links is much smaller

term variations so as to fit the current link loads.

than the number of elements of the traffic matrix, estimated trafficmatrices include estimation errors.

In addition, when the traffic variation trends change, the changes
may cause significant estimation errors if we also use information

Recently, several methods increasing the accuracy of estimation

obtained before the changes, since this information can be very dif-by using additional measurements have been proposed [4, 5]. These

ferent from the current traffic. Therefore, in our method, we checkmethods obtain the additional information by changing the routing

whether the estimated long-term variations match the current link

matrices. To estimate traffic matrices by using the additional infor-

loads. Then, if we detect a mismatch between the estimated long-mation obtained at the different times, we need to consider the timeterm variations and the current link loads, we re-estimate the long-

variations of traffic. Thus, Ref. [5] proposes a method for mod-term variations after removing the traffic information causing theeling traffic variations by using periodic functions and estimatesmismatch, so as to follow the current variations of traffic.

these functions’ parameters. However, when traffic changes unpre-

Fig. 1 shows an overview of the proposed estimation method. Our

dictably, the traffic matrices estimated by this approach cannot fit

method estimates the traffic matrix through the following steps.

the current traffic matrices since it can only estimate the average

Step 1 Estimate the long-term variations of the traffic matrices by

variations of traffic for a period of a day by monitoring link loads

using the link loads monitored the last M times.

for several days.

Step 2 Obtain estimation results of the current traffic matrix by

Therefore, in this paper, we propose a new estimation method,

adjusting the estimated long-term variations so as to fit the current

with which we can accurately estimate current traffic matrices bylink loads.

using the route changes introduced via a TE method. Unlike in

Step 3 Check whether the estimated long-term variations fit theRef. [5], the purpose of our method is to estimate not the long-termcurrent link loads. If they do not match the current link loads, return

variations of traffic but the current traffic matrix, which consists of

to Step 1 after removing the previous information about the end-to-

both long-term variations and short-term variations.

end traffic causing the mismatch. Otherwise, proceed to Step 4.

The rest of this paper is organized as follows. Section 2. describes
the proposed method for estimating current traffic matrices by usingroute changes. Then, in Section 3., we give the results of evaluating our method through simulation. Finally, Section 4. provides aconclusion.

2. Method for estimating current traffic matrix
by using changes in routes

Step 4 Designate the estimation results from Step 2 as the final
estimation results.
In the following subsections, we describe the above steps in detail.
2. 2 Estimating long-term traffic variations
2. 2. 1 Traffic variation model
According to [5], the amount of traffic between each node pairvaries periodically with a certain cycle, such as one day or one week.

2. 1 Overview of estimation method

Therefore, in this paper, we model the traffic amount between nodes

In this paper, we propose a new method for estimating current

i and j as

traffic matrices accurately. We assume that a TE method sometimes
changes routes in the network. Under this condition, we can obtain
additional information, which can be used in estimating the trafficti,j (n) = fi,j (n) + δi,j (n),(2)

where ti,j (n) is the traffic volume between nodes i and j at time n,—2—

fi,j (n) is a function modeling the periodic variation, and δi,j (n) is

we estimate the long-term variations so as to completely fit all thethe variation not included in fi,j (n). In our method, we model themonitored link loads. As a result, estimation results from Eq. (4)long-term variations by fi,j (n) and estimate them by estimating thecan be affected by traffic variations that cannot be modeled by Eq.

parameters of fi,j (n).

(3), making the results very different from the actual traffic.

We model fi,j (n) by applying the model used in [5]. This ap-

To mitigate the impact of δi,j on the estimated long-term varia-

proach models the periodic traffic variation by using sin and cos

tions, in our method, by placing constraints on the variables them-

functions. With this model, the periodic variation is represented asNffi,j (n) =αh,i,j cos
h=0

2πnh
N cycleNf+

αh+Nf ,i,j sin
h=0where N

cycle



2πnh
N cycleselves, we avoid estimating the long-term variations so as to completely fit all the monitored link loads. We thus use the followingequation instead of Eq. (4):
n

.

