回帰構造に基づくテスト環境を考慮した二項ソフトウェア信頼性モデルの拡張 (確率的環境下における数理モデルの理論と応用)

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(1)46. 数理解析研究所講究録第2044巻 2017年 46-51. 回帰構造に基づくテスト環境を考慮した二項ソフトウェア信頼性モデルの拡張 Extended. Binomial‐Type Models. 鳥取大学. with Test Environment Based. 大学院工学研究科. Shinji. Inoue and. Regression Structure. on. 茂. 井上真二,山田. Shigeru Yamada. Graduate School of. Engineering,. Tottori University. Introduction. 1 In. an. process. actual software. depends. on. debugging skills, which affect discrete‐time. factors, such. the software failure. [11] proposed. domain, Shibata et al.. by using. a. probability. that. a. fault detection. or. hazard rate. proportional. a. [2].. reliability growth. coverage, the number of. Okamura et al.. logistic regression approach for incorporating. which represents the. [9]. test‐runs, and In the. phenomenon.. modeling approach. also. proposed. for. extended. an. the effect of software metrics into p_{l},. by the i‐th test case and the parameter in the Furthermore, Kuwa and Dohi [7] proposed seven types of. fault is detected. cumulative Bernoulli trial process model. extended models. testing. as. occurrence. the cumulative Bernoulli trial process model. extending model. it must be natural to consider that the software. testing phase,. the test environment. by applying generalized logic regression. to formulate p_{i} with software metrics in the. cumulative Bernoulli trial process model. We extend the discrete program. [4, 5].. binomial process. size‐dependent software reliability model following. Our extended model enables. ronment factors but also the effect of software. assessment and to. served. More. follows. a. flexibly depict. specifically,. software. a. we assume. to. us. complexity. measures. reliability growth. observed per the i‐th. flexibly describing relationship between the probability. testing period. curve. that discrete software failure. discrete Weibull distribution for. We also consider the. incorporate. not. only. quantitative software reliability. into. described. occurrence. by fault counting. data ob‐. time distribution. basically. the software failure. that. discrete‐time. a. the effect of test envi‐. phenomenon.. occurrence. software failure caused. a. and the related test environment factors. by using. by. a. fault is. generalized. a. linear. modeling approach. Furthermore, goodness‐of‐fit comparisons existing corresponding model and show application examples of our model for software reliability analysis using we. of. conduct. our. models with the. actual data.. 2. Binomial‐Type Software Reliability A discrete. binomial‐type. software. Model. reliability model [4]. is. developed based. on. the. following. basic. as‐. sumptions:. (A1). Whenever and. (A2). no new. software failure is. faults. are. Each software failure discrete. \mathrm{P}\mathrm{r}\{A\} (A3). a. probability. observed, the fault which caused. introduced in the fault‐detection occurs. at. independently. distribution. represent the probability. and. identically distributed. function for I and the. The initial number of faults in the software system,. Now, let \{N(i), i = 0, 1, \} denote detected up to the i‐th. a. testing‐period.. N_{0}(>0). probability ,. discrete stochastic process. From the. immediately,. random times I with the. (i=0,1,2, \cdots). P(i)\displaystyle \equiv \mathrm{P}\mathrm{r}\{I\leq i\}=\sum_{k}^{ $\iota$} mass. it will be detected. procedure.. is. a. ,. where. of event A ,. random. p_{I}(k). and. respectively.. variable, and. is finite.. representing the number of faults. assumptions above,. we. have the. probabihty. mass.

