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29
Probabilistic Models for Reliability Analysis of a System with Three Consecutive Stages of Deterioration
Ibrahim Yusuf
Department of Mathematical Sciences, Bayero University, Kano, Nigeria
[email protected] Ramatu Idris Gatawa Department of Mathematics, North West University, Kano, Nigeria [email protected]
Abstract
In this paper we present availability and mean time to failure estimation of a system where the deterioration rates follow the Weibull distribution. The paper presents modeling and evaluation of availability and mean time to system failure (MTSF) of a consecutive three stage deteriorating system. The system has three possible modes: working with full capacity, deterioration and failure mode. The three stages of deterioration are minor, medium and major deteriorations. Minor and major maintenance are allowed at minor and medium deterioration states and replacement at system failure. Explicit expressions for the availability and mean time to failure of the system are obtained analytically. Graphs have been plotted to determine the behavior of availability and mean time to system failure with respect to time for different values of deterioration, maintenance and replacement rates. Also, high values of the shape parameter decreases mean time to system failure and availability. The system is analyzed using differential difference equations. Keywords: list, keywords, enter, here
I. Introduction
In practical engineering applications, most repairable systems are deteriorative that system failure often cannot be as good as new, it is more reasonable for these deteriorating repairable systems to assume that the successive working times of the system after repair will become shorter and shorter while the consecutive repair times of the system after failure will become longer and longer. Most of these systems are subjected to random deterioration which can result in unexpected failures and disastrous effect on the system availability and the prospect of the economy. Therefore it is important to find a way to slow down the deterioration rate, and to prolong the equipment's life span. Maintenance policies are vital in the analysis of deterioration and deteriorating systems as they help in improving reliability and availability of the systems. Maintenance models can assume minor maintenance, major maintenance before system failure, perfect repair (as good as new), minimal repair (as bad as old), imperfect repair and replacement at system failure.
Several models on deteriorating systems under different conditions have been studied by several researchers such as Berenguer (2008), Frangopol et al (2004), Lam and Zhang (2003), Nicolai et al (2007), Rani and Sukumari (2014), Vinayak and Dharmaraja (2012), Yuan et al (2012), Yuan and Xu (2011). Analysis of reliability and availability model for deteriorating system have been studied under different conditions such as Liu et al. (2011) who investigated reliability analysis of a deteriorating system with delayed vacation of repairman, Tuan et al (2013).A Reliability-based Opportunistic Predictive Maintenance Model for k-out-of-n Deteriorating Systems, Xiao et al (2013) proposed the Bayesian reliability estimation for deteriorating systems with limited samples using the maximum entropy approach, Yusuf et al (2012). Presents modelling the reliability and availability characteristics of a system with three stages of deterioration, Zhang and Wang (2007)
deal with the study of deteriorating cold standby repairable system with priority in use. This paper considers a system with three consecutive stages of deterioration before failure and derived its corresponding mathematical models. Furthermore, we study mean time to system failure and availability using differential difference equation method. The focus of our analysis is primarily to capture the effect of minor maintenance, minor deterioration and shape parameter on mean time to system failure and availability.
The organization of the paper is as follows. Section 2 contains a description of the system under study. Section 3 presents formulations of the models. The results of our numerical simulations are presented and discussed in section 4. Finally, we make some concluding remarks in Section 5.
II. Description and States of the System
In this paper, one unit system is considered. It is assumed that the system most pass through three consecutive stages of deterioration which are minor, medium and major deterioration before failure. The unit is considered to be non repairable. At early state of the system life, the operating unit is exposed to minor deterioration with rate A and this deterioration is rectified through
minor maintenance ax which revert the unit to its earliest position before deterioration. If not
maintained, the unit is allowed to continue operating under the condition of minor deterioration which later changes to medium deterioration with rate X2. At this stage, the strength of the unit is
still strong that it can rectify to early state with rate a2. However, the system can move to major deterioration stage with rate A where the strength of the unit has decreases to the extent that it cannot be reverted to its early state, neither that of minor nor medium deterioration stages. Here the unit is allowed to continue operation until it fails with parameter A and the system is
immediately replaced by with a new one with ratea3. Deteriorating rates follow Weibull distribution f (t) = \ôts \ k = 1,2,3, ô is the shape parameter.
a
a,
Figure 1: Transition diagram of the system Table 1: States of the system_
State
Description
S0 Sx
52
53
54
Initial state, the system is operative.
The system is in minor deterioration mode and is under online minor maintenance, and is operative.
The system is in medium deterioration mode and is under online major maintenance and is operative.
The system is in major deterioration mode and is operative.
