Asymptotic optimality of a joint scheduling–control policy for parallel server queues with multiclass jobs in heavy traffic

To optimize the control of a queuing system with multiple classes of customers and multiple servers, we introduce a novel joint scheduling–control policy that includes customer admission control, service scheduling control, and service rate control. In this policy, any server can serve any class of customers; the service rate control for a server is a unique feature of this policy and is determined by the overall state of the system, not the state of a server or the class of customers it serves. Given the inherent complexity of the system ${}^{\mathrm{\prime }}$s equations and the difficulty of solving them directly, we apply diffusion approximation theory and consider the Halfin–Whitt heavy traffic regime. This approach yields a formally weak limit of the joint scheduling–control problem. This limit problem, which we call the diffusion control problem (DCP), is a stochastic differential equation (SDE). Next, we present the corresponding Hamilton–Jacobi–Bellman (HJB) equation and prove the existence and uniqueness of the solution to this equation. This solution is the optimal Markov policy for the diffusion control problem, and we use this solution to devise a policy for the original joint scheduling–control problem and prove its asymptotic optimality. We designed several experiments to compare the system ${}^{\mathrm{\prime }}$s performance and value functions under different control policies. Our designed joint scheduling–control policy has significant advantages in reducing the system ${}^{\mathrm{\prime }}$s cost and improving service efficiency.