STOCHASTIC BEHAVIOUR OF A SPECIFIC GAS SEPARATOR FOR RELIABILITY MEASURES BY INTRODUCING ALGEBRA OF LOGICS

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ISSN (O): 2249-3905, ISSN(P): 2349-6525 | Impact Factor: 7.196

STOCHASTIC BEHAVIOUR OF A SPECIFIC GAS SEPARATOR FOR RELIABILITY MEASURES BY INTRODUCING ALGEBRA OF LOGICS

1_{Mansi Aggarwal,}2_{Dr R.B. Singh and}3_{Dr Deepankar Sharma}

1_{Research Scholar, Monad University, Hapur}

2_{Prof., Dept. of Maths, Monad University, Hapur}

3_{Prof., Dept. of Maths, Dr K N Modi Institute of Engg & Tech., Modinagar}

ABSTRACT

Present paper is concerned with the stochastic analysis of reliability measures of a specific gas separator system in an industry, which is having a number of components of varying nature. Gas separators are so much useful in many industries and its structure depends on the requirement of the industry. The mathematical analysis has been completed by using Boolean Function Technique by considering general and non-identical lifetime distributions of the components of the system. In particular, Weibull and Exponential time distributions have been taken as lifetime distributions of the components. Some important parameters such as Reliability, MTSF, system’s lifetime distribution, Hazard rate, MRLT (Mean residual life time) have been computed. A graphical study has also been appended at the end to highlight the importance of the results.

KEY WORDS: Stochastic analysis, Reliability measures, Boolean function, Algebra of logics, Mean residual life time.

1. INTRODUCTION

Figure-1 shows the block diagram of considered specific gas-separator on the inlet pipeline. Four pressure shutdown (PSD) valves, PSD1, PSD2, PSD3 and PSD4 have been installed

with voting unit known as Programmable Logic Controller (PLC). To avoid false information the PSD valves are connected in a configuration, 2-out-of-4: G. These valves can close automatically in case of problem in the system. These are held open by hydraulic pressure and when this pressure is bled off, the valves will close by the force of a pre-charged actuator. These PSD valves with PLC unit are connected with separator section in which there are 3 separators connected in configuration, 1-out-of-3: G. To control the pressure in the separator a Pressure Safety Valve (PSV) has been installed. In the case, when pressure in the separator increases beyond a specified high pressure p, then PSV is facilitated with a spring-loaded actuator, which may adjust the pressure at p. Another PSD valve PSD5, similar to previous PSD valves has been installed on the gas outlet from the

separator section.

2. ASSUMPTIONS

(1) Initially, all the main components of the system are good.

(2) The state of every component and the whole system is either operating or failed. (3) The states of all components are statistically independent.

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3. NOTATIONS 10 4 3 2

1,X ,X ,X ,X

X : States of PSD valvesPSD₁,PSD₂,PSD₃,PSD₄ andPSD₅, respectively.

5

X : State of Programmable Logic Controller (PLC).

8 7 6,X ,X

X : States of the separatorsS₁,S₂andS₃, respectively.

9

X : State of pressure safety valve (PSV).

i

X : Negation (complements) ofX_i i.e X_i1X_i.



/ : Conjunction / Disjunction.

) 10 ,

3 , 2 , 1

(i 

X_i : A binary variate taking values 1 or 0 respectively for the device is operating or failed.

4. FORMULATION OF MATHEMATICAL MODEL AND ITS SOLUTION The minimal path and the minimal cut sets for the considered complex system are:

Minimal path sets: Minimal cut sets:



1 2 5 6 9 10



1 



6  P



1 4 5 6 9 10



7 

11 

P



2 3 5 8 9 10



12 

P



2 4 5 6 9 10



13  P



2 4 5 7 9 10



12

x9

PSV

x1 PSD1

x6

x2 PSD2 x5 x10

x7 PSD5

x8

x3 PSD3

1-out of-3:G

x4 PSD4

2-out of-4:G

PSD: Pressure Shut-down Valve

PLC: Programmable Logic Controller PSV: Pressure Safety Valve

Fig-1 (Block diagram of the system under consideration)

PLC

S1

S2

S3

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By using Boolean Function Technique (B.F.T.), the conditions of capability for the successful operation of the system in the form of logical matrix are expressed as:





