  We consider the online scheduling problem of moldable task graphs on
multiprocessor systems for minimizing the overall completion time (or
makespan). Moldable job scheduling has been widely studied in the literature,
in particular when tasks have dependencies (i.e., task graphs) or when tasks
are released on-the-fly (i.e., online). However, few studies have focused on
both (i.e., online scheduling of moldable task graphs). In this paper, we
design a new online scheduling algorithm for this problem and derive constant
competitive ratios under several common yet realistic speedup models (i.e.,
roofline, communication, Amdahl, and a general combination). These results
improve the ones we have shown in the preliminary version of this paper. We
also prove, for each speedup model, a lower bound on the competitiveness of any
online list scheduling algorithm that allocates processors to a task based only
on the task's parameters and not on its position in the graph. This lower bound
matches exactly the competitive ratio of our algorithm for the roofline,
communication and Amdahl's model, and is close to the ratio for the general
model. Finally, we provide a lower bound on the competitive ratio of any
deterministic online algorithm for the arbitrary speedup model, which is not
constant but depends on the number of tasks in the longest path of the graph.
