CC BY
27
Reliability Modelling and Assessment of Multi Standby
Hybrid System
Ibrahim Yusuf, Surajo Mahmud Umar
Department of Mathematical Sciences, Bayero University, Kano, Nigeria iyusuf.mth@buk.edu.ng, smumar.mth@buk.edu.ng
Abstract
Systems connected to an external supporting device for their operations viewed as hybrid systems have been manufactured to meet the demand of industries, economic growth and populace in general. Companies and organizations heavily rely on these systems to conduct their business. The paper deals with the reliability and availability characteristics of four different systems requiring external supporting device for their operation. The system consists of main unit connected to the cold standby supporting devices. The failure and time of both main unit and supporting device are assumed to be exponentially distributed. Markov models are developed and differential difference equations are derived to obtain explicit expressions for the steady-state availability and mean time to failure and perform analytical and numerical comparisons. Comparisons show that system with five cold standby supporting devices is the most reliable system.
Keywords: Availability, mean time to failure, supporting device, single unit
Mathematics Subject Classification (2000): 90B25
1. Introduction
High system reliability and availability play a vital role towards industrial growth as the profit is directly dependent on production volume which depends upon system performance. Thus the reliability and availability of a system may be enhanced by proper design, optimization at the design stage and by maintaining the same during its service life. Because of their prevalence in power plants, manufacturing systems, and industrial systems, many researchers have studied reliability and availability problem of different systems. Hajeeh (2012) deals with availability of a system with different repair options. Hu et al. (2012) presents availability analysis and design optimisation for a repairable series-parallel system with failure dependencies. Jain and Rani (2013) studied the availability analysis for repairable system with warm standby, switching failure and reboot delay. Wang et al. (2012) performed comparative analysis of availability between two systems with warm standby units and different imperfect coverage. Wang and Chen (2009) performed comparative analysis of availability between three systems with general repair times, reboot delay and switching failures.
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 (2014) performed comparative analysis of profit between three dissimilar repairable redundant systems using supporting external device for operation. Yusuf et al (2016) performed reliability computation of a linear consecutive 2-out-of-3 system in the presence of supporting device. Yusuf (2016) presents reliability evaluation of a parallel system with a supporting device and two types of preventive maintenance. The problem considered in this paper is different from the work of discussed authors above. In this paper, a single unit system connected to cold standby external supporting device is considered. The objectives of this paper are: to derive the explicit expressions for the availability and mean time to failure, to determine the optimal system. 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 analytical comparison between the systems are presented in section 4. Numerical examples are presented in section 5. Finally, we make some concluding remarks in Section 6.
2. Description and States of the System
