# Estimation of Reliability for a Two Component Survival Stress-Strength Model

Authors: S. B. Munoli; Rohit R. Mutkekar
Journal: International Journal of Quality, Statistics, and Reliability
 (2011)

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## Abstract

The reliability function for a parallel system of two identical components is derived from a stress-strength model, where failure of one component increases the stress on the surviving component of the system. The Maximum Likelihood Estimators of parameters and their asymptotic distribution are obtained. Further the Maximum Likelihood Estimator and Bayes Estimator of reliability function are obtained using the data from a life-testing experiment. Computation of estimators is illustrated through simulation study.


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## Body

## 1. Introduction

Several authors have considered estimation of system reliability based on stress-strength models. Here are a few references of contributions towards these models. Church and Harris [1] considered estimation of reliability from stress-strength relationship. Downton [2] considered the case of estimation of reliability for a stress-strength model under normal distribution. Wani and Kabe [3] have considered the problem of estimation of system reliability where life time of each component has gamma distribution. Constantine et al. [4] considered estimation of stress-strength relationship under the assumption of stress-strength random variables following gamma distribution with known shape parameters. Bhattacharya and Johnson [5] have considered estimation of reliability in a multicomponent stress-strength model. Kunchur and Munoli [6] have considered estimation of reliability for a multicomponent survival stress-strength model based on exponential distribution. Dan and Krausz [7] have obtained inference for a multistep stress-strength model of parallel system. Kunchur and Munoli [8] have considered estimation of reliability in Freund’s model for a two component system. Hanagal [9] has considered the problem of estimation of system reliability in a two component stress-strength model with cases on distribution of stress as exponential and gamma. Kundu and Gupta [10] have obtained inference of stress-strength relationship for generalized exponential scale family of distribution. Bhattacharya [11] has proposed Bayesian approach to life-testing and reliability estimation.  Draper and Guttman [12] have obtained the Bayes estimator of reliability in a multicomponent stress-strength model.In the present study, we are considering a system of two components. The component survives as long as the stress on it is smaller than its strength. The system survives if at least one component functions (parallel system). Here the stress and strength associated with the components of the system are random variables. To carry out the inference, we assume certain probability distribution for these random variables. Let the strength of the two components beX1 and X2, where X1, X2 are independently and identically distributed gamma random variables with shape parameter γ and scale parameter μ. Let Y1, Y2 be the stress on the two components, respectively. Initially Y1, Y2 are independently and identically distributed as exponential random variables with parameter θ. Statistically dependent failures are typical for modern systems, which involve complicated interactions among component parts, and it is reasonable to assume that the failure of one component does change the stress on surviving components. The strength on the other hand is dependent on various inbuilt qualities of the component such as the technology by which it has been manufactured, the raw materials used, and so forth. Failure of one component of the system need not change these inbuilt qualities of the surviving component of the system. Hence, change in the stress of the surviving component is assumed on failure of other components of the system. That is, in a two component parallel system, the distribution of stress of the surviving component changes upon the failure of the other component. Then the exponential distribution with parameter αθ(α>0) follows. The system fails whenever both the components of the system fail.A block diagram for the proposed model is given in Figure1.Figure 1One can quote several examples for two component parallel systems such as pair of kidneys, eyes, hands, and legs, pair of elevators, and pair of engines in an aircraft. To illustrate the function of the proposed model let us consider the example of a pair of kidneys. Here the function of the kidneys is to purify the blood and thus help to maintain the body in healthy condition. Here the pair of kidneys perform the same function as per their natural built-up mechanism (strength). Failure of one kidney increases the purification work load on the surviving kidney (stress). Here the surviving kidney should carry out the purification function the same as it was when both the kidneys were functioning. Taking this scenario under consideration, we consider that failure of one component of the system changes only the stress and not the strength of the surviving component.The reliability function is derived in Section2.  Life-testing experiment is explained in Section 3. The Maximum Likelihood Estimators (MLEs) are obtained in the same section. Section 4 deals with Bayes estimation of reliability function. Computations of estimators of reliability function (MLE and Bayes) along with the findings of the study are discussed in Section 5. Some results that support the findings of this research paper are proved in the appendix.


