International Journal of Trend in Scientific
Research and Development (IJTSRD)
International Open Access Journal
ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume - 2 | Issue – 5
Reliability Modeling and Analysis of a Parallel Unit System with
Priority to Repair oover Replacement Subject to Maximum
Operation and Repair Times
Reetu
eetu Rathee1, D. Pawar1, S. C. Malik2
1
Assistant Professor, 2Professor
1
Amity Institute of Applied Science, AMITY University, Noida
Noida, Uttar Pradesh,
Pradesh India
2
Department of Statistics, M. D. University, Rohtak, Haryana,, India
ABSTRACT
This paper proposes the study of modelling and
analysis of a two unit parallel system. A constant
failure rate is considered for the units which are
identical in nature. All repair activities like repair,
replacement, preventive maintenance are mended
immediately by a single server. The repair of the unit
is done after its failure and if the fault is not rectified
by the server within a given repair time, called
maximum repair time, the unit replaced by new one.
And, if there is no fault occurs up to a pre
pre-fixed
operation time, called maximum
ximum operation time, the
unit undergoes for the preventive maintenance. The
unit works as new after all repair activities done by
the server. Priority to repair of one unit is given over
the replacement of the other one. All random
variables are statistically
ally independent. The
distribution for the failure, preventive maintenance
and replacement rates are negative exponential
whereas the distribution for all repair activities are
taken as arbitrary with different probability density
functions. Semi-Markov andd regenerative point
techniques are used to derive some reliability
measures in steady state. The variation of MTSF,
availability and profit function has been observed
graphically for various parameters and costs.
Keywords: Parallel system, Preventive Mai
Maintenance,
Replacement, Priority, Reliability Measures
INTRODUCTION
The general purpose of the modern world is to
achieve the require performance level using the
lowest possible cost. And, the parallel system works
not only for maximize the profit but also for minimize
the failure risk as well as cost. Keeping in view of
their practical applications, reliability models of
parallel systems have been developed and analyzed
stochastically by the researchers and reliability
engineers. Kishan and Kumar (2009)
(2009 evaluated
stochastically a parallel system using preventive
maintenance. Further, kumar et al. (2010) and Malik
and Gitanjali (2012) have analyzed cost-benefit
cost
of a
parallel system subject to degradation after repair and
arrival time of the server respectively.
respec
However, to
enhance the profit of the system Reetu and Malik
(2013) and Rathee and Chander (2014) developed
parallel systems using the concept of priority.
Also, the objective of the present paper is to
determine the reliability measures by giving the
priority to one repair activity over the other ones. A
constant failure rate is considered for the units which
are identical in nature. All repair activities are done
immediately by a single server. The repair of the unit
is conducted after its failuree and if the fault is not
rectified by the server within a given repair time, the
unit replaced by new one. And, if there is no fault
occurs up to a pre-fixed
fixed operation time, the preventive
maintenance is conducted. The unit works as new
after all repair activities
ctivities done by the server. Priority to
repair of one unit is given over the replacement of the
other one. All random variables are statistically
independent. The distribution for the failure,
preventive maintenance and replacement rates are
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Aug 2018
Page: 350
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
negative exponential whereas the distribution for all
repair activities are taken as arbitrary with different
probability density functions. Semi-Markov and
regenerative point techniques are used to derive some
reliability measures in steady state. The variation of
MTSF, availability and profit function has been
observed graphically for various parameters and costs.