In this paper, a single unit system connected to an external cold standby supporting devices is considered. It is assumed that the system most work with one supporting device. System I has main unit with five cold standby supporting devices, system II has four cold standby supporting devices, system III has three cold standby supporting devices, system IV has two cold standby supporting devices. It is also assumed that the switching from standby to operation is perfect. Both the 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 ^. Whenever a primary supporting device fails, it is immediately sent to repair with parameter ^ and a standby supporting device is switch to operation. System failure occurs
when the unit has failed with parameter Aq and it is sent for repair with parameter with
parameter ^q or the failure of all supporting device.
Table 1: Transition rate table
So Si S2 S3 s4 s5 Se Sy S8 S9 Sio
So 0 Ä1 0 0 0 X o 0 0 0 0 0
Si i"1 0 Ä1 0 0 0 0 X o 0 0 0
S2 0 i"1 0 Ä1 0 0 0 0 Xo 0 0
S3 0 0 i"1 0 Ä1 0 0 0 0 Xo 0
S4 0 0 0 i"1 0 0 Ä1 0 0 0 Xo
s5 ßo 0 0 0 0 0 0 0 0 0 0
s6 0 0 0 0 i"1 0 0 0 0 0 0
Sy 0 ßo 0 0 0 0 0 0 0 0 0
Se 0 0 ßo 0 0 0 0 0 0 0 0
s9 0 0 0 ßo 0 0 0 0 0 0 0
$10 0 0 0 0 0 0 0 0 0 0 0
3. Formulation of the Models
In order to analyse the system availability of system I, we define p (t) to be the probability that the system at t > 0 is in state s, , i = 0,1,2,3, ...,10 . 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),p1(0),p2(0).....Pio(0)] = [1,0,0,0,0,0,0,0,0,0,0]
Following Trivedi (2007), Wang et al. (2000), and Wang et al. (2006), we obtain the following differential equations from Figure 1: Po' = Uo + *i)Po(t) + ViPi(P) + VoPs(t) Pi (t) = P2 (t) = Ps (t) =
P4 (t) =
Ps'(t) =
Pe (t) = Pi (t) = Ps (t) = P9 (t) =
Pio'(t) =
-(Ao +Ai+ Vi)Pi(t) + AiPo(t) + ^iP2(t) + VoPi(t)
-(Xo +Ai+ hi)P2(t) + XiPi(t) + niPs(t) + noPs(t) -(Xo +Ai+ Hi)Ps(t) + AiP2(t) + nip4(t) + VoP9(t) -(Xo +Ai+ hi)P4(t) + Aip3(t) + niPe(t) + HoPioto -VoPi(t) + ^oPo(t) -^iPe(t) + Aip4(t) -VoPi(t) + ^oPi(t) -^oPs(t) + ¿oP2(t) -VoP9(t) + ^oPs(t) -VoPio(t) + ^oP4(t) This can be written in the matrix form as P = QPr
where Q
— I
— I
— I
— I
— I
— I
— I
— I
(1) (2)
Wo+h) Pi 0 0 0 Mo 0 0 0 0 0
h -(A0 + Ai + Hi) Mi 0 0 0 0 Mo 0 0 0
0 Ai -(A0+Ai+Mi) Mi 0 0 0 0 Mo 0 0
0 0 Ai -(A0+Ai + ^i) Mi 0 0 0 0 Mo 0
0 0 0 Ai -(A0 + Ai+^i) 0 Mi 0 0 0 Mo
0 0 0 0 -Mo 0 0 0 0 0
0 0 0 0 Ai 0 -Mi 0 0 0 0
0 0 0 0 0 0 -Mo 0 0 0
0 0 ¿o 0 0 0 0 0 -Mo 0 0
0 0 0 h 0 0 0 0 0 -Mo 0
0 0 0 0 h 0 0 0 0 0 -Mo-
Let T denote the time-to-failure of the system. We use the following procedure to develop the steady-state availability. 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 AtiM = Po(x) + Pi(x) + P2(x) + Ps(x) + P4(x) =
_MoMi+MoMi^i+MoMi^j+MoMi^i+MoMi^i__^^
M i A-oA-i+V l^oA l^o* i+P l^oA i+P fyo+Vo^ i+VoV i ^i+VoV 2 * i% i+VoV1
To develop the explicit expression for mean time to failure, we use the concept of Trivedi (2002), Wang and Kuo (2000) and Wang et al. (2006) as follows:
The procedures require deleting rows and columns of absorbing states of matrix Q and take the transpose to produce a new matrix, sayM. The expected time to reach an absorbing state is obtained from
E[TP(o^P(absorbing)] = P(0)(-M-1 )(1 , 1, 1, 1, 1) (4)
where the initial conditions are given by P(0) = \po(0),pi(0),p2(0), p3(0),p4(0)] = [1,0,0,0,0] and
/-(A0+A1) A1 0 0
Mi -(¿0+^1+^1) ¿i 0
M = 0 -(A0 + A1 + iu1) A1
0 0 Mi -(Ao+Ai+^i) ( 0 0 0 The explicit expression for is given by MTTfi
\
^ I
(Ac + Ai + Mi)/
MTTF = —
(5)
where
= + 4A0iu? + + ¿2("2 + 6A0Aiiu2 + 6^2 + 6A0A?iMi + + 4A0(Ui + + 4
+ + A1 + 6AgA2) + + 3^0^ + 3A0(ui + 3A0^2 + 4A0Aiiui + 3A0iu2 + A? + + + nl) + + 2A0Ai + 2A0iui +^2 + + M2) + ^i + Mi)
H = (A0iu4 +2A0Aiiul + + 6^2 + + 9^0Ailu2 + 4A0Ai(Ui + 12^^ + 12^^
+ 4^ +
10^ + A-0 + + 5A0Ai + 1020^3 + 5A0Ai)
Special Cases:
Case I: Availability and mean time to failure of system requiring four cold standbys supporting devices
A 1 9-9 - ^
¿r2(~) =
Mi^o^+M^o^+M^o^i+M^o+Mo^i+MoMi^+MoM^i+MoM^i+MoM4 N
M7TF, = —
2 »2
(6) (7)
where
w2 = (A0 + 3A0^ + 3A2^1 + 3A0A2 + 4A0Aliu1 + 3A0iu2 + A? + + + M?)
