International Journal on Mechanical Engineering and Robotics (IJMER) ______

Benefit-Function of Two- Identical Cold Standby Navy Warship System subject to Blasts, explosion and Fire or Human negligence

Ashok Kumar Saini BLJS College, Tosham (Bhiwani) Haryana, Email : [email protected]

Laxmi were unable to detect INS Trishul and prevent the Abstract- An article in India Today reports that since 1990, the has lost one warship in peacetime every side on collision. No casualties were reported. five years. Since 2004, it has lost one naval combatant April 2006: INS Prahar (K98), Veer class corvette, sank every two years. While peacetime losses of warships are not after colliding with the MV Rajiv Gandhi vessel about uncommon (since the World War II, the US Navy has lost 20 nautical miles away from the Goa coast. No 16 warships in accidents; Russia's nuclear Kursk sank in August 2000 after a faulty casualties were reported. The commanding officer of the torpedo exploded during a training exercise), the magazine ship, Lieutenant CommanderYogesh Tripathi was found mentioned that few global navies have such a dubious guilty of negligence by an Indian Navy court- record. According to , while some of martial and dismissed from service. accidents reported since August 2013 was serious, many of them were trivial incidents exaggerated in public. September 2006: INS Dunagiri (F36), Nilgiri class , collided with a Shipping Corporation of These accidents have been attributed to ageing ships in India merchant vessel, the MV Kiti, off the coast need of maintenance, delayed acquisitions by the Ministry of . There were no casualties, but the Dunagiri of Defence, and human error. However naval suffered damage and required extensive repairs. commentators also argue that as India's large navy of 160 ships clocks around 12,000 ship-days at sea every year, in January 2008: INS Sindhughosh (S55), a Kilo-class varied waters and weather, some incidents are inevitable. submarine, collided with a foreign merchant vessel MV Captains of erring ships are dismissed from their Leeds Castlewhile trying to surface in waters north command following an enquiry. The accident on board INS of Mumbai. The submarine was taking part in fleet-level Sindhuratna (S59) led to the resignation of the then Chief of Naval Staff (CNS) Admiral D K Joshi on 26 February war games, when the accident occurred. The Navy 2014, who owned moral responsibility. In this paper we termed it a minor incident with no casualties reported. have taken Blasts, explosion and Fire in the ship or Human  August 2009: A collision of the missile negligence. When the main unit fails due to Human negligence then cold standby system becomes operative. corvette INS Kuthar (P46) with destroyer INS Ranvir Human negligence cannot occur simultaneously in both the (D54) in the was traced to a rudder units and after failure the unit undergoes very costly failure, compounded by a flawed maneuver. repair facility immediately. Applying the regenerative 2010 – present In 2010, three crew men on point technique with renewal process theory the various reliability parameters MTSF, Availability, Busy period, destroyer INS Mumbai (D62) were instantly killed when Benefit-Function analysis have been evaluated. an AK-630 Close-in weapon systemwent off as safety drills were not followed. January 2011: INS Vindhyagiri Keywords: Cold Standby, Blasts, explosion and Fire in the (F42), a Nilgiri-class frigate, capsized after a collision ship or Human negligence, first come first serve, MTSF, with a Cyprus-flagged merchant vessel MV Availability, Busy period, Benefit -Function. Nordlake near the Sunk Rock light house, following INTRODUCTION which a major fire broke out in the ship's engine and boiler room. Everyone on board was evacuated as soon 2000 – 2010 as the fire broke out and hence there were no December 2005: INS Trishul (F43), a Talwar-class casualties. INS Vindhyagiri was later decommissioned. frigate, collided with a commercial vessel, Ambuja  August 2013: Blasts ripped through the torpedo Laxmi, outside the Mumbai harbor, while returning from compartment of the INS Sindhurakshak (S63) while it a training mission. These class of ships use stealth was berthed at the naval dockyard off the Mumbai coast. technologies and a special hull design to ensure a Fifteen Sailors and three officers were killed. Other reduced radar cross section. Radar systems installed by sources state that a small explosion occurred around the port authorities and those on board theAmbuja midnight which then triggered the two larger explosions. The disaster was thought to be the Indian ______ISSN (Print) : 2321-5747, Volume-2, Issue-6,2014 20 International Journal on Mechanical Engineering and Robotics (IJMER) ______navy's worst since the sinking of the frigate INS trials, leading to gas leakage. Since the ship was not Khukri by a Pakistani submarine during the 1971 war. commissioned at the time of the incident, the enquiry into the mishap will be done by Mazagon Dock Limited,  December 2013: INS Konkan (M72), where the ship was constructed. a Pondicherry-class minesweeper under the , caught fire at the naval dockyard  May 2014: INS Ganga (F22) suffered a minor at while undergoing repairs. The fire explosion in the boiler room while undergoing a refit at engulfed much of the ship's interior before it was put off. the Mumbai dockyard. Four people suffered minor No casualties were reported. injuries. There was no fire and no equipment was damaged.  