Scheduling tasks with ready times and deadlines to minimize average error

We consider here the problem of scheduling tasks each of which is logically decomposed into a mandatory subtask and an optional subtask. The mandatory subtask must be executed to completion in order to produce an acceptable result. The optional subtask begins after the mandatory subtask is completed and refines the result in order to reduce the error in the result. If the available processor time is insufficient, the optional subtask can be left incompleted. The error in the result of a task is equal to the processing time of the unexecuted portion of the optional subtask. We describe a preemptive algorithm for scheduling n dependent tasks with rational ready times, deadlines, and processing times on uniprocessor systems. This algorithm is optimal in the sense that it finds feasible schedules meeting ready-time and deadline constraints of all tasks and minimizing the average error of all tasks, whenever feasible schedules exist. The complexity of this algorithm is O(n2log2n). This algorithm can also be used to schedule optimally independent tasks with rational ready times, deadlines and processing times on identical multiprocessor systems. R@Liu, J. W. S., K. J. Lin, and S. Natarajan, "Scheduling real-time, periodic jobs using imprecise results," Proceedings of the IEEE 8th Real-Time Systems Symposium, San Jose, California, December 1987. RChung, J. Y. and J. W. S. Liu, "Performance of algorithms for scheduling periodic jobs to minimize average error," Proceedings of the IEEE 9th Real-Time Systems Symposium, Huntsville, Alabama, December 1988. RChung, J. Y. and J. W. S. Liu, "Performance of algorithms for scheduling periodic jobs to avoid timing faults," to appear in the Proceedings of 22nd Hawaii International Conference on System Sciences, Hawaii, 1989. RLin, K. J., S. Natarajan, J. W. -S. Liu, and T. Krauskopf, "Concord: a system of imprecise computations," Proceedings of the 1987 IEEE Compsac, Japan, October, 1987. RBlazewicz, J. and G. Finke, "Minimizing mean weighted execution time loss on identical and uniform processors", Information Processing Letters, Vol. 24, pp. 259--263, March 1987. RChong, E. K. P. and W. Zhao, Performance evaluation of scheduling algorithms for imprecise computer systems, Technical Report, Department of Computer Science, University of Adelaide, SA 5001, Australia, September 1988. RChong, E. K. P. and W. Zhao, User controlled optimization in task scheduling for imprecise computer systems, Technical Report, Department of Computer Science, University of Adelaide, SA 5001, Australia, October 1988. RLawler, E. L. and J. M. Moore, "A functional equation and its application to resource allocation and scheduling problem," Management Science, Vol. 16, pp. 77--84, 1969. RMcNaughton, R., "Scheduling with deadlines and loss functions," Management Science, Vol. 12, pp. 1--12, 1959. RTarjan, R. E., Data Structures and Network Algorithms, Society for Industrial and Applied Mathematics, Pennsylvania, 1983. RShih, W. K., J. W. S. Liu, J. Y. Chung, and D. W. Gillies, Scheduling tasks with ready times and deadlines to minimize average error, Technical Report, Department of Computer Science, University of Illinois, January 1989.