Single-server queueing systems with mixed traffic are a flexible modeling tool that have been used to analyze polling systems and database striping strategies. They also have applications in manufacturing and telecommunication systems. Ordinary (open) customers arrive according to a Poisson process, receive service, and leave the system. Permanent (closed) customers remain in the system, reentering the queue after receiving service. This work examines the transient behavior of the permanent customers. The cycle times and output/departure process for the permanent customers are described.