A Software Reliability Model with Dependent Failure and Optimal Release Time

Abstract

In the past, because computer programs were restricted to perform only simple functions, the dependence on software was not large, resulting in relatively small losses after a failure. However, with the development of the software market, the dependence on software has increased considerably, and software failures can cause significant social and economic losses. Software reliability studies were previously conducted under the assumption that software failures occur independently. However, as software systems become more complex and extremely large, software failures are becoming frequently interdependent. Therefore, in this study, a software reliability model is developed under the assumption that software failures occur in a dependent manner. We derive the software reliability model through the number of software failure and fault detection rate assuming point symmetry. The proposed model proves good performance compared with 21 previously developed software reliability models using three datasets and 11 criteria. In addition, to find the optimal release time, a cost model using the developed software reliability model was presented. To determine this release time, four parameters constituting the software reliability model were changed by 10%. By comparing the change in the cost model and the optimal release time, it was found that parameter $b$ had the greatest influence.

1. Introduction

Software, one of the main components of a computer, plays an important role in the operation of physical devices. Software was originally developed with the ability to perform extremely small or simple functions. Currently, however, embedded systems that perform multiple functions are being developed. With the rapid development of the software market, technology has also developed, and software is now being used in all fields. Recently, the Internet of Things (IoT) based on the combination of various software, has been commercialized. Furthermore, AIoT (Artificial Intelligence of Things) combined AI (Artificial Intelligence) with IoT (Intelligence of Things) is developing [1]. It means that software has become a very important part not only in the industrial field but also in our daily life.

A software failure is caused by various faults (coding or system errors, etc.). In the past, software failures caused relatively small losses because the degree of software dependence was not as large. However, in today’s world, the degree of dependence on software is extremely high, and thus software failures can cause significant social and economic losses. Therefore, we measured the software reliability, which indicates the ability of a software program to avoid failure for a set period of time and refers to how long the software can be used without such a failure.

Early research on software reliability was conducted based on the assumption that software failures occur independently. Goel and Okumoto proposed the GO model, which is the most basic non-homogeneous Poisson process (NHPP) software reliability growth model [2]. The Hossain, Dahiya, Goel and Okumoto (HDGO) model further extended the GO model [3]. Yamada et al., Ohba, and Zhang et al. [4,5,6] proposed an NHPP S-shaped curve model in which the cumulative number of software failures increases to the S curve. In addition, Yamada et al. [7] proposed a new model in which the test effort invested during phase was reflected in the software reliability model. It is a model that reflects even the resources consumed for testing in the previously developed model. Furthermore, Yamada et al. [8] developed a software reliability model with a constant fault detection rate of $b\left(t\right)=b$, assuming incomplete debugging, in which faults detected during the test phase were corrected and removed.

The model developed by extending the above approach involved a generalized incomplete debugging-error detection rate model. Here, the fault detection rate $b\left(t\right)$ of the model is not a constant but rather a different function [9,10,11,12,13,14]. It started from the software error causing the failure being immediately eliminated and so a new error can be generated [9]. It progressed to that during the fault removal process, whether the fault is removed successfully or not, new faults are generated with a constant probability [13,14]. In addition, because the operating environment of the software is operated differently for each software program, a comparison is difficult to achieve. Therefore, in [15,16,17], a software reliability model was developed considering uncertain factors in the operating environment. Currently, research using non-parametric methods such as deep learning or machine learning is also being conducted [18,19,20,21].

Recently, finding the most optimal model for reliability prediction is an important concern. Through combination of analytic hierarchy method (AHP), hesitant fuzzy (HF) sets and techniques for order of preference by similarity to ideal solution (TOPSIS), Sahu et al., Ogundoyin et al., and Rafi et al. [22,23,24] found the most optimal software reliability model.

However, software failures often occur in a dependent manner because the developed software is composed of extremely complex structures [25]. Here, the dependent failure means that one failure affects other failures or increases the failure probability of other equipment [26,27]. There are two main types of dependent failure. A common cause failure is when several pieces of software fail simultaneously due to a common cause, and a cascading failure is a case in which a part of the system fails and affects other software as well. A software reliability model assuming a dependent failure was developed from the number of software failures and the fault detection rate, which have a dependency relationship in a software reliability model assuming incomplete debugging [28]. In addition, Lee et al. [29] presented a model that assumes that if past software failures are not corrected well, they will continue to have an effect.

In this study, a new software reliability model is developed under the assumption that software failures occur in a dependent manner. It is suitable for a general environment. We show the superiority of the newly developed dependent software reliability model through a comparison under various criteria. In addition, determining the optimal release time of the developed software is also important. If the test period is long, the software will be reliable, but the software development cost will increase. If the test period is short, the reliability of the product may decrease. Therefore, it is important to find a balance between time to market and minimum cost taking into account the installation costs, test costs, and error removal costs, etc. We propose a cost model that combines the proposal software reliability model and the cost coefficient [30,31,32,33]. In addition, among the various parameters of the proposed model, we propose a parameter that has a significant influence on predicting the number of cumulative failures through a variation in the cost model for changes in the parameters [34,35]. Section 2 introduces a new dependent software reliability model and its mathematical derivation. Section 3 introduces the data and criteria, as well as numerical results. Section 4 describes the optimal release time, and finally, in Section 5, we present our conclusions

2. New Dependent Software Reliability Model

Software reliability refers to the probability that the software will not fail a system for a certain period of time under certain conditions. In other words, it evaluates “how long the software can be used without failure”. The reliability function used to evaluate this is as follows:

$$R\left(t\right)=P(T>t)={\int }_t^{\infty }f\left(u\right)du$$

(1)

This denotes the probability of the software operating without failure over a specific time $t$. Here, the probability density function $f\left(t\right)$ assumes the software failure time or lifetime as a random variable $T$. When measuring reliability function $R\left(t\right)$, it is assumed that it follows an exponential distribution with parameter $\lambda$. In addition, it is assumed that the number of failures occurring in given unit time is a Poisson distribution with parameter $\lambda$. When $\lambda$ is a constant, it is the most basic form, and is called a homogeneous Poisson process. Extending this process, many researchers adapt a model where $\lambda$ is an intensity function $\lambda \left(t\right)$ that changes with time rather than a single constant by setting $\lambda$ as a non-homogeneous Poisson process(NHPP) rather than as a homogeneous Poisson process.

$$\mathrm{Pr}\left\{N\left(t\right)=n\right\}=\frac{{\left\{m\left(t\right)\right\}}^n}{n!}{e}^{-m\left(t\right)},n=0,1,2,\cdots ,t\ge 0$$

(2)

In Equation (2), $N\left(t\right)$ is the Poisson probability density function with the time dependent parameter $m\left(t\right)$. The $m\left(t\right)$ is a mean value function which is the integral of $\lambda \left(t\right)$ from $0$ to $t$ in Equation (3). The $\lambda \left(t\right)$ is the intensity function indicating the number of instantaneous failures at time $t$.

$$m\left(t\right)=\mathrm{E}\left[N\left(t\right)\right]={\int }_0^t\lambda \left(s\right)ds$$

(3)

A general class of NHPP software reliability models was proposed by Equation (9) to summarize the existing NHPP models as follows:

$$\frac{dm\left(t\right)}{dt}=b\left(t\right)\left[a\left(t\right)-m\left(t\right)\right]$$

(4)

where, the $m\left(t\right)$ is calculated using the relationship between the number of failures $a\left(t\right)$ at each time point and a fault detection rate $b\left(t\right)$ assuming point symmetry in Equation (4). Various software reliability models have been developed based on the assumption that software failures occur independently.

