JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 11, ISSUE 2, FEBRUARY 2012

1

© 2012 JCSE

www.journalcse.co.uk

Reliability Analysis Based on

Jump Diffusion Model for

an Open Source Software Yoshinobu Tamura and Shigeru Yamada

Abstract—Many open source softwares are developed in all parts of the world, e.g., Firefox, Apache HTTP server, Linux,

Android, etc. Open source software is now attracting attention as the next-generation software development paradigm because

of the cost reduction, quick delivery, work saving. This paper focuses on the irregular fluctuation of version upgrade and the

total track of fault data for open source software. A new approach to software reliability assessment by using the jump diffusion

model based on the stochastic differential equations in order to consider the change of requirements specification and the

irregular fluctuation of version upgrade is presented. Also, actual software fault-count data is analyzed in order to show

numerical examples of software reliability assessment for several open source softwares. In particular, several reliability

assessment measures are derived from our jump diffusion model. Moreover, this paper shows that the proposed the method of

reliability analysis can assist quality improvement for the open source software project.

Index Terms—Reliability, open source software, jump diffusion model, stochastic differential equations.

—————————— � ——————————

1 INTRODUCTION

T present, various open source softwares (OSS's) are developed and released around the world. In recent years, the next-generation software development

environment by using network computing technologies such as a cloud computing is now attracting attention. Also, the successful experience of adopting OSS includes GNU/Linux operating system, Apache HTTP server, and so on[1]. However, the poor handling of the quality and customer support prohibits the progress of OSS. This pa-per focuses on the problems of software quality, which prohibit the progress of OSS. In particular, many OSS's are used because of the cost reduction, quick delivery, and work saving. Also, the development cycle of OSS has been continued without a break. It is important for soft-ware project managers to assess the reliability by using the total record of OSS fault data instead of the data of specified version of OSS. Then, it is need to consider the irregular fluctuation associated with the version upgrade of OSS. Thereby, the software project managers can con-sider the total requirements specification of OSS. In case of considering the effect of the debugging process on en-tire system in the development of a method of reliability assessment for OSS, it is necessary to grasp the situation of registration for bug tracking system, the combination status of OSS's, the degree of maturation of OSS, etc. Moreover, if the size of the software system is large, the number of faults detected during the operating phase becomes large, and the change of the number of faults which are detected and removed through each debugging

becomes sufficiently small compared with the initial fault content at the beginning of operation. Therefore, in such a case, it is appropriate use a stochastic model with conti-nuous state space in order to describe the stochastic be-havior of the fault-detection process such as the total track record of OSS. In particular, the total track record of OSS is treated in order to consider the total domain of fault-prone requirements specification for OSS. For ex-ample, the software managers cannot comprehend it by only using the fault data of specified version for OSS. Many software reliability growth models (SRGM's)[2] have been applied to assess the reliability for quality management and testing-progress control of software development. On the other hand, the effective method of dynamic debugging management for new distributed development paradigm as typified by the open source project has only a few presented[3], [4], [5], [6].

This paper focuses on the total track data of OSS de-veloped under an open source project. Also, a useful me-thod of software reliability assessment considering the version upgrade of OSS is discussed. A software reliabili-ty growth model based on stochastic differential equa-tions in order to consider the total track record of OSS is presented in this paper. Then, it is assumed that the soft-ware fault-detection rate depends on the time, and the software fault-reporting phenomena on the bug tracking system keep an irregular state. Also, actual software fault-count data to show numerical examples of software relia-bility assessment for the OSS is analyzed. In particular, several numerical examples of reliability assessment con-sidering the change of requirements specification and the version upgrade of OSS are shown. Moreover, several reliability assessment measures are derived from the pro-

————————————————

• Yoshinobu Tamura is with the Graduate School of Science and Engineer-ing, Yamaguchi University, Tokiwadai 2-16-1, Ube-shi, Yamaguchi, 755-8611 Japan.

• Shigeru Yamada is with the Graduate School of Engineering,Tottori Uni-versity, Minami 4-101, Koyama, Tottori-shi, 680-8552 Japan.

A

2

posed jump diffusion model. Then, it is shown that the proposed method of reliability analysis can assist quality improvement for OSS's.