10 9 8 5 4 3

10 9 7 5 4 3

10 9 6 5 4 3

10 9 8 5 4 2

10 9 7 5 4 2

10 9 6 5 4 2

10 9 8 5 3 2

10 9 7 5 3 2

10 9 6 5 3 2

10 9 8 5 4 1

10 9 7 5 4 1

10 9 6 5 4 1

10 9 8 5 3 1

10 9 7 5 3 1

10 9 6 5 3 1

10 9 8 5 2 1

10 9 7 5 2 1

10 9 6 5 2 1

10 2

1

, ,

X X X X X X

X X

X

g  

…(1)

Applying algebra of logics, equation (1) may be written as:



X1,X2, X10



X5X9X10 f



X1,X2,X3,X4,X6,X7,X8



g    …(2)

(5)





8 4 3

7 4 3

6 4 3

8 4 2

7 4 2

6 4 2

8 3 2

7 3 2

6 3 2

8 4 1

7 4 1

6 4 1

8 3 1

7 3 1

6 3 1

8 2 1

7 2 1

6 2 1

8 7 6 4 3 2 1

, , , , , ,

X X X

X X X X X X X

f 

… (3)

Substituting the following values in (3),

6 4 1 7

8 3 1 6

7 3 1 5

6 3 1 4

8 2 1 3

7 2 1 2

6 2 1 1

X X X K

      

7 3 2 11

6 3 2 10

8 4 1 9

7 4 1 8

X X X K

   

7 4 3 17

6 4 3 16

8 4 2 15

7 4 2 14

6 4 2 13

8 3 2 12

X X X K

     

and K₁₈ X₃ X₄ X₈

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



18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 9 8 7 6 5 4 3 2 1 13 12 11 10 9 8 7 6 5 4 3 2 1 12 11 10 9 8 7 6 5 4 3 2 1 11 10 9 8 7 6 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 6 5 4 3 2 1 5 4 3 2 1 4 3 2 1 3 2 1 2 1 1 8 7 6 4 3 2

1, , , , , ,

K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K X X X X X X X f                                                                                                                                                           … (4)

Using algebra of logics, one may have the following:

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6 4 2 4 2 2 13 X X X X X X K      , 7 4 2 4 2 2 14 X X X X X X K      , 8 4 2 4 2 2 15 X X X X X X K      , 6 4 3 4 3 3 16 X X X X X X K      , 7 4 3 4 3 3 17 X X X X X X K      , 8 4 3 4 3 3 18 X X X X X X K     

Now, we have:

6 2 1

1 X X X

K  …(5)

7 6 2 1 7 2 1 6 2 1 2 1 1 2 1

X X X X X X X

X X X X X X K

K   

   

 … (6)

7 2 1 2 1 1 6 2 1 2 1 1 2 1 X X X X X X X X X X X X K K          

A

 

say

X X X X X X X       7 6 2 1 2 1 1 8 7 6 2 1 8 2 1 3 2

1K K A X X X X X X X X

K       …(7)

Similarly, we can find the following:

6 3 2 1 4 3 2

1K K K X X X X

K     …(8)

7 6 3 2 1 5 4 3 2

1K K K K X X X X X

K       …(9)

8 7 6 3 2 1 6 5 4 3 2

1K K K K K X X X X X X

K         …(10)

6 4 3 2 1 7 6 5 4 3 2

1K K K K K K X X X X X

K         …(11)

7 6 4 3 2 1 8 7 6 5 4 3 2

1K K K K K K K X X X X X X

K           …(12)

8 7 6 4 3 2 1 9 8 7 6 5 4 3 2

1K K K K K K K K X X X X X X X

K             …(13)

6 3 2 1 10 9 8 7 6 5 4 3 2

1K K K K K K K K K X X X X

K           …(14)

7 6 3 2 1 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K X X X X X

K             …(15)

8 7 6 3 2 1 12 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K K X X X X X X

K               …(16)

6 4 3 2 1 13 12 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K K K X X X X X

K               …(17)

7 6 4 3 2 1 14 13 12 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K K K K X X X X X X

K                 …(18)

8 7 6 4 3 2 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K K K K K X X X X X X X

K                   …(19)

6 4 3 2 1 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K K K K K K X X X X X

