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

Abstract

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.

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

“…Note that the constraints (17) only guarantee that x + n,c = max(0, x n,c ) + k and x − n,c = max(0, −x n,c ) + k for some constant k ≥ 0. However, minimizing the objective function (23) with c + n > 0 and c + n > 0 guarantees that k = 0.…”

“…In this paper, we divide the optimal scheduling problem into two subproblems: the derivation of optimal steady-state trajectories and the derivation of optimal transient trajectories. The current study is an extension of the work in [17]. Similar to [15,17], we formulate both subproblems as Quadratic Programming (QP) problems, with the addition of backlog and setup costs.…”

“…The current study is an extension of the work in [17]. Similar to [15,17], we formulate both subproblems as Quadratic Programming (QP) problems, with the addition of backlog and setup costs. Once the optimal steady-state trajectory is known, we study the best way of reaching it from any initial state, i.e., with minimal costs.…”