Research Journal of Mathematics and Statistics 3(1): 45-50, 2011
ISSN: 2040-7505
© Maxwell Scientific Organization, 2011
Received: November 11, 2010
Accepted: December 02, 2010
Published: February 15, 2011
Profit Estimation for a Gas Separator Plant under Pre-Emptive
Repeat Repair Policy
1
K. Pravesh and 2D. Sharma
1
Singhania University, Rajasthan, India
2
Department of Mathematics, D.J. College of Engineering and Technology, Modinagar, India
Abstract: In this study, the authors have considered a gas separator plant for its profit estimation. This whole
system can fail due to failure of any of its subsystems. On failure of any one gas separator, the whole system
works in degraded state. All failures follow exponential time distribution whereas all repairs follow general time
distribution. Pre-emptive repeat policy has been adopted for repair purpose. The repair of subsystem A, B and
D are pre-empting over the repair of one unit of subsystem C. Also, the system has to wait for repair, in case,
whole subsystem C failed; otherwise repair facilities are always available. Supplementary variables technique
and Laplace transform have been used to formulate and solve the mathematical model of considered system,
respectively. A particular case, when repairs follow exponential time distribution, and long run flow state
probabilities have also been computed to improve practical utility of the model. A numerical computation
together with its graphical illustration has appended at last to highlight important results of the study.
Key words: Gas separator system, Laplace transform, long- run behavior, markovian process, supplementary
variables
C
C
INTRODUCTION
In this considered system, there are four main
subsystems, designated here as A, B, C and D, connected
in series. The subsystem A is a Pressure Shut Down
(PSD) valve and it is responsible to collect input for
separation process. This valve is an automatic valve and
it works by the pressure of input. The second subsystem
B is a Pressure Logic Controller (PLC) system and it
controls the working of PSD valve. The third subsystem
C is a gas separator unit and it separates the useful gas
from the input and exhausted the waste gases. In this
model, the authors have been taken two similar gas
separator units in parallel redundancy (Gupta and
Gupta, 1986) and these are named as C1 and C2. The forth
subsystem D is again a PSD valve. The block-diagram
(Barlow and Proschan, 1965) of considered system has
been shown in Fig. 1a, while Fig. 1b shows the flow of
states (Chung, 1988) of considered system.
MATERIALS AND METHODS
This study was conducted at Department of
Mathematics, D.J. College of Engineering and
Technology, Modinagar, Ghaziabad, India during June
2010.
In this study, the authors have been used
supplementary variables technique (Gupta and
Gupta, 1986) to formulate mathematical model of the
considered system. Various difference-differential
equations have been obtained for the transition states
depicted in Fig. 1b. This set of difference-differential
