Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 9, Issue 1 (June 2014), pp. 13-27
Stochastic Modeling of a Concrete Mixture Plant with
Preventive Maintenance
Ashish Kumar and Monika Saini
Department of Mathematics
Manipal University Jaipur
Jaipur-303007, Raj (India)
S.C. Malik
Department of Statistics
M.D. University
Rohtak-124001, Haryana (India)
akbrk@rediffmail.com
Received: November 26, 2013; Accepted: February 6, 2014
Abstract
In this paper, a stochastic model for concrete mixture plant with Preventive Maintenance (PM) is
analyzed in detail by using a supplementary variable technique. In a concrete mixture plant eight
subsystems are arranged in a series. The system goes under PM after a maximum operation time
and work as new after PM. The time to failure of each subsystem follows a negative exponential
distribution while PM and repair time distributions are taken as arbitrary. A sufficient repair
facility is provided to the system for conducting PM and repair of the system. Repair,
maintenance and switch devices are perfect. All random variables are statistically independent.
Various measures of system effectiveness such as reliability, mean time to system failure
(MTSF), are derived using a supplementary variable technique. The numerical results for
reliability and availability are obtained for particular values of various parameters and costs.
Keywords: Concrete Mixture Plant; Reliability; Availability; Preventive Maintenance and
Supplementary Variable Technique
MSC 2010 No.: 90B25, 60K10
13
14
Gurju Awgichew et. al
1. Introduction
Concrete mixture plants are widely used to produce various kinds of concrete including quaking
concrete and hard concrete, suitable for large or medium scale building works, road and bridge
works and precast concrete plants, etc. Basically, such plants are designed for the production of
all types of concrete, mixed cements, cold regenerations and inertisations of materials mixed
with resin additives. Due to the complexity of modern concrete mixture systems, which involve
high risks, the concept of reliability has become a very important factor in the overall system
design. While dealing with reliability-based design of machines and structures, we can study the
relative importance of mechanical and structural failures from the point of view of loss of human
lives. Reliability analysis of such a system helps us to obtain the necessary information about the
control of various parameters. Arekar et al. (2012), Kharoufeh et al. (2010), Proctor and Singh
(1976), Shakuntla et al. (2011), Malik (2008) and Uematsu and Nishida (1987) have analyzed
single-unit systems under a common assumption that the unit works continuously till failure
without undergoing PM.
The continued operation of the systems may reduce performance and reliability of the system.
Therefore, PM of the unit is necessary after a specific period of time at any stage of operation to
improve the reliability and availability of the system because the cost to repair the system after
its failure is greater than the cost of maintaining the system before its failure. Thus, the method
of preventive maintenance can be adopted to improve the reliability and profit of system. The
concept of preventive maintenance has been used by many researchers such as Malik and Nandal
(2010), Kumar et al. (2012) and Kumar and Malik (2012) while analyzing the redundant systems
with maximum operation time. It is also interesting to note that not much work related to the
reliability modeling of the concrete mixture plant subject to preventive maintenance has been
reported so far in the literature of reliability.
Most of the authors discussed the system possessing Markovian properties. The system having
non-Markovian property can be converted into a system having Markovian nature by introducing
a new variable called a supplementary variable. Initially, Cox (1955) used the supplementary
variable in analyzing a non-Markovian system and presented a systematic solution of reliability
and availability of that system using the supplementary variable technique. Gaver (1963) studied
a parallel redundant system with constant failure and arbitrary repair rates. Since then several
authors have studied the reliability of the various systems using supplementary variable
technique. Singh and Dayal (1991) used supplementary variable technique for problem
formulation. Alfa and Rao (2000) discussed the supplementary variable technique in stochastic
models. The concrete mixture plant with preventive maintenance has not been discussed so far
even though it plays an important role in our daily life and development of the infrastructure.
The assumption of constant failure, maintenance and repair rates may not be practical in any
industry. Keeping this in view, in the present study, we have considered eight-subsystems of the
concrete mixture plant with constant failure and arbitrary repair rates of the subsystems and
discussed the reliability modeling of concrete mixture plant with preventive maintenance using
supplementary variable technique. An attempt has also been made to discuss the availability of
this plant with respect to different failure and repair rates.
AAM: Intern. J., Vol. 9, Issue 1 (June 2014)
15
The paper has been organized as follows: Section 1 is introductory in nature. In Section 2, a
summary of the system and various notations of the subsystems are presented. The basic
assumptions, on which the present analysis is based, are also discussed in Section 2. The
mathematical formulation and solution of the differential-difference equation of Concrete
mixture plant developed using the supplementary variable technique, (assuming constant failure
and variable repair rates) presented in Section 3. Certain conclusions drawn from this analysis
are also discussed in the Section 4.
2. System Description, Notations and Assumptions
Concrete mixing plants are widely used to produce various kinds of concrete including quaking
concrete and hard concrete suitable for large or medium scale building works, road and bridge
works and precast concrete plants, etc. A concrete plant, also known as a batch plant, is a device
that combines various ingredients to form concrete. Some of these inputs include sand,
water, aggregate, fly ash, potash and cement. A concrete plant can have a variety of parts such as
a dosing system, mixer feeding belt conveyor, main chassis superstructure, mixing system,
cement silo, screw conveyor, electrical control system and insulated control cabinet.
In this paper, we consider concrete mixing plant consisting of eight sub-systems namely, a
dosing system, mixer feeding belt conveyor, main chassis superstructure, mixing system, cement
silo, screw conveyor, electrical control system and insulated control cabinet. The complete
description of the systems and their notations required in the mathematical formulation are as
follows:
2.1. System Description
2.1.1. Sub-system A (Dosing system)
It is a storehouse of the aggregates which are controlled by cylinders of two material discharging
hoppers. The size of the two doors is different. The size of the material doors can be adjusted.
2.1.2. Sub-system B (Mixer feeding belt conveyor)
It is a double surface ladder fence. It is jointing with main tower in such a way that it avoids the
shaking of the main tower making the exact weighing.
2.1.3. Sub-system C (Main chassis superstructure)
It is built from color steel sandwich board. Its main functioning is heat preservation and heat
insulation. It is equipped with dust catcher to avoid pollution.
2.1.4. Sub-system D (Mixing system)
In a mixing system various materials such as cement, sand or gravel, and water are combines in a
homogenous manner to form concrete. A typical concrete mixer uses a revolving drum to mix
the components.
2.1.5. Sub-system E (Cement silo)
It is used to store dry, bulk cement.
Gurju Awgichew et. al
16
2.1.6. Sub-system F (Screw conveyor)
It is a duct along which material is conveyed by the rotational action of a spiral vane which lies
along the length of the duct.
2.1.7. Sub-system G (Electrical control system)
It is a collection of electronic devices that manage commands, directs or regulates the behavior
of the whole concrete plant.
2.1.8. Sub-system H (Insulated control cabinet)
In this cabin, electronic devices, fittings, sockets and switches exist.
2.2. Notations
A, B, C, D, E, F, G and H,
indicate that the sub-system is working in full
capacity.
a, b, c, d, e, f, g and h,
indicate the failed state of the sub-system.
αi,
denotes the constant failure rate of the units, where
i = 1,2, …, 8.
αm,
denotes the constant transition rate of the system.
( )
( )
( )
(
denotes the probability that at time t the system is
in good state.
)
(
( )
( )
denote the repair rate of the unit and probability density
function, respectively, for the elapsed repair time ‘x’, where
i = 1, 2, …, 8.
denotes the probability that at time t the system is in failed
state the elapsed repair time lies in the interval ( x, x  ),
where i 1,2, ,8.
)
denotes the probability that at time t the system is under PM,
the elapsed PM time is ‘y’.
( )
denote the preventive maintenance rate of the unit and
probability density function, for the elapsed maintenance
time ‘y’, respectively.
Laplace transform of
( )
AAM: Intern. J., Vol. 9, Issue 1 (June 2014)
( )
( )

