Computation of Reliability and Bayesian Analysis of System Reliability for Mukherjee Islam Failure Model

The Mukherjee Islam Failure Model is considered as a simple model to assess component reliability and may exhibit a better fit for failure data and also provide more appropriate information about hazard rate. This paper presents the reliability computation and Bayesian estimation of system reliability when the applied stress and strength follows the Mukherjee Islam Failure Model. The results obtained in this paper may be applied to semiconductor devices. The main objective of this paper is to compute reliability and Bayesian analysis of system reliab ility. Maximum likelihood estimator and Bayesian methods are used inside the paper.


Introduction
Reliab ility is a broad term that focuses on the ability of a product to perform its intended function. The word "reliability" refers to the ability of a system to perform its stated purpose adequately for a specified period of time under the operational conditions encountered. A system is said to be absolutely reliable if so me undesirable events, called failures, do not occur in the system's operation. Every system has its own set of such undesirable events. The system defined here could be an electronic or mechanical hardware product, a software product, a manufacturing process or even a service. Fo r example, for a mechanical system, a failure is a breakdown of some of its parts or an increase in v ibration above the permitted level. One o f the most dangerous failures of a nuclear reactor is a leak of radioactive material. The reliab ility characteristics are usually expressed in terms of the lifetime. The lifetime is a random time fro m the beginning of the operation until the appearance of a failure, after wh ich further operation is impossible.
The concept of reliab ility is used in a variety of business and industrial settings. In general, the concept of reliability is applied where it is important to achieve the same results over and over again. A manufactu ring process is said to be reliab le when it ach ieves the same result, with in defin ite limits, each time it occurs. An automobile, or other type of product, is reliable if it performs consistently and up to expectations.
Statistical distributions have long been employed in the assessment of semiconductor device and product reliability. The use of the exponential distribution which is frequently preferred over mathemat ically mo re co mp lex d istribution, such as the Weibull and the lognormal among others, suggest that most engineers favor the application of simpler model to obtain failu re rat es and reliab ility figu res qu ickly. It is therefore proposed that the Mukherjee Islam Failu re Model be consid ered as a simp ler altern ative wh ich, in so me circu mstances, may exh ibit a better fit for failure data and provide more appropriate information about reliability and hazard rates. The Mukherjee Islam Failu re model is also used fo r app rop riat e rep resentat ion o f the lo wer tail of t he distribution of random variab le having fixed lower bound.
In the context 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 satisfactorily whenever Y X > . Thus, ( ) Y X > is a measure of co mponent reliability. It has many applications in engineering concepts, deterioration of rocket motors, static fatigue of ceramic co mponents, fatigue failure of aircraft structure and the aging of concrete pressure vessels. Maroof and Islam 1 consider the problem of Bayesian estimation of system reliab ility using the Lo max model. Arulmo zh i 2 gives a simp le and efficient computational method for determining the system reliability of k-out-of-n systems having unequal and equal reliabilities for co mponents. Kucheryavyi and Mil'kov 6 consider the problem of strength reliab ility computation for a beam section of a main gas pipeline.
In this paper we compute the reliability and also obtain the Bayesian estimation of a system's reliability when the for M ukherjee Islam Failure M odel applied stress and strength follow the probability distribution of Mukherjee Islam Failure.

The Model
Let us consider the Mukherjee Islam Failure model with probability density function (pdf) ( ) and the cumulative distribution function (cdf) where θ is the scale parameter and p is the shape parameter. The above distribution main ly appears as the inverse distribution of the Pareto distribution.
The theoretical mean ( ) µ and variance ( ) Time to failure is given by Integrating out by substituting ,

Reliability Computation
Let Y represents the strength of an item with density function ( ) If X and Y independent then probability that Y exceeds X is given by Using eq.(7), we have

Re mark
If stress and strength have same scale parameter θ , then it shows that in defining the life of a co mponent initial value of θ doesn't matter.

Bayesian Analysis of System Reliability
Life time data are impo rtant in reliab ility analysis classical reliability estimation is based on precise lifetime data. Bayesian method has proved to be very useful when the sample size is small. Hierarch ical models are one of the central tools of Bayesian analysis. They after many advantages including the ability to borrow strength to estimate individual parameters and the ability to specify complex models that reflect engineering and physical realities. Broadly, Bayesian models have two parts, first the likelihood function and the other is prior distribution. The likelihood function is constructed from the sampling distribution of the data which describe the probability of observing the data before the experiment is performed. The prior d istribution describes the uncertainty about the parameters of the likelihood function. The prior distribution is then updated to the posterior distribution after observing the data.
Bayesian approach helps the reliability engineers and participators to incorporate a wide variety of forms of informat ion in the estimat ion process and the uncertainties in the parameters due to the lack of knowledge are exp ressed via probability distribution. This is the majo r departure fro m the classical methods, since for classical methods all the parameters are true unknown values. There are uncertain parameters being estimated. In Bayesian Analysis the starting point is the choice of priors. The conjugate and the uniform priors are taken as solution commonly. As a rule, conjugate priors lead to straight forward mathematical calculations and because of this reason it has been applied for reliability estimation.