Reliability estimation in a two-unit hot standby system under classical and Bayesian inferential framework

Abstract

Hot stand-by systems play a critical role in reliability engineering, ensuring uninterrupted operation in applications where system failure is not an option. This study focuses on the reliability assessment of a two-unit hot stand-by system with a perfect switch, using failure time data modeled by the Weighted Exponential-Lindley Distribution (WXLD). Bayesian reliability estimators are proposed under two loss functions: the Squared Error Loss Function (SELF) and the Linear Exponential (LINEX) Loss Function. These estimators are contrasted with Maximum Likelihood Estimators (MLEs) obtained by optimizing the likelihood function. Monte Carlo simulations are utilized to generate synthetic failure time data, facilitating a comparative analysis of the estimators based on their mean squared errors (MSEs). The findings highlight the performance and robustness of the Bayesian estimators relative to the classical MLEs. This comprehensive evaluation contributes to the advancement of reliability estimation techniques, offering practical insights into the effective analysis of hot stand-by systems.

Introduction

Reliability assessment of complex systems is critical across industries such as engineering, healthcare, and manufacturing, where efficiency, safety, and optimal performance are paramount. In industrial applications, ensuring a system’s functionality over a specified mission time $t_0$ defines its reliability, a cornerstone of practical reliability analysis. In modern infrastructure systems, ensuring uninterrupted service is paramount, especially in critical domains such as healthcare, data communication, and transportation. Hot standby systems play a vital role in enhancing system resilience by enabling instantaneous failover in the event of a primary system failure. For example, hospitals rely on hot standby power systems to maintain continuous operation of life-support and surgical equipment during power outages, as even a few seconds of disruption can be life-threatening. Similarly, global data centers and cloud service providers such as AWS and Microsoft Azure implement hot standby configurations to ensure service continuity across regions. Telecommunications providers also deploy such systems to maintain uninterrupted phone and internet services. Another high-stakes application is in railway control systems, where any disruption due to technical failure, cyberattack, or natural disaster can lead to operational breakdowns, safety risks, and financial losses. Companies like Ansaldo STS have developed advanced hot standby disaster recovery solutions for railway signaling and automation. These systems employ continuous synchronization between primary and backup units, redundant safety components, and real-time data transmission to ensure seamless service during failures. Motivated by these real-world demands, this study develops a robust statistical model to analyze and improve the reliability of hot standby systems. The proposed model aims to provide a flexible framework for evaluating system performance under various failure scenarios, with particular attention to applications in mission-critical domains. By bridging the gap between theoretical modeling and practical implementation, this research contributes to the design and assessment of more resilient infrastructure systems. Assessing the reliability, availability and performance of such systems is fundamental to ensuring their optimal functionality, particularly in critical applications. Engineers and analysts employ advanced statistical and probabilistic methods to model and evaluate the behavior of these systems under varying operational conditions.

Reliability analysis of a two-unit hot stand-by system involves estimating key parameters, such as component failure rates, repair times, and overall system reliability. Standby redundancy is typically classified into three types based on failure characteristics: hot standby systems, where components experience the same failure rate regardless of being active or on standby; cold standby systems, where components remain unaffected while on standby; and warm standby systems, where standby components may fail but at a reduced rate compared to active components. This study evaluates the reliability of a two-unit hot stand-by system with a perfect switch mechanism, a design critical for ensuring uninterrupted operation in high-stakes fields such as power generation, telecommunications, aerospace, and healthcare. Such systems mitigate the risk of severe safety hazards and economic losses by seamlessly transitioning between active and standby units, ensuring continuous functionality. To model the failure time data for this system, the study employs WXLD, ¹ a highly flexible distribution well-suited for reliability analysis of systems with complex failure dynamics. The versatility of WXLD in accommodating various failure rate functions enables it to accurately capture the stochastic nature of component failures, outperforming traditional distributions. This adaptability allows for a precise representation of the system’s reliability characteristics, enhancing the practical applicability of the proposed methods for analyzing and improving critical systems.

Numerous research studies have focused on the reliability analysis of hot stand-by systems, employing a variety of statistical modeling methods to address different aspects of system performance. Osaki and Nakagawa ² established reliability calculations for two-unit standby redundant systems with a constant failure rate. Wiens ³ analyzed a 1-out-of-2:G hot-standby system with 2 identical, dependent units and a General Erlang failure time distribution. Fuji and Sandoh ⁴ explored Bayesian estimation for the reliability of a two-unit hot standby redundant system. Shen and Xie ⁵ investigated the impact of standby redundancy both at the system and component levels. Subsequently, Lee et al. ⁶ derived Bayesian reliability estimators for k-unit standby system with perfect switch. Kumar and Kumari ⁷ dealt with a stochastic model for a two-unit hot standby combined hardware-software system in which one unit is operative, and the other is hot standby. Batra and Taneja ⁸ developed three models for a system comprising one operative unit and no/one/two hot standby unit(s). The purpose of their study is to facilitate decision-making concerning the optimal number of hot standby units, with the aim of maximizing profit. Rizwan et al. ⁹ presented a reliability analysis of a two-unit hot standby PLC’s system. Further, Manocha et al. ¹⁰ discussed the stochastic and cost-benefit analysis of two-unit hot standby database system. Kumari and Kumar ¹¹ carried out the comparative study of two-unit hot standby hardware software systems with impact of imperfect fault coverage. More recent studies have further expanded on these foundations. Saini et al. ¹² studied the reliability, availability, and maintainability analysis of hot standby database systems. Malhotra et al. ¹³ evaluated the availability, reliability, and other measures of system effectiveness for two stochastic models between two-unit hot and cold standby redundant systems with varied demand. Oszczypała et al. ¹⁴ discussed the reliability analysis and redundancy optimization of k-out-of-n systems with random variable k using continuous time Markov chain and Monte Carlo simulation. Ge et al. ¹⁵ dealt with the reliability optimization of reliability-redundancy allocation problems based on K-mixed strategy. Peiravi et al. ¹⁶ developed a Continuous-Time Markov Chain model for both mixed and K-mixed strategies and the proposed model estimated reliability under different redundancy strategies. Recently, Chaudhary et al. ¹⁷ explored the comparative research of efficacy of two parallel systems, Model A employing a hot standby system and Model B utilizing a cold standby setup. Munda et al. ¹⁸ used Regeneration point technique for finding various measures of a system comprises three units operative, hot standby, and warm standby—which have varying failure rates following an exponential distribution. The reliability analysis of two-unit hot stand-by systems has been studied in the literature; however, this field remains relatively unexplored, particularly in the context of Bayesian reliability estimation. Most existing studies rely on Semi-Markov processes and regenerative techniques to model system reliability, leaving Bayesian approaches largely unexplored with limited recent research available. The existing studies rely on traditional lifetime distributions, such as the Exponential, Weibull, or Lindley distributions, which may not always provide an optimal fit for failure time data in stand-by systems. These conventional models often assume a constant or monotonically changing failure rate, which might not accurately capture the dynamic nature of system reliability in hot stand-by configurations. For example, Exponential, Weibull, Gamma, and Lindley distributions have been widely used for reliability modeling. However, these distributions often impose restrictive assumptions on failure rates, which may not fully capture the complex failure dynamics of hot stand-by components. For instance, the Exponential distribution assumes a constant failure rate, which is often unrealistic in practical applications where components experience aging or wear-out effects. The Weibull distribution provides more flexibility by allowing monotonically increasing or decreasing hazard rates, but it does not offer the weighted structure needed to model varying operational phases in stand-by systems. The Lindley distribution, though useful for modeling positively skewed failure time data, lacks sufficient adaptability to represent diverse failure patterns. To address this gap, we introduce the WXLD as a novel failure time model for hot stand-by systems. The significance of WXLD lies in its ability to exhibit increasing, decreasing, and constant hazard rate functions, making it more flexible than standard distributions. This characteristic is crucial for modeling components in a hot stand-by system, where different operational phases (active or standby) can influence failure behavior. The weighted structure of WXLD allows it to model components with varying levels of stress and degradation, improving the accuracy of reliability estimations. This makes it a more practical and realistic choice for industries where stand-by redundancy is a critical factor. Compared to conventional distributions, WXLD provides a more adaptable framework for failure time analysis by incorporating an additional weighting mechanism, allowing for better modeling of failure dynamics in complex systems.