NOTATIONS:
E
: Set of regenerative states
𝐸
: Set of non-regenerative states
λ
: Constant failure rate
α0
: The rate by which system undergoes
for preventive maintenance (called
maximum constant rate of operation
time)
β0
: The rate by which system undergoes
for
replacement (called maximum
constant rate of repair time)
FUr /FWr
: The unit is failed and under
repair/waiting for repair
FURp/FWRp : The unit is failed and under
replacement/waiting for replacement
UPm/WPm : The unit is under preventive
maintenance/waiting for preventive
maintenance
FUR/FWR
: The unit is failed and under repair /
waiting for repair continuously from
previous state
FURP/FWRP : The unit is failed and under /waiting
for replacement continuously from
previous state
UPM/WPM : The unit is under/waiting for
preventive maintenance continuously
from previous state
g(t)/G(t)
: pdf/cdf of repair time of the unit
f(t)/F(t)
: pdf/cdf of preventive maintenance
time of the unit
r(t)/R(t)
: pdf/cdf of replacement time of the
unit
qij (t)/Qij (t)
: pdf / cdf of passage time from
regenerative state Si to a regenerative
stateSjor to a failed state Sj without
visiting any other regenerative state in
(0, t]
qij.kr (t)/Qij.kr(t) : pdf/cdf of direct transition time from
regenerative state Si to a regenerative
state Sj or to a failed state Sj visiting
state Sk, Sr once in (0, t]
Mi(t)
: Probability that the system up
initially in state Si E is up
at time t without visiting to any
regenerative state
Wi(t)
: Probability that the server is busy in
the state Si up to time ‘t’ without
making any transition to any other
regenerative state or returning to the
same state via one or more nonregenerative states.
i
: The mean sojourn time in state 𝑆
which is given by
∞
𝜇 = 𝐸(𝑇) =
𝑃(𝑇 > 𝑡) 𝑑𝑡 =
𝑚 ,
where𝑇denotes the time to system failure.
mij:Contribution to mean sojourn time (i) in state Si
when system transits
directlyto state Sjso that i   mij and mij =
j
 tdQ (t )  q
ij
*
ij
'(0)
& 
: Symbol for Laplace-Stieltjes
convolution/Laplace convolution
*/**
: Symbol for Laplace Transformation
/LaplaceStieltjes Transformation
The states S0, S1, S2, S4, S6and S7 are regenerative
while the states S3,S5, S8,S9, S10, S11 and S12 are nonregenerative as shown in figure 1.
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
TRANSITION STATE DIAGRAM
αO
Upm
S2
WPm
f(t)
O
SO
O
2λ
OS
1
FUr
λ
βO
FWr
UPM
f(t)
r(t)
FUR
WPm
αO
r(t)
O
FURp
S11
αO
S5
βO
f(t)
O
UPm
FWr
S3
FUR
g(t)
g(t)
f(t)
S6
λ
λ
g(t)
S4
αO
βO
FUr
FWRp
S10
FURp
WPM
S7
r(t)
βO
g(t)
UPM
WPm
WPm
FURP
S12
S9
Up-State
r(t)
FURp
FWRP
S8
Failed-State
Fig. 1
Regenerative point
Transition Probabilities and Mean Sojourn Times
Simple probabilistic considerations yield the following expressions for the non-zero elements

pij  Qij ()   qij (t )dt as
0
p01 
(1)
0
0
2
, p02 
, p15 
(1  g * (   0   0 )),
2   0
2   0
(   0   0 )

(1  g * (   0   0 )) , p60  f * (   0 ),
(   0   0 )

p10  g * (   0  0 ), p17.3 
(1  g * (  0 ))(1  g * (   0   0 )) ,
(   0   0 )
0

p6,12  p66.12 
(1  f * (   0 )) , p6,11  p61.11 
(1  f * (   0 )) ,
(   0 )
(   0 )
0
p49  p46.9 
(1  r * (   0 )) , p37  p5,10  p78  p74.8  1  g * ( 0 ) ,
(   0 )
0

p14 
(1  g * (   0   0 )) , p47 
(1  r * (   0 )) ,
(   0   0 )
(   0 )
p40  r * (   0 ), p13 
p31  p56  p74  g * (  0 ) , p26  p84  p96  p10,6  p11,1  p12,6  1
It can be easily verify that
p01  p02  p10  p13  p14  p15  p26  p40  p47  p49  p60  p6,11  p6,12  1
p10  p14  p11.3  p16.5  p16.5,10  p17.3  p40  p47  p46.9  1
p60  p61.11  p66.12  p74  p74.8  1
