International Journal of Recent Advances in Engineering & Technology (IJRAET) ______

Cost-Benefit Analysis of Two Similar Warm Standby System subject to Failure due to burst of tyres during takeoff and landing of aircraft

Ashok Kumar Saini BLJS COLLEGE, TOSHAM (BHIWANI) HARYANA, INDIA Email : [email protected]

Airport in Sokoto, but problems were reported shortly Abstract : In this paper we have taken failure due to burst of tyres during takeoff and landing of aircraft. When the after takeoff. The crew attempted to return to the airport main unit fails then warm standby system becomes for an emergency landing, but the aircraft caught fire, operative. Failure due to burst of tyres during landing of experienced an inflight break-up, and crashed 2,875 aircraft cannot occur simultaneously in both the units and metres (9,432 ft) short of runway 34L. When the aircraft after failure the unit undergoes Type-I or Type-II or Type- was about 18 kilometres (11 mi) from the airport and at III repair facility immediately. Applying the regenerative an altitude of 671 metres (2,201 ft), a number of bodies point technique with renewal process theory the various fell from it, indicating that the fire by that time had reliability parameters MTSF, Availability, Busy period, consumed, at least partially, the cabin floor. All 261 Expected number of visits by repairman, Benefit-Function occupants on board—including 247 passengers— analysis have been evaluated. perished in the accident. As of July 2014, the accident Keyword: Failure due to burst of tyres during takeoff and remains the deadliest one involving a Douglas DC-8, as landing of aircraft, MTSF, Availability, Busy Period, well as the second deadliest one taking place on Saudi Expected number of visits by repairman, Profit-Function. Arabian soil, after Saudia Flight 163. INTRODUCTION Cause Flight 2120 was a chartered passenger Prior to departure, the lead mechanic had noticed that flight from Jeddah, Saudi Arabia, to Sokoto, Nigeria on the "#2 and No. 4 tyre pressures were below the 11 July 1991 which crashed shortly after takeoff minimum for flight dispatch" and attempted to inflate from King Abdulaziz International Airport, killing all them, but no nitrogen gas was readily available, and the 247 passengers and 14 crew members on board. The project manager, unwilling to accept a delay, aircraft was a Douglas DC-8 operated by Nationair for disregarded the problem and readied the aircraft for Nigeria Airways. Flight 2120 is the deadliest accident dispatch. As the aircraft was taxiing, the transfer of the involving a DC-8 load from the under-inflated No. 2 tire to the No. 1 tire Aircraft and crew on the same portside axle resulted "in overdeflection, over-heating and structural weakening of the No. 1 The aircraft involved in the accident was a 1968- tyre." "The No. 1 tyre failed very early on the take-off built Douglas DC-8-61, registration C-GMXQ, owned roll", followed almost immediately by the No. 2. The by the Canadian company , and usually operated latter stopped rotating "for reasons not established", and by Nationair; at the time of the accident, it was the subsequent friction of the wheel assembly with the being wet-leased to Nigeria Airways, which in turn sub- runway generated sufficient heat to start a self- leased it to another company to transport sustaining fire. The crew realised there was a problem, Nigerian pilgrims to and from Mecca. William Allan, but not the nature or seriousness of it. The aircraft was the 47-year-old pilot in command, had logged 10,700 not equipped with fire or heat sensors in the wheel flight hours and 1000 hours in type, while Kent assembly. The first officer was recorded remarking, "We Davidge, the 36-year-old first officer, had logged 8,000 gotta flat tire, you figure?" According to Canadian flight hours, of which 550 hours were in type, and Transportation Safety Board members interviewed for Victor Fehr, the 46-year-old flight engineer, had logged an episode ofMayday about the accident, standard 7,500 flight hours, of which 1000 hours were in procedures regarding tire failure during the takeoff roll type. The DC-8 was the primary aircraft type used by on the DC-8 did not then (and still did not as of the the . episode's season 11 airing) include rejecting takeoff for Accident tire or wheel failures, so the captain proceeded with the takeoff. The aircraft departed King Abdulaziz International Airport bound for Sadiq Abubakar III International ______ISSN (Online): 2347 - 2812, Volume-3, Issue -7, 2015 1 International Journal of Recent Advances in Engineering & Technology (IJRAET) ______

