Modelling and evaluation of 1-out-of-3 system reliability using extension of bivariate Freund distribution

Introduction

The dependability of a system is determined not solely by the reliability of its constituent components but also by the configuration of the system, the interdependence among components and the characteristics of their failure time distributions. Among the numerous configurations, the 𝑠-out-of-𝑘 system (𝑠 ≤ 𝑘) serves a pivotal role in augmenting system reliability, particularly within the realm of reliability engineering applications. This model serves to generalize the two fundamental types of systems: the parallel system (when 𝑠 = 1) and the series system (when 𝑠 = 𝑘). Consequently, the 𝑠-out-of-𝑘 configuration offers a versatile framework that reconciles the dichotomies of redundancy and susceptibility in the design of systems. Because, of this flexibility it is widely suitable in various practical scenarios, especially in mission-critical and fault-tolerant applications such as aerospace, communication networks, power systems and cloud computing where redundancy is a key design feature. But it is also true that the workload or pressure on the functioning components will change due to failure of components in the system. So, there will be change in the hazard rate of the functioning components. For instance, in cloud computing environments, data centers often rely on redundant server clusters to ensure high availability and fault tolerance. When a server within a cluster fails, the computational workload and network traffic are redistributed among the remaining active servers. This redistribution of load can lead to increased Central Processing Unit (CPU) utilization, memory consumption and thermal stress, which in turn elevates the failure probability of the surviving servers. Such stress-induced dependence among server lifetimes necessitates the use of models like the Freund distribution, which explicitly accounts for failure rate transitions due to component interdependence. Thus, despite the presence of redundancy, the reliability of the system may deteriorate more rapidly following failures. Freund (1961) [1] has proposed a bivariate exponential model capturing dependent lifetimes and has applied it to real-world systems like engine failures and paired organs. Extensive work has been done using Frend bivariate exponential distribution, and few of them are: Boardman (1968) [2] has developed MLE methods for mixed exponential distributions in life testing. Block and Basu (1974) [3] have introduced the ACBVE model to generalize Freund's distribution with continuous dependency, provided estimation methods and highlighted its practical relevance. Ross (1984) [4] has modeled failure rates that adjust based on the current working set of components, derived system failure distributions and explored repair-based reversibility. Hanagal (1992) [5] has extended Freund's model with large-sample inference for independence and symmetry, confidence intervals under censoring schemes are also obtained. Kunchur and Munoli (1994) [6] have estimated reliability in Freund-based two-component systems. Kim and Kvam (2004) [7] have developed MLEs for systems with unknown load-sharing rules. Kvam and Peña (2005) [8] have proposed a semi-parametric estimator for dynamic loadsharing systems. Park (2010) [9] has derived MLEs and BUEs under exponential and Weibull load-sharing systems. Gurler et al. (2015) [10] have studied mean remaining strength in dependent-component systems under stress. Asha et al. (2018) [11] have modeled load-sharing systems with frailty using bivariate Weibull and positive stable distributions. Zhao et al. (2018) [12] have built a continuous degradation model for load-sharing systems using log-linear stress-degradation links. Katagi and Munoli (2025) [13] have assessed 2-out-of-3 system reliability using Freund's model with MLE and Bayesian methods. Kundu (2025) [14] has proposed a bivariate loadsharing model for dependent failures with ties.

In the present study 1-out-of-3 system, which continues to function until the failure of its last component is considered and analyzed in the context of Freund model. The model formulation is detailed in Section 2. Section 3 provides an analysis of the reliability function, hazard rate and survival function. Parameter estimation based on life testing and simulation studies are presented in Sections 4 and 5, respectively. The concluding remarks are outlined in Section 6.

Model Description

Consider a system of three components arranged in a 1-outof-3 configuration, that is the system functions as long as at least one component is operational. In such a configuration, as individual components experience failure sequentially, the parameter of the life distribution of the remaining operational components will change (without any form of replacement or repair). Specifically, let the lifetimes of the system's components be independent exponential random variables with a common failure rate parameter 𝜆. Upon the failure of any one of the three components, the failure rate of the two remaining components changes to 𝜆 1 . Once the second failure occurs, the failure rate of the single surviving component changes to 𝜆 2 . Depending on the application, the relationship among the failure rates may vary. For example, the condition 𝜆 < 𝜆 1 < 𝜆 2 is typically observed in reliability engineering, where increased stress accelerates component degradation. In contrast, the condition 𝜆 > 𝜆 1 > 𝜆 2 may arise in ecological or competing species scenarios, where the failure (extinction) of one species reduces competition and thereby enhances the survival of others.

Reliability function, hazard rate and mean residual life 3.1 Reliability Function

Reliability of 1-out-of-3 system with above property of changing life distribution upon subsequent failures at time 𝑡 is obtained as below

The first term on the right-hand side corresponds to the event that all three components of the system have survived throughout the interval (0, 𝑡). The second term represents the scenario in which two out of the three components survive until time 𝑡, where the first failure occurs at time 𝑡 1 . The remaining two components operate with failure rate 𝜆 up to time 𝑡 1 , and subsequently with failure rate 𝜆 1 . The third term accounts for the case where only one component survives until time 𝑡, following both the first and second failures. The components initially function with rate 𝜆, transition to rate 𝜆 1 after the first failure at 𝑡 1 , and finally last component operates under rate 𝜆 2 after the second failure at 𝑡 2 . Representing (1) as

Substituting 𝐼 1 and 𝐼 2 of ( 3) and ( 4) into the Equation ( 2) and simplifying, the closed form expression for system reliability is obtained as

𝑅(𝑡) is well-defined, when 3𝜆 ≠ 2𝜆 1 , 𝜆 2 ≠ 2𝜆 1 and 3𝜆 ≠ 𝜆 2 and satisfies the boundary conditions 𝑅(0) = 1, 𝑅(∞) = 0 with 𝑅(𝑡) being non-increasing in 𝑡.