equations has solved by using Laplace transform
(Nagraja et al., 2004). The probabilities of the system
having in different transition states have computed. These
results can be used to obtain various reliability parameters
of the system having similar configurations.
Using probability consideration and limiting
procedure (Sharma et al., 2010), we obtain the following
set of difference-differential equations, governing the
behaviour of considered system, continuous in time and
discrete in space:
The following assumptions have been made for this study:
C
C
C
C
C
Failures are S-independent
The system has to wait for repair, in case, the whole
subsystem C has failed, otherwise repair facilities are
always available
The whole system is new at t = 0
All failures follow exponential time distribution
whereas all repairs follow general time distribution
Nothing can fail from a failed state
Pre-emptive repeat policy has been adopted for repair
Repairs are perfect
Corresponding Author: Dr. D. Sharma, Department of Mathematics, D.J. College of Engineering and Technology, Modinagar,
India
45
Res. J. Math. Stat., 3(1): 45-50, 2011
Fig. 1a: Represents the block-diagram of considered gas separator system
Fig. 1b: Represents the transition of all possible states of considered system
∞
⎡d
⎤
⎢⎣ dt + α A + α B + 2α C + α D ⎥⎦ P0 (t )
∞
=
∫ P ( x, t ) β
A
0
∞
∫
2
+ PCR
( n, t ) βC2 ( n) dn
0
∞
A
( x) dx + ∫ PB ( y, t ) βB ( y) dy
0
∞
∫
∫
0
0
(1)
+ PD ( z, t ) β D ( z ) dz + PC1 ( m, t ) βC1 ( m) dm
46
⎡∂
⎤
∂
⎢ + + β A ( x ) ⎥ PA ( x , t ) = 0
⎣ ∂ x ∂t
⎦
(2)
⎤
⎡∂
∂
⎢ + + β B ( y ) ⎥ PB ( y , t ) = 0
⎦
⎣ ∂ y ∂t
(3)
Res. J. Math. Stat., 3(1): 45-50, 2011
⎡∂
⎤
∂
⎢ + + β D ( y ) ⎥ PD ( z, t ) = 0
⎣ ∂ z ∂t
⎦
(4)
⎡ ∂
⎤
∂
+
+ α A + α B + α C + α D + β C1 ( m)⎥
⎢
⎣ ∂m ∂t
⎦ (5)
PC1 (m, t ) = 0
(16)
1
(0, t ) = α D PC1 (t )
PCD
(17)
2
2
PCR
( 0, t ) = wPCW
(t)
(18)
Initial conditions are:
P0(0) = 1
⎡ ∂
⎤ 1
∂
( x, t ) = 0
+
+ β A ( x )⎥ PCA
⎢
⎣ ∂ x ∂t
⎦
(6)
⎤ 1
⎡ ∂
∂
y, t = 0
+
+ β B y ⎥ PCB
⎢
⎦
⎣ ∂ y ∂t
(7)
⎤ 1
⎡∂
∂
( z, t ) = 0
+ β D ( z )⎥ PCD
⎢ +
⎦
⎣ ∂ z ∂t
(8)
⎡d
⎤ 2
1
⎢ dt + w⎥ PCw ( t ) = α C PC ( t )
⎣
⎦
(9)
⎡∂
⎤ 2
∂
2
⎢ + + βC ( n) ⎥ PCR ( n, t ) = 0
⎣ ∂n ∂t
⎦
(10)
()
1
(0, t ) = α A PC1 (t )
PCA
( )
Otherwise all state probabilities are zero at t = 0.
Taking Laplace Transforms of Eq. (1) through (18)
subjected to initial conditions (19) and then on solving
(Gnedenko et al., 1969) them one by one, we obtain:
P 0 ( s) =
P A ( s) =
P B ( s) =
P D ( s) =
1
B( s)
α A D A ( s)
(11)
PB ( 0, t ) = α B P0 ( t )
(12)
α B D B ( s)
α D D D ( s)
PC1 (0, t ) = 2α C P0 ( t ) +
+
∫
0
( ) ()
1
A( s)
1
CA
A
P CA ( s) =
P CB ( s) =
( x )dx
1
PCB
y , t β B y dy +
∞
∫
(14)
P CD ( s) =
1
1
( z, t )β D ( z)dz
PCD
(24)
B( s)
1
∫ P ( x , t )β
(23)
B ( s)
(13)
0
∞
P C ( s) =
∞
(22)
B( s)
1
PD ( 0, t ) = α D P0 ( t )
(21)
B ( s)
Boundary conditions are:
PA ( 0, t ) = α A P0 ( t )
(20)
α A A( s) D A ( s)
B ( s)
α B A( s) D B ( s)
B ( s)
α D A( s) D D ( s)
B( s)
(25)
(26)
(27)
0
1
(0, t ) = α A PC1 (t )
PCA
P CW ( s) =
2
(15)
47
αC
( s + w)
.
