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J.Soszyska, P.Dziula, M.Jurdziski, K.KoowrockiOn multi-state safety analysis in shipping - RTA # 3-4, 2007, December - Special IssueπbMbpb= lim p b( t)= ,vt→∞∑πMl=1llb = 1,2,...,v,(13)where the probabilities πbof the vector [ πb ] satisfy1xνthe system of equations⎧[π b ] = [ π⎪⎨ v⎪∑ π l = 1.⎩l=1b][ pbl]We assume that the system is composed of ncomponents E , i = 1,2,...,n,the changes of theiprocess Z(t) operation states have an influence on thesystem components E safety and on the system safetyistructure as well. Thus, we denote the conditionalsafety function of the system component E while thesystem is at the operational state zb, b = 1,2,...,v,bys b( )( )( t,⋅)i = [1, s b( )( t,1),s b( )( t,2),i i ..., s b( t,z)i ],where( b)i( b)is ( t,u)= P(T ( u)> t Z(t)= zfor t ∈< 0,∞),b = 1,2,...,v,u = 1,2,...,z,and theconditional safety function of the system while thesystem is at the operational state zb, b = 1,2,...,v,by( b)n b( b)n b( b)n bs ( t,⋅)= [1, s ( t,1),s ( t,2),..., s ( t,z)],n b∈{ 1,2,..., n},b)( b)n bwhere nbare numbers of components in the operationstates z and( b)n bb( b)s ( t,u),= P ( T ( u)> t Z(t)= z )for t ∈< 0,∞),n b∈ { 1,2,..., n},b = 1,2,...,ν ,u = 1,2,...,z.The safety function( )s b( t,u)is the conditionali( )probability that the component E lifetimeiT bi( u)inthe state subset { u , u + 1,..., z}is not less than t, whilethe process Z(t) is at the operation state zb. Similarly,( b)n bthe safety function s ( t,u)is the conditional( )probability that the system lifetime T b ( u)in the statebisubset { u , u + 1,..., z}is not less than t, while theprocess Z(t) is at the operation state z b.In the case when the system operation time is largeenough, the unconditional safety function of the systemis given bys ( t,⋅)= [1, s (t,1),s (t, 2),..., s ( t,z)], t ≥ 0,nwherens ( t,u)= P ( T(u)> t)≅ ∑ pbs( t,u)nnνb=1( b)nbn(14)for t ≥ 0,n b∈ { 1,2,..., n},u = 1,2,...,z,and T (u)is theunconditional lifetime of the system in the safety statesubset { u , u + 1,..., z}.The mean values and variances of the system lifetimesin the safety state subset { u , u + 1,..., z}areν( b)m ( u)= E[T(u)]≅ ∑ p m ( u),u = 1,2,...,z,(15)where [2]m( b)b=1b( u)= ∞ ∫ s ( t,u)dt,n b∈ { 1,2,..., n},(16)0u = 1,2,...,z,and( b)( b)n b2= ∞ 0( b)[ σ ( u)]2∫ts( t,u)dt −[m ( u)], (17)u = 1,2,...,z,nbfor b =1,2,...,ν , and2= ∞ 0[ σ ( u)]2∫ts( t,u)dt −[m(u)], u = 1,2,...,z.(16)nThe mean values of the system lifetimes in theparticular safety states u , are [2]m ( u)= m(u)− m(u + 1), u = 1,2,...,z −1,m ( z)= m(z).(19)5. Ship safety Model in constant operationconditionsWe preliminarily assume that the ship is composed of anumber of main technical subsystems having anessential influence on its safety. There aredistinguished her following technical subsystems:22- 43 -