In the study of the reliability of technical systems in reliability enginerig, coherent systems play a key role. Today, in the context of a coherent system reliability are examined from different perspectives. One of the points of view of integrated systems under different assumptions on lifetime of its components, such as being independent and identically distributed, exchangeable or assume not exchangeable component lifetime distributions. We derive, with use concept of signature mixture representations for reliability residual life functions and new representation for inactivity time of a coherent system, under conditions on the states of components or systems, according to reliability residual life functions or inactivity times of coherent systems. Another view, investigated integrated systems based on different states of its components. A frequently encountered example of systems with shared components occurs in networked computing in which a server (say a file server or Web server) is used in tandem with several individual computers. It is typical that departments within a company or university will store almost all of the files for the department's individual PCs on one central server. If the central server goes down, the PCs with local disks may retain certain limited capabilities, while other PCs may not work at all. The performance of any given pair of PCs will depend on the performance of the shared components (those in the central server) and the performance of its own individual components. An appropriate ion of the situation above is the case of two 'slave' computers linked to a server. The former computers have components with lifetimes U1, U2, …, Un, and V1, V2, …, Vn., respectively, while the server has components with lifetimes W1, W2, …, Wn The lifetime T1 of the first slave computer is thus a function of the Us and the Ws, while the lifetime T2 of the second slave computer depends on the Vs and the Ws. Therefore in section 3, we will focus on the development of exact representations, in quite general settings, of the joint lifetime distribution G of the pair (T1, T2 ) as a function of a pair of matrices (S, S*) we will call the joint signature of the two systems, a distribution-free measure of the designs of the two systems and in section 4, we devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components and a new distribution-free measure, the 'joint bivariate signature', of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems. On the other hand, since the univariate signature can be used to compute the moments of a single coherent system lifetime. Analogously, the bivariate signature can be used to compute product moments and covariance coefficients as follows.