When the landing gear was retracted, "burning rubber passengers disembarked via stairs and were bussed to was brought into close proximity with hydraulic and the terminal. There were no injuries, the aircraft electrical system components", causing the failure of received substantial damage. Debris off the tyre and both hydraulic and pressurisation systems that led to wing was recovered from the departure runway. structural damage and loss of control of the Passenger Ana Arias tweeted that they thought a right aircraft. The Transportation Safety Board later hand wheel exploded just as the aircraft was rotating for concluded, "had the crew left the landing gear extended, takeoff, a panel on top of the right hand wing came off. the accident might have been averted". Fuel, "probably The aircraft returned to Madrid for a safe landing but introduced as a result of 'burn through' of the centre fuel veered off the taxiway. tank", intensified the fire, which eventually consumed the cabin floor. People began falling out of the aircraft The airline confirmed a tyre blew on takeoff, the when their seat harnesses burned through. "Despite the hydraulic problem is under investigation. considerable destruction to the airframe, the aircraft On Dec 11th 2013 the CIAIAC reported the aircraft appeared to have been controllable until just before the crash." blew a right aft tyre during takeoff from runway 36L, debris of which impacted the lower side of the wing and Aftermath causes a hydraulic failure preventing the gear from being retracted. The aircraft landed on runway 32L. A memorial to the incident is located at the head office After landing the aircraft vacated the runway via the last of Greater Airports Authority on the grounds rapid exit taxiway to the left and stopped in a grassy area of Toronto Pearson International Airport in Mississauga, about half way between runway 32L and Terminal T4 Ontario. The construction of the memorial, which with all 8 main tyres being flat. The aircraft received includes a plaque and a cherry tree, was paid for by a substantial damage to the right hand wing including the group of Nationair flight attendants shortly after the loss of internal and external components as well as to accident. both engines due to ingestion of stones during the The aircrash, combined with Nationair's poor reputation excursion from the taxiway. for on-time service and mechanical problems, led to On Jan 12th 2015 Spain's CIAIAC reported in an interim serious problems with public image and reliability report, that the investigation is nearly complete, the draft among tour operators. These difficulties were report is being written for comment and authorization. compounded when Nationair locked out its unionised The CIAIAC stated: "During the takeoff run the aft right flight attendants and proceeded to replace them with wheel on the right main landing gear broke and the strike breakers on 19 November 1991. The lock-out debris from the tire detached and struck the lower wing, lasted 15 months and by the time it ended in early 1993, forming a hole and also rupturing important components Nationair found itself in severe financial trouble. At the in the hydraulic system, which in turn detached and time, Nationair owed the Canadian government millions struck the upper surface of the wing, causing one of the of dollars in unpaid landing fees. Creditors began outer panels to detach. The damage to the hydraulic seizing aircraft and demanded cash up front for services. system made it impossible to retract the landing gear. The company was declared bankrupt in May 1993, The crew detected this immediately during the climb owing CDN$75 million. and declared an emergency, returning to the airport as In 1997, Robert Obadia, owner of Nationair and its instructed by ATC and landing on runway 32L at 12:05. parent company Nolisair, pled guilty to eight counts of At the end of the landing run the aircraft left the runway fraud in relation to the company's activities. via the last rapid-exit taxiway on the left side (L3), coming to a stop outside said taxiway in a grassy area Accident: Delta B763 at Madrid on Dec 5th 2013, located halfway between the end of runway 32L and burst tyre on takeoff, hydraulic failure, taxiway terminal T4." excursion after landing The right hand wing seen in flight (Photo: Ana Arias): By Simon Hradecky, created Thursday, Dec 5th 2013 18:32Z, last updated Monday, Jan 12th 2015 13:44Z A Delta Boeing 767-300, registration N182DN performing flight DL-415 from Madrid,SP (Spain) to New York JFK,NY (USA) with 192 passengers and 11 crew, blew the outboard aft right hand main gear tyre on departure from Madrid's runway 36L, the tyre debris penetrated the right hand wing causing damage to electrical (e.g. transponder) and hydraulic systems as well. The crew climbed to safe altitude, performed an alternate gear extension and returned to Madrid for a safe landing on runway 32L about 25 minutes after departure. While taxiing towards the apron, the aircraft veered left off the taxiway and came to a stop. The ______ISSN (Online): 2347 - 2812, Volume-3, Issue -7, 2015 2 International Journal of Recent Advances in Engineering & Technology (IJRAET) ______

