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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 Issuem−1( m)nR ( , u)= 1 − ∑ ( )nn−it [ R(t,u)][ F(t,u)],ii=0t ∈(-∞,∞), u = 1,2,...,z,or by( m)( m)( m)R ( t,⋅)= [1, R ( t,1),..., R ( t,z)],nwhereR( m )n( t,u)=nmn∑ ( )ii=0m = n − m, u = 1,2,...,z.i[ F(t,u)][ R(t,u)]nin−i, t ∈(-∞,∞),Definition 2.3. A multi-state system is called an „m outof n” system with a hot reserve of its components if itslifetime T (1) (u) in the state subset {u,u+1,...,z} is givenbyT (1) (u) = T ( ), m = 1,2,...,n, u = 1,2,...,z,( n −m+1) uwhere T( n − m+1)( u)is the m-th maximal order statistics inthe sequence of the component lifetimesT i (u) = max{ ( u)},i = 1,2,..,n, u = 1,2,...,z,T ij1≤ j≤2where T i1 (u) are lifetimes of components in the basicsystem and T i2 (u) are lifetimes of reserve components.The reliability function of the homogeneous multi-state„m out of n” system with a hot reserve of itscomponents is given either byIRwhere( 1) ( m)n(1) ( m)( m)( m)(1)(1)(t , ⋅ ) = [1, IR n (1,z),..., IR n ( t,z)],m−1n( )2 i2( n−i)IR n ( t,u)= 1−∑ [1 − ( F(t,u))] [ F(t,u)], (1)ii=0t ∈(-∞,∞), u = 1,2,...,z,or by(1) ( m )(1) ( m )( m )(1)I R n ( t,⋅)= [1, I R n ( t,1),..., I R n ( t,z)],whereIR= ∑(1) ( m )nmii=0( t,u)n2i2 ( n−i)( )[(( t,u))][1 − ( F(t,u))] ,F (2)m = n − m, t ∈(-∞,∞), u = 1,2,...,z.Lemma 2.1.case 1: If(1)(i) ( m)ΙR ( t,u)= − ∑ − 1[( , )]1 m iV t uexp[ −V( t,u)],i=0 i!u = 1,2,...,z, is non-degenerate reliabilityfunction,(1) ( m)(ii) IR n ( t,u)is the reliability function of nondegeneratemulti-state „m out of n” systemwith a hot reserve of its components defined by(16),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m = constant ( m / n → 0,as n → ∞ ),thenlim IR n ( a ( u ) t + b ( u))= ΙR ( t,u),n→∞t ∈CΙR(1) ( m)if and only ifn, u = 1,2,...,z,limn→∞n[1-F2 ( a ( u)t bn( u))u = 1,2,...z,case 2: Ifn(1) ( m)n+ ] = V(t,u), t ∈ C V ,(1) ( µ )−v(t,u)1−(i) ΙR ( t,u)= 1−∫ e2dx ,2π−∞u = 1,2,...,z, is non-degenerate reliabilityfunction,(1) ( m)(ii) IR n ( t,u)is the reliability function ofnon-degenerate multi-state „m out of n”system with a hot reserve of its componentsdefined by (16),(iii) a n (u) > 0, b n (u)∈ (-∞,∞), u = 1,2,...,z,(iv) m / n → µ , 0 < µ < 1, as n → ∞ ,thenn→∞(1) ( m)x2lim IR ( ( ) ((1)n a u t + b u))= ΙR ( t,u),t ∈ C IR , u = 1,2,...,z,if and only ifn( n + 1)[1 − F ( an( u)t + bnlimn→∞m(n − m + 1)n + 1u = 1,2,...,z.2n( µ )( u))]− m=ν ( t,u),case 3: If(1) ( m )mi[ V ( t,u)](i) Ι R ( t,u)= ∑ exp[ −V( t,u)],i=0 i!- 160 -