However, software failures occur not only independently but also dependently. If the failure is not completely fixed, it will continue to affect the next failure. In addition, as the system becomes more complex, the relationship between failure and failure also shows the dependent relationship because of the dependent combination of several software. Therefore, in this study, we assume that failure is dependent on other failures. The mean value function $m\left(t\right)$ based on NHPP software reliability model using the differential equation is as follows:

$$\frac{dm\left(t\right)}{dt}=b\left(t\right)\left[a\left(t\right)-m\left(t\right)\right]m\left(t\right)$$

(5)

In Equation (5), the $m\left(t\right)$ is multiplied once more to assume that the failure occurring from $0$ to $t$ affects another failure. We assume:

$$a\left(t\right)=a\left(1+\alpha t\right), b\left(t\right)=\frac{b}{1+c{e}^{-bt}}$$

(6)

where, $a\left(t\right)$ is the number of software failures at each time point and $b\left(t\right)$ is the fault detection rate. Parameter $a$ is the expected number of faults, $\alpha$ is the increasing rate of the number of faults, $b$ is the shape parameter, and $c$ is the scale parameter. When time $t$ changes, the change according to the values of parameters $b$ and $c$ in the fault detection rate $b\left(t\right)$ are as shown in Figure 1. When $b$ is $1$, it is blue, and when it is $1.5$, it is red. In addition, when $c$ is $1$, it is a dashed line, and when it is $2$, it is a dotted line. It can be seen that the larger $b$ is, the larger the $b\left(t\right)$ is.

When solving the differential equation by substituting $a\left(t\right)$ and $b\left(t\right)$ for $m\left(t\right)$ in Equations (5) and (6), we obtain Equation (7):

$$m\left(t\right)=\frac{{\left(c+e^{bt}\right)}^a{\left(\frac{c+e^{bt}}{c}\right)}^{a\alpha t}{e}^{\frac{a\alpha L{i}_2\left(\frac{e^{bt}+c}{c}\right)}{b}}}{\int \frac{{\left(c+e^{bt}\right)}^a{\left(\frac{c+e^{bt}}{c}\right)}^{a\alpha t}{e}^{\frac{a\alpha L{i}_2\left(\frac{e^{bt}+c}{c}\right)}{b}}b{e}^{bt}}{c+e^{bt}}dt+C}$$

(7)

where $L{i}_s\left(x\right)=\sum _{n=1}^{\infty }\frac{x^n}{n^s}$ is a polylogarithm when $s=2$. At this time, $\alpha =0$ in $a\left(t\right)$.

$$m\left(t\right)=\frac{h{\left(c+e^{bt}\right)}^a}{h\left[{\int }_0^t\frac{{\left(e^{bx}+c\right)}^{a}b{e}^{bx}}{c+e^{bx}}dx\right]+{\left(1+c\right)}^a}$$

(8)

where $h$ is the number of initial failures. In Equation (8), ${\int }_0^t\frac{{\left(e^{bx}+c\right)}^{a}b{e}^{bx}}{c+e^{bt}}dx$ is calculated through an integration using substitution. When $u=c+b{e}^{bx}$ and $du=b{e}^{bx}dx$, it is the same as in Equation (9).

$${\int }_0^t\frac{{\left(e^{bx}+c\right)}^{a}b{e}^{bx}}{c+e^{bt}}dx={\int }_{c+b}^{c+b{e}^{bt}}\frac{u^a}{u}du={\int }_{c+b}^{c+b{e}^{bt}}{u}^{a-1}du=\frac{u^a}{a}=\frac{{\left(c+e^{bt}\right)}^a}{a}$$

(9)

Substituting the result of the substitution integration into Equation (8), the final $m\left(t\right)$ is given by Equation (10).

$$m\left(t\right)=\frac{a}{1+\frac{a}{h}{\left(\frac{1+c}{c+e^{bt}}\right)}^a}$$

(10)

This can be presented as a general model of a dependent failure occurrence in the software reliability model. When $t=0$, $m\left(t\right)$ is $m\left(0\right)=ah/\left(a+h\right)$.

Table 1. Software reliability models.

No.	Model	Mean Value Function	Note
1	Goel-Okumoto (GO) [2]	$m\left(t\right)=a\left(1-e^{-bt}\right)$	Concave
2	Hossain-Dahiya (HDGO) [3]	$m\left(t\right)=\log \left[\frac{\left(e^a-c\right)}{\left(e^{a{e}^{-bt}}-c\right)}\right]$	Concave
3	Yamada et al. (DS) [4]	$m\left(t\right)=a\left(1-\left(1+\mathrm{bt}\right)e^{-bt}\right)$	S-Shape
4	Ohba (IS) [5]	$m\left(t\right)=\frac{a\left(t1-e^{-bt}\right)}{1+\beta e^{-bt}}$	S-Shape
5	Zhang et al. (ZFR) [6]	$m\left(t\right)=\frac{a}{p-\beta }\left[1-{\left(\frac{\left(1+\alpha \right)e^{-bt}}{1+\alpha e^{-bt}}\right)}^{\frac{c}{b}\left(p-\beta \right)}\right]$	S-Shape
6	Yamada et al. (YE) [7]	$m\left(t\right)=a\left(1-e^{-\gamma \alpha \left(1-e^{-\beta t}\right)}\right)$	Concave
7	Yamada et al. (YR) [7]	$m\left(t\right)=a\left(1-e^{-\gamma \alpha \left(1-e^{-\beta t^2/2}\right)}\right)$	S-Shape
8	Yamada et al. (YID 1) [8]	$m\left(t\right)=\frac{ab}{\alpha +b}\left(e^{\alpha t}-e^{-bt}\right)$	Concave
9	Yamada et al. (YID 2) [8]	$m\left(t\right)=a\left(1-e^{-bt}\right)\left(1-\frac{\alpha }{b}\right)+\alpha at$	Concave
10	Pham-Zhang (PZ) [9]	$m\left(t\right)=\frac{\left(\left(c+a\right)\left[1-e^{-bt}\right]-\left[\frac{ab}{b-\alpha }\right]\left(e^{-at}-e^{-bt}\right)\right)}{1+\beta e^{-bt}}$	Both
11	Pham et al. (PNZ) [10]	$m\left(t\right)=\frac{a\left(1-e^{-bt}\right)\left(1-\frac{\alpha }{b}\right)+\alpha at}{1+\beta e^{-bt}}$	Both
12	Teng-Pham (TP) [11]	$m\left(t\right)=\frac{a}{p-q}\left[1-{\left(\frac{\beta }{\beta +\left(p-q\right)\ln \left(\frac{c+e^{bt}}{c+1}\right)}\right)}^{\alpha }\right]$	S-Shape
13	Kapur et al. (KSRGM) [12]	$m\left(t\right)=\frac{A}{1-\alpha }\left[1-{\left(\left(1+bt+\frac{b^2{t}^2}{2}\right)e^{-bt}\right)}^{p\left(1-\alpha \right)}\right]$	S-Shape
14	Roy et al. (RMD) [13]	$m\left(t\right)=a\alpha \left[1-e^{-bt}\right]-\left[\frac{ab}{b-\beta }\left(e^{-\beta t}-e^{-bt}\right)\right]$	Concave
15	Pham (IFD) [14]	$m\left(t\right)=a\left(1-e^{-bt}\right)\left(1+\left(b+d\right)t+bd{t}^2\right)$	Concave
16	Pham (Vtub) [15]	$m\left(t\right)=\mathrm{N}\left[1-{\left(\frac{\beta }{\beta +a^{bt}-1}\right)}^{\alpha }\right]$	S-Shape
17	Chang et al. (TC) [16]	$m\left(t\right)=\mathrm{N}\left[1-{\left(\frac{\beta }{\beta +{\left(at\right)}^b}\right)}^{\alpha }\right]$	Both
18	Song et al. (3P) [17]	$m\left(t\right)=\mathrm{N}\left[1-\left(\frac{\beta }{\beta -\frac{a}{b}\ln \left(\frac{\left(1+c\right)e^{-bt}}{1+c{e}^{-bt}}\right)}\right)\right]$	S-Shape
19	Pham (DP1) [28]	$m\left(t\right)=\alpha \left(1+\beta t\right)\left(\beta t+\left(e^{-\beta t}-1\right)\right)$	Concave, Dependent
20	Pham (DP2) [28]	$m\left(t\right)=m_0\left(\frac{\gamma t+1}{\gamma t_0+1}\right)e^{-\gamma \left(t-t_0\right)}+\alpha \left(\gamma t+1\right)\left(\gamma t-1+\left(1-\gamma t_0\right)e^{-\gamma \left(t-t_0\right)}\right)$	Concave, Dependent
21	Lee et al. (DPF) [29]	$m\left(t\right)=\frac{a}{1+\frac{a}{h}{\left(\frac{b+c}{c+b{e}^{bt}}\right)}^{\frac{a}{b}}}$	S-Shape, Dependent
22	Proposed Model	$m\left(t\right)=\frac{a}{1+\frac{a}{h}{\left(\frac{1+c}{c+e^{bt}}\right)}^a}$	S-Shape, Dependent

