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STOCHASTIC ANALYSIS OF A REPAIRABLE SYSTEM OF NON-IDENTICAL UNITS WITH PRIORITY AND CONDITIONAL FAILURE OF REPAIRMAN
Naveen Kumar1, S.C. Malik2, N. Nandal*
department of Mathematics, SGT University, Gurugram *2Department of Statistics, M.D. University, Rohtak [email protected], 2sc [email protected], *[email protected]
Abstract
Here, we describe the stochastic analysis of a repairable system consisting two non-identical units called the main unit and the other is a duplicate unit. The units have direct complete failure from the operative state. A single repairman has been engaged to carry out the repair activities that can be failed while performing his jobs with the main unit. The repairman does repair activities of the duplicate unit without any problem. Priority for operation and repair to the duplicate unit is given over the main unit. The repairman performs with full efficiency after getting treatment. The distribution for failure rates of the units has been considered as negative exponential while arbitrary distributions have been taken for repair and treatment rates. The use of semi-Markov process and regenerative point technique has been made to study the probabilistic behavior of the system in different possible transition states. The reliability characteristics of the system model have been examined numerically and graphically for particular values of the parameters. The profit of the system has also been analyzed for some fixed values of the repair and other maintenance costs. Keywords: System of Non-identical Units, Priority, Conditional Failure of Repairman and Stochastic Analysis
I. Introduction
Over the years the researchers in the field of the reliability have been struggling to identify the best possible structure of the components and the techniques which can be used to improve the performance of repairable systems. As a result of which some reliability improve techniques for the repairable systems have been emerged as the provision of redundancy, priority in repair discipline and configuration of the components such as series, parallel, series-parallel, parallel-series, k-out-of-n and other mixed mode structures. The technique of cold standby redundancy with different repair policies has been used most frequently during stochastic modeling of repairable systems. Subramanian and Natarajan [10] developed an N-Unit standby redundant system with R repair facilities. Cao and Wu [2] discussed a cold standby system of two unit with replaceable repair facility. Smith [9] highlighted the concept of regenerative stochastic processes. On the other hand, the objective of the manufacturers is not only to produce the systems with considerable reliability but also to launch the products in markets with optimal balance between reliability and the production costs. To cope with this situation it becomes necessary to use systems
Naveen Kumar, S.C. Malik, N. Nandal
STOCHASTIC ANALYSIS OF A REPAIRABLE SYSTEM OF NON-IDENTICAL RT&A, No 1 (67)
UNITS WITH PRIORITY AND CONDITIONAL FAILURE OF REPAIRMAN Volume 17, March 2022 with non-identical units and appropriate repair facilities. The systems with non-identical units have also been studied in the past considering the ideas of priority in repair discipline. Kadyan et al. [4] discussed the stochastic modeling of a system of non-identical units with priority in different mode of failures. Salah and EL-Sherbeny [8] described a two unit non-identical parallel system subject to preventive maintenance and repairs. Kumar et al. [5] analyzed profit of a warm standby non-identical unit system with single server. Kadyan et al. [7] developed system models using the concept of priority. In the field of reliability research it is a common practice that the repair facility called server or repairman cannot fail while performing its assignments i.e. the jobs related to maintenance, repair of the faults and any other precautionary needs of the systems. This assumption on repair facility seems to be unrealistic when repair activities perturb due to the reasons which cause the failure of the service facility or any other catastrophic failure. Chen and Wang [3] analyzed a retrial machine repair problem with warm standbys and a single server with N-policy. Kumar and Nandal [6] developed a system of two non-identical units with conditional failure of repairman. Anuradha et al. [1] analyzed a 1-out-of-2: G System with Priority to Repair and Conditional Failure of Service Facility.