The aircraft off the taxiway (Photo: Ana Arias): Assumptions

1. 1, 2 3 are constant failure rates due to warm standby, when failure due to burst of tyres during takeoff of aircraft, failure due to burst of tyres during landing of aircraft respectively. The CDF of repair time distribution of Type I, Type II and multispecialty repairmen Type-III, IV are G1(t), G2(t) and G3(t), G4(t). 2. The failure due to burst of tyres during landing of aircraft is non-instantaneous and it cannot come simultaneously in both the units. 3. The repair starts immediately after failure due to burst of tyres during takeoff of aircraft and failure The "guilty" tyre (Photo: Ana Arias): due to burst of tyres during landing of aircraft 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 non- operative.

Symbols for states of the System Look back on N182DN while on the bus (Photo: Ana Arias): Superscripts O, WS, BTOF, BTLF, Operative, Warm Standby, failure due to burst of tyres during takeoff of aircraft, failure due to burst of tyres during landing of aircraft respectively Subscripts nbtof, btof, btlf, ur, wr, uR No failure due to burst of tyres during takeoff of aircraft, failure due to burst of tyres during takeoff of aircraft, failure due to burst of tyres during landing of aircraft, 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

In this paper we have taken failure due to burst of tyres 0(Onbtof, WSnbtof) One unit is operative and the other during takeoff and landing of aircraft .When the main unit is warm standby and there is no failure due to burst operative unit fails then warm standby system becomes of tyres during takeoff of aircraft of both the units. operative. Failure due to burst of tyres during landing of 1(BTOFbtof, urI , Onbtof) The operating unit failure due to aircraft cannot occur simultaneously in both the units. burst of tyres during takeoff of aircraft is under repair After failure the unit undergoes repair facility of Type- immediately of Type- I and standby unit starts operating I or Type- II by ordinary repairman, Type III or Type IV with no failure due to burst of tyres during takeoff of by multispecialty repairman immediately when failure aircraft due to burst of tyres during takeoff and landing of aircraft. The repair is done on the basis of first fail first 2(BTLFbtlf, urII , Onbtof) The operative unit failure due to repaired. burst of tyres during landing of aircraft and undergoes repair of Type II and the standby unit becomes operative

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* * with no failure due to burst of tyres during takeoff of p10 = pG1 ( 1)+q G2 ( 2) , aircraft * (4) p14 = p- pG1 ( 1) = p11 , 3(BTLFbtlf, urIII , Onbtof) The first unit failure due to burst * (5) of tyres during landing of aircraft and under Type-III p15 = q- q G1 ( 2) = p12 , multispecialty repairman and the other unit is operative * * p23 = pG2 ( 1)+q G2 ( 2) , with no failure due to burst of tyres during takeoff of * (6) aircraft p26 = p- pG2 ( 1) = p29 , * (7) 4(BTOF btof,uR1 , BTOF btof,wrI) The unit failed due to p27 = q- qG2 ( 2) = p28 , BTOF resulting from failure due to burst of tyres during p = p = p = 1 takeoff of aircraft under repair of Type- I continued 30 82 91 , * * from state 1and the other unit failed due to BTOF p0,10 = pG4 ( 1)+q G4 ( 2) , resulting from failure due to burst of tyres during * (11) takeoff of aircraft is waiting for repair of Type-I. p10,1 = p- pG4 ( 1) = p10,1 , p = q- q G *(  ) = p (11) (1) 5(BTOF btof,uR1 , BTLFbtlf, wrII) The unit failed due to 10,2 4 2 10,2 BTOF resulting from failure due to burst of tyres during We can easily verify that takeoff of aircraft is under repair of Type- I continued from state 1and the other unit fails due to burst of tyres p01 + p02 + p03 = 1, during landing of aircraft is waiting for repair of Type- (4) (5) p10 + p14 (=p11 ) + p15 (=p12 ) = 1, II. (6) (7) p23 + p26 (=p29 ) + p27 (=p28 ) = 1 6(BTLFbtlf, uRII , BTOF btof ,wrI) The operative unit failed due to burst of tyres during landing of aircraft is under p30 = p82 = p91 = 1 repair continues from state 2 of Type –II and the other (11) (12) p10,0+p10,1 (=p10,1)+p10,2 (=p10,2 )= 1 (2) unit failed due to BTOF resulting from failure due to burst of tyres during takeoff of aircraft is waiting under And mean sojourn time is repair of Type-I. µ0 = E(T) = 7(BTLFbtlf ,uRII , BTOFbtof,wrII) The one unit failed due to burst of tyres during landing of aircraft is continued to 3. Mean Time To System Failure be under repair of Type II and the other unit failed due Ø (t) = Q (t)[s] Ø (t) + Q (t)[s] to BTOF resulting from failure due to burst of tyres 0 01 1 02 during takeoff of aircraft is waiting for repair of Type-II. Ø2(t)+ Q0,10(t)[s] Ø10(t)

