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13
Mean Time To System Failure Assessment Of A Single Unit System Requiring Two Types Of Supporting Device
For Operation
Nura Jibrin Fagge* •
Department of Mathematics, Kano State College of Arts, Science and Remedial Studies, Kano, Nigeria e-mail: [email protected]
U.A. Ali •
Department of Mathematics, Federal university, Dutse, Nigeria e-mail: [email protected]
Ibrahim Yusuf •
Department of Mathematical Sciences, Bayero University, Kano, Nigeria e-mail: [email protected]
Abstract
This paper studies the mean time to system failure (MTSF) of single unit system operating with the help of two types of external supporting device. Each type of supporting device has two copies I and II. The system is analyzed using differential difference equation to develop the explicit expression for mean time to system failure. Based on assumed numerical values given to system parameters, graphical illustrations are given to highlight important results.
Keywords: availability, supporting device, probabilistic, single
I. Introduction
Proper maintenance planning plays a role in achieving high system reliability, availability and production output. It is therefore important to keep the equipments/systems always available and to lay emphasis on system availability at the highest order. In real-life situations we often encounter cases where the systems that cannot work without the help of external supporting devices connect to such systems. These external supporting devices are systems themselves that are bound to fail. Such systems are found in power plants, manufacturing systems, and industrial systems. Large volumes of literature exist on the issue relating to prediction of various systems performance connected to an external supporting device for their operations. Yusuf et al (2014) present mathematical modeling approach to analysis of mean time to system failure of two unit
cold standby system with a supporting device. Yusuf et al (2015) performed comparative analysis of MTSF between systems connected to supporting device for operation. Yusuf et al (2014) performed reliability computation of a linear consecutive 2-out-of-3 system in the presence of supporting device.
Existing literatures either ignores the impact of multi-supporting device on system performance. Such works laid emphasis on systems connected to one type of an external supporting device whose failure brings about total breakdown. More sophisticated models of systems connected to multi-external supporting device should be developed to assist in reducing operating costs and the risk of a catastrophic breakdown, to maximize output, system availability, and generated revenue, minimize cost, and assure ongoing quality of the parts being produced. The problem considered in this paper is different from discussed authors above. The purpose of this paper is twofold. The first purpose is to develop the explicit expressions for mean time to system failure. The second is to capture the effect of both failure and repair rates on mean time to system failure based on assumed numerical values given to the system parameters.
The organization of the paper is as follows. Section 2 presents model's description and assumptions. Section 3 presents formulations of the models. Numerical examples are presented and discussed in Section 4. Finally, we make a concluding remark in Section 5.
In this paper, a single unit system connected to two types of supporting device is considered. It is assumed that each type of supporting has a copy on standby and the switching is perfect. It is also assumed that the system work with either two copies of type I supporting device or two copies of type II supporting device or one copy of both type I and II. Both unit and supporting devices are assumed to be repairable. Each of the primary supporting devices fails independently of the state of the other and has an exponential failure distribution with parameter \ and X2 for type I and II respectively. Whenever a primary supporting device fails, it is immediately sent to repair with parameter f and ¡u2 and the standby supporting device is switch to operation. System failure
occur when the unit has failed with parameter X and service rate with parameter with parameter f or the failure of all copies of type I and type II.
