 Open Access
 Total Downloads : 1388
 Authors : N. Swathi
 Paper ID : IJERTV4IS120558
 Volume & Issue : Volume 04, Issue 12 (December 2015)
 DOI : http://dx.doi.org/10.17577/IJERTV4IS120558
 Published (First Online): 12012016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Reliability of Time Dependent StressStrength 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 viceversa 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:
0 
R 

0.1 
0.01 
0.95503 
0.2 
0.01 
0.905285 
0.3 
0.01 
0.856008 
0.4 
0.01 
0.807435 
0.5 
0.01 
0.759787 
0.6 
0.01 
0.71327 
0.7 
0.01 
0.668066 
0.8 
0.01 
0.624337 
0.9 
0.01 
0.58222 
1 
0.01 
0.541824 
1.1 
0.01 
0.503237 
Table 1
Table 3
R 

1 
2 
0.268941 
2 
2 
0.5 
3 
2 
0.731059 
4 
2 
0.880797 
5 
2 
0.952574 
6 
2 
0.982014 
7 
2 
0.993307 
8 
2 
0.997527 
9 
2 
0.999089 
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
0 
R 

0.2 
0.01 
0.905285 
0.2 
0.02 
0.910242 
0.2 
0.03 
0.915204 
0.2 
0.04 
0.92017 
0.2 
0.05 
0.92514 
0.2 
0.06 
0.930114 
0.2 
0.07 
0.935091 
0.2 
0.08 
0.940072 
0.2 
0.09 
0.945055 
0.2 
0.1 
0.950042 
0.2 
0.11 
0.95503 
Table 2
0.2
1 2 3 4 5 6 7 8 9
strength parameter
Table 4
R 

10 
1 
0.999753 
10 
2 
0.999329 
10 
3 
0.998178 
10 
4 
0.995055 
10 
5 
0.986614 
10 
6 
0.964028 
10 
7 
0.905148 
10 
8 
0.761594 
10 
9 
0.462117 
10 
10 
0 
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

Bilikam , J.Edward(1985) : Some stochastic Stress Strength processes , vol .R34 , pp: 269274.

Kecheng Shen(1988) : On the relation between component failure rate and stree strength distributional charecterstics , Micro Electronics Reliability , vol. 28 , pp:801812.

M.N.Gopalan and P.Venkateswarlu(1982) : reliability analysis of time dependent cascade system
with deterministic cycle times , Micro Electronics Reliability , vol. 22, pp:841872.

Dong Shang Chang (1995) : Reliability bounds for the stress strength model , vol.29, pp:1519.

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

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

M.N.Gopalan and Venkateswarlu (1983) : Reliability analysis of time dependent cascade system with random cycle times , vol. 23, pp:355366.

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: 477479, Pergamon Press, OXPORD.