December 2013: In the second incident in the same month, INS Talwar (F40), the lead ship of  November 2014: A Torpedo Recovery Vessel the Talwar class frigate of the Indian Navy, collided of the Astravahini class (A-73) sank 30 NM off with a fishing trawler injuring four of the 27 people on the Vizag coast during a routine mission to recover board the trawler and sinking it. The fishing trawler was torpedoes fired by fleet ships during a routine exercise. operating without lights. The captain of the ship was The accident resulted in the tragic death of one sailor subsequently stripped of command. while four others were reported as missing however 23 other personnel were rescued by SAR teams deployed  December 2013: In the third incident in the right after the incident. same month, INS Tarkash (F50), again a Talwar class frigate, suffered damage to its hull when it hit the jetty Stochastic behavior of systems operating under while docking at the Mumbai naval base. The navy changing environments has widely been studied. . ordered a board of inquiry. Dhillon , B.S. and Natesan, J. (1983) studied an outdoor power systems in fluctuating environment . Kan Cheng  January 2014: INS Betwa (F39), an (1985) has studied reliability analysis of a system in a indigenously built Brahmaputra class guided missile randomly changing environment. Jinhua Cao (1989) has frigate, ran aground and collided with an unidentified studied a man machine system operating under changing object while approaching the Mumbai naval base. The environment subject to a Markov process with two sonar system of the frigate was cracked, leading to states. The change in operating conditions viz. faulty readings and an ingress of saltwater into sensitive fluctuations of voltage, corrosive atmosphere, very low equipment. gravity etc. may make a system completely inoperative.  January 2014: In the second incident in the Severe environmental conditions can make the actual same month, INS Vipul (K46), a veer class corvette of mission duration longer than the ideal mission duration. the elite 22nd Killer Missile Vessel Squadron, was In this paper we have taken Blasts, explosion and Fire detected with a hole in its pillar compartment which in the ship or Human negligence. When the main forced the ship back into the harbor while it was on an operative unit fails then cold standby system becomes operational deployment. operative. Human negligence cannot occur  February 2014: On 3 February, INS Airavat simultaneously in both the units and after failure the unit (L24), a Shardul class amphibious warfare vessel, ran undergoes repair facility of very high cost in case of aground while returning to its home base at Blasts, explosion and Fire in the ship immediately. Visakhapatnam, causing slight damage to its propellers. The repair is done on the basis of first fail first repaired. Following the incident, its commanding officer, Captain Assumptions JPS Virk, was relieved of command pending the . findings of a Board of Inquiry 1. 1 , 2, are failure rates for Blasts, explosion and Fire in the ship , Human negligence respectively.  February 2014: On 26 February, INS The CDF of repair time distribution of Type I and Sindhuratna (S59), a Kilo-class submarine, had a fire Type II are G1(t) and G2(t). detected on board when trials were being conducted which resulted in smoke leading to suffocation and 2. The Human negligence failure is non- death of two officers. Seven sailors were reported instantaneous and it cannot come simultaneously in injured and were airlifted to the naval base hospital in both the units. Mumbai. According to the naval board of inquiry, the 3. The repair starts immediately after the failure due fire was caused due to problems in the cables of the to Blasts, explosion and Fire in the ship or vessel. This particular incident led to the resignation Human negligence works on the principle of first of Chief of Naval Staff (CNS) Admiral D K Joshi on 26 fail first repaired basis. February 2014, who owned moral responsibility for the incidents in the past few months. March 2014: INS 4. The repair facility does no damage to the units and Kolkata, had a malfunction on board which led to a toxic after repair units are as good as new. gas leak killing Commander Kuntal Wadhwa 5. The switches are perfect and instantaneous. instantly. It seems that the ship suffered malfunction in its carbon dioxide unit while undergoing machinery 6. All random variables are mutually independent.

