Research on the maintenance of repairable machines has mostly focused on an open-loop system consisting of an infinite number of identical machines working in a production system, and a repair shop along with a spare parts inventory. The machines used in the production system are subject to failure and are replaced upon failure utilizing a spare part inventory. Meanwhile, the failed parts are repaired in the service center and are sent back to the spare parts inventory. The parts after repair are considered as good as new. In this dissertation it is assumed that the population of machines in the system is finite resulting in a closed-loop system. However, this changes the nature of problem in which the well-known queuing systems and stochastic processes need to be modified accordingly. Moreover, the times in the system are assumed to be generally distributed. For this system, the performance of the overall system is evaluated and the capacities of the repair shop together with the level of spares in inventory which optimize the performance of the system are determined simultaneously. Furthermore, the developed models and algorithms in this scenario, are utilized in order to study and optimize the performance of a healthcare system in which the length of stay of patients are generally distributed in different phases of health care process.