In view of the above observations and facts, the purpose of the present paper is to analyze stochastically a repairable system of non-identical units with the concept of priority and conditional failure of the repairman. The system has one main unit which is initially operative and the other unit is considered as duplicate in cold standby redundancy. The units have direct complete failure from the operative state. A single repairman is engaged to carry out the repair activities that can be failed while performing his jobs with the main unit. The repairman does repair activities of the duplicate unit without any problem. Priority for operation and repair to the duplicate unit is given over the main unit. The repairman performs with full efficiency after getting treatment. The distribution for failure rates of the units has been considered as negative exponential while arbitrary distributions have been taken for repair and treatment rates. The use of semi-Markov process and regenerative point technique has been made to study the probabilistic behavior of the system in different possible transition states. The reliability characteristics of the system model such as MTSF, availability, busy period of the server due to repair of the main and duplicate units, expected number of repairs of the units, expected number of the treatments given to the repairman and finally the profit function have been examined numerically and graphically for particular values of the parameters. The profit of the system has also been analyzed for some fixed values of the repair and other maintenance costs.
II. System Description
1. The system comprises of two non-identical units; one main unit which is initially operative and the other unit is considered as duplicate in cold standby redundancy.
2. The duplicate unit becomes operative after the failure of main unit.
3. A single repairman is engaged to carry out the repair activities that can be failed while performing his jobs with the main unit.
4. The repaired unit works as good as new.
5. Priority for operation and repair to the duplicate unit is given over the main unit.
6. The distribution for failure rates of the units has been considered as negative exponential while arbitrary distributions have been taken for repair and treatment rates
The state transition diagram shown in the figure 1 as:
: Regenerative Point O : Operative State Failed State
Figure 1 -.State Transition Diagram
a) Notations and Abbreviations
0 Operative state
□ Failed State
• Regenerative point
X Failure rate of the repairman
M0 Main unit is Operative and in normal mode
Des Duplicate unit is in cold standby
rt Failure rate of the main unit
r2 Failure rate of the duplicate unit
g(i)/G(i) pdf/cdf of the main unit repair time
htt)/H(t) pdf/cdf of the duplicate unit repair time
a(t)/i4(t) pdf/cdf of the treatment time of the repairman
MFur/MFUR Main unit failed under repair /continuously under repair from previous state
DFur/DFur Duplicate unit failed under repair/continuously under repair from previous state Mwur /MwUR Main unit waiting for repair /continuously waiting for repair from previous state Dwur/Dwur Duplicate unit waiting for repair/continuously waiting for repair from previous state SFur /SFur Repairman failed under treatment/continuously under treatment from previous state
pdf/cdf of transition from regenerative state (or non-regenerative state) SL or to a failed state Sj without visiting any regenerative state in (0,t]
Mean sojourn time (ßt) in state SL when system transits directly to state Sj so that ßi = Hjiriii and my = J0°° tdQij(t) = -qfj(0) The mean sojourn time in state SL
Probability that the system up initially in state Si e E is up at time t without visiting to any regenerative state
Probability that the repairman is busy in the state Sjup to time t without making any transition to any other regenerative state or returning to the same state via one or more non regenerative states
cdf of the first passage time from regenerative state SL to a failed state Probability that the system is in upstate at instant t given that the system entered regenerative state Si at t = 0
Probability that the repairman is busy at instant t given that the system entered regenerative
Rij/Qij
mLj Mi
Mi(t)
mtt)
m
4(0
Bitt)
Naveen Kumar, S.C. Malik, N. Nandal
STOCHASTIC ANALYSIS OF A REPAIRABLE SYSTEM OF NON-IDENTICAL RT&A, No 1 (67)
UNITS WITH PRIORITY AND CONDITIONAL FAILURE OF REPAIRMAN Volume 17, March 2022 state Sj at t = 0
RM* (t) Expected number of repair of the main unit given to the repairman in (0, t] such that the
system entered regenerative state Si at t = 0
RD* (t) Expected number of repair of the duplicate unit given to the repairman in (0, t] such that the
system entered regenerative stated at t = 0
TR* (t) Expected number of Treatment given to the repairman in (0, t] such that the system entered
regenerative state Si at t = 0
Ä Symbol for Stieltjes convolution
© Symbol for Laplace convolution
*/** Symbol for Laplace Transform/ Laplace Stieltjes Transform
P Profit function of the system
Ko Revenue per unit to the system
Ki Cost per unit for which repairman is busy to repair the main unit.
