DOI : 10.17577/IJERTV5IS090405

Text Only Version

1M.Tirumala Devi 1Dept. of Mathematics, Kakatiya University, Warangal, Telangan State

2T. Sumathi Uma Maheswari, 3N. Swathi

2,3Dept. of Mathematics,

University College of Engg. & Tech. For Women, Kakatiya University, Warangal, Telangana State

Abstract:- The cascade reliability model is a special type of redundancy of stress-strength model. The n-cascade system is a hierarchical standby redundancy system, where the standby component taking the place of the failed component for the decreased value of stress and independently distributed strength and it is the cold standby system. Lindley distribution belongs to an exponential family and it can be written as a mixture of an exponential and a gamma distribution with shape parameter 2. In this paper, the general expression for the reliability of cascade system was derived when stress and strength follow lindley distribution and the numerical values (), (), () have been computed for some specific values of the parameters.

Key words: Lindley distribution, cascade system, standby redundancy, stress strength model, Reliability.

INTRODUCTION:

If denotes the strength of the component and is the stress imposed on it, then the reliability of the component is given by [1],

= ( > )

= { ()} ()

0

(or)

= { ()} ()

The cascade system is defined as a special type of standby system with components by Sriwastav et al [2]. The cascade redundancy is defined as a hierarchical standby redundancy where a standby component takes the place of a failed component with a changed stress. This changed stress is times the preceeding stress. is the attenuation factor of stress

. Lindley introduced lindley distribution in the year1958. Lindley distribution belongs to an exponential family and it can be written as a mixture of an exponential and a gamma distribution with shape parameter 2.

STATISTICAL MODEL:

Let 1, 2, 3, , be the strengths of the components 1, 2, 3, , as arranged in order of activation respectively. All the are independent and non-identically distributed random variables with probability density

functions(); i = 1,2, , . Also let 1 be the stress on the first component which is also randomly distributed with the density function (1).

If 1 < 1, the first component which is also randomly distributed stress varies with the density function (1) . If 1 < 1 , the first component 1 works and hence the system survives. 1 1 leads to the failure of 1 ; thus the second component in line viz., 2 ,takes its place and has a strength 2. However, the stress 2 on 2 will normally be different from 1

. Let 2 = 1 , where is the cumulative attenuation factor on the second component and = 12 where by definition

2 2 2

1 = 1.Although the system has suffered the loss of one component , it survives if 2 < 2 and so on .

In general, if the ( 1) component 1 fails then the component , with the strength , gets activated and will be subjected to stress.

= 1 = 1 (1)

= 12 (2)

represents the cumulative attenuation factor on the component .

The system could survive with a loss of the first ( 1) components if and only if i i ; i = 1,2,3, , 1 > . The system totally fails if all the components fail when ; = 1,2, .

The probability () of the system to survive with the first ( 1) components failed and the component

active is

1

() = [{( )} ( > )] (3)

i=1

(2), (3), , () are increments in reliability due to the addition of the 2, 3, , components respectively. Then

() = [1 1, 2 1, , 2 21, 1 11, >

1 2

1

1

] (4)

we can obviously associate the component attenuation factor with 1 .

Since the stress is attenuated, the subsequent stresses 2, 3 are described in terms of 1 . Hence it is necessary to specify the distribution of 1 . Let (1) and () be the probability density function of 1 and ( = 1,2, , ) respectively.

The equation (4) can now be written as

1

1

2

2

1

1

1 1

() = [ 1(1)1 Ã— 2(2)2 Ã— 1(1)1

0 0 0 0

Ã— ()] (1 )1 (5)

1

(or)

= [1(1)2(1) 1( 11)(1)](1)1 (6)

1 2

0

1

(1) = ( )

0

(1) = 1 (1) (7)

Then the system reliability of cascade model is

Reliability computations:

= ()

=1

Let X be the strength and Y be the stress of a system p.d.fs and c.d.fs are

() =

() =

2

1 + (1 + )

2

1 + (1 + )

, , > 0

, , > 0

() = 1 (1 +

+ 1

() = 1 (1 +

) , , > 0

) , , > 0

Reliability for stress strength models:

= ( > )

+ 1

= ()()

0

2

= (1 (1 + + 1)) 1 + (1 + )

0

2

2

= 1 + (1 + ) 1 + (1 + + 1) (1 + )(+)

0 0

2 1 1

2

= 1 1 + [

+ + ( + )2 + (1 + )( + )2 + (1 + )( + )3] (8)

Marginal reliability of the component for =1, 2, 3 is

2 1 1 2

(1) = 1 1 + [

+ + ( + )2 + (1 + )( + )2 + (1 + )( + )3] (9)