shows the value of $m\left(t\right)$ of the existing software reliability model and the model proposed in this study. From models 19–22, it is assumed that a failure occurs in a dependent manner.

3. Numerical Examples

3.1. Data Information

Datasets 1 and 2 are derived from the online communication system (OCS) of ABC Software Co. and uses data accumulated over a $12$-week period. Datasets 1 and 2 show that the cumulative number of failures at $t=1, 2, \cdots , 12$ is $14, 17, \cdots , 81$, and $11, 17, \cdots , 81$, respectively [14]. Dataset 3 is the test data of a medical record system consisting of $188$ software titles and data for one of three releases. It shows that the cumulative number of failures is $90, 107, \cdots , 204$ for $t=1, 2, \cdots , 17$, respectively [36].

Table 2. Cumulative number of software failure datasets.

Index	Dataset 1		Dataset 2		Dataset 3
	Failures	Cumulative Failures	Failures	Cumulative Failures	Failures	Cumulative Failures
1	14	14	11	11	90	90
2	3	17	6	17	17	107
3	4	21	0	17	19	126
4	7	28	5	22	19	145
5	7	35	5	27	26	171
6	18	53	25	52	17	188
7	8	61	10	62	1	189
8	4	65	6	68	1	190
9	2	67	2	70	0	190
10	9	76	10	80	0	190
11	1	77	0	80	2	192
12	4	81	1	81	0	192
13					0	192
14					0	192
15					11	203
16					0	203
17					1	204

shows the accumulated failure data for datasets 1, 2, and 3. We compare the fit between the software reliability models with two failure datasets obtained from OCS and one dataset from Lee et al. [29], which showed good performance as the dependent models (DPF).

3.2. Criteria

This study compares various independent and dependent software reliability models and the proposed model introduced Table 1 using 11 criteria. Based on the difference between the actual observed value and the estimated value, we would like to find a better model by comparing it with criteria reflecting the number of parameters used in each model.

First, the mean squared error (MSE) is defined as the sum of squares of the distance between the estimated value and the actual value when considering the number of parameters and the number of observations [37].

$$\mathrm{MSE}=\frac{{\sum }_{i=1}^n{\left(\widehat{m}\left(t_i\right)-y_i\right)}^2}{n-m}$$

(11)

where $\widehat{m}\left(t_i\right)$ is the estimated value of the model $m\left(t\right)$, $y_i$ is the actual observed value, $n$ is the number of observations, and $m$ is the number of parameters in each model.

Second, the mean absolute error (MAE) defines the difference between the estimated number of failures and the actual value considering the number of parameters and the number of observations as the sum of the absolute values [38].

$$\mathrm{MAE}=\frac{{\sum }_{i=1}^n\left|\widehat{m}\left(t_i\right)-y_i\right|}{n-m}$$

(12)

Third, $\mathrm{Adj}\_{\mathrm{R}}^2$ is the modified coefficient of determination of the regression equation and determines how much explanatory power it has in consideration of the number of parameters [39].

$${\mathrm{R}}^2=1-\frac{{\sum }_{i=1}^n{\left(\widehat{m}\left(t_i\right)-y_i\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y_i}\right)}^2},\mathrm{Adj}\_{\mathrm{R}}^2=1-\frac{\left(1-R^2\right)\left(n-1\right)}{n-m-1}$$

(13)

Fourth, the predictive ratio risk (PRR) is obtained by dividing the distance from the actual value to the estimated value by the estimated value in relation to the model estimation [40].

$$\mathrm{PRR}={\displaystyle \sum }_{i=1}^n{\left(\frac{\widehat{m}\left(t_i\right)-y_i}{\widehat{m}\left(t_i\right)}\right)}^2$$

(14)

Fifth, the predictive power (PP) is obtained by dividing the distance from the actual value to the estimated value by the actual value [41].

$$\mathrm{PP}={\displaystyle \sum }_{i=1}^n{\left(\frac{\widehat{m}\left(t_i\right)-y_i}{y_i}\right)}^2$$

(15)

Sixth, Akaike’s information criterion (AIC) was used to compare likelihood function maximization. This is applied to maximize the Kullback–Leibler level between the probability distribution of the model and the data [42].

$$\begin{gathered} \mathrm{AIC}=-2\log L+2m \\ L={\displaystyle \prod }_{i=1}^n\frac{{\left(m\left(t_i\right)-m\left(t_{i-1}\right)\right)}^{y_i-y_{i-1}}}{\left(y_i-y_{i-1}\right)!}{e}^{-\left(m\left(t_i\right)-m\left(t_{i-1}\right)\right)} \\ \begin{array}{c}logL={\displaystyle \sum _{i=1}^n}\{\left(y_i-y_{i-1}\right)\ln \left(m\left(t_i\right)-m\left(t_{i-1}\right)\right)-\left(m\left(t_i\right)-m\left(t_{i-1}\right)\right)\\ -\ln \left(\left(y_i-y_{i-1}\right)!\right)\}\end{array} \end{gathered}$$

(16)

Seventh, the predicted relative variation (PRV) is the standard deviation of the prediction bias and is defined as [43]

$$\mathrm{PRV}=\sqrt{\frac{{\sum }_{i=1}^n{\left(y_i-\widehat{m}\left(t_i\right)-Bias\right)}^2}{n-1}}$$

(17)

Here, the bias is $\sum _{i=1}^n\left[\frac{\widehat{m}\left(t_i\right)-y_i}{n}\right]$.

The root mean square prediction error (RMSPE) can estimate the closeness with which the model predicts the observation [44]:

$$\mathrm{RMSPE}=\sqrt{Varianc{e}^2+Bia{s}^2}$$

(18)

Ninth, the mean error of prediction (MEOP) sums the absolute value of the deviation between the actual data and the estimated curve and is defined as [38]

$$\mathrm{MEOP}=\frac{{\sum }_{i=1}^n\left|\widehat{m}\left(t_i\right)-y_i\right|}{n-m+1}$$

(19)

Tenth, the Theil statistic (TS) is the average percentage of deviation over all periods with regard to the actual values. The closer the Theil statistic is to zero, the better the prediction capability of the model. This is defined as [45]

$$\mathrm{TS}=100\ast \sqrt{\frac{{\sum }_{i=1}^n{\left(y_i-\widehat{m}\left(t_i\right)\right)}^2}{{\sum }_{i=1}^n{y}_i{}^2}}\%$$

(20)

Eleventh, it takes into account the tradeoff between the uncertainty in the model and the number of parameters in the model by slightly increasing the penalty each time parameters are added to the model when the sample is considerably small [46].

$$\mathrm{PC}=\left(\frac{n-m}{2}\right)\log \left(\frac{{\sum }_{i=1}^n{\left(\widehat{m}\left(t_i\right)-y_i\right)}^2}{n}\right)+m\left(\frac{n-1}{n-m}\right)$$

(21)

Based on the above criteria, we compared the proposed model with the existing NHPP software reliability model. When $\mathrm{Adj}\_{\mathrm{R}}^2$ is closer to $1$, and the other $10$ criteria are closer to $0$, it indicates a better fit. Using R and MATLAB, the parameters of each model were estimated through the LSE method, and the goodness of fit is calculated to compare the superiority. This is a method of estimating parameters through the difference between the model in Table 1 and the actual number of failures in Table 2, and follows $\mathrm{LSE}=\sum _{t=1}^n{\left(y_t-m\left(t\right)\right)}^2$ [47].