K2 Cost per unit for which repairman is busy to repair the duplicate unit.
K3 Cost per unit for repair of main unit
K4 Cost per unit for repair of duplicate unit
K, Cost per unit treatment given to the repairman
TP Transition Probabilities
MSTs Mean Sojourn Times
MTSF Mean Time to System Failure
LT Laplace Transform
LST Laplace Stieltijes Transform
LIT Laplace Inverse Transform
s-MP semi-Markov Process
RPT Regenerative Point Technique
III. Reliability Measures of the System
a) Transition Probabilities
Simple probabilistic considerations yield the following expression for the non-zero elements pij =
1™ Qij(t^= C1ij(t)dt as:
dQ01(tj = r1e~r"tdt, dQ12(tj = g(tje~(r#+ X)t At, dQ14(tj = Ae~(r#+ X)tG(Fjdt, dQ15(tj = r2e~(r#+ X)tG=tjdt, dQ23(tj = r2e~(r#)tdt, dQ30(tj = h(tje~r"t dt, dQ36(tj = r1e~r"t H(t)dt,dQ41(tj = a(tje~r2t dt,dQ48(tj = r+e~r2t A(t)dt dQ51(tj = h(tjdt,dQ61(tj = h(tjdt, dQ71(tj = h(tjdt,dQ87(tj = a(tjdt By taking t ® ^ of the above expressions using pij = Qij (&>j = f%° qij (tjdt, we get
p01 = 1,P12 = 3*(r2 + = ^ [1 - g\r+ + Xj],P1S = ^ [1 - g*(r+ + Xj],
P+3 = 1,Pso = ht(T1j,p36 = [1 - h*(r1j],p41 = a*(r2j,p48 = [1 - a*(r2j], pS1 = h*(0), P71 = h*(0), p61 = h*(0), Pq7 = a*(0j
It is verified that:
P01 = P12 + P11 + P15 = P30 + P36= P11 + P4B= P51 = P61 =P71 = PB7 = 1
b) Mean Sojourn Times
The expected time taken by the system in a particular state before transiting to any other state is known as mean sojourn time or mean survival time in the state. If Ti be the sojourn time in the
Naveen Kumar, S.C. Malik, N. Nandal
STOCHASTIC ANALYSIS OF A REPAIRABLE SYSTEM OF NON-IDENTICAL RT&A, No 1 (67)
UNITS WITH PRIORITY AND CONDITIONAL FAILURE OF REPAIRMAN Volume 17, March 2022 state i, then the mean sojourn time in the state i is
^ = % Pr(Ti > t)orpi = Zj mu But my = - 7- 0)]s=Q
F0 = = ml2 + ™!4 + ™is,F2 = m+3
F3 = m30 + m36,p4 = m4i + m48 F3 = m3o + ™3iUL4 = m4i + m41.B7
c) Reliability and MTSF
Let 0i (t) be the c.d.f. of first passage time from regenerative state Si to a failed state. Regarding the failed state as absorbing state, we have following recursive relations for 0i (t):
№) = Zj Qij(t)®0j(t) + Z< Qik(t) (1)
where Sj is an un-failed regenerative state to which the given regenerative state Si can transit and Sk is a failed state to which the state Si can transit directly. Thus, the following equations are obtained by using (1) as:
0o(t) = Qoi(t) 0 0i(t)