2 MODEL DESCRIPTION

2.1 Stochastic Differential Equation Modeling

Let )(tN be the number of detected faults in the OSS by operational time )0( ≥tt in the bug tracking system. Suppose that )(tN takes on continuous real values. Since latent faults in the OSS are detected and eliminated dur-ing the operational phase, )(tN gradually increases as the operational procedures go on. Thus, under common assumptions for software reliability growth modeling, it is considered the following linear differential equation:

{ },)()()()(

tNtDtbdt

tdN−= (1)

where )(tb is the software fault-detection rate at opera-tional time t and a non-negative function, )(tD , means the amount of change of requirements specification. Also, it is defined )(tD as follows:

,)( tetD βα −= (2)

where α is the number of latent faults in the OSS, and β the changing rate of requirements specification. It is assumed that the fault-prone requirements specification of OSS grows exponentially in terms of t as shown in Figure 1[7], [8]. Thus, the OSS shows a reliability regres-sion trend if β is positive value. On the other hand, the OSS shows a reliability growth trend if β is negative value.

This paper focuses on the total track record of faults to be detected. In this case, the amount of fault data is huge size. Considering the characteristic of such OSS, the soft-ware fault-reporting phenomena keep an irregular state in the early stage of operational phase. Moreover, the ad-dition and deletion of software components are repeated under the development of OSS system, i.e., the OSS re-quirements specification depends on the operation time. Therefore, Eq.(2) is extended to the following stochastic differential equation[9], [10]:

{ }{ },)()()()()(

tNtDttbdt

tdN−+= σγ (3)

where σ is a positive constant representing a magnitude of the irregular fluctuation, and )(tγ a standardized

Gaussian white noise. Eq.(3) is extended to the following stochastic differen-

tial equation of an Ito type:

{ }

{ } ),()()(

)()(2

1)()( 2

tdWtNtD

dttNtDtbtdN

−=

−

−=

σ

σ (4)

where )(tW is a one-dimensional Wiener process which is formally defined as an integration of the white noise

)(tγ with respect to time t . The Wiener process is a Gaussian process and has the following properties:

[ ] ,10)0(Pr ==W (5)

[ ] ,0)(E =tW (6)

[ ] [ ], ,Min)()(E tttWtW ′=′ (7)

where ]Pr[⋅ and ][E ⋅ represent the probability and ex-pectation, respectively.

By using Ito’s formula[9], [10], the solution of Eq.(4) is obtained under the initial condition 0)0( =N as fol-lows[11]:

{ } .)()(exp1)()(0

−−−= ∫ tWdssbtDtN

t

σ (8)

Using solution process )(tN in Eq.(8), several software reliability measures can be derived.

Moreover, the intensity of inherent software failures in case of )()( 1 tbtb ≡ and )()( 2 tbtb ≡ are defined as:

,)(

)(

)(

)(

)(0

1∫ −≅

−=

t

e

e

e

e

tHa

dt

tdH

tNa

dt

tdN

dstb (9)

,)(

)(

)(

)(

)(0

2∫ −≅

−=

t

s

s

s

s

tHa

dt

tdH

tNa

dt

tdN

dstb (10)

where )(tHe and )(tH s mean the exponential SRGM and the delayed S-shaped SRGM, respectively, based on nonhomogeneous Poisson process (NHPP).

Therefore, the transition probability distribution of these two models are obtained as follows:

{ }[ ],)(exp1)()( tWbttDtNe σ−−−= (11)

{ }[ ].)(exp)1(1)()( tWbtbttDtNs σ−−+−= (12)

2.2 Jump-Diffusion Modeling

The jump term can be added to the proposed stochastic differential equation models in order to incorporate the irregular state around the version-upgrade time. Then, the following jump-diffusion process[12] is given as fol-lows.

{ }

{ }

( ) ,1

)()()(

)()(2

1)()(

)(

1

2

−+

−+

−

−=

∑=

λ

σ

σ

tM

i

i

j

jj

Vd

tdWtNtD

dttNtDtbtdN

(13)

Fig. 1. The basic concept of the fault-prone requirements specifica-tion and OSS fault-reporting domain.

3

where )(λtM is a Poisson point process with parameter λ at operation time t . Also, )(λtM the number of oc-curred jumps, λ the jump rate. )(λtM , )(tW , and iV are assumed to be mutually independent. Moreover, iV is i-th jump range.

