Optimal control of queuing systems governed by integro-differential equations

In this research, we advanced the optimal control theory for queuing systems that are characterized by integro-differential equations. Our primary goal was to identify an optimal service rate that minimizes a performance criterion, which is a composite of the system state at the final time and the cost associated with the optimal service rate. The optimal service rate was defined by an optimality system, and this formulation essentially translated the problem into a bilinear control problem within a nonreflexive Banach space, utilizing $L^1$-optimization techniques. We provided a rigorous proof of the existence of an optimal controller and offered a detailed characterization of the optimal control. Additionally, a comparison was made with traditional steady-state results to highlight the differences and improvements. Finally, numerical analysis was conducted on the theoretical results.