II. Description of the System
Figure 1: The State transition diagram of System
III. Formulation of the Model
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 Si. Also let p(t) be the row vector of these probabilities at time t. The initial condition for this problem is:
P(0) = [Po (0), p (0), p2 (0)p0 (0)] = [1, 0,0,0,0,0,0,0,0,0,0]
We obtain the following differential difference equations from Figure 1: p0 (t ) = -(A + 2Ai ) P0 (t ) + V Pi (t ) + MP4 (t )
pi'(t) = -(A + A + A + V )Pi (t) + 2A P0 (t) + № (t) + VP3 (t) + VPe (t)
p2 (t ) = -(A+A+A + V2 ) Pi (t )+A Pi (t ) + V P5 (t )+Vi Pi (t ) + V Pi 0 (t )
P3 (t ) = -(A + 2A2 + V ) P3 (t ) + A Pi (t ) + VP* (t ) + V2P9 (t )
P4' (t) = VP4 (t)+AP0 (t)
P5 (t) = V2P5 (t) + AP2 (t)
Pe(t) = -VP(, (t) + APi (t)
Pi'(t) = -VPi (t) + AP2 (t)
Ps ' (t) = -VPs (t) + AP3 (t)
P9' (t) = V2P9 (t) + 2A2P3 (t)
Pi 0 ' (t ) = -V Pi 0 (t ) + A P2 (t ) (!)
This can be written in the matrix form as
P = TP, (2)
where
T =
M V 0 0 V 0 0 0 0 0 0 1
2\ S2 V2 V 0 0 V 0 0 0 0
0 A S3 0 0 V2 0 V 0 0 V
0 A 0 0 0 0 0 V V2 0
A 0 0 0 -V 0 0 0 0 0 0
0 0 A 0 0 V2 0 0 0 0 0
0 A 0 0 0 0 -V 0 0 0 0
0 0 A 0 0 0 0 -V 0 0 0
0 0 0 A 0 0 0 0 -V 0 0
0 0 0 2A 0 0 0 0 0 V2 0
V 0 0 A 0 0 0 0 0 0 0 -V,
£¡=(2 + 22), S2 = (2 + \+ 2+ fa), S3=(2 + 2+A + ¿u2), 8a = (2 + + fa )
It is difficult to evaluate the transient solutions, hence 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
P(0)^P(absorbing )
P ( 0)(-M )
1 1
v 1y
where the initial conditions are given by P(0) = [Po (0), p (0), p2 (0), P3 (0)] = [1,0,0,0] and
r-( 2M+X) 2\ 0
ff -(X + M+A + M) A 0 f -(X + M+M2 + ft ) v 0 f 0 -(M + 2A + f)
The procedure above is successful because of the following relations
M =
0
A 0
E
T
P(0)^P(absorbing )
] = P ( 0 )J
eMtdt,
(3)
(4)
where
J eMtdt
Mt^ = -m-1
The explicit expression for is given by MTSF
(5)
E
T
P(0)^-P( absorbing )
N
= MTSF = N D
(6)
N = 2AAf + fffM + 2ffA + 2fMA + A + 4f AA + mA + 2m M + 4fAA + 4AA + ViA +
5 A A + MMA+M2A + 3 MA+MAM + 2M3 + 2MA+6XMA + A A + MM + M A + 2AA+4XA + m2M+M2M2+2X(X+X +A+f)(2A+X +m)+2XA(2A+X +M)+2X2 (X+A+X +M)
D = 1 6XA2A + 8A3 A +10A2 A2 + 5AM3 + 2M4 + 4AA + 4X2A2 + MfA2 + 2XA + 2M3M + X3 A + MA+4A2M2+M^X2+2fA+6fMMA+4f2M2M+XMAA + 3MM2XA + 3fXA+f2fX+ 8fXM A + 2MM2AA + 3fAX + 2f2X2X2 + 4p2X2A + f2A2A + 10M2MM + 4fA2A + M2XM + fXA+3 mAMM+2MMA+2mMM2
0
30
0
IV. Numerical Examples
Numerical examples are presented to demonstrate the impact of failure and repair rates on mean time to system failure based on given values of the parameters. For the purpose of numerical example, the following sets of parameter values are used:
ft = 0.3, ft = 0.5, ft = 0.5, A = 0.2, A = 0.3, X( 0.4,0.6,0.8) for Figures 2 - 5.
