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Bivariate change-point modeling for software reliability assessment with uncertainty of testing-environment factor

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Abstract

We observe a phenomenon that the probabilistic characteristic of software failure-occurrence time-interval change notably in an actual testing-phase of a software development process. Testing-time observing such phenomenon is ordinarily called change-point. This phenomenon is treated as one of the factors affecting the accuracy of software reliability assessment based on a software reliability growth model. And regarding software reliability growth modeling, a bivariate software reliability growth model, which describes a software reliability growth process depending on the software reliability growth factor consists of testing-time and testing-effort factors, has been proposed for improving software reliability assessment accuracy based on the software reliability growth process. In this paper, we develop a bivariate software reliability growth model considering with the uncertainty of the change of software failure-occurrence phenomenon at the change-point for improving more the accuracy. Finally we show numerical examples of our model by using actual data.

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Abbreviations

SRGM:

Software reliability growth model

NHPP:

Non-homogeneous Poisson process

MSE:

Mean squared error

\(F( {s,u} )\) :

Bivariate probability distribution function depending on the testing-time \(s\) and the testing-effort \(u\)

\(N_0\) :

Random variable representing the initial fault content

\(N( {s,u} )\) :

Two-dimensional counting process representing the total number of faults detected up to testing-time \(s\) and testing-effort \(u\)

\(X_i\) :

Testing-time factor between the \(i\)th and (\(i-1\))st software failure-occurrence before change-point

\(Z_i\) :

The \(i\)th software failure-occurrence time before change-point

\(M_i\) :

Testing-time factor between the \(i\)th and (\(i-1\))st software failure-occurrence after change-point

\(C_i\) :

The \(i\)th software failure-occurrence time after change-point

\(Y_i\) :

Testing-effort expenditure between the \(i\)th and (\(i-1\))st software failure-occurrence before change-point

\(W_i\) :

Testing-effort factor expended up to the \(i\)th software failure-occurrence time before change-point

\(K_i\) :

Testing-effort expenditure between the \(i\)th and (\(i-1\))st software failure-occurrence after change-point

\(D_i\) :

Testing-effort factor expended up to the \(i\)th software failure-occurrence time after change-point

\(\lambda _s\) :

Testing-environmental coefficient being related to the testing-time factor

\(\lambda _u\) :

Testing-environmental coefficient being related to the testing-effort factor

\({\Lambda }( {s,u})\) :

Mean value function of the two-dimensional NHPP

\({\Lambda }_B \left( {s,u} \right) \) :

Mean value function of the two-dimensional NHPP before change-point

\({\Lambda }_A \left( {s,u} \right) \) :

Mean value function of the two-dimensional NHPP after change-point

\(M\left( {s,u} \right) \) :

Expected number of remaining faults at testing-time \(s\) and testing-effort \(u\)

\(\hbox {E}{[} {\cdot } {]}\) :

Expectation

\(R(\eta |s_e ,u_e )\) :

Operational software reliability function, i.e., the probability that a software failure does not occur in the time-interval \(\left( {s_e ,u_e +\eta } \right) \left( {s_e \ge 0,u_e \ge 0} \right) \) given that the testing has been going up to testing-time \(s_e\) and the testing-effort has been expended up to \(u_e\)

\({\Phi }\) :

Set of software reliability parameter

\(\tau \) :

Set of change-point

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Acknowledgments

This research was partially supported by the Grant-in-Aid for Scientific Research (C), Grant Nos. 25330081 and 25350445, from the Ministry of Education, Culture, Sports, Science and Technology of Japan and The Telecommunications Advancement Foundation.

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Authors and Affiliations

  1. Department of Social Management Engineering, Graduate School of Engineering, Tottori University, Tottori, Japan

    Shinji Inoue, Jun Ikeda & Shigeru Yamada

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  1. Shinji Inoue
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  2. Jun Ikeda
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  3. Shigeru Yamada
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Correspondence to Shinji Inoue.

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Inoue, S., Ikeda, J. & Yamada, S. Bivariate change-point modeling for software reliability assessment with uncertainty of testing-environment factor. Ann Oper Res 244, 209–220 (2016). https://doi.org/10.1007/s10479-015-1869-6

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