(3)

minimize|X(k) − A(k)T̂ est (k)|2(6)

k=n−M +1
2Nf+Φis the number of times monitoring link loads in each

mi,j
i,j

cycle, Nf is a parameter determining the number of terms in Eq.

(αh,i,j − αh,i,j )2,h=0(3), and the αh,i,j are the variables to be estimated by our estima-

where the αh,i,j are the variables estimated the previous time, mi,jtion method. With Nf set to a large value, the traffic variation mod-is the amount of information monitored before, and Φ denotes aeled by Eq. (3) captures more of the short-term variation, but theparameter by which we can set the weight to the constraints on

number of variables to be estimated also increases. In our method,the variables themselves. Using this equation, we estimate all the
αh,i,j (0 <=h<
= 2Nf ) of fi,j (n) so as to fit all the monitored link

we only have to roughly model the traffic variations, because we
can estimate the current traffic matrix by adjusting the roughly es-

loads while keeping the values close to the values estimated the pre-timated long-term variations. That is, in our method, a small Nf isvious time.

sufficient.

When we estimate the long-term variations the first time, how-

2. 2. 2 Method for estimating long-term variations

ever, we have not obtained the αh,i,j . Thus, in such cases, we

In the model described by Eq. (3), the variables αh,i,j determine

set the α0,i,j to the elements of traffic matrices estimated by othermethods [3], and we set the αh,i,j (1 <=h<
= 2Nf ) to 0. By using

the long-term variations. Therefore, our method estimates the longterm variations by estimating the αh,i,j . We estimate the αh,i,j byusing the link loads monitored the last M times. At any time n, thethis approach, we can avoid estimating traffic variations as having
significantly larger values than the actual variations.

link loads and the traffic matrix have a relation described by Eq. (1).

2. 3 Adjustment of estimated long-term variations

Therefore, we estimate all variables so as to satisfy Eq. (1) in any

In described in subsection 2. 2, we estimate the long-term varia-

time. In this paper, we use a least square algorithm to estimate thetions. Because these estimates do not include the δi,j (n) in Eq. (2),variables. That is, when the number of nodes is N , the variables are

however, they do not fit the current link loads. Therefore, we adjust

basically estimated as

the long-term variations estimated as given in subsection 2. 2 so asnminimize|X(k) − A(k)T̂ est (k)|2k=n−M +1

where

T̂

est

(k) =







f0,0 (k)..
.
fi,j (k)..
.
fN,N (k)






(4)

to fit the current link loads.
The adjustment is performed through the following steps. First,
by assigning n to the functions corresponding to the estimatedlong-term variations, we obtain a roughly estimated traffic matrixT̂ est (n). Then, we obtain a traffic matrix T̂ (n) that is close toT̂ est (n) and fits the link loads monitored at time n. That is, we.

(5)

By using Eq. (4), when some routes are changed, we can use additional equations for estimating the variables.
With Eq. (4), however, we may not be able to estimate the long-obtain the estimation results by applying a least square algorithm so
as to satisfy the following conditions:minimize|T̂ (n) − T̂ est (n)|2

(7)

where
A(n)T̂ (n) = X(n).

(8)

term variations accurately because of the effects of traffic variations
that cannot be modeled by Eq. (3). Because the actual traffic variations do include variations that cannot be modeled by Eq. (3) (i.e.,δi,j (n) in Eq. (2)), long-term variations modeled by Eq. (3) cannot
completely fit all the monitored link loads. With Eq. (4), however,

2. 4 Re-estimation of traffic matrix after mismatch of estimated long-term variations
When traffic variation trends change, long-term variations estimated by using all the link loads monitored the last M times can—3—exhibit mismatches with the current traffic. This is because the long-

Too small σi,j causes detection of points where |t̂est
i,j (n)− t̂i,j (n)|term variations are estimated so as to fit the link loads before theis small. We do not, however, need to detect such points, becausechange, which can be very different from the current traffic varia-

the estimated long-term variations there fit the current traffic, since

tions. In such cases of mismatch, we cannot estimate the current|t̂esti,j (n) − t̂i,j (n)| is small. Therefore, to avoid detecting suchtraffic matrices accurately even after adjustment, because the ad-points, we introduce a parameter s and set σi,j to s if σi,j is smaller

justment uses only the current link loads, which are insufficient forthan s.

estimating the traffic matrices accurately.

