This note deals with the scheduling problem of minimizing the sum of job co
mpletion times in a system with n jobs and a single machine. We investigate
the on-line version of the problem where every job has to be scheduled imm
ediately and irrevocably as soon as it arrives, without any information on
the later arriving jobs.
We prove that for any sufficiently smooth, non-negative, non-decreasing fun
ction f(n) there exists an O(f(n))-competitive on-line algorithm for minimi
zing the total completion time if and only if the infinite sum Sigma(n=1)(i
nfinity) 1/(nf(n)) converges.