Estimating Parameters of Software Reliability Growth Models Using Artificial Neural Networks Optimized by the Artificial Bee Colony Algorithm Based on a Novel NHPP

In the fast-paced realm of modern technology, ensuring the reliability of software systems is crucial for maintaining operational efficiency and achieving user satisfaction [1, 2]. Software failures can result in substantial financial repercussions, security vulnerabilities, and a loss of user trust. Consequently, the development of robust Software Reliability Growth Models (SRGMs) is vital for accurately predicting and enhancing software performance throughout its lifecycle [3-6].

In today’s software world, software dependability is a critical objective in software construction. Multi Var Time Series Software Reliability Growth Models Losses occur if software fails to meet the customers’ expectations, poses insecurities, and loses users’ confidence hence the need for developers to incorporate effective SRGMs. As will be discussed, these models are invaluable in the context of software performance prediction and improvement across the software development life-cycle. With an even more complex structure of applications being developed every passing day, there has been growing pressure for even better modeling methods that can capture software failure behavior [7-10].

The Non-Homogeneous Poisson Process, abbreviated as (NHPP) has increasingly become a popular choice for software reliability modeling [11]. This approach enables one to express the failure occurrences with time and enables one to capture changes in failing rates resulting from differences in operating conditions. The new developments in future work have incorporated Order Statistics distributions, namely Rayleigh distribution, in presenting better reliability features to software systems. These distributions help the researchers to understand failure data better and consequently make accurate predictions of the software’s performance [12-14].

Besides statistical methods, the use of new methods like Machine Learning methods, particularly Artificial Neural Networks (ANN) has also been incorporated in software reliability modeling [15, 16]. The integration of ANN with optimization algorithms like the Artificial Bee Colony (ABC) algorithm has proved to improve parameter estimation techniques incorporated in SRGMs. This innovative approach not only enhances the accuracy of the predictors but is also capable of dealing with massive and complicated data and information that are universal in software applications [17-19].

Nevertheless, there are some issues with the application of software reliability modeling even if certain improvements can be seen. While using such models, the assumptions made are often not well adaptable to diverse software systems in their current form. Precisely, the quality and representativeness of failure data play important roles in determining the performance of these models. Finally, situations, when the data is scarce or contaminated, affect the reliability predictions, which proves the importance of developing stronger techniques suitable for software systems.

This research seeks to fill gaps found in existing models of software reliability including the failure to capture intricate patterns that vary with time. Through this work, a new model within the NHPP framework, incorporating the Rayleigh Order Statistics of software reliability, and ANN-ABC principles for parameter estimation has been developed. These more advanced methods improve the capability of the software reliability models and make them more general for many software systems. The results of the study should encourage further development of software engineering research in the aspect of improving quality assurance as well as the prolongation of software systems’ life cycle.

1.1 Literature review

In 2017, the researchers proposed a new NHPP software reliability model with an S-shaped growth curve that accounts for random operating environments. They aimed to improve predictions of software defects, optimize release times, and minimize overall software system costs during the development process [20].  

Furthermore, in 2019, the researchers developed a new software reliability growth model based on the Weibull Order Statistics distribution and the NHPP. They created an algorithm and a MATLAB program for practical implementation, employed maximum likelihood estimation for parameter estimation, and analyzed software failure data to validate the model. Additionally, they suggested future research directions involving advanced optimization techniques to enhance reliability predictions [21].

In 2022, the researchers proposed a hybrid meta-heuristic approach combining Ant Colony Optimization (ACO) with the BAT Algorithm (HACO-BA) to optimize deep learning models for software cost estimation. Their work focused on improving the accuracy of software cost predictions by fine-tuning COCOMO II coefficients and enhancing deep learning model training. By comparing various optimization algorithms, including ACO, BAT, and HACO-BA, the researchers demonstrated that HACO-BA provided better performance in terms of minimizing execution time and improving prediction accuracy [22].

In 2023, all the researchers in this article aimed to apply the NHPP reliability growth model to assess failure data in repairable systems. They explored and estimated various models such as the Crow Power Law model and also conducted tests for goodness of fit. This also involved using a log-linear model when results from the Power Law model were and/or could not adequately fit. Moreover, the utilization of the Weibull Time to Failure Recurrent Neural Network (WTTE-RNN) framework was attempted but not viable for some types of NHPP data [23].