2. 4. 2 Re-estimation of long-term variations after detection

Therefore, in our method, when the estimated long-term varia-

When mismatches between the estimated long-term variations

tions exhibit mismatches with the current traffic, we detect the mis-

and the current traffic are detected, we need to re-estimate the long-matches and re-estimate the long-term variations without using linkterm variations so as to fit the current traffic. Because such mis-loads that do not match the current traffic. In this subsection, we de-matches occur when we estimate the long-term variations by using

scribe how to detect mismatches and identify the end-to-end traffic

previously monitored link loads that are very different from the cur-causing the mismatches, as well as how to re-estimate the long-term

rent traffic variations, we re-estimate the long-term variations by

variations after mismatch detection.

using link loads and routing matrices in which information about

2. 4. 1 Detecting mismatches and identifying end-to-end traffic
causing mismatches

the end-to-end traffic causing the mismatches has been removed.
Our method removes previous information corresponding to the

When the estimated long-term variations are very different fromend-to-end traffics causing mismatches at time n through the fol-the current traffic, the differences between the current link loads andlowing steps. We first remove such information from the routing
matrices A(i)(n − M + 1 <= i < n) by setting elements corre-

the link loads calculated using the estimated long-term variations
are large. In this case, because the results of adjusting T̂ (n) mustsatisfy Eq. (8), while A(n)T̂ est (n) is very different from the curest

sponding to the identified end-to-end traffic to 0. We denote therouting matrix after such replacement as A (i).

(n) − T̂ (n), correspond-

Then, we create a link load matrix X  (i)(n − M + 1 <= i < n)ing to the traffic causing the mismatches, become large. Therefore,

from which information about the identified end-to-end traffic has

we detect mismatches and identify the end-to-end traffic causing the

been removed. The sum of the elements of traffic matrix T corre-

rent link loads X(n), the elements of T̂

mismatches by evaluating T̂est(n) − T̂ (n).

sponding to the identified end-to-end traffic traversing each link at

Because the size of traffic variation that cannot be included in Eq.
(3) depends on the end-to-end traffic [5], if we set a single thresholdfor the elements of T̂est(n) − T̂ (n), traffic with large variations thattime i is calculated as (A(i) − A (i)) T . Therefore, Xi is given byX  (i) = X(i) − A(i) − A (i) T̂ est (i).

(11)



cannot be modeled by Eq. (3) will be erroneously detected as traffic

where T̂ est (i) is the traffic matrix at time i calculated using thecausing mismatches.

estimated long-term variations. In calculating T̂ est (i), we use the

Therefore, we detect mismatches and identify their sources by
est

long-term variations estimated at time n − 1, since the long-term(n) − T̂ (n) with its previous values. Our method

variations estimated at time n can be affected by changing trends.

performs the comparison by using the Smirnov-Grubbs method [6],Next, our method re-estimates the long-term variations by using

comparing T̂

Eq. (12), which is refined from Eq. (6) to use X  (k) and A (k):which can easily detect outliers in sampled data.
Here, we define the elements of T̂ est (n) and T̂ (n) corresponding to the traffic between nodes i and jas t̂esti,j (n) and t̂i,j (n)n−1minimize+|X(n) − A(n)T̂ est (n)|2|t̂esti,j (n) − t̂i,j (n)| is an outlier by calculating
di,j =

σi,j

2Nf

+Φ
,

(9)

where μi,j and σi,j are the average and standard deviation of<<|t̂esti,j (k) − t̂i,j (k)|(n − M + 1 = k = n), respectively. Then,|t̂esti,j (n) − t̂i,j (n)| is detected as an outlier if di,j is larger than thethesholdτ = (M − 1)



2
τθ,M
+2
2
M (M − 2) + M τθ,M
+2

(12)

k=n−M +1

respectively. In the Smirnov-Grubbs method, we detect whether|t̂esti,j (n) − t̂i,j (n)| − μi,j|X  (k) − A (k)T̂ est (k)|2

mi,j
i,j

(αh,i,j − αh,i,j )2.

h=0

2. 4. 3 Re-estimation of traffic matrix after re-estimation of
long-term variations
After re-estimating the long-term variations, we re-estimate thecurrent traffic matrix through the same steps described in subsection 2. 3.

