Reliability of Time Dependent Stress-Strength System for Half Logistic Distribution

Reliability of Time Dependent Stress-Strength System for Half Logistic Distribution

N. Swathi

Department of Mathematics, Kakatiya University, Warangal, Telangana.

Abstract – Failure of a system may occur due to certain type of stresses acting on them. If these stresses do not exceed a certain threshold value the system may work for a long period. On the other hand, if the stresses exceed the threshold they may fail within no time. There is uncertainty about stress and strength random variables at any instant of time and also about the behavior of the variables with respect to time and cycles. Time dependent stress- strength models are considered with repeated application of stress and also the change of the strength with time. Reliability of time dependent stress- strength system is carried out by considering each of stress variables are random- fixed and strength variables are random independent and vice versa and deterministic stress and random independent strength and vice-versa for stress strength follow Half Logistic Distribution. It is observed from the computations that the reliability of the system is depending on the stress parameter and the strength

In the present paper, we have discussed deterministic stress and random fixed strength and vice versa, we had take Half logistic distribution. Reliability computations were done for different cycle lengths .The result is that the system reliability rapidly changes in Rayleigh distribution than the Exponential distribution.

STATISTICAL METHOD

X and Y denote the stress and strength of the system. f

(X) and g(Y) are probability density functions of X and Y . Then the reliability of the system is

parameter and another constant parameters

Key words: Half Logistic distribution, stress strength model, deterministic , random fixed.

INTRODUCTION

Time dependent stress strength models by considering with the repeated application of stress considered as the change of the distribution of strength with time. Stress is used to indicate any agency that tends to induce failure, while Strength indicates any agency resisting failure. Failure is defined to have occurred when the actual stress exceed the actual strength.

There is uncertainty about the stress and strength random variables at any instant of time and also about the

= () [ ()]

Or

= () [ () ]

The reliability computations for deterministic cycle times can take two cases

Case 1: Deterministic stress and random fixed strength

Let the stress be 0, a constant and the strength on

behavior of the random variables with respect to time and cycles .The two terms deterministic and random fixed are used to describe these two uncertainties. In deterministic, the variables assume values that are exactly known a priori. Random fixed refers to the behavior of the

the i th

Where

cycle given by

= 0 , i = 1 2 3 ——-

ai 0 are known constants. Further, the s are

variable with respect to time is fixed or the variable varies in time in a known manner .The failure of components under repeated stresses had been investigated primarily. Repeated stresses are characterized by the time, each load applied and the behavior of time intervals between the applications of loads.

assumed non- decreasing in time. The p.d.f of 0 , 0(0)

is assumed known. Then

The reliability after n cycles to R(t), the reliability at time t , where t is continuous. Simply when cycle times are deterministically known () = , <

But

= 0(0) 0

0+

= [ , , ]

+1,where is instant in time at which the i th cycle

1 2

occurs .The time dependent load was discussed by several researchers. Some of them are Bilikam etal [1]

,Kechengshen[2] , M.N.Gopalan[3] and Dongshang chang[4].

= [12, ] [23 , ]

[1] []

All but the last term in the R.H.S of above

equations are 1s because of restrictions on the ai s which

= 0(0) 0

+

cause the strength

yi to decrease in time. Hence

0

=

20

= [] = 0 (0)0

0+

0+

(1 + 0 )2 0

Case 2: Random fixed stress and deterministic strength Let = 0 + , i = 1,2,3,— denote the stress in cycle i , where bi s are known non negative constants , non

Then

Take 1 + 0 =

1

() = 2

2

decreasing in time . Further let the strength be held constant at 0. The p.d.f. of 0, 0(0) is assumed known.

The restrictions on guarantee non- decreasing. Stress, which in turn ensure that

= []

= ( )

= (0 + 0)

0

= 0(0)0

0

A random variable X is said to have Half logistic distribution if its p.d.f. is given by

1+0+

= 2 [1 1 ]

1 + 0+

20+

= 1 + 0+

For random fixed stress and deterministic strength, the reliability of the system is

R(t)= = []

= ( )

0

= 0(0)0

=

( ) 2

(1+)2

, 0 and

0

0

Similarly a random variable Y is said to have Half logistic distribution if its p.d.f. is given by

20

= (1 + 0 )2 0

0

2

() = (1 + )2 , 0

For deterministic stress and random fixed strength, the system reliability

R(t)= = []

= ( )

Then

Take 1 + 0 =

1+0+

() = 2

2

2

1 0+

= 1 + 0+

Reliability Computations:

Table 1

Table 3

Figure 1

Figure 3

1

1

0.95

0.9

0.85

reliability

0.8

0.75

0.7

0.65

0.9

0.8

reliability

0.7

0.6

0.5

0.4

0.3

0.6

0.55

0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

stress parameter

Table 2

0.2

1 2 3 4 5 6 7 8 9

strength parameter

Table 4

Figure 4

0.96

Figure 2

1

0.9

0.8

0.7

0.6

reliability

0.95

0.5

0.4

0.94

0.3

reliability

0.2

0.93

0.1

0.92

0.91

0.9

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

parameter (an)

0

1 2 3 4 5 6 7 8 9 10

parameter (bn)

CONCLUSION

The stress and strength follow Half Logistic distribution for deterministic stress and random- fixed strength and vice versa . In computations we observed that if the Stress parameter value (), increases the reliability also increases and constant value( x0 ) increases, the reliability decreases. The strength parameter value ( ) is increases, then reliability value decreases and constant value ( y0 ) increases then reliability also increases.

REFERENCES

1. Bilikam , J.Edward(1985) : Some stochastic Stress- Strength processes , vol .R-34 , pp: 269-274.

2. Kecheng Shen(1988) : On the relation between component failure rate and stree strength distributional charecterstics , Micro Electronics Reliability , vol. 28 , pp:801-812.

3. M.N.Gopalan and P.Venkateswarlu(1982) : reliability analysis of time dependent cascade system

   with deterministic cycle times , Micro Electronics Reliability , vol. 22, pp:841-872.

4. Dong Shang Chang (1995) : Reliability bounds for the stress- strength model , vol.29, pp:15-19.

5. Kapur,K.C. and L.R.Lamberson(1977) : Reliability in Engineering Design , Jhon Wiley and sons, Inc., New York.

6. S.C.Gupta and V.K.Kapoor : Fundamentals of Mathematical Statistics.

7. M.N.Gopalan and Venkateswarlu (1983) : Reliability analysis of time dependent cascade system with random cycle times , vol. 23, pp:355-366.

8. T.S.Uma Maheswari (1993) : Reliability comparison of an n- cascade system with the addition of an n- strength system, Micro Electron Reliability, Vol. 33, No. 4, pp: 477-479, Pergamon Press, OXPORD.