International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1400

ISSN 2229-5518

MEASURING RELIABILITY OF n-CASCADE SYSTEM UNDER RANDOM STRESS ATTENTION

By

DIGVIJAY SINGH

Deptt. Of Mathematics, J.P Institute of Engg. & Tech., Meerut (INDIA)

drdigvijay2008@rediffmail.com

ABSTRACT: In this paper the author has considered the system reliability of n-cascade system with stress following normal distribution and strength following exponential distribution. They concluded that even for fewer values of stress and strength parameter a high reliability could be high degree of reliability.

KEYWORD: Cascade system, survival function, standby

system.

h(k i ), i = 1, 2, 3, ..., n . In n-cascade system, if the stress exceeds the strength, then it leads to the failure of the components and the next component in the sequence gets activated. Hence, the system reliability R(n) for the nth component can be evaluated only when the first (n – 1) components fail and the nth component activates and is given by

INTRODUCTION

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The author has considered the system reliability of n-cascade
following exponential distribution. They concluded that even

 n − 1



i i }



n n 

system with stress following normal distribution and strength
for fewer values of stress and strength parameter a high

R(n) = P



(x

i = 1

< y )

 (x

> y )

 .

reliability could be high degree of reliability. The stress
Thus for n = 1, 2, 3 and 4 the reliability expression are
attenuation cascade reliability for a system when both stress and strength are subjected to Rayleigh distribution. They
(a)

R(1) =

P [ x1 > y1 ]

concluded that for lower attenuation factors a high degree of

∞ ∞

reliability could be attained even when the components are
characterized with linearly increasing failure rate.

MATHEMATICAL MODEL

The n-cascade system, a special type of standby system with n components, was defined by Shrivastav and Pandit [1].
(b)

= g(y1 ) dy1

f(x1 ) dx1

y1 ,

Cascade redundancy is such a standby redundancy, where a standby component takes the place of the failed component with changed stress. This changed stress is k times the preceding stress and it is called the attenuation factor.

R(2) =

∞

P [ (x1

< y ) 

y1

(x2

1

> y ) ]

2 ,

∞

Consider a system with n components

= ∫ g(y1 ) dy1 ∫

f(x1 ) dx1 ∫

h(k 2 ) dk 2 ∫

f(x2 ) dx2

C1 , C2 , C3 , ..., Cn

with strengths
(c)

0 0 0 k2 y1

x1 , x2 , x3 , ..., xn

arranged in order of activation. Let

f(xi ), i = 1, 2, 3, ..., n

be the probability density

R(3) =

P [ (x1

< y ) 

(x2

< y )

 (x3

> y ) ]

functions for the independently distributed random variables

x1 , x2 , x3 , ..., xn . Also, let g(y1 ) be the probability

density function for the randomly distributed stress on the ∞

y1 1 k2 y1

first component

y1 . The stress on the components

= g(y1 ) dy1 ∫

f(x1 ) dx1 ∫

h(k 2 ) dk 2 ∫

f(x2 ) dx2

undergoes attenuation at each failure by a random factor k.

k , k , k , ..., k

Their stress attenuation factors 1 2 3 n are defined

l , l , l , ..., l

over intervals 1 2 3 n with probability functions

1

h(k 3 ) dk 3

∞

k 3 k 2 y1

f(x3 ) dx3

,

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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1401

ISSN 2229-5518

(d)

R(4) = P [ (x < y )  (x < y )  (x < y )  (x > y ) ]

β(t + a, b) β(r + s + t + a, b) (r + s + t + 1)

∞ y1

1 k2 y1

 1 − 1 

= g(y1 ) dy1

0 0

f(x1 ) dx1

0

h(k 2 ) dk 2

0

f(x2 ) dx2

 µ r + s + t + 1

(µ +λ ) r + s + t + 1 

 1  .