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
The mean sojourn times (𝝁𝒊 ) is in the state Si are
0  m01  m02 , 1  m10  m13  m14  m15 , 2  m26 , 4  m40  m47  m49 , 6  m60  m6,11  m6,12 ,
1'  m10  m14  m11.3  m16.5  m16.5,10  m17.3 ,
4'  m40  m47  m46.9 , 7  m74  m74.8 ,
Reliability and Mean Time to System Failure (MTSF)
Let  i(t) be the cdf of first passage time from regenerative state Si to a failed state. Regarding thefailed state as
absorbing state, we have the following recursive relations for  i(t):
0 (t )  Q01 (t )& 1 (t )  Q02 (t )
1 (t )  Q10 (t )& 0 (t )  Q14 (t )& 4 (t )  Q13 (t )  Q15 (t )
4 (t )  Q40 (t )& 0 (t )  Q48 (t )  Q49 (t )
(2)
∗∗
Taking LST of above relation (7.4) and solving for Ф (s), we have
1   ** ( s )
*
R (s) 
s
(3)
The reliability of the system model can be obtained by taking Inverse Laplace transform of (3). The mean time
to system failure (MTSF) is given by
1   ** ( s ) N
MTSF = lim
(4)

s0
s
D
Where
N  0  p011  p01 p14 4 and D  1  p01 p10  p01 p14 p40
(5)
Steady State Availability
Let Ai(t) be the probability that the system is in up-state at instant ‘t’ given that thesystementered regenerative
state Si at t = 0.The recursive relations for Ai (t ) are given as:
A0 (t )  M 0 (t )  q01 (t )A1 (t )  q02 (t )A2 (t )
A1 (t )  M 1 (t )  q10 (t ) A0 (t )  q11.3 (t ) A1 (t )  q14 (t ) A4 (t )
 q17.3 (t ) A7 (t )  ( q16.5 (t )  q16.5,10 (t )) A6 (t )
A2 (t )  q26 (t )A6 (t )
A4 (t )  M 4 (t )  q40 (t )A0 (t )  q47 (t )A7 (t )  q46.9 (t )A6 (t )
A6 (t )  M 6 (t )  q60 (t )A0 (t )  q61.11 (t )A1 (t )  q66.12 (t )A6 (t )
A7 (t )  (q74 (t )  q74.8 (t ))A4 (t )
Where
(6)
M 0 (t )  e  (2   0 )t , M 1 (t )  e  (   0   0 )t G (t ) ,
(7)
M 4 (t )  e  (   0 )t R (t ) , M 6 (t )  e  (   0 )t F (t )
Taking LT of above relation (6) and solving for A 0*(s). The steady state availability is given by
N
A0 ()  lim sA0* ( s )  1
(8)
s 0
D1
Where
N1  0 [(1  p47 ){ p60 (1  p11.3 )  p61.11 p10 }  p61.11 p40 ( p14  p17.3 )]
 6 [(1  p47 ){ p01 ( p16.5  p16.5,10 )  p02 (1  p11.3 )}
 p01 p46.9 ( p14  p17.3 )]  1 (1  p47 ){ p01 (1  p66.12 )  p02 p61.11}
(9)
 4 ( p14  p17.3 ){ p01 p60  p61.11}
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
D1  ( 0  2 p02 )[(1  p47 ){ p60 (1  p11.3 )  p61.11 p10 }  p61.11 p40 ( p14  p17.3 )]
 6' [(1  p47 ){ p01 ( p16.5  p16.5,10 )  p02 (1  p11.3 )}  p01 p46.9 ( p14  p17.3 )]
 ( p01 p60  p61.11 ){4' ( p14  p17.3 )  7' ( p14 p47  p17.3 )}
(10)
 1' (1  p47 ){ p01 (1  p66.12 )  p02 p61.11}
Busy Period Analysis for Server
A. Due to Repair
Let BiR (t ) be the probability that the server is busy in repairof the unit at an instant‘t’ given that the system
entered regenerative state Si at t=0.The recursive relations for BiR (t ) are as follows:
B0R (t )  q01 (t )B1R (t )  q02 (t )B2R (t )
B1R (t )  W1 (t )  q10 (t ) B0R (t )  q11.3 (t ) B1R (t )  q14 (t ) B4R (t )
 q17.3 (t ) B7R (t )  ( q16.5 (t )  q16.5,10 (t )) B6R (t )
B2R (t )  q26 (t )B6R (t )
B4R (t )  q40 (t )B0R (t )  q47 (t )B7R (t )  q46.9 (t )B6R (t )
B6R (t )  q60 (t )B0R (t )  q61.11 (t )B1R (t )  q66.12 (t )B6R (t )
B7R (t )  W7 (t )  (q74 (t )  q74.8 (t ))B4R (t )
Where,
(11)
W1 (t )  e(  0  0 )t G (t )  ( e (  0  0 )t 1)G (t )  ( 0e ( 0  0 )t 1)G (t )
and W7 (t )  e 0t G(t )
(12)
Taking LT of above relation (11) and solving for B0R* ( s ) .The time for which serveris busydue to repair is given
by
N
B0R ()  lim sB0R* ( s )  2
(13)
D1
s 0
Where ,
N 2  W1* (0)(1  p47 ){ p01 (1  p66.12 )  p02 p61.11}  W7* (0)( p14 p47  p17.3 ){ p01 p60  p61.11}
and D1 is already mentioned.
