The performance evaluation of hospital logistics is becoming more and more important to guarantee efficient services in health establishments. Therefore, we propose a new discrete-time approach for the steady-state analysis of closed-loop queueing systems with arbitrary topology and generally distributed service times. Based on a finite Markov chain, it is possible to compute the complete cycle time distribution of service systems with a population constraint. In addition, the distribution of the number of customers at each service station can be obtained. The method is applied to the analysis of a sterilization process of medical devices. To verify the method, we compare the results of our discrete-time approach to the results that are obtained by a simulation model.