An NHPP Software Reliability Model with S-Shaped Growth Curve Subject to Random Operating Environments and Optimal Release Time

Abstract

The failure of a computer system because of a software failure can lead to tremendous losses to society; therefore, software reliability is a critical issue in software development. As software has become more prevalent, software reliability has also become a major concern in software development. We need to predict the fluctuations in software reliability and reduce the cost of software testing: therefore, a software development process that considers the release time, cost, reliability, and risk is indispensable. We thus need to develop a model to accurately predict the defects in new software products. In this paper, we propose a new non-homogeneous Poisson process (NHPP) software reliability model, with S-shaped growth curve for use during the software development process, and relate it to a fault detection rate function when considering random operating environments. An explicit mean value function solution for the proposed model is presented. Examples are provided to illustrate the goodness-of-fit of the proposed model, along with several existing NHPP models that are based on two sets of failure data collected from software applications. The results show that the proposed model fits the data more closely than other existing NHPP models to a significant extent. Finally, we propose a model to determine optimal release policies, in which the total software system cost is minimized depending on the given environment.

1. Introduction

‘Software’ is a generic term for a computer program and its associated documents. Software is divided into operating systems and application software. As new hardware is developed, the price decreases; thus, hardware is frequently upgraded at low cost, and software becomes the primary cost driver. The failure of a computer system because of a software failure can cause significant losses to society. Therefore, software reliability is a critical issue in software development. This problem requires finding a balance between meeting user requirements and minimizing the testing costs. It is necessary to know in the planning cycle the fluctuation of software reliability and the cost of testing, in order to reduce costs during the software testing stage, thus a software development process that considers the release time, cost, reliability, and risk is indispensable. In addition, it is necessary to develop a model to predict the defects in software products. To estimate reliability metrics, such as the number of residual faults, the failure rate, and the overall reliability of the software, various non-homogeneous Poisson process (NHPP) software reliability models have been developed using a fault intensity rate function and mean value function within a controlled testing environment. The purpose of many NHPP software reliability models is to obtain an explicit formula for the mean value function, m(t), which is applied to the software testing data to make predictions on software failures and reliability in field environments [1]. A few researchers have evaluated a generalized software reliability model that captures the uncertainty of an environment and its effects on the software failure rate, and have developed a NHPP software reliability model when considering the uncertainty of the system fault detection rate per unit of time subject to the operating environment [2,3,4]. Inoue et al. [5] developed a bivariate software reliability growth model that considers the uncertainty of the change in the software failure-occurrence phenomenon at the change-point for improved accuracy. Okamura and Dohi [6] introduced a phase-type software reliability model and developed parameter estimation algorithms using grouped data. Song et al. [7,8] recently developed an NHPP software reliability model to consider a three-parameter fault detection rate, and applied a Weibull fault detection rate function during the software development process. They related the model to the error detection rate function by considering the uncertainty of the operating environment. In addition, Li and Pham [9] proposed a model accounting for the uncertainty of the operating environment under the condition that the fault content function is a linear function of the testing time, and that the fault detection rate is based on the testing coverage.

In this paper, we discuss a new NHPP software reliability model with S-shaped growth curve applicable to the software development process and relate it to the fault detection rate function when considering random operating environments. We examine the goodness-of-fit of the proposed model and other existing NHPP models that are based on several sets of software failure data, and then determine the optimal release times that minimize the expected total software cost under given conditions. The explicit solution of the mean value function for the new NHPP software reliability model is derived in Section 2. Criteria for the model comparisons and the selection of the best model are discussed in Section 3. The optimal release policy is discussed in Section 4, and the results of a model analysis and the optimal release times are discussed in Section 5. Finally, Section 6 provides some concluding remarks.

2. A New NHPP Software Reliability Model

2.1. Non-Homogeneous Poisson Process

The software fault detection process has been formulated using a popular counting process. The counting process {$\mathrm{N}\left(\mathrm{t}\right),\mathrm{t}\ge 0$} is a non-homogeneous Poisson process (NHPP) with an intensity function $\mathsf{\lambda }\left(\mathrm{t}\right)$, if it satisfies the following condition.

(I)

$\mathrm{N}\left(0\right)=0$

(II)

Independent increments

(III)

${{\displaystyle \int }}_{{\mathrm{t}}_1}^{{\mathrm{t}}_2}\mathsf{\lambda }(\mathrm{t})\mathrm{dt}$, (${\mathrm{t}}_2\ge {\mathrm{t}}_1$): the average of the number of failures in the interval [${\mathrm{t}}_1,{\mathrm{t}}_2$]

Assuming that the software failure/defect conforms to the NHPP condition, $\mathrm{N}\left(\mathrm{t}\right)$($\mathrm{t}\ge 0$) represents the cumulative number of failures up to the point of execution, and $\mathrm{m}\left(\mathrm{t}\right)$ is the mean value function. The mean value function $\mathrm{m}\left(\mathrm{t}\right)$ and the intensity function $\mathsf{\lambda }\left(\mathrm{t}\right)$ satisfy the following relationship.

$$\mathrm{m}\left(\mathrm{t}\right)={{\displaystyle \int }}_0^{\mathrm{t}}\mathsf{\lambda }(\mathrm{s})\mathrm{ds},\frac{\mathrm{dm}\left(\mathrm{t}\right)}{\mathrm{dt}}=\mathsf{\lambda }\left(\mathrm{t}\right).$$

(1)

$\mathrm{N}\left(\mathrm{t}\right)$ is a Poisson distribution involving the mean value function, $\mathrm{m}\left(\mathrm{t}\right)$, and can be expressed as:

$$\mathrm{Pr}\left\{\mathrm{N}\left(\mathrm{t}\right)=\mathrm{n}\right\}=\frac{{\left\{\mathrm{m}\left(\mathrm{t}\right)\right\}}^{\mathrm{n}}}{\mathrm{n}!}\exp \left\{-\mathrm{m}\left(\mathrm{t}\right)\right\},\mathrm{n}=0,1,2,3\dots .$$

(2)

2.2. General NHPP Software Reliability Model

Pham et al. [10] formalized the general framework for NHPP-based software reliability and provided analytical expressions for the mean value function $\mathrm{m}\left(\mathrm{t}\right)$ using differential equations. The mean value function $\mathrm{m}\left(\mathrm{t}\right)$ of the general NHPP software reliability model with different values for $\mathrm{a}\left(\mathrm{t}\right)$ and $\mathrm{b}\left(\mathrm{t}\right)$, which reflects various assumptions of the software testing process, can be obtained with the initial condition $\mathrm{N}\left(0\right)=0$.

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

(3)

The general solution of (1) is

$$\mathrm{m}\left(\mathrm{t}\right)={\mathrm{e}}^{-\mathrm{B}\left(\mathrm{t}\right)}\left[{\mathrm{m}}_0+{{\displaystyle \int }}_{{\mathrm{t}}_0}^{\mathrm{t}}\mathrm{a}\left(\mathrm{s}\right)\mathrm{b}\left(\mathrm{s}\right){\mathrm{e}}^{\mathrm{B}\left(\mathrm{s}\right)}\mathrm{bs}\right]$$

(4)

where $\mathrm{B}\left(\mathrm{t}\right)={{\displaystyle \int }}_{{\mathrm{t}}_0}^{\mathrm{t}}\mathrm{b}(\mathrm{s})\mathrm{ds}$, and $\mathrm{m}\left({\mathrm{t}}_0\right)={\mathrm{m}}_0$ is the marginal condition of (2).

2.3. New NHPP Software Reliability Model

Pham [3] formulated a generalized NHPP software reliability model that incorporated uncertainty in the operating environment as follows:

$$\frac{\mathrm{d}\mathrm{m}\left(\mathrm{t}\right)}{\mathrm{dt}}=\mathsf{\eta }\left[\mathrm{b}\left(\mathrm{t}\right)\right]\left[\mathrm{N}-\mathrm{m}\left(\mathrm{t}\right)\right],$$

(5)

where $\mathsf{\eta }$ is a random variable that represents the uncertainty of the system fault detection rate in the operating environment with a probability density function g; $\mathrm{b}\left(\mathrm{t}\right)$ is the fault detection rate function, which also represents the average failure rate caused by faults; $\mathrm{N}$ is the expected number of faults that exists in the software before testing; and, m(t) is the expected number of errors detected by time t (the mean value function).

Thus, a generalized mean value function, m(t), where the initial condition m(0) = 0, is given by

$$\mathrm{m}\left(\mathrm{t}\right)={{\displaystyle \int }}_{\mathsf{\eta }}^{}\mathrm{N}\left(1-{\mathrm{e}}^{-\mathsf{\eta }{{\displaystyle \int }}_0^{\mathrm{t}}\mathrm{b}\left(\mathrm{x}\right)\mathrm{dx}}\right)\mathrm{dg}\left(\mathsf{\eta }\right).$$

(6)

The mean value function [11] from (4) using the random variable η has a generalized probability density function g with two parameters $\mathsf{\alpha }\ge 0$ and $\mathsf{\beta }\ge 0$ and is given by

$$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{N}{\left(1-\frac{\mathsf{\beta }}{\mathsf{\beta }+{{\displaystyle \int }}_0^{\mathrm{t}}\mathrm{b}\left(\mathrm{s}\right)\mathrm{ds}}\right)}^{\mathsf{\alpha }},$$

(7)

where b(t) is the fault detection rate per fault per unit of time.

