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.
As the complexity of a system increases, its reliability decreases unless compensatory measures are taken. System reliability can be increased by increasing the reliability of its associated components, but sometimes this cannot be achieved beyond certain limits. An alternative way to increase the reliability in such situation is to have redundant configuration of components in the system.
Cascade system is one such special type of standby system. Cascade redundancy is a hierarchical standby redundancy, where an array of components (finite in number) are arranged in the order of activation. Here, the first component is active and the remaining components are at standby. The brunt of attack, in the first instance is borne by the active component. If it survives the attack, the system also survives with no loss and is ready to face the next attack. However, if the active component fails then the next component in the array has to face and withstand the “cushioned” attack on it. The stress acting on the subsequent active component will be “k” times the stress of the previous failed components, where “k” denotes stress attenuation factor.
Research works on reliability modelling and assessment related to cascade model as studied in the literature are quite exhaustive, Pandit and Sriwastav (1975) have featured relevance of geometric distribution in the study of behavior of a cascade system [
Let
The reliability function
where,
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.
To determine reliability function for the model under study, let us consider the strength of the two components (basic and standby) to be
Using results of (1) and (2), we obtain reliability function for the proposed (1 + 1) cascade model as,
To obtain the estimators of “
respectively. The joint probability density function of the random variables
where,
The log-likelihood function of Equation (4) is obtained as,
Differentiating the log-likelihood function given in Equation (5) partially with respect to
Solving Equations ((6) and (7)) simultaneously, we get the Maximum Likelihood Estimator (MLE) of
Similarly, differentiating the log-likelihood function given in Equation (5) with respect to
Solving Equations ((10) and (11)) simultaneously, we get the MLE of
Using the invariance property of MLE, the MLE of reliability function ‘
Here,
We know that,
Also,
On similar grounds we have,
Similarly,
Substituting
Also,
On similar grounds we have,
Substituting the UMVUEs of
To obtain the asymptotic distribution of
where,
Thus, we have the Fisher Information Matrix as,
From the asymptotic properties of MLE under regularity conditions and multivariate central limit theorem we have,
where,
For the
Step 1: Initialize
nential random variable
Step 2: The whole procedure in Step 1 is repeated for
Step 3: Initialize
ponential random variable
Step 4: The whole procedure in Step 3 is repeated for
Step 5: With the help of the statistics
Step 6: With the help of the statistics
From the above results (as shown in
MSE MLE | MSE UMVUE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.25 | 1.00 | 0.810 | 13.412501 | 6.562061 | 6.706251 | 3.124121 | 0.855 | 0.818 | 2.03 × 10−3 | 6.40 × 10−5 |
0.25 | 1.25 | 0.792 | 10.290507 | 2.620377 | 5.145254 | 3.800942 | 0.831 | 0.799 | 1.52 × 10−3 | 4.90 × 10−5 |
0.25 | 1.50 | 0.778 | 09.181533 | 2.998768 | 5.590766 | 3.368335 | 0.816 | 0.784 | 1.44 × 10−3 | 3.60 × 10−5 |
0.50 | 1.00 | 0.867 | 09.778324 | 3.125278 | 4.889162 | 1.915278 | 0.909 | 0.860 | 1.76 × 10−3 | 4.90 × 10−5 |
0.50 | 1.25 | 0.848 | 15.892459 | 6.225975 | 6.912290 | 1.256218 | 0.885 | 0.853 | 1.37 × 10−3 | 2.50 × 10−5 |
0.50 | 1.50 | 0.833 | 12.241618 | 9.066720 | 6.120809 | 3.797080 | 0.868 | 0.837 | 1.23 × 10−3 | 1.60 × 10−5 |
0.75 | 1.00 | 0.897 | 14.691534 | 6.720832 | 5.345767 | 2.380555 | 0.931 | 0.892 | 1.16 × 10−3 | 2.50 × 10−5 |
0.75 | 1.25 | 0.881 | 13.931526 | 7.937316 | 5.965763 | 1.581096 | 0.913 | 0.877 | 1.03 × 10−3 | 1.60 × 10−5 |
0.75 | 1.50 | 0.867 | 15.716419 | 6.075062 | 6.428210 | 1.075062 | 0.898 | 0.870 | 9.61 × 10−4 | 9.00 × 10−6 |
MSE MLE | MSE UMVUE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.25 | 1.00 | 0.810 | 18.036022 | 04.895907 | 09.018011 | 08.965815 | 0.833 | 0.816 | 5.29 × 10−4 | 3.60 × 10−5 |
0.25 | 1.25 | 0.792 | 18.058302 | 04.234054 | 09.029151 | 09.985135 | 0.810 | 0.796 | 3.24 × 10−4 | 1.60 × 10−5 |
0.25 | 1.50 | 0.778 | 15.606312 | 07.447487 | 09.303156 | 15.842461 | 0.795 | 0.781 | 2.89 × 10−4 | 9.00 × 10−6 |
0.50 | 1.00 | 0.867 | 20.403873 | 06.592115 | 10.201937 | 05.012115 | 0.886 | 0.862 | 3.61 × 10−4 | 2.50 × 10−5 |
0.50 | 1.25 | 0.848 | 26.671318 | 12.450741 | 13.335659 | 05.463426 | 0.865 | 0.844 | 2.89 × 10−4 | 1.60 × 10−5 |
0.50 | 1.50 | 0.833 | 22.037254 | 08.622803 | 11.018627 | 07.934204 | 0.849 | 0.835 | 2.56 × 10−4 | 4.00 × 10−6 |
0.75 | 1.00 | 0.897 | 23.010950 | 10.241815 | 11.505475 | 02.827877 | 0.915 | 0.893 | 3.24 × 10−4 | 1.60 × 10−5 |
0.75 | 1.25 | 0.881 | 16.283512 | 08.869339 | 10.941756 | 02.691160 | 0.898 | 0.878 | 2.89 × 10−4 | 9.00 × 10−6 |
0.75 | 1.50 | 0.867 | 21.313683 | 11.199248 | 10.656841 | 07.099248 | 0.883 | 0.865 | 2.56 × 10−4 | 4.00 × 10−6 |
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.
Mutkekar, R.R. and Munoli, S.B. (2016) Estimation of Reliability for Stress-Strength Cascade Model. Open Journal of Statistics, 6, 873-881. http://dx.doi.org/10.4236/ojs.2016.65072