,
17
[ ∫
( )
( )
[ ∫
( )
( )
]
]
denotes the definite integral from 0 to x.
Dosing system
Mixer feeding belt conveyor
Main chassis superstructure
Mixing system
Cement silo
Screw conveyor
Electrical control system
Insulated control cabinet
Figure 1: System Formulation
2.3. Assumptions
(i) Repair and failure rates are independent of each other and their unit is taken as per day.
(ii) Failure and repair rates of the subsystems are taken respectively as constant and variable.
(iii) Performance wise, a repaired unit is as good as new one for a specified duration.
(iv) Sufficient repair facilities are provided.
(v) Service of the subsystem includes repair and/or replacement.
(vi) Switch devices, repairs and preventive maintenances are perfect.
(vii) The distribution of preventive maintenance is considered as arbitrary.
Gurju Awgichew et. al
18
3. Formulation and Solution of Mathematical Model
By probability considerations and continuity arguments, we obtain the following differencedifferential equations governing the behavior of the system:

8 
 8





P
(
t
)

[
P
(
x
,
t
)

(
x
)
dx
]




 Pm ( y, t )  9 ( y )dy.
i
m
0
i
i
 t

i 1
i 1 0


0
(1)



 i ( x )  Pi ( x, t )  0, where i 1,2,...,8.
 
 t x

(2)
 

 9 ( x )  Pm ( y, t )  0.
 
 t y

(3)
The boundary and initial conditions to be satisfied are given below
Boundary Conditions:
Pi (0, t )  i P0 (t ), where i  1,2,...,8,
(4)
Pm (0, t )   m P0 (t ).
(5)
Pi (0)  1, when i  0,
(6)
Initial Conditions:
Pi (0)  0, when i  0.
By taking LT of equations (1)-(5) and using in (6), we get
8
8 
 


s




P
(
s
)

1

P
(
x
,
s
)

(
x
)
dx



i
m
0
i
i



   P9 ( y, s) 9 ( y )dy.
i 1  0
 i 1

 0
(7)
 

  s  i ( x )  Pi ( x, s )  0, where i 1,2,...,8
 x

(8)
 

  s  9 ( y )  Pm ( y, s )  0.
 y

(9)
Pi (0, s)  i P0 ( s), where i  1,2,...,8.
(10)
Pm (0, s)   m P0 ( s).
(11)
Now, integrating equation (8) and further using in equation (10), we get
AAM: Intern. J., Vol. 9, Issue 1 (June 2014)
19
(12)
x
[  sx   i ( x ) dx ]
Pi ( x, s )  Pi (0, s ) e
0
, where i  1,2,...,8.
Integrating equation (9) and further using equation (11), we get
(13)
y
[  sy   9 ( x) dy ]
Pm ( y, s)  Pm (0, s) e
0
.
By using equations (12-13) in equation (7), we get
8
8


 s   i  m  P0 ( s)  1   [i P0 ( s) Si ( s)]  m P0 ( s) S9 ( s).
i 1
 i 1

(14)
8


s

i (1  Si ( s ))  P0 ( s )  0 .


 i 1

(15)
[s  m (1  S9 (s))]P0 (s)  0.
(16)
P0 ( s ) 
(17)
1
,
T ( s)
where
8


T ( s)   s   i (1  Si ( s ))  m (1  S9 ( s ))  .
 i 1

(18)
Now, the Laplace Transformation of the probability that the system is in the failed state is given
by

P1( s )   P1( x, s )dx  1 P0 ( s )
0
1  S1( s)
A ( s)
 1 1
,
s
T ( s)
(19)
where
A1 ( s) 
1  S1 ( s)
.
s
Similarly

Pi ( s )   Pi ( x, s)dx  i P0 ( s )
0
1  Si ( s)
A ( s)
 1 i
.
s
T (s)
(20)
where
Ai ( s) 
1  Si ( s)
, i  2,3,4,5,6,7,8.
s

