Reliability Estimation for the Exponential-Pareto Hybrid System

The reliability of hybrid systems is important in modern technology, specifically in engineering and industrial fields; it is an indicator of the machine's efficiency and ability to operate without interruption for an extended period of time. It also allows for the evaluation of machines and equipment for planning and future development. This study looked at reliability of hybrid (parallel series) systems with asymmetric components using exponential and Pareto distributions. Several simulation experiments were performed to estimate the reliability function of these systems using the Maximum Likelihood method (ML) and the Standard Bayes method (SB) with a quadratic loss (QL) function and two priors: non-informative (Jeffery) and informative (Conjugate). Different sample sizes and parameter values are used in these simulation experiments, and the Mean Squared Error (MSE) was used to compare those experiments. The simulation results showed that the standard Bayes method with Conjugate loss function is better than the results from the maximum likelihood method.


Introduction
The interest in reliability studies in modern research on electronic devices and equipment has increased in the last century.Furthermore, rapid technological advancements and the use of complex systems in various areas of life have increased in this field.As failure to operate results in material losses and decreased production, there are many causes for the interruptions in various types of equipment and machinery.Few studies have been done in this area, and they have all focused on the reliability of simple systems composed of symmetric or asymmetric components with sequential-parallel reliability systems, despite the fact that there are many sources of production composed of complex, manifold, and non-complex work systems.Ali and Woo [1] performed studies in this field using maximum likelihood estimates of the threshold parameter with known parameters for the exponentiated Pareto distribution.Rahim [1] investigated the capabilities of series parallel hybrid reliability systems for carpet production plant machines using the Bayes method.Karam [2] investigated the posterior analysis of five exponentiated distributions (Weibull, Exponential, Inverted Weibull, Pareto, and Gumbel) with different prior distributions and loss functions.Al-Saady [3] modelled the reliability function of the series-parallel hybrid system in the case of asymmetric components.Shafay [4] considered the general form of the underlying distribution and the general conjugate prior, and developed a general procedure for deriving maximum likelihood and Bayesian estimators from an observed generalized Type-II hybrid censored sample.For modelling lifetime data, Maiti and Pramanik [5] proposed a new distribution called the Odd Generalized Exponential-Pareto distribution.Amin [6] discussed the upper record value distribution when the parent distribution is exponential Pareto.Akomolafe, Oladejo, Bello, and Ajiboye [7] looked at some results that describe the generalization of the exponential Pareto and negative Binomial distributions based on their distribution functions and asymptotic properties.Baharith et al. [8] studied the statistical properties of the odd exponential-Pareto IV distribution, which is a member of the odd family of distributions and characterized by decreasing, increasing, and upside down hazard functions.All of these attempts aided in the motivation for this work.
The aim of this study is to provide a reliability analysis of parallel-series systems with asymmetric components, representing the exponential and Pareto distributions.To accomplish this aim, simulation experiments carried out to estimate the reliability function of asymmetric hybrid systems using the maximum likelihood (ML) method and the standard Bayes method with loss function.Section 3 discusses reliability systems, while Section 4 discusses simulation results.

THE EXPONENTIAL AND PARETO DISTRIBUTIONS
The exponential distribution is an effective continuous distribution that is widely used in failure and survival time studies.The probability density function (pdf) for an exponential random variable with a scale parameter is as follows: [9], [10] (; ) =   −  ; ,  > 0 (1) The reliability function can be defined as: (; ) =  −  (2) The Pareto distribution parameter is a continuous and it's used in a variety of applications, particularly in economics.The probability and reliability functions of the random variable (S) for one parameter of the Pareto distribution are as follows: [11], [12] (; ) =   −( +1) ;  ≥ 0 ,  > 0 (3)

Maximum Likelihood (𝐌𝐋)Estimation
It is possible to explain parameter estimation (α) using the Maximum Likelihood (ML).For an independent random sample (t 1 , t 2 , … , t n ) which is drawn from an exponential distributed random variable, the likelihood function of the observations can be expressed as: Then, ML estimator for the parameter α is: Similarly, the MLE for a parameter φ from a Pareto independent random sample (s 1 , s 2 , … , s n ) can be calculated as follows: The reliability function for Exponential and Pareto distributions can then be estimated using the ML method by substituting equation ( 6) in equation ( 2), and equation ( 8) in equation ( 4) as follows: [13], [14]  ̂( ̂(

