Trigonometric function transformation and its application in software reliability modeling

TL;DR

This paper explores transforming trigonometric functions into S-shaped patterns for use in software reliability modeling, offering a new mathematical approach.

Contribution

A novel transformation of the sine function into an S-shaped form for software reliability modeling is introduced and mathematically evaluated.

Findings

Three well-known S-shaped functions were deeply analyzed mathematically.

A trigonometric function was transformed to maintain an S-shaped pattern suitable for software reliability modeling.

The transformed function shows potential for introducing new software reliability models.

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.

Figures

S-shaped as a part of sin(t) function.

Transformation from f(t)=sin(t) to f(t)=bsin(βt+φ)+k.

Numerical illustration of b1(t).

Numerical illustration of b2(t).

Numerical illustration of b3(t) when b > 0 and β >0.

Numerical illustration of b3(t) when b < 0 and β >0.

Numerical illustration of b3(t) when β <0.

Numerical illustration of b4(t).

Numerical illustration of 4 types of S-shaped function.

Equations

Full text

1 Introduction

Over seven decades of affecting all the areas of human life [1], the number of developed software has exceeded 10 million. Software engineering [2], including methodology, programming language, framework, and tool, becomes increasingly complex to meet real-life requirements. Those complicated problems led to the insistence on maintaining the software quality. In the scope of this paper, the regarded topic is non-homogeneous Poisson process (NHPP) software reliability modeling (SRM), that is, the application of probability technique in evaluating one of software quality characteristics [3,4].

Among models belonging to the NHPP SRMs group [5], the S-shaped curves have been widely studied. This term S-shaped is based on the pattern of the letter S, the nineteenth letter of the Latin alphabet. The S-shaped curve increases continuously over time when the growth rate is time-dependent. Geometrically, an S-shaped curve is a merging of convex and concave parts. Some studied S-shaped functions [6–13] verify the usability of this shape. However, previous publications have not thoroughly reviewed those functions analytically. In addition, some well-known trigonometric functions contain some parts that are S-shaped. Notwithstanding, the domain and codomain of trigonometric functions are bounded, so it does not reflect real-life quantities. So this paper (1) deeply analyzes some used S-shaped functions to summarize the advantages and the flaws, and subsequently (2) discusses a new type of S-shaped function, the transformation of the trigonometric function.

The rest of this article is organized into the following sections. The second section discusses the literature on the S-shaped functions in NHPP SRM. The next section introduces the transformation of the trigonometric function. The third section analyses those four S-shaped functions mathematically. Section 3 evaluates the applicability of the transformation in NHPP SRM. The last section summarizes all works and proposes some promising works to extend.

2 Software reliability modelling

2.1 NHPP SRM

The software testing timeline is modeled using the Poisson process. The terminology non-homogeneous is reflected by the parameter of the Poisson distribution, which is a time-dependent function. By time t, the software system has a(t) faults while testers detected and removed m(t) of them. So there are still a(t)–m(t) faults. Consider a short period $[eqn]$ from t to t  +  $[eqn]$ , the number of detected faults is m(t) and $[eqn]$  +  $[eqn]$ , respectively. Therefore, this period detects $[eqn]$ faults. As $[eqn]$ is a short time, assuming that the number of remaining faults, i.e., a(t)–m(t), and the fault detection rate, i.e., b(t), are constant. With notation of $[eqn]$ which $[eqn]$ , there is a relationship between those quantities:

$$[eqn]$$

Let $[eqn]$ , the differential equation is:

$$[eqn]$$

The general solution [10] of Eq (2) is given by:

$$[eqn]$$

where

$$[eqn]$$

In most cases, researchers focus on $[eqn]$ and t0 = 0, so (3) can be shortened to:

$$[eqn]$$

where

$$[eqn]$$

The main obstacle to proposing NHPP SRM is the integral calculation in Eq (5) when specific functions a(t) and b(t) are applied.

2.2 S-shaped functions in SRM

The term S-shaped reflects the growth of the function and can be partitioned into 5 phases:

Phase 1. Firstly, the function increases from a lower bound with a small growth rate.Phase 2. Next, it increases the growth rate; therefore, the function increases more quickly.Phase 3. The function goes through inflection point, which is the point that the first derivation of the function changes from positive to negative quantities.Phase 4. The growth rate decreases, so the function increases more slowly.Phase 5. Lastly, the function converges to the upper bound.

