Reliability Improvement of a Parallel–Series System via Duplication and Reduction Strategies Under the Akshaya Distribution

Abstract

Parallel–series systems are fundamental in many industrial and engineering applications, yet their reliability assessment and improvement remain challenging, particularly when components exhibit non-constant failure rates. This study addresses this challenge by modeling a hybrid parallel–series system whose components follow the Akshaya lifetime distribution, a flexible model that can capture various hazard-rate shapes. For this system, we derive closed-form analytical expressions for key reliability indices, including the system reliability function, mean time to failure (MTTF), reliability equivalence factors (REFs), and $\delta$-fractiles. To enhance system performance, four improvement strategies are formulated and analytically compared: failure-rate reduction, hot duplication, cold duplication with a perfect switch, and cold duplication with an imperfect switch. A comprehensive numerical case study validates the theoretical derivations and demonstrates the effectiveness of each strategy. The results show that cold duplication with a perfect switch yields the highest reliability gain, and the computed REFs provide a quantitative tool for balancing redundancy against component-level improvements. This work provides reliability engineers with a comprehensive analytical framework for the design and enhancement of complex parallel-series systems.

1. Introduction

Modern industrial and engineering systems frequently consist of multiple interconnected components that operate under varying environmental and operational conditions. In many production processes, components are shared across subsystems, and the overall system performance is strongly influenced by their reliability characteristics. These components may exhibit constant or time-dependent failure rates and typically undergo progressive deterioration over time, increasing the probability of failure. Consequently, the modeling of component lifetimes and system reliability, together with the development of effective reliability enhancement strategies, has become a central topic in reliability engineering. Numerous lifetime distributions and improvement techniques—such as failure-rate reduction, hot duplication, and cold duplication—have been widely investigated under both constant and non-constant hazard rate assumptions.

1.1. Classical Reliability Modeling and Enhancement Techniques

Reliability analysis of series, parallel, and series–parallel systems has attracted considerable attention in the literature. Alghamdi and Percy [1] investigated series–parallel systems with independent and identically distributed components following the Weibull distribution and demonstrated system performance improvement through established enhancement methodologies. In a related context, redundancy and failure-rate reduction strategies were analytically compared for series–parallel systems with exponentiated Weibull components, supported by numerical illustrations.

Further contributions addressed systems governed by alternative lifetime models. Baaqeel [2] analyzed a five-component bridge network system whose failure behavior followed the geometric distribution of the unit half-logistic model, showing that both reduction and duplication strategies significantly improve reliability and MTTF. El-Faheem et al. [3] studied a radar system composed of independent Rayleigh components and derived reliability equivalence factors based on redundancy and failure-rate reduction strategies, evaluating their effectiveness using MTTF and $\gamma$—fractiles. Similarly, Ezzati and Rasouli [4] examined compounded series–parallel systems under the linear exponential distribution and demonstrated the impact of warm duplication and component-level improvement on system reliability.

Theoretical foundations of reliability engineering have been comprehensively documented in standard references such as [5], which integrate probability theory, statistical inference, and engineering applications, covering Weibull analysis, redundancy modeling, system safety, Monte Carlo simulation, and reliability testing techniques.

1.2. Reliability Equivalence Analysis and System Comparisons

Reliability equivalence analysis provides a quantitative framework for comparing alternative system designs that achieve equivalent performance levels. Mustafa et al. [6] investigated series–parallel systems with linear exponential components and demonstrated reliability improvements through hot and cold duplication strategies. The concept of reliability equivalence was further extended by Mustafa and El-Faheem [7,8], who introduced survival-based and mean reliability equivalence measures.

Earlier, Mustafa [9] derived reliability equivalence factors for series systems with time-dependent failure rates under Weibull and linear rising failure rate (LIFR) distributions, considering imperfect switching and redundancy. A comprehensive sequence of studies by Sarhan and co-authors [10,11,12,13,14,15] systematically developed reliability equivalence factors for series, parallel, series–parallel, and bridge network systems under exponential lifetime assumptions, employing system reliability and MTTF as performance criteria. These studies established important analytical benchmarks for comparing original and improved system configurations.

1.3. Complex Systems and Recent Developments

With increasing system complexity, recent research has focused on heterogeneous components, storage systems, and bridge networks under flexible lifetime models. Recent developments indicate that integrating heterogeneous components with time-dependent failure behaviors, alongside improvement techniques such as reduction and duplication, can substantially improve storage system performance, particularly with respect to MTTF and reliability equivalence measures [16]. Ramadan et al. [17] evaluated the reliability of bridge network systems with components following the Bilal distribution, deriving $\delta$-fractiles and reliability equivalence factors under multiple improvement strategies.

More recent research has broadened reliability modeling by incorporating performance sharing, performance-excess failures, and storage optimization within series–parallel frameworks, enabling realistic supply–demand balance assessment [18]. Copula-based dependence modeling and Markovian formulations have also been employed to analyze the reliability, availability, and profitability of controlled multi-state systems [19]. Additionally, redundancy optimization techniques using Lagrangean and dynamic programming approaches have demonstrated significant reliability gains in constrained series–parallel systems [20], while algebraic frameworks have shown that symmetric and asymmetric system configurations can be optimal under different architectures [21].

Parallel–series systems are particularly prevalent in practical applications [22], where blocks are connected in parallel, and each block contains series-connected components (see Figure 1). Tolba et al. [23] evaluated bridge structures under the generalized unit half-logistic geometric (GUHLG) distribution. The GUHLG distribution itself was introduced by Nasiru et al. [24] and shown to possess highly flexible density and hazard rate shapes. Xia and Zhang [25] analyzed reliability equivalence factors for parallel systems with gamma-distributed lifetimes.

1.4. Motivation for the Akshaya Distribution

In parallel with these developments, alternative lifetime distributions have been proposed to improve modeling flexibility while maintaining analytical tractability. Ramadan et al. [26] introduced the generalized power Akshaya distribution and investigated system reliability, survival functions, and parameter estimation using both classical and Bayesian approaches. Building on this line of research, the present study adopts the Akshaya distribution for system-level reliability analysis.

The adoption of the Akshaya distribution in this study is driven by both its modeling flexibility and its analytical suitability for system-level reliability analysis. In contrast to classical lifetime distributions such as the Weibull, Gamma, and Generalized Gamma, which typically require multiple shape parameters to represent complex failure mechanisms, the Akshaya distribution achieves a comparable degree of flexibility using a single scale parameter. This parsimony is particularly advantageous in reliability equivalence analysis, where analytical tractability and numerical stability are essential. The Akshaya distribution is capable of capturing a broad spectrum of hazard rate behaviors, including monotonic and non-monotonic patterns, thereby allowing the realistic modeling of component degradation in heterogeneous engineering systems. Furthermore, its closed-form expressions for the probability density, survival, and hazard functions enable explicit derivations of key performance measures such as the mean time to failure (MTTF), $delta$-fractiles, and reliability equivalence factors (REFs) for complex parallel–series configurations. Previous investigations have reported superior goodness-of-fit performance of the Akshaya distribution for lifetime data arising in reliability contexts [27].

1.5. Research Gap and Contributions

Limitations and Motivation: Despite extensive research on reliability enhancement and equivalence analysis, there remains a lack of integrated studies that simultaneously employ a modern, flexible lifetime distribution and systematically compare multiple improvement strategies for hybrid parallel–series systems. In particular, comprehensive analyses under the Akshaya distribution that incorporate practical features such as imperfect switching in cold standby configurations are still scarce.

Contribution: To address this gap, this paper develops a unified analytical framework for a hybrid parallel–series system whose components follow the Akshaya distribution. Closed-form expressions for system reliability, MTTF, $\delta$-fractiles, and reliability equivalence factors are derived. Four improvement strategies—hot duplication, cold duplication with perfect switching, cold duplication with imperfect switching, and failure-rate reduction—are analytically compared using REFs, providing a rigorous quantitative basis for system design and performance evaluation.

The system configuration investigated in this study is illustrated in Figure 1, and the proposed framework enables the systematic assessment of reliability improvement strategies for complex engineering systems.

1.6. Methodological Framework

To provide clarity on the study’s approach, the methodological framework is outlined as follows:

1.

System Modeling: A hybrid parallel-series system (Figure 1) is defined, where each component’s lifetime follows the Akshaya distribution with potentially distinct parameters.

2.

Baseline Analysis: Closed-form expressions for the system’s reliability $R\left(t\right)$ and mean time to failure (MTTF) are derived for the original, unimproved configuration (Section 3).

3.

Improvement Strategies: Four distinct strategies for enhancing system performance are formulated—(i) hot duplication, (ii) cold duplication with a perfect switch, (iii) cold duplication with an imperfect switch, and (iv) failure-rate reduction (Section 4).

4.

Enhanced System Analysis: For each strategy, the reliability function and MTTF for the improved system are derived, accounting for the selected subset of enhanced components.

5.

Comparative Metrics: To enable direct comparison between strategies, two key metrics are derived—(a) Reliability Equivalence Factors (REFs), which quantify the trade-off between reduction and duplication methods (Section 5), and (b) $\delta$-fractiles, which provide a time-based measure of reliability performance (Section 6).

6.

Numerical Illustration & Validation: A specific numerical example is analyzed to demonstrate the application of all derived formulas, compare the effectiveness of each strategy, and interpret the REFs and $\delta$-fractiles (Section 7).

