# Introduction

any maritime disasters happen outside the realms of war. All ships, including those of the military, are vulnerable to problems from weather conditions, faulty design or collision with oil tanker. Some of the disasters occurred in periods of conflict, although their losses were unrelated to any military action.


# Year

Country Description Lives lost 1912 United Kingdom RMS Titanic -A passenger ocean liner and, at the time, the world's largest ship. On 14 April 1912, on her maiden voyage, she struck an iceberg, buckling part of her hull and causing her to sink in the early hours of 15 April. 706 of her 2,223 passengers and crew survived. Her loss was the catalyst for major reforms in shipping safety and is arguably the most famous maritime disaster, being the subject of countless media portrayals.'


# 1,517 1987 Philippines

Doña Paz -On 20 December 1987, the ferry bound for Manila with more than its capacity of unlisted passengers collided with the oil tanker MT Vector in the Tablas Strait, near Marinduque. The resulting fire and sinking left an estimated 4,386 dead which included all but 24 of Doña Paz's passengers, and all but two of Vector's 13-man crew.


# 4,386

In this paper we have taken failure due to struck with iceberg and collision with oil tanker. When the main operative unit fails then warm standby system becomes operative. Failure due to collision with oil tanker cannot occur simultaneously in both the units. After failure the unit undergoes repair facility of Type-I or Type-II by ordinary repairman and Type-III or Type IV by multispecialty repairman immediately when failure due

? The failure due to collision with oil tanker is noninstantaneous and it cannot come simultaneously in both the units.

? The repair starts immediately after failure due to struck with iceberg and failure due to collision with oil tanker 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.

? Repairs are perfect and failure of a unit is detected immediately and perfectly.

? The system is down when both the units are nonoperative.


# III.

Symbols for States of the System a) Superscripts O, CS, SIF, COTF, Operative, Warm Standby, failure due to struck with iceberg failure due to collision with oil tanker respectively.


# Subscripts nsif, sif, cotf, ur, wr, uR

No failure due to struck with iceberg, failure due to struck with iceberg, failure due to collision with oil tanker, 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 b) States of the System 0(O nsif , CS nsif ) One unit is operative and the other unit is warm standby and there is no failure due to struck with iceberg of both the units.


# 1(SIF sif, urI , O nsif )

The operating unit failure due to struck with iceberg is under repair immediately of Type-I and standby unit starts operating with no failure due to struck with iceberg 2(COTF cotf, urII , O nsif ) The operative unit failure due to collision with oil tanker and undergoes repair of Type II and the standby unit becomes operative with no failure due to struck with iceberg 3(COTF cotf, urIII , O nsif ) The first unit failure due to collision with oil tanker and under Type-III multispecialty repairman and the other unit is operative with no failure due to struck with iceberg 4(SIF sif,uR1 , SIF sif,wrI ) The unit failed due to SIF resulting from failure due to struck with iceberg under repair of Type-I continued from state 1and the other unit failed due to SIF resulting from failure due to struck with iceberg is waiting for repair of Type-I. 5(SIF sif,uR1 , COTF cotf, wrII ) The unit failed due to SIF resulting from failure due to struck with iceberg is under repair of Type-I continued from state 1and the other unit fails due to collision with oil tanker is waiting for repair of Type-II. 6(COTF cotf, uRII , SIF sif ,wrI ) The operative unit failed due to collision with oil tanker is under repair continues from state 2 of Type -II and the other unit failed due to SIF resulting from failure due to struck with iceberg is waiting under repair of Type-I. 7(COTF cotf ,uRII , SIF sif,wrII ) The one unit failed due to collision with oil tanker is continued to be under repair of Type II and the other unit failed due to SIF resulting from failure due to struck with iceberg is waiting for repair of Type-II. 8(SIF sif,urIII , COTF cotf, wrII ) The one unit failure due to struck with iceberg is under multispecialty repair of Type-III and the other unit failed due to collision with oil tanker is waiting for repair of Type-II. 9(SIF sif,urIII , COTF cotf, wrI ) The one unit failure due to struck with iceberg is under multispecialty repair of Type-III and the other unit failed due to collision with oil tanker is waiting for repair of Type-I 10(O nsif COTF cotf, urIV ) The one unit is operative with no failure due to struck with iceberg and warm standby unit fails due to collision with oil tanker and undergoes repair of type IV.


# 11(O nsif COTF cotf, uRIV )

The one unit is operative with no failure due to struck with iceberg and warm standby unit fails due to collision with oil tanker and repair of type IV continues from state 10.


