International Journal on Mechanical Engineering and Robotics (IJMER)

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1 Cost-Benefit Analysis of Two Similar Warm Standby system subject to India losing component and failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage Ashok Kumar Saini Assoc. Prof.(Maths), BLJS COLLEGE, TOSHAM (BHIWANI) HARYANA, INDIA drashokksaini2009@gmail.com Abstract : In this paper we have taken failure of Indian component and failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage. When the main unit fails then warm standby system becomes operative. Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage cannot occur simultaneously in both the units and after failure the unit undergoes Type-I or Type-II or Type-III or Type IV repair facility immediately. Applying the regenerative point technique with renewal process theory the various reliability parameters MTSF, Availability, Busy period, Benefit-Function analysis have been evaluated. Keywords: Warm Standby, failure of Indian satellites due to power problems in an imported component, failure (FBTP) of the third stage, first come first serve, MTSF, Availability, Busy period, Benefit -Function. INTRODUCTION 'India losing satellites due to failure of imported components' IANS Updated: Jul 11, :02 IST Even as Indian space scientists are working on the partial restoration of communications satellite INSAT- 4B, they are worried because of the recurring failure of their satellites due to power supply glitches. The reason may be the failure of imported components, according to Indian Space Research Organisation (ISRO) scientists. ISRO has lost two of its satellites earlier -- Chandrayaan in 2009 and INSAT-2D in and INSAT-4B partially now. A big setback to the space agency, which is trying to get a foothold in the global communications satellite building market is the failure of the W2M satellite cobuilt by ISRO and EADS Astrium for Eutelsat Communications in January. "It seems the culprit is imported components for satellite power systems. The Chandrayaan satellite was lost due to power problems in an imported component. It seems the culprit is the imported components used in supplying power to the satellites," an ISRO official told IANS over phone on condition of anonymity. The DC to DC converter in the Chandrayaan satellite failed, which in turn heated up other components/ equipments and stopped their functioning, ultimately forcing ISRO to junk the mission well ahead of its planned life of two years. Another ISRO official, on condition of anonymity, said: "The component is imported as its size is small as the Indian built one is bigger. In space, every additional gram is important. The problem with INSAT-4B may not be connected to DC to DC converter and it is similar to the problem that afflicted W2M satellite." The 3.4-tonne W2M, the heaviest built by ISRO, launched by Ariane5 rocket from French Guyana in December 2008, developed a problem in its power supply sub-systems when it was being transferred to its intended orbit from the test orbit and Eutelsat later said the satellite is not available for service. According to ISRO, the problem with INSAT-4B is a power anomaly in one of the satellite's two solar panels and six Ku-Band and six C-Band transponders were switched off so that there is power for the remaining 12 transponders (six Ku-Band and six C-Band). ISRO officials said the agency imports the solar cells