By using Ito's formula[9], [10], the solution of the fol-lowing equation can be obtained from Eq.(13):

{

.log

)()(exp1)()(

)(

1

0

−

−−−=

∑

∫

=

λ

σ

tM

i

i

t

j

V

tWdssbtDtN

(14)

Therefore, the transition probability distribution of these two models are respectively obtained for the exponential SRGM and the delayed S-shaped SRGM as follows:

{[

,log

)(exp1)()(

)(

1

−

−−−=

∑=

λ

σ

tM

i

i

je

V

tWbttDtN

(15)

{[

,log

)(exp)1(1)()(

)(

1

−

−−+−=

∑=

λ

σ

tM

i

i

js

V

tWbtbttDtN

(16)

3 PARAMETER ESTIMATION

3.1 Method of Maximum-Likelihood

In this section, the estimation method of unknown para-meters α , β , b , and σ in Eq. (8) is presented. The joint probability distribution function of the process

)(tN is denoted as

].0)(|)( , ,)(Pr[

), ; ;, ;,(

011

2211

=≤≤≡ tNytNytN

ytytytP

KK

KK

L

L (17)

Its density of Eq. (17) is denote as

.), ; ;, ;,(

), ; ;, ;,(

21

2211

2211

K

KK

K

KK

yyy

ytytytP

ytytytp

∂∂∂

∂

L

L

L

≡ (18)

Since )(tN takes on continuous values, the likelihood function, l , for the observed data

) , ,2 ,1( ) ,( Kkyt kk L= is constructed as follows:

)., ; ;, ;,( 2211 KK ytytytpl L= (19)

For convenience in mathematical manipulations, the fol-lowing logarithmic likelihood function is used:

.log lL = (20)

The maximum-likelihood estimates α , β , b , and σ are the values making L in Eq. (20) maximize. These can be obtained as the solutions of the following simultane-ous likelihood equations[11]:

.0====∂σ

∂

∂

∂

∂β

∂

∂α

∂ L

b

LLL (21)

3.2 Estimation of Jump-Diffusion Parameters

Generally, it is difficult to estimate the jump-diffusion parameters of stochastic differential equation model. The method of estimation of jump-diffusion parameters are proposed by several researchers. However, the effective method of estimation has only a few presented. A genetic algorithm (GA) in order to estimate the jump-diffusion parameters of the proposed model is used in this section. The procedure of GA algorithm is given in the follow-ing[13].

It is assumed that the proposed jump-diffusion model includes the parameters λ , µ , and τ . µ and τ mean the parameters included in i-th jump range iV .

1. The initial individuals are randomly generated. Also, the set of initial individual to the binary digit is converted.

2. Two parental individuals are selected, and new individuals are produced by the crossover re-combination. The value of fitness is calculated from the evaluated value of each individual. The following value of fitness as the error between the estimated and the actual data is defined in this paper.

{ } ,)(

),( min

0

2∑=

−=n

i

iji

i

yiNF

F θθ

(22)

where )(iN j is the number of detected faults at

operation time i in the proposed jump-diffusion

model, iy the number of actual detected faults.

Also, θ means the set of parameters λ , µ , and

τ .

3. Step.2 and Step.3 are continued until reaching the specific size.

The jump-diffusion parameters λ , µ , and τ are esti-mated by using above mentioned steps.

4 SOFTWARE RELIABILITY ASSESSMENT

MEASURES

4.1 Stochastic Differential Equation Model

Considering the expected number of faults detected up to operation time t , the density function of )(tW is giv-en by:

.2

)(exp

2

1))((

2

−=t

tW

ttWf

π (23)

Information on the cumulative number of detected faults in the system is important to estimate the situation of the progress on the debugging procedures. Since )(tN is a random variable in the proposed model, its expected val-ue can be a useful measure. It can be calculated from Eq.(28) as follows[11]:

4

,2

)(exp1)()(2

0

+−−= ∫ tdssbtDtNt σ

(24)

where )]([E tN is the expected number of faults detected up to time t .