Bayesian estimation is a robust statistical approach that incorporates prior knowledge and uncertainty into parameter estimation. In this study, we apply Bayesian techniques to develop flexible reliability models for a two-unit hot standby system with a perfect switch, using WXLD. This allows us to capture the complex, stochastic nature of component failures, and update prior beliefs with observed data, leading to more accurate reliability assessments. The main motivation of this study is to fill the gap in the literature regarding the use of WXLD for reliability analysis of hot standby systems. Existing research often focuses on constant failure rates, failing to capture the complex failure dynamics. By leveraging the flexibility of WXLD, this study aims to provide a more comprehensive reliability assessment. Additionally, comparing Bayesian estimators with MLE through Monte Carlo simulations will identify the most effective methods for estimating reliability, offering valuable insights for enhancing system performance and decision-making. The structure of the article is organized as follows: Section 2 introduces the notations used throughout the paper along with the detailed description of the model, highlighting the key aspects of the proposed methodology. Section 3 discusses system reliability, including a comprehensive explanation of the foundational assumptions. In Section 4, we derive the MLE for the system’s reliability and develop Bayesian estimators using Lindley’s approximation and the MCMC method under the SELF and LINEX loss functions. Section 5 evaluates the performance of the proposed approach through extensive simulations, using Monte Carlo techniques to compare reliability estimates. The results are systematically presented using tables and graphs. Finally section 6 concludes the study by summarizing key findings and insights.

To effectively model the complexities of hot standby system reliability, we propose the use of WXLD. This model is selected for its ability to accurately capture the varied and intricate failure time patterns that arise from differences in component importance within the system. The PDF and CDF of WXLD, which characterize the failure time behavior, are provided as follows:

$$f\left(t\right)=\frac{4{\theta }^{3}t(2+\theta +t)e^{-2\theta t}}{{(1+\theta )}^2};t\ge 0,\theta >0$$

(1)

$$F\left(t\right)=1-e^{-2\theta t}(\frac{2{\theta }^2{t}^2}{{(1+\theta )}^2}+2\theta t+1)$$

(2)

The reliability function $R(t)$ and hazard function $h(t)$ are as follows:

$$R\left(t\right)=\left(\frac{1+\theta +\theta t+\frac{{(\theta t)}^2}{2}}{1+\theta }\right)e^{-\theta t},t>0,\theta >0$$

$$h\left(t\right)=\frac{4{\theta }^{3}t(2+\theta +t)}{2{\theta }^2{t}^2+2\theta t{(1+\theta )}^2+{(1+\theta )}^2};t>0$$

A random variable T with the PDF given in equation (1) is denoted by WXLD ( $\theta$ ), where $\theta$ is the failure rate (Figure 1).

Figure 1.

Plots of the (a) PDF, (b) CDF, (c) reliability function, and (d) hazard function for some parametric values.

System reliability and assumptions

In the realm of industrial equipment and components operational lifespan, the likelihood that a specific system will operate for a designated “mission” time $t_0$ under defined conditions is referred to as the system’s reliability. In the two-unit hot standby system with perfect switch, we shall assume the following:

1. The system comprises two units that are both independent and identically distributed, along with a switch.

2. While one unit is operational, the other functions as a hot standby.

3. The switch activates instantly in the event of a failure in the currently operational unit.

4. The failure times of both the active and standby units are independent and follow a WXLD with a common failure rate denoted by θ.

5. The unit and the switch operate independently of each other.

6. The switch is failure free.

The reliability of the two-unit hot standby system with a perfect switch at the designated mission time $t_0$ is expressed as follows ¹⁹ :

$$R_H(t_0)=e^{-2\theta t_0}[\frac{2{\theta }^2{t}_0^2}{{(1+\theta )}^2}+2\theta t_0+1](1+\theta t_0),t_0>0$$

(3)

Classical estimation

In this section, we explore the classical approach for evaluating system reliability. To estimate the point value of system reliability $R_H(t_0)$ , we utilize the MLE method.

Maximum likelihood estimation

MLE is a widely used statistical method for estimating the parameters of a probability distribution based on observed data. The principle of MLE is to determine the parameter values that maximize the likelihood function, which represents the probability of obtaining the given data under a specific statistical model. Consider a testing scenario involving a sample of n units in a hot standby system. The testing process concludes as soon as all the units experience failure. The failure times of these units are denoted as $t_1$ , $t_2,\dots ..,t_n$ . The distribution of these failure times is described by a density function, as expressed in equation (1), where the function is dependent on a singular parameter denoted as θ. The Likelihood function $L$ for the data $t_1,t_2,\dots ..,t_n$ is given by:

$$L=L(\theta |t_1,t_2,\dots ..,t_n)=\underset{i=1}{\overset{n}{\Pi }}\frac{4{\theta }^3{t}_i(2+\theta +t_i)e^{-2\theta t_i}}{{(1+\theta )}^2}$$

(4)

Now, the log-likelihood function for a set of observations $t_i$ may be written as

$$\begin{array}{c}\multicolumn{1}{c}{l=\mathrm{nlog}4{\theta }^3-2\mathrm{nlog}(1+\theta )+\sum _{i=1}^n\log t_i}\\ \multicolumn{1}{c}{+\sum _{i=1}^n\log (2+\theta +t_i)-2\theta \sum _{i=1}^n{t}_i}\end{array}$$

(5)

To find the MLE of the parameter θ, the next step involves taking the partial derivative of equation (5) with respect to θ, setting it equal to 0 and solving for θ.

$$\frac{\partial l}{\partial \theta }=\frac{3n}{\theta }-\frac{2n}{(1+\theta )}+\sum _{i=1}^n\frac{1}{(2+\theta +t_i)}-2\sum _{i=1}^n{t}_i=0$$

(6)

To address the absence of a closed-form solution for equation (6), a numerical iteration method has been utilized for estimating the value of the parameter θ.

Having obtained the estimated values of $\hat{\theta }$ and leveraging the Invariance property of the MLE, the ML estimate of $R_H(t_0$ ) can be calculated using equation (3) and expressed as

$${\hat{R}}_H(t_0)=e^{-2\hat{\theta }{t}_0}(\frac{2{\hat{\theta }}^2{t}_0^2}{{(1+\hat{\theta })}^2}+2\hat{\theta }{t}_0+1)[1+\hat{\theta }{t}_0],t_0>0$$

(7)

Bayesian estimation

Bayesian estimation is a statistical approach that incorporates prior knowledge about a parameter along with observed data to obtain updated estimates. Unlike MLE, which relies solely on sample data, Bayesian methods use a prior distribution to represent existing knowledge about the parameter before observing data. A Bayes estimator of $\theta$ is obtained by minimizing a chosen loss function. The Bayes estimate of the system reliability $R_H(t_0)$ is obtained in this section by using SELF and LINEX loss function and by adopting Lindley approximation and MCMC method.

