On-line approximation algorithms for scheduling tasks on identical machines with extendable working time

We study the on-line problem of assigning tasks to identical machines whose regular working time - assumed to be unitary - can be extended. If the tasks assigned to a machine do not exceed the regular working time, then the working time of the machine is considered to be 1; otherwise, it is the completion time of the last task assigned to the machine. Each incoming task has to be assigned immediately to a machine and the assignment cannot be changed later. The objective is to minimize the sum of the working times of the machines. Since the regular working time of the machines can be seen as a given capacity, the problem can also be described through the bin packing terminology: the machines are viewed as bins and the tasks as items. A lower bound of 7/6 on the worst-case relative error of any on-line algorithm is shown. Then it is shown that a list scheduling heuristic which assigns the incoming task to the machine with smallest current load has worst-case error equal to 5/4. The bound is improved to 1.228 by a new algorithm which tends to load the partially loaded machines, as long as this does not cause an increase of the working time by more than a fixed and appropriately chosen quantity x > 0.