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7. When both the units fail, we give priority to repair of type II and the standby unit becomes operative operative unit for repair. with no Human negligence

8. Repairs are perfect and failure of a unit is detected 3(NFhnf,uR , EEFeef,wr) immediately and perfectly. The first unit fails due to Human negligence and under 9. The system is down when both the units are non- very costly Type-1repair is continued from state 1 and operative. the other unit fails due to EEF resulting from Blasts, explosion and Fire in the ship and is waiting for repair of Notations Type -II. 1 , 2 are the failure rates due to Blasts, explosion and 4(NF , NF ) Fire in the ship, Human negligence failure rate, hnf,uR hnf,wr respectively. G1(t), G2(t) – repair time distribution Type The repair of the unit is failed due to NF resulting from -I, Type-II due to Blasts, explosion and Fire in the Human negligence is continued from state 1and the ship or Human negligence failure rate, respectively. other unit failed due to NF resulting from Human negligence is waiting for repair of Type-I. p, q - probability of Blasts, explosion and Fire in the ship or Human negligence failure respectively such 5(EEFeef, uR , EEF eef, wr) that p+ q=1 The operating unit fails due to Blasts, explosion and Fire

Mi(t) System having started from state I is up at time t in the ship (EEF mode) and under repair of Type - II without visiting any other regenerative state continues from the state 2 and the other unit fails also due to Blasts, explosion and Fire in the ship is waiting Ai (t) state is up state as instant t for repair of Type- II. R (t) System having started from state I is busy for i 6(EEF , NF ) repair at time t without visiting any other regenerative eef,uR nhnf,wr state. The operative unit fails due to EEF resulting from Blasts, explosion and Fire in the ship and under repair Bi (t) the server is busy for repair at time t. continues from state 2 of Type –II and the other unit is

Hi(t) Expected number of visits by the server for failed due to NF resulting from Human negligence and repairing given that the system initially starts from under very costly Type-1 regenerative state i Symbols for states of the System 7(O nhnf , NF hnf,ur) Superscripts O, CS, EEF, HNF Operative, Cold The repair of the unit failed due to operative unit fails Standby, Blasts, explosion and Fire in the ship or due to NF resulting from Human negligence is Human negligence respectively completed and there is no Human negligence and the Subscripts nhnf, hnf, eef, ur, wr, uR other unit is failed due to NF resulting from Human negligence is under repair of very costly Type-1 No Human negligence Failure, Human negligence Failure, Blasts, explosion and Fire in the ship failure, 8(O nhnf , EEFeef,ur) under repair, waiting for repair, under repair continued The repair of the unit failed due to NF resulting from from previous state respectively Human negligence is completed and there is no Human Up states – 0, 1, 2, 7, 8 ; negligence and the other unit is failed due to EEF resulting from Blasts, explosion and Fire in the ship is Down states – 3, 4, 5, 6 under repair of Type-II. regeneration point – 0,1,2, 7, 8 Transition Probabilities States of the System Simple probabilistic considerations yield the following

0(Onhnf, CSnhnf) expressions:

One unit is operative and the other unit is cold standby p01 = p, p02 = q, and there is no Human negligence in both the units. * * * p10 = pG1 ( 1)+q G1 ( 2)= p70 , p20 = pG2 ( 1)+q * 1(NF hnf, ur , Onhnf) G2 (2)= p80 , (3) * (4) (5) * The operating unit fails due to Human negligence and p11 = p(1- G1 ( 1))= p14 = p71 ,p28 = q(1- G2 ( 2))= (5) is under repair immediately of very costly Type- I and p25 = p82 (1) standby unit starts operating with no Human negligence We can easily verify that 2(EEFeef, ur , Onhnf) p + p = 1, The operative unit fails due to EEF resulting from 01 02 (4) (3) Blasts, explosion and Fire in the ship and undergoes p10 + p17 (= p14) + p18 (=p13 ) = 1, ______ISSN (Print) : 2321-5747, Volume-2, Issue-6,2014 22 International Journal on Mechanical Engineering and Robotics (IJMER) ______

(5) (6) p80 + p82 + p87 = 1 (2) Taking Laplace Transform of eq. (7-11) and solving for And mean sojourn time is

