Abstract--ZnsammenfassungRandomized Online Algorithms for Maximizing Busy Time Interval Scheduling. We consider the problem of scheduling tasks requiring certain processing times on one machine so that the busy time of the machine is maximized. The problem is to find a probabilistic online algorithm with reasonable worst case performance ratio. We answer an open problem of Lipton and Tompkins concerning the best possible ratio that can be achieved. Furthermore, we extend their results to an m-machine analogue. Finally, a variant of the problem is analyzed, in which the machine is provided with a buffer to store one job. AMS Subject Classification: 68M20Key words: Probabilistic algorithm, online scheduling, interval, busy time.Randomisierte Online-Algorithmen zur Maximierung der Arbeitszeitintervalle. Wir betrachten das Problem der Zuteilung von Aufgaben bestimmter RecherLzeit auf einem Rechner, um so seine Auslastung zu maximieren. Die Aufgabe besteht darin, einen probabilistischen Online-Algorithmus mit verniinftigem worst-case Performance-Verh~iltnis zu fmden. Wir geben die Antwort auf ein offenes Problem von Lipton und Tompkins, das das bestm6gliche Verh~iltnis betrifft. Weiter verallgemeinern wit ihre Ergebnisse auf ein m-Maschinen-Analogon. Schlief31ich wird eine Variante des Problems analysiert, in dem der Rectmer mit einem Zwischenspeicher fiir einen Job versehen ist.

We consider a range of controlled stochastic systems satisfying conservation laws together with a reducibility property that says that appropriate laws continue to hold when access to the system is restricted to a subset of all possible demand (job, customer) types. We show that for linear objectives, the optimum system-wide performance is a nondecreasing submodular (or supermodular) function of the subset chosen and that these properties are inherited from the geometry of the performance space concerned. These results are of considerable interest in their own right, but they also motivate the use of greedy heuristics for the solution of a range of job selection and scheduling problems which have hitherto seemed intractable. Computational experience suggests that such heuristics perform very well.

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We consider controlled stochastic systems in which multiple job types (customers, projects, jobs, etc.) may require processing by one or more servers. The job types are all members of priority classes and admissible controls must respect the constraints imposed by these. For systems which (when unconstrained) satisfy generalised conservation laws, we obtain the performance space and show that linear objectives are optimised by admissible controls which choose job types within each priority class according to a set of Gittins indices. Various degrees of decomposition of the problem are described and illustrated. In the special case of (discounted and undiscounted) branching bandits, the theory is developed further to yield index-based suboptimality bounds for general policies.

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