Estimation of Reliability for Stress-Strength Cascade Model

The study endeavors to provide statistical inference for a (1 + 1) cascade system for exponential distribution under joint effect of stress-strength attenuation factors. Estimators of reliability function are obtained using Maximum Likelihood Estimator (MLE) and Uniformly Minimum Variance Unbiased Estimator (UMVUE) of the parameters. Asymptotic distribution of the parameters is also obtained. Comparison between estimators is made using data obtained through simulation experiment.


Stress-Strength Cascade Model
Let 1 2 , , , n X X X  denote the strengths of n-components in the order of activation and let 1 2 , , , n Y Y Y  be the corresponding stresses acting on them. In a n-cascade system after every failure the stress gets modified by a factor "k" (stress attenuation factor) such that here, 1 k > and we assume that the strength gets modified by a factor "m" (strength attenuation factor) such that The reliability function n R of the system with 'n' components is defined as, Cascade model with more number of standby components is not recommended as the strength goes on depleting with the order of standby which leads to dead investment. In view of this fact, we have considered estimation of reliability for a (1 + 1) cascade model.

Reliability Function for a (1 + 1) Cascade Model
To determine reliability function for the model under study, let us consider the strength of the two components (basic and standby) to be 1 X and 2 X respectively, where 1 2 , X X are independently and identically distributed (i.i.d) exponential random variables with parameter " λ ". Let 1 Y and 2 Y be the stress acting on the two components respectively, where 1 2 , Y Y are i.i.d exponential random variables with parameter ' µ '. To obtain the expression for reliability function, consider, Using results of (1) and (2), we obtain reliability function for the proposed (1 + 1) cascade model as,

Life Testing Experiment
To obtain the estimators of " 2 R ", suppose "n" systems whose reliability function is defined as in expression (3) where, The log-likelihood function of Equation (4) is obtained as,

Estimators of Reliability Function (MLE & UMVUE)
Differentiating the log-likelihood function given in Equation (5) partially with respect to λ , m and equating it to zero, we get, Solving Equations ( (6) and (7)) simultaneously, we get the Maximum Likelihood Estimator (MLE) of λ and m as, Similarly, differentiating the log-likelihood function given in Equation (5) with respect to µ , k and equating it to zero, we get, Solving Equations ( (10) and (11)) simultaneously, we get the MLE of µ and k as, Using the invariance property of MLE, the MLE of reliability function '  2 R ' is obtained by substituting the MLEs of , , , m k λ µ in Equation (3) and is given by, Here,  2 R denotes the estimator of reliability function obtained through MLE of the parameters. Further, estimator of the reliability function " 2 R * " attained through the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of the parameters is obtained as follows.
We know that, On similar grounds we have, [result as mentioned in Equation (15)], we get, Substituting the UMVUEs of , , , m k λ µ in Equation (3), we get estimator of the reliability function " 2 R * " obtained through the UMVUE of the parameters.

Asymptotic Distribution
To obtain the asymptotic distribution of , , , From the asymptotic properties of MLE under regularity conditions and multivariate central limit theorem we have,

Simulation Experiment
For the th i system, the random variables 1 2 , i i x x (with respect to strength) and random variables 1 2 , i i y y (with respect to stress) are generated independently as follows: Step for the 2 nd component of the th i system.
Step 2: The whole procedure in Step 1 is repeated for , , x x y ′ ′ ′ and 2 y′ the MLE of parameters , , , m k λ µ of the model are obtained. Using these MLEs in the expression of reliability function, the MLE of reliability function is obtained.
Step 6: With the help of the statistics 1 2 1 , , x x y ′ ′ ′ and 2 y′ the UMVUE of parameters , , , m k λ µ are obtained. Using these UMVUEs in the expression of reliability function, estimator of the reliability function based on UMVUE of the parameters is obtained. Table 1 and Table 2 give the results of the above simulation experiment for different values of , , , m k λ µ and n.

Conclusion
From the above results (as shown in Table 1 and Table 2), we observe that reliability of  the system improves for larger values of strength attenuation factor (m) and for lower values of stress attenuation factor (k). Here, we also observed the estimates of reliability improves for larger value of the sample size "n". This indicates that reliability of a system can be enhanced by strengthening the inbuilt mechanism of the system, which ultimately withstands the effects of the external environment in which it operates. Further, on comparing the efficiencies of MLE of reliability function with reliability estimator obtained using UMVUEs of the parameters, we observed reliability estimator obtained from the UMVUEs of the perform better than the MLE of reliability function in terms of Mean Square Error (MSE) for the given data set. This emphasizes the need to strengthen the processes such that they are least affected by effects of the variation factors which intern boost the reliability of the operating system.