In 2024, the researchers propose a finite failure software reliability model based on the extended log-logistic distribution within the framework of the NHPP. They derived the model's properties, applied the maximum likelihood estimation method to estimate its parameters using the Newton-Raphson method, and evaluated the model with three real software reliability datasets. The results show that the proposed model outperformed several well-known NHPP models in terms of goodness-of-fit and predictive performance across all datasets [24].

Future trends for SRGM are aimed mainly at the improvement of the accuracy of prediction, flexibility of models, and applicability of the modeling approaches in practice. These include the application of artificial intelligence and machine learning for enhanced prediction as well as the use of real-time failure data to enhance model responsiveness. It has been observed that modern mixed models that integrate statistical and machine learning methods have been evolving. Further, the focus has shifted to the collection of user-oriented reliability measures, like user satisfaction level and user experience. Finally, efforts are made to get more empirical evidence to prove models on different software systems and environments which remains a challenge for developing more advanced SRGMs to fulfill the demands of modern software engineering.

1.2 Limitations of the NHPP model for Rayleigh Order Statistics in SRGM

Model assumptions: The proposed reliability growth model is based on specific assumptions inherent to the Rayleigh process and the NHPP. These assumptions may not hold for all software systems or failure scenarios, potentially limiting the model's applicability in diverse contexts.

Data dependency: The effectiveness of the Weibull order statistics distribution in modeling software reliability is contingent upon the quality and quantity of failure data available. In cases where data is sparse or not representative of the software's operational environment, the model's predictions may be less reliable.

Complexity of real-world systems: Real-world software systems often exhibit complex behaviors that may not be fully captured by the proposed model. Factors such as user interactions, environmental conditions, and varying usage patterns can influence software reliability in ways that are difficult to quantify.

Computational resources: The implementation of the algorithm and MATLAB program may require significant computational resources, especially when dealing with large datasets or complex models. This could pose challenges for practitioners with limited access to advanced computational tools.

Limited scope of optimization techniques: Although the paper suggests the integration of advanced optimization techniques like ANN and ABC for future research, the current study does not explore these methods in depth. The potential benefits of these techniques remain theoretical until empirically validated in the context of software reliability.

Generalizability: The findings and methodologies presented may be specific to the datasets used in the study. Generalizing the results to other software systems or industries may require further validation and testing.

Focus on quantitative metrics: The model primarily emphasizes quantitative reliability metrics, potentially overlooking qualitative aspects of software reliability, such as user experience and satisfaction, which are also critical for assessing overall software performance.

By tackling the pressing need for dependable software systems, this research contributes significantly to the ongoing advancements in software engineering, particularly in enhancing quality assurance practices and prolonging the operational lifespan of software applications. The insights and methodologies presented in this work are intended to serve as a valuable resource for both researchers and practitioners, promoting further progress in the domain of software reliability.

This paper introduces an innovative approach to assessing software reliability by formulating a growth model based on the Rayleigh Order Statistics distribution, anchored in the NHPP. The Rayleigh process is esteemed for its adaptability in modeling diverse failure rates, making it particularly suitable for addressing the intricacies of software reliability. By utilizing order statistics, this model establishes a comprehensive framework for analyzing failure data and estimating key reliability metrics.

Moreover, the paper advocates for the incorporation of advanced optimization techniques, such as the combination of ANN and ABC, in future research endeavors. This synergistic approach holds promise for enhancing parameter estimation processes, thereby yielding more precise predictions of software reliability.

1.3 Theoretical foundations of Software Reliability Growth Model

Software reliability growth modeling is an important sub-discipline of software engineering that deals with the prediction and enhancement of the reliability of software systems in the software development life-cycle stages. SRGMs apply statistical assessments to the failure data collected, allowing the developer to predict the remaining defective items and the reliability level of the software in use. SRGMs can also serve as a guide to defining the improvement of reliability when defects are found and fixed to facilitate better planning of release times, improvement of QA measures, and, consequently, the minimization of expenses connected with software failures. With the developing complexity of the software systems, operational effectiveness and user satisfaction calls for efficient SRGMs.

