We study scheduling problems motivated by recently developed techniques for microprocessor thermal management at the operating systems level. The general scenario can be described as follows. The microprocessor temperature is controlled by the hardware thermal management system that continuously senses the chip temperature and automatically reduces the processor's speed as soon as the thermal threshold is exceeded. Some tasks are more CPU-intensive than other and thus generate more heat during execution. The cooling system operates nonstop, reducing (at an exponential rate) the deviation of the processor's temperature from the ambient temperature. As a result, the processor's temperature, and thus the performance as well, depends on the order of the task execution. Given a variety of possible underlying architectures, models for cooling and for hardware thermal management, as well as types of tasks, this gives rise to a plethora of interesting and never studied scheduling problems. We focus on scheduling real-time jobs in a simplified model for cooling and thermal management. A collection of unit-length jobs is given, each job specified by its release time, deadline and heat contribution. If, at some time step, the temperature of the system is t and the processor executes a job with heat contribution h, then the temperature at the next step is (t + h)/2. The temperature cannot exceed the given thermal threshold t. The objective is to maximize the throughput, that is, the number of tasks that meet their deadlines. We prove that in the offline case computing the optimum schedule is NP-hard, even if all jobs are released at the same time. In the online case, we show a 2-competitive deterministic algorithm and a matching lower bound.
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