We propose an NHPP software reliability model including the random operating environment using Equations (3)–(5) and the following assumptions [7,8]:

(a)	The occurrence of a software failure follows a non-homogeneous Poisson process.
(b)	Faults during execution can cause software failure.
(c)	The software failure detection rate at any time depends on both the fault detection rate and the number of remaining faults in the software at that time.
(d)	Debugging is performed to remove faults immediately when a software failure occurs.
(e)	New faults may be introduced into the software system, regardless of whether other faults are removed or not.
(f)	The fault detection rate $\mathrm{b}\left(\mathrm{t}\right)$ can be expressed by (6).
(g)	The random operating environment is captured if unit failure detection rate $\mathrm{b}\left(\mathrm{t}\right)$ is multiplied by a factor $\mathsf{\eta }$ that represents the uncertainty of the system fault detection rate in the field

In this paper, we consider the fault detection rate function $\mathrm{b}\left(\mathrm{t}\right)$ to be as follows:

$$\mathrm{b}\left(\mathrm{t}\right)=\frac{{\mathrm{a}}^2\mathrm{t}}{1+\mathrm{at}},\mathrm{a}>0,\mathrm{a},\mathrm{b}>0,$$

(8)

We obtain a new NHPP software reliability model with S-shaped growth curve subject to random operating environments, m(t), that can be used to determine the expected number of software failures detected by time $\mathrm{t}$ by substituting function b(t) above into (5) so that:

$$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{N}{\left(1-\frac{\mathsf{\beta }}{\mathsf{\beta }+\mathrm{at}-\ln \left(1+\mathrm{at}\right)}\right)}^{\mathsf{\alpha }}.$$

(9)

3. Criteria for Model Comparisons

Theoretically, once the analytical expression for mean value function $\mathrm{m}\left(\mathrm{t}\right)$ is derived, then the parameters in $\mathrm{m}\left(\mathrm{t}\right)$ can be estimated using parameter estimation methods (MLE: the maximum likelihood estimation method, LSE: the least square estimation method); however, in practice, accurate estimates may not be obtained by the MLE, particularly under certain conditions where the mean value function $\mathrm{m}\left(\mathrm{t}\right)$ is too complex. The model parameters to be estimated in the mean value function $\mathrm{m}\left(\mathrm{t}\right)$ can then be obtained using a MATLAB program that is based on the LSE method. Six common criteria; the mean squared error (MSE), Akaike’s information criterion (AIC), the predictive ratio risk (PRR), the predictive power (PP), the sum of absolute errors (SAE), and R-square (R²) will be used for the goodness-of-fit estimation of the proposed model, and to compare the proposed model with other existing models, as listed in

Table 1. NHPP software reliability models.

No.	Model	$\mathbf{m}\left(\mathbf{t}\right)$
1	GO Model [14]	$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{a}\left(1-{\mathrm{e}}^{-\mathrm{bt}}\right)$
2	Delayed S-shaped Model [15]	$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{a}\left(1-\left(1+\mathrm{bt}\right){\mathrm{e}}^{-\mathrm{bt}}\right)$
3	Inflection S-shaped Model [16]	$\mathrm{m}\left(\mathrm{t}\right)=\frac{\mathrm{a}\left(1-{\mathrm{e}}^{-\mathrm{bt}}\right)}{1+{\mathsf{\beta }\mathrm{e}}^{-\mathrm{bt}}}$
4	Yamada ImperfectDebugging Model [17]	$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{a}\left[1-{\mathrm{e}}^{-\mathrm{bt}}\right]\left[1-\frac{\mathsf{\alpha }}{\mathrm{b}}\right]+\mathsf{\alpha }\mathrm{at}$
5	PNZ Model [10]	$\mathrm{m}\left(\mathrm{t}\right)=\frac{\mathrm{a}\left[1-{\mathrm{e}}^{-\mathrm{bt}}\right]\left[1-\frac{\mathsf{\alpha }}{\mathrm{b}}\right]+\mathsf{\alpha }\mathrm{at}}{1+{\mathsf{\beta }\mathrm{e}}^{-\mathrm{bt}}}$
6	PZ Model [18]	$\mathrm{m}\left(\mathrm{t}\right)=\frac{\left(\left(\mathrm{c}+\mathrm{a}\right)\left[1-{\mathrm{e}}^{-\mathrm{bt}}\right]-\left[\frac{\mathrm{ab}}{\mathrm{b}-\mathsf{\alpha }}\left({\mathrm{e}}^{-\mathsf{\alpha }\mathrm{t}}-{\mathrm{e}}^{-\mathrm{bt}}\right)\right]\right)}{1+{\mathsf{\beta }\mathrm{e}}^{-\mathrm{bt}}}$
7	Dependent Parameter Model [19]	$\mathrm{m}\left(\mathrm{t}\right)={\mathrm{m}}_0\left(\frac{\mathsf{\gamma }\mathrm{t}+1}{{\mathsf{\gamma }\mathrm{t}}_0+1}\right){\mathrm{e}}^{-\mathsf{\gamma }\left(\mathrm{t}-{\mathrm{t}}_0\right)}+\mathsf{\alpha }\left(\mathsf{\gamma }\mathrm{t}+1\right)(\mathsf{\gamma }\mathrm{t}-1+\left(1-{\mathsf{\gamma }\mathrm{t}}_0\right){\mathrm{e}}^{-\mathsf{\gamma }\left(\mathrm{t}-{\mathrm{t}}_0\right)}$
8	Testing Coverage Model [4]	$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{N}\left[1-{\left(\frac{\mathsf{\beta }}{\mathsf{\beta }+{\left(\mathrm{at}\right)}^{\mathrm{b}}}\right)}^{\mathsf{\alpha }}\right]$
9	Three parameter Model [7]	$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{N}\left[1-\left(\frac{\mathsf{\beta }}{\mathsf{\beta }-\frac{\mathrm{a}}{\mathrm{b}}\ln \left(\frac{\left(1+\mathrm{c}\right){\mathrm{e}}^{-\mathrm{bt}}}{1+{\mathrm{ce}}^{-\mathrm{bt}}}\right)}\right)\right]$
10	Proposed New Model	$\mathrm{m}\left(\mathrm{t}\right)=\mathrm{N}{\left(1-\frac{\mathsf{\beta }}{\mathsf{\beta }+\mathrm{at}-\ln \left(1+\mathrm{at}\right)}\right)}^{\mathsf{\alpha }}$

. These criteria are described as follows.

The MSE is

$$\mathrm{MSE}=\frac{{{\displaystyle \sum }}_{\mathrm{i}=0}^{\mathrm{n}}{\left(\hat{\mathrm{m}}\left({\mathrm{t}}_{\mathrm{i}}\right)-{\mathrm{y}}_{\mathrm{i}}\right)}^2}{\mathrm{n}-\mathrm{m}}.$$

(10)

AIC [12] is

$$\mathrm{AIC}=-2\log \mathrm{L}+2\mathrm{m}.$$

(11)

The PRR [13] is

$$\mathrm{PRR}={{\displaystyle \sum }}_{\mathrm{i}=0}^{\mathrm{n}}{\left(\frac{\hat{\mathrm{m}}\left({\mathrm{t}}_{\mathrm{i}}\right)-{\mathrm{y}}_{\mathrm{i}}}{\hat{\mathrm{m}}\left({\mathrm{t}}_{\mathrm{i}}\right)}\right)}^2.$$

(12)

The PP [13] is

$$\mathrm{PP}={{\displaystyle \sum }}_{\mathrm{i}=0}^{\mathrm{n}}{\left(\frac{\hat{\mathrm{m}}\left({\mathrm{t}}_{\mathrm{i}}\right)-{\mathrm{y}}_{\mathrm{i}}}{{\mathrm{y}}_{\mathrm{i}}}\right)}^2.$$

(13)

The SAE [8] is

$$\mathrm{SAE}={{\displaystyle \sum }}_{\mathrm{i}=0}^{\mathrm{n}}\left|\hat{\mathrm{m}}\left({\mathrm{t}}_{\mathrm{i}}\right)-{\mathrm{y}}_{\mathrm{i}}\right|.$$

(14)

The correlation index of the regression curve equation (${\mathrm{R}}^2$) [9] is

$${\mathrm{R}}^2=1-\frac{{{\displaystyle \sum }}_{\mathrm{i}=0}^{\mathrm{n}}{\left(\hat{\mathrm{m}}\left({\mathrm{t}}_{\mathrm{i}}\right)-{\mathrm{y}}_{\mathrm{i}}\right)}^2}{{{\displaystyle \sum }}_{\mathrm{i}=0}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{i}}}\right)}^2}.$$

(15)

Here, $\hat{\mathrm{m}}\left({\mathrm{t}}_{\mathrm{i}}\right)$ is the estimated cumulative number of failures at ${\mathrm{t}}_{\mathrm{i}}$ for $\mathrm{i}=1,2,\cdots ,\mathrm{n}$; ${\mathrm{y}}_{\mathrm{i}}$ is the total number of failures observed at time ${\mathrm{t}}_{\mathrm{i}}$; $\mathrm{n}$ is the actual data which includes the total number of observations; and, $\mathrm{m}$ is the number of unknown parameters in the model.