The reliability of the system is given by

1 k3k2 y1 1 ∞

The expressions for the survival function as follows

h(k 3 ) dk 3

0 0

.

f(x3 ) dx3

0

h(k 4 ) dk 4

k 4 k3k2 y1

f(x4 )dx4

q(1) = λ

EXPONENTIAL STRESS, STRENGTH AND BETA ATTENUATION FACTORS

with the probability density functions

λ + µ

1 ,

∞ r r

∑ 

 r + 1 r + 1 

(a)

−λi xi

β(a, b) r =0  r

,

∞ ∞

 λ

r + s + 1 r s

(λ + µ1 )



f(xi )

= λ e , i = 1, 2 , 3, ..., n

i ,

q(3) =

λ ∑∑  (−1)

µ µ β(r + s + a, b)β(r + a, b)

(b)

g(y1 )

= µ e

−µy1

.

{β(a, b)}

r =0 s =1  r s

 1

(r + s + 1)

1 

−

k a −1 (1 − k )b −1

 r + s + 1 r + s + 1 

h = i i

, i = 1, 2, 3, ..., n , a,b > 0

 λ (λ + µ1 )



( ki )

β (a, b)

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. q(4) = λ

∞ ∞ ∞

 (−1)

r + s + t + 2 µ r µ s µ t

2 3 4 β(s + t + a, b)

Then the expressions for reliability are

µ

{β(a, b)} 3

∑∑∑ 

r =0 s =0 t =1  r s

R(1) =

µ +λ

β(t + a, b) β(r + s + t + a, b) (r + s + t + 1)

(a) 1 ,

 1 − 1 

 λ r + s + t + 1

(λ +µ ) r + s + t + 1 

(b)

 1

The survival of the system is given by

 .

µ ∞ (−1) r λ r β(r + a, b)

 1 1 

q =

q1 (1 − q2 )(1 − q3 )(1 − q 4 )

= ∑  2

+  − 

R(2) (r 1)

r +1 r +1

β(a, b) r =0  r

©

 µ (µ + λ1 )



µ= 1 λ = i λ

Table 1 : Reliability for and i 1

µ ∞ ∞  (−1) r + s λ r λ s β(r + s + a, b)β(s + a, b)

R(3) =

{β(a, b)} 2

∑ ∑  2 3

r =1 s =0  r s

 1

(r + s + 1)

1 

−

 µ r + s + 1

(µ +λ ) r + s + 1 

(d)



µ ∞ ∞ ∞

1

 (−1) r +s + t λ r λ s λ t

 ,

R(4) =

{β(a, b)} 3

∑∑∑ 

r =1 s =1 t =0 

2 3 4 β(s + t + a, b)

r s t

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Table 2 : Reliability for

µ = 5

λ = i λ

and i 1

Table 5 : Survival function for

λ = 1

µ = i µ

and i 1

Table 3 : Reliability for

µ= 20

λ = i λ

and i 1

Table 6 : Survival function for

λ = 5

µ = i µ

and i 1

Table 4 : Reliability for

µ = 25

λ = i λ

and i 1

Table 7 : Survival function for

λ = 20

µ = i µ

and i 1

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International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1403

ISSN 2229-5518

Table 8 : Survival function for

λ = 25

µ = i µ

and i 1

Interpretation of Result

Numerical results are calculated both for reliability

λ

(for i

= i λ λ

1 and 1 ranging from 0.01 to 0.10 in step of

µ

0.01) and the survival function for ( i

= i µ

and 1
ranging from 0.01 to 0.10) for different stress and strength

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parameter.
From the above results, we conclude that R

λ

decreases with the increase in 1
increase in values of parameter µ .

q

but increases due to
Similarly,

λ

also increases with the increase of

µ

value of 1 but decreases as 1 range from 0.01 to 0.10.

REFERENCES

[1] Srivastav G. L. and Pandit S. N. (1975) : Studies on cascade reliability – I, IEEE Trans. Reliability, 53-

60.

[2] Raghava Char, A. C. N., KeshavRao B. and PanditS. N. N. (1988) : Survival functions of a component

under strength attenuation, IAPQR Trans.,13.

[3] Evans J. R. and Lindsay W. M. (2000) : The management and control of quality, 5th Ed., South Western Publishing, Cincinnati, OH.

[4] Gunes M. and Deveci I. (2002) : Reliability of service systems and an application in student office, International Journal of Quality and Reliability, Vol. 19, 206-14.

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ISSN 2229-5518

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