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Profitfunction of twonon identical cold standby system under the influence of rainfall and humidity
Ashok Kumar Saini
Associate Professor, Department of Mathematics, BLJS College, Tosham, Bhiwani, Haryana, INDIA.
Email: [email protected]
Research Article
Abstract Introduction: Reliability is a measure of how well a system performs or meets its design requirements. It is hence the prime concern of all scientists and engineers engaged in developing such a system.. In this paper we have taken FH failure due to humidity.. When the main unit fails due to Rain Fall then cold standby system becomes operative. Rain Fall cannot occur simultaneously in both the units and after failure the unit undergoes very costly repair facility immediately. Applying the regenerative point technique with renewal process theory the various reliability parameters MTSF, Availability, Busy period, ProfitFunction analysis have been evaluated.
Keywords: Cold Standby, FHfailure due to humidity, Rain Fall, first come first serve, MTSF, Availability, Busy period, Profit Function.
INTRODUCTION
Stochastic behavior of systems operating under changing environments has widely been studied. Jinhua Cao (1989) has studied a man machine system operating under changing environment subject to a Markov process with two states. Dhillon, B.S. and Natesan, J. (1983) studied an outdoor power systems in fluctuating environment. Kan Cheng 1985) has studied reliability analysis of a system in a randomly changing environment. The change in operating conditions viz. fluctuations of voltage, corrosive atmosphere, gravity etc. may make a system completely inoperative. Severe environmental conditions can make the actual mission duration longer than the ideal mission duration. In this paper we have taken FH  Failure due to Humidity and other failure due to Rain Fall. When the main operative unit fails due to Rain Fall then cold standby system becomes operative. Rain Fall cannot occur simultaneously in both the units and after failure the unit undergoes repair facility of very high cost in case of failure due to Rain Fall immediately. Failure due to Humidity may be destructive. The repair is done on the basis of first fail first repaired.
ASSUMPTIONS
 F_{1}(t) and F_{2}(t)_{ }are general failure time distributions due to Rain Fall and Humidity. The repair is of two types Type I, TypeII with repair time distributions as G _{1}(t) and G _{2}(t) respectively.
 The Rain Fall are noninstantaneous and it cannot come simultaneously in both the units.
 Whenever the Rain Fall occur within specified limit of the unit, it works as normal as before. But as soon as there occur Rain Fall of magnitude beyond specified limit of the unit the operation of the unit stops automatically.
 The repair starts immediately after the Rain Fall of beyond specified limit of the unit are over and works on the principle of first fail first repaired basis.
 The repair facility does no damage to the units and after repair units are as good as new.
 The switches are perfect and instantaneous.
 All random variables are mutually independent.
 When both the units fail, we give priority to operative unit for repair.
 FH Failure due to Humidity when there is humidity in the atmosphere beyond specified limit.
 Repairs are perfect and failure of a unit is detected immediately and perfectly.
 The system is down when both the units are nonoperative.
SYMBOLS FOR STATES OF THE SYSTEM
F_{1}(t)_{ }and F_{2}(t)_{ }are the failure rates due to Rain Fall and Humidity respectively
G_{1}(t), G_{2}(t) – repair time distribution Type I, TypeII due to Rain Fall and Humidity respectively
Superscripts: O, CS, RF, FH
Operative, Cold Standby, Rain Fall, Failure due to Humidity respectively
Subscripts: nrf, rf, hum, ur, wr, uR
No Rain Fall, Rain Fall, Humidity, under repair, waiting for repair, under repair continued from previous state respectively
Up states: 0, 1, 2;
Down states: 3, 4
Regeneration point: 0, 1, 2
Notations
M_{i}(t) System having started from state I is up at time t without visiting any other regenerative state
A_{i }(t) state is up state as instant t
R_{i }(t) System having started from state I is busy for repair at time t without visiting any other regenerative state.
B_{i }(t) The server is busy for repair at time t.
H_{i}(t) Expected number of visits by the server for repairing given that the system initially starts from regenerative state i
By Rain Fall we mean Rain Fall beyond the specified limit
States of the System
0(O_{nrf}, CS_{nrf})
One unit is operative and the other unit is cold standby and there are no Rain Fall in both the units.
1(SORF _{rf, ur}, O_{nrf})
The operating unit fails due to Rain Fall and is under repair immediately of very costly Type I and standby unit starts operating with no Rain Fall.
2(FH _{nrf,hum, ur}, O_{nrf})
The operative unit fails due to FH resulting from Humidity and undergoes repair of type II and the standby unit becomes operative with no Rain Fall.
3(RF_{rf,uR}, FH _{nrf, hum,wr})
The first unit fails due to Rain Fall and under very costly Type! repair is continued from state 1 and the other unit fails due to FM resulting from Humidity and is waiting for repair of Type II.
4(RF _{rf,uR}, RF_{rf,wr})
The one unit fails due to Rain Fall is continues under repair of very costly Type  I from state 1 and the other unit also fails due to Rain Fall. is waiting for repair of very costly Type I.
