DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

LI i
wi
( t ) ≥ ω
k( i )
for all i ∈ I and t ≥ 0
LB
(7)
T
Proof. A component’s instantaneous resource availability RA i (t), which is the available fraction
of a resource when the component requests the resource at time t, is more than or equal to
assigned weight proportion as:
w i( t )
RA i( t ) ≥ for t ≥ 0 . (8)
ω
k( i )
Service time S i (t) is the time taken to process a task at time t and has a relationship with RA i (t)
as:
∫
t+ Si
( t)
RAi
( τ ) dτ
= Pi
. (9)
t
Suppose a component i receives its tasks at a constant interval of T LB /L i . Then, under the
resource allocation in (7), S i (t) is less than or equal to T LB /L i over time as shown in (10).
i
Pi
=
∫
RA i ( τ )dτ
≥
∫
T
⇒
L
LB
i
t+
S ( t )
t
≥ S ( t )
i
t+
S (
t
i
t )
wi
( t ) LI
dτ
≥
ω T
k(
i )
i
LB
S ( t )
i
(10)
So, any component can complete by T LB and generate tasks at a constant interval of T LB /L i
from t=T LB /L i (first task generation time) under the resource allocation in (7) when it receives
tasks at a constant interval of T LB /L i from t=0 (first task arrival time). As tasks are infinitesimal
and root tasks increase task availability, each component can receive infinitesimal tasks at a
constant interval in 0≤t≤T LB or more preferably, and complete at less than or equal to T LB . So,
the network completes at T LB .
So, a network can achieve a performance close to T LB under this resource allocation in the
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