This structured framework ensures a comprehensive analysis, from foundational modeling to the derivation of practical decision-making tools.

2. Fundamental Functions of Akshaya Lifetime Distribution

The Akshaya distribution is a single-parameter lifetime model developed to provide superior fitting performance compared to several existing distributions in reliability analysis and statistical data modeling. Its applications have demonstrated improved reliability measures, establishing it as a useful tool for analyzing lifetime data in various contexts (see [27]). The Akshaya distribution is characterized by the following probability density function (PDF) and cumulative distribution function (CDF), given, respectively, by 

$$f(x;\theta )={\displaystyle \frac{{\theta }^4}{{\theta }^3+3{\theta }^2+6\theta +6}}{(1+x)}^3{e}^{-\theta x};x>0,\theta >0,$$

(1)

$$F(x;\theta )=1-\left[1+{\displaystyle \frac{{\theta }^3{x}^3+3{\theta }^2(\theta +1)x^2+3\theta ({\theta }^2+2\theta +2)x}{{\theta }^3+3{\theta }^2+6\theta +6}}\right]e^{-\theta x}.$$

(2)

and the corresponding hazard rate function is expressed as

$$h(x;\theta )={\displaystyle \frac{{\theta }^4{(1+x)}^3}{{\theta }^3{x}^3+3{\theta }^2(\theta +1)x^2+3\theta ({\theta }^2+2\theta +2)x+({\theta }^3+3{\theta }^2+6\theta +6)}},$$

(3)

and the associated reliability function can be written as

$$R(x;\theta )=\left[1+{\displaystyle \frac{{\theta }^3{x}^3+3{\theta }^2(\theta +1)x^2+3\theta ({\theta }^2+2\theta +2)x}{{\theta }^3+3{\theta }^2+6\theta +6}}\right]e^{-\theta x}.$$

(4)

As seen in Figure 2, the PDF of the Akshaya distribution and the hazard rate function exhibit a variety of shapes, demonstrating its adaptability.

3. MTTF and Reliability for the Primary System

As illustrated in Figure 1, the considered configuration is a mixed parallel series network consisting of n parallel subsystems, where each subsystem i $(i=1,2,\dots ,n)$ contains $m_i$ components connected in series. Each component lifetime is assumed to follow the Akshaya distribution. Let $R_{ij}\left(t\right)$ denote the reliability of the jth component in the ith subsystem $(j=1,2,\dots ,m_i)$, $R_i\left(t\right)$ represent the reliability of the subsystem i, and $R\left(t\right)$ describe the overall reliability of the system.

Based on Equation (4), the expression for the reliability of component j in subsystem i can be written as

$$R_{ij}\left(t\right)=\left[1+{\displaystyle \frac{{\theta }_{ij}^3{t}^3+3{\theta }_{ij}^2({\theta }_{ij}+1)t^2+3{\theta }_{ij}({\theta }_{ij}^2+2{\theta }_{ij}+2)t}{{\theta }_{ij}^3+3{\theta }_{ij}^2+6{\theta }_{ij}+6}}\right]e^{-{\theta }_{ij}t}.$$

(5)

Since the components within a subsystem are connected in series, the reliability of subsystem i is the product of the reliabilities of its components:

$$R_i\left(t\right)=\prod _{j=1}^{m_i}{R}_{ij}\left(t\right).$$

(6)

Then, because the n subsystems are arranged in parallel, the overall system reliability is given by

$$R\left(t\right)=1-\prod _{i=1}^n\left(1-R_i\left(t\right)\right).$$

(7)

The mean time to failure (MTTF) of the system is obtained by integrating the reliability function from zero to infinity:

$$MTTF\left(t\right)={\int }_0^{\infty }R\left(t\right)dt,$$

(8)

4. Methods of Performance Enhancement

This section examines performance enhancement techniques applied to system components, with a focus on duplication and reduction methods.

4.1. Enhancement Strategies

The improvement approaches considered comprise the hot redundancy scheme, in which an identical spare component operates concurrently with the primary unit in an active (hot) standby mode, and the cold redundancy scheme, where a backup component remains inactive and is engaged only after the failure of the original unit, with the switching mechanism assumed to be either perfect or imperfect.

Assume G is the set of enhanced components under the chosen enhancement method, where $\left|G\right|=e$, $0<e\le N$, and $N={\sum }_{i=1}^n{m}_i$ (it is noted that when $e=0$, it gives the original system). Furthermore, let $G_i$ represent the set of enhanced components over subsystem i, with $|G_i|=e_i$, $0<e_i\le m_i$, and ${\bigcup }_{i=1}^n{G}_i=G$ and ${\sum }_{i=1}^n{e}_i=e$.

4.1.1. Hot Enhancement Strategy

In this approach, the reliability of the system is increased by equipping selected components with identical counterparts that function simultaneously in an active (hot) standby arrangement. An illustrative example of this configuration, featuring a single main component paired with one hot standby unit, is depicted in Figure 3.

Let $R_{ij}^H\left(t\right)$ be the reliability of the jth component in the ith subsystem when the hot redundancy scheme is applied. Similarly, let $R_i^H\left(t\right)$ denote the reliability of the subsystem i under this improvement method and $R_G^{H;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)$ the overall system reliability corresponding to the same strategy. Thus, we have

$$\begin{array}{c}{R}_{ij}^H\left(t\right)=1-{\left(1-R_{ij}\left(t\right)\right)}^2\hfill \\ =1-{\left(1-\left[1+{\displaystyle \frac{{\theta }_{ij}^3{t}^3+3{\theta }_{ij}^2({\theta }_{ij}+1)t^2+3{\theta }_{ij}({\theta }_{ij}^2+2{\theta }_{ij}+2)t}{{\theta }_{ij}^3+3{\theta }_{ij}^2+6{\theta }_{ij}+6}}\right]e^{-{\theta }_{ij}t}\right)}^2,\hfill \end{array}$$

(9)

$$\begin{array}{c}{R}_i^H\left(t\right)=1-\prod _{j\in \overline{G_i}}\left(1-R_{ij}\left(t\right)\right)\prod _{j\in G_i}\left(1-R_{ij}^H\left(t\right)\right)\hfill \\ =1-\prod _{j\in \overline{G_i}}\left(1-\left[1+{\displaystyle \frac{{\theta }_{ij}^3{t}^3+3{\theta }_{ij}^2({\theta }_{ij}+1)t^2+3{\theta }_{ij}({\theta }_{ij}^2+2{\theta }_{ij}+2)t}{{\theta }_{ij}^3+3{\theta }_{ij}^2+6{\theta }_{ij}+6}}\right]e^{-{\theta }_{ij}t}\right)\hfill \\ \times \prod _{j\in G_i}\left(1-\left[1-{\left(1-\left[1+{\displaystyle \frac{{\theta }_{ij}^3{t}^3+3{\theta }_{ij}^2({\theta }_{ij}+1)t^2+3{\theta }_{ij}({\theta }_{ij}^2+2{\theta }_{ij}+2)t}{{\theta }_{ij}^3+3{\theta }_{ij}^2+6{\theta }_{ij}+6}}\right]e^{-{\theta }_{ij}t}\right)}^2\right]\right),\hfill \end{array}$$

(10)

and

$$\begin{array}{c}{R}_G^{H;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)=\prod _{i=1}^n{R}_i^H\left(t\right)\hfill \\ =\prod _{i=1}^n\left[1-\prod _{j\in \overline{G_i}}\left(1-\left[1+{\displaystyle \frac{{\theta }_{ij}^3{t}^3+3{\theta }_{ij}^2({\theta }_{ij}+1)t^2+3{\theta }_{ij}({\theta }_{ij}^2+2{\theta }_{ij}+2)t}{{\theta }_{ij}^3+3{\theta }_{ij}^2+6{\theta }_{ij}+6}}\right]e^{-{\theta }_{ij}t}\right)\right.\hfill \\ \left.\times \prod _{j\in G_i}\left(1-\left[1-{\left(1-\left[1+{\displaystyle \frac{{\theta }_{ij}^3{t}^3+3{\theta }_{ij}^2({\theta }_{ij}+1)t^2+3{\theta }_{ij}({\theta }_{ij}^2+2{\theta }_{ij}+2)t}{{\theta }_{ij}^3+3{\theta }_{ij}^2+6{\theta }_{ij}+6}}\right]e^{-{\theta }_{ij}t}\right)}^2\right]\right)\right],\hfill \end{array}$$

(11)

where $\overline{G}$ represents the collection of components that remain without improvement under this scheme.

The MTTF of the system is then computed as

$$MTT{F}_G^H\left(t\right)={\int }_0^{\infty }{R}_G^{H;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)dt,$$

(12)

where $R_G^{H;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)$ is denoted in Equation (11).

4.1.2. Cold Enhancement with Imperfect Switch Strategy

This approach improves system reliability by adding standby replicas of chosen components in a cold redundancy setup, with a switching mechanism that may fail, as depicted in Figure 4. Here, the switch is considered imperfect, meaning it carries a non-zero probability of malfunction during activation.