(14)
B. Due to Replacement
Let BiRp (t ) be the probability that the server is busy in replacement of the unit at an instant ‘t’ given that the
system entered regenerative state Si at t=0.The recursive relations for BiRp (t ) are as follows:
B0Rp (t )  q01 (t )B1Rp (t )  q02 (t )B2Rp (t )
B1Rp (t )  q10 (t ) B0Rp (t )  q11.3 (t ) B1Rp (t )  q14 (t ) B4Rp (t )
 q17.3 (t ) B7Rp (t )  ( q16.5 (t )  q16.5,10 (t )) B6Rp (t )
B2Rp (t )  q26 (t )B6Rp (t )
B4Rp (t )  W4 (t )  q40 (t )B0Rp (t )  q47 (t )B7Rp (t )  q46.9 (t )B6Rp (t )
B6Rp (t )  q60 (t )B0Rp (t )  q61.11 (t )B1Rp (t )  q66.12 (t )B6Rp (t )
B7Rp (t )  ( q74 (t )  q74.8 (t ))B4Rp (t )
Where,
(15)
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
W4 (t )  e (   0 )t R (t )  ( 0e (  0 )t 1) R (t )
(16)
B0Rp* ( s ) .The
Taking LT of above relation (15) and solving for
time for which server is busy due to replacement
is given by
N
B0Rp ()  lim sB0Rp* ( s)  3
(17)
D1
s 0
Where,
N3  W4* (0)( p14  p17.3 )( p01 p60  p61.11 ) and D1 is already mentioned.
(18)
C. Due to Preventive Maintenance
Let BiP (t ) be the probability that the server is busy in preventive maintenance of the unit at an instant‘t’ given
that the system entered regenerative state Si at t=0.The recursive relations for BiP (t ) are as follows:
B0P (t )  q01 (t )B1P (t )  q02 (t )B2P (t )
B1P (t )  q10 (t )B0P (t )  q11.3 (t )B1P (t )  q14 (t )B4P (t )
 q17.3 (t ) B7P (t )  ( q16.5 (t )  q16.5,10 (t )) B6P (t )
B2P (t )  W2 (t )  q26 (t ) B6P (t )
B4P (t )  q40 (t )B0P (t )  q47 (t )B7P (t )  q46.9 (t )B6P (t )
B6P (t )  W6 (t )  q60 (t )B0P (t )  q61.11 (t )B1P (t )  q66.12 (t )B6P (t )
B7P (t )  ( q74 (t )  q74.8 (t ))B4P (t )
Where,
(19)
W6 (t )  e(  0 )t F (t )  ( 0e(  0 )t 1) F (t )  ( e(  0 )t 1) F (t ) and W2 (t )  F (t )
(20)
Taking LT of above relation (19) and solving for B0P* ( s ) .The time for which server is busy due to preventive
maintenance is given by
N
B0P ()  lim sB0P* ( s )  4
(21)
D1
s 0
Where,
N 4  W2* (0) p02 [(1  p47 ){ p60 (1  p11.3 )  p61.11 p10 } p61.11 p40 ( p14  p17.3 )]
 W6* (0)[(1  p47 ){ p01 ( p16.5  p16.5,10 )  p02 (1  p11.3 )}  p01 p46.9 ( p14  p17.3 )]
and D1 is already mentioned.