0i(P) = Qi+(t) 0 0+(P) + Qii(t) 0 0i(t) + Qi5(t) 0+(t) = Q23(t)0 03(t) 03(t) = Q30(t) 0 0o(t) + Q36(t) 0i(t) = Q4i(t)0 0i(t) + Qisit) Taking LST of above relations to obtain 0*o*(s) using this, we have
r*(S)=
-
Taking LIT of R*(s), we can obtain the reliability R(t) of the system model. The MTSF is given by where,
MTSF =limn^0 R*(s) =
Di
Ni = V0(Pi2 + Pis + PuPia) + Fi + F2Pi2 + F3Pi2 + P4Pi4r D± = (1 - P12P30 - PiiPii)
d) Steady State Availability
Let Ai(t) be the probability that the system is in up-state at epoch 't' given that the computer system entered regenerative state Si at t = 0. The recursive relations for Ai(t) are given as
Ai (t) = Mi (t) + Y" qf (t)©Aj (t) (2)
where Sj is any successive regenerative state to which the regenerative state Si can transit through n transitions. Thus, the following equations are obtained by using (2) as:
A= (t) = M0 (t) + q0i©Ai(t) A((t)= Mi(t)+ qi2(ty©A+(t) + qi4(ty©Ai(t) + qis(t)©As(t) A+(t) = M+(t)+ q+3©^3(t) A3(t)= M3(t)+ q3o(t)©A=(t) + q3i,6(t)©Ai(t)
Ai(t) = Mi(t) + q4i(t)©Ai(t) + qii . a7(t)©Ai(t) As(t) = qsi(t)©Ai(t)
where
M0(t) = e~r"t,Mi(t) = e~(r#+X)tG(t),M2(t) = e~r#t,M4(t) = e~r#tA=), M3(t) = e~r"tH(t)
Taking L.T of above expressions and calculate the value of A*0 (s), we have
* N+
A0 = limits A0(t) = lims^0 (s) = —r
U2
where,
N2 = F0 - F0Pi+P36 - F0Pi4 - F0Pis + Fi + F2Pi2 + V3Pi2 - P4Pii D+ = V0Pi2P30 + Fi + F2Pi2 + ^3Pi2 + V-4Pi4 + FsPis
e) Busy Period of the Repairman Due to Repairs
Naveen Kumar, S.C. Malik, N. Nandal
STOCHASTIC ANALYSIS OF A REPAIRABLE SYSTEM OF NON-IDENTICAL RT&A, No 1 (67)
UNITS WITH PRIORITY AND CONDITIONAL FAILURE OF REPAIRMAN Volume 17, March 2022 Let B* (t) be the probability that server is busy in repairing the unit at epoch 't' given that the system entered state Si at t = 0. The recursive relations for B* (t) are given as:
BB (t) = W* (t) + Y" qf (t)©B}H (t) (3)
where Sj is any successive regenerative state to which the regenerative state Si can transit through n transitions. Thus, the following equations are obtained by using (3) as:
i) Repair of Main Unit
BD(t) = q01©BD(t) B*(t) = W±(t) + q12(t)©BD(t) + q115(t)©BD (t) + q14(t)©BD(t) B+D(t) = q+3©BD(t) BD (t) = q3o(t)©BD(t) + q3i.6(t)©BD (t) BD(t) = q4i(t)©BD(t) + q4i.87(t)©BD (t) BD(t) = q5i©BD(t)
where
Wi(t) = e-(r#- x)tG(t) Taking L.T. of above expressions and calculate the value of B% (s), we have
N3
BD = lims^0 sB0*(s) =
U2
Where,N3 = цland D2 is already defined.