The main models in Software Reliability Growth Modeling (SRGM) include:

NHPP: This model takes into consideration different failure rates at different times to accommodate for dynamism in environments such as software.

Weibull distribution models: These models effectively provide options for the portrayal of various failure behaviors by holding the capability to portray rising and declining failure rates.

Logarithmic and exponential models: These have been widely adopted since they are easy to apply but often, they lack the capability of modeling complex failure functions.

Hybrid models: These models integrate standard and advanced statistical methods with MLE’s and contribute to improved prognosis and adaptability for increased variable difficulty and data licenses.

Meta-heuristic approaches: Two classes of metaheuristic methods, including Genetic Algorithms and Particle Swarm Optimization, are applied to enhance parameter estimation of SRGMs.

All of these models have some advantages and disadvantages, the decision on which model to use when depends on the nature of the software being analyzed.

The NHPP, which is assumed by the majority of SRGMs, is described by the following equation:

$p[N(t)=y]=\frac{[m(t)]^y e^{-m(t)}}{y!}, y=1,2,3, \ldots$                        (1)

It describes the total number of failures up to a specific execution time $t$, shown as $N(t)(t>0)$. The predicted cumulative number of failures at time $t$ is represented by the mean value function $m(t)$, as follows:

$m(t)=\int_0^t \lambda(u) d u, 0<\tau<\infty$                        (2)

with $m(t)$ the NHPP-based dependability function may be stated as follows [24]. The likelihood of no failures in the time interval $(0, t)$ is defined as the reliability function $R(t)$, which is provided by:

$R(t)=p\{N(t)=0\}=e^{-m(t)}$                        (3)

Reliability $R\left(\frac{y}{t}\right)$ generally indicates the likelihood that there won't be any failures during the period. $[t, t+y]$ is given by:

$R\left(\frac{y}{t}\right)=p\{N(t+y)-N(t)=0\}=e^{-[m(t+y)-m(t)]}$                        (4)

Eq. (4) is called the SRGM or software reliability based on NHPP. The probability density functions as follows:

$f(y)=\lambda(\mathrm{t}+\mathrm{y}) e^{-[m(t+y)-m(t)]}$                        (5)

3.1 Mathematical formulations

Software reliability is best described as the ability of a software system to perform without encountering any error over a given duration of operation given certain conditions. Reliability analysis must also be done and quality maintenance of software is highly important in order that the system have the right performance and dependability that is needed. Software Reliability Growth Models therefore stand as key drivers for assessing performance and achieving rigorous debugging in order to improve the durability of the system [29].

A previous model has been developed based on the Rayleigh Order Statistics distribution fit within the NHPP model. This integration enables the model to capture failure rates to increase monotonically with time as observed in realistic software systems. The foundation of the model is the mean value function $m_1(t)$ which is mathematically:

$m_1(t)=\int_0^t \lambda(u) d u$                                          (6)

where, $\lambda(t)$ denotes the failure intensity function. For a random variable $X$ following a Rayleigh process, the occurrence rate is determined by:

$\lambda(t)=\frac{1}{b^2} t, t \geq 0, b>0$                                          (7)

where, $b$ represents scale parameter. If we substitute Eq. (7) into Eq. (6) and simplify, then we obtain:

$m_1(t)=\frac{1}{2 b^2} t^2$                                          (8)

For the stochastic process, the inter-arrival times are determined by Hussain et al. [29]:

$f(t)=\lambda(t) e^{-m\left(t_0\right)}$                                          (9)

For the Rayleigh process, we may express the probability density function as follows:

$f(t)=\frac{1}{b^2} t e^{-\frac{1}{2 b^2} t_0^2}, t>0$                                          (10)

The cumulative distribution function for New Process is:

$F(t)=1-e^{-\frac{1}{2 \mathrm{~b}^2} \mathrm{t}_0^2}$                                          (11)

Suppose that $\left(X_1, \ldots, X_n\right)$, there are n random variables with joint distribution. The $X_{i / s}$ is arranged in ascending order, which represents the order of the corresponding statistics. Thus $X_{1: n} \leq X_{2: n} \leq \cdots \leq X_{n: n}$. An independent, identically distributed sample from a continuous distribution of absolute density $f(x)$ has the joint density function of the order statistics as [30]:

$\begin{gathered}f_{X_{1: n}, \ldots, X_{n: n}}\left(x_1, x_2, \ldots, x_n\right)=n!\prod_{i=1}^n f\left(x_i\right), \\ \infty<x_1<x_2<\cdots<x_n<\infty\end{gathered}$                                          (12)

NHPP models or fault counting models can be classified into finite or infinite failure models, depending on the specification. In this model, the number of failures follows the distribution of a Non-Homogeneous Poisson Process, and the failure severity function is determined according to this distribution $\lambda_1(x)$ is defined as $\lambda_1(x)=a f(x)$ where $a$ is the number of failures expected and $f(x)$ is the probability density function of $X$. Based on NHPP assumptions, mean value function is $m(x)=a F(x)$, where $F(x)$ is the cumulative distribution function of $X$ and $a=\frac{m(x)}{F(x)}$ [31].

3.2 Properties of the model

The proposed model exhibits the following properties:

Time-dependent failure analysis: The model integrates failure rates that tend to rise over time, factors considered when analyzing software stress and operational usage with the help of the NHPP framework.

Order statistics integration: Order statistics are incorporated into the model in order to model the prospective time until the k-th failure, a parameter, which is essential if the system needs to be repaired more than once.

Adaptability: The model is flexibly designed for accommodating different current operational conditions and different failure patterns making it to suit many scenarios of software reliability.

3.3 Assumptions

The proposed model is based on the following assumptions:

•The software failure process follows Rayleigh distribution of the NHPP model.

•The failure rates rise with time and this is consistent with the behavior experienced with software reliability data.

•Failure data needs to be of high quality and preferably representative of the actual system for accurate identification of the parameters and subsequent prediction of the system.

3.4 New process order statistics growth model

Let $X_1, X_2, \ldots, X_n$ random variables represent a sample of the cumulative time intervals between failures. And let $X_{1: n}, \ldots, X_{n: n}$ the original random variables so that $X_{1: n} \leq$ $X_{2: n} \leq \cdots \leq X_{n: n}$ are called the order statistics. The probability density function of Rayleigh process $r^{\text {th }}$ order statistics is given by:

$\begin{aligned}f_{r: n}(x)= & r\binom{n}{r} \frac{x}{b^2}\left[1-e^{\frac{-x_0}{2 b^2}}\right]^{r-1} e^{-(n-r+1) \frac{x_0 2}{2 b^2}} \\& x \in[0, \infty) ; b>0 ; 1 \leq r \leq n\end{aligned}$                                          (13)

The $c d f$ is:

$F_{r: n}(x)=\sum_{i=r}^n\binom{n}{i}\left[1-e^{\frac{-x_0{ }^2}{2 b^2}}\right]^r e^{-(n-i) \frac{x_0^2}{2 b^2}}$                                          (14)

Simulation is a scenario designed to compare any system with the real world, and is defined as the attempt to simulate a particular process under specific circumstances using artificial methods that resemble natural conditions. This includes building a smaller model that is an identical copy of the real model and performing tests on the miniature model examining the results and generalizing them to the original model, or computer simulation by writing a program for the methods to be chosen under realistic programming conditions and then observing the results obtained with the program and drawing a conclusion based on them [37, 39].

There are different simulation methods, namely the (analog method), the (mixed method) and the (Monte Carlo method). The Monte Carlo method is one of the most important and widely used simulation methods, in which a random sample of the phenomenon is generated, that corresponds to the behavior of a certain probability distribution that the phenomenon has. To achieve this, the probability distribution of the phenomenon it has (CDF) it is known that the set of samples random in this way possesses the property of independence because random samples in this method are by applying the mathematical method to each sample separately [38-40].

To put the previously discussed ideas into practice, the practical part of the research focused on the estimators of the suggested model for both of the approaches used, utilizing a simulation method. The objective was to apply the RMSE statistical criteria to various sample sizes in order to assess the optimality of these estimators. The purpose of the simulation model was to provide a comparison study of the approaches that were evaluated. By showing how the estimate techniques affect the following variables, this strategy seeks to determine the best technique for estimating parameters inside the interval of the new process distribution [39].