Pm ( s )   Pm ( x, s)dx  m P0 ( s )
0
1  S9 ( s)
A ( s)
 m m
.
s
T ( s)
where
Am ( s) 
1  Sm ( s )
.
s
(21)
Gurju Awgichew et. al
20
It is worth noting that
8
 Pi ( s )  Pm ( s ) 
i 0
1
.
s
(22)
Evaluation of Laplace transforms of up and down state probabilities
The Laplace transforms of the probabilities that the system is in up (i.e., good) and down (i.e.,
failed) state at time ‘‘t’’ are as follows
Pup ( s )  P0 ( s ) 
1
,
T ( s)
8
Pdown ( s )   Pi ( s )  Pm ( s ) 
i 1
8
 Ai ( s )  Am ( s )
i 1
.
(23)
T ( s)
Steady-State Probabilities
Using Abel’s Lemma in Laplace transforms, viz.
lim sZ ( s)  lim Z (t )  Z (say )
s0
t 
Provided the limit on the right hand side exists, the following time independent probabilities
have been obtained.
Pup 
1
,
8
[1    i Si' (0)   m S9' (0)]
i 1
8
   i Si' (0)
i 1
Pdown 
8
[1    i Si' (0)   m S9' (0)]
i 1
(24)
.
Reliability Indices
In order to obtain system reliability, consider repair rates (i.e., ( ) ) equal to zero. Using the
method similar to that in section 2, the differential–difference equations are:
 8

 t   i  m  P0 (t )  0.
i 1


Theorem 1.
The reliability of the system is given by
(25)
AAM: Intern. J., Vol. 9, Issue 1 (June 2014)
21
8
R(t )  e
(  i  m )t
i 1
(26)
.
Proof:
The proof of the Theorem 1 is given in the appendix.
Corollary 1.
The mean time to system failure (MTSF) is:
MTSF 
1
8
 i
i 1
.
(27)
 m
Proof:
∫
Calculating
( )
implies the result ‘*’ given in the appendix.
Special Case (Availability)
When repair rates follows exponential time distribution. Setting S9 ( s) 
m
s  m
and
i
, where i , i=1, 2, 3, ..., 8, are constant repair rates. Putting these values in equation
s  i
(17), we get
Si ( s ) 
8
s ( s   m )  ( s  i ) 
i 1
8
 ( s  i )( s   m )
i 1
.
8 
8
8
 