(10)

3. Evaluation

where M is the number of samples, θ is a parameter specifying the

3. 1 Metrics

detection sensitivity, and τθ,M is a value corresponding to the top

In this section, we describe an evaluation of our method by sim-

θ/M % points of the T distribution with M − 2 degrees of freedom.

ulation. In the simulation, we evaluated our method by two general
—4—

5

14

25

8
0

Our method with re-estimation
Our method without re-estimation
Simple additional equation method
Tomogravity

15

13
1

20

7
16
10

4

12

9

3

RMSE

6

2

17

11

15
10

18

5

Fig. 2 EON topology

0metrics: (1) the accuracy of estimation, and (2) the performance of

0

50

100

150

200

Time

a TE method using the estimated traffic matrices.

Fig. 3 Time variation of RMSE (when some traffic variations change)

To evaluate the accuracy, we used a specific metric – the root

3. 3 Accuracy of the estimation

mean squared error (RMSE).
RMSE =

In our method, we obtain estimation results by adjusting the esti1(t̂i,j (n) − ti,j (n))2
N 2 1<i,j<N
=

(13)

=

To evaluate the performance of a TE method using the estimatedtraffic matrices, we investigated whether the purpose of the TE
method was achieved. The next subsection describes the purposeof the TE method used in our simulation.
3. 2 Environment used in evaluation
In our method, we assume that a TE method changes routes sometimes. In this evaluation, we used the optical layer TE as an example of a TE method. The optical layer TE establishes opticallayer paths between two IP routers over a physical network consisting of IP routers and optical cross-connects (OXCs). A set of
optical layer paths forms a virtual network topology (VNT). Traffic between two routers is carried over the VNT by using IP layer
routing. Under these conditions, the optical layer TE accommodates
traffic that fluctuates widely by dynamically reconfiguring the VNT.
In our simulation, we used the European Optical Network (EON)(19 nodes, 37 links) shown in Fig. 2 as the physical topology andexecuted the optical layer TE method proposed in [4] once an hour.
The purpose of this method is to keep the maximum link utilization
under the threshold TH by adding or deleting optical layer paths
with a limitation on the number of optical layer paths reconfigured
at one time. In this simulation, we set the maximum number of optical layer paths reconfigured at one time to 30, TH to 0.7 and TL to
0.4.
In the simulation, we investigate the accuracy of the estimation
when the some traffic change suddenly. Therefore, we generate
end-to-end traffic by adding sudden changes to the traffic generated
by adding variations to sin functions whose amplitudes and phases
were randomly generated. We added sudden changes to the traffic
from nodes 2 to 4, 9 to 1, and 0 to 12 at times 70, 110, and 140, respectively. The rates of the sudden traffic changes from nodes 2 to4, 9 to 1, and 0 to 12 were, respectively, 120 % , 150 %, and 160 %of the maximum rate of traffic before the addition.
In our estimation method, we use parameters M , Nf , Φ, θ and s.
In this simulation, we set M to 160, Nf to 2, Φ to 0.01, θ to 0.01,
and s to 1.

mated long-term variations so as to fit the current link loads. In addition, when the trends of traffic variations change and the estimatedlong-term variations do not match the current traffic, our method detects mismatches and identifies the end-to-end traffic causing them,after which it re-estimates the long-term variations.
Therefore, we investigated the effectiveness of adjusting the estimated long-term variations and effectiveness of re-estimation, bycomparing the accuracy of our estimation method with the accuracies of the following methods:•

A method using only the current link loads. For this method,

we used the tomogravity method with the simple gravity model [3].
•A method using the link loads monitored at previous times

but not considering the time variations of traffic [4].
•Our method without re-estimation.