K                  …(20)

7 6 4 3 2 1 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K K K K K K K X X X X X X

(8)

…(21)

8 7 6 4 3 2 1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

1K K K K K K K K K K K K K K K K K X X X X X X X

K                     

…(22)

Now making use of equations (5) through (22) in equation (4), we obtain





Q

 

say

X X X X X X X

X X X X X X

X X X X X

X X X X X X X

X X X X X X

X X X X X

X X X X X X

X X X X X

X X X X

X X X X X X X

X X X X X X

X X X X X

X X X X X X

X X X X X

X X X X

X X X X X

X X X X

X X X

X X X X X X X

f 

  



 

  



 



 

  

 



  



 

  

 



  



8 7 6 4 3 2 1

7 6 4 3 2 1

6 4 3 2 1

8 7 6 4 3 2 1

7 6 4 3 2 1

6 4 3 2 1

8 7 6 3 2 1

7 6 3 2 1

6 3 2 1

8 7 6 4 3 2 1

7 6 4 3 2 1

6 4 3 2 1

8 7 6 3 2 1

7 6 3 2 1

6 3 2 1

8 7 6 2 1

7 6 2 1

6 2 1

8 7 6 4 3 2 1

, , , , , ,

…(23)

Now in view of equations (2) and (23), we have



X X X



X X X Q

(9)

10 9 8 7 6 5 4 3 2 1 10 9 7 6 5 4 3 2 1 10 9 6 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1 10 9 7 6 5 4 3 2 1 10 9 6 5 4 3 2 1 10 9 8 7 6 5 3 2 1 10 9 7 6 5 3 2 1 10 9 6 5 3 2 1 10 9 8 7 6 5 4 3 2 1 10 9 7 6 5 4 3 2 1 10 9 6 5 4 3 2 1 10 9 8 7 6 5 3 2 1 10 9 7 6 5 3 2 1 10 9 6 5 3 2 1 10 9 8 7 6 5 2 1 10 9 7 6 5 2 1 10 9 6 5 2 1 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X                                            … (24)

5. RELIABILITY ANALYSIS

Finally, the probability of successful operation (i.e. reliability) of the system is given by







₁, ₂,., ₁₀ 1



Pg X X X

R_S



1 2 6 1 2 6 7 1 2 6 7 8 1 2 3 6 1 2 3 6 7 1 2 3 6 7 8

10 9

5R R RR R RRQ R RRQQ R RQ RR RQ RQ R RQ RQ Q R

R     

 6 3 2 1 8 7 6 4 3 2 1 7 6 4 3 2 1 6 4 3 2

1QQR R RQQ RQ R RQQ RQQ R QR R R

R   

 7 6 4 3 2 1 6 4 3 2 1 8 7 6 3 2 1 7 6 3 2

1R RQ R QR RQ Q R QRQ R R QR Q RQ R

Q   

 6 4 3 2 1 8 7 6 4 3 2

1R QR QQ R QQ R R R

Q 

 Q₁Q₂R₃R₄Q₆R₇Q₁Q₂R₃R₄Q₆Q₇R₈





1 2 7 1 2 6 1 3 6 1 2 8 1 3 7 1 3 8 1 4 6 1 4 7

10 9

5R R RR R RR R RRR RR R RR R RR R RR R RR R

R       

 7 4 3 6 4 3 8 4 2 7 4 2 6 4 2 8 3 2 7 3 2 6 3 2 8 4

1R R R R R R R R R R R R R R R R R R R R R R R R R R

R        



8 4 3R R

R

 R₁R₂R₆R₇R₁R₂R₆R₈R₁R₂R₇R₈2R₁R₂R₃R₆2R₁R₂R₃R₇R₁R₃R₆R₇

8 3 2 1 2RR R R

 R₁R₃R₆R₈R₁R₃R₇R₈2R₁R₂R₄R₆2R₁R₃R₄R₆2R₁R₂R₄R₇

7 4 3 1 2RR R R

 R₁R₄R₆R₇2R₁R₂R₄R₈2R₁R₃R₄R₈R₁R₄R₆R₈R₁R₄R₇R₈R2R3R6R7

8 4 3 2 7 6 4 2 7 4 3 2 6 4 3 2 8 7 3 2 8 6 3

2R R R R R R R 2R R R R 2R R R R R R R R 2R R R R

R     

 8 7 6 2 1 8 7 4 3 8 6 4 3 7 6 4 3 8 7 4 2 8 6 4

2R R R R R R R R R R R RR R R R R R R RR R R R

R     

(10)