Hazard Rate

Hazard rate is the probability that a unit that has functioned without failure up to the instant 𝑡 will fail in the interval (𝑡, 𝑡 + ∆𝑡) and is denoted by ℎ(𝑡)

With 𝑓(𝑡) being probability density function (PDF) of the system lifetime, which is given by -𝑑𝑅(𝑡)

𝑑𝑡

which is valid under the same conditions imposed for Equation (5), ensuring that all denominators of Equation ( 7) are non-zero.

Hazard rate of the model is derived as:-

The behavior of the ℎ(𝑡) is analyzed under two distinct sets of parameter values, and the corresponding results are illustrated graphically in Figure 1.

Mean Residual Life

The mean residual life (𝑀𝑅𝐿) at time 𝑡, denoted by 𝑀𝑅𝐿(𝑡), represents the expected remaining lifetime of the system, conditioned on survival up to time 𝑡. It is given by

Let the integrated reliability function be 𝑅 * (𝑡) = ∫ 𝑅(𝑥) 𝑑𝑥 ∞ 𝑡 .

Using the reliability function 𝑹(𝒙) from Equation ( 5), the expression for 𝑹 * (𝒕) is obtained as follows:-

Thus, the 𝑴𝑹𝑳 function reduces to:-

The variation in 𝑀𝑅𝐿 is analyzed for two distinct parameter sets and presented graphically in Figure 1.

Estimation of Parameters by Life Testing.

A system with the features described in Section 2 is set into life testing and is observed until all components have failed.

Let 𝑈, 𝑉 and 𝑊 respectively denote the times at which the first, second and third failures occur.  The probability that the first failure occurs at time 𝑈 = 𝑢 and other two components remain operational is given by 𝜆𝑒 -𝜆𝑢 𝑒 -𝜆𝑢 𝑒 -𝜆𝑢 = 𝜆𝑒 -3𝜆𝑢  The probability that the second failure takes place at time 𝑉 = 𝑣, with the one component still functioning, given that the first failure occurred at 𝑈 = 𝑢 is given by 𝜆 1 𝑒 -𝜆 1 (𝑣-𝑢) 𝑒 -𝜆 1 (𝑣-𝑢) = 𝜆 1 𝑒 -2𝜆 1 (𝑣-𝑢)  The probability that the third failure is observed at time 𝑊 = 𝑤, given that the first and second failures respectively occurred at times 𝑈 = 𝑢, 𝑉 = 𝑣 is given by 𝜆 2 𝑒 -𝜆 2 (𝑤-𝑣)

The joint pdf of 𝑼, 𝑽 and 𝑾 is as follows 𝑓 𝑈,𝑉,𝑊 (𝑢, 𝑣, 𝑤) = 3! 𝜆𝑒 -3𝜆𝑢 𝜆 1 𝑒 -2𝜆 1 (𝑣-𝑢) 𝜆 2 𝑒 -𝜆 2 (𝑤-𝑣) , 0 < 𝑢 < 𝑣 < 𝑤 < ∞.

For all values of 𝜆, 𝜆 1 and 𝜆 2 defined on the interval (0, ∞) and 0 < 𝑢 < 𝑣 < 𝑤 < ∞, the Equation ( 12) is non-negative and satisfies the condition

When 𝑛 such systems subjected to life test and joint likelihood function is given by 𝐿(𝜆, 𝜆 1 , 𝜆 2 ) = ∏ 𝑓 𝑈,𝑉,𝑊 (𝑢 𝑖 , 𝑣 𝑖 , 𝑤 𝑖 )

Using the natural logarithm function to both sides of the Equation ( 13), followed by computing the partial derivatives concerning each parameter, and equating these derivatives to zero, the maximum likelihood estimators (MLE) of 𝜆, 𝜆 1 , 𝜆 2 are:

, 𝜆 ̂2 = As MLEs hold invariance property, the MLE 𝑹 ̂(𝒕) of 𝑹(𝒕) is obtained as

Simulation Study

The failure times of the components of the system are generated synthetically in order to estimate the MLEs of parameters.

 2 is that for any fixed 𝑡 the mean absolute bias of 𝑅 ̂(𝑡) decreases as the 𝑛 increases. This is expected, as larger values of 𝑛 typically result in improved estimation precision due to reduced sampling variability.

Conclusions

This study presented a reliability analysis of a 1-out-of-3 system using the Freund bivariate exponential model, which incorporates dependency among component lifetimes resulting from progressive system stress. The expressions for the reliability function, hazard rate and mean residual life (MRL) were derived and examined under two distinct sets of failure rates: increasing and decreasing. The analysis revealed that while the reliability function consistently decreases over time, the behavior of the hazard rate and MRL depends significantly on the pattern of failure rate progression. In particular, the hazard rate increases and MRL decreases in the case of increasing failure rates, whereas the reverse trend is observed when the failure rates decrease, demonstrating the sensitivity of system behavior to the underlying stress dynamics.

Parameter estimation was carried out using maximum likelihood estimation based on data obtained from life testing of the system. A simulation study was conducted to evaluate the performance of the estimators, with results showing that the mean absolute bias decreases as sample size increases, indicating estimator consistency and precision. The findings validate the applicability of the Freund model in load-sharing environments and highlight its relevance in systems where operational stress dynamically alters component failure behavior. This framework offers practical insights for engineers and reliability analysts in designing and evaluating fault-tolerant systems across domains such as mechanical engineering, communication networks and cloud computing infrastructure.

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