In this paper we propose new approximate methods for the performance analysis of closed-loop production systems with unreliable machines and random processing times. Each approximate method decomposes the production system consisting of K machines into a set of K subsystems, each subsystem consisting of two machines separated by a finite buffer. Then, each subsystem is analyzed by three different analyzing methods in isolation. The population constraint of the closed-loop production system is taken into account by prescribing that the sum of average buffer levels in the subsystems is equal to the number of customers in the closed-loop production system. We establish a set of equations that characterize unknown parameters of the servers in the subsystems. An iterative procedure is then used to determine the unknown parameters. Experimental results show that these methods provide a good estimation of the throughput.