Insertion of a random task in a schedule: a real-time approach

We consider the case of a single resource. A given schedule (possibly optimal) is evaluated by means of the sum of the delays of the tasks. A task appears in the system at a random time. The duration of this task is random, as well as its due date. The goal is to complete the task at the latest by its due date while increasing as little as possible the sum of the delays of the initial tasks. We have to find an algorithm that reduces as most as possible the amount of computation to be performed in real time at the expenses of the amount of computation to be performed off-line.