I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 3 Issue. 1

Cascade Reliability for Generalized Exponential Distribution 1,

T.sumathi UmaMaheswari, 2,N.Swathi.

1,2,

Department of Mathemat ics, Kakatiya Un iversity, Warangal, Andhra Pradesh

Abstract Cascade reliability model is a special type of Stress - Strength model. The n - Cascade system is a h ierarch ical standby redundancy system, where the standby component taking the place of failed component with decreased value of stress and independently distributed strength. This paper deals with the generalized exponential distribution with cascade system.

Key Words : Stress â€“ Strength model, Cascade system, generalized exponential distribution. 1. Introduction The two- parameter generalized exponential (GE) distribution has been proposed by the Debasis Kundu and Rameshwar D.Gupta[1]. It has been extensively studied by Gupta and Kundu [2] and Kundu, Gupta and [3]. Note that the generalized exponential distribution is a sub- model of the exponential Weibull distribution introduced by Mudholkar and Srivastava [4] and later studied by Mudholkar, Srivastava and Freimer [5].An - cascade system is defined as a special type of standby system with components by Sriwastav et al ., . Cascade redundancy is defined as a hierarchical standby redundancy where a standby component takes the place of a failed co mponent with a changed stress. This changed stress is times the preceding stress. is the attenuation factor. Sriwastav and Pandit[6] derived the expressions for reliability of an n-cascade system when stress and strength follow exponential d istribution. They co mputed reliability values for a 2 -cascade system with gamma and normal stress and strength distributions. Raghava Char et al [7] studied the reliability of a cascade system with normal stress and strength distribution. T.S.Uma Maheswari et al [9] studied the reliability co mparison of ncascade system with addition of n-strengths system when stress and strength follow exponential distribution.

2. Statistical Model If the r.v .X denotes the strength and the r.v. Y denotes the stress of the component, then the reliability of the component is given by

Let be the strengths of the components as arranged in order of act ivation respectively. A ll the are independently distributed random variab les with probability density functions . Also let Y be the stress acted on the components which is also randomly distributed with the density function If , the first component works and hence the system survives. leads to the failure of ; thus the second component in line viz., ,takes its place and has a strength ..Although the system has suffered the loss of one component, it survives if and so on . In general, if the component fails then the co mponent , with the strength , gets activated and will be subjected to the stress Y. The

system

could

survive

with

a .

The probability

loss of the first components if and only if The system totally fails if all the components fail when of the system to survive with the first components failed and the

component active is

are the increments in reliability due to the addition of

components

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Then we can obviously associate the co mponent attenuation factor with Let and be the probability density function of and The equation can now be written as

. respectively.

(o r)

The two- parameter GE distribution has the following density functions

And the distribution function

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2- Cascade system

3- Cascade system

In general

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3. Reliability Computations:

Figure 2

Figure 1

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1

R(1) 0.9

R(1)

0.9

0.8

0.7

0.7

R(1)

R(1)

0.8

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0.5

0.5

0.4

0.4

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3

4

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6 Alpa

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5

6 Beta

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I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 3 Issue. 1

Table 5 α 1 2 3 4 5 6 7 8 9 10 11

Table 6 β 2 2 2 2 2 2 2 2 2 2 2

R(3) 0.1 0.083333 0.068182 0.057143 0.04902 0.042857 0.038043 0.034188 0.031034 0.028409 0.02619

α 5 5 5 5 5 5 5 5 5 5

β 1 2 3 4 5 6 7 8 9 10

R(3) 0.028409 0.04902 0.064103 0.075188 0.083333 0.089286 0.093583 0.096618 0.098684 0.1

4. Conclusion The reliab ility of n- cascade system when stress and strength follow generalized exponential distribution. In this paper we find out the formula for n-cascade system. Fro m co mputations Reliab ility increases in 1-cascade system whenever strength parameter α increases and reliability decreases whenever stress parameter β increases and vice-versa for 2-cascade, 3-cascade,---n-cascade system.

References [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8].

Rameshwar D.Gupta & Debasis Kundu (1999): Generalized Exponential Distribution, Austral. & New Zealand J.Statist. 41(2), 173-188. Gupta, R.D. &Kundu D. (2001): Generalizes exponential distribution, different method of estimations, Journal of Statistical Co mputation and Simulat ion 69, 315-338. Gupta, R.D. & Kundu, D. (2002): Generalized exponential distribution, statistical inferences, Journal of Statistical theory and Applications 1, 101-118. Gov ind S. Mudholkar & Srivastava (1996) : Generalizat ion of the weibull distribution with Application to the Analysis of survival data , journal of the Americal Statistical Association , Dec 1996, vol 91, No. 436, 1575- 1583. Gov ind S. Mudholkar , Srivastava and D.K. Freimerm (1995) : The exponential weibull family ; a reanalysis of the bus motor failure data , Technometrics, 37(4), 436- 445 S.N.Narahari Pandit and G.L.Sriwastav (1975). Studies in Cascade Reliab ility –I, IEEE Transactions on Reliability , Vo l.R-24, No.1, pp.53-57. A.C.N.Raghava char, B.Kesava Rao and S.N.Narahari Pandit(Sept.1987).The Reliability of a Cascade system with Normal Stress and Strength distribution, ASR, Vol. No.2, pp. 49-54. T.S.Uma Maheswari(1993). Reliability of cascade system with normal stress and exponential strength, Micro Electron Reliability, Pergamon Press, OXFORD, Vo l.33, pp: 927-936.

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