8(BTOFbtof,urIII , BTLFbtlf, wrII) The one unit failure due Ø1(t) = Q10 (t)[s] Ø0(t) + Q14(t) +Q15(t) to burst of tyres during takeoff of aircraft is under multispecialty repair of Type-III and the other unit failed Ø2(t) = Q23 (t)[s] Ø3(t) + Q26(t) + Q27(t) due to burst of tyres during landing of aircraft is waiting Ø3(t) = Q30(t)[s] Ø0(t) for repair of Type-II. Ø10(t) = Q10,0(t)[s] Ø10(t) + Q10,1(t)[s]Ø1(t)+ Q10,2(t)[s] 9(BTOFbtof,urIII, BTLFbtlf, wrI) The one unit failure due Ø2(t) (3-6) to burst of tyres during takeoff of aircraft is under multispecialty repair of Type-III and the other unit We can regard the failed state as absorbing failed due to burst of tyres during landing of aircraft is Taking Laplace-Stiljes transform of eq. (3-6) and waiting for repair of Type-I solving for * 10(Onbtof , BTLFbtlf, urIV ) The one unit is operative with ø0 (s) = N1(s) / D1(s) (7) no failure due to burst of tyres during takeoff of aircraft and warm standby unit fails due to burst of tyres during where * * * * * landing of aircraft and undergoes repair of type IV. N1(s) = {Q01 + Q0,10 Q10,1 } [ Q14 (s) + Q15 (s) ] + * * * * * {Q02 + Q0,10 Q10,2 } [ Q26 (s) + Q27 (s) ] 11(Onbtof , BTLFbtlf, uRIV ) The one unit is operative with * * * * * * no failure due to burst of tyres during takeoff of aircraft D1(s) = 1 - {Q01 + Q0,10 Q10,1 } Q10 - {Q02 + Q0,10 * * * * * and warm standby unit fails due to burst of tyres during Q10,2 } Q23 Q30 - Q0,10 Q10,0 landing of aircraft and repair of type IV continues from Making use of relations (1) & (2) it can be shown that state 10. * ø0 (0) =1 , which implies that ø0 (t) is a proper Transition Probabilities distribution. Simple probabilistic considerations yield the following expressions: MTSF = E[T] = (s) s=0 p01 = 1 / 1 + 2 +3, p02 = 2 / 1 + 2 +3 , p0,10 = 3 / ’ ’ 1 + 2 +3 = (D1 (0) - N1 (0)) / D1 (0) ______ISSN (Online): 2347 - 2812, Volume-3, Issue -7, 2015 4 International Journal of Recent Advances in Engineering & Technology (IJRAET) ______

(4) (7 = ( + ( p01 + p0,10 p10,1) +( p02 + p0,10 p10,2)( D2(s) = {1 - 11 }{1- 28 82 }-

+ µ3)+ µ10 p0,10 / (1 - (p01 + p0,10 p10,1) p10 - (p02 + (5) (6) 12 29 91 -{ 01+ 0,10 p0,10 p10,2) p23 ) - p0,10 p10,0 (11) }[ {1 – (7) } + where 10,1 10 28 82 (5) 휇0 = 휇01+ 휇02 +µ0,10 , 12 23 30 ] – { 02 + 0,10 (4) (5) 휇1 = 휇10 + 휇11 + 휇12 , (11) (4) 10,2 }{[ 23 30 {1 – 11 }+ (7) (6) 휇2 = 휇23+휇28 + 휇29 , (6) 29 91 10] µ10= µ10,0 + µ10,1+ µ10,2 (Omitting the arguments s for brevity) 4. Availability analysis The steady state availability Let Mi(t) be the probability of the system having started from state i is up at time t without making any other A0 = = = regenerative state. By probabilistic arguments, we have