6 4 3 2 1 8 7 6 3 1 8 7 3 2 1 8 6 3 2 1 7 6 3 2

1 2 2 3

2RR R R R  RR RR R  RR R R R RR R R R  RR R R R



8 6 4 2 1 8 4 3 2 1 7 6 4 3 1 7 6 4 2 1 7 4 3 2

1 2 2 3 2

3RR R R R  RR R R R  RR R R R  RR RR R  RR R R R



8 7 6 3 2 8 7 6 4 1 8 7 4 3 1 8 7 4 2 1 8 6 4 3

1 2 2

2RR R R R  RR R R R  RR R R R RR R R R R R R R R



8 7 6 4 3 8 7 6 4 2 8 7 4 3 2 8 6 4 3 2 7 6 4 3

2 2 2

2R R R R R  R RR R R  R R R R R R R R R R R R R R R



8 7 4 3 2 1 8 6 4 3 2 1 7 6 4 3 2 1 8 7 6 3 2

1 3 3 3

2RR R R R R  RR R R R R  RR R R R R  RR R R R R





8 7 6 4 3 2 1 8 7 6 4 3 2 8 7 6 4 3 1 8 7 6 4 2

1 2 2 3

2RR R R R R  RR R R R R  R R R R R R  RR RR R R R

 …(25)

where, R_i be the reliability of ith state x_i and Q_i 1R_i be the corresponding unreliability, 10

1 i

 

 .

6. SOME PARTICULAR CASES

Case 1: If all the components are identical: In this case, the reliability of every component is taken as R. Then the equation (25) yields

S

R R6



1842R39R217R33R4



…(26)

Case 2: When life time distribution of each component is Weibull: In this case, if random

variable Ti denotes the life time of ithcomponent, then its life time probability density

function (p.d.f.) is given by:

 

_ _ 



 t

i i

i

e t t

g  1 

where, _i be the rate of failure of ith component and  is shape parameter. Then reliability of ith component is given by

_R_i

 

_t _eit; i1,2.---10,  0,t0,_i 0

and the system’s reliability at time t is given by

 

_

 



 _

 59

1 60

1 j

t d i

t c S

j

i _e

e t

R

 

…(27)

where,

8 6 4 3 10

1

1 



    



i i

c , ₃ ₄ ₇ ₈

10

1

2 



    



i i

c ,

7 6 4 3 10

1

3 



    



i i

c , ₂ ₄ ₇ ₈

10

1

4 



    



i i

c ,

8 6 4 2 10

1

5 



    



i i

c , ₂ ₄ ₆ ₇

10

1

6 



    



i i

c ,

8 7 3 2 10

1

7 



    



i i

c , ₂ ₃ ₆ ₈

10

1

8 



    



i i

c ,

7 6 3 2 10

1

9 



    



i i

c , ₁ ₄ ₇ ₈

10

1

10 



    



i i

c ,

8 6 4 1 10

11 



    

 i

c , ₁ ₄ ₆ ₇

10

12 

  



i i c

c

c ,

8 6 10

1 32 31

30   



  



i i c

c

c , ₃ ₈

10

1 34

33  



  



i i c

c ,

8 2 10

1 36

35  



  



i i c

c , ₆ ₇

10

1 39 38

37   



  



i i c

c

c ,

7 3 10

1 41

40  



  



i i c

c , ₂ ₇

10

1 43

42  



  



i i c

c ,

6 3 10

1 45

44  



  



i i c

c , ₂ ₆

10

1 47

46  



  



i i c

c , ₂ ₃

10

1

48 



  



i i

c ,

4 1 10

1

49 



  



i i

c , ₁ ₈

10

1 51

50  



  



i i c

c , ₁ ₇

10

1 53

52  



  



i i c

c ,

6 1 10

1 55

54  



  



i i c

c , ₁ ₃

10

1

56 



  



i i

c , ₁ ₂

10

1

57 



  



i i

c ,





 

