Research article · 2019 · Full text
International Journal of Statistics and Applied MathematicsReliability computation of system reliability of a stress-strength model with new Rayleigh Pareto distribution
Volume 4(5) · Pages 112–116
Research areasProbabilistic and Robust Engineering DesignStatistical Distribution Estimation and ApplicationsAdvanced Multi-Objective Optimization Algorithms
Abstract
Author-writtenIn this paper presents the reliability computation of system reliability when the applied stress and strength follows the New Rayleigh Pareto distribution. The New Rayleigh Pareto Distribution (NRPD) is considered as a simple model to make component reliability and may place a good fit for failure data and also provide more appropriate information about hazard rate. The results may be applied to semiconductor devices. Maximum likelihood estimator are used here.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Modern society relies heavily on the continuous operation of complex machinery and equipment for safety, security, and economic stability. View evidence ↗
- Statistical distributions are standard tools used to assess the reliability of products and semiconductor devices. View evidence ↗
- Engineers often prefer simple models, like the exponential distribution, over more complex ones like Weibull or lognormal to obtain failure rates and reliability results more quickly. View evidence ↗
The question
- The study aims to compute the reliability of a system where both the applied stress and the strength follow the New Rayleigh Pareto distribution (NRPD). View evidence ↗
What they did
- The authors use Maximum Likelihood Estimators (MLE) to estimate the parameters of the distribution. View evidence ↗
- The research employs a stress-strength model where a component fails when the random stress (X) exceeds its random strength (Y). View evidence ↗
- The study considers an r-out-of-n system, which remains operational as long as at least r of its n components are functioning. View evidence ↗
What they found
- In the specific case where r=n, the system is a series system, meaning it only functions if every single component is operational. View evidence ↗
- In the case where r=1, the system is a parallel system, which only fails if all n components fail. View evidence ↗
What it means
- The authors conclude that the NRPD is superior to the exponential distribution because its hazard rate is not constant, making it more useful for forecasting early-stage component life. View evidence ↗
- The authors recommend using the New Rayleigh Pareto distribution when analyzing failure data for electronic devices. View evidence ↗
Visual summary
1 visualsIllustrations explain the paper; they do not show study data.
Stress-Strength Failure Concept
AI-generated conceptual illustration (a mechanical steel bolt under tension). It explains the idea and does not show study data.
View source evidence ↗Introduction
Many lifetime data used for statistical analysis follows a particular statistical distribution. Recent days technological world nearly everyone depends on the continued functioning a broad collection of complex machinery and equipment for our every day to day life security, safety, ability to move easily and economic welfare. We expect our electronic devices, lights, hospital monitoring control, next generation aircraft, nuclear power plants ect., to function whenever we need them. It will be fail, the results can be disastrous illness or even loss of life. Statistical distributions have long been employed in the assessment of semiconductor device and product reliability. The use of the exponential distribution which more preferred over mathematically very difficult distribution, such as the Weibull and the lognormal among others, suggest that most of the engineers interest the application of simple model to find out failure rates and reliability results quickly. It is therefore proposed that the new Rayleigh Pareto distribution be considered as a simpler alternative which in some situations , may place a good fit for failure data and provide more appropriate information about reliability and hazard rates. The New Rayleigh Pareto distribution is also used to appropriate representation of the lower tail of the distribution of random variable having fixed lower bound. In the concept of 'strength-reliability', the stress-strength model describes the life of a component, which has a random strength Y and is subjected to a random stress X . The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfies whenever X Y . Thus, X Y is a measure of component reliability. It has number of applications in engineering concepts, deterioration of rocket motors ect., the reliability estimation of a single component stress-strength version has been considered by several authors assuming various lifetime distributions for the stressstrength random variates. Enis and Geisser (1971) [6] , Downtown (1973) [4] , Awad and Gharraf (1986) [1] , McCool (1991) [9] , Nandi and Aich (1994) [11] , Kundu and Gupta (2005, 2006) [7,8] Arulmozhi (2003) [6] . In this paper we compute the reliability of a system's reliability when the applied stress and strength follow the probability distribution of New Rayleigh Pareto distribution.
The model
Let us consider the New Rayleigh Pareto distribution with probability density function (pdf)
The cumulative distribution function (cdf )of the NRPD is given by
The theoretical mean and variance are derived as;
Variance
The corresponding survival function is
be a random sample of size n . We suppose that the lifetime of these components follows the New Rayleigh Pareto distribution and then the likelihood function is given by
, partially differentiate with respect to and equating to zero and partially differentiate with respect to and equating to zero , partially differentiate with respect to and equating to zero respectively; 0 ln 2
From ( 5) we obtain MLE of as a function of is given by
Reliability computation
Let Y represents the strength of an item with density function
are the scale parameter and 1 is the shape parameter and the probability density function of stress X is given by 0 If X and Y are independent the probability that Y (strength), X (stress) then the reliability of the component is given by
x y e e y x (8) Using (8) , we have dy e e y R y y
On simplification , the reliability of the system is
Example
This system is operational if and only if at least r components out of the system's n components are operational. An example might be a radar network having n radar control stations which can be cover a certain area under control and only if at least r of its service stations are operational. The structure of the system function is given by
A particular case of this system when r=n is called a series system. It is operational if and only if all of its elements are above. One can verify easily that the structure function for a series system can be represented as
Here we can observe that of a series system as an information transmission system having n "stations" in which the information is transmitted properly only if all the stations are operational. Other extreme case of r-out-of-n-system is r=1. Such a system is known as parallel system. We can observe that a parallel system as a system with one main operating unit and (n-1) extra units in stand by. The system is down if and only iff all n units are not operational. The structure of the system function for a parallel system is given by
The procedure for considering reliability system when applied the stress-strength follows the New Rayleigh Pareto distribution may be used as a single model to assess component reliability. The advantage or merit of this distribution is that the hazard rate, as opposed to the exponential distribution's is not be a constant and may be more useful in forecasting the original stages of component's life. It is suggested that this distribution be given consideration when analyzing the electronic devices failure data.
References
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From OpenAlex, OpenCitations and reference lists read by Scinovex
- On Estimating Reliability of a Stress – Strength Model in Case of Rayleigh Pareto Distribution ↗
Emad Sh. M. Haddad, Feras Sh. M. Batah · Iraqi Journal of Science · 2021
The stress – strength model is one of the models that are used to compute reliability. In this paper, we derived mathematical formulas for the reliability of the stress – strength model that follows Rayleigh Pareto (Rayl. – Par) distribution. Here, the model has a single component, where strength Y is subjected to a stress X, represented by moment, reliability…
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