Coupled task scheduling with convex resource consumption functions

We study a single machine scheduling problem with coupled tasks under limited resource availability. Each job comprises of two tasks which are separated by an exact amount of delay. We assume that the processing time of the initial task and the duration of the delay period are equal and the same for all jobs. The processing time of the completion task, however, is job-dependent and modelled as a convex function of the amount of resource the job is allocated. The scheduling objective consists of minimizing the makespan, subject to an upper-bound on the resource availability. We provide several properties of an optimal solution and develop an O(n²) time algorithm for the problem.