Generally, the 2nd and 4th phases are the main parts of the S-shaped function, which are the convex and concave functions, respectively. There are some types of S-shaped functions in NHPP SRM that have been used as fault detection rates. First form [6,7] is Eq (7):

$$[eqn]$$

Substitute into Eq (5):

$$[eqn]$$

Inflection form [8–11] is in Eq (11):

$$[eqn]$$

Substitute into Eq (5):

$$[eqn]$$

And extended inflection form [12–14] is in Eq (11):

$$[eqn]$$

Substitute into Eq (5):

$$[eqn]$$

There are some studied models which use S-shaped functions in Table 1.

Table 1: Some existing NHPP SRMs.

3 Trigonometric function transformation

3.1 Trigonometric function

Originally, trigonometric functions were the ratios between angles of the right triangle. The first analysis consideration [15] of a trigonometric function is in 1669 by the derivation of the power series for the sine function of Isaac Newton. There are three famous trigonometric functions sine $[eqn]$ , cosine $[eqn]$ , and tangent $[eqn]$ . Their reciprocals are the cosecant, the secant, and the cotangent, respectively. Each of those six functions has a corresponding inverse function (e.g. $[eqn]$ or $[eqn]$ or $[eqn]$ ) and hyperbolic functions ( $[eqn]$ , $[eqn]$ , $[eqn]$ , $[eqn]$ , $[eqn]$ , and $[eqn]$ ). The angle, i.e., the unit of the input of a trigonometric function, can be measured by degrees in geometry and radians in calculus, in which the value milestones are 360^∘^ and $[eqn]$ and their multiples, respectively. Trigonometric functions are periodic functions which period is $[eqn]$ or $[eqn]$ , i.e. $[eqn]$  +  $[eqn]$  +  $[eqn]$ and $[eqn]$  +  $[eqn]$  +  $[eqn]$ . The periodical of $[eqn]$ is illustrated at the top of Fig 1.

S-shaped as a part of sin(t) function.

The rest of this manuscript studies the most well-known trigonometric function $[eqn]$ . As shown at the bottom of Fig 1, a part of the $[eqn]$ function in $[eqn]$ has an S-shaped pattern. $[eqn]$ increase in all of $[eqn]$ while growth rate in $[eqn]$ and $[eqn]$ are positive and negative, respectively.

3.2 Trigonometric function transformation

$[eqn]$ has an S-shaped form, but the lower bound, upper bound, and growth rate are fixed [16]. In the period $[eqn]$ of length $[eqn]$ , the range of codomain is 2 when $[eqn]$ and $[eqn]$ and the inflection point is $[eqn]$ . To apply this trigonometric function, we need to transform it. Aspects that need to be transformed include (1) the domain range in which $[eqn]$ remains an S-shaped pattern, (2) the codomain as a couple of (min, max), and (3) the position of the inflection point as a moment when $[eqn]$ changes its behavior. As illustrated in Fig 2, those aspects can be met by the mathematical transformation of $[eqn]$ , which includes 4 steps:

Step 1. From $[eqn]$ , it is scaled vertically to change the range of codomain, i.e., $[eqn]$ :

$$[eqn]$$

Step 2. From Eq (13), it is scaled horizontally to change the length of the period in which $[eqn]$ remains in an S-shaped pattern:

$$[eqn]$$

Step 3. From Eq (14), it is shifted horizontally to change the bounds of the function domain:

$$[eqn]$$

Step 4. From Eq (15), it is shifted vertically to change the bounds of the function codomain:

$$[eqn]$$

Transformation from f(t)=sin(t) to f(t)=bsin(βt+φ)+k.

Table 2 shows the function b4(t) when some of variables vanishes. Because of their appearance in multiplication, b and $[eqn]$ vanish when they take a value of 1. On the other hand, $[eqn]$ and k appear in addition, so they vanish when taking a value of 0. For example, when $[eqn]$ and $[eqn]$ , b4(t) becomes $[eqn]$ .

Table 2: Function b4(t)=bsin(βt+φ)+k when some variables are vanished.