= N2(s) / D2(s) (12) µ0 = E(T) = where Mean Time To System Failure (3) (6) (5) (6) (3) N2(s) = 0 (1 - 78 - 87 )- 82 ( 27 78 + Ø0(t) = Q01(t)[s] Ø1(t) + Q02(t)[s] Ø2(t) (5) (4) 28 - 71 Ø1(t) = Q10 (t)[s] Ø0(t) + Q13(t) + Q14(t) (4) (6) (3) (4) (5) (4) (6) Ø2(t) = Q20 (t)[s] Ø0(t) + Q25(t) + Q26(t) (3-5) ( 17 + 87 18 )+ 71 82 ( 17 - 27 (3) We can regard the failed state as absorbing 18 )]+ 01 [ 1(1 – (3) (6) (4) (3) (3) Taking Laplace-Stiljes transform of eq. (3-5) and 78 87 ) + 71 ( 7 + 78 8)+ 18 ( (6) solving for 7 87 - 8)- * ø0 (s) = N1(s) / D1(s) (6) (5) (6) (3) (5) (4) 82 ( 1( 27 78 + 28 )+ 17 (- 2( where (3) (5) 78 + 7 28 )- * * * * * N1(s) = Q01 [ Q13 (s) + Q14 (s) ] + Q02 [ Q25 (s) + Q26 (3) (6) (3) * (s) ] 18 ( 2+ 7 27 )}] 02[ 2(1 – 78 (6) (6) * * * * 87 ) + 27 ( D1(s) = 1 - Q01 Q10 - Q02 Q20 (3) (5) (6) (4) Making use of relations (1) & (2) it can be shown that 7 + 78 8)+ 28 ( 7 87 + 8) - 71 ( * (6) (5) ø0 (0) =1 , which implies that ø0 (t) is a proper 1(- 27 - 28 + distribution. (6) (4) (5) (3) (6) 87 )+ 17 ( 2+ 28 8)- 18 (- 2 87 + (6) 8 27 )}] (s) MTSF = E[T] = (3) (6) 18 ( 2+ 7 27 )}] s=0 (3) (6) (5) (6) (3) ’ ’ D2(s) = (1 - 78 - 87 ) - 82 ( 27 78 + = (D1 (0) - N1 (0)) / D1 (0) (5) (4) 28 )- 71

= ( +p01 + p02 ) / (1 - p01 p10 - p02 p20 ) (4) (6) (3) (4) (5) (4) (5) ( 17 + 87 18 )+ 71 82 ( 17 28 - where (3) 18 )]+ 01[- 10 (1 – (4) (3) (6) μ0 = μ01+ μ02 , μ1 = μ01 + μ17 + μ18 , μ2 = μ02+μ27 + (3) (6) (4) (3) (3) (5) 78 87 ) - 71 ( 70+ 78 80)- 18 ( 70 μ28 (6)- )- Availability analysis 87 80 (5)( - ( (6) (3) + (5))+ (4) ( ( Let Mi(t) be the probability of the system having started 82 10 27 78 28 17 20 (3) (5) from state i is up at time t without making any other 78 - 70 28 )+ regenerative state. By probabilistic arguments, we have (3)( + (6))}] [- (1 – (3) (6)) − t − t - (   ) 18 20 70 27 02 20 78 87 M0(t) = e 1 e 2 , M1(t) =p G1(t) e 1+ 2 = M7(t) (6) (3) - 27 ( 70 + 78 80 )- - (   ) M2(t) =q G2(t) e 1+ 2 = M8(t) (5) (6) (4) (6) (5) 28 ( 70 87 + 80 ) - 71 ( 10 ( 27 + 28 The point wise availability Ai(t) have the following (6) (4) recursive relations 87 )- 17 ( 20- (5) (3) (6) (6) A0(t) = M0(t) + q01(t)[c]A1(t) + q02(t)[c]A2(t) 28 80 )- 18 ( 20 87 + 80 27 )}] (3) A1(t) = M1(t) + q10(t)[c]A0(t) + q18 (t)[c]A8(t)+ (Omitting the arguments s for brevity) q (4)(t)[c]A (t) , 17 7 The steady state availability (5) A2(t) = M2(t) + q20(t)[c]A0(t) + [q28 (t)[c] A8(t) + (6) q27 (t)] [c]A7(t) A0 = = = (4) A7(t) = M7(t) + q70(t)[c]A0(t) + [q71 (t)[c] A1(t) + (3) q78 (t)] [c]A8(t) A8(t) = M8(t) + q80(t)[c]A0(t) (5) (6) +[q82 (t)[c] A2(t) + q87 (t)] [c]A7(t) (7-11) Using L’ Hospitals rule, we get

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A0 = = (13) (t) = So that

The expected up time of the system in (0,t] is The expected number of visits by the repairman for repairing the identical units in (0,t] (t) = H0(t) = Q01(t)[s][1+ H1(t)] + Q02(t)[s][1+ H2(t)] (3) (4) So that (14) H 1 (t) = Q 10 (t)[s]H 0 (t)] + Q18 (t)[s] H8(t) + Q17 (t)] [s]H7(t) , The expected down time of the system in (0,t] is (5) (6) H2(t) = Q20(t)[s]H0(t) + Q28 (t) [s] H8(t) +Q27 (t)] (t) = t- (t) [c]H7(t) H (t) = Q (t)[s]H (t) + Q (4)(t) [s] H (t) +Q (3)(t)] So that (15) 7 70 0 71 1 78 [c]H8(t) (5) (6) H8(t) = Q80(t)[s]H0(t) + Q82 (t) [s] H2(t) +Q87 (t)] [c]H (t) (23-27) The expected busy period of the server when there is 7 EEF-failure resulting from Blasts, explosion and Fire Taking Laplace Transform of eq. (23-27) and solving for in the ship or NF- failure due to Human negligence in (0,t]