The MSE measures the distance of a model estimate from the actual data that includes the total number of observations and the number of unknown parameters in the model. AIC is measured to compare the capability of each model in terms of maximizing the likelihood function (L), while considering the degrees of freedom. The PRR measures the distance of the model estimates from the actual data against the model estimate. The PP measures the distance of the model estimates from the actual data. The SAE measures the absolute distance of the model. For five of these criteria, i.e., MSE, AIC, PRR, PP, and SAE, the smaller the value is, the closer the model fits relative to other models run on the same dataset. On the other hand, ${\mathrm{R}}^2$ should be close to 1.

We use (8) below to obtain the confidence interval [13] of the proposed NHPP software reliability model. The confidence interval is described as follows;

$$\hat{\mathrm{m}}\left(\mathrm{t}\right)\pm {\mathrm{Z}}_{\mathsf{\alpha }/2}\sqrt{\hat{\mathrm{m}}\left(\mathrm{t}\right)},$$

(16)

where, ${\mathrm{Z}}_{\mathsf{\alpha }/2}$ is $100\left(1-\mathsf{\alpha }\right)$, the percentile of the standard normal distribution.

Table 1 summarizes the different mean value functions of the proposed new model and several existing NHPP models. Note that models 9 and 10 consider environmental uncertainty.

4. Optimal Software Release Policy

In this section, we next discuss the use of the software reliability model under varying situations to determine the optimal software release time, and to determine the optimal software release time, T*, which minimizes the expected total software cost. Many studies have been conducted on the optimal software release time and its related problems [20,21,22,23,24]. The quality of the system will normally depend on the testing efforts, such as the testing environment, times, tools, and methodologies. If testing is short, the cost of the system testing is lower, but the consumers may face a higher risk e.g., buying an unreliable system. This also involves the higher costs of the operating environment because it is much more expensive to detect and correct a failure during the operational phase than during the testing phase. In contrast, the longer the testing time, the more faults that can be removed, which leads to a more reliable system; however, the testing costs for the system will also increase. Therefore, it is very important to determine when to release the system based on test cost and reliability. Figure 1 shows the system development lifecycle considered in the following cost model: the testing phase before release time T, the testing environment period, the warranty period, and the operational life in the actual field environment, which is usually quite different from the testing environment [24].

The expected total software cost $\mathrm{C}\left(\mathrm{T}\right)$ [24] can be expressed as

$$\mathrm{C}\left(\mathrm{T}\right)={\mathrm{C}}_0+{\mathrm{C}}_1\mathrm{T}+{\mathrm{C}}_2\mathrm{m}\left(\mathrm{T}\right){\mathsf{\mu }}_{\mathrm{y}}+{\mathrm{C}}_3\left(1-\mathrm{R}\left(\mathrm{x}|\mathrm{T}\right)\right)+{\mathrm{C}}_4\left[\mathrm{m}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)-\mathrm{m}\left(\mathrm{T}\right)\right]{\mathsf{\mu }}_{\mathrm{w}}$$

(17)

where, ${\mathrm{C}}_0$ is the set-up cost of testing, ${\mathrm{C}}_1\mathrm{T}$ is the cost of testing, ${\mathrm{C}}_2\mathrm{m}\left(\mathrm{T}\right){\mathsf{\mu }}_{\mathrm{y}}$ is the expected cost to remove all errors detected by time $\mathrm{T}$ during the testing phase, ${\mathrm{C}}_3\left(1-\mathrm{R}\left(\mathrm{x}|\mathrm{T}\right)\right)$ is the penalty cost owing to failures that occurs after the system release time $\mathrm{T}$, and ${\mathrm{C}}_4\left[\mathrm{m}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)-\mathrm{m}\left(\mathrm{T}\right)\right]{\mathsf{\mu }}_{\mathrm{w}}$ is the expected cost to remove all of the errors that are detected during the warranty period [$\mathrm{T},\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}$]. The cost that is required to remove faults during the operating period is higher than during the testing period, and the time that is needed is much longer.

Finally, we aim to find the optimal software release time, T*, with the expected minimum in the environment as follows:

$$\mathrm{Minimize}\mathrm{C}\left(\mathrm{T}\right).$$

(18)

5. Numerical Examples

5.1. Data Information

Dataset #1 (DS1), presented in

Table 2. Dataset #1 (DS1) : real time command and control system (RTC&CS) data set.

Hour Index	Failures	Cumulative Failures	Hour Index	Failures	Cumulative Failures
1	27	27	14	5	111
2	16	43	15	5	116
3	11	54	16	6	122
4	10	64	17	0	122
5	11	75	18	5	127
6	7	83	19	1	128
7	2	84	20	1	129
8	5	89	21	2	131
9	3	92	22	1	132
10	1	93	23	2	134
11	4	97	24	1	135
12	7	104	25	1	136
13	2	106	-	-	-

, was reported by Musa [25] based on software failure data from a real time command and control system (RTC&CS), and represents the failures that were observed during system testing (25 hours of CPU time). The number of test object instructions delivered for this system, which was developed by Bell Laboratories, was 21,700.

Dataset #2 (DS2), as shown in

Table 3. DS2 : medical record system (MRS) data set.

Week Index	Failures	Cumulative Failures	Week Index	Failures	Cumulative Failures
1	90	90	10	0	190
2	17	107	11	2	192
3	19	126	12	0	192
4	19	145	13	0	192
5	26	171	14	0	192
6	17	188	15	11	203
7	1	189	16	0	203
8	1	190	17	1	204
9	0	190	-	-	-

, is the second of three releases of software failure data collected from three different releases of a large medical record system (MRS) [26], consisting of 188 software components. Each component contains several files. Initially, the software consisted of 173 software components. All three releases added new functionality to the product. Between three and seven new components were added in each of the three releases, for a total of 15 new components. Many other components were modified during each of the three releases as a side effect of the added functionality. Detailed information of the dataset can be obtained in the report by Stringfellow and Andrews [26].

Dataset #3 (DS3), as shown in

Table 4. DS3 : Tandom Computers (TDC) data set.

Time Index (CPU hours)	Cumulative Failures	Time Index (CPU hours)	Cumulative Failures	Time Index (CPU hours)	Cumulative Failures
519	16	4422	58	8205	96
968	24	5218	69	8564	98
1430	27	5823	75	8923	99
1893	33	6539	81	9282	100
2490	41	7083	86	9641	100
3058	49	7487	90	10,000	100
3625	54	7846	93	-	-

, is from one of four major releases of software products at Tandom Computers (TDC) [27]. There are 100 failures that are observed within testing CPU hours. Detailed information of the dataset can be obtained tin the report by Wood [27].

5.2. Model Analysis

Table 5. Model parameter estimation and comparison criteria from RTC&CS data set (DS1). Least-squares estimate (LSE); mean squared error; Akaike’s information criterion (AIC); predictive ratio risk (PRR); predictive power (PP), sum absolute error (SAE), correlation index of the regression curve equation (${\mathrm{R}}^2$ ).

Model	LSE’s	MSE	AIC	PRR	PP	SAE	R2
GOM	$\widehat{\mathrm{a}}$ = 136.050, $\hat{\mathrm{b}}$ = 0.138	33.822	121.878	0.479	0.262	118.530	0.972
DSM	$\hat{\mathrm{a}}$ = 124.665, $\widehat{\mathrm{b}}$ = 0.356	134.582	210.287	12.787	1.181	239.335	0.889
ISM	$\hat{\mathrm{a}}$ = 136.050, $\hat{\mathrm{b}}$ = 0.138 $\hat{\mathsf{\beta }}$ = 0.0001	35.363	123.878	0.479	0.262	118.532	0.972
YIDM	$\widehat{\mathrm{a}}$ = 81.252, $\hat{\mathrm{b}}$ = 0.340 $\hat{\mathsf{\alpha }}$ = 0.0333	9.435	116.403	0.035	0.031	60.842	0.993
PNZM	$\widehat{\mathrm{a}}$ = 81.562, $\widehat{\mathrm{b}}$ = 0.337 $\hat{\mathsf{\alpha }}$ = 0.033, $\hat{\mathsf{\beta }}$ = 0.00	9.888	118.388	0.037	0.032	60.877	0.993
PZM	$\hat{\mathrm{a}}$ = 0.01, $\hat{\mathrm{b}}$ = 0.138 $\widehat{\mathsf{\alpha }}$ = 800.0, $\widehat{\mathsf{\beta }}$ = 0.00, $\widehat{\mathrm{c}}$ = 136.04	38.895	127.878	0.479	0.262	118.530	0.972
DPM	$\hat{\mathsf{\alpha }}$ = 28650, $\hat{\mathsf{\beta }}$ = 0.003 ${\mathrm{t}}_0$ = 0.00, ${\mathrm{m}}_0$ = 71.8	274.911	382.143	0.857	3.568	304.212	0.792
TCM	$\hat{\mathrm{a}}$ = 0.000035, $\widehat{\mathrm{b}}$ = 0.734, $\widehat{\mathsf{\alpha }}$ = 0.29, $\widehat{\mathsf{\beta }}$ = 0.002, $\widehat{\mathrm{N}}$ = 427	7.640	116.932	0.019	0.019	47.304	0.995
3PFDM	$\hat{\mathrm{a}}$ = 1.696, $\widehat{\mathrm{b}}$ = 0.001 $\hat{\mathrm{c}}$ = 6.808, $\hat{\mathsf{\beta }}$ = 1.574 $\widehat{\mathrm{N}}$ = 173.030	17.827	119.523	0.137	0.100	81.313	0.987
New Model	$\widehat{\mathrm{a}}$ = 0.277, $\widehat{\mathsf{\alpha }}$ = 0.328 $\widehat{\mathsf{\beta }}$ = 17.839, $\widehat{\mathrm{N}}$ = 228.909	7.361	114.982	0.022	0.022	47.869	0.994

,

Table 6. Model parameter estimation and comparison criteria from MRS data set (DS2).

