This paper presents a mathematical study of a time-sharing system with a single server and many queues. In each queue a customer receives one quantum of service and is then sent to the end of the next queue, provided his service demand has not been completely satisfied. When a quantum of service is completed, the server attends to the first customer in the lowest-index nonempty queue. The discipline of the highest indexed queue is “round-robin,” while all other queues obey the FIFO rule. Newly arrived customers join the end of the first queue. The paper derives mathematical expressions for the main performance measures and illustrates them by graphical means.