Squared error loss function

The SELF holds considerable prominence in assessing estimator performance within statistical estimation and decision theory. It assumes equal penalization for overestimation and underestimation. It is widely used in Bayesian analysis due to its simplicity and ease of interpretation. For hot stand-by systems, SELF ensures a balanced estimation of reliability parameters without bias toward conservative or optimistic predictions. It finds widespread application when the objective is to minimize the mean squared discrepancy between the estimated and true values. In mathematical terms, SELF is expressed as

$$L(\theta ,\hat{\theta })={(\theta -\hat{\theta })}^2$$

Here, θ represents the true value or parameter being estimated, and δ represents the estimated value or parameter.

The Bayes estimator of $\theta$ under SELF is:

$${\theta }_{\mathrm{SELF}}^B=E_{\pi }(\theta |t)$$

(8)

where $E_{\pi }$ indicated to the expectation of the posterior distribution.

LINEX loss function

The LINEX loss function is an asymmetric loss function that accounts for situations where overestimation and underestimation have different consequences. In hot stand-by systems, underestimating reliability may lead to unnecessary maintenance costs, while overestimating it can result in unexpected system failures. It provides a more realistic assessment of reliability by incorporating such asymmetry, making it particularly useful for decision-making in system maintenance and risk management.

The LINEX loss function is defined as ²⁰

$$L(\theta ,\hat{\theta })=e^{q(\hat{\theta }-\theta )}-q(\hat{\theta }-\theta )-1$$

where $\hat{\theta }$ is the estimate of the unknown parameter $\theta$ and $q\ne 0$ is the known loss parameter. The extent and sign of the loss parameter $q$ indicate the level of asymmetry and directionality, respectively. Overestimation is of greater concern when $q$ is positive, while the scenario is reversed for a negative $q$ . The Bayes estimate under the LINEX loss function is expressed as follows:

$${\theta }_{\mathrm{LINEX}}^B=-\frac{1}{q}\log E[e^{-q\theta }|t_1,t_2,\dots ..,t_n]$$

(9)

where $E[e^{-q\theta }|t_1$ , $t_2,\dots ..,t_n]$ is the posterior expectation which exists and finite.

Let $\theta$ has independent gamma prior with hypermeter ( $\alpha$ , $\beta$ ). The density for $\theta$ is given by

$$g\left(\theta \right)=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\theta }^{\alpha -1}{e}^{-\beta \theta },\theta >0,\alpha ,\beta >0$$

(10)

The gamma prior is chosen due to its conjugacy with the likelihood function, which facilitates closed-form posterior distributions, making the Bayesian estimation process more computationally efficient. Additionally, the gamma distribution is highly flexible and can model different levels of prior knowledge about the failure rate parameters of the system. The choice of prior significantly influences the posterior estimates. A weakly informative gamma prior ensures that the inference is primarily driven by the observed data, minimizing prior influence. In contrast, an informative gamma prior incorporates expert knowledge or historical data, improving estimation accuracy when limited failure data are available.

The posterior distribution of $\theta$ is given by

$$\begin{array}{c}\pi (\theta |t_1,t_2,\dots ..,t_n)=\frac{L(\theta |t_1,t_2,\dots ..,t_n)g\left(\theta \right)}{{\int }_0^{\infty }L(\theta |t_1,t_2,\dots ,t_n)g\left(\theta \right)d\theta }\hfill \\ =K^{-1}\frac{{\theta }^{3n+\alpha -1}}{{(1+\theta )}^{2n}}\exp [-\theta (\beta +2\sum _{i=1}^n{t}_i)]\underset{i=1}{\overset{n}{\Pi }}{t}_i(2+\theta +t_i)\hfill \end{array}$$

(11)

where $K^{-1}$ is the normalizing constant can be computed as

$$K^{-1}=\underset{0}{\overset{\infty }{\int }}\frac{{\theta }^{3n+\alpha -1}}{{(1+\theta )}^{2n}}\exp [-\theta (\beta +2\sum _{i=1}^n{t}_i)]\underset{i=1}{\overset{n}{\Pi }}{t}_i(2+\theta +t_i)d\theta$$

Equation (11) indicates that obtaining a closed-form solution for the Bayes estimate ${\theta }^B$ of θ is not feasible. To address this challenge, we employ various approximation techniques, including the Lindley approximation and MCMC methods. These approaches are utilized to assess the desired estimates, as outlined below.

Lindley approximation

To derive the Bayes estimate, we initiate our exploration by employing Lindley’s approximation, a prevalent numerical method widely employed for this purpose. For more in-depth information, refer to Lindley. ²¹ Let $u(\theta )$ be any arbitrary function, then its posterior expectation is expressed as,

$$E(u(\theta )/t)=\frac{{\int }^u\left(\theta \right)v\left(\theta \right)e^{l(\theta )}d\theta }{{\int }^v(\theta )e^{l(\theta )}d\theta }$$

where $u\left(\theta \right)$ is the function of $\theta$ only, $v(\theta )$ : prior density function and $l\left(\theta \right)=$ log likelihood function.

Using Lindley’s approximation, $E(u(\theta )/t)$ approximately estimated by

$$\begin{array}{c}\multicolumn{1}{c}{E(u\left(\theta \right)/t)=[u+0.5\sum _i\sum _j(u_{\mathrm{ij}}+2u_i{\rho }_j){\sigma }_{\mathrm{ij}}}\\ \multicolumn{1}{c}{+0.5\sum _i\sum _j\sum _k\sum _l{L}_{\mathrm{ijk}}{\sigma }_{\mathrm{ij}}{\sigma }_{\mathrm{kl}}{u}_l]+o\left(\frac{1}{n^2}\right)}\end{array}$$

Here $i,j,k,l=1,2,\dots .,n;\theta =({\theta }_1,{\theta }_2,\dots .,{\theta }_n);u_i=\frac{\partial u}{\partial {\theta }_i};u_{\mathrm{ij}}=\frac{{\partial }^{2}u}{\partial {\theta }_i\partial {\theta }_j}$ ;

$L_{\mathrm{ij}}=\frac{{\partial }^{2}l}{\partial {\theta }_i{\theta }_j},{\rho }_i=\frac{\partial \rho }{\partial {\theta }_i}$ where $\rho$ is the logarithm of prior distribution.