ii) Repair of Duplicate unit
B$(t) = qoi©B? (t) B?(t) = qio(t)©B% (t) + qi4(t)©B° (t) + qu . 5(t)©BD(t) B+A(t) = q+3©BA (t) B°(t) = W3(t) + q3o(t)©B° (t) + q3i . 6(t)©B? (t) B?(t) = qii(t)©BC (t) + q4i . 87(t)©BA(t) B£(t) = W5(t) + qSi©Bl (t)
where
W3 (t) = e-(r" )tJ=(F) +[ri er" &©1]H(F) W5(t) = H==,
Taking L.T. of above expressions and calculate the value of B% (s), we have
N3D
B% = lims^o sB%*(s) = --r-U2
where, N3D= W3^(0)pi2 and D2 is already defined. f) Expected Number of Repairs of the Main Unit
Let Rf (t) be the expected number of repairs of the unit by the repairman in (0, t] such that the system entered regenerative statei at t = 0. The recursive relation forflf (t) are given as:
Rf (t) = Qff (t) (E{Sj + Rf (t)] (4)
Where j is any regenerative state to which the given regenerative state i transits and Sj = 1 if j is the regenerative state where the repairman does job afresh, otherwise, Sj = O.Thus, the following equations are obtained by using (4) as:
Rf(t) = Q01 ®Rf (t) Ri(t) = Q12(t) 0[1 + Rf(t)] + Q14(t) 0 Rf (t) + Q1±.5(t) 0 Rf (t) R?(t) = Q23 0 R3 (t) RH(t) = Q3O(t)0 Rf(t) + Q3(. 6(t) 0 Rf(t) RH(t) = Qn(t) 0 Rf (t) + Q11 . 87(t) 0 r33(t) Rf(t) = QSi 0 R!_(t)
Taking L.S.T. of above expressions and calculating for Rf (s), we have
RDD = ^0 sH==(s), = ^
where,N4= pi2 and D2 is already defined.
g) Expected Number of Repairs of the Duplicate Unit
Let Rf (t) be the expected number of repairs of the unit by the repairman in (0, t] such that the system entered regenerative statei at t = 0. The recursive relation fori? (t) are given as:
R?(t) = Y.j Qff(tMSj + R? (t)] (5)
Where j is any regenerative state to which the given regenerative state i transits and Sj = 1 if j is the regenerative state where the repairman does job afresh, otherwise, Sj = O.Thus, the following equations are obtained by using (5) as:
R?(t) = Q0i® R? (t) R?(t) = Qi2(t) ® R?(t) + Qu(t) ® R?(t) + Qn.s(t) ® R?(t) R?(t) = Q20 ®R? (t) R?(t) = Q3o(t) ®[1 + R?(0] + Q3i.6(t) ®[1 + R?(0] R?(t) = Qn(t) ® R?(t) + Qn.a7(t) ®[1 + R?(0] R?(t) = Qsi ®[l+_R?(t)] Taking L.S.T. of above expressions and calculating fori? (s), we have
— Ns R? = ^0 sRg(s) =
U2
where,Ns= pi2 and D2 is already defined.
h) Expected Number of Treatment Given to the Repairman
Let TR (t) be the expected number of repairs of the unit by the repairman in (0, t] such that the system entered regenerative statei at t = 0. The recursive relation for TR (t) are given as:
TlR(t) = Yj QtfwGJSj + TR (t)] (6)
Where j is any regenerative state to which the given regenerative state i transits and Sj = 1 if j is the regenerative state where the repairman does job afresh, otherwise, Sj = O.Thus, the following equations are obtained by using (6) as:
TR (t) = Q0i® TR (t) TR(t) = Qi2(t) ® TR(t) + Qi4(t)® TR(t) + Qn.s(t) ® TR(t)
TR (t) = Q23 ® TR (t) TR(t) = Q30(t) ® TR(t) + Q3i.e(t) ® TR(t) TR (t) = Q4i(t) ®[1+ TR (t)] + Q4i.57(t) ® [1 + TR(t)] TR (t) = Qsi ® TR (t) Taking L.S.T. of above expressions and calculating for TR (s),we have
TR = lims^0 sTR(s) = , where, N6=pi4 and D2 is already defined.