(22)
Expected Number of Repairs
Let Ri (t ) be the expected number of repairs by the server in (0, t] given that the system entered the regenerative
state Si at t = 0. The recursive relations for Ri (t ) are given as:
R0 (t )  Q01 (t ) & R1 (t )  Q02 (t )& R2 (t )
R1 (t )  Q10 (t ) & [1  R0 (t )]  Q11.3 (t ) & [1  R1 (t )]  Q14 (t ) & R4 (t )
 Q17.3 (t ) & R7 (t )  Q16.5 (t ) & [1  R6 (t )]  Q16.5,10 (t ) & R6 (t )
R2 (t )  Q26 (t ) & R6 (t )
R4 (t )  Q40 (t )& R0 (t )  Q47 (t ) & R7 (t )  Q46.9 (t ) & R6 (t )
R6 (t )  Q60 (t ) & R0 (t )  Q61.11 (t )& R1 (t )  Q66.12 (t )& R6 (t )
R7 (t )  Q74 (t ) & [1  R4 (t )]  Q74.8 (t ) & R4 (t )
(23)
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Taking LST of above relations (23) and solving for R0** ( s ) .The expected no. of repairs per unit time by the
server are giving by
N
R0 ()  lim sR0** ( s )  5
(24)
D1
s 0
Where,
N5  ( p10  p11.3  p16.5 )(1  p47 ){ p01 (1  p66.12 )  p02 p61.11}
 p74 ( p14 p47  p17.3 )( p01 p60  p61.11 )
and D1 is already mentioned.
(25)
Expected Number of Replacements
Let Rpi (t ) be the expected number of replacements by the server in (0, t] given that the system entered the
regenerative state Si at t = 0. The recursive relations for Rpi (t ) are given as:
Rp0 (t )  Q01 (t )& Rp1 (t )  Q02 (t ) & Rp2 (t )
Rp1 (t )  Q10 (t ) & Rp0 (t )  Q11.3 (t ) & Rp1 (t )  Q14 (t ) & Rp 4 (t )
 Q17.3 (t ) & Rp7 (t )  Q16.5,10 (t ) & [1  Rp6 (t )]  Q16.5 (t ) & Rp6 (t )
Rp2 (t )  Q26 (t ) & Rp6 (t )
Rp4 (t )  Q40 (t )& [1  Rp0 (t )]  Q47 (t )& Rp7 (t )  Q46.9 (t ) & [1  Rp6 (t )]
Rp6 (t )  Q60 (t ) & Rp0 (t )  Q61.11 (t )& Rp1 (t )  Q66.12 (t ) & Rp6 (t )
Rp7 (t )  Q74 (t )& Rp4 (t )]  Q74.8 (t ) & [1  Rp4 (t )]
(26)
Taking LST of above relations (26) and solving for Rp0** ( s ) .The expected no. of replacements per unit time by
the server are giving by
N
Rp0 ()  lim sRp0** ( s )  6
(27)
D1
s 0
Where,
N 6  p16.5,10 (1  p47 ){ p01 (1  p66.12 )  p02 p61.11}
 ( p40  p 46.9 p )( p14  p17.3 ){ p01 p60  p61.11}
 p74.8 ( p14 p47  p17.3 )( p01 p60  p61.11 )
and D1 is already mentioned.
(28)
Expected Number of Preventive Maintenances
Let Pi (t) be the expected number of preventive maintenance by the server in (0, t] given that the system entered
the regenerative state Siat t = 0. The recursive relations for Pi (t) are given as:
P0 (t )  Q01 (t ) & P1 (t )  Q02 (t ) & P2 (t )
P1 (t )  Q10 (t ) & P0 (t )  Q11.3 (t ) & P1 (t )  Q14 (t ) & P4 (t )
 Q17.3 (t ) & P7 (t )  {Q16.5 (t )  Q16.5,10 (t )}& P6 (t )
P2 (t )  Q26 (t ) & [1  P6 (t )]
P4 (t )  Q40 (t )& P0 (t )  Q47 (t ) & P7 (t )  Q46.9 (t ) & P6 (t )
P6 (t )  Q60 (t ) & [1  P0 (t )]  Q61.11 (t ) & [1  P1 (t )]  Q66.12 (t ) & [1  P6 (t )]
P7 (t )  {Q74 (t )  Q74.8 (t )}& P4 (t )
(29)
Taking LST of above relations (29) and solving for P0** ( s ) .The expected no. of preventive maintenances per
unit time by the server are giving by
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
P0 (  )  lim sP0** ( s ) 
s 0
N7
D1
(30)
Where,
N 7  p02 [(1  p47 ){ p60 (1  p11.3 )  p61.11 p10 }  p61.11 p40 ( p14  p17.3 )]
 [(1  p47 ){ p01 ( p16.5  p16.5,10 )  p02 (1  p11.3 )}  p01 p46.9 ( p14  p17.3 )]
and D1 is already mentioned.