In the scenario under consideration, we find that

$${\rho }_1=\frac{(\alpha -1)}{\theta }-\beta$$

$$L_{11}=\frac{{\partial }^{2}l}{\partial {\theta }^2}=\frac{-3n}{{\theta }^2}+\frac{2n}{{(1+\theta )}^2}-\sum _{i=1}^n\frac{1}{{(2+\theta +t_i)}^2}$$

$$L_{111}=\frac{6n}{{\theta }^3}-\frac{4n}{{(1+\theta )}^3}+2\sum _{i=1}^n\frac{1}{{(2+\theta +t_i)}^3}$$

${\sigma }_{\mathrm{ij}},i,j=1,2$ are obtained by using $L_{\mathrm{ij}},i,j=1,2$ .

$${\sigma }_{11}={[-L_{11}]}^{-1}={[\frac{-3n}{{\theta }^2}+\frac{2n}{{(1+\theta )}^2}-\sum _{i=1}^n\frac{1}{{(2+\theta +t_i)}^2}]}^{-1}$$

Here $u=u\left(\theta \right)=R_H,$

$$\begin{array}{c}\multicolumn{1}{c}{u_1=\frac{\partial u}{\partial \theta }=t_0{e}^{-2\theta t_0}[\frac{2{\theta }^2{t}_0^2}{{(1+\theta )}^2}+2\theta t_0+1]}\\ \multicolumn{1}{c}{[1-2(1+\theta t_0)]+(1+\theta t_0)[e^{-2\theta t_0}\frac{2t_0^2}{{(1+\theta )}^4}+2t_0]}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{u_{11}=\frac{{\partial }^{2}u}{{\partial }^2\theta }=1-2(1+\theta t_0)[e^{-2\theta t_0}\left(\frac{4\theta t_0^2}{(1+{\theta }^3}\right)+2t_0]}\\ \multicolumn{1}{c}{-2t_0{e}^{-2\theta t_0}[\frac{2{\theta }^2{t}_0^2}{{(1+\theta )}^2}+2\theta t_0+1][t_0+(1-2(1+\theta t_0]}\\ \multicolumn{1}{c}{+t_0[e^{-2\theta t_0}\times \frac{2t_0^2}{{(1+\theta )}^4}+2t_0]}\\ \multicolumn{1}{c}{-(1+\theta t_0)\left[\frac{4t_0^2}{{(1+\theta )}^5}{e}^{-2\theta t_0}\{t_0(1+\theta )+2\}\right]}\end{array}$$

The Bayes estimate ${\left(R_H^B\right)}_{\mathrm{LIN}}$ using the Lindley approximation under SELF is given by

$${\left(R_H^B\right)}_{\mathrm{LIN}}=R_H+\frac{1}{2}\left[u_{11}{\sigma }_{11}\right]+u_1{\rho }_1{\sigma }_{11}+\frac{1}{2}\left[L_{111}{u}_1{\sigma }_{11}^2\right]$$

(12)

All the expressions in the above equations are obtained using ML estimates of the parameters.

Now, let us consider the LINEX loss function, specifically designed for scenarios where the cost of overestimation is more significant than that of underestimation. Zellner ²² discussed Bayesian estimation and prediction using LINEX loss. Equation (9) provides the Bayes estimate under LINEX loss.

The posterior expectation $E[e^{-q\theta }|t_1$ , $t_2,\dots ..,t_n]$ for LINEX loss function is

$$E[e^{-q\theta }|t_1,t_2,\dots ..,t_n]=\frac{{\int }_0^{\infty }{e}^{-q\theta }L(\theta |t_1,t_2,\dots ..,t_n)g\left(\theta \right)d\theta }{{\int }_0^{\infty }L(\theta |t_1,t_2,\dots ..,t_n)g(\theta )d\theta }$$

(13)

The integrals in the given equation resist analytical solution. Consequently, we resort again the Lindley’s approximation to derive the Bayesian estimator for system reliability. The Bayesian estimate for system reliability $R_H$ using Lindley’s approximation under the asymmetric LINEX loss function is expressed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\left(R_H^B\right)}_{\mathrm{LIN}}=-\left(\frac{1}{q}\right)\log \left(e^{-q{R}_H}\right)+\frac{1}{2}\left[u_{11}{\sigma }_{11}\right]+u_1{\rho }_1{\sigma }_{11}}\\ \multicolumn{1}{c}{+\frac{1}{2}\left[L_{111}{u}_1{\sigma }_{11}^2\right]}\end{array}$$

(14)

Here,

$$u=e^{-q{R}_H}$$

$$\begin{array}{c}\multicolumn{1}{c}{u_1=\frac{\partial u}{\partial \theta }=e^{-2\theta t_0}[\frac{2{\theta }^2{t}_0^2}{{(1+\theta )}^2}+2\theta t_0+1]}\\ \multicolumn{1}{c}{\{(1+\theta t_0)+t_0(1-2(1+\theta t_0)\}+(1+\theta t_0)}\\ \multicolumn{1}{c}{[e^{-2\theta t_0}\times \frac{2t_0^2}{{(1+\theta )}^4}+2t_0]}\\ \multicolumn{1}{c}{u_{11}=\frac{{\partial }^{2}u}{{\partial }^2\theta }=e^{-2\theta t_0}[\frac{2{\theta }^2{t}_0^2}{{(1+\theta )}^2}+2\theta t_0+1]}\\ \multicolumn{1}{c}{\times (1+t_0-2t_0^2)+\{(1+\theta t_0)+t_0(1-2(1+\theta t_0)\}}\\ -(1+\theta t_0)\left[\frac{4t_0^2}{{(1+\theta )}^5}{e}^{-2\theta t_0}\{t_0(1+\theta )+2\}\right]\hfill \end{array}$$

The remaining values have been previously provided. All the expressions have been derived utilizing ML estimates of the parameters.

Markov Chain Monte Carlo methods

In this section, we explore the computation of the Bayesian estimate for $R_H$ and the related credible interval through the utilization of the MCMC technique. The idea behind this method is to use posterior conditional density functions to generate posterior samples of parameters of interest. The Metropolis–Hastings (M–H) algorithm proves valuable in calculating Bayesian estimates of unknown parameters particularly in scenarios where obtaining analytically tractable solutions is challenging in practical applications. For an in-depth exploration of the MCMC method and the M-H algorithm, refer to the comprehensive discussions by Metropolis et al. ²³ and Hastings. ²⁴ Equation (11) generates the posterior density function for the parameter of interest. The posterior conditional density functions for $\theta$ can be articulated by applying this equation:

$${\pi }^{*}(\theta |\underset{¯}{t})\propto \frac{{\theta }^{3n+\alpha -1}}{{(1+\theta )}^{2n}}\exp [-\theta (\beta +2\sum _{i=1}^n{t}_i)]*\underset{i=1}{\overset{n}{\Pi }}{t}_i(2+\theta +t_i)$$

(15)

The conditional density function of $\theta$ , as demonstrated by (15), cannot be derived in the familiar form of well-known density functions. In such instances, the M-H technique proves advantageous for generating random samples from the posterior density of $\theta$ , employing a normal proposal distribution.

The steps of associated M-H algorithm are given below:

• (1) Set the initial value ${\theta }^{\left(0\right)}=\hat{\theta }$

• (2) Set $l=1$ .

• (3) Using the following M-H algorithm, generate ${\theta }^{(l)}$ from ${\pi }^{*}({\theta }^{\left(l\right)}|\underset{¯}{t})$ with the normal proposal distribution $N({\theta }^{(l-1)},V\left(\theta \right))$ , where $V(\theta )$ can be obtained from the main diagonal in the inverse Fisher information matrix.

• (4) Generate a proposal $\tilde{\theta }$ from $N({\theta }^{(l-1)},V(\theta ))$ .

• (i) Evaluate the acceptance probability

$${\eta }_{\theta }=\min [1,\frac{{\pi }^{*}(\widetilde{\theta |}\underset{¯}{t})}{{\pi }^{*}({\tilde{\theta }}^{t-1}|\underset{¯}{t)}}]$$

• (ii) Generate a sample $u_1$ from uniform (0,1) distribution.

