مقایسه تصادفی عمر گذشته و عمر بقای سیستم¬های(n-k+1) ¬ از n با مولفه¬های مستقل و وابسته

The
 −  + ١-out-of- System is a system consisting of n components and works if and only if at least  −  + ١ out of  components are operating ( ≤ ). This system is very popular and is applied in industrial and survival analysis systems. The classical theory of systems assumes that components of the system are independent and identically distributed. But In real situations, however, there may be a structural dependence and nonidentical among components of the system. In this thessis, we are investigated this systems whit independent and identically, independent and nonidentical and dependent components. We obtaine lifetime distribution in the case of nonidentical by permanent and in the case of dependence by an Archimedean copula with a completely monotone generator. In the case of independency, (identical and nonidentical) obtaine expectation of past lifetime of the th component 
,
 = 
 − :
| :
≤  and showe this is a decreasing function of  and increasing function of  and when , the number of the components of the system, increases 
,
 also increases. In the case of dependency, we denote past lifetime of the components by
 − :
| :
≤  and we show that past lifetime of the th component is stochastically non increasing with respect to  and non decreasing with respect to  and when , the number of the components of the system, increases past lifetime also increases. We denote residual lifetime of the components by :
− | :
>  and we show that residual lifetime of the th component is stochastically non decreasing with respect to  and non increasing with respect to  and and when , the number of the components of the system, increases past lifetime also decreases. Also, are representationed The concept of Again. Finally, is compared lifetimes of two
 −  + ١-out-of- systems with heterogeneous dependent components.