(31)
Profit Analysis
The profit incurred to the system model in steady state can be obtained as
P  K0 A0  K1B0R  K2 B0Rp  K3 B0P  K4 R1  K5 Rp0  K6 P0
(32)
Where,
P = Profit of the system model after reducing cost per unit time busy of the server and
cost per repair activity per unit time
K 0 = Revenue per unit up-time of the system
K1 =Cost per unit time for which server is busy due to repair
K 2 =Cost per unit time for which server is busy due to replacement
K 3 =Cost per unit time for which server is busy due to preventive maintenance
K 4 = Cost per repair per unit time
K 5 = Cost per replacement per unit time
K 6 = Cost per preventive maintenance per unit time
CONCLUSION
After solving the equations of MTSF, availability and profit function for the particular case g(t) = 𝜃𝑒 , r(t) =
𝛽𝑒
and f(t) = 𝛼𝑒 , we conclude that
 These reliability measures are increasing as the repair rate θ, replacement rate β increases while their values
decline with the increase of failure rate λ and the rate 𝛼 by which the unit undergoes for preventive
maintenance.
 The MTSF and availability keep on upwards with the increase of the rate 𝛽 by which unit undergoes for
replacement but profit decreases.
 The system model becomes more profitable when we increase the preventive maintenance rate α.
GRAPHS FOR PARTICULAR CASE
MTSF Vs Failure Rate (λ)
5.02
α0=0.2,β0=3,θ=2.1,α=5,β=10
β0=1
θ=4.1
α0=0.201
α=7
β=5
5
4.98
4.96
4.94
MTSF
4.92
4.9
4.88
4.86
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Failure Rate (λ)
Fig.2
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Availability
Availability Vs Failure Rate (λ)
0.973
0.971
0.969
0.967
0.965
0.963
0.961
0.959
0.957
0.955
α0=0.2,β0=3,θ=2.1,α=5,β=10
β0=1
θ=4.1
α0=0.201
α=7
β=5
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Failure Rate (λ)
Fig.3
Profit (P)
Profit Vs Failure Rate (λ)
13950
13850
13750
13650
13550
13450
13350
13250
13150
13050
α0=0.2,β0=3,θ=2.1,α=5,β=10
β0=1
θ=4.1
α0=0.201
α=7
β=5
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Failure Rate (λ)
REFERENCES
1. R. Kishan and M. Kumar (2009), “Stochastic
analysis of a two-unit parallel system with
preventive maintenance”, Journal of Reliability
and Statistical Studies, vol. 22, pp. 31- 38.
2. Kumar, Jitender, Kadyan, M.S. and Malik, S.C.
(2010): Cost-benefit analysis of a two-unit parallel
system subject to degradation after repair. Journal
of Applied Mathematical Sciences, Vol.4 (56),
pp.2749-2758.
3. Malik S.C. and Gitanjali (2012). Cost-Benefit
Analysis of a Parallel System with Arrival Time
of the Server and Maximum Repair Time.
International Journal of Computer Applications,
46 (5): 39-44.
Fig.4
4. Reetu and Malik S.C. (2013),A Parallel System
with Priority to Preventive Maintenance Subject to
Maximum Operation and Repair Time. American
journal of Mathematics and Statistics, Vol. 3(6),
pp. 436-440.
5. Rathee R. and Chander S. (2014), A Parallel
System with Priority to Repair over Preventive
Maintenance Subject to Maximum Operation and
Repair Time. International Journal of Statistics
and Reliability Engineering, Vol. 1(1), pp. 57-68.
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