• (iii) If $u_1$ < ${\eta }_{\theta }$ , accept the proposal and set ${\theta }^{(l)}=\tilde{\theta }$ else set ${\theta }^{(l)}={\theta }^{(l-1)}$ .

• (5) Compute $R_H^{(l)}$ at ${\theta }^{(l)}$ .

• (6) Set $l=l+1$ .

• (7) Repeat Steps (3)–(6), $N$ times and obtain ${\theta }^{(l)}$ and $R_H^{(l)},l=1,2,\dots N$ .

This sample is employed for the computation of the Bayes estimate and the construction of the highest posterior density (HPD) credible interval for $R_H$ . To ensure convergence and mitigate the impact of initial value selection, the initial $M$ simulated variants are excluded. Subsequently, the chosen samples are denoted as $R_H^{(j)},j=M+1,\dots N$ , for sufficiently large $N$ .

Based on the SELF, the approximate Bayes estimates of $R_H$ is given by

$${\left(R_H^B\right)}_{\mathrm{MC}}=\frac{1}{N-M}\sum _{j=M+1}^N{R}_H^{(j)}$$

where $M$ is the burn-in period.

The approximate Bayes estimates for $R_H$ under LINEX loss function, from equation (9) is

$${\left(R_H^B\right)}_{\mathrm{MC}}=\frac{-1}{q}\log \left[\frac{1}{N-M}\sum _{j=M+1}^N{e}^{-{\mathrm{qR}}_H^{(j)}}\right]$$

The HPD $100\left(1-\gamma \right)\%$ credible interval of $R_H$ is obtained by the method of Chen and Shao. ²⁵

Simulation

In this section, we use R software to generate random samples from the WXLD distribution. The simulation experiment is conducted to evaluate the reliability coefficient and compare the performance of the proposed estimation methods.

Simulation study

This section presents evaluation of ML and Bayes estimators for system reliability $R_H$ through an extensive Monte-Carlo simulation study using different sample sizes and $\theta$ values. Comparisons are conducted utilizing both point and interval estimation approaches. In point estimation, evaluations are made based on Bias and MSE criteria. In the realm of interval estimation, we compute credible intervals for $R_H$ . The results are based on 2000 replications. We calculate the MSE based-on replication (N) as follows:

$$\mathrm{MSE}\left(R_H\right)=\frac{1}{N}\sum _{j=1}^N{[{\hat{R}}_H^{(i)}-R_H]}^2$$

(16)

$$\mathrm{and}\mathrm{Bias}\left(R_H\right)=\frac{1}{N}\sum _{i=1}^N({\hat{R}}_H^{(i)}-R_H)$$

(17)

The MLE of $R_H$ and its corresponding MSEs are computed for sample sizes $n=$ 10 and 20 with 2000 replications. Further, we set the parameters $\theta =$ 0.5, 1.0 and 2.0. The chosen parameter values represent different failure rate scenarios, covering a range from lower to higher failure intensities, which helps in assessing the performance of the estimators under varying conditions. $\theta =$ 0.5 represents a highly reliable system with a lower failure rate, $\theta =$ 1.0 corresponds to a moderate failure rate, and $\theta =$ 2.0 accounts for more frequent failures, mimicking real-world hot stand-by systems. In the Bayesian framework, we examine two distinct gamma priors: GAM (1, 2) and GAM (1, 4). Tables 1 and 2 present ML estimates, Bayes estimates under the SELF and LINEX loss functions, utilizing Lindley approximation and the MCMC method. The analysis is conducted for sample sizes of $n=$ 30 and 50, considering various values of θ and a specified mission time, $t_0$ at 0.5, 1.0, 2.0, 3.0, and 4.0. Different mission times are considered to evaluate system reliability at both short-term and long-term operational periods. This choice reflects real-world applications where a system’s reliability is assessed over different durations, helping practitioners determine optimal maintenance schedules and risk assessment strategies. Shorter mission times (e.g. $t_0=$ 0.5, 1.0) help analyze early-life reliability, which is critical for high-risk industries where immediate failure risks must be minimized. Longer mission times (e.g. $t_0=$ 3.0, 4.0) simulate extended operational periods, helping to assess long-term performance and maintenance planning. Table 1 corresponds to GAM (1, 2) prior, while Table 2 employs GAM (1, 4) prior. The tables include estimates along with Bias and MSE metrics for comprehensive evaluation. We initiated the MCMC chain with distinct initial values and carried out a total of 10,000 iterations. To alleviate the impact of the initial distribution, we excluded the initial 5000 iterations, often denoted as the burn-in period. Tables 3 and 4 provide the HPD credible intervals, AL and CP for the Bayes estimates under GAM (1, 2) and GAM (1, 4) priors respectively.

Table 1.

ML and Bayes estimates of reliability $R_H$ under SELF and LINEX loss function across different sample sizes and $\theta$ values, along with MSE and |Bias| using GAM (1,2) prior.