5(FH _{nrf, hum, uR}, RF _{rf, wr})
The operating unit fails due to Humidity (FH mode) and under repair of Type  II continues from the state 2 and the other unit fails due to Rain Fall is waiting for repair of very costly Type I.
6(FH _{nrf,hum,uR}, FH _{nrf,hum,wr})
The operative unit fails due to FH resulting from Humidity and under repair continues from state 2 of Type –II and the other unit is also failed due to FH resulting from Humidity and is waiting for repair of TypeII and there is no Rain Fall.
Figure 1: The State Transition Diagram
Transition Probabilities
Simple probabilistic considerations yield the following expressions:
p_{01 }=_{ }_{,}_{ }p_{02 }=_{ }_{ }
p_{10 }= _{,}_{ }p_{13} =p_{11}^{(3)}=_{ }p_{11}^{(4)}=_{ }_{ }
p_{25} = p_{22}^{(5)}=_{ }p_{22}^{(6) }=_{ }_{ }
clearly
p_{01 }+ p_{02 }= 1,
p_{10 }+ p_{13} =(p_{11}^{(3) }) +_{ }p_{14 }= (_{ }p_{11}^{(4)})^{ }= 1,
p_{20 }+ p_{25} = (p_{22}^{(5)})^{ }+ p_{26 }=(p_{22}^{(6)})^{ }= 1 (1)
And mean sojourn time are
µ_{0 }= E(T) = (2)
Mean Time To System Failure
Ø_{0}(t) = Q_{01}(t)[s] Ø_{1}(t) + Q_{02}(t)[s] Ø_{2}(t)
Ø_{1}(t) = Q_{10} (t)[s] Ø_{0}(t) + Q_{13}(t) + Q_{14}(t)
Ø_{2}(t) = Q_{20} (t)[s] Ø_{0}(t) + Q_{25}(t) + Q_{26}(t) (35)
We can regard the failed state as absorbing
Taking LaplaceStiljes transform of eq. (35) and solving for
ø_{0}^{*}(s) = N_{1}(s) / D_{1}(s) (6)
Where
N_{1}(s) = Q_{01}^{*}[ Q_{13 }^{* }(s) + Q_{14 }^{* }(s) ] + Q_{02}^{*}[ Q_{25 }^{* }(s) + Q_{26 }^{* }(s) ]
D_{1}(s) = 1  Q_{01}^{* }Q_{10}^{*}  Q_{02}^{* }Q_{20}^{*}
Making use of relations (1) and (2) it can be shown that ø_{0}^{*}(0) =1, which implies that ø_{0}^{*}(t) is a proper distribution.
MTSF = E[T] = ^{ (s)} = (D_{1}^{’}(0)  N_{1}^{’}(0)) / D_{1} (0)
s=0
= ( +p_{01} + p_{02} ) / (1  p_{01} p_{10 } p_{02} p_{20} )
where
= _{1 }+ _{2, }_{1}= _{0 }+ _{3 }+ _{4}
++
AVAILABILITY ANALYSIS
Let M_{i}(t) be the probability of the system having started from state I is up at time t without making any other regenerative state belonging to E. By probabilistic arguments, we have
The value of M_{0}(t), M_{1}(t), M_{2}(t) can be found easily.
The point wise availability A_{i}(t) have the following recursive relations
A_{0}(t) = M_{0}(t) + q_{01}(t)[c]A_{1}(t) + q_{02}(t)[c]A_{2}(t)
A_{1}(t) = M_{1}(t) + q_{10}(t)[c]A_{0}(t) + q_{11}^{(3)}(t)[c]A_{1}(t)+ q_{11}^{(4)}(t)[c]A_{1}(t),
A_{2}(t) = M_{2}(t) + q_{20}(t)[c]A_{0}(t) + [q_{22}^{(5)}(t)[c]+ q_{22}^{(6)}(t)] [c]A_{2}(t) (79)
Taking Laplace Transform of eq. (79) and solving for
= N_{2}(s) / D_{2}(s) (10)
Where
N_{2}(s) = _{0}(s)(1  _{11}^{(3)}(s)  _{11}^{(4)}(s)) (1 _{22}^{(5)}(s) _{22}^{(6)}(s)) + (s)_{1}(s)
[ 1 _{22}^{(5)}(s) _{22}^{(6)}(s)] + _{02}(s) _{2}(s)(1 _{11}^{(3)}(s)  _{11}^{(4)}(s))
D_{2}(s) = (1  _{11}^{(3)}(s) _{11}^{(4)}(s)) { 1  _{22}^{(5)}(s)  _{22}^{(6)}(s)^{ })[1( _{01}(s) _{10} (s))(1
_{11}^{(3)}(s) _{11}^{(4)}(s))]
The steady state availability
A_{0} = = =
Using L’ Hospitals rule, we get
A_{0} = = (11)
The expected up time of the system in (0,t] is
(t) = So that (12)
The expected down time of the system in (0,t] is
(t) = t (t) So that (13)
The expected busy period of the server when there is FHfailure resultlng from Humidity in (0,t]
R_{0}(t) = q_{01}(t)[c]R_{1}(t) + q_{02}(t)[c]R _{2}(t)
R_{1}(t) = S_{1}(t) + q_{01}(t)[c]R_{1 }(t) + [q_{11}^{(3)}(t) + q_{11}^{(4)}(t)[c]R_{1}(t),
R_{2}(t) = q_{20}(t)[c]R_{0}(t) + [q_{22}^{(6)}(t)+q_{22}^{(5)}(t)][c]R_{2}(t) (1416)
Taking Laplace Transform of eq. (1416) and solving for
= N_{3}(s) / D_{3}(s) (17)
Where
N _{3}(s) = _{01}(s) _{1}(s) and
D _{3}(s)= (1  _{11}^{(3)}(s) _{11}^{(4)}(s)) – _{01}(s) is already defined.