Figure 3 shows the RMSE when we added these sudden traffic changes. The results show that the errors for the tomogravity
method are the largest. This is because the tomogravity method
uses only the current link loads, which is an insufficient amount of
information.
The errors for the additional equation method are also large. Thisis because that method does not consider traffic variations but assumes instead that the true traffic matrix does not change during TE
execution. Therefore, this method cannot estimate traffic matrices
accurately when traffic varies, even while monitoring the link loads
a sufficient number of times.
On the other hand, the errors for our methods are relatively small.
That is, by including the link loads monitored at previous times inconsidering the time variations of traffic, we can estimate trafficmatrices accurately. However, the RMSE for our method without
re-estimation increases after time 70, whereas the RMSE for our
method with re-estimation remains small after time 70.
To investigate the impact of sudden changes in detail, we compared the estimation results obtained for traffic with sudden changes
added. Figure 4 shows the estimation results for our method withand without re-estimation.
This figure shows that both methods can accurately estimate all
the traffic amounts before adding the sudden changes. After adding
the changes, however, the traffic rate estimated by our method with-

—5—

1
200
150
100
50

Maximum link utilization

250
Estimated amount
Actual amount

Estimated amount
Actual amount

200

Traffic amounts

Traffic amounts

250

150
100
50

0

0
0

50

100

150

200

0

50

Time

100

150

200

0.9
0.8
0.7
Our method with re-estimation
Our method without re-estimation
Additional equation method
Tomogravity

0.6
0.5

Time

0.4
0

(a) Our method with re-estimation

50(b) Our method without re-estimation

100

150

200

Time

Fig. 5 Variation in maximum link utilization after TE execution
Fig. 4 Estimation results of traffic between nodes 9 and 1

4. Concluding remarksout re-estimation cannot capture the changes. This is because thatIn this paper, we have proposed a method for estimating current

method also uses the link loads monitored before adding the sudden

traffic matrices by using route changes introduced by a TE method.

changes, which are very different from the current traffic variations.

In this method, we first estimate the long-term variations of traffic

On the other hand, our method with re-estimation can estimate

by using the link loads monitored the last M times. Then, we ad-

the traffic amounts accurately even after adding the sudden changes.

just the estimated long-term variations so as to fit the current link

This is because by re-estimating the long-term variations after re-loads. In addition, when the traffic variation trends change and themoving information about the end-to-end traffic causing the mis-

estimated long-term variations cannot match the current traffic, ourmatches between the estimated long-term variations and the current

method detects mismatches. Then, so as to capture the current traf-

traffic, we avoid the impact of information that is very different fromfic variations, the method re-estimates the long-term variations after

the current traffic variations.

removing information about the end-to-end traffic causing the mis-

3. 4 Impact on performance of TE methods

matches. For this paper, we evaluated our method through simula-

Finally, we evaluate the performance of TE methods using traffiction. The results show that our method can estimate current trafficmatrices estimated by our method. The TE method used in our sim-

matrices accurately even when some end-to-end traffic changes sud-ulations configured the VNT and routes over the VNT so as to keep

denly.

the maximum link utilization under the threshold TH . Therefore, inthis evaluation, we investigated the maximum link utilization after
TE was performed. For this simulation, we used the same traffic
described in the previous subsection.

Our future work will include optimally setting parameters such as
M , Φ and Nf .

Acknowledgement

Figure 5 shows the results of this simulation. The figure shows

This work was partially supported by Japan Society for the Pro-

that when using the tomogravity method or the additional equation

motion of Science (JSPS), Grant-in-Aid for Young Scientists (Start-

method, the maximum link utilization becomes significantly largerup) 19800023.

than the threshold TH . This is because the estimation errors of thesemethods are large, as described above.
This figure also shows that the maximum link utilizations in thecase of using our method without re-estimation sometimes become
significantly larger than the threshold, as well. This is causedby significant underestimation of the traffic including the suddenchanges. As shown in Fig. 4, our method without re-estimation
cannot capture the added sudden changes and significantly underestimates their amounts. Because of such underestimates, when the
TE method changes the routes of the underestimated traffic, it does
not reserve enough bandwidth.
On the other hand, in the case of using our method with reestimation, we can reduce the maximum link utilization to around
TH at all times. This is because, with re-estimation, our method canestimate traffic matrices accurately even when the traffic changes
suddenly.

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in GMPLS-based multiregion networks,” IEEE Journal on Selected
Areas in Communications, vol. 21, pp. 1254–1262, Oct. 2003.
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