Model	LSE’s	MSE	AIC	PRR	PP	SAE	R2
GOM	$\hat{\mathrm{a}}$ = 197.387, $\widehat{\mathrm{b}}$ = 0.399	80.678	184.331	0.170	0.101	104.403	0.939
DSM	$\hat{\mathrm{a}}$ = 192.528, $\hat{\mathrm{b}}$ = 0.882	232.628	331.857	1.291	0.333	142.544	0.823
ISM	$\hat{\mathrm{a}}$ = 197.354, $\hat{\mathrm{b}}$ = 0.399 $\widehat{\mathsf{\beta }}$ = 0.000001	86.440	186.334	0.171	0.101	104.370	0.939
YIDM	$\hat{\mathrm{a}}$ = 182.934, $\hat{\mathrm{b}}$ = 0.464 $\widehat{\mathsf{\alpha }}$ = 0.0071	78.837	157.825	0.128	0.087	100.617	0.944
PNZM	$\widehat{\mathrm{a}}$ = 183.124, $\hat{\mathrm{b}}$ = 0.463 $\hat{\mathsf{\alpha }}$ = 0.007, $\hat{\mathsf{\beta }}$ = 0.00	84.902	159.873	0.128	0.087	100.608	0.944
PZM	$\hat{\mathrm{a}}$ = 195.990, $\widehat{\mathrm{b}}$ =0.3987 $\hat{\mathsf{\alpha }}$ = 1000.00, $\hat{\mathsf{\beta }}$ = 0.00, $\hat{\mathrm{c}}$ = 1.390	100.989	190.332	0.172	0.102	104.354	0.939
DPM	$\widehat{\mathsf{\alpha }}$ = 26124.0, $\widehat{\mathsf{\gamma }}$ = 0.0044 ${\mathrm{t}}_0$ = 0.00, ${\mathrm{m}}_0$ = 147.00	769.282	480.341	0.415	0.712	334.128	0.494
TCM	$\hat{\mathrm{a}}$ = 0.053, $\widehat{\mathrm{b}}$ = 0.774, $\hat{\mathsf{\alpha }}$ = 181.0, $\hat{\mathsf{\beta }}$ = 38.6, $\hat{\mathrm{N}}$ = 204.1	72.283	158.933	0.052	0.048	103.196	0.956
3PFDM	$\hat{\mathrm{a}}$ = 0.028, $\hat{\mathrm{b}}$ = 0.210 $\widehat{\mathrm{c}}$ = 9.924, $\hat{\mathsf{\beta }}$ = 0.005 $\hat{\mathrm{N}}$ = 206.387	81.090	163.797	0.073	0.061	106.341	0.951
New Model	$\hat{\mathrm{a}}$ = 0.008, $\widehat{\mathsf{\alpha }}$ = 0.275, $\widehat{\mathsf{\beta }}$ = 0.001, $\widehat{\mathrm{N}}$ = 207.873	60.623	151.156	0.043	0.041	98.705	0.960

and

Table 7. Model parameter estimation and comparison criteria from MRS data set (DS3).

Model	LSE’s	MSE	AIC	PRR	PP	SAE	R2
GOM	$\hat{\mathrm{a}}$ = 133.835, $\hat{\mathrm{b}}$ = 0.000146	8.620	86.136	0.556	0.242	42.166	0.991
DSM	$\hat{\mathrm{a}}$ = 101.918, $\widehat{\mathrm{b}}$ = 0.000507	45.783	117.316	22.692	1.318	101.659	0.951
ISM	$\hat{\mathrm{a}}$ = 133.835, $\widehat{\mathrm{b}}$ = 0.000146 $\hat{\mathsf{\beta }}$ = 0.000001	9.127	88.136	0.556	0.242	42.166	0.991
YIDM	$\hat{\mathrm{a}}$ = 130.091, $\widehat{\mathrm{b}}$ = 0.00015 $\hat{\mathsf{\alpha }}$ = 0.000003	9.084	88.267	0.561	0.243	42.052	0.991
PNZM	$\widehat{\mathrm{a}}$ = 121.178, $\hat{\mathrm{b}}$ = 0.000163 $\hat{\mathsf{\alpha }}$ = 0.000009, $\widehat{\mathsf{\beta }}$ = 0.00	9.532	90.326	0.530	0.234	41.538	0.991
PZM	$\widehat{\mathrm{a}}$ = 122.259, $$ = 0.0002 $\hat{\mathsf{\alpha }}$ = 9955.597, $\hat{\mathsf{\beta }}$ = 0.305 $\widehat{\mathrm{c}}$ = 0.569	11.491	92.020	0.643	0.268	44.848	0.990
DPM	$\hat{\mathsf{\alpha }}$ = 123.193, $\hat{\mathsf{\gamma }}$ = 0.0001 ${\mathrm{t}}_0$ = 0.0001, ${\mathrm{m}}_0$ = 38.459	156.480	212.867	0.917	2.879	196.360	0.851
TCM	$\hat{\mathrm{a}}$ = 0.000013, $\hat{\mathrm{b}}$ = 0.78, $\hat{\mathsf{\alpha }}$ = 141.399, $\widehat{\mathsf{\beta }}$ = 54.71, $\widehat{\mathrm{N}}$ = 254.707	7.090	90.758	0.091	0.068	37.880	0.9937
3PFDM	$\hat{\mathrm{a}}$ = 0.016, $\widehat{\mathrm{b}}$ = 0.07 $\widehat{\mathrm{c}}$ = 0.00001, $\widehat{\mathsf{\beta }}$ = 157.458 $\widehat{\mathrm{N}}$ = 205.025	9.410	92.360	0.420	0.200	39.909	0.992
New Model	$\hat{\mathrm{a}}$ = 0.064, $\hat{\mathsf{\alpha }}$ = 0.731, $\widehat{\mathsf{\beta }}$ = 2509.898, $\widehat{\mathrm{N}}$ = 337.765	6.336	88.885	0.086	0.066	36.250	0.9940

summarize the results of the estimated parameters of all 10 models in Table 1 using the LSE technique and the values of the six common criteria: MSE, AIC, PRR, PP, SAE, and ${\mathrm{R}}^2$. We obtained the six common criteria at $\mathrm{t}=1,2,\cdots ,25$ from DS1 (Table 2), at $\mathrm{t}=1,2,\cdots ,17$ from DS2 (Table 3), and at cumulative testing CPU hours from DS3 (Table 4). As can be seen in Table 5, when comparing all of the models, the MSE and AIC values are the lowest for the newly proposed model, and the PRR, PP, SAE, and ${\mathrm{R}}^2$ values are the second best. The MSE and AIC values of the newly proposed model are 7.361, 114.982, respectively, which are significantly less than the values of the other models. In Table 6, when comparing all of the models, all criteria values for the newly proposed model are best. The MSE value of the newly proposed model is 60.623, which is significantly lower than the value of the other models. The AIC, PRR, PP, and SAE values of the newly proposed model are 151.156, 0.043, 0.041, and 98.705, respectively, which are also significantly lower than the other models. The value of R² is 0.960 and is the closest to 1 for all of the models. In Table 7, when comparing all of the models, all the criteria values for the newly proposed model are best. The MSE value of the newly proposed model is 6.336, which is significantly lower than the value of the other models. The PRR, PP, and SAE values of the newly proposed model are 0.086, 0.066, and 36.250, respectively, which are also significantly lower than the other models. The value of R² is 0.9940 and is the closest to 1 for all of the models.

Figure 2, Figure 3 and Figure 4 show the graphs of the mean value functions for all 10 models for DS1, DS2, and DS3, respectively. Figure 5, Figure 6 and Figure 7 show the graphs of the 95% confidence limits of the newly proposed model for DS1, DS2, and DS3.

Table A1. 95% Confidence interval of all 10 models (DS1).