to make the solar panels that supply power to the satellite. Queried about ISRO tightening its quality control processes, an official said the agency is now focusing on the component quality. For ISRO, loss of satellites means loss of revenue opportunities - that is higher than the cost of satellite and the rocket that launched it. Now eyes are on the successful operation of the to-be launched Hylas communication satellite built by ISRO/EADS Astrium for Britain-based Avanti Communications. Its success is expected to clear any doubts of India's capability in satellite manufacturing. 9

2 "We have taken care of the power supply glitches in that," an ISRO official said. FAILURE: Indian GSLV fails during launch with GSAT-5P satellite December 25, 2010 by Chris Bergin The Indian Space Research Organization (ISRO) GSLV- F06 launch vehicle has suffered another failure, when it exploded around 60 seconds into ascent. The vehicle was carrying the GSAT-5P telecommunications satellite on Christmas Day, lifting off at 10:34am GMT from the Satish Dhawan Space Centre, Shriharikota. The launch was originally rescheduled due to a leak in the Russian cryogenic engine on the third stage of the vehicle. Indian Launch: The Geosynchronous Satellite Launch Vehicle (GSLV) is capable of placing the INSAT-II class of satellites (2000 2,500 kg) into Geosynchronous Transfer Orbit (GTO). The standard GSLV is a three stage vehicle GSLV is 49 m tall, with 414 t lift off weight. It has a maximum diameter of 3.4 m at the payload fairing. First stage comprises S125 solid booster with four liquid (L40) strap-ons. The second stage (GS2) is liquid engine and the third stage (GS3) is a cryo stage. The vehicle develops a lift off thrust of 6573 km. The first flight of GSLV took place from SHAR on April 18, 2001 by launching 1540 kg GSAT-1. It was followed by four more launches; GSLV-D2 on May 8, 2003 (GSAT kg), GSLV-F01 on September 20, 2004 (EDUSAT 1950 kg), GSLV-F02 on July 10, 2006, GSLV-F04 on September 2, 2007 (INSAT-4CR 2130 kg) and GSLV-D3 on April 15, Two of its last three flights have ended unsuccessfully, most recently with the April 15 launch of GSAT-4 (FBTP) of the third stage. Saturday s failure leaves the ISRO with three failures from their last four launches, an extended record of four failures, one partial success and two successes from their previous seven launches. The vehicle launching GSAT-5P was taller by two metres and heavier by four tonnes as compared to its standard configuration. The Russian made cryogenic engine was powered with 15.2 tonnes of fuel (liquid hydrogen as fuel and liquid oxygen as oxidizer), an increase of around three tonnes, and the engine s length also increased. The rocket had a larger fairing four-metres in diameter and made of fibre reinforced plastic (FRP) as compared to the standard configuration of 3.4-metre diameter fairing made from aluminium alloy metal. Although the current GSLVs are mixed in their fortunes, the GSLV Mk III which is scheduled to debut in This vehicle is designed to make ISRO fully self reliant in launching heavier communication satellites of INSAT-4 class, which weigh 4500 to 5000 kg, via mission launch capability for GTO, LEO, Polar and intermediate circular orbits. GSLV-Mk III is designed to be a three stage vehicle, with 42.4 m tall with a lift off weight of 630 tonnes. First stage comprises two identical S200 Large Solid Booster (LSB) with 200 tonne solid propellant, that