3.3. Results of Dataset 1

Table 3. Parameter estimation of model from dataset 1.

No.	Model	Estimation
1	GO	$\widehat{a}=191.3881$, $\widehat{b}=0.0483$
2	HDOG	$\widehat{a}=191.3880$, $\widehat{b}=0.04832$, $\widehat{c}=1.3929$
3	DS	$\widehat{a}=92.0916$, $\widehat{b}=0.3034$
4	IS	$\widehat{a}=88.9815$, $\widehat{b}=0.3274$, $\widehat{\beta }=3.9383$
5	ZFR	$\widehat{a}=14.6285$, $\widehat{b}=0.21179$, $\widehat{\alpha }=33.5808$ $\widehat{\beta }=0.0304$, $\widehat{c}=15.1085$, $\widehat{p}=0.2085$
6	YE	$\widehat{a}=212.1517$, $\widehat{\alpha }=0.2021$ $\widehat{\beta }=0.00568$, $\widehat{\gamma }=38.0032$
7	YR	$\widehat{a}=101.8036$, $\widehat{\alpha }=0.5271$ $\widehat{\beta }=0.0276$, $\widehat{\gamma }=3.3321$
8	YID1	$\widehat{a}=181.0676$, $\widehat{b}=0.05131$, $\widehat{\alpha }=0.00114$
9	YID2	$\widehat{a}=140.9842$, $\widehat{b}=0.06636$, $\widehat{\alpha }=0.0126$
10	PZ	$\widehat{a}=27.3845$, $\widehat{b}=0.41396$, $\widehat{\alpha }=0.1349$ $\widehat{\beta }=4.5437$, $\widehat{c}=63.7781$
11	PNZ	$\widehat{a}=27.3845$, $\widehat{b}=0.0218$ $\widehat{\alpha }=0.0000332$, $\widehat{\beta }=0.0000684$
12	TP	$\widehat{a}=0.8506$, $\widehat{b}=0.38022$, $\widehat{\alpha }=0.6721$, $\widehat{\beta }=0.000333$ $\widehat{c}=114.6551$, $\widehat{p}=0.0122$, $\widehat{q}=0.00326$
13	KSRGM	$\widehat{A}=3.2070$, $\widehat{b}=8.76921$ $\widehat{\alpha }=0.9764$, $\widehat{p}=0.4157$
14	RMD	$\widehat{a}=78.5350$, $\widehat{b}=0.21915$ $\widehat{\alpha }=1.3358$, $\widehat{\beta }=0.2192$
15	IFD	$\widehat{a}=7.6597$, $\widehat{b}=0.84452$, $\widehat{d}=0.00171$
16	Vtub	$\widehat{a}=1.8954$, $\widehat{b}=0.70887$, $\widehat{\alpha }=6.7593$ $\widehat{\beta }=62.2968$, $\widehat{N}=83.1687$
17	TC	$\widehat{a}=0.1736$, $\widehat{b}=1.33331$, $\widehat{\alpha }=11.5457$ $\widehat{\beta }=18.8347$, $\widehat{N}=105.1851$
18	3P	$\widehat{a}=1.4640$, $\widehat{b}=0.3299$, $\widehat{\beta }=0.6008$ $\widehat{N}=94.1037$, $\widehat{c}=37.8421$
19	DP1	$\widehat{\alpha }=0.1104$, $\widehat{\beta }=2.5829$
20	DP2	$\widehat{\alpha }=46197.046$, $\widehat{\gamma }=0.00451$ $\widehat{t_0}=3.7402$, $\widehat{m_0}=30.01197$
21	DPF	$\widehat{a}=80.3065$, $\widehat{b}=0.06122$ $\widehat{c}=13.9314$, $\widehat{h}=9.8182$
22	Proposed model	$\widehat{a}=80.0907$, $\widehat{b}=0.07231$ $\widehat{c}=15.9288$, $\widehat{h}=9.8182$

shows the estimated values for the parameters of each model obtained using dataset 1. Each parameter of the proposed model is represented by $\widehat{a}=80.0907$, $\widehat{b}=0.07231$, $\widehat{c}=15.9288$, and $\widehat{h}=9.8182$. Figure 2 shows the result of calculating the estimated value of $m\left(t\right)$ at each time point based on the cumulative number of failures at each time point in dataset 1 and each model equation. The black dotted line represents the actual data, and the dark red solid line represents the predicted failure value at each time point of the proposed model. Compared with other models, it shows the predicted value closest to the actual value.

Table 4. Comparison of all criteria from dataset 1.

No.	Model	MSE	MAE	Adj_R2	PRR	PP	AIC	PRV	RMSPE	MEOP	TS	PC
1	GO	21.1918	4.4966	0.9627	0.4187	0.2758	80.5308	4.3889	4.3892	4.0878	7.6254	16.5565
2	HDOG	23.5465	4.9962	0.9581	0.4187	0.2758	82.5308	4.3889	4.3892	4.4966	7.6254	16.5875
3	DS	22.4994	3.8718	0.9604	9.4838	0.7292	92.7861	4.4119	4.5135	3.5198	7.8572	16.8558
4	IS	16.8528	3.8882	0.9700	1.4452	0.3769	80.9153	3.6786	3.7104	3.4994	6.4512	15.0824
5	ZFR	24.2415	5.8035	0.9540	1.2042	0.3441	86.8823	3.6068	3.6338	4.9744	6.3174	18.4848
6	YE	26.5773	5.6309	0.9519	0.4182	0.2753	84.5796	4.3963	4.3965	5.0052	7.6380	16.9984
7	YR	33.2210	5.1600	0.9398	23.6437	0.9460	103.5558	4.6865	4.8967	4.5867	8.5395	17.8909
8	YID1	23.6037	4.9679	0.9579	0.4140	0.2764	82.4537	4.3944	4.3946	4.4711	7.6347	16.5984
9	YID2	23.8374	5.0079	0.9575	0.4070	0.2771	82.5743	4.4162	4.4163	4.5071	7.6724	16.6427
10	PZ	217.3085	17.6277	0.5983	0.5816	1.2389	84.1256	4.7893	11.3435	15.4243	20.4300	24.8053
11	PNZ	4000.777	68.0533	−6.2441	2.6223	10.4504	142.6741	25.7721	52.1779	60.4918	93.7126	37.0551
12	TP	31.0301	7.0496	0.9387	1.6004	0.3946	89.3412	3.7102	3.7518	5.8747	6.5247	21.7987
13	KSRGM	26.0975	5.4865	0.9527	1.5345	0.4198	88.8357	4.3440	4.3556	4.8769	7.5688	16.9255
14	RMD	21.9129	5.0225	0.9604	1.1967	0.3659	84.6918	3.9784	3.9909	4.4644	6.9355	16.2264
15	IFD	29.9616	5.5723	0.9467	0.653	0.3059	87.6572	4.9344	4.9498	5.0151	8.6017	17.6717
16	Vtub	18.9012	4.8111	0.9650	0.7591	0.2784	82.2273	3.4511	3.4667	4.2097	6.0252	16.2579
17	TC	26.6474	5.8343	0.9507	1.8283	0.4312	89.1447	4.0938	4.1159	5.1050	7.1541	17.4601
18	3P	21.7357	5.0238	0.9599	1.4395	0.3767	84.9059	3.6859	3.7164	4.3958	6.4613	16.7470
19	DP1	361.825	19.1467	0.3631	448.095	3.2745	174.8811	14.9748	17.8944	17.4061	31.5087	30.7442
20	DP2	113.5394	11.4623	0.7945	0.5996	1.0133	109.7792	9.0867	9.0870	10.1888	15.7870	22.8067
21	DPF	9.9490	3.2278	0.9819	0.0689	0.0606	73.2850	2.6894	2.6899	2.8692	4.6732	13.0680
22	Proposed model	9.9274	3.2140	0.9821	0.0647	0.0577	73.4605	2.6866	2.6870	2.8569	4.6682	13.0594

