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J. Soszyska Systems reliability analysis in variable operation conditions - RTA # 3-4, 2007, December - Special Issue( b)( b)( b)= lim k l F ( a ( u)t + b ( u),u)= 0n→∞n nu = 1,2,...,z,b = 1,2,...,v,andnnfor t < 0,( )V b ( b)( b)( b)( t,u)= lim k l F ( a ( u)t + b ( u),u)n→∞n n= lim k l ( 1−exp[ − ])n→∞n nntk ln n= lim k l ( 1−1+− o())n→∞n ntk ln nntk ln n= t for t ≥ 0,u = 1,2,...,z,b = 1,2,...,v,which means that according Lemma 3 the limitreliability function of that system is given by (39)-(40).The next auxiliary theorem is proved in [7].Lemma 4. If(i) lim k = k,k > 0, 0 < m ≤ k,lim l = ∞ ,n→∞nvnn→∞(ii) R ( t,u)= ∑ p R ( t,u)is a non-degenerateb=1b( b)reliability function,( m)( m)( m)(iii) ( t,⋅)[1, R ( t,1),...,R ( t,)],RRk n , l=,,znkn lnkn lnt ∈ ( −∞,∞),where( m)knlnv,( t)≅ ∑ pb[R ( t)]b=1( m)knln( b)is the reliability function of a homogeneous regularmulti-state series- “m out of k ” system, where[ R=( m)knln,( t,u)](kn)−11 −m ∑ ii=0( b)[ R( b)( t,u)]lnin[1 − [ R( b)t∈(-∞,∞), u = 1,2,...,z,b = 1,2,...,v,( t,u)]ln]kn −iis its reliability function at the operational statethenR ( t,⋅)= [1, R ( t,1),...,R ( t,z)],t ∈ ( −∞,∞),zb,lim[ Rn→∞( b)( a( b)n( u)t + b( b)n( u),u)]ln= R( b)0for t ∈ C , u = 1,2,...,z,b =1,2,...,v,( b)R ( u)0( b)( t,u)(42)where R0( t,u), u = 1,2,...,z,is a non-degeneratereliability function andR ( t,u)v m−1⎛k ⎞ ( b)i ( b)k −i= 1 − ∑ pb∑ ⎜ ⎟[R0( t,u)][1 − R0( t,u)](43)b=1 i=0⎝i ⎠for t ∈(-∞,∞), u = 1,2,...,z.Proposition 4. If components of the multi-statehomogeneous, regular series- “m out of kn” system atthe operational state z b(i) have exponential reliability functions,( )R b ( t,u)= 1for t < 0,( b)( b)R ( t,u)= exp[ −λ ( u)t]for t ≥ 0,(44)u = 1,2,...,z,b =1,2,...,v,(ii) k n → k, k > 0 , l n → ∞, m = const ,( b)1 ( b)(iii) an( u)= , b ( u)= 0( )n,bλ ( u)lnu = 1,2,...,z,b = 1,2,...,v,then( m)( m)( m)R9( t,⋅ ) = [1, R9( t,1),...,R9( t,z)],(45)t ∈(-∞,∞),where( m)R9( t,u)⎧1,⎪v m−1⎪ ⎛k⎞1 − ∑ pb∑ ⎜ ⎟ [exp[ −t]]≅ ⎨ b=1 i=0⎝i ⎠⎪⎪⎪k −i⎩⋅[1 − exp[ −t]],it < 0,t ≥ 0,(46)is the multi-state limit reliability function of thatsystem , i.e. for n large enough we haveR( m)9( t,u)is the multi-state limit reliability function of thatsystem if and only if [7]- 195 -