• Change in sample size.
• Change in model parameter values.

5.1 Stages of building a simulation experience 

First stage: Parameter value selection: Many of the default values were selected based on prior studies and experimental tests ($a=1.9 ; 0.9 ; 1.2 ; 1.1$ and $b=0.9 ; 1.2 ; 1.1 ; 1.9$), with statistical methods such as modified maximum likelihood estimation and an enhanced neural network optimized using the ABC algorithm employed to determine the optimal parameter values. It is the most important stage on which the program’s steps and procedures depend. Below are the steps for this stage:

Choose default values for the parameters of the new process. Several default values were chosen for the failures expected parameter a and the scale parameter b for the new process by reviewing previous studies and experimenting with many default values for the parameters, which led us to choose the best of these values, as follows above.

Second stage: Choose sample sizes: Several different sample sizes (small, medium, large) were chosen as follows: ($n=20 ; 50$).

Third stage: Number of replications: The simulation was repeated 1000 times to obtain reliable estimates and to account for the variability in the data to achieve a balance between computational efficiency and statistical reliability.

Fourth stage: Performance metrics: Root Mean Square Error (RMSE) measure was used. To evaluate the accuracy of parameter estimates. It is calculated as

$R M S E=\sqrt{\frac{\sum_{i=1}^Q\left(\widehat{\gamma}_l-\gamma\right)^2}{Q}}$

where,

$\widehat{\gamma}_l$: represents value of the parameter estimated in iteration $i$.

$\gamma$: represents the real parameter value.

$Q$: represents number of iterations.

In the context of the study, RMSE quantifies the difference between the observed software failure data and the values predicted by the proposed model. The smaller the RMSE, the closer the predictions are to the actual data, indicating better model performance.

Fifth stage: Data generation: At this stage, random data is generated using the inverse transformation method and according to the new process, as follows:

Generating a random variable $u_i$ that follows a uniform distribution with the interval $(0,1)$ using the cumulative distribution function with the help of the Rand.

$u_i \sim U(0,1), i=0,1,2, \ldots \ldots, n$                             (22)

where, $u_i$ represents a continuous random variable that follows a uniform distribution.

Convert the data generated in step (first) that follows a uniform distribution into data that follows a new mixed distribution using the inverse function (CDF) transformation method and according to Eq. (17) and as in the following formula.

$t_i=\sqrt{2 b^2 * \ln \left(1-\frac{u}{a}\right)}, i=0,1,2, \ldots, n$                             (23)

Sixth stage: At this stage, parameters are estimated over the period for the new process and for all methods, which are MMLE, and ANN-ABC.

Seventh stage: The experiment is repeated (1000) times.

5.2 Simulation result

After conducting the simulation experiment by implementing the program in the electronic calculator, the results of the parameter estimate for the new mixed Rayleigh distribution were obtained using the two estimation methods described in the above sections and the RMSE. The results of the simulation estimations were presented in Table 1 in order to reach the best estimate. Comparison between the studied estimation methods. In simulating the proposed model, parameters a and b were chosen based on empirical evidence from prior studies and sensitivity analysis to balance model accuracy and computational efficiency. The values different a, and b were selected for their effectiveness in minimizing RMSE and capturing observed failure characteristics.

Table 1. Simulated RMSE comparison of MMLE and ANN-ABC estimates for new process parameters

The first differences of the mean value function are presented in Tables 4 and 5, m(x) of the model suggested in this paper well compared with the Weibull model [21], estimated based on the ANN-ABC. Table 3 illustrates the effectiveness of the proposed model compared to Table 4 and the Weibull model that can unavailable failure numbers Table 4 shows the comparison of the proposed model in terms of the average time length with the result presented in Table 5 and Figure 3. This is because the mean value function rises with failure and this indicates huge differences. For example, the mean value function increases from 1.0604 when the system fails number 3 to 6.0667 when the system fails number 4 suggesting that the system can have multiple failures as a result of perhaps inherent software flaws. On the other hand, the latter failures possess more stable values of the successive differences which indicates a possibility of the reliability growth curve flattening which should be studied further [45]. This analysis therefore supports the utility of advanced modeling methods like ANN-ABC in developing greater insights or otherwise, of the patterns of software reliability and to assist in the formulation of good approaches towards improving the performance of software systems and systems.

