DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

Manuscript for IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 9
problem is NP-hard on the number of components in general. The increase of the number of
tasks and consideration of alternative algorithms make the problem even harder. Moreover, there
can be large number of nodes in our networks.
Though it may be possible to use some available heuristic algorithms from the job shop
scheduling problem by taking into account alternative algorithms, our scheduling problem has a
particular characteristic, i.e., the number of tasks for each component can be large. Though the
increase of the number of tasks adds more complexity, it can also lead us to develop an efficient
heuristic programming model. In this section, we characterize an optimal resource allocation by
analyzing the impacts of the largeness and subsequently build a mathematical programming
model solvable in polynomial time.
3.1 Optimal resource allocation
Consider current time t=0 and assume that each component uses a value mode common to all
the tasks (i.e. pure strategy). We will discuss the optimality of the pure strategy later in this
subsection. We define Load Index LI i which represents component i’s total CPU time required to
process its tasks. As a component needs to process its own root tasks as well as incoming tasks
from its predecessors, its number of tasks L i is identified as in (3), where i denotes the immediate
predecessors of component i. Then, LI i is represented as in (4).
∑
L i = rti
+ La
(3)
a∈i
LI = L f v )
(4)
i
To provide theoretical foundation of optimal resource allocation policy, we convert a network
into a network with tasks having infinitesimal processing times. Each root task is divided into r
infinitesimal tasks and each f i (v i ) is replaced with f i (v i )/r. Then, the load index of each component
i
i ( i