  s j ( s   m )  ( s  i )    s m  ( s  i ) 
j 1 
i  j ,i 1
i m,i 1

 
(28)
4. Numerical Analysis
Table-1: Effect of failure rate ( 1 ) on Reliability (R(t))
Time
α2
α3
α4
α5
α6
α7
1
2
3
4
5
6
7
8
9
10
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
α8
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
αm
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
R(t) for
α1=.001
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
R(t) for
α1=.005
0.874328
0.764449
0.668379
0.584382
0.510942
0.44673
0.390589
0.341503
0.298585
0.261061
R(t) for
α1=.05
0.835855
0.698654
0.583973
0.488117
0.407995
0.341025
0.285047
0.238258
0.199149
0.16646
Gurju Awgichew et. al
22
Table 2. Effect of failure rate (  2 ) on Reliability (R(t))
Time
α1
α3
α4
α5
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
α6
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
α7
α8
αm
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
R(t) for
α2=.002
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
R(t) for
α2=.008
0.872581
0.761397
0.664381
0.579726
0.505858
0.441402
0.385159
0.336082
0.293259
0.255892
R(t) for
α2=.08
0.811963
0.659285
0.535315
0.434656
0.352925
0.286562
0.232678
0.188926
0.153401
0.124556
Table 3. Effect of failure rate (  3 ) on Reliability (R(t))
Time
α1
α2
α4
α5
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
α6
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
α7
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
α8
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
αm
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
R(t) for
α3=.003
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
R(t) for
α3=.009
0.872581
0.761397
0.664381
0.579726
0.505858
0.441402
0.385159
0.336082
0.293259
0.255892
R(t) for
α4=.004
R(t) for
α4=.007
R(t) for
α3=.06
0.829195
0.687564
0.570125
0.472745
0.391997
0.325042
0.269523
0.223487
0.185315
0.153662
Table 4. Effect of failure rate (  4 ) on reliability (R(t))
Time
α1
α2
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
α3
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
α5
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
α6
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
α7
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
α8
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
αm
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
0.875202
0.765979
0.670387
0.586724
0.513503
0.449419
0.393332
0.344246
0.301285
0.263685
R(t) for
α4=.07
0.821766
0.675299
0.554937
0.456028
0.374749
0.307955
0.253067
0.207962
0.170896
0.140436
AAM: Intern. J., Vol. 9, Issue 1 (June 2014)
23
T Table 5. Effect of failure rate (  5 ) on reliability (R(t))
Time
α1
α2
α3
α4
α6
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
α7
α8
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
αm
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
R(t) for
α5=.05
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
R(t) for
α5=.09
0.843412
0.711343
0.599955
0.506009
0.426774
0.359946
0.303583
0.256046
0.215952
0.182136
R(t) for
α5=.15
0.794295
0.630905
0.501125
0.398041
0.316162
0.251126
0.199468
0.158437
0.125846
0.099959
R(t) for
α6=.03
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
R(t) for
α6=.093
0.824235
0.679363
0.559954
0.461534
0.380412
0.313549
0.258438
0.213013
0.175573
0.144713
R(t) for
α6=.12
0.802278
0.64365
0.516386
0.414285
0.332372
0.266655
0.213931
0.171632
0.137697
0.110471
R(t) for
α7=.0001
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
R(t) for
α7=.003
0.87529
0.766133
0.670588
0.586959
0.51376
0.449689
0.393608
0.344521
0.301556
0.263949
R(t) for
α7=.012
0.867448
0.752466
0.652725
0.566204
0.491153
0.426049
0.369576
0.320588
0.278093
0.241231
T Table 6. Effect of failure rate (  6 ) on reliability (R(t))
T
Time
α1
α2
α3
α4
α5
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
α7
α8
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
α8
αm
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
αm