shows the results of calculating the criteria of each model using the parameters obtained through dataset 1. As a result, the values of $\mathrm{MSE}$, $\mathrm{MAE}$, $\mathrm{PRR}$, $\mathrm{PP}$, $\mathrm{PRV}$, $\mathrm{RMSPE}$, $\mathrm{MEOP}$, $\mathrm{TS}$, $\mathrm{PC}$ of the proposed model show the smallest values of $9.9274$, $3.2140$, $0.0647$, $0.0577$, $2.6866$, $2.6870$, $2.8569$, $4.6682$, and $13.0594$, respectively. The $\mathrm{AIC}$ shows the second smallest value at $73.4605$. In addition, $\mathrm{Adj}\_{\mathrm{R}}^2$ is 0.9821, which is the closest to $1$. The DPF model shows the highest value with $\mathrm{AIC}=73.2850$, and the second highest result for the other criteria. The model with the third highest criterion is the Vtub model.

3.4. Results of Dataset 2

Table 5. Parameter estimation of model from dataset 2.

No.	Model	Estimation
1	GO	$\widehat{a}=518.0607$, $\widehat{b}=0.0156$
2	HDOG	$\widehat{a}=518.0607$, $\widehat{b}=0.01561,\widehat{c}=15.4479$
3	DS	$\widehat{a}=105.5701$, $\widehat{b}=0.2548$
4	IS	$\widehat{a}=85.7894$, $\widehat{b}=0.4918$, $\widehat{\beta }=13.4498$
5	ZFR	$\widehat{a}=4.3416$, $\widehat{b}=0.3506$, $\widehat{\alpha }=67.6487$ $\widehat{\beta }=0.00128$, $\widehat{c}=57.1077$, $\widehat{p}=0.0548$
6	YE	$\widehat{a}=583.6809$, $\widehat{\alpha }=0.0700$ $\widehat{\beta }=0.00224$, $\widehat{\gamma }=88.6105$
7	YR	$\widehat{a}=0.8619$, $\widehat{\alpha }=5.0727$ $\widehat{\beta }=0.0000000232$, $\widehat{\gamma }=0.3849$
8	YID1	$\widehat{a}=369.1037$, $\widehat{b}=0.02206$, $\widehat{\alpha }=0.00446$
9	YID2	$\widehat{a}=174.7977$, $\widehat{b}=0.04618$, $\widehat{\alpha }=0.0292$
10	PZ	$\widehat{a}=80.3516$, $\widehat{b}=0.5195$, $\widehat{\alpha }=4.0448$ $\widehat{\beta }=15.4266$, $\widehat{c}=4.4447$
11	PNZ	$\widehat{a}=82.6869$, $\widehat{b}=0.5259$ $\widehat{\alpha }=0.0013$, $\widehat{\beta }=15.5683$
12	TP	$\widehat{a}=0.2772$, $\widehat{b}=0.5328$, $\widehat{\alpha }=0.5551$, $\widehat{\beta }=0.000359$ $\widehat{c}=76.4359$, $\widehat{p}=0.5610$, $\widehat{q}=0.5583$
13	KSRGM	$\widehat{A}=85.4315$, $\widehat{b}=0.3829$ $\widehat{\alpha }=0.0353$, $\widehat{p}=1.5904$
14	RMD	$\widehat{a}=102.8360$, $\widehat{b}=0.2355$ $\widehat{\alpha }=1.0649$, $\widehat{\beta }=0.2355$
15	IFD	$\widehat{a}=21.4202$, $\widehat{b}=0.2758$, $\widehat{d}=0.00001004$
16	Vtub	$\widehat{a}=2.1725$, $\widehat{b}=0.7383$, $\widehat{\alpha }=48.1784$ $\widehat{\beta }=961.5799$, $\widehat{N}=81.1633$
17	TC	$\widehat{a}=0.1524$, $\widehat{b}=1.9761$, $\widehat{\alpha }=21.6626$ $\widehat{\beta }=22.4388$, $\widehat{N}=87.7121$
18	3P	$\widehat{a}=2.3685$, $\widehat{b}=0.4823$, $\widehat{\beta }=1.1973$ $\widehat{N}=94.4688$, $\widehat{c}=59.8790$
19	DP1	$\widehat{\alpha }=0.01104$, $\widehat{\beta }=8.2449$
20	DP2	$\widehat{\alpha }=4.1249$, $\widehat{\gamma }=0.0000000161$ $\widehat{t_0}=0.0000168$, $\widehat{m_0}=0.000213$
21	DPF	$\widehat{a}=79.1447$, $\widehat{b}=0.1928$ $\widehat{c}=69.3637$, $\widehat{h}=9.2699$
22	Proposed model	$\widehat{a}=79.1444$, $\widehat{b}=0.2001$ $\widehat{c}=72.3208$, $\widehat{h}=9.3327$

shows the estimated values for the parameters of each model obtained using dataset 2. Each parameter of the proposed model is represented as $\widehat{a}=79.1444$, $\widehat{b}=0.2001$, $\widehat{c}=72.3208$, and $\widehat{h}=9.3327$. Figure 3 shows the results of calculating the estimated value of $m\left(t\right)$ for each point in time based on the cumulative number of failures at each time point in dataset 2 and each model equation. Here, the black dotted line represents the actual data, whereas the dark red solid line represents the predicted failure value at each time point of the proposed model. Compared with the other models, the predicted value is closest to the actual value.

Table 6. Comparison of all criteria from dataset 2.