## 2. Reliability Function

In order to find the reliability function of the model discussed in Section1, let us consider that U is the minimum (Y1,Y2) and V is the maximum (Y1,Y2) and let W=V-U.HereU follows exponential distribution with parameter (2θ) and W follows exponential distribution with parameter (αθ).The reliability function of this system is given by(1)R=P[Max⁡(X1,X2)>Max⁡(Y1,Y2)]=1+2⋅μγΓγ⋅[2(α-2)∑i=γ∞(μ)ii!⋅Γ(i+γ)(αθ+2μ)(i+γ)-α(α-2)⋅∑i=γ∞(μ)ii!⋅Γ(i+γ)(2θ+2μ)(i+γ)].
The details of these derivations are given in Lemma A.1.


## 3. Life-Testing Experiment

In order to obtain the estimators ofR, suppose that “n” systems whose life distribution is characterized by the reliability function derived in Section 2 are put on life-testing experiment. Here, X1i, X2i(i=1,2⋯n) are observed and X1i, X2i are independently and identically distributed gamma random variables with shape parameter γ and scale parameter μ. Also, the data of stress Uj=Min(Y1j,Y2j) and Vj=Max(Y1j,Y2j), (j=1,2⋯m) are obtained separately from a simulation of conditions of the operating environment. Uj, Wj(j=1,2,3⋯m) are exponential random variables with parameters 2θ and αθ, respectively.Now, the joint probability density function of the random variablesX1i, X2i(i=1,2⋯n), Uj, Wj(j=1,2⋯m) is given by(2)L=(μγΓγ)2n⋅{∏i=1n(x1i)γ-1}⋅{∏i=1n(x2i)γ-1}⋅e-μ(x1′+x2′)⋅(2θ)m⋅(αθ)m⋅e-2θu′⋅e-αθw′,
where(3)x1′=∑i=1nx1i,x2′=∑i=1nx2i,u′=∑j=1muj,w′=∑j=1mwj.
The MLEs (the estimators that maximize the likelihood function) of parameters γ, μ, θ, α are given in the following expressions, respectively,
(4)γ̂=n2n[A]-[∑i=1nlog⁡⁡(x1i)+∑i=1nlog⁡⁡(x2i)],(5)μ̂=2n2(x1′+x2′)[2n{A}-{∑i=1nlog⁡⁡(x1i)+∑i=1nlog⁡⁡(x2i)}],(6)θ̂=m2u′,(7)α̂=2u′w′,
where 𝒜 denote log(x1′+x2′)-log(2n)Using  the invariance property of MLEs, the MLE of reliability functionR̂ is obtained by substituting the MLEs of parameters γ, μ, θ,  and α in expression (1), that is,(8)R̂=1+2⋅μ̂γ̂Γγ̂⋅[2(α̂-2)∑i=γ̂∞(μ̂)ii!⋅Γ(i+γ̂)(α̂θ̂+2μ̂)(i+γ̂)-α̂(α̂-2)⋅∑i=γ̂∞(μ̂)ii!⋅Γ(i+γ̂)(2θ̂+2μ̂)(i+γ̂)].The details of these findings are given in the form of LemmaA.2.From the asymptotic properties of MLEs under regularity conditions and multivariate central limit theorem, we have[n(γ̂-γ),n(μ̂-μ),m(θ̂-θ),m(α̂-α)]→N4[0,A(γ,μ,θ,α)],  where A(γ,μ,θ,α) are the corresponding elements of the inverse of Fisher Information Matrix “I(γ,μ,θ,α)” as(9)I(γ,μ,θ,α)=[n(2γ+1)2γ2-2nμ00-2nμ2nμ200002mθ2mαθ00mαθmα2]
and its inverse is given by(10)I-1=[2n[(2γ+1)/γ2-4]2μn[(2γ+1)/γ2-4]002μn[(2γ+1)/γ2-4]μ2(2γ+1)n[(2γ+1)/γ2-4]0000θ2m-αθm00-αθm2α2m].


