J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012)
GERT Analysis of a Three Unit Cold Standby System with Single Repair Facility K. P. JOSE Department of Mathematics St. Peter’s College, Kolenchery-682 311, Kerala, India (Received on: 31st December, 2011) ABSTRACT A three-unit cold standby system has been modeled and analyzed through GERT (Graphical Evaluation Review Technique) approach. Various performance characteristics of the system have been derived analytically. Various plots are also given to highlight the importance of the results obtained. Keywords: Graphical Evaluation Review Technique, Flow Graph Theory, W-function.
1. INTRODUCTION The network modeling techniques represents a valuable aid in the analysis and synthesis of systems. Networks and network analysis are playing an increasingly important role in the descriptions of and improvement of operational systems. The ease with which the systems can be modeled in the network forms is the fundamental reason for this significant increase in the use of networks. Graphical Evaluation Review Technique (GERT) has been initiated by Prisker and Happ4,5, pritsker and Whitehouse7 as new graphical procedure which combines the disciplines of Flow graph theory, Moment generating function and PERT (Programme Evaluation Review Technique) for analysis of stochastic networks having logical nodes
and directed branches. Recently, Shanker and Sahani1, investigated some standby systems with repair facility through GERT approach. However the study of standby systems with more than two units, though very important, has received much less attention, possibly because of the built-in difficulties and analyzing them. In this paper an attempt has been made and analyzes a three unit cold standby system through GERT approach. The following reliability characteristics of system interest to system designers and operation managers have been derived. i) Mean time to system failure ii) Steady-state availability of the system iii) Busy-time, idle-time and busyness of the service facility iv) Steady-state probability of the system
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
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K.P. Jose, J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012)
2. BRIEF REVIEW OF GERT GERT analysis is applied to stochastic network having the following features. a) Each network consists of logical nodes and directed branches. b) A branch has probability that the activity with it will be performed. c) Other parameters describe the activities represented by the branches. In this paper the reference will be made only to the time parameter. The time associated with a branch is characterized by a moment generating function (m.g.f.) of the form ∞
∞
where denotes the density of and , any real variable. The probability that any branch is realized is multiplied by the m.g.f.to yield a -function such that This function is used to obtain the information of relationship which exists between the nodes. Furthermore, the equivalent probability of realization of the
network, equivalent m.g.f. and mean time of realization of the network, respectively are given as ; and
3. DESCRIPTION OF THE SYSTEM The system consists of three independent and identical units. At time t = 0, two units are switched on-line and other is kept as cold standby. The life time and repair time of the units are independent random variables. The system fails when all the independent units are failed. There is a single repair facility and discipline is FIFO. The following assumptions are made: i)
The units are completely rejuvenated after repair. ii) All the switch over times are negligible and switching is perfect iii) The life time and repair time of the units are exponentially distributed with parameters , respectively.
4. GERT ANALYSIS OF THE SYSTEM W21
1
W43
W12 S1
W23 S2
W34 S3
S4
W32 Figure1. GERT network of the tree-unit cold standby system Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
K.P. Jose, J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012)
The possible states of the tree unit cold standby system mentioned above can be defined as follows. : initial state of the system. : state in which two units are operating and is in standby. : state in which two units are operating and other one is being repaired. : state in which one unit is operating, one unit is being repaired and other is non-operating. : state in which all the units are nonoperating.
2 2
3
4.1 Determination of MTSF
4 0
any node to
1 will be
probability to move to ! 1 is
The mean time to system failure (MTSF) is defined as the time until the system is completely inoperative. This is accomplished by finding the W-function from the initial state to the terminal state by applying Mason’s3 rule in the figure 1as follows
where
Using these state dependents and , it can be observed Whitehouse7, that the time to move from any node to 1 or node ! 1 is an exponential distribution with mean and probability to move from
1 ,
1 , if j
At start up the system is assumed to be in state in state .
The failure and repair time density function of a unit is given as and , respectively. The dependent ′s and ′s will thus be 1 2 0
57
while the
.
Therefore, the above states enable us to construct the GERT representation of the system as shown in figure 1, where
# 2 2
$1 ! % 2 2
$1 ! %
$1 ! % 2 2
$1 ! %
"1 !
and "1 ! #
Therefore,
2 1 1 1 2 2 2
2
1 1 1 1 1 2
2 2
2 2
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
(1)
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K.P. Jose, J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012)
Now probability of realization of the network 1 and hence & '
4.2 Determination of System Availability The system availability ( , is defined as the probability that the system is in operating state at time . The steady-state availability is defined as , - , - ./01 - &2' &2& &&3
( lim ( ∞
Now, we derive &2' and &&3 of the system as follows.
From the definition of states it is evident that time between failure occurs between the states and . Therefore, taking m.g.f. equal to 1 for all paths emanating from state , i.e., from to and to , we have ;
(2)
Since the system becomes inoperative in state , after the failure of third unit, therefore, the equivalent W-function is obtained from equations (1) and (2) as follows.
# "1 ! # 2 2
2
1! "1 ! # "1 ! # ! "1 ! # 2 2 2 2 2
2 2
Thus, &2' 4
4 5
In order to determine &&3, the equivalent W-function is
$1 ! % Here 1, therefore &&3 4
1 5
"1 !