Let $R_{ij}^{Im}\left(t\right)$ denote the reliability of the jth component in the ith subsystem under this improvement scheme. Assume that the switch reliability, denoted by $R\left(x\right)$, follows the Akshaya distribution with scale parameter $\alpha$. The corresponding expression is given by

$$R_{ij}^{lm}\left(t\right)=R_{ij}\left(t\right)+{\displaystyle \frac{{\theta }_{ij}^4}{{\theta }_{ij}^3+3{\theta }_{ij}^2+6{\theta }_{ij}+6}}\times I(t;{\theta }_{ij},\alpha ),$$

(13)

where $\alpha$ is the scale parameter of the switch, $R_{ij}\left(t\right)$ is given in Equation (5), and $I(t;{\theta }_{ij},\alpha )$ is the integral expression evaluated in Appendix A.

Let $R_i^{Im}\left(t\right)$ represent the reliability of the subsystem i under this enhancement scheme, and let $R_G^{Im;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)$ denote the overall system reliability achieved through this approach. Thus, we can write

$$R_i^{Im}\left(t\right)=1-\prod _{j\in \overline{G_i}}\left(1-R_{i,j}^{Im}\left(t\right)\right)\prod _{j\in G_i}\left(1-R_{ij}^{Im}\left(t\right)\right),$$

(14)

and consequently, the overall system reliability can be expressed as

$$R_G^{Im;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)=\prod _{i=1}^n{R}_i^{Im}\left(t\right).$$

(15)

The MTTF can be determined as

$$MTT{F}_G^{Im}\left(t\right)={\int }_0^{\infty }{R}_G^{Im;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)dt,$$

(16)

where $R_G^{Im;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)$ is given in Equation (6).

4.1.3. Cold Enhancement Strategy

In this strategy, the system’s performance is enhanced due to upgrading selected components through the addition of cold standby units that operate identically to the originals. The switch here is assumed to be perfect (i.e., $P(failure=0$). Figure 5 provides a basic illustration of this scheme, where the configuration includes a primary component supported by a single cold standby unit. Let $R_{ij}^C\left(t\right)$ be the reliability of the jth component in the ith subsystem under this redundancy approach. Similarly, let $R_i^C\left(t\right)$ denote the reliability of subsystem i, and $R_G^{C;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)$ the overall system reliability achieved under this scheme. Accordingly, the expression for the reliability of the improved component j in subsystem i is formulated as

$$\begin{array}{c}{R}_{i,j}^C\left(t\right)=R\left(t\right)+{\int }_0^{t}f\left(x\right)R(t-x)dx\hfill \\ =R_{ij}\left(t\right)+{\displaystyle \frac{{\theta }_{ij}^{4}t{e}^{-{\theta }_{ij}t}}{140{(6+{\theta }_{ij}(6+{\theta }_{ij}(3+{\theta }_{ij})))}^2}}\times \hfill \\ \left[\begin{array}{c}\left(210(2+t)\left(2+t(2+t)\right)\right)+42{\theta }_{ij}\left(20+t\left(40+t\left(30+t(10+t)\right)\right)\right)\hfill \\ +7{\theta }_{ij}^2(2+t)(30+t(60+t(40+t(10+t))))\hfill \\ +{\theta }_{ij}^3\left(140+t(420+t(490+t(280+t(84+t(14+t)\left)\right)\left)\right))\right)\hfill \end{array}\right],\hfill \end{array}$$

(17)

where $R_{ij}\left(t\right)$ is given in Equation (5). The reliability function of subsystem i according to this method is

$$R_i^C\left(t\right)=1-\prod _{j\in \overline{G_i}}\left(1-R_{ij}\left(t\right)\right)\prod _{j\in G_i}\left(1-R_{ij}^C\left(t\right)\right),$$

(18)

and as a consequence, the reliability function of the system is given by

$$R_G^{C;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)=\prod _{i=1}^n{R}_i^C\left(t\right),$$

(19)

and the MTTF can be obtained as

$$MTT{F}_G^C\left(t\right)={\int }_0^{\infty }{R}_G^{C;{\left\{G_i\right\}}_{i=1}^n}dt,$$

(20)

where $R_G^{C;{\left\{G_i\right\}}_{i=1}^n}$ is given in Equation (19).

4.2. Reduction Enhancement Strategy

Another reliability improvement strategy is known as the reduction technique. In this method, the hazard rate of a selected component is scaled down by a constant factor, $\rho$, where $0<\rho <1$, so that its reliability becomes comparable to that of components improved via redundancy-based schemes.

Let A denote the collection of all components enhanced through the reduction technique, with cardinality $\left|A\right|=r$, where $0<r\le N$ and $N={\sum }_{i=1}^n{m}_i$. Define $A_i$ as the subset of improved components belonging to subsystem i, with $|A_i|=r_i$, where $0<r_i\le m_i$, satisfying ${\bigcup }_{i=1}^n{A}_i=A$ and ${\sum }_{i=1}^n{r}_i=r$.

Let $h_{ij}^{\rho }\left(t\right)$ represent the hazard rate of the jth component in the subsystem i after applying this reduction strategy. Similarly, denote by $R_{ij}^{\rho }\left(t\right)$ the reliability of this component, by $R_i^{\rho }\left(t\right)$ the reliability of subsystem i, and by $R_A^{\rho ;{\left\{A_i\right\}}_{i=1}^n}\left(t\right)$ the overall system reliability obtained under this approach. Thus, we can write

$$h_{ij}^{\rho }\left(t\right)={\displaystyle \frac{\rho \left(6+6\theta +3{\theta }^2+{\theta }^3\right)\left(\theta +t^2\right)}{\theta \left(2+{\theta }^3\right){\left(1+t\right)}^3}},$$

(21)

$$R_{ij}^{\rho }\left(t\right)=exp\left(-{\displaystyle \frac{\rho \left(6+\theta \left(6+\theta (3+\theta )\right)\right)\left(\frac{t\left(-2-3t+\theta \left(2+t\right)\right)}{2{\left(1+t\right)}^2}+ln\left(1+t\right)\right)}{\theta \left(2+{\theta }^3\right)}}\right),$$

(22)

$$R_i^{\rho }\left(t\right)=1-\prod _{j\in \overline{A_i}}\left(1-R_{ij}\left(t\right)\right)\prod _{j\in A_i}\left(1-R_{ij}^{\rho }\left(t\right)\right),$$

(23)

and

$$R_A^{\rho ;{\left\{A_i\right\}}_{i=1}^n}\left(t\right)=\prod _{i=1}^n{R}_i^{\rho }\left(t\right),$$

(24)

where $\overline{A}$ denotes the collection of components that remain unmodified under this technique.

• The mean time to failure (MTTF) is then determined as

$$MTT{F}_A^{\rho }\left(t\right)={\int }_0^{\infty }{R}_A^{\rho ;{\left\{A_i\right\}}_{i=1}^n}\left(t\right)dt,$$

(25)

where $R_A^{\rho ;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)$ is given in Equation (24).

5. Reliability Equivalence Factors

The reliability equivalence factors (REFs) for hot-, cold-, and imperfect-switch redundancy, denoted as ${\rho }^D\left(\gamma \right),D\in \{H,Im,C\}$, quantify the scaling parameters needed to adjust the failure rate of the baseline system so that it becomes equivalent to that of the improved configuration achieved through duplication-based strategies. In general, the values of ${\rho }^D\left(\gamma \right),D\in \{H,Im,C\}$ are obtained by solving the following system of equations:

$$R_A^{\rho ;{\left\{A_i\right\}}_{i=1}^n}\left(t\right)=\delta ,R_G^{D;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)=\delta ,D=H,Im,C,$$

(26)

where $R_A^{\rho ;{\left\{A_i\right\}}_{i=1}^n}\left(t\right)$ and $R_G^{D;{\left\{G_i\right\}}_{i=1}^n}\left(t\right)$ are given in Equations (24), (11), (15), and (19), respectively.

Solving these systems of equations generally necessitates the use of numerical techniques. In this study, all numerical solutions were obtained using the FindRoot function in Mathematica (version 13.1), which implements a hybrid of Newton-Raphson and secant methods.

6. $\mathit{\delta }$-Fractiles

The $\delta$-fractiles of the original system can be computed by determining the value of L that satisfies the following equation:

$${\int }_L^{\infty }f\left(t\right)dt=\delta ,$$

which can equivalently be expressed as

$$R\left(L\right)=\delta ,$$

(27)

substituting Equation (7) into Equation (27), the $\delta$-fractiles of the baseline system can be evaluated through suitable numerical procedures.

Similarly, the $\delta$-fractiles corresponding to systems enhanced using duplication-based strategies are derived by finding the value of L that satisfies the following equations:

$$R^D\left(L\right)=\delta ,D\in \{H,Im,C\}.$$

(28)

By substituting Equations (11), (15) and (19) into Equation (28), the $\delta$-fractiles for systems improved via duplication-based schemes can likewise be determined through appropriate numerical techniques.

7. Numerical Analysis

7.1. Case Study: Reliability Enhancement of a Safety Monitoring Module

This section demonstrates the practical applicability of the proposed reliability enhancement framework through a realistic industrial case study. We consider a safety monitoring module deployed in a processing plant, responsible for continuously tracking critical operating parameters such as temperature and pressure. System failure may lead to severe safety hazards; therefore, redundancy and reliability optimization are essential design objectives.

