Optimal steady-state and transient trajectories of multi-queue switching servers with a fixed service order of queues

The optimal scheduling problem of a system with two fluid queues attended by a switching server is addressed from two angles, the optimal steady-state and the optimal transient problem. The considered system includes features, such as setup times, setup costs, backlog and constraints on queue contents, cycle times and service times. First, the steady-state problem is formulated as a quadratic problem (QP), given a fixed cycle time. Evaluation of the QP problem over a range of cycle times results in the optimal steady-state trajectory, minimizing the total cycle costs or time average costs. Second, given initial conditions, we derive the optimal transient trajectory that leads to the optimal steady-state trajectory in a finite amount of time at minimal costs. For systems with backlog, we introduce additional costs on the number of cycles required to reach the steady-state trajectory in order to simplify the transient trajectory. The transient switching behavior and optimal initial modes are also addressed. Furthermore, we show by means of an example that the method can be extended to multi-queue switching servers.