The system is inoperative._
III. Formulation of the Models
In order to analyze the system availability of the system, we define p (t) to be the probability that the system at t > 0 is in state S.. Also let p(t) be the row vector of these probabilities at time t. The initial condition for this problem is:
P(0) = [A (0), A (0), p (0), p3 (0), p4(0)] = [1,0,0,0,0]
We obtain the following differential difference equations from Figure 1: d P0 (t) = ~\Sts-lp0 (t) + aiPi (t) + a2p2 (t) + «3^4 (t)
d dt d
(t ) = -(«i + ^StS-1 ) Pi(t) + \5ts-lp0(t )
-p2 (t) = -(a2 + AStS-1)p2(t) + A2StS1p1(t)
dps (t) = S1 p3(t) + A5tSs lp2(t)
d p4 (t) = asp A (t) + AAStS-lps (t)
This can be written in the matrix form as
P = TPr
where
(1) (2)
i
-\StS-1 a a2 0 a
\StS-i -(a+^StS- 1 ) 0 0 0
0 ^SStS-i -(a2+\StS-1 ) 0 0
0 0 A^StS-x -A4StS-x 0
0 0 0 A4StS-1 a
\
T =
3 J
The steady-state availability (the proportion of time the system is in a functioning condition or equivalently, the sum of the probabilities of operational states) is given by
AV (œ) = P0 (<») + Pi (œ) + P2 (œ) + P3 (<») = N
(3)
D
where
N = a3A4(a + SA2t(s-1) ) (a2 + S^t(s-1) ) + a3SAi\t(s-1) (a2 + S^t(s-1) ) + a3S2AlA2A3t(s-1) + a3S2 (t(s-1) )
D = S3^^A>A4 ts 1) + a3S2Ai^^(t{S 1)) + a3S2\^A4(t{S 1)) + a3a2Ss^A4 tS 1 + a3S2^A>A4 t(t( 1)) + axa2a3AA + axaSKKtS-1) + a2a3S^A4éS-1) + a3S2A2^A4 (tS-1) )
Following Trivedi (2002), Wang and Kuo (2000), Wang et al. (2006) to develop the explicit for MTSF. The procedures require deleting rows and columns of absorbing states of matrix T and take the transpose to produce a new matrix, say M. The expected time to reach an absorbing state is obtained from
E
T
P(0)^P (absorbing )
P ( 0)(-M 1 )
m 1 1
v1/
M =
0
where the initial conditions are given by P(0) = [p (0), A (0), p (0), p (0)] = [1,0,0,0] and
(-\8t5-1 \St3-1 0
a -(a+XSt3-1 ) x-^st3-1 0
a 0 -(a+xst3-1 ) Xst3-1
v 0 0 0 -x4st3-1
The explicit expression for is given by MTSF
(4)
MTSF =
x(a+sxt(s-1) ) (a+sxt(s-1) )+sxx(a+sxt(s-1) )+s2 XXX (t(s-1) ) + s2 XXX (t(s-1) )
s3xXXX(t(s-1))3
(5)
IV. Discussion
Numerical examples are presented to demonstrate the impact of repair and failure rates on steady-state availability and net profit of the system based on given values of the parameters. For the purpose of numerical example, the following sets of parameter values are used: dx = 0.1, (X2= 0.3,
a3= 0.5, \ = 0.4, 4= 0.5, 4 = 0.6, XA = 0.2,5 = 0.9, 0 < t < 10
<D
30r
25-
-w
(fr 20-O
-l-J
<D
E
15
/' / >•'
/ / //
?
10 L
5
Time(t)
ai
=0.3
ai
=0.4
a1
=0.5
10
Figure 2: Mean time to system failure against for different values of (Xx (0.3,0.4,0.5)
0
0.94
to
>
<C
0.92
0.9
0.86
s
//y
/ v
hi \it
!f
4 x- 5 m 6 Time(t)
10
Figure 3: Availability against for different values of a (0 3,0.4,0.5)
0
0
2
3
7
8
9
0 Z5
tc
-w o
-l-J
<d
_J_L
4 5
Tlme(t)
10
Figure 4: Mean time to system failure against for different values of A(0.4,0.5,0.6)
0
1
£
1
03
>
<
0.75
4 T- 5 6 Time(t)
-----------
9 10
Figure 5: Availability against time for different values of 4(0.4,0.5,0.6)
0
1
2
3
7
8
40
30
tfr 20
10
\\
V
4 -r- 5 ... 6 Time(t)
_._,g=1.2
_._,g=1.3 _._,8=14
10
0
0
Figure 6: Mean time to system failure against for different values of 4(0.4,0.5,0.6)
0.95 0.9 ^ 0.85 0.8
03
< 0.75 0.7 0.65
0 1 2 3 4 5 6 7 8 9 10
Time(t)
Figure 7: Availability against time for different values of £(1.2,1.3,1.4)
Numerical results of mean time to system failure and availability with respect to time are depicted in Figures 2 and 3 for different values of minor maintenance rates a. In these Figure the mean
time to system failure and availability increases as time increases when ax increases from 0.3 to
0.5. This sensitivity analysis implies that minor maintenance to the system should be invoked to increase the life span of the system. On the other hand, simulations in Figures 4 and 5, depicts the impact of time on mean time to system failure and availability for different values of minor deterioration JJ . From these Figures, the mean time to system failure and availability decreases as
time increases for different values of JJ . The above sensitivity analysis depicted the effect of minor
maintenance and deterioration rates on mean time to system failure and availability. It can be observe that minor maintenance played a significant role in increasing the mean time to system failure and availability whereas minor deterioration slow down the mean time to system failure. From simulations depicted in Figures 6 and 7, it is evident that the choice of the shape parameter 5 influences the time taken for the system to reach the failure state. The higher the value of this shape parameter 5, the less the values of mean time to system failure and availability.
V. Conclusion
This paper studied a one unit system with three consecutive stages of deterioration before failure. Explicit expressions for the mean time to system failure and availability are derived. The numerical simulations presented in Figures 2 - 7 provide a description of the effect of maintenance and deterioration rates on mean time to system failure. On the basis of the numerical results obtained for particular cases, it is suggested that the system availability can be improved significantly by:
• Adding more cold standby units.
• Increasing the maintenance rate.
• Exchange the system at major deterioration with new one before failure.
References
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