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B. Kwiatuszewska-Sarnecka On asymptotic approach to reliability improvement of multi-state systemswith components quantitative and qualitative redundancy: „m out of n” systems - RTA # 3-4, 2007, December - Special Issues2[1− F(a( u)t + b= ν ( t,u),u = 1,2,..., z.nn( u))∗ F(anm(n − m + 1)n + 1( u)t + bn( u))]]− mcase 3: If(3) ( m )mi[ V ( t,u)](i) Ι R ( t,u)= ∑ exp[ −V( t,u)],i=0 i!m = n − m, u = 1,2,...,z, is non-degeneratreliability function,( m )(ii)(3)I R n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a mixed reserve of its componentsdefined by (31),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) n - m = m = constant ( m / n →1as n → ∞),thenlim ( ( ) ((3)I R n a u t + b u))= Ι R ( t,u),n→∞t ∈CIR(3) ( m )if and only ifn, u = 1,2,...,z,lim n[s1 [ F(an( u)t + bn( u))]n→∞+ s2[F(a= V ( t,u)t ∈ C ,Vn( u)t + bnu = 1,2,...,z.n2( u))∗ F(an( m )( u)t + bn( u))]]Proposition 2.3. If components of the homogeneousmulti-state „m out of n” system with a mixed reserve ofits components have multi-state exponential reliabilityfunctionsandcase 1 m = constant,a nthen1 exp[ λ(u)bn( u)]( u)= , = n,u = 1,2,...,z,λ(u)2s+ s λ(u)b ( u)1(3) ( m)m−1exp[−it]ΙR ( t,u)= 1- ∑ exp[-exp[-t]],i=0 i!t ∈ (-∞,∞), u = 1,2,...,z,2n,case 2 m / n → µa n( u)=λ(u)0 < µ < 1, n → ∞ ,µ / 2,(2s1 + s2)( n + 1)1 2(1 − µ )b n( u)=, u = 1,2,...,z,λ(u)2s 1+ s 2thenx2−2(3) ( µ )1tΙR ( t,u)= 1 - ∫ e dx ,2π−∞t ∈ (-∞,∞), u = 1,2,...,z,case 3 n - m = m = constant ( m / n → 1,n → ∞ ),2a n ( u)= , b n( u)= 0,u = 1,2,...,z,λ ( u)(2ss ) nthen(3) ( m )1 +Ι R ( t,u)= 1, t < 0,(3) ( m )Ι R ( t,u)= ∑ −i=nm022iis its limit reliability function.Proof:case 1: Since for all fixed u, we haveandti! exp[-t2 ], t ≥ 0 , u = 1,2,...,z,an ( u)t + bn( u)→ ∞ as n → ∞ for t ∈ ( −∞,∞),1 – [ F(a ( u)t b ( u))]n+= 2 exp[ −λ(u)(a ( u)t b ( u))]nn+− exp[ −2λ ( u)(a ( u)t b ( u))], t ∈ ( −∞,∞),n+1 – F( a ( u)t + b ( u))∗ F(a ( u)t b ( u))= [1 + λ(u)(aexp[ −λ(u)(aThereforen nn +nn( u)t + b( u)t + bnn2nn( u))]( u))],t ∈(−∞,∞).n- 167 -