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. 2025 Dec 31;20(12):e0339286.
doi: 10.1371/journal.pone.0339286. eCollection 2025.

Trigonometric function transformation and its application in software reliability modeling

Dai-Nghia Vy  1 , Van-Thuan Nguyen  1 , Quyet-Thang Huynh  2 , Trung-Nghia Phung  3 , Hung-Cuong Nguyen  1
Affiliations

Trigonometric function transformation and its application in software reliability modeling

Dai-Nghia Vy et al. PLoS One. .

Abstract

Context: Many models based on S-shaped functions demonstrate their advantages in non-homogeneous Poisson process software reliability modeling. However, three well-known types have been used without deep mathematical evaluation. Furthermore, some other promising S-shaped functions should be aimed at.

Objectives: (1) Transform the sine function to change the domain and co-domain without losing the S-shaped pattern, and (2) Evaluate four types of S-shaped functions mathematically, including three well-known types and this newly considered type of transformation form of the trigonometric function.

Methods: The transformation is taken by a four-step process, including horizontal and vertical shift, horizontal and vertical scale, to maintain the S-shaped form. The mathematical evaluation is performed using numerical analysis techniques in four aspects of function: special cases, domain, range, and limitations.

Results: The first contribution is a deep mathematical appreciation of three well-known S-shaped functions. The second is the mathematical transformation of the trigonometric function to meet the real assumption. The last are the advantages and the applicability of this transformation in software reliability modeling.

Conclusion: Besides three well-used S-shaped functions, the transformation of the trigonometric function has a new advantage. Most importantly, there is a possibility of using the trigonometric function to introduce a new software reliability model.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. S-shaped as a part of sin(t) function.
Fig 2
Fig 2. Transformation from f(t)=sin(t) to f(t)=bsin(βt+φ)+k.
Fig 3
Fig 3. Numerical illustration of b1(t).
Fig 4
Fig 4. Numerical illustration of b2(t).
Fig 5
Fig 5. Numerical illustration of b3(t) when b > 0 and β >0.
Fig 6
Fig 6. Numerical illustration of b3(t) when b < 0 and β >0.
Fig 7
Fig 7. Numerical illustration of b3(t) when β <0.
Fig 8
Fig 8. Numerical illustration of b4(t).
Fig 9
Fig 9. Numerical illustration of 4 types of S-shaped function.
See this image and copyright information in PMC

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