  This paper introduces a parallel scheduling problem where a directed acyclic
graph modeling $t$ tasks and their dependencies needs to be executed on $n$
unreliable workers. Worker $i$ executes task $j$ correctly with probability
$p_{i,j}$. The goal is to find a regimen $\Sigma$, that dictates how workers
get assigned to tasks (possibly in parallel and redundantly) throughout
execution, so as to minimize the expected completion time. This fundamental
parallel scheduling problem arises in grid computing and project management
fields, and has several applications.
  We show a polynomial time algorithm for the problem restricted to the case
when dag width is at most a constant and the number of workers is also at most
a constant. These two restrictions may appear to be too severe. However, they
are fundamentally required. Specifically, we demonstrate that the problem is
NP-hard with constant number of workers when dag width can grow, and is also
NP-hard with constant dag width when the number of workers can grow. When both
dag width and the number of workers are unconstrained, then the problem is
inapproximable within factor less than 5/4, unless P=NP.