(2) 47. function that. m. faults. detected up to the i‐th testing‐period. are. as. \displaystyle \mathrm{P}\mathrm{r}\{N(i)=m\}=\sum_{n}\left(\begin{ar ay}{l} n\ m \end{ar ay}\right) \{P(i)\}^{m}\{1-P(i)\}^{n-m}\mathrm{P}\mathrm{r}\{N_{0}=n\} (m=0,1,2, \cdots n) Eq. (1),. In a. we. consider the. that the. case. probability distribution of. (K, $\lambda$). binomial distribution with parameters. which is. given. (1). .. the initial fault content, N_{0} , follows. as. \mathrm{P}\mathrm{r}\{N_{0}=n\}= \left(\begin{ar ay}{l} K\ n \end{ar ay}\right)$\lambda$^{n}(1- $\lambda$)^{K-n} (0< $\lambda$<1; n=0,1, \cdots , K). (2). .. Eq. (2) has the following physical assumptions:. (a). The software system consists of K lines of code. (b). Each code has. (c). Each software failure caused. a. fault with. a. constant. by. a. (LOC). at the. beginning. of the. testing‐phase.. $\lambda$.. probability. fault remaining in the software system. occurs. independently. and. randomly. These. assumptions. are. useful to. apply. binomial distribution to the. a. initial fault content in the software system, and to. reliability. assessment. [6].. influences the software. Substituting Eq. (2) detected up to the i‐th. Pr{N(の. The program size is. reliability growth into. Eq. (1),. probability. mass. function of the. the effect of the program size into software. of the important metrics of software complexity which. process in the. we can. testing‐period. one. incorporate. derive the. testing‐phase.. probability. mass. function of the number of faults. as. =m\displayst le\}=\sum_{n=m}^{K}\left(\begin{ar y}{l n\ m \end{ar y}\right) \left(bgin{ar y}{l K\ n \ed{ar y}\ight) \left(bgin{ar y}{l K\ m \end{ar y}\ight) \displayst le\{$\lambda$P(i)\}^{m}\sum_{n=m}^{K}\left(\begin{ar y}{l K-m\ n-m \end{ar y}\right) = \left(\begin{ar ay}{l} K\\ m \end{ar ay}\right) \{ $\lambda$ P(i)\}^{m}\{1- $\lambda$ P(i)\}^{K-m} (m=0,1,2, \cdots K) {P(の \} m{1 —P(の \}. $\lambda$^{n}(1- $\lambda$)^{K-n}. n一. \{ $\lambda$(1-\mathrm{P}( の )\}^{n-m}(1- $\lambda$)^{K-n}. =. (3). .. Eq. (3), several types of discrete software reliability model with the effect of program size can be developed by giving suitable probability distributions for the software failure‐occurrence times, respec‐ From. tively. 3. Extension of Our Model We need to. failure. give. occurrence. suitable. (3). distribution [8]. framework in Weibull. a. Eq.. probability. mass. time distribution in order to. to the software failure. P(i)=1-(1-p_{l})^{i^{ $\gamma$}}. so. testing period, and. due to the. forth.. Eq. (3), P(i) that represents the software ,. specific. model from based. reliability growth modeling,. occurrence. given. a. time distribution. The. we. on. our. apply. modeling a. discrete. probability distribution. as. (4). ,. where p_{l} (0<p_{\dot{l}} < 1) represents the. period. develop. For flexible discrete software. function of the discrete Weibull distribution is. the i‐th. function in. probability that. $\gamma$ denotes the. a. software failure caused. shape parameter. The probability. maturity of testing skill, changing fault target,. by p_{ $\iota$}. a. fault is observed per. depends. existence of fault prone. the testing modules, and. on.

(3) 48. In this. And. we. research,. we assume. consider the. fohowing. In the. factors. given by. be. p_{$\iota$}=\displaystyle\frac{1}{1+\exp[-$\alpha\beta$_{i}^{\mathrm{T} ]} In. (5), $\beta$_{i}. Eq.. testing period, In the. =. the test environment factors at the i‐th. logistic, probit. ($\alpha$_{0}, $\alpha$_{1}, \cdots , $\alpha$_{n}). represents the. n. the. relationship. is. A^{\mathrm{T} the transposed matrix of matrix A.. given by. (6). p_{\mathrm{i}}=1-\exp[-\exp\{ $\alpha \beta$_{i}^{\mathrm{T}}\}]. p_{l} is. And, $\Phi$^{-1}(p_{\mathrm{t} ). = $\alpha \beta$_{i}^{\mathrm{T}. (7). ,. Software. \mathrm{E}[N(i)]. ,. .. hand, the behaviors of. Reliability. reliability. software failure. can. these functions. are. assessment. are. measures. reliability. phenomenon. well‐known metrics for assessment. Eq. (1).. in. The. =\mathrm{E}[N_{0}]P(i). \mathrm{V}\mathrm{a}\mathrm{r}[N(i)]. A discrete software. in. quantitative software reliability. under the basic. assumptions of the. expectation of the number of detected. faults,. ,. \{P(i)\}^{z}\{1-P(i)\}^{n-z}Pr\{N_{0}=n\} (8). .. is also