= N4(s) / D3(s) (28) R0(t) = q01(t)[c]R1(t) + q02(t)[c]R 2(t) ’ (3) In the long run , H0 = N4(0) / D3 (0) (29) R1(t) = S1(t) + q10(t)[c]R0 (t) + q18 (t)[c] R8 (t) + (4) q17 (t)[c]R7(t) Benefit- Function Analysis (5) R2(t) = S2(t) + q20(t)[c]R0(t) + q28 (t) R8(t) The Benefit-Function analysis of the system considering (6) +q27 (t)][c]R7(t) mean up-time, expected busy period of the system under (4) Human negligence Failure or Blasts, explosion and Fire R7(t) = S7(t) + q70(t)[c]R0(t) + Q71 (t) R1(t) (3) in the ship , expected number of visits by the repairman +q78 (t)][c]R8(t) for unit failure. (5) R8(t) = S8(t) + q80(t)[c]R0(t) + Q82 (t) R2(t) (6) The expected total Benefit-Function incurred in (0,t] is +q87 (t)][c]R7(t) (16-20) Taking Laplace Transform of eq. (16-20) and solving for C(t) = Expected total revenue in (0,t] - expected busy period of the system under failure due to Human negligence Failure or failure due to = N3(s) / D2(s) (21) Blasts, explosion and Fire in the ship for repairing the where units in (0,t ] - expected number of visits by the repairman for (3) (6) (4) (3) N 3(s) = 01[ S1(1 – 78 87 ) + 71 ( S7 + 78 repairing of identical the units in (0,t] S )+ (3)( S 8 18 7 The expected total cost per unit time in steady state is (6) (5) (6) (3) (5) (4) 87 - S 8)]- 01 82 ( S 1 27 78 + 28 )+ 17 (3) (5) C = = (S 2 78 + S 7 28 )- = K1A0 - K 2R0 - K 3H0 (3) (6) (3) (6) 18 ( S 2+ S 7 27 )]+ 02[S 2(1 – 78 87 ) + (6) (3) (5) (6) (4) where 27 ( S 7 + 78 S 8)+ 28 ( S 7 87 + S 8) - 02 71 ( (6) (5) (6) (4) (5) (3) K1 - revenue per unit up-time, S 1(- 27 - 28 87 17 (S 2+ 28 S 8)- 18 (- (6) (6) K - cost per unit time for which the system is under S 2 87 + S 8 27 )] 2 repair of type- I or type- II and K3 - cost per visit by the repairman for units repair. D 2(s) is already defined. CONCLUSION (Omitting the arguments s for brevity) After studying the system , we have analyzed

In the long run, R0 = (22) graphically that when the failure rate due to Blasts, explosion and Fire in the ship or failure rate due to The expected period of the system under EEF-failure Human negligence Failure increases, the MTSF and resulting from Blasts, explosion and Fire in the ship or steady state availability decreases and the Benefit- NF- failure due to Human negligence in (0,t] is function decreased as the failure increases.

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REFERENCES environment, Microelectron. Reliab. ,1983; 23, 867-881. [1] Barlow, R.E. and Proschan, F., Mathematical theory of Reliability, 1965; John Wiley, New [4] Kan, Cheng, Reliability analysis of a system in a York. randomly changing environment, Acta Math. Appl. Sin. 1985, 2, pp.219-228. [2] Gnedanke, B.V., Belyayar, Yu.K. and Soloyer , A.D. , Mathematical Methods of Relability [5] Cao, Jinhua, Stochatic Behaviour of a Man Machine Theory, 1969 ; Academic Press, New York. System operating under changing environment subject to a Markov Process with two states, [3] Dhillon, B.S. and Natesen, J, Stochastic Anaysis of Microelectron. Reliab. ,1989; 28, pp. 373-378. outdoor Power Systems in fluctuating

5(EEFeef, uR , 8(O nhnf , 3(NF , EEF ) hnf,uR eef, wr EEFeef,ur EEFeef,wr) )

2(EEFeef, 0(Onhnf, 1(NFhnf, ur , Onhnf) CSnhnf) ur , Onhnf)

6(EEF eef,uR , 4(NF hnf,uR , NF nhnf,wr) 7(O nhnf , NF hnf,ur) NF hnf,wr)

Fig. The State Space Diagram

Up state down state

 Regeration point

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