The module consists of two statistically independent subsystems connected in parallel. Subsystem 1 contains a single high-precision sensor, whereas Subsystem 2 comprises two lower-cost sensors arranged in series for cross-validation purposes. This architecture corresponds to the hybrid parallel–series structure depicted in Figure 1, with parameters $n=2$, $m_1=1$, and $m_2=2$.

7.1.1. Component Reliability Modeling

Based on historical failure records of comparable sensing devices, component lifetimes are assumed to follow the Akshaya distribution. The model parameters are selected to reflect heterogeneous reliability characteristics across components:

• Component 1 (Subsystem 1): High-reliability sensor with a gradually increasing hazard rate, ${\theta }_1=1.8$.
• Component 2 (Subsystem 2): Standard sensor with moderate degradation behavior, ${\theta }_2=1.5$.
• Component 3 (Subsystem 2): Sensor from a distinct production batch exhibiting higher early-life failure tendency, ${\theta }_3=0.7$.
• For cold standby strategies with imperfect switching, the switch lifetime follows an Akshaya distribution with parameter $\alpha =0.04$.

7.1.2. Baseline Performance and Improvement Strategies

Using the analytical expressions derived in Section 3, Section 4, Section 5 and Section 6, the mean time to failure (MTTF) of the original system are computed as $\mathrm{MTTF}=2.1944$. To assess improvement effectiveness, duplication-based strategies are applied to the following component subsets:

$$G_1=\left\{1\right\},G_2=\left\{2\right\},G_3=\{1,2\},G_4=\{2,3\},G_5=\{1,2,3\}.$$

Table 1. MTTF values under different duplication strategies for selected component subsets.

Subset G	Hot	Cold (Imperfect)	Cold (Perfect)	Baseline
Baseline	2.1944	2.1944	2.1944	2.1944
$G_1$	2.3000	2.5535	2.5536	2.1944
$G_2$	2.7406	3.4204	3.4205	2.1944
$G_3$	2.7945	3.5801	3.5804	2.1944
$G_4$	3.0974	4.5502	4.5516	2.1944
$G_5$	3.1323	4.6221	4.6240	2.1944

reports the resulting MTTF values under hot duplication, cold duplication with imperfect switching, and cold duplication with a perfect switch.

Applying cold duplication with a perfect switch to all components ($G_5$) increases the MTTF by approximately $110.7\%$, confirming the substantial benefit of redundancy in safety-critical systems.

7.2. Reliability Equivalence Factor (REF) Analysis

While duplication improves reliability, it may incur a high cost. Reliability Equivalence Factors (REFs) provide a quantitative comparison between system-level redundancy and component-level improvement through failure-rate reduction.

Table 2. REFs for direct equivalence scenarios ($A=G$).

	$\mathit{\delta }=0.1$			$\mathit{\delta }=0.5$			$\mathit{\delta }=0.9$
Set	H	Im	C	H	Im	C	H	Im	C
$\left\{1\right\}$	NE	NE	NE	0.8138	0.8137	0.8137	0.6454	0.4358	0.2956
$\left\{2\right\}$	0.3866	0.3594	0.3594	0.1788	0.1429	0.1429	0.0520	0.0312	0.0312
$\{1,2\}$	NE	NE	NE	0.4155	0.3938	0.3938	0.1637	0.1493	0.1160
$\{2,3\}$	0.2229	0.2146	0.2117	0.1069	0.0945	0.0847	0.0356	0.0280	0.0159
$\{1,2,3\}$	NE	NE	NE	0.4155	0.3938	0.3938	0.1637	0.1493	0.1160

shows the REFs for direct equivalence scenarios ($A=G$). Also, Table 3, Table 4, Table 5, Table 6 and Table 7 present detailed REF values across different reliability targets $\delta \in (0,1)$.

Three key observations emerge:

1.

Cold duplication consistently dominates hot duplication, requiring smaller failure-rate reductions to achieve equivalent performance;

2.

As $\delta$ increases, REF values decrease sharply, indicating diminishing feasibility of component-only improvement at high reliability targets;

3.

The frequent occurrence of “NE” demonstrates scenarios where redundancy yields reliability gains unattainable through any admissible failure-rate reduction.

7.3. Sensitivity Analysis

A sensitivity study is conducted to examine robustness with respect to component parameters ${\theta }_i$ and switch parameter $\alpha$. Results confirm that the relative ranking

$$\mathrm{Cold}\left(\mathrm{Perfect}\right)>\mathrm{Cold}\left(\mathrm{Imperfect}\right)>\mathrm{Hot}>\mathrm{Baseline}$$

is preserved across low-, medium-, and high-reliability regimes. Furthermore, cold-standby performance deteriorates rapidly as switch quality decreases, emphasizing the critical role of reliable switching mechanisms in redundancy design.

• Engineering Implications

The numerical findings provide clear guidance for system designers. Cold standby redundancy offers substantial reliability benefits, but only when supported by sufficiently reliable switching. In high-reliability systems, even moderate switch degradation can significantly offset the expected gains, potentially rendering hot duplication a more cost-effective alternative.

From these results, several important observations can be drawn:

• The ranking of MTTF values for all configurations is

  $$\mathrm{Cold}\left(\mathrm{Perfect}\right)>\mathrm{Cold}\left(\mathrm{Imperfect}\right)>\mathrm{Hot}>\mathrm{Main}\mathrm{System}.$$
• Improvements are consistently higher when duplication is applied to a larger set of components (e.g., $G_5$ outperforms $G_4$, $G_3$, $G_2$, and $G_1$).
• The $\delta$-fractile results demonstrate that employing cold duplication with a perfect switching mechanism yields the greatest improvement in system reliability.
• Among the evaluated enhancement strategies, cold duplication proves to be the most effective, followed by cold duplication with an imperfect switch. Although hot duplication is comparatively less impactful, it still delivers noticeable reliability gains over the baseline system.

Table 3. $\delta$-fractile values for the baseline system and its enhanced configurations.

$\delta$	$L\left(\delta \right)$	$L_{G_1}^H$	$L_{G_2}^H$	$L_{G_3}^H$	$L_{G_4}^H$	$L_{G_5}^H$
0.1	16.9540	17.0594	19.6493	19.6811	21.6082	21.6217
0.2	13.7314	13.9335	16.4524	16.5179	18.3032	18.3336
0.3	11.6630	11.9586	14.3570	14.4609	16.1130	16.1650
0.4	10.0644	10.4503	12.7025	12.8498	14.3688	14.4477
0.5	8.7066	9.1801	11.2663	11.4629	12.8443	12.9563
0.6	7.4741	8.0336	9.9334	10.1866	11.4218	11.5748
0.7	6.2861	6.9309	8.6171	8.9367	10.0120	10.2162
0.8	5.0553	5.7856	7.2133	7.6151	8.5065	8.7775
0.9	3.6106	4.4247	5.4940	6.0090	6.6666	7.0335
$\delta$	$L\left(\delta \right)$	$L_{G_1}^{Im}$	$L_{G_2}^{Im}$	$L_{G_3}^{Im}$	$L_{G_4}^{Im}$	$L_{G_5}^{Im}$
0.1	16.9540	18.0763	25.0062	25.1122	31.8593	31.8681
0.2	13.7314	15.1478	20.9243	21.1527	27.1342	27.1633
0.3	11.6630	13.2563	18.1897	18.5560	23.9327	23.9962
0.4	10.0644	11.7736	15.9964	16.5131	21.3373	21.4533
0.5	8.7066	10.4908	14.0715	14.7486	19.0330	19.2240
0.6	7.4741	9.3013	12.2738	13.1192	16.8522	17.1468
0.7	6.2861	8.1257	10.4978	11.5175	14.6648	15.0988
0.8	5.0553	6.8701	8.6189	9.8158	12.3111	12.9310
0.9	3.6106	5.3312	6.3692	7.7365	9.4438	10.3130
$\delta$	$L\left(\delta \right)$	$L_{G_1}^C$	$L_{G_2}^C$	$L_{G_3}^C$	$L_{G_4}^C$	$L_{G_5}^C$
0.1	16.9540	18.0765	25.0093	25.1152	31.8735	31.8822
0.2	13.7314	15.1480	20.9263	21.1547	27.1441	27.1731
0.3	11.6630	13.2565	18.1911	18.5575	23.9401	24.0035
0.4	10.0644	11.7737	15.9975	16.5142	21.3431	21.4589
0.5	8.7066	10.4909	14.0723	14.7494	19.0375	19.2284
0.6	7.4741	9.3013	12.2744	13.1198	16.8556	17.1500
0.7	6.2861	8.1258	10.4982	11.5180	14.6673	15.1012
0.8	5.0553	6.8702	8.6191	9.8161	12.3128	12.9327
0.9	3.6106	5.3312	6.3693	7.7367	9.4447	10.3140

Table 3. $\delta$-fractile values for the baseline system and its enhanced configurations.