derived. as. \mathrm{V}\mathrm{a}\mathrm{r}[N(i)]=\mathrm{E}[N(i)^{2}]-(\mathrm{E}[N(i)])^{2} =\mathrm{V}\mathrm{a}\mathrm{r}[N_{0}]\{P(i)\}^{2}+\mathrm{E}[N_{0}]P(i)\{1-P(i)\}. i‐th. measures. as. \displayst le\mathrm{E}[N(i)]=\sum_{z=0}^{n}z\sum_{n}\left(\begin{ar y}{l n\ z \end{ar y}\right). in the time interval. p_{l}=1 and p_{ $\iota$}=0.. different each other around. Assessment Measures. derive software. occurrence. is derived. Its variance,. Furthermore, the comple‐. following relationship: \log\{-\log(1-p_{i})\}= $\alpha \beta$_{i}^{\mathrm{T}} It is worth mentioning that the logistic, probit and complementary log‐log functions are mostly same around p_{i} =0.5.. behaviors of the. Software. .. given by. where there exists the. assessment. We. p_{ $\iota$} and the test environment. ). mentary log‐log regression approach,. 4. testing period.. complementary log‐. kinds of test environment factors at the i‐th. is the coefficient vector, and. denotes the standard normal distribution.. On the other. and. (5). probit regression approach,. where $\Phi$. on. .. (1, $\beta$_{1, $\iota$}, $\beta$_{2,i}, \cdots , $\beta$_{n,i}). $\alpha$=. p_{\mathrm{z} = $\Phi$( $\alpha \beta$_{i}^{\mathrm{T} ). depends. three types of functions for p_{l} :. logistic regression approach, the relationship between. log functions. can. that p_{ $\iota$}. reliability. function is defined. the. as. (i, i+h] (i, h=0,1,2, \cdots) given that. (9). .. probability the. testing. that or. a. the. software failure does not. operation has continued. occur. to the. testing period. Then, the discrete software reliability function, R(i, h) under the basic assumption ,. Eq. (1) R (i ) h ). is derived. as. =\displaystyle \sum_{k}\mathrm{P}\mathrm{r}\{N(i+h)=k|N(i)=k\}\mathrm{P}\mathrm{r}\{N(i)=k\}. =\displaystyle\sum_{k} [\{P(i)\}^{k}\{1-P(i+h)\} More. specifically,. we can. 一ん. .. \displaytle\sum_{n}\left(begin{ar y}{l n\ k \end{ar y}\right). derive the discrete software. content fohows the binomial distribution in. R(i, h)=[1- $\lambda$\{P(i+h)-P(i)\}]^{K}. Eq. (2). \{1-P(i+h)\}^{n}\cdot \mathrm{P}\mathrm{r}\{N_{0}=n\}. reliability. function in the. case. ].. (10). that the initial fault. as. (11).

(4) 49. Furthermore, discrete. instantaneous and cumulative. MTBF_{I} (i) and MTBFc(i),. also derived. are. mean. time between software failures. (MTBFs),. as. MTBF_{I}(i)=1/(\mathrm{E}[N(i+1)]-\mathrm{E}[N(i)]) MTBFc(i) =i/\mathrm{E}[N(i)]. (12). ,. (13). ,. respectively. Parameter Estimation Method. 5. We compare the. sponding model,. performance of. models for software. our. collected from actual software testing. details of the data DS1. are. shown. (t_{\mathrm{t} , y_{\mathrm{i} , c_{\mathrm{t} ) ( i=1,2,. :. as. 22; t22. \cdots. ,. phases. The data. sets. are. existing. corre‐. two data sets. factors, by using. respectively. =22 , y22 =212 ). called DS1 and DS2. The. c_{22}=0.9198 ) where t_{x} is measured. on. the basis of. on. the basis of. 10^{5} (LOC) [3],. (t_{\mathrm{z}}, y_{l}, c_{l})(i=1,2, \cdots , 24; t_{24}=24, y_{24}=296, c_{24}=0.9095) 10^{5} (LOC) [3],. :. assessment with the. follows:. weeks and program size K=1.630\times. DS2. reliability. which does not consider the effect of test environment. where t_{i} is measured. weeks and program size K=1.972\times. where y_{l} represents the number of faults detected up to t_{i} and c_{i} is the CO. t_{l}. to. In this model. .. comparison. the software failure. the actual. fohows. treat the CO. testing. fault detection. occurrence or. coverage. as. attained up. testing‐coverage. the test environment factors. phenomenon. Thus,. we assume. $\beta$_{i}. \equiv. c_{l}. affecting. Regarding. .. data, DS1 shows. The. curve.. we. an exponential software reliability growth curve and DS2 shows an \mathrm{S} ‐shaped existing corresponding model assumes that the software failure occurrence time distribution. P(i)=1-(1-p)^{x^{ $\gamma$}}(i=0,1,2, \cdots). failure caused. by. a. fault is observed per. in. one. Eq. (3), where p represents the probability that a software testing period and $\gamma$ is the shape parameter of the discrete. Weibuh distribution. For. quantitative comparisons. (abbreviated. errors. MSE. as. in terms of. MSE) [10],. we. data,. we use mean. square. mean. (14). square. errors. actual values for all of the observed data the MSE. From Table. though. to the actual. as. =\displayst le\frac{1}N}\sum_{k=1}^{N}[y_{k} —Ê[N(tk)]2.. The MSE represents the. on. fitting performance. which is calculated. 