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This paper deals with online scheduling of tasks to partially reconfigurable devices. Such devices are able to execute several tasks in parallel. All tasks share the reconfigurable surface as a single resource which leads to highly dynamic allocation situations. To manage such devices at runtime, we propose a reconfigurable operating system that splits into three main modules: scheduler, placer, and loader. The main characteristic of the resulting online scheduling problem is the strong nexus between scheduling and placement. We discuss a fast online placement technique and then focus on scheduling real-time tasks. We devise guarantee-based schedulers for two scenarios, namely tasks with arbitrary and synchronous arrival times. The schedulers exploit the knowledge about task properties to improve the system's performance. The experiments show that the developed schedulers lead to substantial performance gains at an acceptable runtime overhead.