Queues Served in Cyclic Order

We consider a system of queues served in cyclic order by a single server. The ith queue is characterized by general service time distribution function Hi(-) and Poisson input with parameter V We study two variations of this model. In the first, called the exhaustive seivice model, the process begins with the arrival of a unit at some queue, say A, when the system is otherwise empty. The server begins on this unit immediately, and continues to serve queue A until for the first time the server becomes idle and there are no units waiting in queue A. The server then looks at the next queue in the cyclic order, queue .4 + 1, and serves those units, if any, that have accumulated during the serving period of queue A. The server continues to serve queue A + 1 until for the first time the server * The R A N D Corporation. 675 676 T H E BELL SYSTEM T E C H N I C A L J O U R N A L , MARCH 1969 becomes idle and there are no units waiting in queue A + 1. The process continues in this manner, with the queues being served in cyclic order, until for the first time the system becomes completely empty. The process is then re-initiated by the arrival of the next unit. No time is required to switch from one queue to the next. The second variation, called the gating model, differs from the first in the following way: When the server moves to a queue with at least one waiting unit, the server accepts only those units that were waiting when the server arrived, deferring service of all subsequent arriving units until the next cycle.