− t − t − t M0(t) = 푒 1 푒 2 푒 3 , -  t Using L’ Hospitals rule, we get M1(t) =p G1(t) e 1 -  t M2(t) =q G2(t) e 2 , A0 = = (17)

M3(t) = G3(t), The expected up time of the system in (0,t] is (t) = M (t) = G (t) e -  t 10 4 3 The point wise availability Ai(t) have the following recursive relations So that (18)

A0(t) = M0(t) + q01(t)[c]A1(t) + The expected down time of the system in (0,t] is (t) q02(t)[c]A2(t) + q0,10(t)[c]A10(t) = t- (t) A1(t) = M1(t) + q10(t)[c]A0(t) + (5) (4) q12 (t)[c]A2(t)+ q11 (t)[c]A1(t) , So that (19)

A2(t) = M2(t) + q23(t)[c]A3(t) + Similarly, we can find out (7) (6) q28 (t)[c] A8(t) + q29 (t)] [c]A9(t) 1.The expected busy period of the server when there is

A3(t) = M3(t) + q30(t)[c]A0(t) , failure due to burst of tyres during takeoff of aircraft, and burst of tyres during landing of aircraft in (0,t]-R0 A8(t) = q82(t)[c]A2(t) 2.The expected number of visits by the repairman Type- A9(t) = q91(t)[c]A1(t), I or Type-II for repairing the identical units in (0,t]-H0 A10(t) = M10(t) + q10,0(t)[c]A0(t) + 3.The expected number of visits by the multispecialty (11) (11) repairman Type-III or Type-IV for repairing the q10,1 (t)[c]A1(t)+ q 10,2 (t)[c]A2(t) (8-15) identical units in (0,t]-W0, Y0. Taking Laplace Transform of eq. (8-15) and solving for Benefit-Function

The Benefit-Function analysis of the system considering = N2(s) / D2(s) (16) mean up-time, expected busy period of the system under where failure due to burst of tyres during takeoff of aircraft, and burst of tyres during landing of aircraft, expected

N2(s) ={ 0,10 10+ 0 } [{1 – number of visits by the repairman for unit failure. The expected total Benefit-Function incurred in (0,t] is (4) (7 (5) (6) 11 }{1- 28 82 }- 12 29 C = = = (11) (7) 91 ] + { 01+ 0,10 10,1 }[ 1{1 – 28 82} K1A0 - K 2R0 - K 3H0 - K 4W0 - K 5 Y0 (5) (11) + 12 23 3+ 2]+{ 02 + 0,10 10,2 } [{ 23 where 3}{1 – K1 - revenue per unit up-time, K2 - cost per unit time for (4) (6) which the system is busy under repairing, K - 11 }+ 29 91 1] 3

______ISSN (Online): 2347 - 2812, Volume-3, Issue -7, 2015 5 International Journal of Recent Advances in Engineering & Technology (IJRAET) ______cost per visit by the repairman type- I or type- II for environment, Microelectron. Reliab. ,1983; 23, units repair, 867-881.

K4 - cost per visit by the multispecialty repairman [2] Kan, Cheng, Reliability analysis of a system in a Type- III for units repair, randomly changing environment, Acta Math. Appl. Sin. 1985, 2, pp.219-228. K5 - cost per visit by the multispecialty repairman Type- IV for units repair [3] Cao, Jinhua, Stochatic Behaviour of a Man Machine System operating under changing CONCLUSION environment subject to a Markov Process with After studying the system, we have analyzed graphically two states, Microelectron. Reliab. ,1989; 28, pp. that when the failure rate due to burst of tyres during 373-378. takeoff of aircraft and due to burst of tyres during [4] Barlow, R.E. and Proschan, F., Mathematical landing of aircraft increases, the MTSF, steady state theory of Reliability, 1965; John Wiley, New availability decreases and the Profit-function decreased York. as the failure increases. [5] Gnedanke, B.V., Belyayar, Yu.K. and Soloyer , REFERENCES A.D. , Mathematical Methods of Relability Theory, 1969 ; Academic Press, New York. [1] Dhillon, B.S. and Natesen, J, Stochastic Analysis of outdoor Power Systems in fluctuating

Fig. The State Transition Diagram Up-State Down-State

regeneration point 

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