Estimation of Reliability for Stress-Strength Cascade Model

DOI: 10.4236/ojs.2016.65072    1,165 Downloads   1,616 Views   Citations
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.

1. Introduction

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.

2. Estimation of Reliability for a (1 + 1) Cascade Model

2.1. Stress-Strength Cascade Model

Let denote the strengths of n-components in the order of activation and let 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, and we assume that the strength gets modified by a factor “m” (strength attenuation factor) such that here, .

The reliability function of the system with ‘n’ components is defined as, where, for 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.

2.2. 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 and respectively, where are independently and identically distributed (i.i.d) exponential random variables with parameter “ ”. Let and be the stress acting on the two components respectively, where are i.i.d exponential random variables with parameter ‘ ’. To obtain the expression for reliability function, consider,

(1)

(2)

Using results of (1) and (2), we obtain reliability function for the proposed (1 + 1) cascade model as,

(3)

2.3. Life Testing Experiment

To obtain the estimators of “”, suppose “n” systems whose reliability function is defined as in expression (3) are put on life testing experiment. Here,

are observed and are i.i.d exponential random variables with parameters “” and “” respectively. Also, the data of stress are obtained separately from simulation of conditions of the operating environment and are i.i.d exponential random variables with parameter “” and “

respectively. The joint probability density function of the random variables and is given by,

(4)

where,

The log-likelihood function of Equation (4) is obtained as,

(5)

2.4. Estimators of Reliability Function (MLE & UMVUE)

Differentiating the log-likelihood function given in Equation (5) partially with respect to, and equating it to zero, we get,

(6)

(7)

Solving Equations ((6) and (7)) simultaneously, we get the Maximum Likelihood Estimator (MLE) of and as,

(8)

(9)

Similarly, differentiating the log-likelihood function given in Equation (5) with respect to, and equating it to zero, we get,

(10)

(11)

Solving Equations ((10) and (11)) simultaneously, we get the MLE of and as,

(12)

(13)

Using the invariance property of MLE, the MLE of reliability function ‘’ is obtained by substituting the MLEs of in Equation (3) and is given by,

(14)

Here, denotes the estimator of reliability function obtained through MLE of the parameters. Further, estimator of the reliability function “” attained through the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of the parameters is obtained as follows.

We know that, , implies

(15)

Also, , implies

On similar grounds we have,

(16)

Similarly, , implies

Substituting [result as mentioned in Equation (15)], we get,

(17)

Also, , implies

On similar grounds we have,

(18)

Substituting the UMVUEs of in Equation (3), we get estimator of the reliability function “” obtained through the UMVUE of the parameters.

2.5. Asymptotic Distribution

To obtain the asymptotic distribution of, let us denote the Fisher Information Matrix of as.

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,

is inverse of Fisher Information Matrix “

where,

3. Simulation Experiment

For the system, the random variables (with respect to strength) and random variables (with respect to stress) are generated independently as follows:

Step 1: Initialize for the 1st and 2nd component of the system. Uniform random numbers is generated from. Further, expo-

nential random variable is obtained for the 1st component of the system. Another uniform random numbers is generated from. Further, exponential random variable is obtained for the 2nd component of the system.

Step 2: The whole procedure in Step 1 is repeated for number of systems and the statistics are obtained.

Step 3: Initialize for the 1st and 2nd component of the system. Uniform random numbers are generated from. Further, ex-

ponential random variable is obtained for the 1st component of the system. Another uniform random numbers are generated from. Further, exponential random variable is obtained for the 2nd component of the system.

Step 4: The whole procedure in Step 3 is repeated for number of systems and the statistics are obtained.

Step 5: With the help of the statistics and the MLE of parameters 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 and the UMVUE of parameters 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 and n.

4. 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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Mutkekar, R. and Munoli, S. (2016) Estimation of Reliability for Stress-Strength Cascade Model. Open Journal of Statistics, 6, 873-881. doi: 10.4236/ojs.2016.65072.

  Pandit, S.N.N. and Sriwastav, G.L. (1975) Studies in Cascade Reliability—1.IEEE Transactions on Reliability, R-24, 53–57. http://dx.doi.org/10.1109/TR.1975.5215330  Raghavachar, A.C.N., Kesava Rao, B. and Ramacharyulu Pattabhi, N.C. (1983) Survival Function under Stress Attenuation in Cascade Reliability. Opsearch, 20, 190-207.  Maheshwari, U., Rekha, T.S., Anjan Rao, E. and Raghava Char, A.C.N. (1993) Reliability of a Cascade System with Normal Stress and Exponential Strength. Microelectronics Reliability, 33, 929-936. http://dx.doi.org/10.1016/0026-2714(93)90289-B  Rekha, A. and Shyam Sundar, T. (1997) Reliability of a Cascade System with Exponential Strength and Gamma Stress. Microelectronics Reliability, 37, 683-685. http://dx.doi.org/10.1016/S0026-2714(97)87650-0  Rekha, A. and Chenchu Raju, V.C. (1999) Cascade System Reliability with Rayleigh Distribution. Botswana Journal of Technology, 8, 14-19.  Shyam Sundar, T. (2012) Case Study of Cascade Reliability with Rayleigh Distribution. International Journal of Computer Technology & Electronics Engineering, 2, 78-87.  Hanagal, D.D. (1996) Estimation of System Reliability in a Two Component Stress-Strength Models. Economic Quality Control, 11, 145-154.  Kunchur, S.H. and Munoli, S.B. (1994) Estimation of Reliability for a Multi Component Survival Stress-Strength Model Based on Exponential Distribution. Communications in Statistics—Theory & Methods, 23, 3273-3283. http://dx.doi.org/10.1080/03610929408831446  Kundu, D. and Gupta, R.D. (2005) Estimation of P(Yhttp://dx.doi.org/10.1007/s001840400345  Munoli, S.B. and Mutkekar, R.R. (2011) Estimation of Reliability for a Two Component Survival Stress-Strength Model. International Journal of Quality, Statistics & Reliability, 2011, Article ID: 721962. http://dx.doi.org/10.1155/2011/721962 