Optimal steady-state and transient trajectories of a two queue switching server

We consider two fluid queues attended by a switching server and address the optimal steady-state and transient trajectory problems. The steady-state problem is formulated as a quadratic problem (QP), given a fixed cycle time. Evaluating the QP problem over a range of cycle times results in the optimal steady-state trajectory. We minimize the holding costs, backlog costs and setup costs, allow setup times and allow constraints on queue contents, cycle times and service times. Second, given initial conditions, we derive the optimal transient trajectory that leads to the optimal steady-state trajectory in a finite amount of time with minimal costs. The transient switching behavior and optimal initial modes are also addressed.