In this paper, we present a model for the broadcast star network with collision avoidance switch and with multiple priority classes of messages. Also, we propose a service discipline which is easy to implement and exhibits relatively low variance of message delay. The proposed service discipline, which is used to select a message among messages with the same class-index, is a combination of first-come first-served (FCFS) and random-order of service (ROS) disciplines, and it uses the priority fields in the message header. Then, the ready stations can be considered to form a distributed queue, and the entire system can be considered to form a priority M/G/1 queueing system with multiple server vacations. We analyze the model to obtain distributions of message delay for the proposed service discipline, and compare the results with that for the FCFS discipline. From numerical results, we show that the variance or message delay for the proposed service discipline is almost the same as that for the FCFS discipline for any distribution of message length and also for any value of normalized propagation delay.