Table 7. Effect of failure rate (  7 ) on reliability (R(t))
Time
α1
α2
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
α3
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
α4
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
α5
α6
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
Gurju Awgichew et. al
24
T Table 8. Effect of failure rate ( 8 ) on reliability (R(t))
Time
α1
α2
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
α3
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
α4
α5
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
α6
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
α7
αm
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
R(t) for
α8=.0002
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
R(t) for
α8=.004
0.874503
0.764755
0.66878
0.58485
0.511453
0.447267
0.391136
0.342049
0.299123
0.261584
R(t) for
α8=.122
0.777167
0.603989
0.4694
0.364802
0.283512
0.220336
0.171238
0.133081
0.103426
0.080379
.0002
.0002
.0002
.0002
.0002
.0002
.0002
.0002
.0002
.0002
R(t) for
αm=.04
0.877832
0.770589
0.676448
0.593808
0.521263
0.457582
0.40168
0.352607
0.30953
0.271715
R(t) for
αm=.12
0.810341
0.656653
0.532113
0.431193
0.349413
0.283144
0.229443
0.185927
0.150664
0.12209
R(t) for
αm=.32
0.663451
0.440167
0.29203
0.193747
0.128542
0.085281
0.05658
0.037538
0.024905
0.016523
T Table 9. Effect of Transition rate (  m ) on reliability (R(t))
Time
α1
α2
1
2
3
4
5
6
7
8
9
10
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
α3
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
α4
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
α5
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
α6
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
α7
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
α8
AAM: Intern. J., Vol. 9, Issue 1 (June 2014)
25
Table-10: Availability of Concrete Mixture Plant w.r.t. failure rate (α1).
Set 1: α2=.002, α3=.003, α4=.004, α5=.05, α6=.03, α7=.0001, α8=.0002, αm=.04, β1=.05, β2=.9, β3=.3, β4=.59,
β5=.9, β6=.98, β7=.7, β8=.9, βm=1.2
α1
In set 1
In set 1
In set 1
In set 1
In set 1
In set 1
In set 1
Replace
Replace
Replace
Replace
Replace
Replace
Replace
Set 1
α4=.004
α7=.0001
α2=.002
α3=.003
α5=.05 by α6=.03 by
α8=.0002
by
by
by α2=.07 by α3=.05
α5=.5
α6=.3
by α8=.02
α2=.04
α7=.005
0.001 0.876526
0.822082
0.770692 0.832026
0.609434
0.706026
0.87118
0.859943
0.002 0.874992
0.820733
0.769506 0.830644
0.608692
0.705031
0.869665
0.858466
0.003 0.873463
0.819388
0.768324 0.829266
0.607951
0.704038
0.868155
0.856995
0.004
0.87194
0.818047
0.767145 0.827893
0.607213
0.703048
0.86665
0.855529
0.005 0.870422
0.816711
0.76597 0.826525
0.606477
0.702061
0.865151
0.854067
0.006 0.868909
0.815379
0.764798 0.825161
0.605742
0.701076
0.863656
0.852611
0.007 0.867402
0.814051
0.76363 0.823801
0.605009
0.700095
0.862167
0.851159
0.008
0.8659
0.812728
0.762465 0.822446
0.604278
0.699116
0.860683
0.849713
0.009 0.864403
0.811409
0.761305 0.821096
0.603548
0.69814
0.859204
0.848271
0.01 0.862911
0.810095
0.760147 0.819749
0.602821
0.697166
0.85773
0.846835
In set 1
Replace
αm=.04 by
αm=.19
0.789972
0.788726
0.787483
0.786245
0.785011
0.78378
0.782553
0.781331
0.780111
0.778896
Table-11: Availability of Concrete Mixture Plant w.r.t. repair rate (β1).
Set 2: α1=.001,α2=.002, α3=.003, α4=.004, α5=.05, α6=.03, α7=.0001, α8=.0002, αm=.04, β2=.9, β3=.3, β4=.59,
β5=.9, β6=.98, β7=.7, β8=.9, βm=1.2
β1
In set 2
In set 2
In set 2
In set 2
In set 2
In set 2
In set 2
Replace
Replace
Replace
Replace
Replace
Replace
Replace
Set 2
β2=.9 by
β3=.3 by β4=.59 by
β5=.9 by
β6=.98 by
β7=.7 by
β8=.9 by
β2=1.9
β3=1.3
β4=1.42
β5=2.5
β6=2.3
β7=1.7
β8=1.9
0.01 0.870422 0.871309
0.876289
0.873435
0.89822
0.883939 0.870486
0.870511
0.02 0.874227 0.875121
0.880145
0.877266 0.902273
0.887863 0.874291
0.874316
0.03 0.875502
0.8764
0.881438
0.87855 0.903631
0.889179 0.875567
0.875592
0.04 0.876142
0.87704
0.882086
0.879194 0.904312
0.889839 0.876206
0.876231
0.05 0.876526 0.877425
0.882476