Sample size $(n=30)$
ML estimates						Bayes estimates
Priors	Parameters	$\hat{\theta }$	$t_0$	$R_H(t_0)$	${\hat{R}}_H(t_0)$	SELF		LINEX
						Lindley	MCMC	Lindley	MCMC
			0.5	0.97350	0.95631	0.94562	0.94852	0.91652	0.93589
					0.00648	0.00532	0.00613	0.00552	0.00611
					0.01719	0.02788	0.02498	0.05698	0.03761
			1.0	0.90979	0.88452	0.87234	0.87598	0.85432	0.86521
					0.00467	0.00411	0.00441	0.00420	0.00454
					0.02527	0.03745	0.03381	0.05547	0.04458
	$\theta =0.5$	1.5475	2.0	0.73575	0.71852	0.69472	0.70668	0.68746	0.69852
					0.12297	0.00347	0.00372	0.00658	0.00721
					0.01723	0.04103	0.02907	0.04829	0.03723
			3.0	0.55782	0.53781	0.52287	0.53125	0.50219	0.52338
					0.13066	0.00278	0.00302	0.00812	0.00684
					0.02001	0.03495	0.02657	0.05563	0.03444
			4.0	0.40600	0.39742	0.38647	0.38921	0.34785	0.36458
					0.09502	0.00237	0.00256	0.00578	0.00685
					0.00858	0.01953	0.00821	0.04957	0.03284
			0.5	0.90979	0.89647	0.88602	0.88845	0.84336	0.85647
GAM (1, 2)					0.00779	0.00511	0.00652	0.00731	0.00765
					0.01332	0.02377	0.02134	0.06643	0.05332
			1.0	0.73575	0.72114	0.71456	0.71648	0.67874	0.69854
	$\theta =1.0$				0.04600	0.00387	0.00429	0.00735	0.00808
					0.01461	0.02119	0.01927	0.05701	0.03721
		1.8556	2.0	0.40600	0.39556	0.38563	0.38749	0.36745	0.37509
					0.01735	0.00287	0.00315	0.00801	0.00867
					0.01044	0.02037	0.01851	0.03855	0.03091
			3.0	0.19914	0.18745	0.17698	0.17952	0.14526	0.16497
					0.00301	0.00226	0.00216	0.00242	0.00264
					0.01169	0.02216	0.01962	0.05388	0.03417
			4.0	0.09157	0.08762	0.07642	0.07965	0.06527	0.07215
					0.00403	0.00195	0.00195	0.00341	0.00371
					0.00395	0.00115	0.00292	0.00263	0.00267
			0.5	0.73575	0.71036	0.70471	0.70876	0.67456	0.69452
					0.02955	0.03125	0.03265	0.00856	0.00953
	$\theta =$ 2.0				0.02539	0.02104	0.02299	0.02119	0.02253
		1.9641	1.0	0.40600	0.38260	0.37445	0.37869	0.32569	0.35728
					0.10113	0.04321	0.04812	0.00871	0.00976
					0.02340	0.03155	0.02731	0.08031	0.04872
			2.0	0.08157	0.06214	0.05214	0.05748	0.04526	0.05125
					0.10495	0.00336	0.00375	0.01126	0.01265
					0.01943	0.02943	0.02409	0.03631	0.03032
			3.0	0.06426	0.04670	0.03212	0.03781	0.02369	0.03412
					0.04438	0.00274	0.00310	0.01046	0.02514
					0.01756	0.03214	0.02645	0.04057	0.03014
			4.0	0.02578	0.02572	0.01637	0.01785	0.01215	0.01532
					0.01441	0.00165	0.00264	0.01178	0.00254
					0.00006	0.00941	0.00793	0.01363	0.01046
Sample size $(n=50)$
Priors	Parameters	$\hat{\theta }$	$t_0$	$R_H(t_0)$	${\hat{R}}_H(t_0)$	SELF LINEX
						Lindley	MCMC	Lindley	MCMC
			0.5	0.97350	0.94542	0.92364	0.93402	0.89741	0.91521
					0.01075	0.00632	0.00853	0.00752	0.00871
					0.02808	0.04986	0.03948	0.07609	0.05829
			1.0	0.90979	0.87413	0.86415	0.86715	0.84524	0.85212
					0.00531	0.00447	0.00495	0.00452	0.00517
					0.03566	0.04564	0.04264	0.06455	0.05767
	$\theta =$ 0.5	1.5475	2.0	0.73575	0.69745	0.65662	0.68810	0.64321	0.66475
					0.00352	0.00347	0.00350	0.00338	0.00341
Sample size $(n=50)$
Priors	Parameters	$\hat{\theta }$	$t_0$	$R_H(t_0)$	${\hat{R}}_H(t_0)$	SELF LINEX
						Lindley	MCMC	Lindley	MCMC
					0.03830	0.07913	0.04765	0.09254	0.07100
			3.0	0.55782	0.52107	0.51481	0.52254	0.48544	0.51411
GAM (1, 2)					0.00278	0.00262	0.00265	0.00271	0.00273
					0.03675	0.04301	0.03528	0.07238	0.04371
			4.0	0.40600	0.37854	0.36511	0.37742	0.32201	0.34175
					0.00237	0.00218	0.00225	0.00230	0.00231
					0.02746	0.04089	0.02858	0.08399	0.06425
			0.5	0.90979	0.82569	0.81701	0.82635	0.83665	0.84502
	$\theta =$ . 1.0				0.01075	0.00511	0.00652	0.00386	0.00512
					0.08410	0.09278	0.08344	0.07314	0.06477
		1.8556	1.0	0.73575	0.69772	0.66213	0.68405	0.68110	0.68792
					0.00531	0.00388	0.00423	0.00245	0.00342
					0.03803	0.07362	05170	0.05465	0.04783
			2.0	0.40600	0.36651	0.35239	0.36794	0.36219	0.36695
					0.00352	0.00289	0.00315	0.00185	0.00256
					0.03949	0.05361	0.03806	0.04381	0.03905
			3.0	0.19914	0.16754	0.14665	0.14610	0.14652	0.15224
					0.00262	0.00226	0.00247	0.00156	0.00205
					0.03160	0.05249	0.05304	0.05262	0.04690
			4.0	0.09157	0.07563	0.05412	0.06745	0.06501	0.07069
					0.00218	0.00195	0.00212	0.00122	0.00174
					0.01594	0.03745	0.02412	0.02656	0.02088
			0.5	0.73575	0.70336	0.69416	0.69845	0.68455	0.68988
					0.00844	0.00761	0.00810	0.00814	0.00816
					0.03239	0.04159	0.03730	0.05120	0.04587
		1.9641	1.0	0.40600	0.36541	0.34485	0.35212	0.31369	0.34523
					0.00648	0.00375	0.00385	0.00215	0.00353
					0.04059	0.06115	0.05388	0.09231	0.06077
	$\theta =$ 2.0		2.0	0.08157	0.05523	0.04894	0.05570	0.03665	0.04712
					0.00346	0.00170	0.00285	0.00149	0.02753
					0.02634	0.03263	0.02587	0.04492	0.03445
			3.0	0.06426	0.03296	0.03117	0.03365	0.02165	0.02786
					0.00458	0.00393	0.00425	0.00086	0.00669
					0.03130	0.03309	0.03061	0.04261	0.03640
			4.0	0.02578	0.02364	0.01538	0.01675	0.01023	0.01456
					0.00445	0.00167	0.00037	0.00078	0.00047
					0.00214	0.01040	0.00903	0.01555	0.01122

The second row denotes the MSEs associated with the respective estimates and the third column signifies the |Bias| of the estimates.

Table 2.

ML and Bayes estimates of reliability $R_H$ under SELF and LINEX across different sample sizes and $\theta$ values, along with MSE and |Bias| using GAM (1, 4) prior.