Table 4. Successive differences of the mean value function for the new model obtained using ANN-ABC optimization based on mean square error

Table 5. Successive differences of the mean value function for the Weibull model obtained using ANN-ABC optimization based on mean square error

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Figure 3. Histogram of cumulative failures with logarithmic function representation

Table 6. RMSE comparison of estimation methods for the proposed order statistics growth model and the Weibull model

As shown in Table 6, ANN trained using ABC algorithm outperformed the MMLE method, achieving the lowest RMSE for the proposed model compared to the Weibull model.

7.1 Accuracy evaluation

The comparison of the execution time of the ANN combined with the ABC algorithm with the MMLE method is likewise dependent on the size of data set, the complexity of the developed model, and the procedure of algorithm implementation. ANN-ABC, despite being potentially more accurate because of high flexibility due to non-parametric belonging, trained iteratively and optimized, highly computational and slower in large data mining exercises. On the other hand, MMLE is more efficient and the method which goes straight to calculate the maxima of the likelihood function, but it is not suitable for complex and non-linear data forms [46]. Deciding between these is a function of the user’s particular needs when it comes to maximizing accuracy versus computational time with ANN-ABC scaling in favor of accuracy over speed and MMLE appropriate for the speed it offers over other methods.

7.2 Discussion

This paper presents the incorporation of Rayleigh Order Statistics distribution into a new service time distribution model anchored on NHPP. This approach allows for a more reactive approach to different failure rates, thus coping with the unbounded failure of real-life software systems. This incorporation of Artificial Neural Networks is in combination with optimization methods such as the ABC algorithm that leads to more precise identification of parameters which in turn provides much more likelihood of correct predictions regarding the performance of the given software. These contributions have real-world relevance for software developers and engineers because the model proposed in this paper can enhance quality control processes and extend the deployment time of software solutions. Thus, given the higher understanding of failure patterns and reliability growth, the model helps make successful maintenance decisions and avoid resource wastage. However, the study has its shortcomings, which include the quality and quantity of failure data, failure of the method to represent real-world system complexity, and computational difficulty in the implementation of this method.

Regarding its originality, additional focus should be made on comparing the paper’s proposed methodologies to the traditional NHPP models or other statistical methods to achieve a clearer perspective of the strengths and the weaknesses of the proposed work as a contribution to the state of the art in software reliability analysis. Perhaps the deconstruction of these aspects would add more depth to these arguments, and, therefore, the study; more so, because the research findings appear to have applicability for both researchers and practitioners.

This article is a major breakthrough in software reliability modeling owing to the development of the growth model formulated from the Rayleigh Order Statistics distribution within the context of the NHPP model. This approach improves the feasibility of modeling various failure rates of software systems since failure rates differ from one project to another. The feature of the proposed model presents clear advantages in reliability assessment over conventional techniques since it can provide more reliable estimation of failure behavior and other crucial measures affecting reliability to enhance the objectives of quality in software engineering disciplines.

In addition, the use of ANN together with optimization algorithms like ABC algorithm opens another prospect for improvement in parameters identification as well as prediction accuracy especially in regard to large and intricate datasets characteristic of software systems. This paper also puts forward the direction for the future empirical research with reference to the proposed methodologies for the software systems and industries after establishing the effectiveness of the proposed measures through quantitative analysis the proposed measures should also incorporate other qualitative measures such as the satisfaction level of the end user for the proposed measures for the software systems.

The experimental results indicated and have clearly proved that the proposed model enhanced the reliability of the software. Thus, by presenting the results for increased reliability indicators, the work meets the goal of improving the software reliability growth model. Just as in the integration of ANNs and swarm intelligence through the use of the ABC algorithm, the optimization of parameter estimation and enhancement of predictive error can be revealed. This research also enhances knowledge in software reliability modeling and emphasizes a more pragmatic facet of employing new computation approaches to deal with complicated failure data in enhancing software systems dependency.