No.	Model	MSE	MAE	Adj_R2	PRR	PP	AIC	PRV	RMSPE	MEOP	TS	PC
1	GO	52.7922	6.9705	0.9252	0.4713	0.6631	113.9369	6.9157	6.9267	6.3369	11.8896	21.1202
2	HDOG	58.6580	7.7450	0.9159	0.4713	0.6631	115.9369	6.9157	6.9267	6.9705	11.8896	20.6949
3	DS	40.0324	5.9373	0.9433	8.5312	1.0273	116.8063	5.9998	6.0299	5.3975	10.3536	19.7368
4	IS	30.7961	5.1688	0.9559	5.1137	0.8444	102.4472	4.9413	5.0132	4.6519	8.6150	17.7954
5	ZFR	42.3215	7.0918	0.9353	3.2756	0.7195	104.7204	4.7496	4.8001	6.0787	8.2459	20.1564
6	YE	66.0197	8.6990	0.9038	0.4695	0.6677	117.922	6.9138	6.9280	7.7325	11.8923	20.6380
7	YR	4668.125	73.375	−5.7994	4.00 × 1017	12.000	2807.273	28.0112	56.369	65.2222	100.000	37.6722
8	YID1	58.7470	7.7242	0.91571	0.4677	0.6707	115.8113	6.9207	6.9319	6.9518	11.8987	20.7017
9	YID2	58.9848	7.7955	0.91544	0.4730	0.6551	116.2140	6.9390	6.9463	7.0159	11.9227	20.7199
10	PZ	42.3248	6.7234	0.9371	11.0678	1.0170	110.5663	5.0554	5.1787	5.8830	8.9070	19.0795
11	PNZ	35.2459	5.7358	0.9486	6.5289	0.9046	104.9515	4.8955	5.0492	5.0985	8.6893	18.1275
12	TP	63.8102	10.1380	0.8983	4.8099	0.8346	112.9315	5.0230	5.3563	8.4484	9.2430	23.6011
13	KSRGM	50.5240	6.8779	0.9265	114.740	1.4819	135.0029	5.8715	6.0461	6.1137	10.4035	19.5679
14	RMD	49.4957	7.4822	0.9279	3.9839	0.8882	116.9778	5.9847	5.9985	6.6509	10.297	19.4857
15	IFD	54.1681	7.6923	0.9223	0.7680	0.6392	116.0510	6.6572	6.6573	6.9231	11.4255	20.3365
16	Vtub	35.2724	6.0531	0.9476	2.4728	0.6598	101.2283	4.6866	4.7335	5.2965	8.1311	18.4415
17	TC	50.7521	7.5092	0.9245	21.0460	1.1721	121.4836	5.5793	5.6745	6.5706	9.7535	19.7150
18	3P	40.5560	6.8976	0.9397	4.5115	0.8180	107.3639	5.0151	5.0748	6.0354	8.7189	18.9300
19	DP1	319.4078	17.5031	0.5477	219.278	2.8473	190.4276	14.6847	16.8565	15.9119	29.2453	30.1207
20	DP2	4668.094	73.3747	−5.7994	8.23 × 1011	11.9998	5,039.181	28.0112	56.3689	65.2219	99.9997	37.6722
21	DPF	19.0466	4.3652	0.9722	0.1630	0.1495	92.2767	3.7212	3.7218	3.8802	6.3876	15.6657
22	Proposed model	18.9722	4.3544	0.9723	0.1615	0.1482	92.2155	3.7139	3.7145	3.8706	6.3751	15.6500

shows the results of calculating the criteria of each model using the parameters obtained through dataset 2. As a result, the values of $\mathrm{MSE}$, $\mathrm{MAE}$, $\mathrm{PRR}$, $\mathrm{PP}$, $\mathrm{AIC}$, $\mathrm{PRV}$, $\mathrm{RMSPE}$, $\mathrm{MEOP}$, $\mathrm{TS}$, and $\mathrm{PC}$ of the proposed model are $18.9722$, $4.3544$, $0.1615$, $0.1482$, $92.2155$, $3.7139$, $3.7145$, $3.8706$, $6.3751$, and $15.6500$, respectively, which show the smallest criteria. $\mathrm{Adj}\_{\mathrm{R}}^2$ is $0.9723$, which is the closest to 1. The model with the second highest criterion is DPF, and Vtub is the third best-fitting model.

3.5. Results of Dataset 3

Table 7. Parameter estimation of model from dataset 3.

No.	Model	Estimation
1	GO	$\widehat{a}=197.387$, $\widehat{b}=0.399$
2	HDOG	$\widehat{a}=197.3858$, $\widehat{b}=0.3985$, $\widehat{c}=0.00088$
3	DS	$\widehat{a}=192.528$, $\widehat{b}=0.882$
4	IS	$\widehat{a}=197.354$, $\widehat{b}=0.399$, $\widehat{\beta }=0.000001$
5	ZFR	$\widehat{a}=198.0864$, $\widehat{b}=0.0038$, $\widehat{\alpha }=1545.538$ $\widehat{\beta }=3.7206$, $\widehat{c}=603.6647$, $\widehat{p}=4.7236$
6	YE	$\widehat{a}=248.808$, $\widehat{\alpha }=0.00797$ $\widehat{\beta }=0.2253$, $\widehat{\gamma }=208.4032$
7	YR	$\widehat{a}=206.0833$, $\widehat{\alpha }=1.0937$ $\widehat{\beta }=0.1427$, $\widehat{\gamma }=2.3984$
8	YID1	$\widehat{a}=183.4522$, $\widehat{b}=0.4620$, $\widehat{\alpha }=0.0066$
9	YID2	$\widehat{a}=182.934$, $\widehat{b}=0.464$, $\widehat{\alpha }=0.0071$
10	PZ	$\widehat{a}=195.990$, $\widehat{b}=0.3987$, $\widehat{\alpha }=1000.0$, $\widehat{\beta }=0.0000$, $\widehat{c}=1.390$
11	PNZ	$\widehat{a}=183.125$, $\widehat{b}=0.463$ $\widehat{\alpha }=0.007$, $\widehat{\beta }=0.0001$
12	TP	$\widehat{a}=21.2071$, $\widehat{b}=0.3086$, $\widehat{\alpha }=0.9415$, $\widehat{\beta }=0.0209$ $\widehat{c}=1.8073$, $\widehat{p}=0.2950$, $\widehat{q}=0.1959$
13	KSRGM	$\widehat{A}=61.8904$, $\widehat{b}=53.2238$ $\widehat{\alpha }=0.6853$, $\widehat{p}=0.0256$
14	RMD	$\widehat{a}=5.4873$, $\widehat{b}=0.3986$ $\widehat{\alpha }=35.9711$, $\widehat{\beta }=208.4596$
15	IFD	$\widehat{a}=23.5220$, $\widehat{b}=0.6188$, $\widehat{d}=0.000000002$
16	Vtub	$\widehat{a}=1.2170$, $\widehat{b}=2.9515$, $\widehat{\alpha }=0.0595$ $\widehat{\beta }=0.000006$, $\widehat{N}=194.7808$
17	TC	$\widehat{a}=0.053$, $\widehat{b}=0.774$, $\widehat{\alpha }=181.0$ $\widehat{\beta }=38.600$, $\widehat{N}=204.140$
18	3P	$\widehat{a}=0.0307$, $\widehat{b}=0.2038$, $\widehat{\beta }=0.000581$ $\widehat{N}=203.8418$, $\widehat{c}=100.8152$
19	DP1	$\widehat{\alpha }=0.1105$, $\widehat{\beta }=3.1046$
20	DP2	$\widehat{\alpha }=0.0290$, $\widehat{\gamma }=6.0596$ $\widehat{t_0}=0.7697$, $\widehat{m_0}=0.0387$
21	DPF	$\widehat{a}=194.766$, $\widehat{b}=0.304$ $\widehat{c}=304.566$, $\widehat{h}=135.464$
22	Proposed model	$\widehat{a}=194.7684$, $\widehat{b}=0.3062$ $\widehat{c}=307.0805$, $\widehat{h}=135.5641$

shows the estimated values for the parameters of each model obtained using dataset 3. Each parameter of the proposed model is represented through $\widehat{a}=194.7684$, $\widehat{b}=0.3062$, $\widehat{c}=307.0805$, and $\widehat{h}=135.5641$. Figure 4 shows the results of calculating the estimated value of $m\left(t\right)$ at each point in time based on the number of cumulative failures at each time point in dataset 3 and for each model equation. The black dotted line indicates the actual data, and the dark red solid line is the predicted failure value at each time point for the proposed model. Compared with other models, the proposed model shows the predicted value closest to the actual value.

Table 8. Comparison of all criteria from dataset 3.