3 4
Finally, Availability (, is given by
ŕ°°ŕ´Šŕ°Ž ŕ°śŕ°Żŕ´Šŕ°śŕ´‹ŕ°Ž ŕ°°ŕ´Šŕ°Ż ŕ°°ŕ´Šŕ°Ž ŕ°śŕ°Żŕ´Šŕ°śŕ´‹ŕ°Ž ŕ°
ŕ´‹ ŕ°°ŕ´Šŕ°Ż
ŕ°Ž ŕ°Ž ŕ°Ż ŕ°Ż ŕ°Ž ŕ°Ž ŕ°Ż
(3)
The plots of equation (3) are shown in figure 2. These plots indicate that for specified values of , the availability decreases for corresponding increasing values of .
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K.P. Jose, J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012)
4.3 Determination of busy-time and idle-time for service facility When process is in state , all units and spare are in workable condition and the service facility is idle. Idle is the consecutive time that the service facility has no work. This path is epresented by
"1 ! # .
Therefore , 61 7 - The complement of this quantity is the consecutive time, the service facility is busy. It is called the busy time, equivalent Wfunction is . In the system under consideration:
FAILURE RATE (Îť) Figure2. Steady state availability vs. failure rate
1
Therefore,
1 1
1 1 1 1 2
2
2
1 1 1 1 2
2
61 89 : -
Finally, the busyness of the service facility may be defined as 29 :1
61 89 : - 61 89 : - 61 7 -
(4)
The plots of the equation (4) are shown in figure 3. These plots show that for specified value of , the busyness increases for corresponding increasing values of . Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
60
K.P. Jose, J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012) 1 µ =1 µ =2 µ =3
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10
12
FAILURE RATE (λ) Figure3 Busyness vs. failure rate
4.4 Determination of steady- state probabilities
W21’
W43
S1’
S1
S2
W12
S3
W23
S4
W34
W32 Figure 4
The steady state probabilities are the probabilities that at any time after the system has reached steady state, the process will be at a given state. The system has a regenerative nature that all returns to a given
point in the system have a common distribution. The steady state probabilities for a given state are equal to the expected total time of the regeneration.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
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K.P. Jose, J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012)
The expected time of regeneration is equal to the first moment of m.g.f. representing the time to return to any node in the system. Thus, the mean recurrence time ′
1
from the state to the state ′ (say) is found from the representation in figure 4. The W-function from the state to the state ′ is given by
′ 1 1
1 1 1 1 1 2
2
2
2
1 1 1 1 2
2
Mean time of regeneration is obtained as
4
4 ; â&#x20AC;˛ <5
To find the portion of time spent in each state, set the m.g.f. equal to one for all paths which do not emanate from the node of interest. (1) Time spent in state : the time spent in state is obtained from equation (5) by taking ; â&#x20AC;˛ ;
; ;
1
1 2
2 2
1 2
Therefore,
â&#x20AC;˛
Hence, mean time spent in state
4 2 2
(2) Time spent in state : the time spent in state is obtained from equation (5) by taking
;
; 1; 1
Hence, mean time spent in state
(3) Time spent in state : the time spent in state is obtained from equation (5) by taking
; â&#x20AC;˛
; 1; 1
Hence, mean time spent in state
(4) Time spent in state : the time spent in state is obtained from equation (5) by taking ; â&#x20AC;˛ ; 1; 1
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
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K.P. Jose, J. Comp. & Math. Sci. Vol.3 (1), 55-62 (2012)
Hence, mean time spent in state
The steady state probabilities of different states are given by ŕŻ&#x153;
! "# $ŕŻ&#x153; 1,2,3,4 %&# ' ( ) * ! )+ , - , )
Finally, steady-state probabilities of the process in different state can be expressed as follows (i)
Steady state probability of state is given by = 4 4 2
(ii) Steady state probability of state is given by = 4 4 2 (iii) Steady state probability of state is given by 4 4 4 2 (iv) Steady state probability of state is given by 4 = 4 4 2 =
CONCLUDING REMARKS
standby
In this paper a three-unit cold system has been successfully
modeled and analyzed through GERT method. The procedure developed here may also be used for analyzing other complex reliability systems with variable failure and repair rates. To highlight the results through plots one may use MATLAB package2. REFERENCES 1. Gauri Shankar and Vandana Shahani, GERT analysis of a two-unit cold standby system with repair, Microelectron, Reliab., 35(5),837-840 (1995). 2. Ke Chen, Peter J. Giblin, Alan Irving, Mathematical Explorations with MATLAB, Cambridge University Press (1999). 3. Mason S.J., Some properties of signal flow graphs, Proc.IRE 41, 1144-1156 (1953). 4. Pritsker A. A.B. and Happ W.W., GERT: Graphical Evaluation and Review Technique-I, Fundamentals, J. Ind. Eng.,17, 267-276 (1966). 5. Pritsker A. A. B. and Happ W.W., GERT-II: Probabilistic Industrial Engineering Applications, J.Ind.Eng.,17, 293-301 (1966). 6. Ravichandran N., Stochastic methods in reliability theory, Wiley, New York (1989). 7. Whitehouse G.E., System analysis and design using network techniques, Prentice Hall, Englewood Cliffs, New Jersey (1973).
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)