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We consider the online parallel machine scheduling problem of minimizing the makespan under eligibility constraints that restrict each job to be processed only on one of its eligible machines. The greedy approach known as AW is known to be optimal for this problem when the number of machines, m, is a power of 2, i.e. m=2^k. However, in other cases, the gap between the best known competitive ratio and its lower bound can be as large as 1. In this paper, we construct new competitive ratio and its lower bound whose gap is no more than an irrational number which is approximately 0.1967 and establish optimality for the cases when the number of machines can be written as a sum of two powers of 2, i.e. [Formula: see text] for k≠k^′. We further analyze the case with seven machines showing that their gap is no more than 1/180(≈0.00556). Moreover, we present new lower bounds of the competitive ratio for the …