 10

1 60 59 58

i i c

c

c  , ₃ ₄ ₈

10

1

1 



   



i i

d , ₃ ₄ ₇

10

1

2 



   



i i

d ,

6 4 3 10

1

3 



   



i i

d , ₄ ₇ ₈

10

1 5

4  



   



i i d

d ,

8 6 4 10

1 7

6  



   



i i d

d , ₂ ₄ ₈

10

1

8 



   



i i

d ,

7 6 4 10

1 10

9  



   



i i d

d , ₂ ₄ ₇

10

1

11



   



i i

d ,

6 4 2 10

1

12 



   



i i

d , ₃ ₇ ₈

10

1 14

13  



   



i i d

d ,

8 7 2 10

1 16

15  



   



i i d

d , ₃ ₆ ₈

10

1 18

17  



   



i i d

(12)

8 6 2 10

1 20

19  



   



i i d

d , ₂ ₃ ₈

10

1

21 



   



i i

d ,

7 6 3 10

1 23

22  



   



i i d

d , ₂ ₆ ₇

10

1 25

24  



   



i i d

d ,

7 3 2 10

1

26 



   



i i

d , ₂ ₃ ₆

10

1

27 



   



i i

d ,

8 4 1 10

1

28 



   



i i

d , ₁ ₄ ₇

10

1

29 



   



i i

d ,

6 4 1 10

1

30 



   



i i

d , ₁ ₇ ₈

10

1 32

31 



   



i i d

d ,

8 6 1 10

1 34

33  



   



i i d

d , ₁ ₃ ₈

10

1

35 



   



i i

d ,

7 6 1 10

1 37

36  



   



i i d

d , ₁ ₃ ₇

10

1

38 



   



i i

d ,

6 3 1 10

1

39 



   



i i

d , ₁ ₂ ₈

10

1

40 



   



i i

d ,

7 2 1 10

1

41 



   



i i

d , ₁ ₂ ₆

10

1

42 



   



i i

d ,

4 10

1 44

43  

 



i i d

d , ₂

10

1 57

56  



 



i i d

d and ₁

10

1 59

58  



 



i i d

d .

The expression for mean time to system failure (MTSF), in this case, is given by

 





 0

. . .

.T SF R t dt

M _S

dt e

e

j t d i

t

ci j







 

   

 

 

0

59

1 60

1

 

 

_  

  

 

    





 

59

1 1 60

1 1

j j i

i d

c  



 …(28)

Case 3: When life time distribution of each component is exponential: Now since

exponential distribution is a particular case of Weibull distribution for 1, so by putting 1



(13)

 

_

 



 _

 59

1 60

1 j

t d i

t c S

j

i _e

e t

R …(29)

and



 



 59

1 60

1

1 1

j j i ci d

MTSF …(30)

where, ci’s and dj’s have mentioned earlier.

7. SYSTEM’S LIFE TIME DISTRIBUTION

Here we want to obtain system’s life time probability density function (p.d.f.), hazard rate and moments when the component’s life time distributions are Weibull with different parameters as discussed in case 2.

Let random variable T denotes the system’s life time. Then the p.d.f. of T is given by:

 

R

 

t

dt d t

f  _S

   

 







 

 59

1 60

1 1

j

t d j i

t c i

j

i _d _e

e c

t  

 …(31)

Again, the system’s hazard rate will be

 

_{ }

   

 



   

 

 





 

  

59

1 60

1

59

1 60

1 1

j t d i

t c

j

t d j i

t c i

S _i _j

j i

e e

e d e

c t

t R

t f t r

 



…(32)

The first moment of T about origin is given by

 



_

 

0

t dt f t T

E







 

   

 

 

0

59

1 60

1

dt e

d e

c t

j

t d j i

t c i

j i

 



 

_  

  

 

    





 

59

1 1 60

1 1

j j i

i d

c  



 …(33)

Second moment of T about origin is given by

 

_  

  

 

    





 

59

1 2 60

1 2

2 2 2 1 1

j j i

i d

c T

E

 



In general, rth moment of T about origin is given by

 

_  

  

 

    





 

59

1 60

1

1 1

j

r j i

r i r

d c

r r T E

 



 …(34)

_  

  

 

    





 

59

1 2 60

1 2

1 1

2 2

j j i

i d

c  



_{ }

 