3.3 Applying trigonometric function transformation in NHPP SRM

Substitute Eq (16) into (6), we have:

$$[eqn]$$

Eq (5) becomes:

$$[eqn]$$

To get a solution of Eq (18), we should choose one of the following ways:

A suitable function $[eqn]$ .Considering the vanishment of some parameters of set $[eqn]$ .

or apply both on one single use. Substitute each case of b4(t) from Table 2 into Eq (18), m4(t) are in Table 2.

4 Numerical analysis

Let $[eqn]$ be a sign function of x. $[eqn]$ is an AND logic operator.

4.1 First form

Let consider:

$$[eqn]$$

If b = 0, b1(t) becomes a constant function of 0. If $[eqn]$ , we have:

$$[eqn]$$ $$[eqn]$$

The domain of b1(t), $[eqn]$ , and $[eqn]$ are:

$$[eqn]$$

The range of b1(t), $[eqn]$ , and $[eqn]$ are:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Fig 3 illustrates b1(t) in some cases of parameters: green line in case of b = 1.5, light blue line in case of b = 0.5, red line in case of b = 1.0, and orange line in case of b = −1.0. Pure lines are in rational cases and dashed lines in others.

Numerical illustration of b1(t).

Limitations of b1(t) and $[eqn]$ are in Table 3. So variation table of b1(t) when b > 0 and b < 0 are in Table 4 and Table 5, respectively. We can see that this form is not fully S-shaped because it only contains the two last phases of the S-shaped curve as described in sub-section 2.2. b1(t) does not have an early, quick growth period and an inflection point. In $[eqn]$ , it grows slowlier because the $[eqn]$ is negative, then $[eqn]$ is decreased. Furthermore, $[eqn]$ reflects the convergence of this function to an upper bound as illustrated in Fig 3.

Table 3: Limitations of b1(t) and ∂∂tb1(t).

Table 4: Variation table of b1(t) when b > 0.

Table 5: Variation table of b1(t) when b < 0.

4.2 Second form

Let consider:

$$[eqn]$$

We have some special cases:

If $[eqn]$ , b2(t) is indeterminate.If b = 0 and $[eqn]$ , b2(t) becomes constant function of 0.If $[eqn]$ , b2(t) becomes constant function of b.

In other cases in next subsub-sections when $[eqn]$ and $[eqn]$ , we have:

$$[eqn]$$ $$[eqn]$$

The domain of b2(t), $[eqn]$ , and $[eqn]$ are:

$$[eqn]$$

The range of b2(t), $[eqn]$ , and $[eqn]$ are:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

$[eqn]$ , then inflection point of b2(t) is $[eqn]$ when $[eqn]$ . Limitations of b2(t), $[eqn]$ , and $[eqn]$ are in Table 6.

Table 6: Limitations of b2(t) and ∂∂tb2(t).

So variation table of b2(t) are in Tables 7, 8, 9, and 10. Fig 4 illustrates b2(t) in some cases of parameters: blue line in case of $[eqn]$ , red line in case of $[eqn]$ , green line in case of $[eqn]$ , orange line in case of $[eqn]$ , and purple line in case of $[eqn]$ . Pure lines are in rational cases and dashed lines in others. The triangle with the corresponding filled color is the inflection point of each curve.

Numerical illustration of b2(t).

Table 7: Variation table of b2(t) when b > 0 and β >0.

Table 8: Variation table of b2(t) when b > 0 and β <0.

Table 9: Variation table of b2(t) when b < 0 and β >0.

4.3 Third form

Let consider:

$$[eqn]$$

We have some special cases:

If $[eqn]$ , b2(t) is indeterminate.If b = 0 and $[eqn]$ , b3(t) becomes constant function of 0.If $[eqn]$ , b3(t) becomes constant function of b.If k = 1, b3(t) becomes constant function of b.If k = 0, b3(t) becomes b2(t).

In other cases in next subsub-sections when $[eqn]$ , $[eqn]$ , and $[eqn]$ , we have:

$$[eqn]$$ $$[eqn]$$

The domain of b3(t), $[eqn]$ , and $[eqn]$ are:

$$[eqn]$$

The range of b3(t), $[eqn]$ , and $[eqn]$ are:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

So $[eqn]$ , then inflection point of b3(t) is $[eqn]$ when $[eqn]$ . Table 11 are limitations of b3(t), $[eqn]$ , and $[eqn]$ . So variation table of b3(t) when b > 0, $[eqn]$ , and k>1 are in Tables 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25.