No.	Model	MSE	MAE	Adj_R2	PRR	PP	AIC	PRV	RMSPE	MEOP	TS	PC
1	GO	80.6779	6.9602	0.9301	0.1705	0.1013	184.3314	8.6734	8.6955	6.5252	4.7492	34.1231
2	HDOG	86.4370	7.4526	0.9247	0.1714	0.1015	186.2332	8.6705	8.6951	6.9558	4.7491	33.2854
3	DS	232.628	9.5029	0.7982	1.2915	0.3330	331.8567	14.6423	14.7605	8.9090	8.0644	42.0654
4	IS	86.4395	7.4550	0.9247	0.1706	0.1013	186.3337	8.6711	8.6953	6.9580	4.7492	33.2856
5	ZFR	111.138	9.4352	0.9010	0.1837	0.1047	193.0813	8.7023	8.7388	8.6489	4.7734	32.2423
6	YE	79.3698	8.1539	0.9304	0.0993	0.0738	167.3203	8.0202	8.0298	7.5715	4.3853	31.6111
7	YR	378.666	12.8708	0.6679	2.7655	0.4685	523.3189	17.3551	17.5296	11.9514	9.5785	41.7676
8	YID1	78.8312	7.1635	0.9313	0.1282	0.0868	157.6388	8.2937	8.3046	6.6860	4.5353	32.6406
9	YID2	78.8367	7.1869	0.9313	0.1276	0.0866	157.8252	8.2915	8.3047	6.7078	4.5355	32.6411
10	PZ	100.990	8.6962	0.9108	0.1719	0.1017	190.3321	8.6767	8.7014	8.0272	4.7525	32.2669
11	PNZ	84.9077	7.7388	0.9256	0.1281	0.0867	159.8744	8.2915	8.3049	7.1860	4.5356	32.0493
12	TP	98.6971	10.573	0.9113	0.0740	0.0626	166.0911	7.8528	7.8540	9.6118	4.2889	31.5071
13	KSRGM	111.132	8.1738	0.9025	0.2521	0.1297	NA	9.4631	9.5000	7.5900	5.1890	33.7990
14	RMD	93.1051	8.0261	0.9184	0.1714	0.1015	188.2567	8.6714	8.6960	7.4528	4.7496	32.6486
15	IFD	3691.538	60.2705	−2.2183	24.7762	2.5036	466.1166	51.3584	56.5265	56.2524	31.0359	59.5661
16	Vtub	28.6529	5.2313	0.9747	0.0094	0.0090	Inf	4.6356	4.6357	4.8289	2.5315	24.7084
17	TC	72.2812	8.5966	0.9361	0.0521	0.0479	158.9319	7.3621	7.3628	7.9353	4.0207	30.2602
18	3P	81.0554	8.8835	0.9284	0.0731	0.0614	164.5417	7.7935	7.7967	8.2001	4.2577	30.9476
19	DP1	11,068.48	101.169	−8.6006	9,218.87	6.8842	1224.361	78.7966	100.6555	94.8460	55.6271	71.0335
20	DP2	12,760.68	116.690	−10.191	11,546.76	6.8801	1248.593	78.7816	100.6144	108.3548	55.6039	64.6312
21	DPF	26.8104	4.9195	0.9765	0.0096	0.0092	NA	4.6673	4.6673	4.5682	2.5487	24.5565
22	Proposed model	26.8047	4.9209	0.9765	0.0096	0.0092	NA	4.6668	4.6668	4.5694	2.5484	24.5551

shows the results of calculating the criteria of each model using the parameters obtained through dataset 3. As a result, $\mathrm{MSE}$ and $\mathrm{PC}$ of the proposed model show the smallest values of $26.8047$ and $24.5551$, respectively, and $\mathrm{Adj}\_{\mathrm{R}}^2$ shows the closest value to $1$ at $0.9765$. In addition, $\mathrm{MAE}$, $\mathrm{PRR}$, $\mathrm{PP}$, $\mathrm{PRV}$, $\mathrm{RMSPE}$, $\mathrm{MEOP}$, and $\mathrm{TS}$ are $4.9209$, $0.0096$, $0.0092$, $4.6668$, $4.6668$, $4.5694$, and $2.5484$, respectively, showing the second smallest values. Figure 4 shows the estimated failure values at each time point using the developed models. The Vtub model shows the most suitable criteria of $0.0094$, $0.0090$, $4.6356$, $4.6357$, and $2.5315$ in $\mathrm{PRR}$, $\mathrm{PP}$, $\mathrm{PRV}$, $\mathrm{RMSPE}$, and $\mathrm{TS}$, and DPF shows the most suitable criteria of $4.9195$ and $4.5682$ with $\mathrm{MAE}$ and $\mathrm{MEOP}$. However, in calculating the AIC of the Vtub model, DPF, KSRGM, and the newly proposed model, a value indicating $t=14$ is shown, indicating that the calculation is no longer being applied. In the process of calculating the AIC value, if there is no difference between the value at a specific point in time and the next point in time, the denominator is 0, so the AIC calculation can not be performed.

4. Optimal Release Time

When releasing software, it is very important that find the optimal release time. In order to find that, we need to find a time that minimizes the cost. We apply $m\left(t\right)$ proposed in Section 2 to the cost model to find the optimal time point between time to market and the minimum cost. The optimal time is suggested based on the cost model that reflects the software installation cost, software test cost, operation cost, software removal cost, and risk cost when the software failure occurs. Figure 5 describes the software field environment from the software installation of the software cost model. The expected software cost model follows Equation (22) [30,31].

$$C\left(T\right)=C_0+C_{1}T+C_{2}m\left(T\right)+C_3\left(1-R\left(x|T\right)\right)$$

(22)

where $C_0$ is the installation cost for system testing, $C_1$ is the system test cost per unit time, $C_2$ is the error removal cost per unit time during the test phase, and $C_3$ is the penalty cost owing to a system failure. In addition, $x$ represents the time the software was used. In addition, in the cost model equation, $R\left(x|T\right)$ follows (23) [32,33].

$$R\left(x|T\right)=e^{-\left[m\left(t+x\right)-m\left(t\right)\right]}$$

(23)

In this section, we propose a cost model using dataset 1 based on the proposed software reliability model and find the optimal time point between time to market and the minimum cost by changing the cost coefficients from $C_0$ to $C_3$.

4.1. Results of the Optimal Release Time

For the parameters of the cost model, $a,b,c,\mathrm{and}h$ calculated through numerical examples described in Section 3 were used. The cost coefficient of the cost model aims to find the optimal release time with the lowest cost by finding the optimal value through the changes in several values. The baseline value of the cost coefficient is as follows:

$$C_0=500,C_1=20,C_2=50,C_3=5000,x=6$$

Here, baseline denotes to the reference value for confirming the change of the cost coefficient. The total cost value obtains as a reference value is $4888.856$, and the optimal release time $T$ at this time is $18.3$.

Table 9. Optimal release time of expected total cost according to baseline.

Base	x = 2		x = 4		x = 6		x = 8		x = 10
	T*	C(T)	T*	C(T)	T*	C(T)	T*	C(T)	T*	C(T)
	18.2	4886.985	18.3	4888.735	18.3	4888.856	18.3	4888.863	18.3	4888.863

changes the cost coefficient of each reference value, checks the minimum cost $C\left(T\right)$ and optimal release time $T^{*}$, and then checks the changing trend to find the most optimal release time $T^{*}$. When $x=2$, the smallest total cost value obtains $4886.985$ at $T^{*}=18.2$. When $x=4$, the smallest total cost value shows $4888.735$ at $T^{*}=18.3$. When $x=6$, the smallest total cost value shows $4888.856$ at $T^{*}=18.3$. When $x$ is $8$ and $10$, the smallest total cost value shows $4888.863$ at $T^{*}=18.3$.

Here, $C_0$ is the setup cost, and as the value increases, the cost, which is directly proportional, increases as well; thus, the lower the setup cost is, the lower the cost.

