Recently , interest has grown in the study of multi-state systems because some practical systems and components have several different performance levels . In addition , each component may have more than two different failure modes which each mode has a different effect on the system level performance . Thus , such systems should be investigated as multi-state systems . The multi - state system is a system with the property that the system and each component have i, (i= ? , ? , … , M) possible states , where ? is the complete failure state and m is the perfect functioning state . In recent years , there has been some interest in the multi-state k-outof-n systems and its properties . The system is said to have multi-state k-out-of- n structure if the system is in state j or above if there exists an integer value l (j lt; l lt; M) such that at least kl components are in state l or above . Multistate k-out-of-n systems have many applications for describing engineering systems such as oil supply system , manufacturing system and lighting system . El-Neweihi et . al . (????) introduced the first definition of multi-state k-out-of-n systems . According to this definition , a system has a performance level in state j or above if at least k components are in state j or above . A generalized multi-state k -out-of- n : G system defined by Huang et . al . (????) which in this system $ k $ can have different values for different system state levels . Zuo and Tian (????) obtained algorithms for reliability evaluation of such system . Afterwards , Tian et al (????) studied a new multi-state k-out-of-n system , and made a algorithm for its reliability evaluation . Kolowrocki and Soszyska (????) , Kolowrocki and Kwiatuszewska-Sarnecka (????) , and Eryilmaz (????) considered a multi-state k -out-of- n system with M amp;#??; ? possible states , then the lifetime of the system in state j or above is actually the time spent by the system with a minimum performance level of j for j = ? , ... , M and investigated the time spent by the system in specific state subsets or specific states . Barlow and Wu (????) discussed the coherent systems with multi-state components by using the concepts of path and cut sets from multi-state case . Recently , the multi-state k-out-of-n systems and its properties have been extensively discussed by several authors . See for example Tian et . al . (????) , Zhao and Cui (????) , Chatuverdi et al(????) , Proschan et al (????) , Eryilmaz (????) , Zuo et al (????) . In many practical systems , the systems consisting of n independent element that each element has two or more dependent components . In addition , this system and its components may have several level performance , therefore , such systems should be studied as multi-state systems . This research is a study on the multi-state (r , s) -out-of- n systems consisting of $ n $ independent elements that each has two or more dependent components . we introduce the system and prepare the assumptions and notations that will be needed throughout the paper . The abtained reliability function of the system and the mean residual lifetime of multi-state (r , s) -out-of- n systems depends on the distributions of bivariate order statistics . کلیدواژه فارسی کلیدواژه انگلیسی