Table 10: Variation table of b2(t) when b < 0 and β <0.

Table 11: Limitations of b3(t) and ∂∂tb3(t).

Table 12: Variation table of b3(t) when b > 0, β >0, and k > 1.

Table 13: Variation table of b3(t) when b > 0, β >0, and 0 < k < 1.

Table 14: Variation table of b3(t) when b > 0, β >0, and –1 < k < 0.

Table 15: Variation table of b3(t) when b > 0, β >0, and k<–1.

Figs 5, 6, and 7 illustrate b3(t) in cases of parameters. Fig 5 illustrates when b > 0 and $[eqn]$ which are green line in case of $[eqn]$ , blue line in case of $[eqn]$ , red line in case of $[eqn]$ , and orange line in case of $[eqn]$ .

Numerical illustration of b3(t) when b > 0 and β >0.

Numerical illustration of b3(t) when b < 0 and β >0.

Numerical illustration of b3(t) when β <0.

Table 16: Variation table of b3(t) when b > 0, β <0, and k > 1.

Table 17: Variation table of b3(t) when b > 0, β <0, and 0 < k < 1.

Table 18: Variation table of b3(t) when b > 0, β <0, and k < 0.

Table 19: Variation table of b3(t) when b < 0, β >0, and k > 1.

Table 20: Variation table of b3(t) when b < 0, β >0, and 0 < k < 1.

Table 21: Variation table of b3(t) when b < 0, β >0, and –1 < k < 0.

Table 22: Variation table of b3(t) when b < 0, β >0, and k<–1.

Table 23: Variation table of b3(t) when b < 0, β <0, and k > 1.

Table 24: Variation table of b3(t) when b < 0, β <0, and 0 < k < 1.

Table 25: Variation table of b3(t) when b < 0, β <0, and k < 0.

Fig 6 corresponds with b < 0 and $[eqn]$ . The parameters are similar to Fig 5 except b is negative: green line in case of $[eqn]$ , blue line in case of $[eqn]$ , red line in case of $[eqn]$ , and orange line in case of $[eqn]$ .

The last illustration of this third form is Fig 7 when $[eqn]$ : green line in case of $[eqn]$ , blue line in case of $[eqn]$ , red line in case of $[eqn]$ , orange line in case of $[eqn]$ , purple line in case of $[eqn]$ , and black line in case of $[eqn]$ . The triangle with the corresponding filled color is the inflection point of each curve. Pure lines are in rational cases and dashed lines in others.

4.4 Fourth form

Let consider:

$$[eqn]$$

So

$$[eqn]$$

And

$$[eqn]$$

Special cases of b4(t) are listed in Table 2. Because $[eqn]$ is a periodic function, it is necessary to evaluate the rational domain and the codomain of this function. Its period is $[eqn]$ , so $[eqn]$ is considered initially. After it is scaled and shifted horizontally, the domain becomes $[eqn]$ . After $[eqn]$ is scaled and shifted vertically, the codomain becomes $[eqn]$ . $[eqn]$ and $[eqn]$ are continuous function, so limitations of b4(t), $[eqn]$ , and $[eqn]$ at any point are those values at this point. Let consider:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

Therefore, without loss of generality, let’s focus on the case of $[eqn]$ instead of cases of $[eqn]$ , $[eqn]$ , and $[eqn]$ . From those analysis, b4(t) will be studied in $[eqn]$ when $[eqn]$ . Let t* is the solution of $[eqn]$ in $[eqn]$ , i.e. $[eqn]$ . As mentioned before, $[eqn]$ , so the range of b4(t), $[eqn]$ , and $[eqn]$ are:

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

So $[eqn]$ , then inflection points of b4(t) are $[eqn]$ , $[eqn]$ , $[eqn]$ . The variation table of b4(t) are in Tables 26, 27, and 28.

Table 26: Variation table of b4(t) when k > b or k<–b.