2

2

(2) = 1 (1){1 2(1)}(1)1

0

[1 11 (1 +

2 2 1

2 2 1

=

11

1 + 1

)] {1 [1 (1 +

21

2

2

2 + 1

)]}

2 1

0 1 (1 + 1)11

[ 1 + 1 ]

2

2

= 1 [(1 +

+ 2 2 1 + 2 2 1 ) (2+1)1

1 + 1

0

1 2 + 1

2

2

+ 1

2

2

+ (11 1 11 2 111 2 211 ) (22+1)1

2 + 1

2 + 1 1

1

1

1 2

+ ( 111

211 2 211

1 + 1

1

2

1 + 1 1

1 + 1 2 + 1 1

2 311 ) (22+1)1 ] 1

1

1

1

1

2

= 1 + 1 [(2 + 1)

1 + 1 2 + 1 1

+ 1 1

(2 + 1)2 (2 + 1 + 1)

1 +

(2 + 1 + 1)2

2

2

2

(2 + 1)(2 + 1)2

2 2 2

2

2

2

2

2 2 + 2 2 2 2

(2 + 1)(2 + 1 + 1)2 (2 + 1)(2 + 1)3 (2 + 1)(2 + 1 + 1)3

2 2

2

2

1 2 1

2 1 2

(1 + 1)(2 + 1 + 1)2 (1 + 1)(2 + 1 + 1)3 (1 + 1)(2 + 1)(2 + 1 + 1)3

2 2 2

6

2 1 2 ] (10)

2

2

(1 + 1)(2 + 1)(2 + 1 + 1)4

(3) = 1 (1)2( 1){1 3(1 )}(1)1

2

0

11

3

21

31

[1 11 (1 +

=

1 + 1

)] [1 221 (1 +

2

2

2 + 1

)] {1 [1 331 (1 +

3

3 + 1

)]}

1

0 1 (1 + 1)11

[ 1 + 1 ]

2

= 1 {[1 +

+ 3 3

+ 3 3

2] (3+1)1 [1 11

3

3

1 11 ] [1 21

2

2

1 + 1

0

1 3 + 1 1

3 + 1 1

1 + 1 1

2

2 1221 ]} 1

2 + 1

1

1

2

= 1 + 1

1

[(3 + 1)

1

+ (3 + 1)2

1

(3 + 1 + 1)

1

(3 + 1 + 1)2

1

(2 + 3 + 1)

3 3 3 3

1 + 1 +

2 3

3

3

1 3

+

(2 + 3 + 1)2 (2 + 3 + 1 + 1)

(2 + 3 + 1 + 1)2 (3 + 1)(3 + 1)2

2 3

2 3 2 3

3

3 3 3 3 + 3 3

(3 + 1)(3 + 1 + 1)2 (3 + 1)(2 + 3 + 1)2 (3 + 1)(2 + 3 + 1 + 1)2

3

2

2

2

(2 + 1)(2 + 3 + 1)2

2 3

2

2

2

+ (2 + 1)(2 + 3 + 1 + 1)2

2 3

3

3

23

+ (3 + 1)(3 + 1)3

2 3 2 3 3

233

(3 + 1)(3 + 1 + 1)3

233

(3 + 1)(2 + 3 + 1)3

+ 233

(3 + 1)(2 + 3 + 1 + 1)3

3 2 3 2 3

2

2

222

(2 + 1)(2 + 3 + 1)3 +

22

(2 + 1)(2 + 3 + 1 + 1)3

1

(1 + 1)(3 + 1 + 1)2

2 3

21

2 3

+ 1

3

+ 21

(1 + 1)(3 + 1 + 1)3

(1 + 1)(2 + 3 + 1 + 1)2 (1 + 1)(2 + 3 + 1 + 1)3

3

2

2 3 2 3

2

3 1 3 + 3 1 3

(1 + 1)(3 + 1)(3 + 1 + 1)3 (1 + 1)(3 + 1)(2 + 3 + 1 + 1)3

3

212

2 3

223

+ 2 2 3

(1 + 1)(2 + 1)(2 + 3 + 1 + 1)3 (2 + 1)(3 + 1)(2 + 3 + 1)3

2 3

223

2 3

613

+ 2 3 3

(2 + 1)(3 + 1)(2 + 3 + 1 + 1)3 (1 + 1)(3 + 1)(3 + 1 + 1)4

2 3

3

3

613

+ (1 + 1)(3 + 1)(2 + 3 + 1 + 1)4

3

2

2

612

+ (1 + 1)(2 + 1)(2 + 3 + 1 + 1)4

2 3

6

2 3

6

3

3

2 2 3

(2 + 1)(3 + 1)(2 + 3 + 1)4 +

2 3 23

(2 + 1)(3 + 1)(2 + 3 + 1 + 1)4

2 3 2 3

+ 623123

(1 + 1)(2 + 1)(3 + 1)(2 + 3 + 1 + 1)4

2 3

24123

+ 2 3

] (11)