## 4. Bayes Estimation of Reliability Function

In order to obtain Bayes estimator of reliability functionRB̂, assume that parameters γ and α are known (i. e., γ=γ0 and α=α0) and consider the prior distribution of parameters μ and θ [13] as(11)g(μ)=1Γp⋅e-μ⋅μp-1,p≥0,0<μ<∞,g(θ)=1Γq⋅e-θ⋅θq-1,q≥0,0<θ<∞.The Bayes estimator of reliability function is obtained as the posterior expectation ofR and is given by(12)RB̂=∬0∞R⋅f2(μ,θ∣x1i,x2i,uj,wj)dμdθ=∬0∞[∑i=γ0∞(μ)ii!1+2⋅μγ0Γγ0⋅[2(α0-2)∑i=γ0∞(μ)ii!⋅Γ(i+γ0)(α0θ+2μ)(i+γ0)-α0(α0-2)⋅∑i=γ0∞(μ)ii!⋅Γ(i+γ0)(2θ+2μ)(i+γ0)]]⋅f2(μ,θ∣x1i,x2i,uj,wj)⋅dμdθ=1+I1-I2,
where(13)I1=∬0∞4⋅μγ0(α0-2)⋅Γγ0⋅∑i=γ0∞(μ)ii!⋅Γ(i+γ0)(α0θ+2μ)(i+γ0)×f2(μ,θ∣x1i,x2i,uj,wj)⋅dμdθ=4.μγ0(α0-2)⋅Γγ0⋅Γ(2nγ0+p)⋅Γ(2m+q)⋅∑i=γ0∞Γ(i+γ0)i!⋅∬0∞A1dμdθ
with(14)A1=(e-μ(x1′+x2′+1)⋅μ2nγ0+p+i-1⋅e-θ(2u′+α0w′+1)(α0θ+2μ)(i+γ0))⋅(θ(2m+q-1)⋅(x1′+x2′+1)(2nγ0+p)⋅(2u′+α0w′+1)(2m+q)),I2=∬0∞2⋅α0⋅μγ0(α0-2)⋅Γγ0⋅∑i=γ0∞(μ)ii!⋅Γ(i+γ0)(2θ+2μ)(i+γ0)×f2(μ,θ∣x1i,x2i,uj,wj)⋅dμdθ=2⋅α0⋅μγ0(α0-2)⋅Γγ0⋅Γ(2nγ0+p)⋅Γ(2m+q)⋅∑i=γ0∞Γ(i+γ0)i!⋅∬0∞A2dμdθ
with(15)A2=(e-μ(x1′+x2′+1)⋅μ2nγ0+p+i-1⋅e-θ(2u′+α0w′+1)(2θ+2μ)(i+γ0))⋅(θ(2m+q-1)⋅(x1′+x2′+1)(2nγ0+p)⋅(2u′+α0w′+1)(2m+q)).
Substituting values of expressions (13), and (14) in expression (12) we obtain the Bayes estimator of reliability functionRB̂.The details of these findings are given in the form of LemmaA.3.


