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Reliability
and CostEffectiveness of a Redundant System with Three Types of
Failures and Repairs
Dharmvir Singh
Vashisth^{1}, A. S. Gousia Banu^{2}
^{1}Department of Mathematics, R.N. Engg.
College, Rohtak (Haryana), INDIA.
^{2}Department of Management, Narayana
Engineering College (Nec) Nellore (A.P), INDIA
Correspondence addresses
[email protected] ,
[email protected]
Research Article
Abstract:The goal of this paper is to
carry out the reliability and cost effectiveness of a two unit cold
standby system with allowed downtime and correlated failure and repair
times. For this purpose, a stochastic Model is developed in which one
unit is operative and other is kept as cold standby. Each unit has
three modes of failure viz. normal, degraded and total failure. The
time taken in replacement of a failed unit is negligible but it is a
random variable. The failure rates are constant and system repair time
are arbitrary distributed. The numerical results for reliability and
costeffectiveness are obtained using supplementary variable
technique. The repair and failure times are correlated .
Introduction: In most of the
reliability models of dissimilar units studied so far with three
modes, viz, normal, partial and total failure have been considered.
Gupta et al. [1] have studied two unit stand by system under partial
failure and preemplive repair priority. They have all assumed that
failure and repair are independent but it looks in practice that they
have certain correspondence. Goel et al. [2] have considered that
failure and repair time follow a behaviour exponential distribution.
Sharma [4] has initiated the study of availability of the system which
consist of two identical cold stand by units with constant failure
rates. Initially one unit is operative while the other remains as
stand by. Each of the units has three modes, i.e., normal, degraded
and total failure. The system fails when both the units fail and may
also fail due to common cause failure. The time taken in replacement
of a failed unit by a stand by unit is not negligible but it is a
random variable. Goel et al. [2] also presented a mathematical model
for predicting a two identical active units redundant system with
three types of failures namely mechanical, catastrophic and human
failure. Any units may fail either partially or completely. And the
system is only required when all the units fail including the helping
unit. The failure rates are constant and system repair times are
arbitrarily distributed. He obtained state probabilities by Laplace
transforms.
In this paper, we are interested
to analyse the two unit cold stand by system with allowed downtime and
correlated failure and repair times. Using this concept of correlation
into consideration the joint distribution of failure and repair times
is taken to be bivariate exponential distributed. The numerical
results for reliability and costeffectiveness are obtained to improve
the importance of the study.
Notations:
(a)
(b) : Constant failure rate
due to catastrophic failure.
(c) : Constant failure rate
due to human failure.
(d) : Constant failure rate due to mechanical
failure. (1)
(e) : Constant waiting due to
mechanical failure of both the units.
(f) (at time t the system is in state
(g) (the system is in
state at time t due to partial mechanical failure
of
helping unit and
elapsed repair time lies in the interval
(h) (the system is in state
at time t due to the complete mechanical
failure of
helping unit and elapsed repair time lies in the interval
(i) (the system is in the state
at time t due to catastrophic failure and
elapsed repair
time lies in the interval
(j) (the system is in the state
at time t and elapsed repair time in the
interval
(k) (the system is in state
at time t due to human error and elapsed
repair time lies
in the interval
(l) (at time t, the system is in state
due to partial failure of maintenance
helping unit)
(m) (at time t, the system is in state
due to partial failure of maintenance
and complete
failure of helping unit).
(n) (the system is in state
at time t due to complete failure of unit)
(o) First order probabilities that failed / failed /
operable / waited.
(p):system repaired in time
(q) ,
unless otherwise stated.
Transition of diagram
Formulation of Mathematical Models:
Probabilistic consideration and preceding
stated procedures yield the following equations:
......(2)
Initial conditions
If a final state, it is ………..So otherwise 0.
Mathematical Analysis:
Taking Laplace of the equation (2) we obtain
(3)
Gupta, Singh and Kishore [2] also obtained the
up and down state Probabilitie
where, (4)
Ergodic Behavior of the System:
Goel, Singh and Kishore [2] using base, we have
has obtained values as follows:
(5)
Also,
(6)
(7)
where,
Particular Cases:
(1) Constant Rate Repair: When repair
rate is an exponential setting by
in relations (6) one may get
Goel, Singh and Kishore [2] the following value
(8)
Non Repairable System:
The Laplace system of transform of reliability
when all repairs of the system are zero, i.e., given by
where,
,
The reliability of the system is
(9)
where,
Discussion:
We look parameters as follows as Goel, Singh
and Kishore [2].
Let
(10)
The analyses for availability is obtained as
follows:
(11)
Calculating values of
T 
0 
1 
2 
3 
4 
5 


0.99 
0.94 
0.83 
0.77 
0.67 
0.61 
The values obtained above show that
decreases as time increases. It can also be shown
that increase of also causes decrease in
Let us consider
(12)
On account of (12), (8) and (9), we get
and also
R(t)
T 
0 
1 
2 
3 
4 
5 
R(t) 
1.153 
0.552 
0.233 
0.153 
0.132 
0.128 
We draw a graph in R(t), i.e., reliability time.
It is found that reliability decreases with time initially decreases
is sharp but gradually it is a uniform decrease.
Cost Effectiveness of the system:
Assuming that the service facility is
always available and remains busy for time t during interval ] 0, t [.
Let be revariance cost per unit time and service cost
per unit times.
The profit function Profit(t)
Let,
the expected profit are shown in the following
table, we get respectively
Particular values of
we calculate values of
a profit function, we give them in this table
Table : 1
Sr.No 
Time 
If
G(t) 
If
G(t) 
If
G(t) 
1
2
3
4
5
6
7 
0
1
2
3
4
5
10 
0.067
0.0922
0.148
0.267
0.7611
1.125
3.933 
0.067
0.9012
0.733
1.043
1.247
1.378
1.605 
0.067
0.867
1.533
2.237
2.848
3.374
3.606 
This table shows the expected
profit during this interval (0, t) for the fixed value per unit of time.
This shows that expected profit v/s time decreases rapidly where service
cost and increases if
References:
[1]
Gupta P. P., Singh S. B. and Geol C. K., Stand by
redundant system involving the concepts of multifailure human failure
under head of line repair policy, Bulletin of Pure & Applied Sciences,
20, pp. 34535, 2001.
[2]
Goel C. K., Singh S. B. and Jai Kishore, Reliability
analysis of two units redundant system with three types of failure under
waiting time to repair. Acta Ciencia Indica, XXIX,pp. 677688, 2003.
[3]
Mokaddis G. S. and Tawfak M. L., Some characteristics of
two dissimilar unit cold stand by redundant with three modes.
Microelectron and Reliability, 38, pp. 487504, 1996.
[4]
Sharma Viresh: Availability analysis of two unit by system
with delayed replacement under perfect switching.