$\delta$	$L\left(\delta \right)$	$L_{G_1}^H$	$L_{G_2}^H$	$L_{G_3}^H$	$L_{G_4}^H$	$L_{G_5}^H$
0.1	16.9540	17.0594	19.6493	19.6811	21.6082	21.6217
0.2	13.7314	13.9335	16.4524	16.5179	18.3032	18.3336
0.3	11.6630	11.9586	14.3570	14.4609	16.1130	16.1650
0.4	10.0644	10.4503	12.7025	12.8498	14.3688	14.4477
0.5	8.7066	9.1801	11.2663	11.4629	12.8443	12.9563
0.6	7.4741	8.0336	9.9334	10.1866	11.4218	11.5748
0.7	6.2861	6.9309	8.6171	8.9367	10.0120	10.2162
0.8	5.0553	5.7856	7.2133	7.6151	8.5065	8.7775
0.9	3.6106	4.4247	5.4940	6.0090	6.6666	7.0335
$\delta$	$L\left(\delta \right)$	$L_{G_1}^{Im}$	$L_{G_2}^{Im}$	$L_{G_3}^{Im}$	$L_{G_4}^{Im}$	$L_{G_5}^{Im}$
0.1	16.9540	18.0763	25.0062	25.1122	31.8593	31.8681
0.2	13.7314	15.1478	20.9243	21.1527	27.1342	27.1633
0.3	11.6630	13.2563	18.1897	18.5560	23.9327	23.9962
0.4	10.0644	11.7736	15.9964	16.5131	21.3373	21.4533
0.5	8.7066	10.4908	14.0715	14.7486	19.0330	19.2240
0.6	7.4741	9.3013	12.2738	13.1192	16.8522	17.1468
0.7	6.2861	8.1257	10.4978	11.5175	14.6648	15.0988
0.8	5.0553	6.8701	8.6189	9.8158	12.3111	12.9310
0.9	3.6106	5.3312	6.3692	7.7365	9.4438	10.3130
$\delta$	$L\left(\delta \right)$	$L_{G_1}^C$	$L_{G_2}^C$	$L_{G_3}^C$	$L_{G_4}^C$	$L_{G_5}^C$
0.1	16.9540	18.0765	25.0093	25.1152	31.8735	31.8822
0.2	13.7314	15.1480	20.9263	21.1547	27.1441	27.1731
0.3	11.6630	13.2565	18.1911	18.5575	23.9401	24.0035
0.4	10.0644	11.7737	15.9975	16.5142	21.3431	21.4589
0.5	8.7066	10.4909	14.0723	14.7494	19.0375	19.2284
0.6	7.4741	9.3013	12.2744	13.1198	16.8556	17.1500
0.7	6.2861	8.1258	10.4982	11.5180	14.6673	15.1012
0.8	5.0553	6.8702	8.6191	9.8161	12.3128	12.9327
0.9	3.6106	5.3312	6.3693	7.7367	9.4447	10.3140

Figure 6 visualizes the reliability functions for different improvement strategies and subsets, illustrating the consistent superiority of duplication-based methods, especially those involving a perfect switch.

Similarly, Figure 7 illustrates the reliability curves of the baseline configuration alongside those of the systems enhanced using hot and cold duplication schemes for the scenarios (a) $G=\left\{1\right\}$ and (b) $G=\left\{2\right\}$.

Furthermore, Figure 8 presents the corresponding reliability profiles for the original setup and its improved versions when duplication is applied to (a) $G=\{1,2\}$, (b) $G=\{2,3\}$, and (c) $G=\{1,2,3\}$.

Table 4, Table 5, Table 6, Table 7 and Table 8 summarize the reliability equivalence factor (REF) values required to achieve equivalence between the original system and the systems enhanced via duplication strategies, considering all selected subsets of B.

Table 4. Reliability equivalence factor (REF) values for the case $A=\left\{1\right\}$ with improvement subsets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	NE	NE	NE	0.8305	0.6454	0.4383
$G_2$	NE	NE	0.9340	0.8050	0.6944	0.5932	0.4952	0.3941	0.2783
$G_3$	NE	NE	0.9205	0.7864	0.6706	0.5640	0.4605	0.3541	0.2337
$G_4$	NE	0.9203	0.7648	0.6466	0.5473	0.4584	0.3741	0.2887	0.1927
$G_5$	NE	0.9182	0.7613	0.6412	0.5398	0.4485	0.3615	0.2733	0.1751
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	0.9579	0.8138	0.6840	0.5607	0.4358	0.295
$G_2$	NE	0.7912	0.6565	0.5578	0.4782	0.4093	0.3457	0.2821	0.2095
$G_3$	NE	0.7835	0.6428	0.5372	0.4497	0.3722	0.2997	0.2276	0.1487
$G_4$	0.8896	0.6646	0.5279	0.4280	0.3483	0.2810	0.2214	0.1658	0.1086
$G_5$	0.8895	0.6643	0.5271	0.4265	0.3456	0.2767	0.2148	0.1562	0.0958
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	0.9578	0.8137	0.6840	0.5606	0.4358	0.2956
$G_2$	NE	0.7912	0.6564	0.5578	0.4782	0.4093	0.3456	0.2821	0.2095
$G_3$	NE	0.7834	0.6428	0.5372	0.4497	0.3722	0.2997	0.2276	0.1487
$G_4$	0.8894	0.6645	0.5278	0.4279	0.3482	0.2809	0.2214	0.1657	0.1086
$G_5$	0.8893	0.6642	0.5270	0.4264	0.3456	0.2766	0.2148	0.1562	0.0958

Table 4. Reliability equivalence factor (REF) values for the case $A=\left\{1\right\}$ with improvement subsets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	NE	NE	NE	0.8305	0.6454	0.4383
$G_2$	NE	NE	0.9340	0.8050	0.6944	0.5932	0.4952	0.3941	0.2783
$G_3$	NE	NE	0.9205	0.7864	0.6706	0.5640	0.4605	0.3541	0.2337
$G_4$	NE	0.9203	0.7648	0.6466	0.5473	0.4584	0.3741	0.2887	0.1927
$G_5$	NE	0.9182	0.7613	0.6412	0.5398	0.4485	0.3615	0.2733	0.1751
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	0.9579	0.8138	0.6840	0.5607	0.4358	0.295
$G_2$	NE	0.7912	0.6565	0.5578	0.4782	0.4093	0.3457	0.2821	0.2095
$G_3$	NE	0.7835	0.6428	0.5372	0.4497	0.3722	0.2997	0.2276	0.1487
$G_4$	0.8896	0.6646	0.5279	0.4280	0.3483	0.2810	0.2214	0.1658	0.1086
$G_5$	0.8895	0.6643	0.5271	0.4265	0.3456	0.2767	0.2148	0.1562	0.0958
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	0.9578	0.8137	0.6840	0.5606	0.4358	0.2956
$G_2$	NE	0.7912	0.6564	0.5578	0.4782	0.4093	0.3456	0.2821	0.2095
$G_3$	NE	0.7834	0.6428	0.5372	0.4497	0.3722	0.2997	0.2276	0.1487
$G_4$	0.8894	0.6645	0.5278	0.4279	0.3482	0.2809	0.2214	0.1657	0.1086
$G_5$	0.8893	0.6642	0.5270	0.4264	0.3456	0.2766	0.2148	0.1562	0.0958

Table 5. Reliability equivalence factor (REF) values for the configuration $A=\left\{2\right\}$ with component improvement sets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.3866	0.3078	0.2556	0.2143	0.1788	0.1466	0.1160	0.0853	0.0520
$G_2$	0.32089	0.2435	0.1940	0.1563	0.1251	0.0981	0.0736	0.0507	0.0281
$G_3$	0.3202	0.2421	0.1917	0.1531	0.1209	0.0928	0.0672	0.0430	0.0192
$G_4$	0.2774	0.2038	0.1573	0.1223	0.0937	0.0693	0.0476	0.0278	0.0093
$G_5$	0.2771	0.2031	0.1562	0.1208	0.0917	0.0666	0.0442	0.0236	0.0042
Sets	$D=I$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.3594	0.2749	0.2200	0.1780	0.1429	0.1123	0.0843	0.0578	0.0312
$G_2$	0.2094	0.1533	0.1186	0.0928	0.0721	0.0547	0.0396	0.0261	0.0136
$G_3$	0.2074	0.1491	0.1121	0.0840	0.0610	0.0412	0.0238	0.0085	NE
$G_4$	0.0898	0.0488	0.02503	0.0086	NE	NE	NE	NE	NE
$G_5$	0.0896	0.0483	0.0241	0.0069	NE	NE	NE	NE	NE
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.3594	0.2749	0.2200	0.1779	0.1429	0.1123	0.0843	0.0578	0.0312
$G_2$	0.2093	0.1533	0.1185	0.0928	0.0721	0.0547	0.0396	0.02601	0.0136
$G_3$	0.2073	0.1491	0.1121	0.0840	0.0609	0.0412	0.0238	0.0085	NE
$G_4$	0.0895	0.0486	0.0249	0.0085	NE	NE	NE	NE	NE
$G_5$	0.0894	0.0482	0.0240	0.0068	NE	NE	NE	NE	NE

