Cascade reliability of stress-strength system for the new Rayleigh-Pareto distribution

Introduction

Cascade reliability model is a special type of Stress-Strength model. Reliability may be used as a measure of the system's success in providing its function properly. Mathematically, reliability ) (t R is that a system will be successful in the interval ) , 0

, where T is a random variable denoting the failure time. Cascade systems were first proposed and studied by [1] they have studied an n-cascade reliability for exponential distribution. They evaluate reliability values for a two-cascade system with gamma and normal stress and strength distribution [2] . studied the reliability of a cascade system with normal stress and strength distribution [3] . Rekha and Shyam Sundar have derived an expression of the reliability of an n-cascade system by considering attenuation factor with the same parameter value. For their study, they considered exponential strength and gamma stress distribution [4] . has done a case study of cascade reliability using Weibull distribution [5] , has done a case study of cascade reliability with Rayleigh distribution [6] have studied cascade reliability of stress-strength system considering strength follows mixed exponential distribution [7] . has done comparison of an n-cascade system by considering normal stress and exponential strength distribution [8] . studied the reliability analysis of a redundant cascade system by using Markovian approach assuming n-components which are arranged in the hierarchical order. The n-cascade system survive with loss of m components by k number of attacks [9] . studied about estimation of reliability for stress-strength cascade model by comparison between estimators made using data obtained through simulation experiment [912] has done a cascade and system reliability for Exponential distribution. In this paper we determined a statistical model of New Rayleigh-Pareto strength-stress distribution which are considered for computations of marginal and system reliabilities supported by reliability computations with numerical and graphical study along with various values.

Statistical Model

Let us consider the New Rayleigh Pareto distribution with probability density function (pdf)

The cumulative distribution function (cdf) of the NRPD is given by Rn is given by

The marginal reliability n R is the reliability of the system for the n th component is given by

Reliability Computation

Let us consider the New Rayleigh Pareto distribution with probability density functions (pdf's) of X and Y .

Let X represents the strength of an component with density function

 are the scale parameters and 1  is the shape parameter and its distribution function is given by

and the probability density function of stress Y is given by

Then from   6 , we evaluate cascade reliability.

In general an n-cascade reliability,

Simulation and Graphical Study

In this section, the simulation work and graphical presentation by determining cascade and system reliability is carried out. In the table 1 strength parameter is constant i.e., 4 and stress parameter is steadily increasing it from 1 to 7.5. In the table 2 stress parameter is constant i.e., 4and the strength parameter steadily increasing it from 1 to 7.5. In the table 3 strength parameter is increasing it from 1 to 7.5 and stress parameter decreasing it from 7.5 to 1. In table 4 stress parameter is increasing it from 1 to 7.5 and strength parameter is decreasing it from 7.5 to 1.In table 5 stress parameter increasing it from 1 to 7.5 and by taking scale parameters 12 0 .1 5 8 1, 0 .2 5   strength parameter is steadily increasing from 1 to 7.5.. In table 6 strength parameter is constant and by taking scale parameters 12 0 .5 , 0 .2 5   stress parameter is increasing from 1 to 7.5 . Now we determine the cascade reliability by using (11) and then next compute the system reliability using (4) . Next we plot the graphs by taking the progressive values of strength and stress against cascade and system reliability respectively.

Conclusion

In this paper the reliability formulae for an n-cascade system using new rayleigh-pareto strength-stress distribution is found out. We observe that the system reliability s R reaches to 0.99 i.e., unity (one) if we choose strength parameter constant and stress parameter steadily increasing. In addition, if we increase strength parameter and decrease stress parameter the cascade reliability little bit increases and then decreases but the system reliability steadily increases and reaches to unity (one). Hence we conclude that the cascade and system reliability is found to be constant if strength and stress parameters are steadily increasing.

References

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