This paper analyzes a queuing model consisting of a system of queues that are attended periodically (in a given order) by a single server. The server empties each queue before proceeding to the next queue in sequence. A changeover time is required whenever the server switches from one queue to another. For a stationary process expressions are obtained for the Laplace-Stieltjes transforms of the waiting-time and intervisit-time distributions at each queue. It is also shown how the moments of these distributions can be calculated.