Model	Time Index	1	2	3	4	5		6	7			8		9
GOM	LCL	9.329	21.586	32.810	42.823	51.671		59.452	66.277			72.253		77.479
	$\hat{\mathrm{m}}\left(t\right)$	17.537	32.814	46.121	57.713	67.811		76.607	84.269			90.944		96.758
	UCL	25.745	44.041	59.431	72.602	83.950		93.761	102.261			109.635		116.037
DSM	LCL	1.352	11.192	24.289	37.792	50.271		61.120	70.191			77.571		83.458
	$\hat{\mathrm{m}}\left(t\right)$	6.253	19.945	36.058	51.914	66.220		78.484	88.644			96.861		103.387
	UCL	11.154	28.698	47.827	66.035	82.170		95.847	107.097			116.150		123.316
ISM	LCL	9.328	21.584	32.808	42.820	51.668		59.449	66.274			72.250		77.476
	$\hat{\mathrm{m}}\left(t\right)$	17.535	32.811	46.118	57.709	67.807		76.603	84.266			90.941		96.755
	UCL	25.743	44.038	59.428	72.599	83.947		93.758	102.258			109.631		116.034
YIDM	LCL	14.263	28.936	40.449	49.466	56.637		62.468	67.333			71.506		75.185
	$\hat{\mathrm{m}}\left(t\right)$	23.831	41.573	54.982	65.305	73.433		79.998	85.451			90.111		94.209
	UCL	33.399	54.211	69.515	81.144	90.228		97.528	103.568			108.717		113.232
PNZM	LCL	14.191	28.840	40.360	49.400	56.598		62.455	67.343			71.534		75.227
	$\hat{\mathrm{m}}\left(t\right)$	23.741	41.460	54.879	65.230	73.389		79.984	85.462			90.143		94.255
	UCL	33.291	54.080	69.399	81.059	90.179		97.513	103.581			108.752		113.283
PZM	LCL	9.329	21.586	32.810	42.823	51.671		59.452	66.277			72.253		77.479
	$\hat{\mathrm{m}}\left(t\right)$	17.537	32.813	46.121	57.713	67.811		76.607	84.269			90.944		96.758
	UCL	25.745	44.041	59.431	72.602	83.950		93.761	102.261			109.635		116.037
DPM	LCL	55.306	55.649	56.223	57.028	58.068		59.344	60.859			62.614		64.611
	$\hat{\mathrm{m}}\left(t\right)$	71.929	72.316	72.964	73.874	75.047		76.485	78.189			80.162		82.403
	UCL	88.551	88.984	89.706	90.720	92.026		93.626	95.520			97.710		100.195
TCM	LCL	17.974	30.408	39.981	47.851	54.561		60.419	65.621			70.302		74.555
	$\hat{\mathrm{m}}\left(t\right)$	28.423	43.306	54.443	63.465	71.086		77.695	83.535			88.768		93.508
	UCL	38.872	56.204	68.905	79.080	87.611		94.971	101.449			107.234		112.461
3PFDM	LCL	12.011	25.458	36.751	46.227	54.252		61.123	67.065			72.251		76.816
	$\hat{\mathrm{m}}\left(t\right)$	20.991	37.452	50.708	61.611	70.737		78.487	85.151			90.942		96.021
	UCL	29.970	49.447	64.665	76.995	87.221		95.851	103.237			109.633		115.227
New	LCL	18.358	30.526	39.950	47.745	54.426		60.283	65.501			70.208		74.492
	$\hat{\mathrm{m}}\left(t\right)$	28.893	43.444	54.407	63.344	70.933		77.542	83.401			88.663		93.438
	UCL	39.428	56.363	68.864	78.944	87.440		94.801	101.300			107.118		112.384
Model	Time Index	10	11	12	13		14		15		16			17
GOM	LCL	82.045	86.033	89.514	92.552		95.201		97.512		99.527			101.284
	$\hat{\mathrm{m}}\left(t\right)$	101.823	106.235	110.078	113.426		116.342		118.882		121.095			123.023
	UCL	121.600	126.436	130.641	134.300		137.483		140.253		142.663			144.762
DSM	LCL	88.083	91.672	94.431	96.534		98.127		99.326		100.225			100.895
	$\hat{\mathrm{m}}\left(t\right)$	108.498	112.457	115.494	117.807		119.557		120.874		121.861			122.596
	UCL	128.914	133.241	136.557	139.080		140.988		142.423		143.497			144.298
ISM	LCL	82.043	86.031	89.512	92.550		95.200		97.511		99.526			101.283
	$\hat{\mathrm{m}}\left(t\right)$	101.820	106.232	110.076	113.424		116.340		118.881		121.094			123.022
	UCL	121.597	126.434	130.639	134.298		137.481		140.251		142.662			144.761
YIDM	LCL	78.512	81.588	84.485	87.257		89.939		92.558		95.133			97.678
	$\hat{\mathrm{m}}\left(t\right)$	97.905	101.316	104.523	107.586		110.546		113.433		116.267			119.064
	UCL	117.298	121.044	124.561	127.916		131.153		134.307		137.401			140.451
PNZM	LCL	78.562	81.642	84.540	87.310		89.987		92.600		95.168			97.703
	$\hat{\mathrm{m}}\left(t\right)$	97.960	101.376	104.584	107.645		110.600		113.479		116.305			119.092
	UCL	117.359	121.110	124.628	127.980		131.212		134.358		137.442			140.481
PZM	LCL	82.045	86.033	89.514	92.552		95.201		97.512		99.527			101.284
	$\hat{\mathrm{m}}\left(t\right)$	101.823	106.235	110.078	113.426		116.342		118.882		121.095			123.023
	UCL	121.600	126.436	130.641	134.300		137.483		140.253		142.663			144.762
DPM	LCL	66.855	69.346	72.087	75.081		78.331		81.838		85.606			89.636
	$\hat{\mathrm{m}}\left(t\right)$	84.916	87.701	90.759	94.093		97.704		101.593		105.762			110.212
	UCL	102.977	106.055	109.432	113.105		117.078		121.348		125.918			130.788
TCM	LCL	78.453	82.050	85.387	88.500		91.416		94.157		96.744			99.191
	$\hat{\mathrm{m}}\left(t\right)$	97.840	101.828	105.521	108.959		112.174		115.193		118.038			120.726
	UCL	117.227	121.606	125.654	129.418		132.933		136.229		139.332			142.261
3PFDM	LCL	80.863	84.475	87.718	90.647		93.303		95.724		97.940			99.974
	$\hat{\mathrm{m}}\left(t\right)$	100.512	104.512	108.096	111.326		114.253		116.917		119.352			121.586
	UCL	120.162	124.549	128.473	132.006		135.203		138.110		140.764			143.198
New	LCL	78.423	82.051	85.417	88.555		91.491		94.248		96.844			99.296
	$\hat{\mathrm{m}}\left(t\right)$	97.806	101.829	105.554	109.019		112.257		115.293		118.148			120.841
	UCL	117.189	121.607	125.690	129.484		133.024		136.338		139.452			142.387
Model	Time Index	18	19	20	21		22		23		24			25
GOM	LCL	102.8153	104.15	105.3133	106.3271		107.2106		107.9804		108.6512			109.2357
	$\hat{\mathrm{m}}\left(t\right)$	124.7022	126.165	127.4392	128.5491		129.516		130.3582		131.0919			131.731
	UCL	146.5892	148.1799	149.565	150.7711		151.8214		152.736		153.5326			154.2263
DSM	LCL	101.3931	101.7621	102.0346	102.2353		102.3828		102.4909		102.57			102.6278
	$\hat{\mathrm{m}}\left(t\right)$	123.1427	123.5475	123.8463	124.0664		124.2281		124.3466		124.4333			124.4967
	UCL	144.8924	145.3328	145.658	145.8975		146.0734		146.2023		146.2967			146.3656
ISM	LCL	102.8143	104.1492	105.3126	106.3265		107.21		107.9799		108.6508			109.2353
	$\hat{\mathrm{m}}\left(t\right)$	124.7012	126.164	127.4383	128.5484		129.5154		130.3577		131.0914			131.7306
	UCL	146.588	148.1789	149.5641	150.7703		151.8207		152.7354		153.5321			154.2259
YIDM	LCL	100.2015	102.7107	105.2103	107.7037		110.1934		112.6811		115.1679			117.6547
	$\hat{\mathrm{m}}\left(t\right)$	121.8354	124.5875	127.3263	130.0555		132.7779		135.4956		138.2097			140.9215
	UCL	143.4693	146.4644	149.4423	152.4073		155.3625		158.31		161.2516			164.1883
PNZM	LCL	100.2169	102.7154	105.2037	107.6855		110.1632		112.6385		115.1129			117.587
	$\hat{\mathrm{m}}\left(t\right)$	121.8523	124.5927	127.3191	130.0356		132.7449		135.4491		138.1497			140.8477
	UCL	143.4877	146.47	149.4345	152.3856		155.3266		158.2597		161.1866			164.1084
PZM	LCL	102.8153	104.15	105.3133	106.3271		107.2106		107.9804		108.6512			109.2357
	$\hat{\mathrm{m}}\left(t\right)$	124.7022	126.165	127.4392	128.5491		129.516		130.3582		131.0919			131.731
	UCL	146.5892	148.1799	149.565	150.7711		151.8214		152.736		153.5326			154.2263
DPM	LCL	93.93128	98.49414	103.3268	108.4316		113.8108		119.4664		125.4007			131.6157
	$\hat{\mathrm{m}}\left(t\right)$	114.9445	119.961	125.2629	130.8518		136.7288		142.8956		149.3535			156.1038
	UCL	135.9577	141.4278	147.199	153.2719		159.6469		166.3248		173.3063			180.5919
TCM	LCL	101.5124	103.7205	105.8252	107.8352		109.7585		111.6019		113.3714			115.0726
	$\hat{\mathrm{m}}\left(t\right)$	123.2736	125.6944	127.9996	130.1994		132.3026		134.3169		136.2493			138.1057
	UCL	145.0349	147.6682	150.174	152.5635		154.8467		157.032		159.1271			161.1389
3PFDM	LCL	101.8497	103.5836	105.1914	106.6864		108.0801		109.3824		110.6019			111.7464
	$\hat{\mathrm{m}}\left(t\right)$	123.6436	125.5443	127.3057	128.9424		130.4673		131.8914		133.2244			134.4748
	UCL	145.4374	147.505	149.4199	151.1983		152.8544		154.4004		155.8469			157.2032
New	LCL	101.6161	103.8169	105.9085	107.8999		109.799		111.6129		113.3476			115.009
	$\hat{\mathrm{m}}\left(t\right)$	123.3873	125.8	128.0908	130.2702		132.3469		134.3289		136.2233			138.0364
	UCL	145.1586	147.783	150.2732	152.6404		154.8947		157.045		159.099			161.0638