A( s)
B ( s)
(28)
Res. J. Math. Stat., 3(1): 45-50, 2011
P CR ( s) =
2
wα C A( s) DC2 ( s)
( s + w) B ( s)
where, K = s + "A + "B +"C +"D
A( s) =
(29)
PD =
(31)
PC1 =
{
( K)
A( 0)
B ′( 0)
1
PCA
=
}
and
B( s) = K + α C − α A S A ( s) − α B S B ( s)
wα C A( s)
{
( s + w)
S C ( s)
2
− 2α C + A( s) α A S A ( s) + α B S B ( s)
}]
2
PCR
=
(33)
+ α D S D ( s) S C ( K )
1
(38)
α A A( 0) M A
(39)
B ′ ( 0)
α B A( 0) M B
1
PCB
=
B ′ ( 0)
α D A(0) M D
1
PCD
=
B ′ ( 0)
α C A(0)
1
PCW
=
wB ′ ( 0)
1 − α A S A ( s) + α B S B ( s) + α D S D ( s) DC1 ( K )
− α D S D ( s) −
(37)
B ′( 0)
(32)
2α C DC1
[
αD MD
α C A( 0)
B ′( 0)
(40)
(41)
(42)
M C2
(43)
where
2α C ⎡1 − S C ( K0 ) ⎤
⎢⎣
⎥⎦
1
A( 0) =
It is worth noticing that:
P 0 ( s) + P A ( s) + P B ( s) + P D ( s) +
P C ( s) + P CA ( s) + P CB ( s) + P CD ( s) + (33)
1
1
1
P CW ( s) + P CR ( s) =
2
2
1
1
s
PB =
αAMA
B ′( 0)
αB MB
B ′( 0)
(45)
⎡d
⎤
B ′( 0) = ⎢ B( s) ⎥
⎣ ds
⎦ s=0
(46)
Availability and profit function analysis: Laplace
transform of availability of the considered system is given
by:
s→ 0
provided the limit on L.H.S. exists, we have the following
long-run probabilities
of various flow-states
(Sharma et al., 2005) from Eq. (20) through (29):
PA =
K0 = "A + "B + "C + "D
and Mi = mean time to repair ith failure = − S i ( 0)
lim P( t ) = lim sP( s) = P( say )
1
P0 =
B ′( 0)
1
′
Long-run probabilities of various flow-states: Using
final value theorem in Laplace transforms viz,
t →∞
(44)
α C + (α A + α B + α D ) S C ( K0 )
P up ( s) =
1
s + α A + α B + 2α C + α D
⎡
⎤
2α C
⎢1 +
⎥
⎣ s +α A +α B +αC +α D ⎦
(34)
On taking inverse Laplace transform, we get:
(35)
{
Pup ( t ) = 2 exp − (α A + α B + α C + α D ) t
{
− exp − (α A + α B + 2α C + α D ) t
(36)
48
}
}
(48)
Res. J. Math. Stat., 3(1): 45-50, 2011
Also,
1.02
Series 1
1.00
Pdown ( t ) = 1 − Pup ( t )
(49)
0.98
0.96
Pup (t)
Again, the profit function for the considered system is
given by:
0.94
0.92
0.90
0.88
t
∫
G( t ) = C1 Pup ( t ) dt − C2 t
0.86
0.84
0
0.82
1
where, C1 and C2 are the revenue per unit up time and
repair cost per unit time, respectively. So here,
{
1− e (
}
} ⎥⎥ − C t
) ⎤
− α A + α B + 2α C + α D t
α A + α B + 2α C + α B ⎥
3
4
5
6
t
7
8
9
10
11
Fig. 2: The way availability of the considered system decreases
with the increase in time
(50)
G(t)
{
⎡
− (α A + α B + α C + α D )t
⎢ 2 1− e
G( t ) = C1 ⎢
⎢ α A + α B + αC + α B
⎣
2
2
⎦
50
45
40
35
30
25
20
15
10
5
0
G(t)
1
Numerical computation: For a numerical computation,
let us consider the values:
2
3
4
5
6
t
7
8
9
10
11
Fig. 3: The way profit function of the considered system
increases with the increase in time
"A = 0.004, "B = 0.002, "C = 0.008, "D = 0.005, C1 = Rs.
7.00, C2 = Rs.. 2.00 and t = 0, 1, 2, ---- .
than previous one. By making use of pre-emptive repeat
repair, we have done a better analysis of practical
situations.
Figure 2 shows the values of availability function
w.r.t. time t. Analysis of Fig. 2 reveals that availability of
considered system decreases approximately in constant
manner. There are no sudden jumps in the values of
availability function.
Figure 3 shows the values of profit function w.r.t.
time t. This graph represents that profit function of
considered system increases catastrophically in the
beginning but thereafter it increases smoothly.
By using these values in Eq. (48) and (50), one can draw
the graphs shown through Fig. 2 and 3.