are strapped on to the second stage, the L110 re-startable liquid stage. The third stage is the C25 LOX/LH2 cryo stage. The large payload fairing measures 5 m in diameter and can accommodate a payload volume of 100 cum. GSAT-5P with 24 C-band transponders and 12 extended C-band transponders was meant for augmenting communication services currently provided by Indian National Satellite System (INSAT). It is meant to boost TV, telemedicine and tele-education, and telephone services. The satellite, developed by ISRO Satellite Centre, Bangalore, was the fifth in the GSAT series. It had a designed mission life of 12 years. Stochastic behavior of systems operating under changing environments has widely been studied. 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. Jinhua Cao (1989) has studied a man machine system operating under changing environment subject to a Markov process with two states. The change in operating conditions viz. fluctuations of voltage, corrosive atmosphere, very low 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 failure of Indian satellites due to power problems in an imported component and Pump (FBTP) of the third stage. When the main operative unit fails then warm standby system becomes operative. Failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage cannot occur simultaneously in both the units and after failure the unit undergoes repair facility of Type- II by ordinary repairman or Type III, Type IV by multispecialty repairman immediately when failure of Indian satellites due to power problems in an imported component and Pump (FBTP) of the third stage. The repair is done on the basis of first fail first repaired. Assumptions 1. 1, 2 3 are constant failure rates when failure of warm standby, failure of Indian satellites due to power problems in an imported component and failure caused by an anomaly on the Fuel Booster ISRO are still pressing ahead with the development of 10

3 Turbo Pump (FBTP) of the third stage respectively. The CDF of repair time distribution of Type I, Type II and multispecialty repairmen Type-III, IV are G 1 (t), G 2 (t) and G 3 (t) G 4 (t). 2. The failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage is non-instantaneous and it cannot come simultaneously in both the units. 3. The repair starts immediately after failure of Indian component and failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage 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. 4. The switches are perfect and instantaneous. 5. All random variables are mutually independent. 6. When both the units fail, we give priority to operative unit for repair. 7. Repairs are perfect and failure of a unit is detected immediately and perfectly. 8. The system is down when both the units are nonoperative. Symbols for states of the System Superscripts O, WS, SPPF, FBTPF, Operative, Warm Standby, failure of Indian satellites due to power problems in an imported component, Pump (FBTP) of the third stage respectively. Subscripts nsppf, sppf, fbtpf, ur, wr, ur No failure of Indian satellites due to power problems in an imported component, failure of Indian satellites due to power problems in an imported component, failure (FBTP) of the third stage, under repair, waiting for repair, under repair continued from previous state respectively Up states 0, 1, 2, 3, 10 ; Down states 4, 5, 6, 7,8,9,11 regeneration point 0,1,2, 3, 8, 9,10 States of the System 0(O nsppf, WS nsppf ) One unit is operative and the other unit is warm standby and there is no failure of Indian component of both the units. 