+ A^ + + +^2 + + ^2) + A2(A0 + Ax + + A? #2 = + 4^0^ + 3A0^ + + 6A0Aliu1 + + 4A0A? + 3A0A2^ + 2A0Aliu2 + ^0^2 + ¿4
H
1
Case II: Availability and mean time to failure of system requiring three cold standbys supporting devices
A ✓ _ _MoMi+MoM^i+MoMi^2__/ON
r3( ) ,,. 2 _ J^M2 J ,2 . J-:/3 J _J.||_ J?.,. ,,. ,,2 2 .J.,:. ,,3 ( )
¿0 + 2AQA1 + 2AOMI + 2A2+AiMi+^2+A1(AO+A1+Mi)
Case III : Availability and mean time to failure of system requiring two cold standbys supporting devices
MoMi+MoMi^i
Mi^o^i+M2^o+Mo^i+MoMi^i+MoM2 2^+Ao+Mi
/1 f^N _ _MoMl+MoMl^l__/-i
= „_,_, , ,,2j . „„J2 . „.„. J .a. „.„2 ( )
M77F4 = 12 (11)
4. Comparison between the systems
MAPLE software package was used to program the analytical comparison in this study. The results are presented below.
¿ri(~)-^2(~)=^ (12)
^M-^iK^^ (13)
(14)
M77Fi - M7TF2 =
At (AJ +4A0A! + 4A0^i + 6A0A2 + 9A0Ai^i + 6A0^2 + 4A0Ai + 6A0Ai^i + 6A0A1M2 + 4A0^? + A4 + Ai^ + A^2 + A^ + ^4)
H H
Ibrahim Yusuf, Surajo Mahmud Umar
RELIABILITY MODELLING AND ASSESSMENT OF MULTI STANDBY RT&A, No 3 (54)
Volume 14, September 2019 HYBRID SYSTEM r
2 3 = H2(xo + 3X:0^ + 2Xo^ + 3XoXl + 2Xo^^ + Xo^0+Xi) ( )
(17)
MTTF3 - MTTF4
+ 2.1+ + A^l + n3)
(A^ + 3Ao^ + 2Ao^ + 3AoA2 + 2AoA^ji^ + AojJ.\ + Al)(Ao + 2AoA^ + Aofa + T^) where
D4 = ^±AoA± + y.\Ao + + MoiMi + VoV-l D3 = ^AoA\ + li\AoAr + n\Ao + ^3 + Vo^l + VoV-lh + VoV-l D2 = ^AoA\ + &AoAl + li\AoAr + ^Ao + ^Af + Vo^l + ^oV-l^l + VoV-lh + VoP4 D^ = ^AoA4 + ^AoA3 + AoA2 + ^AoA^ + + fto^-i + + ^o^i^1 + ftoV-l^2 + +
MoMlFrom (12) to (17)
AT1(™) > An(^) > ATs(™) > AT4(™) MTTF1 (<*>) > MTTF2(~>) > MTTF3(<*>) > MTTF4(™)
> 0
5. Numerical example
Numerical examples are presented to demonstrate the impact of repair and failure rates on steady-state availability and mean time to failure of the system based on given values of the parameters. MATLAB software package was used to program the numerical comparison in this study. The results are presented below. For the purpose of numerical example, the following sets of parameter values are used:Ax = 0.3, Ao = 0.2, ^ = 0.6, ^ = 0.6 for Figures 2 and 3 and A^ = 0.4,Ao = 0.1, ^ = 0.05 for Figures 4 and 5.
■(5
>
<
0.02 0.04 0.06 0.
0.1
Figure 1: Availability against supporting device failure rateAi
t 0.4
'TO 0.3 <
-
AV4 -
AV3
- ^^^^ AV2 -
AV1
0.1
Figure 2: Availability against supporting device repair rate ni
0.76
0.75
CO 0.73
0 72
0 7
0.7
0.6
0.5
0.2
0 1
0
0
0 02
0 04
0 06
0 12
0 14
0 16
0. 18
0 2
Figures 1 and 3 show the results availability and mean time to failure for the four systems against the failure rate \. It is clear from the figures that system I (system with five standby supporting device) has higher availability and mean time to failure as compared to the other three systems. Similar observation is also depicted in Figures 2 and 4 with respect to repair rate . It is evident from these figures that system I (system with five standby supporting device) has higher availability and mean time to failure as compared to the other three systems. These tend to suggest that system I is better than the other systems.
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 steady-state availability was derived. Comparative analysis was performed analytically along with numerically example in this study. It is enough to mention first that the optimal system is system with five cold standbys supporting devices.
Thus,
¿ri(~) > ¿rz(~) > ¿T3(~) > MTTFi (k) > MTTF2 (K) > MTTF? (<*>) > MTTF4(-)
On the basis of the analytical and numerical results obtained f, it is suggested that the system reliability 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.
The system can further be developed into system with more standbys in solving reliability and availability problems.
The present study is important to system designers, engineers, maintenance managers and plant management for proper maintenance analysis, decision and safety of the system as a whole. The study will also assist engineers, decision makers and plant management to avoid an incorrect reliability assessment and consequent erroneous decision-making, which may lead to unnecessary expenditures, incorrect maintenance scheduling and reduction of safety standards.
Conflict of Interests
The authors declare that there is no conflict of interests.
References
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