2 59

1 1 60

1 1 2

1 1

    

  

 

   

  

     







 j

j i

i d

c  



 …(35)

8. MEAN RESIDUAL LIFE TIME (MRLT)

MRLT is the stochastic expectation of remaining life of the system, after surviving for a period t and can be obtained as:

 

t E



X t : Xt



 =

_{   }





t

dx x R t R

1

_{ }



 



 t

S t dt

R t R

1

 

_

_{ }



    

  

            







60

1 1

i ct

i

c t

R  

 



 



  

    

  

            







59

1 1

1

j d t

j

d  



 

_{ }

     

 

      

      

     

 

    







60

1 1

1 1 1

i

t c

i

c t

R

  

 



 

    

 

      

      

     







59

1 1

j

t d

j

d

 



 …(36)

where, 



 

x n t xn e t dt

0 1

is incomplete gamma function. The value of incomplete gamma

function can be seen from the table.

9. NUMERICAL COMPUTATION

For a more concrete study of the system’s behaviour, we calculate the values of reliability, failure time distribution and hazard rate of the system with respect to mission time t for different values of, keeping the other parameters fixed as:

005 . 0 ,

004 . 0 ,

003 . 0 ,

002 . 0 , 001 .

0 ₅ ₆ ₇ ₈ ₉ ₁₀

4 3 2

1             

 .

By using these values, the reliability Rs

 

t and failure time distribution f

 

t of the system

are given by equations (27)and (31), respectively.

 



 _t _t _t _t _t

t

s t e e e e e e

R ( )18 .016 24 .020 6 .022 9 .018 3 .024 18 .019

24e.017t 8e.023t 9e.021t …(27)



    



 t t t t t

e e

e t

t

f( ) 10.288 .016 0.48 .020 0.132 .022 0.162 .018 0.072 .024

0.342e.019t 0.408e.017t 0.184e.023t 0.189e.021t



…(31) The changes in Rs(t), f(t) and r(t) have been represented by graphs shown in figures 2, 3 and

4, respectively.

(15)

The figure-2 represents the reliability of system with respect to time when some fixed values have. From the graph we conclude that the reliability of the system approaches to zero more rapidly when we increase the value of.

Figure-3 gives the system’s failure time density with respect to time when  has some fixed values. The density function increases initially for bigger value of  and after certain period of time it decreases rapidly for value of. For higher values of  curves become more peaked.

Figure-4 shows the trends of the system’s hazard rate with respect to time when  has some fixed values. The hazard rate of the system increases as the values of  increases.

Fig-2: Reliability vs time

0

0.2

0.4

0.6

0.8

1

1.2

0

6

16

26

36

46

56 t --->

R

s

(t

)

--->

B=.25

B=.75

B=1

B=1.25

(16)

Fig-3: Failure time density vs time

Fig-4: Hazard rate vs time 0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 1 6 11 16 21 26 31 36 41 46 51 56

t

f(t)

B=.25

B=.75

B=1

B=1.25

B=1.75

0.0000000 0.1000000 0.2000000 0.3000000 0.4000000 0.5000000 0.6000000

1 6 11 16 21 26 31 36 41 46 51 56 61

t --->

r(

t)

-

-->

(17)

REFERENCES

1. Cluzeau, T.; Keller, J.; Schneeweiss, W. (2008): “An Efficient Algorithm for Computing the Reliability of Consecutive-k-Out-Of-n:F Systems”, IEEE TR. on Reliability, Vol.57 (1), 84-87.

2. Gupta P.P., Agarwal S.C. (1983): “A Boolean Algebra Method for Reliability Calculations”, Microelectron. Reliability, Vol.23, 863-865.

3. Lai C.D., Xie M., Murthy D.N.P. (2005): “On Some Recent Modifications of Weibull Distribution”, IEEE TR. on Reliability, Vol.54 (4), 563-569.

4. Tian, Z.; Yam, R. C. M.; Zuo, M. J.; Huang, H.Z.(2008): “Reliability Bounds for Multi-State k-out-of- n Systems”, IEEE TR. on Reliability, Vol.57 (1), 53-58. 5. Zhimin He., Han T.L., Eng H.O. (2005): “A Probabilistic Approach to Evaluate the