## 5. Computation of Estimators

For theith system, the random variables x1i, x2i (with respect to strength) and random variables ui,wi (with respect to stress) are generated independently as follows.Step 1.
Initializej1=1, x1i=0.0, x2i=0.0, n=n0, γ=γ0 for the 1st and 2nd components of the system. Uniform random numbers U1[j1], V1[j1]  are generated from U(0,1). Further for a given value of =μ0, exponential random variables U2[j1]=(-1/μ0)·ln(1-U1[j1]), V2[j1]=(-1/μ0)·ln(1-V1[j1]) are obtained for the 1st and the 2nd components of the ith system, respectively. Now, j1 is incremented by 1 and the process of generating the exponential random variables is repeated for both the components of the ith system. This repetition process is continued until  j1≤γ0 and the subsequent exponential random variable values generated are noted for both the components of the ith system. Here x1i=x1i+U2[j1] and x2i=x2i+V2[j1]  forj1=1,2⋯γ0. Here x1i,x2i denote gamma random variables with shape parameter  γ=γ0 and scale parameterμ=μ0. We also compute the values ln(x1i) and ln(x2i).Step 2.
The whole procedure in Step1 is repeated for n=n0 number of systems, and the statistics x1′=∑i=1nx1i,x2′=∑i=1nx2i, ∑i=1nlog(x1i), ∑i=1nlog(x2i) are computed.Step 3.
Initializei=1, ui=0.0, wi=0.0, m=m0. Uniform random numbers U3[i], U4[i]are generated from U(0,1). Further for a given value of θ=θ0, α=α0 exponential random variables ui=(-1/θ0)·ln(1-U3[i]),wi=(-1/α0·θ0)·ln(1-U4[i]) are obtained. The value of i is incremented by 1, and the above process of generated exponential random variables is repeated. This repetition process is continued until  i≤m and the subsequent exponential random variable values generated ui, wifor i=1,2⋯mare computed. The statistics u′=∑i=1mui, w′=∑i=1mwi are computed.Step 4.
With the help of the statisticsx1′, x2′, ∑i=1nlog(x1i), ∑i=1nlog(x2i), u′ and w′, the MLEs of parameters γ, μ, θ, α of the model are obtained. Using these MLEs in the expression of reliability function, the MLE of reliability function is obtained.
For the parameter valuesγ=γ0, μ=μ0, θ=θ0, and α=α0  the value of reliability function is also obtained.
The Bayes estimator of reliability function is obtained using the simulated values ofx1′, x2′, ∑i=1nlog(x1i), ∑i=1nlog(x2i), u′ and w′ for given values of p and q.Tables1, 2, and 3 give the results of the above simulation experiment for different values of n and m.Table 1
γ=3, μ=1.0, θ=0.5, n=5, m=5, p=10, and q=10.
αRx1′x2′∑i=1nlog(x2i)∑i=1nlog(x2i)u′w′R̂RB̂Bias (MLE)Bias (Bayes)0.50.47613.0174413.699453.946464.048234.881188.934510.6190.5622.05 × 10−27.40 × 10−31.00.67615.0085321.812654.649737.076593.291567.663880.7980.7441.49 × 10−24.62 × 10−32.00.77416.1364515.411865.423994.941805.010723.712950.8660.7928.46 × 10−33.24 × 10−4Table 2
γ=3, μ=1.0, θ=0.5, n=6, m=6, p=11, and q=11.
αRx1′x2′∑i=1nlog(x2i)∑i=1nlog(x2i)u′w′R̂RB̂Bias (MLE)Bias (Bayes)0.50.47612.8899617.436034.1193185.502262.3025020.48920.5740.5569.61 × 10−36.40 × 10−31.00.67614.8203616.229164.5780785.222553.7614510.48760.7590.7316.89 × 10−33.03 × 10−32.00.77410.1749616.288892.7508965.102114.328174.353860.8220.7872.31 × 10−31.69 × 10−4Table 3
γ=3, μ=1.0, θ=0.5, n=7, m=7, p=12, and q=12.
αRx1′x2′∑i=1nlog(x2i)∑i=1nlog(x2i)u′w′R̂RB̂Bias (MLE)Bias (Bayes)0.50.47623.0979517.293466.9418475.7548794.1998711.83710.5120.4941.29 × 10−33.24 × 10−41.00.67612.5014034.217162.20349610.3634291.6169814.57100.7470.7245.04 × 10−32.30 × 10−32.00.77420.3707225.067687.0691347.4476184.012976.649310.8070.7801.09 × 10−33.60 × 10−5


## 6. Conclusion

The reliability function for the proposed model is evaluated in terms of stress-strength relationship rather than considering the time factor, as it is realistic to observe reliability of a system functioning under the influence of external factors (stress) as compared to longevity of working time associated with the system.Though the expression for reliability function involves sum of infinite series, it is observed that for given values of the parameters the value of the reliability function stabilizes at the 15th value(i=15) of the running variable involved in the sums.Stress and strength associated with the components of the system possess different physical properties; thereby data on stress and strength in the life-testing experiment is observed separately based on their corresponding operative environments.The MLEs are sufficient, efficient and also maximizes the likelihood of the joint distribution function. Further, using the invariance property of MLE, it is easy to obtain the MLE of reliability function.Bayes estimator is based on the prior information obtained through certain pilot study that helps in synthesizing the information to be generated for the system under function. Hence, Bayes estimator of reliability function is obtained by considering certain prior information for the parameters. But the process of obtaining Bayes estimator of reliability function is quite tedious as it involves lengthy numerical calculations.From Tables1, 2, and 3, it is clear that for greater values of “m” and “n” (large sample size) both MLE and Bayes estimators perform better. In the majority of the cases both the estimators overestimate the true value of reliability function “R.” Here we observe that Bayes estimator is a better estimator in terms of bias for the given data set.



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*Source: 721962-2011-08-18.xml*