0.879581 0.904721
0.890235
0.87659
0.876615
0.06 0.876782 0.877682
0.882735
0.879839 0.904994
0.890499 0.876846
0.876872
0.07 0.876965 0.877865
0.882921
0.880023 0.905189
0.890688 0.877029
0.877055
0.08 0.877102 0.878003
0.88306
0.880161 0.905336
0.890829 0.877167
0.877192
0.09 0.877209
0.87811
0.883168
0.880269
0.90545
0.89094 0.877274
0.877299
0.1 0.877294 0.878196
0.883255
0.880355 0.905541
0.891028 0.877359
0.877385
In set 2
Replace
βm=1.2 by
βm=2.2
0.882055
0.885962
0.887272
0.887929
0.888323
0.888586
0.888774
0.888915
0.889025
0.889113
5. Conclusion
The results and system reliability are shown in Tables (1-10) which indicates that the reliability
of the system decreases with the increase of failure rates (  i ) and transition rate  m w.r.t. time
and for fixed values of other parameters. Also, it is analyzed that there are sudden jumps in the
values of reliability function and over a long period of time the system becomes less and less
reliable. Table 10, shows that are availability of the system decreases with the increase of the
failure rate ( 1 ). Table 11, shows the behavior of steady state availability with respect to repair
rate ( 1 ) and observed that availability of the system increase with the increases of the repair rate
and preventive maintenance rate.
Gurju Awgichew et. al
26
REFERENCES
Cox, D.R. (1955). Analysis of Non Markovian stochastic processes by the inclusion of
Supplementary variables, Proc. Comb. Phill. Soc., Vol. 51, pp. 433–441.
Gaver, D.P. (1963). Time to failure and availability of parallel system with repair, IEEE T.
Reliab. R-12, pp.30–38.
Proctor, C.L., Singh, B. (1976). A repairable 3- state device, IEEE Trans. Reliab., R-25, pp.
210- 211.
Uematsu, K, Nishida, T. (1987). One-unit system with a failure rate depending upon the
degree of repair, Mathematica Japonica,Vol. 32, No.1, pp.139-147.
Singh,J., Dayal,B. (1991). A 1-out of-N: G system with common cause failure and critical
human errors, Microelectron Reliab. Vol.31, pp.101–104.
Alfa, A.S., Rao, T.S.S. (2000). Supplementary variable technique in stochastic models,
Probab. Eng. Inform. Sci. Vol.14, pp. 203–218.
Malik, S.C. (2008): Reliability modeling and profit analysis of a single-unit system with
inspection by a server who appears and disappears randomly, Journal of Pure and Applied
Mathematika Sciences, Vol. LXVII, No. 1-2, pp. 135-146.
Kharoufeh, J.P., Solo, C.J. and Ulukus, M.Y. (2010). Semi-Markov models for degradationbased reliability, IIE Transactions. Vol. 42, pp. 599–612.
Malik, S. C. and Nandal, P. (2010). Cost- Analysis of Stochastic Models with Priority to
Repair Over Preventive Maintenance Subject to Maximum Operation Time, Edited Book,
Learning Manual on Modeling, Optimization and Their Applications, Excel India
Publishers, pp.165-178.
Shakuntla S., Lal, A.K., Bhatia, S.S. and Singh, J. (2011). Reliability analysis of polytube
industry using supplementary variable technique, Applied Mathematics and Computation,
Vol.218, pp. 3981-3992.
Arekar, K., Ailawadi, S. and Jain, R. (2012). Reliability modeling for wear out failure period
of a single unit system, Journal of Statistical and Econometric Methods, Vol.1 (1), pp. 3341.
Kumar, A., Malik, S. C. (2012). Stochastic Modeling of a Computer System with Priority to
PM over S/W Replacement Subject to Maximum Operation and Repair Times.
International Journal of Computer Applications, Vol.43 (3), pp. 27-34.
Kumar, A., Malik, S.C. and Barak, M.S. (2012). Reliability Modeling of a Computer System
with Independent H/W and S/W Failures Subject to Maximum Operation and Repair
Times, International Journal of Mathematical Achieves, Vol.3, No. 7, pp.2622-2630.
APPENDIX
Derivation of Equations (1)-(3)
Assuming failure rates of the system are constant and repair rates are variable. By applying
supplementary variable technique, we develop the following differential difference equations
)
associated with the state transition diagram (fig. 1) of the system at time(
) and (
8
8 

