  Online algorithm has been an emerging area of interest for researchers in
various domains of computer science. The online $m$-machine list scheduling
problem introduced by Graham has gained theoretical as well as practical
significance in the development of competitive analysis as a performance
measure for online algorithms. In this paper, we study and explore the
performance of Graham's online \textit{list scheduling algorithm(LSA)} for
independent jobs. In the literature, \textit{LSA} has already been proved to be
$2-\frac{1}{m}$ competitive, where $m$ is the number of machines. We present
two new upper bound results on competitive analysis of \textit{LSA}. We obtain
upper bounds on the competitive ratio of $2-\frac{2}{m}$ and
$2-\frac{m^2-m+1}{m^2}$ respectively for practically significant two special
classes of input job sequences. Our analytical results can motivate the
practitioners to design improved competitive online algorithms for the
$m$-machine list scheduling problem by characterization of real life input
sequences.
