  We study scheduling control of parallel processing networks in which some
resources need to simultaneously collaborate to perform some activities and
some resources multitask. Resource collaboration and multitasking give rise to
synchronization constraints in resource scheduling when the resources are not
divisible, that is, when the resources cannot be split. The synchronization
constraints affect the system performance significantly. For example, because
of those constraints, the system capacity can be strictly less than the
capacity of the bottleneck resource. Furthermore, the resource scheduling
decisions are not trivial under those constraints. For example, not all static
prioritization policies retain the maximum system capacity and the ones that
retain the maximum system capacity do not necessarily minimize the delay (or in
general the holding cost). We study optimal scheduling control of a class of
parallel networks and propose a dynamic prioritization policy that retains the
maximum system capacity and is asymptotically optimal in diffusion scale and
conventional heavy-traffic regime with respect to the expected discounted total
holding cost objective.
