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 T1, T2, ..., T(k) of tasks is a sequence s1, s2, ..., s(k) of states where si is the state in which T1 is processed; the cost of a schedule is the sum of all task processing costs and state transition costs incurred. An on-line scheduling algorithm is one that chooses s(i) only knowing T1T2 ... 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 Absolute value of S - 1 for every task system in which d is symmetric, and w(S, d) = O(Absolute value of S2) 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 wBAR(S, d) O(log Absolute value of S).
