DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

denotes the immediate predecessors of component i. Then, by denoting CPU time per task in the
given topology as P i , LI i is represented as in (5).
∑
L i = rti
+ La
(4)
i
a∈i
LI = L P . (5)
To provide theoretical foundation of optimal resource allocation, we convert a network into a
network with infinitesimal tasks. Each root task is divided into r infinitesimal tasks and each P i is
replaced with P i /r. Then, the load index of each component is the same as the original network
but tasks are infinitesimal. We denote the completion time of the network with infinitesimal
tasks as T´. Also, we define a term called task availability as an indicator of relative preference
for task arrival patterns. A component’s task availability for an arrival pattern is higher than for
another if cumulative number of arrived tasks is larger or equal over time. A component prefers a
task arrival pattern with higher task availability as it can utilize more resource. Consider a
network and reconfigure it such that all components have their tasks in their queues at t=0. Each
component has maximal task availability in the reconfigured network and the completion time of
the reconfigured network forms the lower bound T LB of a network’s completion time T given by:
i
i
T
LB
ωk
= Max ∑ LI i
k∈K
a . (6)
ω
k i∈
S I
[ k ]
Then, assuming a hypothetical weighted round-robin server 2 for CPU scheduling, T´ equals to
T LB when each machine allocates resource to the residing components according to (7), where
k(i) denotes a machine in which component i resides.
2 The hypothetical server has idealized fairness as the CPU time received by each thread in a round is infinitesimal
and proportional to the weight of the thread. This assumption is reasonable because quantum size is relatively
infinitesimal compared to working horizon in reality.
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