i 1
i 1 0
0
P0 (t  t )  P0 (t )[1   i   m ]   [  Pi (t ) i ( x)dxt ]   Pm (t ) m ( x)dxt  o(t ).
AAM: Intern. J., Vol. 9, Issue 1 (June 2014)
27
Pi (t  t , x  x)  Pi (t , x)[1  i ( x)x]  o(t , x) where i  1,2,...,8.
Pm (t  t , y  y)  Pm (t , y)[1  m ( y)y]  o(t , y).
Proof of Theorem 1:
Taking Laplace transform of (25) and using (6) we get
8
( s   i   m ) P0 (s)  1
i 1
Using the initial conditions, the solution can be written as
P0 ( s) 
R( s) 
1
8
s   i
i 1
,
 m
1
8
s   i
i 1
,
 m
Taking inverse Laplace transform, we get
(*)
8
R(t )  e
(  i  m )t
i 1
.
aBCDEFGH
P1(x,t)
ABCDEFGh
P8(x,t)
AbCDEFGH
P2(x,t)
α2
β8(x)
α8
α1 β1(x)
β2(x)
ABCDEFgH
P7(x,t)
ABCDEfGH
P6(x,t)
α7
β7(x)
β6(x)
α6
α3
ABCDEFGH
P0(t)
β3(x)
αm
β5(x)
α5
βm(y
)
ABCDeFGH
P5(x,t)
ABcDEFGH
P3(x,t)
α4
β4(x)
ABCdEFGH
P4(x,t)
abcdefgh
Pm(x,t)
Operative State
Figure 2. State transition Diagram
Failed State