IV Profit Analysis
The following expression can be used to obtain Profit of the system model:
PA = K0A0 - KrBD(t) - K2B°(t) - K3R$(t) - K4R?(t) - KsTR(t) (7)
V. Particular Cases
Let us take
g(t) = ae-at, h(t) = |3e-^, a(t) = Ye-rt,
RT&A, No 1 (67) Volume 17, March 2022
Pl2 =
12 (a+A+r2)
Pl5 =
r2
h = ", hi =
-, h = ~
(a+A+r2) r 2 r2
(a+A+r2) 1
h =
Pl4 =
A
P48 =
14 (a+A+r2)'H41=—' ^B 1 11 P4 =
(ß+r2)
r2 „ = ß
(Y+r2), (ß+ri)
<P
36=
(Y+r2)' h ß,
(ß+ri)
' — r2 1-1 1-1 ' _ if ri . P4 = (YTT2) [ß + ß], P3 = ß Vr^
MTSF = —,Availability = %
Di 3 D2
N3
Busy period of the repairman (B) =—-
D2
Expected number of repairs of main unit (RM0) =—,
N4
Expected number of repairs of duplicate unit (RD0) =—
D2
Expected number of Treatment given to the Repairman (TR0) =—,
D2
where,
"1 = ^ [■
+
+
+
1 +
+
+
ri L(a+A+r2) (a+A+^ (a+A+^XY+^J (a+A+r2) r2 (a+A+r2) (ß+^) (a+A+r2) (a+A+r2)(Y+r2)7
ß AY
D1 = 1----, w v
1 (a+A+r2) (ß+ri) (a+A+r2)(Y+r2)
1 ri 1 a 1 A 1 r2 1 1 a a 1
n2 =---i—---------—^ +-+ ---- +--
2 ri (ß+ri) ri 0^+^ ri (a+i+r2) ri (a+A+r2) (a+Ä+r2) r2 (a+A+r2) (a+Ä+r2) (ß+ri)
A
(a+Ä+r2) (Y+r2)
N3 =
D+ =
(a+/+r2)
', N3D =
7^4
■ + ^
¡,---N,- =---Nf,=---
(a+-+r2)' 1 (a+-+r2) 2 (a+-+r2) b (a+-+r2)
r2
■ + 1
+ 1,
+
/
(a+/+r2) (a+-+r2) (ß+rt) r2 (a+-+r2) ß (a+-+r2) (ß+rt) (Y+r2) (a+/+r2)
Ji + 1Z + 1.
Lr ßj ß (a+-+r2)
VI. Graphical Presentation
The graphical representation of MTSF, availability and profit function has been shown in figures 2,3 and 4 respectively to check their behavior with respect to the values of the parameters associated with failure and repair rates. From Figure 2, it is observed that the MTSF of the system decreases when failure rate of main unit is increased from 0.01 to 0.1. Also, MTSF increases with an increase inrepair rate of main unit, duplicate unit and treatment rate of the repairman.
10000 9000 -8000 -7000 6000 5000 4000 3000 2000 1000 o
- A.005,ct=2,ß= 1.5,Y=5,r2 = .02 -A.005,a=3,ß=1.5,Y=5,r2=.02 -A.005,a=2,ß=3,Y=5,r2 = .02 X.005,ct=2,ß=1.5,T=5,r2=.05
i
i
i
i
i
i
i
i
i
—A.007,a=2,ß=1.5,Y=5,r2 = .02 -•—X.005,a=2,ß=1.5,Y=7,r2 = .02
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Failure Rate of Main Unit ri
Figure 2: MTSF Vs Failure Rate of Main Unit
From Figure 3, it is clearly seen that the availability of the system decreases rapidly with increase of failure rate of main unit. Also, availability of the system increases with an increase inrepair rate of main unit, duplicate unit and treatment rate of the repairman.
a
Figure 3: Availability Vs Failure Rate of Main Unit
From Figure 4, it is observed that the profit decreases when failure rate of the main unit increases. Also,the profit of the system is increases with an increase inrepair rate of main unit, duplicate unit and treatment rate of the repairman.
14990 -14980 -14970 14960 I 14950 -o 14940 14930 14920 -14910 -14900
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Failure Rate of Main Unit n
-A.005,a=2,P=1.5,Y=5,r2=.02 -X.005,a=3,(3=1.5,Y=5,r2=.02 -X.005,a=2,(3=3,Y=5,r2=.02 X.005,a=2,3=1.5,Y=5,r2=.05 -\.007,a=2,P=1.5,Y=5,r2=.02 -A.005,a=2,(3=1.5,Y=7,r2=.02
Figure 4: Profit Vs Failure Rate of Main Unit
VII. Conclusion
The idea of priority for repair and operation of the duplicate unit has been used to determine reliability characteristics of a stochastic model developed for a system of non-identical units with failure of repairman. The failure of repairman is called conditional failure as it fails only during the repair of the main unit. In this study reliability measures such as MTSF, availability and profit function are obtained and their behavior is shown respectively figures: Figure 2, Figure 3 and Figure 4. It is observed that MTSF, availability and profit function decline when failure rate increases. On the other hand, these measures increase with the increase of repair rate of main unit, duplicate unit and treatment rate of the repairman. Further, the study reveals that profit of the system model can be increased by increasing the repair rate of the duplicate unit.