RESULTS AND DISCUSSION
In this study, the author has considered a gas
separator system to compute its availability and profit
function. Parallel redundancy has been used at design
stage, to improve performance of considered system. Preemptive repeat repair policy has been adopted for repair
the failed components of the system. Supplementary
variables technique and Laplace transform have been
utilized to formulate and solve the mathematical model of
gas separator, respectively. Long-run flow-state
probabilities have been obtained to improve practical
utility of the model. A numerical example and its
graphical illustration have appended at last to highlight
important results.
Pandey and Jacob (1995), Sharma and Sharma (2005)
have evaluated the reliability of complex redundant
systems but no care was given to repair disciplines and
therefore the results obtained in this study are much better
CONCLUSION
This study is significant to priority repairs and
redundancy concept. In conclusion, we observed that we
can improve system’s overall performance by using
parallel redundancy and priority repairs. Availability and
profit function remains better as compared to simple
system. This study is very useful for the practical systems
having similar configurations. The results obtained in this
study can be directly implemented to similar systems.
49
Res. J. Math. Stat., 3(1): 45-50, 2011
ACKNOWLEDGEMENT
PCi1 (j, t)) Pr (System is failed due to failure of ith
subsystem while one unit of subsystem C
has been already failed}.Elapsed repair time
for ith subsystem lies in the interval (j, j +
)).
The authors want to convey their sincere thanks to
Prof. B.D. Sharma, Department of Mathematics, JIIT,
Noida, India for their able guidance and encouragements
during the preparations of this study.
REFERENCES
NOTATIONS
Barlow, R.E. and F. Proschan, 1965. Mathematical
Theory of Reliability. John Wiley, New York.
Chung, W.K., 1988. A K-out-of-n:G redundant system
with dependant failure rates and common cause
failures. Microelectron. Reliab., U.K, 28: 201-203.
Gnedenko, B.V., Y.K. Belayer and Soloyar, 1969.
Mathematical Methods of Reliability Theory.
Academic Press, New York.
Gupta, P.P. and R.K. Gupta, 1986. Cost analysis of an
electronic repairable redundant system with critical
human errors. Microelectron. Reliab., U.K, 26:
417-421.
Nagraja, H.N., N. Kannan and N.B. Krishnan, 2004.
Reliability. Springer Publication.
Pandey, D. and M. Jacob, 1995. Cost analysis, availability
and MTTF of a three state standby complex system
under common-cause and human failures.
Microelectronic. Reliab., U.K, 35: 91-95.
Sharma, S.K., D. Sharma and M. Masood, 2005.
Availability estimation of urea manufacturing
fertilizer plant with imperfect switching and
environmental failure. J. Comb. Info. Sys. Sci.,
29(1-4): 135-141.
Sharma, D. and J. Sharma, 2005. Estimation of reliability
parameters for telecommunication system. J. Comb.
Info. Sys. Sci., 29(1-4): 151-160.
Sharma, D., S. Gupta and N. Awasthi, 2010. Cost
estimation for ATM with pre-emptive resume repair.
Int. J. Comput. Intell. Info. Secur., Australia, 1(9):
119-127.
Notations used in this study are as follows:
"i
w
$i(j))
Si(j)
Failure rate of ith subsystem
Waiting rate for repair
First order probability that the ith failure will
be repaired in time interval (j, j + )),
conditioned that it was not repaired up to
time j
$i (j) exp{-I$i (j)dj}
P i(s)
Laplace transform of function Pi(t)
Di(j)
P0(t)
Pi(j, t))
[1- S i(j)] / j, œ i and j
Pr {System is operable}.
Pr {System suffers with ith failure}. Elapsed
repair time lies in the interval (j, j + )).
PC1 (m,t)) Pr {System suffers with failure of one unit
of system C}. Elapsed repair time lies in the
interval (m, m + )).
2
(t)
PCW
Pr {system is failed due to failure of two
units of subsystem C and is waiting for
repair}.
2
(n, t)) Pr {system is failed due to failure of two
PCR
units of subsystem C and is ready for
repair}. Elapsed repair time lies in the
interval (n, n + )).
50