Cascade System Reliability with Stress and Strength Follow Lindley Distribution

Cascade System Reliability with Stress and Strength Follow Lindley Distribution

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.

Sriwastav and Pandit[2] derived expressions for the reliability of cascade system when stress and strength follow exponential distribution. They computed reliability values for a 2-cascade system with gamma and normal stress and strength distributions. Raghavachar et al [3] studied the reliability of a cascade system with normal stress and strength distribution. Uma Maheswari et al [4] studied the reliability of single strength under stresses with stress and strength follow exponential, normal and gamma distributions. They concluded that when stresses acted on a single strength component with stress and strength follow exponential distribution, then the reliability of the system is same as the series system. Uma Maheswari et al [5] studied the reliability comparison of cascade system with the addition of an strengths system when stress and strength follow exponential distribution. Uma Maheswari et al [6] studied the reliability of a cascade system with normal stress and exponential strength. Uma Maheswari et al [7] studied the reliability of single stress under strengths with stress and strength follow exponential, gamma and normal distributions. They concluded that when strengths acted on a single stress component with stress and strength follow exponential distribution, then the reliability of the system is same as the parallel system. Tirumala Devi et al [8] studied the reliability of a system with stresses and strengths. Chumchum et al [9] studied the cascade system with Pr( < < ). Ghitany et al [10] studied the properties and applications of lindley distribution.

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 × 2(2)2 × 1(1)1

0 0 0 0

× ()] (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)

Reliability for cascade model:

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:

1. K C Kapur and L R Lamberson, Reliability in Engineering Design, John Wiley and Sons, Inc., New York, 1977.

2. S.N.Narahari Pandit and G.L.Sriwastav, Studies in Cascade Reliability I, IEEE Transactions on Reliability, R-28, Iss 5, 1979, pp 411-414.

3. A.C.N.Raghava Char, B.Kesava Rao and S.N. Narahari Pandit, The Reliability of a Cascade system with Normal Stress and Strength distribution, ASR, vol. no.2, 1987, pp 49-54.

4. T.S.Uma Maheswari, Reliability of single strength under

   stresses, Micro Electron Reliability, Pergamon Press, OXFORD, vol 32, no 10, 1992, pp 1475-1478.

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6. T.S.Uma Maheswari, Reliability of a cascade system with normal stress and exponential strength, Micro Electron Reliability, Pergamon Press, OXFORD, vol 33, no 4, 1993, pp 927-936.

7. T.S.Uma Maheswari, Reliability of single stress under

   strengths of life distribution, Micro Electron Reliability, Pergamon Press, OXFORD, vol 34, no 3, 1994, pp 569-572.

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9. Chumchum Doloi, Munindra Borah and Jonali Gogoi, Cascade system with Pr( < < ), Journal of Informatics and Mathematical Sciences, vol 5, no 1, 2013, pp 37-47.

10. M.E.Ghitany, B.Atieh and S.Nadarajah, Lindley distribution and its application, Mathematics and Computers in Simulation, 78, 2008, pp 493-506.