,

Table A2. 95% Confidence interval of all 10 models (DS2).

Model	Time Index	1	2		3		4		5		6		7		8		9
GOM	LCL	49.147	88.100		114.753		132.788		144.944		153.123		158.620		162.312		164.791
	$\hat{\mathrm{m}}\left(t\right)$	64.942	108.518		137.757		157.376		170.540		179.373		185.300		189.276		191.945
	UCL	80.737	128.935		160.761		181.963		196.135		205.623		211.980		216.241		219.099
DSM	LCL	29.754	81.610		119.319		141.607		153.581		159.671		162.660		164.090		164.762
	$\hat{\mathrm{m}}\left(t\right)$	42.537	101.340		142.735		166.930		179.867		186.433		189.651		191.191		191.914
	UCL	55.320	121.071		166.151		192.253		206.153		213.194		216.643		218.291		219.066
ISM	LCL	49.138	88.084		114.731		132.764		144.918		153.095		158.591		162.282		164.761
	$\hat{\mathrm{m}}\left(t\right)$	64.931	108.500		137.733		157.349		170.511		179.343		185.269		189.245		191.913
	UCL	80.725	128.915		160.736		181.935		196.104		205.590		211.946		216.207		219.065
YIDM	LCL	51.989	90.820		116.125		132.604		143.452		150.736		155.770		159.386		162.108
	$\hat{\mathrm{m}}\left(t\right)$	68.171	111.517		139.254		157.176		168.926		176.797		182.228		186.125		189.057
	UCL	84.354	132.215		162.383		181.748		194.400		202.858		208.686		212.864		216.007
PNZM	LCL	51.946	90.780		116.107		132.609		143.476		150.772		155.811		159.427		162.145
	$\hat{\mathrm{m}}\left(t\right)$	68.123	111.474		139.234		157.181		168.952		176.835		182.272		186.169		189.097
	UCL	84.300	132.168		162.361		181.753		194.428		202.899		208.733		212.912		216.049
PZM	LCL	49.064	88.017		114.677		132.724		144.892		153.080		158.585		162.284		164.769
	$\hat{\mathrm{m}}\left(t\right)$	64.848	108.425		137.674		157.306		170.483		179.327		185.263		189.247		191.921
	UCL	80.631	128.834		160.672		181.888		196.074		205.573		211.940		216.210		219.074
DPM	LCL	123.469	124.167		125.336		126.981		129.105		131.715		134.816		138.411		142.507
	$\hat{\mathrm{m}}\left(t\right)$	147.252	148.012		149.283		151.071		153.379		156.212		159.574		163.470		167.904
	UCL	171.036	171.857		173.230		175.161		177.652		180.709		184.333		188.529		193.301
TCM	LCL	60.749	93.551		114.901		129.743		140.446		148.355		154.305		158.844		162.346
	$\hat{\mathrm{m}}\left(t\right)$	78.066	114.526		137.919		154.071		165.673		174.225		180.648		185.542		189.313
	UCL	95.384	135.501		160.936		178.399		190.901		200.096		206.991		212.239		216.281
3PFDM	LCL	57.549	93.474		115.850		130.807		141.307		148.941		154.636		158.970		162.319
	$\hat{\mathrm{m}}\left(t\right)$	74.461	114.441		138.954		155.226		166.605		174.858		181.005		185.677		189.284
	UCL	91.374	135.408		162.058		179.645		191.903		200.776		207.374		212.384		216.249
New	LCL	62.346	93.042		114.265		129.443		140.426		148.466		154.433		158.930		162.375
	$\hat{\mathrm{m}}\left(t\right)$	79.861	113.965		137.225		153.745		165.652		174.345		180.786		185.634		189.345
	UCL	97.377	134.889		160.184		178.048		190.878		200.224		207.138		212.338		216.315
Model	Time index	10		11		12		13.		14		15		16		17
GOM	LCL	166.455		167.572		168.321		168.824		169.162		169.389		169.541		169.643
	$\hat{\mathrm{m}}\left(t\right)$	193.735		194.937		195.743		196.284		196.647		196.890		197.054		197.163
	UCL	221.016		222.302		223.164		223.743		224.132		224.392		224.567		224.684
DSM	LCL	165.073		165.216		165.280		165.309		165.322		165.328		165.331		165.332
	$\hat{\mathrm{m}}\left(t\right)$	192.249		192.402		192.472		192.503		192.517		192.523		192.526		192.527
	UCL	219.424		219.588		219.663		219.696		219.711		219.718		219.721		219.722
ISM	LCL	166.425		167.542		168.291		168.794		169.131		169.358		169.510		169.612
	$\hat{\mathrm{m}}\left(t\right)$	193.703		194.904		195.710		196.251		196.614		196.857		197.021		197.130
	UCL	220.981		222.267		223.129		223.708		224.096		224.357		224.532		224.649
YIDM	LCL	164.269		166.076		167.661		169.107		170.464		171.767		173.035		174.281
	$\hat{\mathrm{m}}\left(t\right)$	191.383		193.328		195.033		196.587		198.047		199.446		200.809		202.147
	UCL	218.498		220.580		222.405		224.068		225.629		227.126		228.583		230.014
PNZM	LCL	164.298		166.095		167.669		169.101		170.445		171.733		172.986		174.217
	$\hat{\mathrm{m}}\left(t\right)$	191.415		193.349		195.041		196.581		198.025		199.410		200.756		202.078
	UCL	218.531		220.602		222.413		224.061		225.606		227.087		228.526		229.940
PZM	LCL	166.437		167.557		168.309		168.814		169.152		169.380		169.532		169.635
	$\hat{\mathrm{m}}\left(t\right)$	193.716		194.921		195.729		196.272		196.636		196.881		197.045		197.155
	UCL	220.995		222.285		223.150		223.731		224.120		224.382		224.558		224.675
DPM	LCL	147.109		152.223		157.853		164.006		170.687		177.901		185.654		193.951
	$\hat{\mathrm{m}}\left(t\right)$	172.880		178.401		184.474		191.101		198.286		206.034		214.349		223.235
	UCL	198.650		204.580		211.094		218.195		225.885		234.167		243.045		252.519
TCM	LCL	165.073		167.214		168.906		170.251		171.327		172.192		172.889		173.455
	$\hat{\mathrm{m}}\left(t\right)$	192.249		194.552		196.371		197.818		198.974		199.903		200.653		201.260
	UCL	219.424		221.890		223.837		225.384		226.621		227.614		228.416		229.065
3PFDM	LCL	164.940		167.013		168.668		170.001		171.083		171.968		172.697		173.303
	$\hat{\mathrm{m}}\left(t\right)$	192.105		194.335		196.115		197.548		198.711		199.662		200.446		201.097
	UCL	219.270		221.658		223.563		225.096		226.340		227.357		228.195		228.891
New	LCL	165.057		167.175		168.872		170.249		171.380		172.319		173.107		173.773
	$\hat{\mathrm{m}}\left(t\right)$	192.231		194.510		196.335		197.815		199.031		200.040		200.886		201.602
	UCL	219.405		221.845		223.797		225.381		226.682		227.761		228.666		229.431

and

Table A3. 95% Confidence interval of all 10 models (DS3).