Table 10. Optimal release time of expected total cost according to $C_0$.

C0	x = 2		x = 4		x = 6		x = 8		x = 10
	T*	C(T)	T*	C(T)	T*	C(T)	T*	C(T)	T*	C(T)
300	18.2	4686.985	18.3	4688.735	18.3	4688.856	18.3	4688.863	18.3	4688.863
500	18.2	4886.985	18.3	4888.735	18.3	4888.856	18.3	4888.863	18.3	4888.863
700	18.2	5086.985	18.3	5088.735	18.3	5088.856	18.3	5088.863	18.3	5088.863

compares the changes when the coefficients of are $300$, $500$, and $700$. It is found that the higher the value is, the higher the total cost value, whereas the optimal time does not change. Therefore, it appears that $C_0$ does not help determine the optimal release point. However, because the setup cost for a system stabilization is required, the appropriate $C_0$ cost coefficient is set to $500$. Figure 6 shows a graph of the results according to the change in $C_0$.

Table 11. Optimal release time of expected total cost according to $C_1$.

${\mathit{C}}_1$	$\mathit{x}=2$		$\mathit{x}=4$		$\mathit{x}=6$		$\mathit{x}=8$		$\mathit{x}=10$
	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$
10	18.9	4702.044	19.0	4702.818	19.0	4702.864	19.0	4702.866	19.0	4702.866
20	18.2	4886.985	18.3	4888.735	18.3	4888.856	18.3	4888.863	18.3	4888.863
30	17.8	5066.820	17.9	5069.652	17.9	5069.861	17.9	5069.873	17.9	5069.874

compares the changes when the coefficients of $C_1$ are $10$, $20$, and $30$. The results show that when $C_1$ is $10$, the total cost is the minimum at approximately $18.9$ to $19.0$, and when $C_1$ is $20$, the minimum value is at $18.2$ to $18.3$, and when it is $30$, the total cost shows the minimum value at approximately $17.8$ to $17.9$. As the cost coefficient $C_1$ increases, the optimal release time is gradually pushed back. Figure 7 shows a graph of the results according to the changes in $C_1$.

Table 12. Optimal release time of expected total cost according to $C_2$.

${\mathit{C}}_2$	$\mathit{x}=2$		$\mathit{x}=4$		$\mathit{x}=6$		$\mathit{x}=8$		$\mathit{x}=10$
	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$
30	18.2	3285.254	18.3	3286.995	18.3	3287.116	18.3	3287.123	18.3	3287.123
40	18.2	4086.120	18.3	4087.865	18.3	4087.986	18.3	4087.993	18.3	4087.993
50	18.2	4886.985	18.3	4888.735	18.3	4888.856	18.3	4888.863	18.3	4888.863
60	18.2	5687.851	18.3	5689.604	18.3	5689.726	18.3	5689.733	18.3	5689.733

compares the changes when the coefficients of $C_2$ are $30$, $40$, $50$, and $60$. It can be seen that the cost coefficient $C_2$ does not change from $18.2$ to $18.3$ at the optimal release time as the value changes. Figure 8 shows a graph of the results according to the change in $C_2$.

Table 13. Optimal release time of expected total cost according to $C_3$.

${\mathit{C}}_3$	$\mathit{x}=2$		$\mathit{x}=4$		$\mathit{x}=6$		$\mathit{x}=8$		$\mathit{x}=10$
	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$
5000	18.2	4886.985	18.3	4888.735	18.3	4888.856	18.3	4888.863	18.3	4888.863
7000	18.5	4893.210	18.6	4894.846	18.6	4894.958	18.6	4894.964	18.6	4894.964
10,000	18.9	4899.648	19.0	4901.186	19.0	4901.277	19.0	4901.281	19.0	4901.282
15,000	19.2	4906.802	19.3	4908.208	19.3	4908.297	19.3	4908.301	19.3	4908.301

compares the changes when the coefficients of $C_3$ are $5000$, $7000$, $\mathrm{10,000}$, and $\mathrm{15,000}$. The results show that when $C_3$ is $5000$, the total cost is the minimum at approximately $18.2$ to $18.3$; when it is $7000$, it shows the minimum value at $18.5$ to $18.6$; when it is $\mathrm{10,000}$, the total cost shows the minimum value at approximately $18.9$ to $19.0;$ and when it is $\mathrm{15,000}$, the total cost shows the minimum value at approximately $19.2$ to $19.3$. This indicates that the optimal release time gradually increases as the cost coefficient $C_3$ increases. Figure 9 shows a graph of the results according to the changes in $C_3$.

4.2. Results of Variation in Cost Model for Changes in Parameter

In this section, we check whether the optimal release time is affected by the change in the cost model according to the change in the parameters of the proposed model. The parameters $a,b,c,$ and $h$ of the proposed model are set at −20%, −10%, 0%, 10%, and 20%, respectively, in 10% increments, and the coefficient of the cost model is fixed at the baseline value in Section 4.1. Thus, the minimum cost value is calculated depending on changes in the parameters, and it derives appropriate release time. In

Table 14. Optimal release time of cost according to parameter change.

	$-20\mathit{\%}$		$-10\mathit{\%}$		$0\mathit{\%}$		$10\mathit{\%}$		$20\mathit{\%}$
	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$	${\mathit{T}}^{*}$	$\mathit{C}\left(\mathit{T}\right)$
$a$	20.4	4129.849	19.2	4507.887	18.3	4888.856	17.6	5272.215	16.8	5657.258
$b$	22.6	4979.801	20.0	4929.426	18.3	4888.856	16.8	4855.545	15.4	4827.546
$c$	16.4	4847.806	17.4	4869.089	18.3	4888.856	19.2	4907.319	20.0	4924.648
$h$	18.6	4892.931	18.4	4890.757	18.3	4888.856	18.2	4887.130	18.2	4885.583

, 0% is the same as the value suggested in Table 9 by substituting the parameter estimates described in Section 3 and the coefficient values of the cost model proposed in Section 4.

From Table 14 and Figure 10, Figure 11, Figure 12 and Figure 13, the value of the cost model $C\left(T\right)$ increases as the change in parameter $a$ increases, whereas the optimal release time $T^{*}$ decreases. As the values of parameters $b$ and $h$ increase, the cost model $C\left(T\right)$ increases, and the release time $T^{*}$ is shown to decrease; in addition, it is found that the change in parameter $h$ had a very slight effect on the optimal release time compared to parameter $b$. As the value of parameter $c$ increases, the cost model $C\left(T\right)$ and release time $T$* increase together. Based on this, it is found that parameter $a$ had a very large minimum width of the cost model compared with the changes of the other parameters, and parameter $b$ had the greatest influence on determining the optimal release time.

5. Conclusions

In this study, a new software reliability model was developed under the assumption that software failures occur in a dependent manner. We used three datasets for our evaluations. The first and second datasets showed the best fit, and the third dataset showed better results compared with many previously proposed models. The proposed model showed better results than DP1, DP2, and DPF, which are previously developed software-dependent failure occurrence models.

In addition, based on the proposed model, the optimal release time according to the change in the cost coefficient was suggested, and the total cost was analyzed accordingly. When the test cost was increased, the release time gradually increased, as did the overall cost; therefore, the optimal release time can be achieved when $C_1$ is 20. In the proposed model, fault detection rate $b$ was found to be the most important parameter for determining the optimal release time.

In the past, studies were conducted by assuming independence in the case of software failures; however, in a real environment, the software execution environment is extremely diverse and complex. Therefore, it is necessary to develop a model that assumes a dependent failure occurrence and propose a model that considers the actual operating environment. We plan to conduct a study using machine learning and deep learning for the proposed software-dependent failure occurrence in the future work.