In the long run, R_{0} = (18)
The expected period of the system under FHfailure resulting from Humidity in (0,t] is
(t) = So that
The expected Busy period of the server when there is Rain Fall in (0,t]
B_{0}(t) = q_{01}(t)[c]B_{1}(t) + q_{02}(t)[c]B_{2}(t)
B_{1}(t) = q_{01}(t)[c]B_{1}(t) + [q_{11}^{(3)}(t)+ q_{11}^{(4)}(t)] [c]B_{1}(t),
B_{2}(t) = T_{2}(t) + q_{02}(t)[c] B_{2}(t) + [q_{22}^{(5)}(t)+ q_{22}^{(6)}(t)] [c]B_{2}(t)
T_{2}(t) = e^{ λ}_{1}^{t }G_{2}(t) (19 21)
Taking Laplace Transform of eq. (1921) and solving for
= N_{4}(s) / D_{2}(s) (22)
Where
N_{4}(s) = _{02}(s) _{2}(s))
And D_{2}(s) is already defined.
In steady state, B_{0} = (23)
The expected busy period of the server for repair in (0,t] is
(t) = So that (24)
The expected number of visits by the repairman for repairing the nonidentical units in (0,t]
H_{0}(t) = Q_{01}(t)[s][1+ H_{1}(t)] + Q_{02}(t)[s][1+ H_{2}(t)]
H_{1}(t) = Q_{10}(t)[s]H_{0}(t)] + [Q_{11}^{(3)}(t)+ Q_{11}^{(4)}(t)] [s]H_{1}(t),
H_{2}(t) = Q_{20}(t)[s]H_{0}(t) + [Q_{22}^{(5)}(t) +Q_{22}^{(6)}(t)] [c]H_{2}(t) (2527)
Taking Laplace Transform of eq. (2527) and solving for
= N_{6}(s) / D_{3}(s) (28)
In the long run, H_{0} = (29)
PROFITFUNCTION ANALYSIS
The ProfitFunction analysis of the system considering mean uptime, expected busy period of the system under Rain Fall when the units stops automatically, expected busy period of the server for repair of unit under Humidity, expected number of visits by the repairman for unit failure.
The expected total ProfitFunction incurred in (0,t] is
C(t) = Expected total revenue in (0,t]  expected total repair cost in (0,t] due to Rain Fall failure
 expected total repair cost due to FH failure resulting from Humidity for repairing the units in (0,t ]
 expected busy period of the system under Rain Fall when the units automatically stop in (0,t]
 expected number of visits by the repairman for repairing of nonidentical the units in (0,t]
The expected total cost per unit time in steady state is
C = =
= K_{1}A_{0}  K _{2}R_{0 } K _{3}B_{0}  K _{4}H_{0 }
Where
K_{1}: revenue per unit uptime,
K_{2}: cost per unit time for which the system is under repair of type I
K_{3}: cost per unit time for which the system is under repair of typeII
K_{4}: cost per visit by the repairman for units repair.
CONCLUSION
After studying the system, we have analysed graphically that when the failure rate due to Humidity and failure rate due to Rain Fall increases, the MTSF and steady state availability decreases and the Profitfunction decreased as the failure increases.
REFERENCES
 Barlow, R.E. and Proschan, F., Mathematical theory of Reliability, 1965; John Wiley, New York.
 Gnedanke, B.V., Belyayar, Yu.K. and Soloyer, A.D., Mathematical Methods of Relability Theory, 1969 ; Academic Press, New York.
 Dhillon, B.S. and Natesen, J, Stochastic Anaysis of outdoor Power Systems in fluctuating environment, Microelectron. Reliab.. 1983; 23, 867881.
 Cao, Jinhua, Stochatic Behaviour of a Man Machine System operating under changing environment subject to a Markov Process with two states, Microelectron. Reliab., 1989; 28, pp. 373378.
 Kan, Cheng, Reliability analysis of a system in a randomly changing environment, Acta Math. Appl. Sin. 1985, 2, pp.219228.