Model	Time Index	519	968	1430	1893	2490	3058	3625	4422	5218	5823
GOM	LCL	3.641	9.407	15.376	21.176	28.274	34.589	40.464	48.022	54.803	59.482
	$\hat{\mathrm{m}}\left(t\right)$	9.767	17.639	25.218	32.318	40.791	48.196	55.000	63.660	71.359	76.641
	UCL	15.892	25.870	35.060	43.460	53.309	61.803	69.535	79.298	87.916	93.799
DSM	LCL	−0.409	3.059	8.744	15.540	24.802	33.366	41.233	50.791	58.514	63.256
	$\hat{\mathrm{m}}\left(t\right)$	2.966	8.910	16.770	25.422	36.671	46.770	55.885	66.811	75.550	80.883
	UCL	6.342	14.760	24.796	35.305	48.540	60.174	70.537	82.832	92.586	98.510
ISM	LCL	3.641	9.407	15.376	21.176	28.273	34.589	40.464	48.022	54.802	59.482
	$\hat{\mathrm{m}}\left(t\right)$	9.767	17.639	25.218	32.318	40.791	48.196	55.000	63.660	71.359	76.641
	UCL	15.892	25.870	35.060	43.460	53.309	61.803	69.535	79.298	87.916	93.799
YIDM	LCL	3.631	9.384	15.337	21.122	28.202	34.503	40.367	47.916	54.698	59.387
	$\hat{\mathrm{m}}\left(t\right)$	9.751	17.608	25.171	32.254	40.707	48.096	54.888	63.539	71.241	76.534
	UCL	15.871	25.832	35.004	43.385	53.213	61.689	69.409	79.162	87.784	93.680
PNZM	LCL	3.701	9.506	15.493	21.294	28.373	34.657	40.493	47.993	54.724	59.378
	$\hat{\mathrm{m}}\left(t\right)$	9.853	17.767	25.363	32.460	40.909	48.275	55.033	63.627	71.271	76.523
	UCL	16.005	26.028	35.234	43.627	53.445	61.893	69.573	79.261	87.817	93.668
PZM	LCL	3.458	9.130	15.085	20.933	28.150	34.608	40.627	48.354	55.238	59.940
	$\hat{\mathrm{m}}\left(t\right)$	9.499	17.277	24.856	32.024	40.646	48.218	55.187	64.039	71.852	77.157
	UCL	15.539	25.424	34.628	43.116	53.141	61.828	69.747	79.723	88.466	94.373
DPM	LCL	26.407	26.678	27.161	27.873	29.166	30.827	32.943	36.756	41.628	46.096
	$\hat{\mathrm{m}}\left(t\right)$	38.581	38.903	39.475	40.318	41.845	43.799	46.275	50.713	56.340	61.461
	UCL	50.754	51.128	51.789	52.763	54.523	56.770	59.608	64.671	71.051	76.827
TCM	LCL	5.929	11.851	17.436	22.634	28.871	34.406	39.605	46.448	52.816	57.386
	$\hat{\mathrm{m}}\left(t\right)$	12.993	20.787	27.763	34.075	41.497	47.982	54.009	61.863	69.110	74.278
	UCL	20.058	29.723	38.090	45.516	54.122	61.559	68.413	77.279	85.404	91.170
3PFDM	LCL	3.990	9.962	16.016	21.804	28.789	34.941	40.631	47.938	54.522	59.105
	$\hat{\mathrm{m}}\left(t\right)$	10.271	18.361	26.012	33.076	41.400	48.605	55.191	63.564	71.041	76.216
	UCL	16.552	26.759	36.008	44.348	54.011	62.270	69.752	79.190	87.561	93.327
New	LCL	5.981	12.096	17.800	23.076	29.375	34.946	40.168	47.027	53.403	57.974
	$\hat{\mathrm{m}}\left(t\right)$	13.066	21.099	28.210	34.606	42.091	48.611	54.658	62.525	69.774	74.941
	UCL	20.151	30.102	38.621	46.135	54.807	62.276	69.148	78.023	86.146	91.909
Model	Time index	6539	7083	7487	7846	8205	8564	8923	9282	9641		10,000
GOM	LCL	64.535	68.049	70.489	72.543	74.495	76.350	78.112	79.786	81.376		82.886
	$\hat{\mathrm{m}}\left(t\right)$	82.318	86.251	88.977	91.267	93.441	95.504	97.461	99.319	101.081		102.754
	UCL	100.101	104.454	107.465	109.992	112.387	114.658	116.810	118.851	120.786		122.621
DSM	LCL	67.773	70.526	72.248	73.576	74.736	75.748	76.627	77.391	78.053		78.626
	$\hat{\mathrm{m}}\left(t\right)$	85.943	89.018	90.939	92.418	93.710	94.834	95.812	96.660	97.395		98.031
	UCL	104.112	107.510	109.629	111.260	112.683	113.921	114.997	115.930	116.738		117.437
ISM	LCL	64.535	68.049	70.489	72.543	74.495	76.350	78.112	79.786	81.376		82.886
	$\hat{\mathrm{m}}\left(t\right)$	82.318	86.251	88.977	91.267	93.441	95.504	97.461	99.319	101.081		102.754
	UCL	100.100	104.454	107.465	109.992	112.387	114.658	116.810	118.851	120.786		122.621
YIDM	LCL	64.460	67.996	70.457	72.532	74.508	76.389	78.181	79.887	81.511		83.059
	$\hat{\mathrm{m}}\left(t\right)$	82.234	86.192	88.941	91.255	93.455	95.547	97.537	99.430	101.231		102.945
	UCL	100.007	104.388	107.425	109.978	112.403	114.706	116.894	118.974	120.951		122.831
PNZM	LCL	64.417	67.936	70.389	72.462	74.440	76.328	78.131	79.854	81.500		83.074
	$\hat{\mathrm{m}}\left(t\right)$	82.186	86.125	88.865	91.177	93.380	95.480	97.483	99.394	101.218		102.962
	UCL	99.954	104.314	107.342	109.892	112.320	114.631	116.834	118.934	120.937		122.850
PZM	LCL	64.953	68.387	70.742	72.704	74.547	76.279	77.905	79.430	80.860		82.201
	$\hat{\mathrm{m}}\left(t\right)$	82.786	86.629	89.260	91.447	93.499	95.425	97.231	98.924	100.510		101.995
	UCL	100.619	104.872	107.777	110.189	112.451	114.571	116.558	118.418	120.160		121.789
DPM	LCL	52.288	57.681	62.083	66.287	70.771	75.541	80.600	85.954	91.607		97.562
	$\hat{\mathrm{m}}\left(t\right)$	68.511	74.610	79.566	84.280	89.292	94.604	100.222	106.148	112.385		118.937
	UCL	84.734	91.540	97.049	102.274	107.812	113.668	119.843	126.341	133.163		140.312
TCM	LCL	62.528	66.257	68.936	71.255	73.519	75.730	77.891	80.003	82.069		84.090
	$\hat{\mathrm{m}}\left(t\right)$	80.065	84.247	87.243	89.832	92.355	94.815	97.216	99.560	101.849		104.086
	UCL	97.603	102.237	105.550	108.408	111.190	113.900	116.541	119.116	121.629		124.082
3PFDM	LCL	64.115	67.651	70.138	72.257	74.295	76.256	78.144	79.963	81.717		83.410
	$\hat{\mathrm{m}}\left(t\right)$	81.847	85.806	88.586	90.949	93.218	95.399	97.497	99.515	101.459		103.333
	UCL	99.578	103.961	107.033	109.641	112.142	114.543	116.849	119.067	121.202		123.257
New	LCL	63.115	66.844	69.522	71.840	74.104	76.315	78.476	80.590	82.657		84.680
	$\hat{\mathrm{m}}\left(t\right)$	80.725	84.903	87.897	90.484	93.006	95.465	97.866	100.210	102.500		104.739
	UCL	98.334	102.963	106.272	109.128	111.907	114.615	117.255	119.830	122.343		124.797

in Appendix A list the 95% confidence intervals of all 10 NHPP software reliability models for DS1, DS2, and DS3. In addition, the relative error value of the proposed software reliability model confirms its ability to provide more accurate predictions as it remains closer to zero when compared to the other models (Figure 8, Figure 9 and Figure 10).

5.3. Optimal Software Release Time

Factor η captures the effects of the field environmental factors based on the system failure rate as described in Section 2. System testing is commonly carried out in a controlled environment, where we can use a constant factor η equal to 1. The newly proposed model becomes a delayed S-shaped model when η = 1 in (7). Thus, we apply different mean value functions $\mathrm{m}\left(\mathrm{t}\right)$ to the cost model $\mathrm{C}\left(\mathrm{T}\right)$ of (8) when considering the three conditions described below. We apply the cost model to these three conditions using DS1 (Table 2). Using the LSE method, the parameters of the delayed S-shaped model and the newly proposed model are obtained, as described in Section 5.2.

