Asymptotic Optimal Scheduling of V-Systems with Deadlines and Customer Abandonment

We consider a V-structured queueing system with two classes of customers: class 1 customers cannot abandon but have a waiting-time deadline while class 2 customers may abandon. The objective is to minimize the number of abandonments of class 2 customers while meeting the deadline for class 1 customers. We consider the problem in an asymptotic framework, and prove that under the heavy traffic regime, the threshold policy that gives priority to class 1 customers if the age of the head-of-the-line class 1 customer exceeds a threshold is asymptotically optimal.