In practice, almost all dynamic systems require decisions to be made on-line, without full knowledge of their future impact on the system. A general model for the processing of sequences of tasks is introduced, and a general on-line decision algorithm is developed. It is shown that, for an important class of special cases, this algorithm is optimal among all on-line algorithms.
Specifically, a task system (
S,d
) for processing sequences of tasks consists of a set
S
of states and a cost matrix
d
where
d
(
i, j
is the cost of changing from state
i
to state
j
(we assume that
d
satisfies the triangle inequality and all diagonal entries are 0). The cost of processing a given task depends on the state of the system. A schedule for a sequence
T
1
,
T
2
,…,
T
k
of tasks is a sequence
s
1
,
s
2
,…,
s
k
of states where
s
i
is the state in which
T
i
is processed; the cost of a schedule is the sum of all task processing costs and the state transition costs incurred.
An on-line scheduling algorithm is one that chooses
s
i
only knowing
T
1
T
2
…
T
i
. Such an algorithm is
w
-competitive if, on any input task sequence, its cost is within an additive constant of
w
times the optimal offline schedule cost. The competitive ratio
w
(
S
,
d
) is the infimum
w
for which there is a
w
-competitive on-line scheduling algorithm for (
S
,
d
). It is shown that
w
(
S
,
d
) = 2|S|–1
for every task system
in which
d
is symmetric, and
w
(
S, d
) =
O
(|
S
|
2
) for every task system. Finally, randomized on-line scheduling algorithms are introduced. It is shown that for the uniform task system (in which
d
(
i,j
) = 1 for all
i,j
), the expected competitive ratio
w¯
(
S,d
) =
O
(log|S|).