Table 27: Variation table of b4(t) when b > k > 0.

Table 28: Variation table of b4(t) when 0>k>–b.

Fig 8 illustrates b4(t) in some cases of parameters: red line in case of $[eqn]$ , blue line in case of $[eqn]$ , green line in case of $[eqn]$ , and orange line in case of $[eqn]$ . The triangle with the corresponding filled color is the inflection point of each curve.

Numerical illustration of b4(t).

4.5 Rational range of four S-shaped functions

Table 29 summarizes the properties of four S-shaped functions. The first column is the following aspects, which are divided into four groups by thicker lines. 172 Special cases of parameters of a function take on certain values, and the function could become a simpler form. For example, when a = 0, quadratic polynomial $[eqn]$  +  2x  +  3 becomes linear polynomial f1(x) = 2x + 3. 173 Domain is the set of legal inputs of function.

Table 29: Rational range of four S-shaped functions.

The next aspect to consider is whether the function has an S-shaped pattern or not. As mentioned in Sect 2.2, the 2nd phase, as convex, and the 4th phase, as concave, of the function are often focused on. So 174 Part S-shaped is the case where the function has a later part of an S-shaped curve. It is later instead of the earlier part because this part satisfies the real condition "the ability of the tester has a limitation, so it converges to a specific value after a long time". And then, 175 Fully S-shaped is when the function has the whole form of S-shaped.

Certainly, the fault detection rate, i.e., the ratio of detected errors to total system errors, has to reach a positive value. So it should analyze whether the S-shaped part of the function is positive or not. 176 Positive part S-shaped is in case S-shaped part of 174 is positive. 177 Positive 2 phases S-shaped relates to the newly perspective. As mentioned in Sect 2.2, the 2nd phase of convex and 4th phase of concave are the main parts of an S-shaped curve, so 177 evaluates the positive part of the curve that contains both two phases or just one of them. Similar to 176, 178 Positive fully S-shaped is in case the S-shaped curve of 175 is positive.

The next variable is time t, which should be analyzed in the positive range. 179 *Positive part S-shaped when $[eqn]$ *, 180 *Positive 2 phases part S-shaped when $[eqn]$ *, and 181 *Positive fully S-shaped when $[eqn]$ * are corresponded cases of 176, 177, and 178 when $[eqn]$ , respectively.

This tabular summary shows the advantages of the transformation of the trigonometric function over the previous S-shaped functions. The first, transformation of a trigonometric function, is the only function that can reach a fully S-shaped form when $[eqn]$ , which is presented in the last row. The second, this fourth S-shaped form b4(t) can reach all cases from 174 to 180 more easily than other forms. Furthermore, the conditions for achieving them are similar.

4.6 Numerical illustration

Fig 9 illustrates the form of 4 S-shaped functions b1(t) in Eq (7), b2(t) in Eq (9), b3(t) in Eq (11), and b4(t) in Eq (16). The red line is b1(t) when b = 1. The blue line is b2(t) when b = 2 and $[eqn]$ . With b3(t), the green line is b = 2, $[eqn]$ , and k = 0.2 while orange line is b = 2, $[eqn]$ , and k = 1.5. The purple line is b4(t) when b = 1.5, $[eqn]$ , $[eqn]$ , and k = 0.5.

Numerical illustration of 4 types of S-shaped function.

5 Conclusions and future works

This article evaluates the S-shaped characteristic of the most famous trigonometric function $[eqn]$ . The transformation of this function is studied to verify that it will be suitable for real conditions. The transformation methods include scaling vertically, scaling horizontally, shifting horizontally, and shifting vertically. The transformed function has been proven to have certain advantages and is suitable for NHPP SRM. The remaining work of SRM introduction is proposing a total fault function a(t) to substitute into Eq (18).

Some long-used S-shaped functions in Eqs (7), (9), and (11) and newly considered function in Eq (16) are analyzed mathematically. The considered characteristics are the domain, codomain, pattern, and the increment process of functions. These mathematical analyses support the derivation of the application domain of the functions, which ensures conformity with real-life conditions and maintains the S-shaped property.

In the next works, there are some promising approaches, such as finding a(t) or evaluating some simple forms of b4(t) by vanishing some parameters or both, to propose a new software reliability model.