  We consider a system of parallel queues where tasks are assigned (dispatched)
to one of the available servers upon arrival. The dispatching decision is based
on the full state information, i.e., on the sizes of the new and existing jobs.
We are interested in minimizing the so-called mean slowdown criterion
corresponding to the mean of the sojourn time divided by the processing time.
Assuming no new jobs arrive, the shortest-processing-time-product (SPTP)
schedule is known to minimize the slowdown of the existing jobs. The main
contribution of this paper is three-fold: 1) To show the optimality of SPTP
with respect to slowdown in a single server queue under Poisson arrivals; 2) to
derive the so-called size-aware value functions for
M/G/1-FIFO/LIFO/SPTP/SPT/SRPT with general holding costs of which the slowdown
criterion is a special case; and 3) to utilize the value functions to derive
efficient dispatching policies so as to minimize the mean slowdown in a
heterogeneous server system. The derived policies offer a significantly better
performance than e.g., the size-aware-task-assignment with equal load (SITA-E)
and least-work-left (LWL) policies.