Priors	Sample size $(n=30)$
	ML estimates					Bayes estimates
	Parameters	$\hat{\theta }$	$t_0$	$R_H(t_0)$	${\hat{R}}_H(t_0)$	SELF		LINEX (q = 0.5)
						Lindley	MCMC	Lindley	MCMC
			0.5	0.97350	0.95631	0.91236	0.92625	0.90363	0.92146
					0.00648	0.00456	0.00541	0.00432	0.00545
					0.01719	0.06114	0.04725	0.06987	0.05204
			1.0	0.90979	0.88452	0.85456	0.86589	0.83698	0.84789
					0.00417	0.00325	0.00404	0.00361	0.00411
					0.02527	0.05523	0.04390	0.07281	0.06190
	$\theta =$ 0.5	1.5475	2.0	0.73575	0.71852	0.64589	0.72796	0.61378	0.65456
					0.12297	0.00310	0.00456	0.00578	0.00687
					0.01723	0.08986	0.00779	0.12197	0.08119
			3.0	0.55782	0.53781	0.50112	0.52130	0.50256	0.51456
					0.13066	0.00242	0.00411	0.00647	0.00712
					0.02001	0.05670	0.03652	0.05526	0.04326
			4.0	0.40600	0.39742	0.36569	0.37459	0.33475	0.35710
					0.09502	0.00218	0.00316	0.00897	0.00664
					0.00858	0.04031	0.03141	0.07125	0.04890
			0.5	0.90979	0.89647	0.85639	0.83692	0.82145	0.84756
GAM (1, 4)					0.00779	0.00416	0.00321	0.00645	0.00615
					0.01332	0.05340	0.07287	0.08834	0.06223
			1.0	0.73575	0.72114	0.75471	0.70256	0.65274	0.67896
	$\theta =$ 1.0				0.04600	0.00402	0.00436	0.00547	0.00707
					0.01461	0.01896	0.03319	0.08301	0.05679
		1.8556	2.0	0.40600	0.39556	0.35963	0.37841	0.36102	0.36256
					0.01735	0.00369	0.00405	0.00636	0.00725
					0.01044	0.04637	0.02759	0.04498	0.04344
			3.0	0.19914	0.18745	0.15456	0.16397	0.13256	0.15678
					0.00301	0.00239	0.00331	0.00789	0.00612
					0.01169	0.04458	0.03517	0.06658	0.04236
			4.0	0.09157	0.08762	0.05789	0.06478	0.05442	0.06478
					0.00403	0.00154	0.00249	0.00385	0.00391
					0.00395	0.03368	0.02679	0.03715	0.02679
			0.5	0.73575	0.71036	0.68520	0.69743	0.64215	0.67489
					0.02955	0.02014	0.02189	0.00642	0.00874
					0.02539	0.05055	0.03832	0.0936	0.06086
	$\theta =$ 2.0	1.9641	1.0	0.40600	0.38260	0.35477	0.36459	0.30696	0.32145
					0.10113	0.03159	0.03874	0.00630	0.00810
					0.02340	0.05123	0.04141	0.09904	0.08455
			2.0	0.08157	0.06214	0.05017	0.05325	0.03694	0.04213
					0.10495	0.00219	0.00310	0.01025	0.01128
					0.01943	0.03140	0.02832	0.04463	0.03944
			3.0	0.06426	0.04670	0.02158	0.02697	0.02879	0.02884
					0.04438	0.00178	0.00254	0.01136	0.01574
					0.01756	0.04268	0.03729	0.03547	0.03542
			4.0	0.02578	0.02572	0.02017	0.01975	0.02169	0.01714
					0.01441	0.00145	0.00216	0.01247	0.00174
					0.00006	0.00561	0.00603	0.00403	0.00864
Sample size $(n=$ 50)
Priors	Parameters	$\hat{\theta }$	$t_0$	$R_H(t_0)$	${\hat{R}}_H(t_0)$	SELF		LINEX (q = 0.5)
						Lindley	MCMC	Lindley	MCMC
			0.5	0.97350	0.94542	0.90156	0.91545	0.88475	0.89752
					0.01075	0.01052	0.00741	0.00433	0.00641
					0.02808	0.07194	0.05805	0.08875	0.07598
			1.0	0.90979	0.87413	0.84581	0.83259	0.83665	0.84336
					0.00531	0.00156	0.00274	0.00428	0.00432
					0.03566	0.06398	0.07720	0.07314	0.06643
GAM (1, 4)	$\theta$ =0.5	1.5475	2.0	0.73575	0.69745	0.63478	0.67964	0.57891	0.61248
					0.00352	0.00268	0.00285	0.00315	0.00347
Sample size $(n=$ 50)
Priors	Parameters	$\hat{\theta }$	$t_0$	$R_H(t_0)$	${\hat{R}}_H(t_0)$	SELF		LINEX (q = 0.5)
						Lindley	MCMC	Lindley	MCMC
					0.03830	0.10097	0.05611	0.15684	0.12327
			3.0	0.55782	0.52107	0.49263	0.50289	0.47592	0.50287
					0.00262	0.00126	0.00205	0.00254	0.00260
					0.03675	0.06519	0.05493	0.08190	0.05495
			4.0	0.40600	0.37854	0.34771	0.35225	0.33147	0.33748
					0.00718	0.00645	0.00618	0.00236	0.00329
					0.05746	0.02829	0.03375	0.04453	0.04685
			0.5	0.90979	0.82569	0.79661	0.80179	0.82115	0.83669
	$\theta$ =1.0				0.01075	0.00511	0.00633	0.00478	0.00512
					0.08410	0.11318	0.01080	0.08864	0.07310
		1.8556	1.0	0.73575	0.69772	0.64336	0.66478	0.67584	0.68564
					0.00531	0.00388	0.00521	0.00480	0.00475
					0.03803	0.09239	0.07097	0.02188	0.05011
			2.0	0.40600	0.36651	0.32898	0.35702	0.35974	0.36123
					0.00352	0.00346	0.00348	0.00324	0.00312
					0.03949	0.07702	0.04898	0.04626	0.04477
			3.0	0.19914	0.16754	0.13669	0.14325	0.14510	0.14887
					0.00262	0.00226	0.00245	0.00217	0.00239
					0.06160	0.03245	0.05589	0.05404	0.05027
			4.0	0.09157	0.07563	0.03226	0.04891	0.05247	0.06387
					0.00268	0.00195	0.00218	0.00256	0.00227
					0.01594	0.05931	0.04266	0.03910	0.02770
			0.5	0.73575	0.70336	0.67821	0.68741	0.67845	0.68213
					0.00709	0.00623	0.00642	0.00654	0.00691
					0.03239	0.05754	0.04834	0.05730	0.05362
		1.9641	1.0	0.40600	0.36541	0.32567	0.34589	0.41796	0.32669
					0.00948	0.00623	0.00921	0.00559	0.00669
					0.04059	0.08033	0.06011	0.01196	0.07931
	$\theta =$ 2.0		2.0	0.08157	0.05523	0.03692	0.04481	0.05897	0.04712
					0.01046	0.00913	0.00889	0.00452	0.00753
					0.02634	0.04465	0.03676	0.02260	0.03445
			3.0	0.06426	0.03296	0.03147	0.04556	0.03102	0.02633
					0.01158	0.00082	0.00974	0.00485	0.00647
					0.03130	0.00149	0.01870	0.03324	0.03793
			4.0	0.02578	0.02364	0.01357	0.01574	0.02369	0.01369
					0.01245	0.00847	0.00827	0.00364	0.00612
					0.00214	0.01221	0.01004	0.00209	0.01209

The second row denotes the MSEs associated with the respective estimates and the third column signifies the |Bias| of the estimates.

Table 3.

The HPD credible interval, AL and CP for Bayes estimates of Reliability $R_H$ at various values of parameter $\theta$ under GAM (1, 2) prior.

$\theta$	$n$	HPDCI	AL	CP
0.5	10	(0.35781, 0.73722)	0.37941	0.9850
	20	(0.40546, 0.69550)	0.29004	0.9800
	30	(0.43297, 0.67644)	0.24347	0.9805
	40	(0.44929, 0.66329)	0.21400	0.9715
	50	(0.45884, 0.65202)	0.19318	0.9755
1.0	10	(0.23346, 0.57352)	0.34006	0.9815
	20	(0.27641, 0.53401)	0.25760	0.9755
	30	(0.29723, 0.51288)	0.21565	0.9725
	40	(0.31067, 0.49973)	0.18906	0.9670
	50	(0.31957, 0.48997)	0.17040	0.9645
1.5	10	(0.60649, 0.90212)	0.29563	0.9915
	20	(0.64652, 0.87631)	0.22979	0.9850
	30	(0.66436, 0.86021)	0.19585	0.9850
	40	(0.67925, 0.85135)	0.17210	0.9815
	50	(0.68742, 0.84368)	0.15626	0.9735
2.0	10	(0.44736, 0.78896)	0.34160	0.9895
	20	(0.48748, 0.75329)	0.26581	0.9840
	30	(0.50889, 0.73437)	0.22548	0.9805
	40	(0.52241, 0.72162)	0.19921	0.9755
	50	(0.53123, 0.71161)	0.18038	0.9750

The HPDCI represents a 95% confidence interval for Bayes estimates.

Table 4.