Standard Bayes (𝐒𝐁) Estimation
The Bayes method assumes that the estimated parameters are random variables.These parameters represented in a p.d.f.known as the prior distribution, and then a p.d.f.known as the posterior distribution can be obtained by combining the likelihood function and this prior using the Bayes inversion formula.By using a random sample (v 1 , v 2 , … , v n ) of size n with p.d.f f(v; β), the inversion formula is: [13], [15], [16] Where: is the Prior distribution, and π(β|V) is the posterior distribution.
Applying the formula (11) to Exponential and Pareto distributions respectively, we get:

Prior Distributions
There are several forms of prior distributions, in the absence of sufficient primary information on the estimated parameter, or when none is available, the prior distribution is referred to as a little informative prior.When information from previous experiments about this parameter is available, the prior distribution is known as the conjugate prior or informative prior.[13], [10], [15], [16]

Jeffery Prior
If there is insufficient or no information about the parameter, the prior distribution can be calculated using Jefferys' formula, which is dependent on the parameter's zone.Given that the parameter field is (0, ∞), the prior distribution based on fisher information I(β), which is stated as follows: [17], [15], [16] The Jeffery prior is obtained by substituting () to the equation ( 14): Applying the prior distribution based on the fisher information I() to the exponential distribution gives the following: , ℎ   ( 16) Substituting equation (17) into equation ( 16), obtaining: And for the Pareto distribution, the following results are obtained: Equation ( 21) is obtained by substituting equation (20) into equation (19).

Systems Reliability
According to the definition of system reliability, it is the probability that a system will continue to function properly after a specified period of time () or that it won't fail during the period [0, ].A number of components make up the system, and the nature of the connections between those components determines the importance of the system's reliability.It is important to know the behaviour pattern of the components and their impact on the system behaviour.

Types of Systems
Systems can be classified into four types: a) Series System: The components in this system are connected respectively, and the failure of any component causes the entire system to fail.If the system has (m) components (devices), then the reliability of the series system is expressed mathematically as follows: b) Parallel System: A system in which the components are interconnected so that a failure does not cause the system to fail.The mathematical formula for a parallel reliability is: (36) c) Hybrid system: A system contains many partial systems and each partial system contains many compounds within each partial system, and a system made up of several partial systems connected in series or parallel, and within each partial system.The number of compounds that are linked in series or parallel.The series-parallel hybrid system, on the other hand, will be investigated in this study.d) Hybrid (Parallel-Series) System: This hybrid system is made up of (l) partial systems connected in series, and within each partial system, there are (m) partial systems that are connected in parallel.The system continues to function even when only one component is active, whereas the system ceases to function if one of the partial systems fails and figure 1 shows an example of this system in action.2004).When the system consists of two series interconnected partial systems and each partial system consists of two components; the first follows an exponential distribution with parameter (α), and the second follows a Pareto distribution with parameter (φ), then the reliability function of each component in the partial system is defined by equations ( 2) and ( 4) respectively, and the reliability function of the partial system according to equation ( 37) is obtained as: The reliability function of the parallel-series hybrid system based on equation (36) would be:

Reliability estimators for the parallel-series hybrid system
The ML method is used to estimate the reliability function of the parallel-series hybrid system by substituting equations ( 9) and ( 10) in (38) as follows: And the reliability function of the parallel-series hybrid system using the Bayes method with Jeffery prior and quadratic loss function is obtained by substituting equations ( 31

Simulations and Results
Monte Carlo Simulation Experiments are used to estimate the reliability function of the seriesparallel hybrid system and are based on the R 3.5.1 program as follows: [19], [20] I.Select the default values for the parameters of the Exponential and the Pareto distributions, as well as the parameters of the natural accompanying function as shown in Table 1.Table 2 displays the simulation results for estimating and comparing the parallel-series hybrid system reliability function estimators by MLE method and Bayes method with the quadratic loss function and two priors.Standard Bayes method with the Jeffery prior, and then the ML method, whereas the MSE converge when the sample size is increased, also the MSE for the reliability estimators reduces as the parameter and time values increases.

Table 1 :
PARAMETER VALUES OF SIMULATION EXPERIMENTS

Table 2 :
Reliability Estimators of (Parallel-Series) Hybrid System and MSE