Table 5. Reliability equivalence factor (REF) values for the configuration $A=\left\{2\right\}$ with component improvement sets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.3866	0.3078	0.2556	0.2143	0.1788	0.1466	0.1160	0.0853	0.0520
$G_2$	0.32089	0.2435	0.1940	0.1563	0.1251	0.0981	0.0736	0.0507	0.0281
$G_3$	0.3202	0.2421	0.1917	0.1531	0.1209	0.0928	0.0672	0.0430	0.0192
$G_4$	0.2774	0.2038	0.1573	0.1223	0.0937	0.0693	0.0476	0.0278	0.0093
$G_5$	0.2771	0.2031	0.1562	0.1208	0.0917	0.0666	0.0442	0.0236	0.0042
Sets	$D=I$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.3594	0.2749	0.2200	0.1780	0.1429	0.1123	0.0843	0.0578	0.0312
$G_2$	0.2094	0.1533	0.1186	0.0928	0.0721	0.0547	0.0396	0.0261	0.0136
$G_3$	0.2074	0.1491	0.1121	0.0840	0.0610	0.0412	0.0238	0.0085	NE
$G_4$	0.0898	0.0488	0.02503	0.0086	NE	NE	NE	NE	NE
$G_5$	0.0896	0.0483	0.0241	0.0069	NE	NE	NE	NE	NE
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.3594	0.2749	0.2200	0.1779	0.1429	0.1123	0.0843	0.0578	0.0312
$G_2$	0.2093	0.1533	0.1185	0.0928	0.0721	0.0547	0.0396	0.02601	0.0136
$G_3$	0.2073	0.1491	0.1121	0.0840	0.0609	0.0412	0.0238	0.0085	NE
$G_4$	0.0895	0.0486	0.0249	0.0085	NE	NE	NE	NE	NE
$G_5$	0.0894	0.0482	0.0240	0.0068	NE	NE	NE	NE	NE

Table 6. Reliability equivalence factor (REF) values for the case $A=\{1,2\}$ with improvement subsets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	0.6131	0.5257	0.4484	0.3756	0.3018	0.2175
$G_2$	NE	0.8089	0.6641	0.5598	0.4759	NE	NE	NE	NE
$G_3$	NE	0.8076	0.6620	0.5568	0.4719	0.3980	0.3292	0.2605	0.1832
$G_4$	NE	0.7744	0.6310	0.5280	0.4458	0.3749	0.3096	0.2448	0.1723
$G_5$	NE	0.7739	0.6301	0.5267	0.4439	0.3723	0.3062	0.2404	0.1669
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.8368	0.6880	0.5801	0.4929	0.4169	0.3462	0.2755	0.1961
$G_2$	0.9654	0.7335	0.5979	0.5016	0.4256	0.3608	0.3016	0.2430	0.1771
$G_3$	0.9639	0.7303	0.5926	0.4940	0.4154	0.3480	0.2862	0.2251	0.1573
$G_4$	0.8851	0.6619	0.5300	0.4364	0.3634	0.3022	0.2476	0.1950	0.1377
$G_5$	0.8850	0.6616	0.5294	0.4353	0.3614	0.2991	0.2431	0.1887	0.1292
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.8221	0.6676	0.5539	0.4610	0.3792	0.3026	0.2259	0.1411
$G_2$	0.9006	0.6574	0.5116	0.4061	0.3221	0.2507	0.1870	0.1272	0.0674
$G_3$	0.9005	0.6572	0.5115	0.4059	0.3219	0.2506	0.1869	0.1272	0.0674
$G_4$	0.7923	0.5663	0.4310	0.3334	NE	NE	NE	NE	NE
$G_5$	NE	NE	0.8363	0.6741	0.5454	0.4361	0.3381	0.2449	0.1479

Table 6. Reliability equivalence factor (REF) values for the case $A=\{1,2\}$ with improvement subsets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	NE	NE	0.6131	0.5257	0.4484	0.3756	0.3018	0.2175
$G_2$	NE	0.8089	0.6641	0.5598	0.4759	NE	NE	NE	NE
$G_3$	NE	0.8076	0.6620	0.5568	0.4719	0.3980	0.3292	0.2605	0.1832
$G_4$	NE	0.7744	0.6310	0.5280	0.4458	0.3749	0.3096	0.2448	0.1723
$G_5$	NE	0.7739	0.6301	0.5267	0.4439	0.3723	0.3062	0.2404	0.1669
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.8368	0.6880	0.5801	0.4929	0.4169	0.3462	0.2755	0.1961
$G_2$	0.9654	0.7335	0.5979	0.5016	0.4256	0.3608	0.3016	0.2430	0.1771
$G_3$	0.9639	0.7303	0.5926	0.4940	0.4154	0.3480	0.2862	0.2251	0.1573
$G_4$	0.8851	0.6619	0.5300	0.4364	0.3634	0.3022	0.2476	0.1950	0.1377
$G_5$	0.8850	0.6616	0.5294	0.4353	0.3614	0.2991	0.2431	0.1887	0.1292
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.8221	0.6676	0.5539	0.4610	0.3792	0.3026	0.2259	0.1411
$G_2$	0.9006	0.6574	0.5116	0.4061	0.3221	0.2507	0.1870	0.1272	0.0674
$G_3$	0.9005	0.6572	0.5115	0.4059	0.3219	0.2506	0.1869	0.1272	0.0674
$G_4$	0.7923	0.5663	0.4310	0.3334	NE	NE	NE	NE	NE
$G_5$	NE	NE	0.8363	0.6741	0.5454	0.4361	0.3381	0.2449	0.1479

Table 7. Reliability equivalence factor (REF) values for the configuration $A=\{2,3\}$ with component improvement sets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.2229	0.1785	0.1494	0.1265	0.1069	0.0890	0.0720	0.0547	0.0356
$G_2$	0.2037	0.1589	0.1300	0.1077	0.0891	0.0725	0.0573	0.0425	0.0269
$G_3$	0.2035	0.1585	0.1293	0.1068	0.0878	0.0710	0.0553	0.0401	0.0242
$G_4$	0.1924	0.1485	0.1203	0.0986	0.0806	0.0646	0.0500	0.0360	0.0216
$G_5$	0.1923	0.1483	0.1200	0.0983	0.0801	0.0640	0.0492	0.0350	0.0204
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.2146	0.1680	0.1377	0.1143	0.0945	0.0770	0.0607	0.0448	0.0280
$G_2$	0.1768	0.1370	0.1114	0.0919	0.0755	0.0612	0.0481	0.0356	0.0227
$G_3$	0.1764	0.1361	0.1101	0.0900	0.0732	0.0584	0.0447	0.0317	0.0185
$G_4$	0.1547	0.1180	0.0948	0.0771	0.0624	0.0495	0.0377	0.0266	0.0154
$G_5$	0.1546	0.1180	0.0946	0.0768	0.0620	0.0490	0.0370	0.0257	0.0143
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.2117	0.1631	0.1311	0.1060	0.0847	0.0658	0.0483	0.0319	0.0159
$G_2$	0.1588	0.1169	0.0905	0.0707	0.0546	0.0410	0.0291	0.0185	0.0089
$G_3$	0.1588	0.1169	0.0904	0.0706	0.0546	0.0410	0.0291	0.0185	0.0088
$G_4$	0.1309	0.0943	0.0718	0.0555	0.0424	0.0316	0.0222	0.0140	0.0067
$G_5$	0.6244	0.3048	0.2122	0.1560	0.1156	0.0840	0.0580	0.0360	0.0169

Table 7. Reliability equivalence factor (REF) values for the configuration $A=\{2,3\}$ with component improvement sets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.2229	0.1785	0.1494	0.1265	0.1069	0.0890	0.0720	0.0547	0.0356
$G_2$	0.2037	0.1589	0.1300	0.1077	0.0891	0.0725	0.0573	0.0425	0.0269
$G_3$	0.2035	0.1585	0.1293	0.1068	0.0878	0.0710	0.0553	0.0401	0.0242
$G_4$	0.1924	0.1485	0.1203	0.0986	0.0806	0.0646	0.0500	0.0360	0.0216
$G_5$	0.1923	0.1483	0.1200	0.0983	0.0801	0.0640	0.0492	0.0350	0.0204
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.2146	0.1680	0.1377	0.1143	0.0945	0.0770	0.0607	0.0448	0.0280
$G_2$	0.1768	0.1370	0.1114	0.0919	0.0755	0.0612	0.0481	0.0356	0.0227
$G_3$	0.1764	0.1361	0.1101	0.0900	0.0732	0.0584	0.0447	0.0317	0.0185
$G_4$	0.1547	0.1180	0.0948	0.0771	0.0624	0.0495	0.0377	0.0266	0.0154
$G_5$	0.1546	0.1180	0.0946	0.0768	0.0620	0.0490	0.0370	0.0257	0.0143
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	0.2117	0.1631	0.1311	0.1060	0.0847	0.0658	0.0483	0.0319	0.0159
$G_2$	0.1588	0.1169	0.0905	0.0707	0.0546	0.0410	0.0291	0.0185	0.0089
$G_3$	0.1588	0.1169	0.0904	0.0706	0.0546	0.0410	0.0291	0.0185	0.0088
$G_4$	0.1309	0.0943	0.0718	0.0555	0.0424	0.0316	0.0222	0.0140	0.0067
$G_5$	0.6244	0.3048	0.2122	0.1560	0.1156	0.0840	0.0580	0.0360	0.0169

