An optimal online algorithm for single machine scheduling to minimize total general completion time

We study the online problem of single machine scheduling to minimize total general completion time. General completion time is defined as C-j(alpha) = (C-j)(alpha), where C-j denotes the completion time of job J(j) and alpha >= 1 is a constant integer. Total general completion time characterizes the feather in service that when a customer is served later in time, his dissatisfaction increases in a manner of power function. The objective function Sigma(C-j)(alpha) can also be viewed as a total weighted completion time, but the "weight" is no longer a constant number. Our purpose to minimize customers' total dissatisfaction. The problem is online in the sense that all jobs arrive over time. Each job's processing time becomes known at its arrival time. Preemption is not allowed. For this online problem, we show that a lower bound on competitive ratio is 2(alpha) and prove that D-SPT (delayed shortest processing time) algorithm is optimal with a competitive ratio 2(alpha).