(1 + 1)(2 + 1)(3 + 1)(2 + 3 + 1 + 1)5

2 3

In general

() =

2

1

1

1

1

[

+ 1 1

1

1 1

1 + 1

( + 1) ( + 1)2

( + 1 + 1)

( + 1 + 1)2

( + + 1)

1

1

1

=2

1

+ 1

+ 1

=2

(

+

+ 1)2

=2

(

+

+ 1

+ 1)

=2

(

+

+ 1

+ 1)2

+

1

( + 1)( + 1)2

( + 1)( + 1 + 1)2

( + 1)( + + 1)2

1

+

1

=2

( + 1)( + + 1 + 1)2 ( + 1)( + + 1)2

=2

1

=2

2

2

+ +

( + 1)( + + 1 + 1)2

( + 1)( + 1)3

( + 1)( + 1 + 1)3

=2

1

2

1

+

2

( + 1)( + + 1)3 ( + 1)( + + 1 + 1)3

=2

1

2

=2

1

2

+ 1

( + 1)( + + 1)3

( + 1)( + + 1 + 1)3

(1 + 1)( + 1 + 1)2

=2

=2

1

21

(1 + 1)( + 1 + 1)3

+ 1

(1 + 1)

1

( + + 1 + 1)2

+ 21

1

1

=2

21

(1 + 1)

( + + 1 + 1)3

(1 + 1) ( + 1)( + 1 + 1)3

+ 21

=2

1

+ 21

1

(1 + 1)

( + 1)( + + 1 + 1)3

(1 + 1)

( + 1)( + + 1 + 1)3

1

=2

2

1

=2

2

+

( + 1)( + 1)( + + 1)3 ( + 1)( + 1)( + + 1 + 1)3

=2

6

6

=2

1

1

(1 + 1) ( + 1)( + 1 + 1)4

+ 1

(1 + 1)

( + 1)( + + 1 + 1)4

6

1

=2

1

+ 1

6

(1 + 1)

( + 1)( + + 1 + 1)4

( + 1)( + 1)( + + 1)4

1

=2

=2

+ 6

( + 1)( + 1)( + + 1 + 1)4

=2

6

1

+ 1

+

(1 + 1) ( + 1)( + 1)( + + 1 + 1)4

=2

+ 1 ( + 1)!

(1 + 1) ( + 1)( + 1 + + 2 + 1 + 1)+2

=2 1 2

Total reliability of the cscade system is

= (1) + (2) + (3) + + () (13)

Numerical Calculations:

Table 1 ( = )

 R(1) R(2) R(3) 3 0.3 0.01 0.1 5 0.003121 0.002621 0.008866 0.014608 0.4 0.01 0.1 5 0.00182 0.001624 0.017581 0.021025 0.5 0.01 0.1 5 0.001199 0.001106 0.030321 0.032626 0.6 0.01 0.1 5 0.000855 0.000805 0.047809 0.049469 0.7 0.01 0.1 5 0.000645 0.000614 0.070803 0.072062 0.8 0.01 0.1 5 0.000506 0.000487 0.100101 0.101094 0.9 0.01 0.1 5 0.000409 0.000396 0.136545 0.13735 1 0.01 0.1 5 0.00034 0.000331 0.18102 0.181691

Table 2( = )

 R(1) R(2) R(3) 3 0.5 0.01 0.1 5 0.001199 0.001106 0.030321 0.032626 0.5 0.01 0.2 5 0.001199 0.000939 0.027745 0.027851 0.5 0.01 0.3 5 0.001199 0.000774 0.025713 0.027686 0.5 0.01 0.4 5 0.001199 0.000632 0.024095 0.027642 0.5 0.01 0.5 5 0.001199 0.000517 0.022799 0.024515 0.5 0.01 0.6 5 0.001199 0.000424 0.021754 0.023377 0.5 0.01 0.7 5 0.001199 0.00035 0.020911 0.02246 0.5 0.01 0.8 5 0.001199 0.000291 0.020228 0.021718 0.5 0.01 0.9 5 0.001199 0.000244 0.019677 0.02112 0.5 0.01 1 5 0.001199 0.000206 0.019232 0.020637

CONCLUSION:

The general expression for the reliability of cascade system was derived when stress and strength follow lindley distribution and the numerical values

(1), (2), (3) 3 was computed for some specific values of the parameters. From the tables, it is observed that as the stress parameter increases the system reliability increases and the system reliability decreases if the strength

parameter increases when stress and strength follow lindley distribution.

REFERENCES:

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