It is important for software managers to assess the number of latent faults according to the change of specifi-cation of OSS. Considering the change of requirements specification, the number of latent faults is given by

.)( tetD βα= (25)

Also, the number of remaining faults considering the change of requirements specification can be obtained as follows:

)].()([E)]([E tNtDtNr −= (26)

Moreover, the mean time between software failures is useful to measure the property of the frequency of soft-ware failure-occurrence (or fault-detection). Then, the cumulative MTBF(denoted by CMTBF ) is approximate-ly given by:

.)]([E

1)(

tNtMTBFC = (27)

4.2 Jump-Diffusion Model

Similarly, the cumulative number of detected faults in the system is important to estimate the situation of the progress on the software debugging procedures. Since

)(tN j is a random variable in the proposed model, it is calculated as Eq.(14)[11].

Also, it is important for software managers to assess the number of latent faults according to the change of specification of OSS. the number of remaining faults based on the jump-diffusion model considering the change of requirements specification can obtain as fol-lows:

{.log

)()(exp)(

)()()(

)(

1

0

−

−−⋅=

−=

∑

∫

=

λ

σ

tM

i

i

t

jrj

V

tWdssbtD

tNtDtN

(28)

Also, the mean time between software failures is useful to measure the property of the frequency of software fail-ure-occurrence. Then, CMTBF is approximately given by:

.)(

1)(

tNtMTBF

j

Cj = (27)

5 NUMERICAL ILLUSTRATIONS

5.1 Data for Numerical Illustrations

The successful experience of OSS's include Firefox, OpenOffice, Ubuntu, Android, etc. In this way, the OSS is closely watched from the point of view of the cost reduc-tion and the quick delivery.

There are many open source projects around the world. In particular, we focus on several OSS's in terms of the application software, server one, and embedded one in order to evaluate the performance of the proposed me-thod. The numerical examples by using the data sets for several OSS's are shown.

This paper focuses on Android mobile phone OS[14], Apache HTTP server[15], Firefox Webbrowser[16], and Thunderbird mailer[17]. The fault-count data used in this paper are collected in the bug tracking system on the website of each open source project. Table 1 shows the OSS versions for the total record of actual fault data used in this paper.

5.2 Reliability Assessment Results

The estimated expected cumulative numbers of de-tected faults in Eqs. (24), )]([E tN , in case of

)()( 1 tbtb ≡ and )()( 2 tbtb ≡ in Eqs. (9) and (10) for each OSS are shown in Figures 2~5, respectively. Also, the sample paths of the estimated numbers of detected faults based on the proposed jump-diffusion model in Eqs. (15) and (16), )]([E tN je and )]([E tN js , in case of

)()( 1 tbtb ≡ and )()( 2 tbtb ≡ for each OSS are shown in Figures 6~13, approximately.

From Figures 2~13 are shown that Eq. (11) fits better than Eq. (12) on the whole. In particular, Android mobile phone OS has the characteristic growth curve. However, the proposed model fits into the characteristic growth curve of Android. Also, above mentioned results show that the proposed model can cover the OSS growth curve by using either Eq. (11) or Eq. (12).

TABLE 1 OSS VERSIONS FOR THE TOTAL TRACK RECORD OF

ACTUAL FAULT DATA.

5

Fig. 3. The estimated cumulative number of detected faults,

)]([E tN , in case of )()( 1 tbtb ≡ and )()( 2 tbtb ≡ , (Apache

HTTP server).

Fig. 4. The estimated cumulative number of detected faults,

)]([E tN , in case of )()( 1 tbtb ≡ and )()( 2 tbtb ≡ , (Firefox

Web browser).

Fig. 5. The estimated cumulative number of detected faults,

)]([E tN , in case of )()( 1 tbtb ≡ and )()( 2 tbtb ≡ , (Thun-

derbird mailer).

Fig. 6. The sample path of the estimated number of detected faults,

)(ˆ tNe , in case of )()( 1 tbtb ≡ (Android mobile phone OS).

Fig. 7. The sample path of the estimated number of detected faults,

)(ˆ tNs , in case of )()( 2 tbtb ≡ (Android mobile phone OS).

Fig. 2. The estimated cumulative number of detected faults,

)]([E tN , in case of )()( 1 tbtb ≡ and )()( 2 tbtb ≡ , (An-

droid mobile phone OS).

6

Fig. 8. The sample path of the estimated number of detected faults,

)(ˆ tNe , in case of )()( 1 tbtb ≡ (Apache HTTP server).

Fig. 9. The sample path of the estimated number of detected faults,

)(ˆ tNs , in case of )()( 2 tbtb ≡ (Apache HTTP server).

