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F.Grabski, A. Zaska-FornalThe model of non-renewal reliability systems with dependent time lengths of componentsRTA # 3-4, 2007, December - Special Issue= G1 ( t)G2( t)G2( t)G3(t)KGn ( t)Gn( t)Gn+1(t)= G t)G ( t)KGn ( t)R()(12)2( 3tThus, for t ≥ 0~R(t)≤ R(t)holds.The inequality means that the reliability of a seriessystem with dependent (in the considering sense) lifelengths of components is greater than (or equal) to thereliability of that system with independent life lengthsof components and the same distributions as themarginals of T 1 , T 2 , …, T n .Accepting the assumption about independence of thelife lengths of the components even though the randomvariables describing the life lengths are dependent, wemake an obvious mistake but that error is „safe”because the real series system has a greater reliability.That estimation is very conservative.Example 1.Assume that a non-negative random variable Ui,describing time to failure of the component caused bythe shock from source z i has a Weibull distributionwith parametersα i, λ i, i = 1,2,..., n + 1for u i > 0−λG ( u ) = P(U > u ) = ei u i, i = 1, 2,..., n + 1.iiiiThe reliability function of a series system withdependent components satisfiesR(t)= P(T > t)= G= eFor n = 3 andG−(λ1ta1+ ... + λn+1t a n+1 )ttαiGG1( )2( ) ...n( )n+1()ttR(t)10.80.60.40.20.5 1 1.5 2 tFigure 1. The graph of the series reliability functionwith dependent components~The reliability function R ( t),t ≥ 0 of the series systemwith independent life lengths of components, the samemarginals satisfies~ −(0.1t1.2+ 0.4t2+ 0.2t2.2+ 0.2 t 3)R(t)= P(T> t)= e3. Reliability of the object of the parallelstructureThe life length of the object of a parallel structure is arandom variable defined byT = max(T1 , T2,..., Tn) . (13)Let us compute the reliability function of the object:R(t)= P(T > t)= 1 − P(T ≤ t)= 1 − P(T= P({T11≤ t,T2> t}∪{T≤ t,...,T2n≤ t)> t}∪ ... ∪{Tn.> t}).(14)Using the formula of probability of a sum of events weobtainR(t)= P(T > t)= ∑ P(Tni=1in> t)− ∑ P(T > t,T > t)i,j=1i t)−...we get+ ( −1)n+1P(T1> t,T2> t,...,Tn> t) .R(t)= P(T> t)= e−(0.1t1.2+ 0.2t2+ 0.1t2.2+ 0.2t3)Hence and from (4), (6), (7) we getThe graph of the function is presented in Figure 1.- 85 -