DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

Manuscript for IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 16
w
*
i
=
T
p∈K
*
i
∑
T
n(
i)
*
p
ω
n(
i)
(22)
Component’s decision
v
*
i
*
1 ( Ti
− Ri
( t)
= f
− i ( )
(23)
L ( t)
i
4.2 Analysis
Resource manager’s bidding function b n (T) in (19) can be composed referring to the solution
algorithm of fractional knapsack problem. In the fractional knapsack problem, there are multiple
items that can be broken into fractions. Given unit weight and unit value of each item, the
problem is to determine the amount of each item so as to maximize total value subject to a
weight capacity. The fractional knapsack problem can be easily solved by a greedy algorithm,
i.e., take as much as possible of the item that is the most valuable per unit weight until the
capacity is reached. Similarly, b n (T) can be composed using a greedy algorithm. As b i (T i ) in (18)
is a piecewise-linear increasing concave function, take the most valuable piece per unit T i among
the first available pieces until all pieces are taken. This greedy algorithm leads building the
resource manager’s bidding function in O(|K n | 2 ), where |X| denotes the cardinality of set X.
Similarly, resource manager’s decision problem in (21) can be solved in O(|K n | 2 ), using the
greedy algorithm except that (fractional) pieces are taken until a capacity is reached. So, the
complexity of all resource managers’ local problems is O(|A| 2 ) in the worst case when |N|=1.
The seller’s decision problem in (20) is simply a single variable problem, which can be solved
using diverse search methods depending on the structure of objective function. As each b n (T) is
piecewise-linear increasing concave function, ∑b n (T) is also a piecewise-linear increasing
concave function and its number of pieces is proportional to |A|. To compose ∑b n (T) from each