The HPD credible interval, AL and CP for Bayes estimates of Reliability $R_H$ at various values of parameter $\theta$ under GAM (1, 4) prior.

$\theta$	$n$	HPDCI	AL	CP
0.5	10	(0.33547, 0.76875)	0.43328	0.9010
	20	(0.39143, 0.70635)	0.31492	0.9410
	30	(0.42414, 0.68014)	0.25600	0.9405
	40	(0.43909, 0.66392)	0.22483	0.9400
	50	(0.44878, 0.65025)	0.20147	0.9385
1.0	10	(0.21149, 0.59719)	0.38570	0.9240
	20	(0.26396, 0.54011)	0.27615	0.9330
	30	(0.28727, 0.51394)	0.22667	0.9360
	40	(0.30173, 0.49853)	0.19680	0.9420
	50	(0.31122, 0.48997)	0.17875	0.9410
1.5	10	(0.76872, 0.96174)	0.19302	0.9915
	20	(0.79545, 0.94809)	0.15264	0.9850
	30	(0.80839, 0.93951)	0.13112	0.9850
	40	(0.81684, 0.93355)	0.11671	0.9815
	50	(0.82263, 0.92894)	0.10631	0.9735
2.0	10	(0.63088, 0.90126)	0.27038	0.9895
	20	(0.65921, 0.87618)	0.21697	0.9840
	30	(0.67481, 0.86145)	0.16264	0.9805
	40	(0.68715, 0.85327)	0.15612	0.9755
	50	(0.69618, 0.84693)	0.15075	0.9750

The HPDCI represents a 95% confidence interval for Bayes estimates.

From Tables 1 and 2, it is evident that under GAM (1, 2) and GAM (1, 4) prior

• The ML estimate of system reliability ${\hat{R}}_H$ , exhibits a decreasing trend as the mission time ( $t_0$ ) extends from 0.5 to 4.0, while maintaining a constant sample size of $n=$ 30. Also, as the parameter $\theta$ experiences an increase, there is a corresponding decrease in the ML estimate value.

• The bias of the ML estimate is nearly negligible. However, the MSEs exhibit variability, being higher at certain points and decreasing at others, with a consistent downward trend observed as the values of $t$ and θ increase.

• The Bayes estimate, under both SELF and LINEX loss functions, demonstrates a decrease as the mission time ( $t_0$ ) increases. Additionally, it is noteworthy that the Bayes estimate obtained through the MCMC method outperforms the Lindley approximation, providing superior results.

• Increasing the parameter $\theta$ leads to a decrease in the Bayes estimate, as observed through both Lindley and MCMC methods, under the considered loss functions.

• The comparison of MSE between ML estimates and Bayes estimates, employing both Lindley and MCMC methods, consistently demonstrates superior performance of Bayes estimates over ML estimates under both loss functions for both GAM (1,2) and GAM (1,4) priors.

• Upon comparing the Bayes estimate using the Lindley and MCMC methods, it becomes evident that the Lindley approximation consistently provides superior estimates of system reliability. This superiority is observed in terms of lower MSEs under both loss functions and for both GAM (1,2) and GAM (1,4) priors.

• With an increase in the sample size from $n=$ 30 to 50, the reliability estimates consistently decrease as mission time $\left(t_0\right)$ values increase. This trend holds true under both considered priors.

• In the case of $n=$ 50, Bayes estimates outperform ML estimates, demonstrating superior performance in terms of MSEs. Furthermore, under the Lindley approximation, we observe more favorable results compared to the MCMC method.

• When the sample size increases from $n=$ 30 to 50, the Bayesian estimate, as determined by the Lindley method, outperforms the estimate obtained through the MCMC method in terms of MSEs.

From Tables 3 and 4, one can make the following observations:

• Except at $\theta =$ 2.0, AL consistently decreases, as we increase the parameter $\theta$ within the range of 0.5 to 1.5. This trend is observed under both the GAM (1, 2) and GAM (2, 4).

• CP exhibits a continuous decrease as we increase the parameter $\theta$ from 1.0 to 2.0, with the exception occurring at $\theta =$ 0.5.

• As the sample size increases from $n=$ 10 to 50, the average length of the credible interval consistently decreases in both considered cases.

• Upon comparing the average length of intervals between Tables 3 and 4, it becomes evident that, under the GAM (1, 4) model the average length consistently decreases for all values of $\theta$ except when $\theta =$ 2.0. This observation implies a general trend of narrowing intervals with varying $\theta$ with the notable exception at $\theta =$ 2.0.

The graphical representation of the simulation study findings is depicted in Figures 2 to 9.

Figure 2.

ML and Bayes reliability estimates under SELF and LINEX when $\theta =0.5,n=30$ and prior GAM (1, 2).

Figure 3.

ML and Bayes reliability estimates under SELF and LINEX when $\theta =1.0,n=30$ and prior GAM (1, 2).

Figure 4.

ML and Bayes reliability estimates under SELF and LINEX when $\theta =0.5,n=30$ and prior GAM (1, 4).

Figure 5.

ML and Bayes reliability estimates under SELF and LINEX when $\theta =1.0,n=30$ and prior GAM (1, 4).

Figure 6.

MSEs of reliability estimates when θ = 0.5, n = 50 and prior GAM (1, 2).

Figure 7.

MSEs of reliability estimates when θ = 1.0, n = 50 and prior GAM (1, 2)

Figure 8.

MSEs of reliability estimates when $\theta =0.5,n=$ 50 and prior GAM (1, 4).

Figure 9.

MSEs of reliability estimates when $\theta =1.0,n=50$ and prior GAM (1, 4).

Conclusion

Hot stand-by systems are integral to reliability engineering, providing uninterrupted functionality in critical applications. This study addresses the challenge of estimating the reliability of a two-unit hot stand-by system with a perfect switch, utilizing failure time data modeled by WXLD. Prior research in reliability estimation has primarily focused on constant failure rates and simpler models, leaving a significant gap in exploring advanced distributions such as WXLD for complex system dynamics. This paper bridges that gap by introducing both frequentist and Bayesian estimation approaches to assess system reliability. The frequentist method employs MLE for point estimates, while the Bayesian approach incorporates Lindley’s approximation and MCMC techniques under SELF and LINEX loss functions. Bayesian estimates are further supported by HPD credible intervals. A detailed simulation study provides a comparative analysis of these methods, demonstrating the superior performance of Bayesian estimators, particularly when leveraging informative priors such as GAM (1, 2). The results emphasize the robustness and precision of Bayesian methods over MLE, with smaller MSEs and improved interval estimates under the Bayesian framework. While this study focuses on a two-unit system, future research can extend these methodologies to more complex configurations, including multi-unit hot stand-by systems with multiple standby components and varying switching mechanisms. Additionally, incorporating alternative failure distributions, such as the generalized Lindley or weighted Weibull distributions, may improve model flexibility and applicability. From a computational perspective, advanced Bayesian techniques like Variational Bayes or Hybrid MCMC could enhance estimation efficiency, particularly for large-scale reliability assessments. Due to the limited literature on Bayesian methods in this domain, this study does not incorporate real-life failure data for empirical validation. As Bayesian reliability analysis of hot stand-by systems remains largely unexplored, future research can focus on applying these methods to actual failure data. This would enable a more comprehensive evaluation of reliability estimates and other system characteristics, further validating the practical applicability of Bayesian approaches in engineering reliability studies.