VIII. Application
The oxygen supply system which is shown in figure 5 can be considered as a direct application of the present study. An acute shortage of medical oxygen and oxygen cylinders has been observed during COVID-19 pandemic situation everywhere throughout the World. The oxygen therapy was in dire need for the survival of patients during this pandemic. The scarcity of oxygen cylinders has also pushed up the demand for oxygen concentrators. Today, oxygen concentrators are in great demand after devices for oxygen therapy in home isolation. Therefore, the present study has been designed to analyze the oxygen supply system comprises oxygen concentrator as a main unit and the oxygen cylinder as its duplication. In case of electricity failure, it becomes necessary to give priority for operation and repair of the oxygen cylinder to cover the risk. Thus, it is a non-identical system of two units in which the concepts of priority and the failure of repairman have been considered to examine some important reliability characteristics so that the users of the oxygen supply system may take appropriate decision to minimize the risk.
Duplicate Unit
Figure 5: Oxygen Supply System
References
[1] Anuradha, Malik, S.C. and Jha, P.C. (2021). Profit analysis of a 1-out-of-2: G system with priority to repair and conditional failure of service facility. International Journal of Statistics and Reliability Engineering, 8(2): 264-271.
[2] Cao, J. and Yanhong, W. (1989). Reliability of a two unit cold standby system with replaceable repair facility. Microelectron. Reliab., 29(2):145 -150.
[3] Chen, W.L. and Wang, K.H. (2018). Reliability analysis of a retrial machine repair problem with warm standbys and a single server with N-policy. Reliability Engineering & System Safety, 180: 476-486.
[4] Kadyan, M.S., Chander, S. and Grewal, A.S. (2004). Stochastic analysis of non-identical units reliability models with priority and different modes of failure. Decision and Mathematical Sciences, 9(1-3):59 - 82.
[5] Kumar, A., Pawar, D. and Malik, S.C. (2019). Profit analysis of a warm standby non-identical unit system with single server performing in normal/abnormal environment. Life Cycle
Naveen Kumar, S.C. Malik, N. Nandal
STOCHASTIC ANALYSIS OF A REPAIRABLE SYSTEM OF NON-IDENTICAL RT&A, No 1 (67)
UNITS WITH PRIORITY AND CONDITIONAL FAILURE OF REPAIRMAN Volume 17, March 2022 Reliability and Safety Engineering, 8:219-226.
[6] Kumar, N. and Nandal, N. (2020). Stochastic modeling of a system of two non-identical units with priority for operation and repair to main unit subject to conditional failure of repairman. International Journal of Statistics and Reliability Engineering, 7(1):114-122.
[7] Kadyan, S., Barak, M.S. and Gitanjali (2020). Stochastic analysis of a non-identical repairable system of three units with priority for operation and simultaneous working of cold standby units. International Journal of Statistics and Reliability Engineering, 7(2):269-274.
[8] Salah and Sherbeny, E.L. (2012). Stochastic analysis of a two non-identical unit parallel system with different types of failures subject to preventive maintenance and repairs. Mathematical Problems in Engineering, 2013:1-10.
[9] Smith W.L. Regenerative Stochastic Processes, Proceeding Royal Society, London, 1955.
[10] Subramanian, R. and Natarajan, R. (1982). An N-Unit Standby Redundant System with R Repair Facilities and Preventive Maintenance. Microelectronics Reliability, 22 (3):367 - 377.
