Before revision:   We show that zero queueing delay for such parallel jobs can be achieved using  a variant of the  \ policy, the batch-filling policy, with a probe overhead d = \omega\left(\frac{1}{(1-\lambda)\log k}\right)  , where k is the number of tasks in each job  .
After revision:  We show that zero queueing delay for such parallel jobs can be achieved using a variant of the  \ policy, the batch-filling policy, with a probe overhead d = \omega\left(\frac{1}{(1-\lambda)\log k}\right) in the sub-Halfin-Whitt heavy-traffic regime , where k is the number of tasks in each job .