# IV.


# Transition Probabilities

Simple probabilistic considerations yield the following expressions: 4) , p 15 = q-q G 1 * ( ? 2 ) = p 12 (5) ,  (11) ,p 10,2 = q-q G 4 * ( ? 2 ) = p 10,2
p 01 = ? 1 / ? 1 + ? 2 +? 3 , p 02 = ? 2 / ? 1 + ? 2 +? 3 , p 0,10 = ? 3 / ? 1 + ? 2 +? 3 p 10 = pG 1 * ( ? 1 )+q G 2 * ( ? 2 ) , p 14 = p-pG 1 * ( ? 1 ) = p 11(p
(

We can easily verify that p 01 + p 02 + p 03 = 1, p 10 + p 14 (=p 11 (4) ) + p 15 (=p 12 (5) ) = 1, p 23 + p 26 (=p 29 (6) ) + p 27 (=p 28 (7) ) = 1 p 30 = p 82 = p 91 = 1 p 10,0 + p 10,1 (11) (=p 10,1 ) + p 10,2 (12) (=p 10,2 ) = 1

And mean sojourn time is µ 0 = E(T) =

V.

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 10,1 (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 transiform of eq. (3-6) and solving for
ø 0 * (s) = N 1 (s) / D 1 (s)(7)
where 


# Conclusion

After studying the system, we have analyzed graphically that when the failure rate due to struck with iceberg and due to collision with oil tanker increases, the MTSF, steady state availability decreases and the Profitfunction decreased as the failure increases.  
21![+q G 2 * ( ? 2 ) , p 26 = p-pG 2 * ( ? 1 ) = p 29 (6) ,p 27 = q-qG 2 * ( ? 2 ) = p 28 (7) , p 30 = p 82 = p 91 = 1 , p 0,10 = pG 4 * ( ? 1 )+q G 4 J e XV Issue II Version I p 10,1 = p-pG 4 * ( ? 1 ) = p 10,1](image-2.png "23 = pG 2 *( ? 1 )")
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 VIII.](image-3.png "K 1 -")
![Fig. The State Transition Diagram](image-4.png "")
			© 2015 Global Journals Inc. (US) Global Journal of Researches in Engineering ( ) Volume XV Issue
		
		
			

## MTSF = E[T] =

where ?? 0 = ?? 01 + ?? 02 +µ 0,10 , ?? 1 = ?? 10 + ?? 11 (4) + ?? 12 (5) , ?? 2 = ?? 23 +?? 28 (7) + ?? 29 (6) , µ 10 = µ 10,0 + µ 10,1 + µ 10,2

VI.


## 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

Profit-Function of Two Similar Warm Standby Navy Ship System Subject to Failure Due to Struck with The point wise availability A i (t) have the following recursive relations

A 10 (t) = M 10 (t) + q 10,0 (t)[c]A 0 (t) + q 10,1 (Omitting the arguments s for brevity)

The steady state availability

Using L' Hospitals rule, we get

The expected up time of the system in (0,t] is

The expected down time of the system in (0,t] is

Similarly, we can find out

? The expected busy period of the server when there is failure due to struck with iceberg and collision with oil tanker in (0,t]-R 0 .

? The expected number of visits by the repairman Type-I or Type-II for repairing the identical units in (0,t]-H 0 .

? The expected number of visits by the multispecialty repairman Type-III or Type-IV for repairing the identical units in (0,t]-W 0 , Y 0 .

VII.


## Benefit-Function

The Benefit-Function analysis of the system considering mean up-time, expected busy period of the system under failure due to struck with iceberg and collision with oil tanker, expected number of visits by the repairman for unit failure. The expected total Benefit-Function incurred in (0,t] is
			
			

				


* 
	
		Stochastic Anaysis of outdoor Power Systems in fluctuating environment
		
			BSDhillon
		
		
			JNatesen
		
	
	
		Microelectron. Reliab
		
			23
			
			1983
		
	




* 
	
		Reliability analysis of a system in a randomly changing environment
		
			ChengKan
		
	
	
		Acta Math. Appl. Sin
		
			2
			
			1985
		
	




* 
	
		Stochatic Behaviour of a Man Machine System operating under changing environment subject a Markov Process with two states
		
			JinhuaCao
		
	
	
		Microelectron. Reliab
		
			28
			
			1989
		
	




* 
	
		Mathematical theory of Reliability
		
			REBarlow
		
		
			FProschan
		
		
			1965
			John Wiley
			New York
		
	




* 
	
		
		
			BVGnedanke
		
		
			YuKBelyayar
		
		
			ADSoloyer
		
	
	
		Mathematical Methods of Relability Theory
		
			1969
			Academic Press