Table 8. Reliability equivalence factor (REF) values for the case $A=\{1,2,3\}$ with improvement subsets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.8239	0.6548	0.5327	0.4383	0.3612	0.2941	0.2310	0.1637
$G_2$	NE	0.7769	0.6128	0.4947	NE	0.3303	0.2671	NE	0.1472
$G_3$	NE	0.7758	0.6112	0.4926	NE	NE	NE	NE	0.1410
$G_4$	NE	0.7482	0.5879	0.4725	0.3836	0.3118	0.2503	0.193	0.1342
$G_5$	NE	0.7478	0.5872	0.4715	0.3823	0.3101	0.2481	0.1909	0.1309
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.7999	0.6307	0.5091	0.4155	0.3395	0.2741	0.2133	0.1493
$G_2$	0.9594	0.7139	0.5630	0.4541	0.3702	0.3027	NE	NE	0.1371
$G_3$	0.9580	0.7112	0.5590	0.4488	0.3636	0.2946	NE	0.1815	NE
$G_4$	0.8826	0.6521	0.5109	0.4086	NE	0.2663	NE	NE	0.1141
$G_5$	0.8825	0.6519	0.5104	0.4078	0.3285	0.2645	0.2102	0.1607	0.1096
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.7877	0.6154	0.4906	0.3938	0.3146	0.2459	0.1820	0.1160
$G_2$	0.8975	0.6482	0.4963	0.3870	0.3028	0.2352	0.1784	0.1280	0.0789
$G_3$	0.8974	0.6480	0.4962	0.3869	0.3027	0.2351	0.1784	0.1280	0.0790
$G_4$	0.7917	0.5646	0.4282	0.3307	0.2559	0.1963	0.1470	0.1041	0.0633
$G_5$	NE	0.9850	0.7434	0.5771	0.4522	0.3526	0.2687	0.1936	0.1198

Table 8. Reliability equivalence factor (REF) values for the case $A=\{1,2,3\}$ with improvement subsets $G=\{G_1,G_2,G_3,G_4,G_5\}$.

Sets	$D=H$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.8239	0.6548	0.5327	0.4383	0.3612	0.2941	0.2310	0.1637
$G_2$	NE	0.7769	0.6128	0.4947	NE	0.3303	0.2671	NE	0.1472
$G_3$	NE	0.7758	0.6112	0.4926	NE	NE	NE	NE	0.1410
$G_4$	NE	0.7482	0.5879	0.4725	0.3836	0.3118	0.2503	0.193	0.1342
$G_5$	NE	0.7478	0.5872	0.4715	0.3823	0.3101	0.2481	0.1909	0.1309
Sets	$D=Im$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.7999	0.6307	0.5091	0.4155	0.3395	0.2741	0.2133	0.1493
$G_2$	0.9594	0.7139	0.5630	0.4541	0.3702	0.3027	NE	NE	0.1371
$G_3$	0.9580	0.7112	0.5590	0.4488	0.3636	0.2946	NE	0.1815	NE
$G_4$	0.8826	0.6521	0.5109	0.4086	NE	0.2663	NE	NE	0.1141
$G_5$	0.8825	0.6519	0.5104	0.4078	0.3285	0.2645	0.2102	0.1607	0.1096
Sets	$D=C$
	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$	$0.7$	$0.8$	$0.9$
$G_1$	NE	0.7877	0.6154	0.4906	0.3938	0.3146	0.2459	0.1820	0.1160
$G_2$	0.8975	0.6482	0.4963	0.3870	0.3028	0.2352	0.1784	0.1280	0.0789
$G_3$	0.8974	0.6480	0.4962	0.3869	0.3027	0.2351	0.1784	0.1280	0.0790
$G_4$	0.7917	0.5646	0.4282	0.3307	0.2559	0.1963	0.1470	0.1041	0.0633
$G_5$	NE	0.9850	0.7434	0.5771	0.4522	0.3526	0.2687	0.1936	0.1198

7.4. Summary of Reliability Equivalence Factor Trends

To distill the key patterns from the extensive data in Table 3, Table 4, Table 5, Table 6 and Table 7,

Table 9. Summary of reliability equivalence factors (REFs) for the direct equivalence case: duplication applied to the same component set as reduction.

	$\mathit{\delta }=0.1$			$\mathit{\delta }=0.5$			$\mathit{\delta }=0.9$
Component	Duplication Strategy			Duplication Strategy			Duplication Strategy
Set ($\mathit{A}$)	H	Im	C	H	Im	C	H	Im	C
$\left\{1\right\}$	NE	NE	NE	NE	0.8138	0.8137	0.6454	0.4358	0.4358
$\left\{2\right\}$	0.3866	0.3594	0.3594	0.1788	0.1429	0.1429	0.0520	0.0312	0.0312
$\{1,2\}$	NE	NE	NE	NE	0.4155	0.3938	0.1637	0.1493	0.1160
$\{2,3\}$	0.2229	0.2146	0.2117	0.1069	0.0945	0.0847	0.0356	0.0280	0.0159
$\{1,2,3\}$	NE	NE	NE	NE	0.4155	0.3938	0.1637	0.1493	0.1160

summarizes the REF values for the direct equivalence case, where the set of components improved by reduction (A) is the same as the set enhanced by duplication (G). This comparison provides the most straightforward insight into the trade-off between component-level improvement and redundancy for a given set of components. The table displays REFs for three representative reliability levels ($\delta =0.1,0.5,0.9$) across all component sets and duplication strategies.

Table 2 reveals several consistent trends. Firstly, for a given component set and $\delta$, the REF is generally lowest for cold-perfect duplication (C), followed by cold-imperfect (Im), and highest for hot duplication (H). This confirms that matching the performance of more effective redundancy strategies requires a smaller reduction in component failure rates (i.e., the redundancy is relatively more advantageous). Secondly, as $\delta$ increases, the REF typically decreases, meaning that at higher system reliability levels, achieving equivalence via component improvement alone demands a more substantial (often impractical) reduction in failure rate. Finally, the prevalence of “NE” (No Equivalence) for smaller $\delta$ values, particularly for larger component sets, underscores that duplication can provide unique gains that cannot be matched by any feasible reduction ($0<\rho <1$) at those reliability targets. This summary reinforces the primary insight that redundancy, especially cold standby, is a uniquely powerful strategy for achieving high system reliability.

• In addition, some observations can be drawn from Figure 6 and Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8:

• The ordering of system reliability under the hot duplication strategy is

  $$R_{\{1,2,3\}}^H>R_{\{1,2\}}^H>R_{\left\{1\right\}}^H>R_{\{2,3\}}^H>R_{\left\{2\right\}}^H.$$
• Under cold enhancement with an imperfect switch, the reliability ranking becomes

  $$R_{\{1,2,3\}}^{Im}>R_{\{1,2\}}^{Im}>R_{\left\{1\right\}}^{Im}>R_{\{2,3\}}^{Im}>R_{\left\{2\right\}}^{Im}.$$
• For cold duplication with a perfect switch, the ordering is

  $$R_{\{1,2,3\}}^C>R_{\{1,2\}}^C>R_{\left\{1\right\}}^C>R_{\{2,3\}}^C>R_{\left\{2\right\}}^C.$$
• A value of $\rho =0.5$, for instance, indicates that halving the hazard rate (i.e., doubling the mean lifetime) of the component(s) in set A provides the same system-level improvement as implementing the specified duplication on set G. Thus, smaller REF values ($\rho \to 0$) imply that the reduction method is highly effective relative to duplication; a modest improvement in component reliability yields an equivalent system gain. Conversely, values closer to 1 ($\rho \to 1$) suggest that duplication is the more potent strategy, as achieving the same benefit via reduction would require an impractically large enhancement of the component(s).
• Applying hot duplication to component $\left\{1\right\}$ increases $L\left(0.8\right)$ from $5.0553$ to $5.57856$ (Table 3). An equivalent effect on $L\left(0.9\right)$ can be achieved by reducing the failure rate of component $\left\{1\right\}$ by

  1.

  $\rho =0.6454$ in $G_1$ (Table 4),

  2.

  $\rho =0.0853$ in $G_2$ (Table 5),

  3.

  $\rho =0.3018$ in $G_3$ (Table 6),

  4.

  $\rho =0.0547$ in $G_4$ (Table 7),

  5.

  $\rho =0.2310$ in $G_5$ (Table 8).

• Cold duplication with an imperfect switch for component $\left\{2\right\}$ raises $L\left(0.5\right)$ from $8.7066$ to $14.0715$ (Table 3). An equivalent increase in $L\left(0.8\right)$ occurs when the failure rate of component $\left\{2\right\}$ is reduced by

  1.

  $\rho =0.4782$ in $G_1$ (Table 4),

  2.

  $\rho =0.0721$ in $G_2$ (Table 5),

  3.

  $\rho =0.4256$ in $G_3$ (Table 6),

  4.

  $\rho =0.0755$ in $G_4$ (Table 7),

  5.

  $\rho =0.3702$ in $G_5$ (Table 8).

• Cold duplication of all three components $\{1,2,3\}$ increases $L\left(0.3\right)$ from $11.6630$ to $24.0035$ (Table 3). An equivalent gain in $L\left(0.8\right)$ can be obtained by lowering their failure rates by

  1.

  $\rho =0.5270$ in $G_1$ (Table 4),

  2.

  $\rho =0.0240$ in $G_2$ (Table 5),

  3.

  $\rho =0.8363$ in $G_3$ (Table 6),

  4.

  $\rho =0.2122$ in $G_4$ (Table 7),

  5.

  $\rho =0.7434$ in $G_5$ (Table 8).