Fig. 10. The sample path of the estimated number of detected faults,

)(ˆ tNe , in case of )()( 1 tbtb ≡ (Firefox Web browser).

Fig. 11. The sample path of the estimated number of detected faults,

)(ˆ tNs , in case of )()( 2 tbtb ≡ (Firefox Web browser).

Fig. 12. The sample path of the estimated number of detected faults,

)(ˆ tNe , in case of )()( 1 tbtb ≡ (Thunderbird mailer).

Fig. 13. The sample path of the estimated number of detected faults,

)(ˆ tNs , in case of )()( 2 tbtb ≡ (Thunderbird mailer).

Moreover, it is important for software managers to as-

sess the irregular fluctuation in terms of the number of detected faults according to the version upgrade of OSS. Then, Figures 6~13 show that the irregular fluctuation of

7

the proposed model occur around the time of version upgrade, respectively. In particular, these results mean that the number of detected faults in terms of Android mobile phone OS growth as operating procedures go on. In particular, Figures 6~13 show that the number of de-tected faults decrease in spots at the operation time, be-cause of the jump diffusion state. This means the pheno-mena in terms of the imperfect debugging, the change of specification, etc.

Above mentioned results, the software project manag-ers can confirm the frequency of irregular fluctuation by using the sample path of the estimated number of de-tected fault based on the proposed jump diffusion model. Also, the software project managers can comprehend the stability of OSS from Figures 6~13.

6 CONCLUDING REMARKS

The jump diffusion model in order to consider around the time of version upgrade of OSS has been proposed. Also, this paper focused on the requirements specification of OSS, and discussed the method of reliability assessment based on the stochastic differential equation model. At present, a new paradigm of distributed development typified by such an open source project will evolve at a rapid pace in the future. In particular, it is difficult for the software project managers to assess the reliability for the total record of OSS. The pro-posed method may be useful as the method of reliability assessment for the OSS.

A software reliability analysis based on stochastic diffe-rential equations in order to consider the change of require-ments specification of OSS has proposed in this paper. Then, it has been assumed that the software fault-detection rate depends on the time, and the software fault-reporting phe-nomena on the bug tracking system keep an irregular state. Also, actual software fault-count data has analyzed in order to show numerical examples of software reliability assess-ment for the total track record of OSS. Moreover, several reliability assessment measures have been derived from the proposed model.

In particular, it is important for software managers to as-sess the irregular fluctuation in terms of the number of de-tected faults according to the version upgrade of OSS. Then, this paper has shown that the proposed model can compre-hend the irregular fluctuation occurring around the time of version upgrade. Thereby, the software project managers can confirm the frequency of irregular fluctuation by using the sample path of the estimated number of detected faults based on the proposed jump diffusion model. Also, the pro-posed model can describe both the reliability growth trend and the reliability regression trend.

ACKNOWLEDGMENT

This work was supported in part by the Grant-in-Aid for Scientific Research (C), Grant No. 22510150 from the Min-istry of Education, Culture, Sports, Science, and Technol-ogy of Japan.

REFERENCES

[1] E-Soft Inc., Internet Research Reports,

http://www.securityspace.com/s_survey/data/

[2] S. Yamada, Elements of Software Reliability-Modeling Approach-, Kyoritsu--

Shuppan, Tokyo, 2011.

[3] A. MacCormack, J. Rusnak, and C.Y. Baldwin, “Exploring the structure

of complex software designs: an empirical study of open source and

proprietary code,“ Informs Journal of Management Science, vol. 52, no. 7,

pp. 1015-1030, 2006.

[4] G. Kuk, “Strategic interaction and knowledge sharing in the KDE de-

veloper mailing list,“ Informs Journal of Management Science, vol. 52, no. 7,

pp. 1031-1042, 2006.

[5] Y. Zhoum and J. Davis, “Open source software reliability model: an

empirical approach,“ Proceedings of the Workshop on Open Source

Software Engineering (WOSSE), vol. 30, no. 4, pp. 67-72, 2005.

[6] P. Li, M. Shaw, J. Herbsleb, B. Ray, and P. Santhanam, “Empirical eval-

uation of defect projection models for widely-deployed production

software systems,“ Proceeding of the 12th International Symposium on

the Foundations of Software Engineering (FSE-12), pp. 263-272, 2004.

