Bounds for Scheduling in Non-Identical Uniform Multiprocessor Systems

With multiprocessors and multicore processors becoming ubiquitous, focus has shifted from research on uniprocessors to that on multiprocessors. Results derived for the uniprocessor case unfortunately do not always directly extend to the multiprocessor case in a straightforward manner. This necessitates a paradigm shift in the approach used to design and analyse the behaviour of such processors. In the case of Real-time systems, that is, systems characterised by explicit timing constraints, analysis and performance guarantees are more important, as failure is unacceptable. Scheduling algorithms used in Real-time systems have to be carefully designed as the performance of the system depends critically on them. Efficient tests for determining if a set of tasks can be feasibly scheduled on such a computing system using a particular scheduling algorithm thus assumes importance. Traditionally, the ‘task utilization’ parameter has been used for devising such tests. Utilization based tests for Earliest Deadline First(EDF) and Rate Monotonic(RM) scheduling algorithms are known and are well understood for uniprocessor systems. In our work, we derive limits on similar bounds for the multiprocessor case. Our work differs from previous literature in that we explore the case when the individual processors constituting the multiprocessor need not be identical. Each processor in such a system is characterised by a capacity, or speed, and the time taken by a task to execute on a processor is inversely proportional to its speed. Such instances may arise during system upgradation, when faster processors may be added to the system, making it a non-identical multiprocessor, or during processor design, when the different cores on the chip may have different processing power to handle dynamic workloads. We derive results for the partitioned paradigm of multiprocessor scheduling, that is, when tasks are partitioned among the processors, and interprocessor migration after a part of execution is completed is not allowed. Results are derived for both fixed priority algorithms(RM) and dynamic priority algorithms (EDF) used on individual processors. A maximum and iii