• Consider the value $\rho =0.6454$ in Table 3 for $A=\left\{1\right\}$, $D=H$, $G_1=\left\{1\right\}$ at $\delta =0.8$. This indicates that adding a hot standby to component 1 achieves the same system reliability improvement as reducing component 1’s failure rate to $64.54\%$ of its original value (a $35.46\%$ reduction). An engineer can use this to decide whether improving the component’s inherent reliability or adding redundancy is more feasible given cost, weight, and complexity constraints.
• The notation “NE” (No Equivalence) signifies that for the given $\delta$ and strategy combination, no reduction factor $0<\rho <1$ can achieve system reliability equal to that provided by the duplication method. This highlights scenarios where redundancy is uniquely advantageous.
• The REF tables enable a quantitative cost-benefit analysis. A reliability engineer can compare the estimated cost and complexity of implementing a duplication scheme against the cost and feasibility of achieving the requisite component improvement (e.g., via higher-grade materials, improved manufacturing, or derating) indicated by the REF, thereby supporting optimal system design decisions.

7.5. Parameter Sensitivity Analysis

To assess the robustness and generalizability of the improvement strategies, a comprehensive parameter sensitivity analysis is conducted. This analysis examines the effects of varying both the component reliability parameters (${\theta }_i$) and the switch reliability parameter ($\alpha$) on system performance metrics, particularly the mean time to failure (MTTF).

• Variation in Component Reliability Parameters

Three distinct component parameter sets are defined to represent different inherent reliability levels:

• Low-Reliability Set: Components with higher hazard rates (${\theta }_1=2.5$, ${\theta }_2=3.0$, ${\theta }_3=3.5$).
• Medium-Reliability Set: Original baseline parameters (${\theta }_1=1.8$, ${\theta }_2=1.5$, ${\theta }_3=0.7$).
• High-Reliability Set: Components with lower hazard rates (${\theta }_1=0.5$, ${\theta }_2=0.8$, ${\theta }_3=1.0$).

Table 10. System MTTF for different component parameter sets under various improvement strategies (all improvements applied to $G_5=\{1,2,3\}$).

Component Parameter Set	Baseline MTTF	Hot Duplication	Cold (Perfect) Duplication	Cold (Imperfect, $\mathit{\alpha }=\mathbf{0.04}$)
Low-Reliability	0.6421	0.9427	1.4345	1.4321
Medium-Reliability	2.1944	3.1323	4.6240	4.6221
High-Reliability	7.8532	11.0246	15.8927	15.8893

presents the MTTF for the baseline system and for systems improved via hot duplication, cold perfect duplication, and cold imperfect duplication (with switch parameter $\alpha =0.04$) applied to component set $G_5=\{1,2,3\}$ across these three parameter sets.

The results in Table 10 demonstrate that the relative performance ranking of improvement strategies remains consistent across all component reliability levels: Cold (Perfect) > Cold (Imperfect) > Hot > Baseline. The absolute MTTF gains are larger for systems with higher inherent reliability, but the percentage improvements relative to the baseline are remarkably stable. For instance, cold perfect duplication yields approximately a 123–125% MTTF improvement over the baseline across all three parameter sets, confirming the robustness of this strategy’s superiority.

• Variation in Switch Parameter

The performance of the cold standby strategy with an imperfect switch depends critically on the reliability of the switching mechanism, characterized by the parameter $\alpha$ of its Akshaya distribution.

Table 11. Sensitivity of system MTTF to the switch parameter $\alpha$ for cold imperfect duplication ($G_5$) across different component parameter sets.

Component	MTTF for Cold Imperfect Duplication (${\mathit{G}}_5$)
Parameter Set	$\mathbf{\alpha }=\mathbf{0.01}$	$\mathbf{\alpha }=\mathbf{0.04}$	$\mathbf{\alpha }=\mathbf{0.50}$	$\mathbf{\alpha }=\mathbf{1.00}$
Low-Reliability	1.4343	1.4321	1.1083	0.8827
Medium-Reliability	4.6238	4.6221	3.4562	2.7418
High-Reliability	15.8921	15.8893	11.7245	9.2576
Perfect Switch Limit	MTTF for Cold Perfect Duplication ($G_5$)
Low-Reliability	1.4345
Medium-Reliability	4.6240
High-Reliability	15.8927

extends the analysis by examining the MTTF under the cold imperfect duplication strategy for the component set $G_5$ across the full range of component parameter sets and for varying $\alpha$ values.

Table 11 reveals several important insights:

1.

As $\alpha$ increases (the switch becomes less reliable), the MTTF decreases monotonically for all component sets.

2.

The performance degradation is relative to the perfect switch limit; for $\alpha =0.01$, the MTTF is nearly identical to the perfect switch case across all component sets.

3.

The absolute impact of switch unreliability is more pronounced in systems with higher inherent component reliability. However, the percentage degradation relative to the perfect switch case is similar across component sets (e.g., approximately 40% reduction when $\alpha =1.00$).

4.

For poorly reliable switches ($\alpha \ge 0.50$), the advantage of cold imperfect duplication diminishes significantly, approaching the performance of hot duplication or even the baseline system for very high $\alpha$.

Practical Implications: This sensitivity analysis underscores two critical design principles: (1) the superiority of cold standby strategies is robust across different component reliability levels, and (2) the benefits of cold standby can be severely compromised by a low-quality switch. Therefore, system designers must ensure that switch reliability is commensurate with the target system performance; otherwise, the investment in cold standby redundancy may not yield the expected returns. These findings provide quantitative guidance for specifying switch reliability requirements during system design.

8. Conclusions and Future Work

This work analyzed the reliability characteristics of a parallel-series system in which all components follow the Akshaya lifetime distribution. Four different reliability enhancement techniques were formulated and evaluated: hot duplication, cold duplication with a perfect switch, cold duplication with an imperfect switch, and failure-rate reduction. Closed-form expressions for key reliability metrics, including system reliability, mean time to failure (MTTF), $\delta$-fractiles, and reliability equivalence factors (REFs), were derived for each strategy.

The comparative analysis, supported by a detailed numerical example, consistently demonstrated a clear performance ranking among the duplication strategies: cold duplication with a perfect switch yielded the highest reliability improvement, followed by cold duplication with an imperfect switch, and then hot duplication. This ranking can be critically explained by the fundamental operational principles of each method:

• Cold duplication (perfect switch) ensures that the standby component does not degrade until activation, effectively offering a “fresh” component upon failure of the primary. This eliminates simultaneous wear and maximizes the system’s lifetime, provided the switch is perfectly reliable.
• Cold duplication (imperfect switch) incorporates the risk of switch failure during activation. While the standby component itself remains pristine, the non-zero probability of switch failure introduces an additional point of potential system failure, thus slightly diminishing the overall reliability gain compared to a perfect switch.
• Hot duplication operates with both the primary and standby components active and subject to wear from time zero. This concurrent exposure to operational stresses leads to a higher cumulative failure rate over time, making it less effective than cold standby strategies in scenarios where the standby component’s dormant state prevents degradation.

The extensive tables of reliability equivalence factors (REFs) provide a quantitative bridge between the failure-rate reduction method and the three duplication strategies. The REF values, which generally decrease as the target reliability level $\delta$ increases, indicate that achieving high system reliability through component improvement alone often requires disproportionately large reductions in failure rates. The frequent occurrence of “No Equivalence” (NE) entries, particularly for higher $\delta$ values or when comparing reduction to more effective duplication methods, underscores the inherent limitation of the reduction strategy and highlights scenarios where redundancy is indispensable for achieving stringent reliability targets.

The choice of the Akshaya distribution, with its flexible hazard rate shape, allowed for the realistic modeling of component degradation. The observed results are therefore particularly relevant for systems whose components exhibit similar non-constant failure rates. However, the general methodological framework—deriving REFs to compare enhancement strategies—remains applicable to other lifetime distributions.

In summary, this study provides a comprehensive analytical toolkit for reliability engineers designing parallel–series systems. The critical insight is that while all strategies improve performance, cold standby redundancy with a highly reliable switch is the most potent option for significantly extending system life. The REFs offer a practical decision-making tool, enabling designers to weigh the cost and complexity of adding redundancy against the feasibility of achieving equivalent gains through component-level reliability enhancement. Future work could extend this analysis to systems with dependent component failures or explore optimization models that incorporate economic costs alongside these reliability metrics.

• Future Work: While this study provides a general analytical framework, several directions merit further investigation.

• Application to Case Studies: Applying this framework to a specific real-world system (e.g., a power electronics module or communication network) using field-fitted Akshaya parameters would demonstrate the complete workflow from parameter estimation to strategy selection.
• Incorporating Dependence: Extending the model to account for dependent component failures, perhaps using copula-based approaches, would increase its applicability to systems where components share loads or environmental stresses.
• Economic Optimization: Integrating cost models with the reliability metrics to solve optimization problems that balance system performance against economic constraints (e.g., minimizing total cost subject to a reliability target) would be a valuable practical extension.
• Alternative Distributions: Investigating the performance of the same improvement strategies under other flexible lifetime distributions (e.g., exponentiated Weibull, generalized Lindley) would help generalize the comparative insights.
• Multi-State and Degradation Models: Extending the analysis to multi-state systems or systems where component degradation follows a stochastic process could address more complex real-world scenarios.

In summary, this work offers reliability engineers a comprehensive toolkit for analyzing and improving hybrid parallel–series systems. The derived formulas and comparative metrics facilitate informed design trade-offs, supporting the development of more reliable and cost-effective systems.