(1) The expected total software cost with controlled environmental factor (η = 1) is

$${\mathrm{C}}_1\left(\mathrm{T}\right)={\mathrm{C}}_0+{\mathrm{C}}_1\mathrm{T}+{\mathrm{C}}_2\mathrm{m}\left(\mathrm{T}\right){\mathsf{\mu }}_{\mathrm{y}}+{\mathrm{C}}_3\left(1-\mathrm{R}\left(\mathrm{x}|\mathrm{T}\right)\right)+{\mathrm{C}}_4\left[\mathrm{m}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)-\mathrm{m}\left(\mathrm{T}\right)\right]{\mathsf{\mu }}_{\mathrm{w}}$$

(19)

where

$$\mathrm{m}\left(\mathrm{T}\right)=\mathrm{a}(1-\left(1+\mathrm{bT}\right){\mathrm{e}}^{-\mathrm{bT}}),\mathrm{m}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)=\mathrm{a}(1-\left(1+\mathrm{b}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)\right){\mathrm{e}}^{-\mathrm{b}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)}).$$

(20)

(2) The expected total software cost with a random operating environmental factor (η = f(x)) is

$${\mathrm{C}}_2\left(\mathrm{T}\right)={\mathrm{C}}_0+{\mathrm{C}}_1\mathrm{T}+{\mathrm{C}}_2\mathrm{m}\left(\mathrm{T}\right){\mathsf{\mu }}_{\mathrm{y}}+{\mathrm{C}}_3\left(1-\mathrm{R}\left(\mathrm{x}|\mathrm{T}\right)\right)+{\mathrm{C}}_4\left[\mathrm{m}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)-\mathrm{m}\left(\mathrm{T}\right)\right]{\mathsf{\mu }}_{\mathrm{w}}$$

(21)

where

$$\mathrm{m}\left(\mathrm{T}\right)=\mathrm{N}{\left(1-\frac{\mathsf{\beta }}{\mathsf{\beta }+\mathrm{aT}-\ln \left(1+\mathrm{aT}\right)}\right)}^{\mathsf{\alpha }},\mathrm{m}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)=\mathrm{N}{\left(1-\frac{\mathsf{\beta }}{\mathsf{\beta }+\mathrm{a}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)-\ln \left(1+\mathrm{a}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)\right)}\right)}^{\mathsf{\alpha }}.$$

(22)

(3) The expected total software cost between the testing environment (η = 1) and field environment (η = f(x)) is

$${\mathrm{C}}_3\left(\mathrm{T}\right)={\mathrm{C}}_0+{\mathrm{C}}_1\mathrm{T}+{\mathrm{C}}_2{\mathrm{m}}_1\left(\mathrm{T}\right){\mathsf{\mu }}_{\mathrm{y}}+{\mathrm{C}}_3\left(1-\mathrm{R}\left(\mathrm{x}|\mathrm{T}\right)\right)+{\mathrm{C}}_4\left[{\mathrm{m}}_2\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)-{\mathrm{m}}_1\left(\mathrm{T}\right)\right]{\mathsf{\mu }}_{\mathrm{w}}$$

(23)

where

$${\mathrm{m}}_1\left(\mathrm{T}\right)=\mathrm{a}\left(1-\left(1+\mathrm{bT}\right){\mathrm{e}}^{-\mathrm{bT}}\right),{\mathrm{m}}_2\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)=\mathrm{N}{\left(1-\frac{\mathsf{\beta }}{\mathsf{\beta }+\mathrm{a}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)-\ln \left(1+\mathrm{a}\left(\mathrm{T}+{\mathrm{T}}_{\mathrm{w}}\right)\right)}\right)}^{\mathsf{\alpha }}.$$

(24)

We consider the following coefficients in the cost model for the baseline case:

$${\mathrm{C}}_0=100,{\mathrm{C}}_1=20,{\mathrm{C}}_2=50,{\mathrm{C}}_3=2000,{\mathrm{C}}_4=400,{\mathrm{T}}_{\mathrm{w}}=10,\mathrm{x}=20,{\mathsf{\mu }}_{\mathrm{y}}=0.1,{\mathsf{\mu }}_{\mathrm{w}}=0.2$$

(25)

The results of the baseline case are listed in

Table 8. Optimal release time T* subject to the warranty period.

Warrnaty Period	C1(T)	T*	C2(T)	T*	C3(T)	T*
Tw = 2	1173.41	14.2	1403.78	11.6	599.88	10.5
Tw = 5	1286.95	14.9	1928.63	22.8	1334.72	11.3
Tw = 10(basic)	1338.70	15.1	2398.24	34.7	2263.33	12.3
Tw = 15	1348.88	15.2	2702.33	42.7	2969.88	13.0

, and the expected total cost for the three conditions above is 1338.70, 2398.24, and 2263.33, respectively. For the second condition, the expected total cost and the optimal release time are high. The expected total cost is the lowest for the first condition, and the optimal release time is shortest for the third condition.

To study the impact of different coefficients on the expected total cost and the optimal release time, we vary some of the coefficients and then compare them with the baseline case. First, we evaluate the impact of the warranty period on the expected total cost by changing the value of the corresponding warranty time and comparing the optimal release times for each condition. Here, we change the values of T_w from 10 h to 2, 5, and 15 h, and the values of the other parameters remain unchanged. Regardless of the warranty period, the optimal release time for the third condition is the shortest, and the expected total cost for the first condition is the lowest overall. Figure 11 shows the graph of the expected total cost for the baseline case. Figure 12, Figure 13 and Figure 14 show the graphs of the expected total cost subject to the warranty period for the three conditions.

Next, we examine the impact of the cost coefficients, C₁, C₂, C₃, and C₄ on the expected total cost by changing their values and comparing the optimal release times. Without loss of generality, we change only the values of C₂, C₃, and C₄, and keep the values of the other parameters C₀ and C₁ unchanged, because different values of C₀ and C₁ will certainly increase the expected total cost. When we change the values of C₂ from 50 to 25 and 100, the optimal release time is only changed significantly for the second condition. As can be seen from

Table 9. Optimal release time T* according to cost coefficient C_2.

Cost Coefficient C2	C1(T)	T*	C2(T)	T*	C3(T)	T*
C2 = 25	1036.02	15.2	2013.25	37.5	1972.06	12.5
C2 = 50 (basic)	1338.70	15.1	2398.24	34.7	2263.33	12.3
C2 = 100	1943.64	15.1	3141.20	29.1	2843.35	12.1

, the optimal release time T* is 37.5 when the value of C₂ is 25, and 29.1 when the value of C₂ is 100. When we change the value of C₃ from 2000 to 500 and 4000, the optimal release time is only changed significantly for the first condition. As

Table 10. Optimal release time T* according to cost coefficient C_3.

Cost Coefficient C3	C1(T)	T*	C2(T)	T*	C3(T)	T*
C3 = 500	1270.65	16.5	2398.24	34.7	2262.96	12.4
C3 = 2000 (basic)	1338.70	15.1	2398.24	34.7	2263.33	12.3
C3 = 4000	1376.26	14.6	2398.24	34.7	2263.77	12.3

shows, the optimal release time T* is 16.5 when the value of C₃ is 500, and 14.6 when the value of C₃ is 4000. When we change the value of C₄ from 400 to 200 and 1000, the optimal release time is changed for all of the conditions. As can be seen from

Table 11. Optimal release time T* according to cost coefficient C_4.

Cost Coefficient C4	C1(T)	T*	C2(T)	T*	C3(T)	T*
C4 = 200	1183.14	14.3	1859.29	20	1590.02	11.6
C4 = 400 (basic)	1338.70	15.1	2398.24	34.7	2263.33	12.3
C4 = 1000	1680.99	16.3	3272.23	61	4253.45	12.8

, the optimal release time T* is 14.3 for the first condition when the value of C₄ is 200, and 16.3 when the value of C₄ is 1000. In addition, the optimal release time T* is 20.0 for the second condition when the value of C₄ is 200, and 61.0 when the value of C₄ is 1000. The optimal release time T* is 11.6 for the third condition when the value of C₄ is 200, and 12.8 when the value of C₄ is 1000. Thus, the second condition has a much greater variation in optimal release time than the other conditions. As a result, we can confirm that the cost model of the first condition does not reflect the influence of the operating environment, and that the cost model of the second condition does not reflect the influence of the test environment. Figure 15 shows the graph of the expected total cost according to the cost coefficient C₂ in the 2nd condition. Figure 16, Figure 17 and Figure 18 show the graphs of the expected total cost according to cost coefficient C₄ in the three conditions.

6. Conclusions

Existing well-known NHPP software reliability models have been developed in a test environment. However, a testing environment differs from an actual operating environment, so we considered random operating environments. In this paper, we discussed a new NHPP software reliability model, with S-shaped growth curve that accounts for the randomness of an actual operating environment. Table 5, Table 6 and Table 7 summarize the results of the estimated parameters of all ten models that are applied using the LSE technique and six common criteria (MSE, AIC, PRR, PP, SAE, and R²) for the DS1, DS2, and DS3 datasets. As can be seen from Table 5, Table 6 and Table 7, the newly proposed model displays a better overall fit than all of the other models when compared, particularly in the case of DS2. In addition, we provided optimal release policies for various environments to determine when the total software system cost is minimized. Using a cost model for a given environment is beneficial as it provides a means for determining when to stop the software testing process. In this paper, faults are assumed to be removed immediately when a software failure has been detected, and the correction process is assumed to not introduce new faults. Obviously, further work in revisiting these assumptions is worth the effort as our future study. We hope to present some new results on this aspect in the near future.