[7] T. Fujiwara and S. Yamada, “A testing-domain dependent software

reliability growth model for practical application,“ Proceedings of the

Second World Congress for Software Quality, pp. 821-826, 2000.

[8] S. Yamada and T. Fujiwara, “Testing-domain dependent software

reliability growth models and their comparisons of goodness-of-fit,“ In-

ternational Journal of Reliability, Quality and Safety Engineering, vol. 8, no. 3,

pp. 205-218, 2001.

[9] L. Arnold, Stochastic Differential Equations-Theory and Applications, John

Wiley & Sons, New York, 1974.

[10] E. Wong, Stochastic Processes in Information and Systems, McGraw-Hill,

New York, 1971.

[11] S. Yamada, M. Kimura, H. Tanaka, and S. Osaki, “Software reliability

measurement and assessment with stochastic differential equations,“

IEICE Transactions on Fundamentals, vol. E77-A, no. 1, pp. 109-116, 1994.

[12] R.C. Merton, “Option pricing when underlying stock returns are dis-

continous,“ Journal of Financial Economics, vol. 3, pp. 125-144, 1976.

[13] J.H. Holland, Adaptation in natural and artificial systems, University of

Michigan Press, 1975.

[14] Open Handset Alliance, Android, http://www.android.com/

[15] The Apache HTTP Server Project, The Apache Software Foundation,

http://httpd.apache.org/

[16] Firefox Project, Mozilla Foundation,

http://www.mozilla.org/projects/firefox/

[17] Thunderbird Project, Mozilla Foundation,

http://www.mozilla.org/thunderbird/ Yoshinobu Tamura was received the B.S.E., M.S., and Ph.D. de-grees from Tottori University in 1998, 2000, and 2003, respectively. From 2003 to 2006, he was a Research Assistant at Tottori Universi-ty of Environmental Studies. From 2006 to 2009, he was a Lecturer and Associate Professor at Faculty of Applied Information Science of Hiroshima Institute of Technology, Hiroshima, Japan. Since 2009, he has been working as a Associate Professor at the Graduate School of Science and Engineering, Yamaguchi University, Ube, Japan. His research interests include reliability assessment for open source software. He is a regular member of the Information Processing Society of Japan, the Operations Research Society of Japan, the Society of Project Management of Japan, and the IEEE. Dr. Tamura received the IEEE Reliability Society Japan Chapter Awards in 2007 and the Research Leadership Award in Area of Reliability from the ICRITO in 2010.

8

Shigeru Yamada was received the B.S.E., M.S., and Ph.D. degrees from Hiroshima University in 1975, 1977, and 1985, respectively. From 1988 to 1993, he was an associate professor at the Faculty of Engineering of Hiroshima University, Japan. Since 1993, he has been working as a professor at the Department of Social Manage-ment Engineering, Graduate School of Engineering, Tottori Universi-ty, Tottori-shi, Japan. He has published numerous technical papers in the area of software reliability models, project management, relia-bility engineering, and quality control. He has authored several books entitled such as Introduction to Software Management Model (Kyoritsu Shuppan, 1993), Software Reliability Models: Fundamen-tals and Applications (JUSE, Tokyo, 1994), Statistical Quality Control for TQM (Corona Publishing, Tokyo, 1998), Software Reliability: Model, Tool, Management (The Society of Project Management, 2004), Quality-Oriented Software Management (Morikita Shuppan, 2007), and Elements of Software Reliability (Kyoritsu Shuppan, 2011). Dr. Yamada received the Best Author Award from the Infor-mation Processing Society of Japan in 1992, the TELECOM System Technology Award from the Telecommunications Advancement Foundation in 1993, the Paper Award from the Reliability Engineer-ing Association of Japan in 1999, the International Leadership Award in Reliability Engg. Research from the ICQRIT/SREQOM in 2003, the Best Paper Award at the 2004 International Computer Sympo-sium, the Best Paper Award from the Society of Project Management in 2006, the Leadership Award from the ISSAT in 2007, and the International Leadership and Pioneering Research Award in Soft-ware Reliability Engineering from the SREQOM/ICQRIT in 2009. He is a regular member of the IPSJ, the ORSJ, the Japan SIAM, the REAJ, JIMA, the JSQC, the SPM Japan, and the IEEE.