1,. we can. the actual data shows. conducted. a. an. say. our. models. exponential. goodness‐of‐comparison. of the number of detected faults between the estimated and. points. Table or. based. smaller MSE is not sufficient to conclude that. model and the. degree of freedom of. AIC =-2\mathrm{M}\mathrm{L}\mathrm{L}+2 $\phi$. our. 1 shows the results of model. keep high fitting performance. \mathrm{S} ‐shaped software on. reliability growth. the Akaike information criterion. our. approach. models became. is better than the. higher.. comparisons based. to the actual data curve.. even. Furthermore,. (AIC) [1]. because. a. existing corresponding. The AIC is calculated. as. (15). ,. where MLL represents the maximum that. a. model. \log likelihood and $\phi$ indicates the number of parameter. We judged indicating smaller AIC fits better to the actual data. Table 1 also shows the results of model. comparisons based Table. the. on. the MLL and AIC. From the results of model comparisons based. be said that. 1, degree of freedom of it. We show. can. our. our. models possess better. fitting performance. even. on. the AIC in. case. considering. models.. examples of the application of software rehability analysis based. actual data set DS2.. in the. Especially,. we. show the numerical. examples. in. case. on our. model. by using the. that p_{i} follows the. probit.

(5) 50. Table 1. Results of model comparisons based. :. the MSE and AIC.. on. \overline{\overline{\mathrm{D}\mathrm{S}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}27.308-2 40.314 8 .62}\mathrm{M}\mathrm{S}\mathrm{E}\mathrm{M}\mathrm{L}\mathrm{L}\mathrm{A}\mathrm{I}\mathrm{C} probit. 26. 958. -2240.27. 4499.534. complementaly log-\log. 31. 105. -2240.70. 4489.40. \displaystyle\frac{\mathrm{E}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}28.256-2 51.594509.17}{\mathrm{D}\mathrm{S}21\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}3 .954-3052.1361 2. 6}. \underline{\frac{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{t}{\mathrm{c}\mathrm{o}\m3a9th7rm1{3m}\m4a7th5r3m4{p.}1\m5athrm{l}\math-r3m0{e5}\4m.9a7th-r3m0{m5}2\.m1a3th-rm0{5e}2\m07athrm{n}\mathr6m1{t}\2m.a1t4hr6m1{a}\2ma7th6r1m{r5}\.9m4athrm{y} log‐log Existing model. \infty\infty0 \inftymathr{o}\mathr{o}. $\varpi0tlde{mho}ga\$tilde{mhro}. \simequvorln{\math}$vrpiw D\supetmahr{E}bs$\Pileftarow. \tilde{frac$omgq\Phi}{supetmarE. \mathrm{o}\supset. 目. \infty\mathr {o}\mathr {o} \underli {\mathr {m}\cir \underlin{\mathrm{o}\cir} @. 10 12 14 \uparrow 6 18 20 22 24. Testing. Fig. 1. :. Estimated. regression approach.. Ê[N(i)],. We obtain the parameter estimates. ). parameter estimates of $\lambda$, $\alpha$_{0}, as. (number of weeks). expected number of faults detected,. \hat{ $\lambda$}=1.5167\times 10^{-3} \hat{ $\alpha$}_{0}=-2.2289, \hat{$\alpha$}_{1} estimated. time. K\times\hat{ $\lambda$}\simeq 29\mathrm{S}. $\alpha$_{1}. ,. =3.9569\times. and $\gamma$ ,. and the 95% confidence limits for DS2.. the method of maximum likelihood. by. \hat{$\lambda$}, \hat{ $\alpha$}_{0}, \hat{$\alpha$}_{1} \hat{$\gam a$} expected number of initial faults. 10^{-1} and \hat{ $\gamma$}=1.5755 where ,. ,. respectively. Then,. because K=1.972\times 10^{5}. (LOC). the. in DS2.. The 100 $\gamma$ % confidence limits for. 100(1+ $\gamma$)/2 6. Ê[N(i)]. are. derived. as. Figure. Ê[N(i)],. dependent behavior of the expected number of faults detected,. 1. depicts. are. as. the. can. be. the estimated time‐. and its 95% confidence limits.. \hat{\mathrm{E} [N(i)]\pm K_{ $\gamma$}\sqrt{\overline{\mathrm{V}\mathrm{a}\mathrm{r} [N(i)]}. percent point of the standard normal distribution. ,. and. ,. where. K_{ $\gamma$}. indicates the. [12].. Conclusion We. proposed. an. extended. environmental factors occurrence. on. binomial‐type software reliability. the software. reliability growth. process.. model with the effect of the testing‐ Especially, the discrete software failure‐. time distribution follows the discrete Weibull distribution. parameter estimation method of with that of. model, existing corresponding model. performance. of. our. model with. our. existing. and conducted. basically. Further,. comparisons. of the. in terms of MSE. In future. models. [7, 9, 11] by using. a. we. performance. studies,. we. lot of software. discussed. of. our. a. model. need to check the. fault‐counting. data.

