In this paper we shall generalize Graham's result so as to include a parameter characterizing the number of tasks assigned to processors by the LPT rule. The new result will show that the worst-case performance bound for LPT sequencing approaches unity approximately as 1+1/k, where k is the least number of tasks on any processor, or where k is the number of tasks on a processor whose last task terminates the schedule. Thus, we shall have a result very similar to the parameterized bounds for bin-packing heuristics [JDUGG]. We shall also obtain out of the analysis an alternate proof of Graham's result.