1(SPPF sppf, uri, O nsppf ) The operating unit failure of Indian component is under repair immediately of Type- I and standby unit starts operating with no failure of Indian component 2(FBTPF FBTPF, urii, O nsppf ) The operative unit failure (FBTP) of the third stage and undergoes repair of type II and the standby unit becomes operative with no failure of Indian satellites due to power problems in an imported component 3(FBTPF FBTPF, uriii, O nsppf ) The first unit failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage and under Type-III multispecialty repairman and the other unit is operative with no failure of Indian satellites due to power problems in an imported component 4(SPPF sppf,ur1, SPPF sppf,wri ) The unit failed due to SPPF resulting from failure of Indian satellites due to power problems in an imported component, under repair of Type- I continued from state 1and the other unit failed due to SPPF resulting from failure of Indian component is waiting for repair of Type-I. 5(SPPF sppf,ur1, FBTPF FBTPF, wrii ) The unit failed due to SPPF resulting from failure of Indian satellites due to power problems in an imported component, is under repair of Type- I continued from state 1and the other unit failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage is waiting for repair of Type- II. 6(FBTPF FBTPF, urii, SPPF sppf,wri ) The operative unit Pump (FBTP) of the third stage is under repair continues from state 2 of Type II and the other unit failed due to SPPF resulting from failure of Indian satellites due to power problems in an imported component, is waiting under repair of Type-I. 7(FBTPF FBTPF,uRII, SPPF sppf,wrii ) The one unit failure (FBTP) of the third stage is continued to be under repair of Type II and the other unit failed due to SPPF resulting from failure of Indian satellites due to power problems in an imported component is waiting for repair of Type- II. 8(SPPF sppf,uriii, FBTPF FBTPF, wrii ) The one unit failure of Indian satellites due to power problems in an imported component is under multispecialty repair of Type-III and the other unit failure caused by an anomaly stage is waiting for repair of Type-II. 9(SPPF sppf,uriii, FBTPF FBTPF, wri ) The one unit failure of Indian component is under multispecialty repair of Type-III and the other unit failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage is waiting for repair of Type-I 10(O nsppf FBTPF FBTPF, uriv ) The one unit is operative with no failure of Indian component and warm standby unit failure caused by an 11

4 anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage and undergoes repair of type IV. 11(O nsppf FBTPF FBTPF, uriv ) The one unit is operative with no failure of Indian component and warm standby unit failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage and repair of type IV continues from state 10. Transition Probabilities Simple probabilistic considerations yield the following expressions: p 01 = 1 / , p 02 = 2 / , p 0,10 = 3 / p 10 = pg 1 ( 1 )+q G 2 ( 2 ), p 14 = p- pg 1 ( 1 ) = p (4) 11, p 15 = q- q G 1 ( 2 ) = p (5) 12, p 23 = pg 2 ( 1 )+q G 2 ( 2 ), p 26 = p- pg 2 ( 1 ) = p (6) 29, p 27 = q- qg 2 ( 2 ) = p 28, p 30 = p 82 = p 91 = 1 p 0,10 = pg 4 ( 1 )+q G 4 ( 2 ) p 10,1 = p- pg (11) 4 ( 1 ) = p 10,1 p 10,2 = q- q G (11) 4 ( 2 ) = p 10,2 (1) We can easily