(6) 51. with software metrics.. Acknowledgement This research. was. 16\mathrm{K}00098 , from the. in part. supported. Ministry. of. by. (C),. the Grant‐in‐Aid for Scientific Research. Education, Culture, Sports,. Science and. Grant No.. of Japan.. Technology. References. [1]. AC‐19,. [2]. A. Akaike,. H.. T.. new. Dohi,. Osaki, Software reliability assessment models based Comput. Modelling, Vol. 38, pp. 1177‐1184, 2003.. Math.. Fujiwara and S. Yamada, A. T.. software. on a. reliability growth. model. [5]. IEEE Trans.. S. Inoue and S.. Sci.,. [6]. Vol.. measure. —. —,. 71‐75,. Syst.. Man. Cybern.. —. Part A :. goodness‐of‐fit comparisons,. E90‐A). information. metrics based. Symposium. on. No.. 12,. on. Syst. Hum.,. Vol.. 37, No. 2,. pp.. 170‐179,. program. 2007.. modeling,. assessment:. IEICE Trans. Fundam. Electron. Commun.. Comput.. 2891‐2902, 2007.. pp.. initial fault. Software. 2002.. Kimura, S. Yamada, H. Tanaka and S. Osaki, Software reliability. M.. cumulative Bernoulli. testing‐domain. Proc. 13th IEEE International. Yamada, Discrete program‐size dependent software reliability. and. estimation,. pp.. coverage. on. Yamada, Generalized discrete software reliability modeling with effect of. S. Inoue and S.. size,. testing‐path. new. Reliability Engineering (ISSRE02),. [4]. Control, Vol.. (1974). K. Yasui and S.. trial processes. [3]. IEEE Trans. Autom.. look at the statistical model identification. pp. 716‐723. content,. Trans.. Inf.. Process. Soc.. measurement with. Jpn.) Vol. 34,. No.. 7,. pp.. prior‐. 1601‐1609,. 1993.. [7]. D. Kuwa and T.. ceedings of SAC. [8]. T.. 5,. [9]. 2013),. pp.. Nakagawa pp.. H.. Dohi, Generalized logit‐based software reliability modeling with. the 37th Annual International. 246‐255,. and S.. IEEE. Osaki,. Okamura,. (ISSRE10),. [11]. H. K.. The discrete Weibuh distribution. Y. Etani and T.. pp.. IEEE Trans.. Reliab., Vol. R‐24,. No.. 31‐40,. Dohi, A multi‐factor software reliability model based. 21st IEEE International. IEEE. CPS,. Symposium. on. on. logistic. re‐. Software Reliability Engineering. 2010.. Pham, Software Reliability, Springer‐Verlag, Singapore, 2000.. Shibata,. homogeneous. K.. Rinsaka and T.. Poisson. S. Yamada, Software. Tokyo,. 2013.. Dohi,. Metrics‐Uased software. processes, Proceedings of. liability Engineering (ISSRE06),. [12]. 2013.. 300−301, 1975.. gression, Proceedings of the. [10]. CPS,. Pro‐. metrics data. Computer Software and Applications Conference (COMP‐. pp.. reliability. The 17th International. models. Symposium. on. using. non‐. Software. Re‐. 52‐61, IEEE CPS, 2006.. Reliability Modeling. —. Fundamentals and. Applications. —,. Springer‐Verlag,.

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