32 1.8
0.1 0.2 0.3 0.4 0.5 0.6
0.8 0.9
Figure 2: Availability against type I supporting device repair rate fa for different values of 2(0.4,0.6,0.8)
i
08
— - — 2=0.4
— ■ — 2=06
— - — 2=0.8
0 1 02 0 3 04
05
2„
06 0 7 0 8 0 9
Figure 3: Availability against type I supporting device failure rate 2 for different values of 2(0.4,0.6,0.8)
— ■ — 2=0.4
- ■ - 2=0.6
- ■ - 2=0.8
0.1 0.2 0.3 0.4 0.5 0.6
0.8 0.9
Figure 4: Availability against type II supporting device repair rate fa for different values of 2(0.4,0.6,0.8)
2.6
2.4
a) 2
.6
.4
0
0.7
1
fa
0
1
2.6
2.4
2.2
2
aj 1.8
.6
.4
0
0.7
1
fa
2
2 2
! 2
"ro
EE 2
aj
bô
oo
o
a}
H 1
£=
a} 1
3 8 6
4 2 2 8 6
1 .4 1 .2 1
— — 2=04
— ■ — 2=06
— - — 2=0.8
V
s.
0.5
Figure 5: Availability against type II supporting device failure rate 22 for different values of
2(0.4,0.6,0.8)
0
0.1
0.2
0.3
0.4
0.6
0.7
0.8
O. 9
1
IV. Discussion
Numerical results of availability with respect to type k, k — I, II supporting devices repair fa
and failure rates 2i, i —1,2 for different values of2(0.4,0.6,0.8)are depicted in Figures 2 - 5
respectively. In Figures 2 and 4, the mean time to system failure increases as fa and fa for
different values of unit failure rate 2 . This sensitivity analysis implies that major maintenance to the unit and supporting devices should be invoked to improve and maximize the mean time to system failure, production output as well as the profit. On the other hand, Figures 3 and 5 show that the availability decreases as 2 and 22 increases for different values of unit failure rate 2.
This sensitivity analysis implies that major maintenance should be invoked to the unit and supporting devices to minimize the failure of the system in order to improve and maximize the mean time to system failure, production output as well as the profit.
V. Conclusion
This paper studied a single system connected to two types of supporting device type I and II for its operation. Explicit expression for the mean time to system failure was derived. The numerical simulations presented in Figures 2 - 5 provide a description of the effect of failure rate and repair rate on mean time to system failure for different values of unit failure rate 2 . On the basis of the numerical results obtained for particular cases, it is suggested that the system mean time to system failure can be improved significantly by:
(i) Adding more cold standby units.
(ii) Increasing the repair rate.
(iii) Reducing the failure rate of the system by hot or cold duplication method.
(iv) Exchange the system when old with new one before failure.
References
[1] K.S. Trivedi. Probability and Statistics with Reliability, Queuing and Computer Science Applications, 2nd edition, John Wiley and Sons, New York (2002).
[2] K. -H. Wang and C. -C. Kuo . Cost and probabilistic analysis of series systems with
mixed standby components, Applied Mathematical Modelling, 24: (2000), 957-967.
[3] K. -H. Wang, C.H. Hseih, and C.H Liou. Cost benefit analysis of series systems with cold standby components and a repairable service station, Quality technology and quantitative management, 3(1), (2006), 77-92.
[4] I. Yusuf, B. Yusuf, B and A.M. Lawan. Mathematical modeling approach to analyzing mean time to system failure of two unit cold standby system with a supporting device and repairable service station, OJMMO , 2(2): (2014), 7-15. DOI: 10.12966/ojmmo.05.01.2014
[5] I. Yusuf, U.A.Ali, M. Babagana and B. Gimba. Probabilistic models for MTSF comparison between systems connected to supporting device for operation, American Journal of Computational and Applied Mathematics 2015, 5(5): (2015) 140-146 DOI:10.5923/j.ajcam.20150505.02
[6] I.. Yusuf, B. Yusuf and S.I. Bala. Mean time to system failure analysis of a linear consecutive 3-out-of-5 warm standby system in the presence of common cause failure, Journal of Mathematics and computational Sciences, 4(1), (2014),58-66.
