Cs. Chang et al., SCHEDULING PARALLEL PROCESSORS - STRUCTURAL-PROPERTIES AND OPTIMAL POLICIES, Mathematical and computer modelling, 23(11-12), 1996, pp. 93-114
Consider a set of parallel processors operating in a distributed fashi
on, which prohibits task migration among processors. A given set of jo
bs, each of which consists of a number of tasks, is to be processed by
the parallel processors. Finding an optimal schedule in this context
is a difficult combinatorial problem in general. Our focus here is on
identifying key properties of the problem so as to provide insight to
the structure of optimal policies. We demonstrate that these propertie
s are all rooted in the subadditivity, submodularity, convexity, and S
chur convexity of two operators, max and plus, which relate task times
to the flow time (expected job completion time). We show that in gene
ral the optimal policy has a threshold structure and a sequential ''ta
il'': there exists a threshold, such that once the number of jobs alre
ady scheduled exceeds this threshold, all remaining jobs must be sched
uled sequentially, i.e., each job with the entirety of its tasks is as
signed to one processor only. In the special case of two processors, w
e further develop a recursive algorithm that generates the complete op
timal schedule. For the case of three or more processors, we focus on
a class of policies that we call ''fully parallel or fully sequential'
' (FPFS), and identify optimal policies within this class.