verify that p 01 + p 02 + p 03 = 1, p 10 + p 14 (=p (4) (5) 11 ) + p 15 (=p 12 ) = 1, p 23 + p 26 (=p (6) 29 )+p 27 (=p 28 ) = 1 p 30 = p 82 = p 91 = 1 p 10,0 + p (11) 10,1 (=p 10,1 ) + p (12) 10,2 (=p 10,2 ) = 1 (2) And mean sojourn time is µ 0 = E(T) = Mean Time To System Failure Ø 0 (t) = Q 01 (t)[s] Ø 1 (t) + Q 02 (t)[s] Ø 2 (t)+ Q 0,10 (t)[s] Ø 10 (t) Ø 1 (t) = Q 10 (t)[s] Ø 0 (t) + Q 14 (t) + Q 15 (t) Ø 2 (t) = Q 23 (t)[s] Ø 3 (t) + Q 26 (t) + Q 27 (t) Ø 3 (t) = Q 30 (t)[s] Ø 0 (t) Ø 10 (t) = Q 10,0 (t)[s] Ø 10 (t) + Q 10,2 (t)[s] Ø 1 (t)+ Q 10,2 (t)[s] Ø 2 (t) (3-6) We can regard the failed state as absorbing Taking Laplace-Stiljes transform of eq. (3-7) and solving for ø 0 (s) = N 1 (s) / D 1 (s) N 1 (s) = {Q 01 + Q 0,10 Q 10,1 } [ Q 14 (s) + Q 15 (s) ] + {Q 02 + Q 0,10 Q 10,2 } [ Q 26 (s) + Q 27 (s) ] D 1 (s) = 1 - {Q 01 + Q 0,10 Q 10,1 } Q 10 - {Q 02 + Q 0,10 Q 10,2 } Q 23 Q 30 - Q 0,10 Q 10,0 MTSF = E[T] = Making use of relations (1) & (2) it can be shown that ø 10,1 (11) }[ 10 { } + 0 (0) =1, which implies that ø 0 (t) is a proper 12 (5) ] { 02 + distribution. 0,10 12 (s) s=0 = (D 1 (0) - N 1 (0)) / D 1 (0) = ( + ( p 01 + p 0,10 p 10,1 ) +( p 02 + p 0,10 p 10,2 )( + µ 3 )+ µ 10 p 0,10 / (1 - (p 01 + p 0,10 p 10,1 ) p 10 - (p 02 + p 0,10 p 10,2 ) p 23 ) - p 0,10 p 10,0 μ 0 = μ 01 + μ 02 +µ 0,10, μ 1 = μ 10 + μ 11 (4) + μ 12 (5), μ 2 = μ 23 +μ 28 + μ 29 (6), µ 10 = µ 10,0 + µ 10,1 + µ 10,2 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. By probabilistic arguments, we have M 0 (t) = e 1 t e 2 t e 3 t, M 1 (t) =p G 1 (t) e - 1 t M 2 (t) =q G 2 (t) e - 2 t, M 3 (t) = G 3 (t), M 10 (t) = G 4 (t) e - 3 t 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) + q 0,10 (t)[c]a 10 (t) A 1 (t) = M 1 (t) + q 10 (t)[c]a 0 (t) + q 12 (5) (t)[c]a 2 (t)+ q 11 (4) (t)[c]a 1 (t), A 2 (t) = M 2 (t) + q 23 (t)[c]a 3 (t) + q 28 (t)[c] A 8 (t) + q 29 (6) (t)] [c]a 9 (t) A 3 (t) = M 3 (t) + q 30 (t)[c]a 0 (t) A 8 (t) = q 82 (t)[c]a 2 (t) A 9 (t) = q 91 (t)[c]a 1 (t) A 10 (t) = M 10(t) + q 10,0(t)[c]A 0(t) + q 10,1 (11) (t)[c]a 1 (t)+ q 10,2 (11) (t)[c]a 2 (t) (8-15) Taking Laplace Transform of eq. (7-15) and solving for = N 2 (s) / D 2 (s) (16) N 2 (s) ={ 0, } [{1 11 (4) }{1-28 (7 82 }- 12 (5) 29 (6) 91 ] + { 01+ 0,10 10,1 (11) }[ 1 {1 28 (5) 82} ]+{ ,10 10,2 (11) } [{ 23 3}{1 11 (4) }+ 29 (6) 91 1] D 2 (s) = {1-11 (4) }{1-28 (7 82 }- 12 (5) 29 (6) 91 -{ 01+ 0,10

5 10,2 (11) }{[ {1 11 (4) }+ 29 (6) 91 10] The steady state availability A 0 = = = Using L Hospitals rule, we get A 0 = = (17) Where N 2 (0) ={p 0,10 10 (0)+ 0 (0) } [{1 p 11 (4) }{1- p 28 }- p 12 (5) p 29 (6) ] + { p 01 + p 0,10 p 10,1 (11) }[ 1 (0) {1 p 28 } +p 12 (5) p 23 3 (0)+ 2(0)]+{ p 02 +p 0,10 p 10,2 (11) } [{ p 23 3(0)+ 2(0) }{1 p 11 (4) }+ p 29 (6) 1(0)] D 2 (0) =µ 0 [p 10 (1- p 28 }+ p 12 (5) p 23 ]+ µ 1 [p 29 (6) + p 01 p 23 - p 0,10 {p 10,0 {1- p 28 }+p 23 p (11) 10,2 p 23 }]+ µ 2 [(1-p (4) 11 ) - p 01 p 10 -p 0,10 (p 10 - p 10 (11) (5) p 10,2 + p 12 p 10,0 )] } + µ 3 [p 23 [p (5) 12 {p 01 + p 0,10 p (11) 10,1 }+(1 p (4) 11 }{ p 02 + p 0,10 p (11) 10,2 }]+ µ 8 [p 28 (1- p 0,10 p 10,0 - p 10 { p 01 + p 0,10 p (11) 10,1 })] + µ 9 [p (6) 29 { p (5) 12 (1- p 0,10 p 10,0 +( p 02 + p 0,10 p (11) 10,2 })] + µ 10 [p (6) 29 { p (5) 12 (1- p 0,10 p 10,0 +( p 02 + p 0,10 p (11) 10,2 })] and µ 3 = µ 30, µ 9 = µ 91, µ 8 = µ 81 The expected up time of the system in (0,t] is (t) = So that (18) The expected down time of the system in (0,t] is (t) = t- (t) So that (19) The expected busy period of the server when there is failure of Indian satellites due to power problems in an imported component and failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage in (0,t]-R 0 R 0 (t) = q 01 (t)[c]r 1 (t) + q 02 (t)[c]r 2 (t) + q 0,10 (t)[c]r 10 (t) R 1 (t) = S 1 (t) + q 10 (t)[c]r 0 (t) + q 12 (5) (t)[c] R 2 (t) + q 11 (4) (t)[c]r 1 (t) R 8 (t) +q 29 (6) (t)][c]r 9 (t) R 3 (t) = S 3 (t) + q 30 (t)[c]r 0 (t) R 8 (t) = S 8 (t) + q 82 (t)[c]r 2 (t) R 9 (t) = S 9 (t) + q 91 (t)[c]r 1 (t) R 10 (t) = S 10 (t) + q 10,0 (t)[c]r 0 (t) + q 10,1 (11) (t)[c]r 1 (t)+ q 10,2 (11) (t)[c]r 2 (t) (20-26) S 1 (t) =p G 1 (t) e - 1 t, S 2 (t) =q G 2 (t) e - 2 t S 3 (t) = S 8 (t)= S 9 (t) = G 3 (t) S 10 (t) = G 4 (t) (27) Taking Laplace Transform of eq. (16-26) and solving for = N 3 (s) / D 2 (s) (23) N 3 (s) ={ ,10 10,1 (11) }[ S 1 ( } + 12 (5) [ S S3+ 28 S8+ 29 (6) S9)]]+ { ,10 10,2 (11) } [ { S S S 8 + S (6) 9 29 )(1-11 (4) (6) )+ S ] + 0,10 S 10 [{ }{1-11 (4) (6) } (5) ] and D 2 (s) is already defined. In the long run, R 0 = (24) Where N 3 (0) ={p 01 +p 0,10 p (11) 10,1 }[ S 1 (1 p 28 } +p (5) 12 [ S 2 +p 23 S3+p 28 S8+p (6) 29 S9)]]+ {p 02 +p 0,10 p (11) 10,2 } [ { S 2 + p 23 S 3 +p 28 S 8 + S 9 p (6) 29 )(1- p (4) 11 )+ S 1 p (6) 29 ] + p 0,10 S 10 [{1-p 28 }{1- p (4) 11 }- p (6) 29 p (5) 12 ] and D 2 (0) is already defined. The expected busy period of the server when there is failure of Indian satellites due to power problems in an imported component and failure caused by an anomaly stage in (0,t] is R 2 (t) = S 2 (t) + q 23 (t)[c]r 3 (t) + q 28 (t) H 8 (t) +Q (6) 29 (t)] [c]h 9 (t) 13 (t) = So that Q 01 The expected number of visits by the repairman Type-I or Type-II for repairing the identical units in (0,t]-H 0 H 0 (t) = Q 01 (t)[s][1+ H 1 (t)] + Q 02 (t)[s][1+h 2 (t)]+q 0,10 (t)[s] H 10 (t)] H 1 (t) = Q 10 (t)[s]h 0 (t)] + Q 12 (5) (t)[s] H 8 (t) + Q 11 (4) (t)] [s]h 1 (t), H 2 (t) = Q 23 (t)[s]h 3 (t) + Q 28 (t) [s]

6 H 3 (t) = Q 30 (t)[s]h 0 (t) H 8 (t) = Q 82 (t)[s]h 2 (t) H 9 (t) = Q 91 (t)[s]h 1 (t) H 10 (t) = Q 10,0 (t)[s]h 10 (t)] + Q 10,1 (11) (t)[s]h 1 (t)]+q 10,2 (11) (t)[s] H 2 (t)] (25-30) Taking Laplace Transform of eq. (25-30) and solving for = N 4 (s) / D 3 (s) (31) N 4 (s) = { Q 01 + Q 02 }[ { 1 Q 11 (4) }{1-Q 28 Q 82 } Q 12 (5) Q 29 (6) Q 91 ] And D 3 (s) = {1 Q (4) 11 } { 1- Q 28 Q 82 } Q (5) (6) 12 Q 29 Q 91 ](1- Q 0,10 Q 10,0 )-{ Q 01 + Q 0,10 Q (11) 10,1 }[ Q 10 { 1 (5) Q 28 Q 82 }+ Q 12 Q 23 Q 30 ] {Q 02 + Q 0,10 Q (11) 10,2 }[ Q 23 Q 30 {1 Q (4) 11 }+ Q (6) 29 Q 91 Q 10 ] In the long run, H 0 = N 4 (0) / D 3 (0) (32) N 4 (0) ={1 p 0,10 }[ {1 p 11 (4) } { 1- p 28 } p 12 (5) p 29 (6) ] The expected number of visits by the multispecialty repairman Type-III for repairing the identical units in (0,t]-W 0 W 0 (t)=q 01 (t)[s]w 1 (t)+ Q 02 (t)[s] W 2 (t) + Q 10,0 (t)[s] W 10 (t) W 1 (t) = Q 10 (t)[s]w 0 (t)] + Q 12 (5) (t)[s] W 2 (t) + Q 11 (4) (t)] [s]w 1 (t), W 2 (t) = Q 23 (t)[s]w 3 (t) + Q 28 (t) [s] W 8 (t) +Q 29 (6) (t)] [c]w 9 (t) W 3 (t) = Q 30 (t)[s][1+w 0 (t) ] W 8 (t) = Q 82 (t)[s][1+w 2 (t) ] W 9 (t) = Q 91 (t)[s][1+w 1 (t) ] W 10 (t)=q 10,0 (t)[s]w 0 (t)+ Q (11) 10,1 (t)[s] W 1 (t) + Q (12) 10,2 (t)[s] W 2 (t) (33-39) Taking Laplace Transform of eq. (33-39) and solving for = N 5 (s) / D 3 (s) (40) N 5 (s) = {Q 01 + Q 0,10 Q 0,10 (11) }[Q 12 (5) [ Q 23 Q 30 + Q 28 (5) Q 82 + Q 29 (6) Q 91 ] + {Q 02 + Q 0,10 Q 10,2 (11) }[ [ Q 23 Q 30 + Q 28 (5) Q 82 + Q 29 (6) Q 91 {1 Q 11 (4) }] In the long run, W 0 = N 5 (0) / D 3 (0) (41) N 5 (0) = {p 01 + p 0,10 p 10,1 (11) } p 12 (5) + { p 02 + p 0,10 p 10,2 (11 } {1 p 11 (4) }] The expected number of visits by the multispecialty repairman Type-III for repairing the identical units in (0,t]-Y 0 Y 0 (t)=q 01 (t)[s]y 1 (t)+ Q 02 (t)[s] Y 2 (t) + Q 0,10 (t)[s] [1+Y 10 (t)] Y 1 (t) = Q 10 (t)[s]y 0 (t) + Q 12 (5) (t)[s] Y 2 (t) + Q 11 (4) (t)] [s]y 1 (t), Y 2 (t) = Q 23 (t)[s]y 3 (t) + Q 28 (t) [s] Y 8 (t) +Q 29 (6) (t)] [c]y 9 (t) Y 3 (t) = Q 30 (t)[s][1+y 0 (t) ] Y 8 (t) = Q 82 (t)[s]y 2 (t) Y 9 (t) = Q 91 (t)[s]y 1 (t) Y 10 (t)=q 10,0 (t)[s]y 0 (t)+ Q 10,1 (11) (t)[s] Y 1 (t) + Q 10,2 (12) (t)[s] Y 2 (t) (42-48) Taking Laplace Transform of eq. (42-48) and solving fory 0 (s),we get Y 0 (s) = N 6 (s) / D 3 (s) (49) N 6 (s) = Q 0,10 [{1 Q (4) (5) 11 }(1- Q 28 Q 82 } - Q (5) (6) 12 Q 29 Q 91 {1- Q 0,10 Q,10,0 }+{Q 02 + Q 0,10 Q (11) 10,2 }[ [ Q 23 Q 30 {1 Q (4) 11 }+ Q (6) 10 Q 29 Q 91 ] In the long run, W 0 = N 6 (0) / D 3 (0) (50) N 6 (0) = p 0,10 [{1-p 11 (4) }{1- p 28 }- p 12 (5) p 29 (6) ] p 12 (5) + { p 02 + p 0,10 p 10,2 (11 } {1 p 11 (4) }] Benefit- Function Analysis The Benefit-Function analysis of the system considering mean up-time, expected busy period of the system under failure of Indian satellites due to power problems in an imported component and failure caused by an anomaly stage, expected number of visits by the repairman for unit failure. The expected total Benefit-Function incurred in (0,t] is C(t) = Expected total revenue in (0,t] - expected busy period of the server when there is failure of Indian satellites due to power problems in an imported component and failure caused by an anomaly stage in (0,t] - expected number of visits by the repairman Type- I or Type- II for repairing of identical the units in (0,t] 14

7 - expected number of visits by the multispecialty repairman Type- III for repairing of identical the units in (0,t] - expected number of visits by the multispecialty repairman Type- IV for repairing of identical the units in (0,t] C = = = K 1 A 0 - K 2 R 0 - K 3 H 0 - K 4 W 0 K 5 Y 0 K 1 - revenue per unit up-time, K 2 - cost per unit time for which the system is busy under repairing, K 3 - cost per visit by the repairman type- I or type- II for units repair, K 4 - cost per visit by the multispecialty repairman Type- III for units repair K 5 - cost per visit by the multispecialty repairman Type- IV for units repair CONCLUSION After studying the system, we have analyzed graphically that when the failure rate due to failure of Indian component, failure caused by an anomaly on the Fuel Booster Turbo Pump (FBTP) of the third stage increases, the MTSF, steady state availability decreases and the Profit-function decreased as the failure increases. REFERENCES [1] Dhillon, B.S. and Natesen, J, Stochastic Analysis of outdoor Power Systems in fluctuating environment, Microelectron. Reliab.,1983; 23, [2] Kan, Cheng, Reliability analysis of a system in a randomly changing environment, Acta Math. Appl. Sin. 1985, 2, pp [3] 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 [4] Barlow, R.E. and Proschan, F., Mathematical theory of Reliability, 1965; John Wiley, New York. [5] Gnedanke, B.V., Belyayar, Yu.K. and Soloyer, A.D., Mathematical Methods of Relability Theory, 1969 ; Academic Press, New York. Fig. The State Transition Diagram Up-State Down-State regeneration point 15