Periodic Task Scheduling Using Group Theory
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Solution Overview
Problem
Scheduling systems face challenges in computing collision-free schedules for periodic tasks across resources with different service rates, particularly in distributed environments like packet-switched networks, where computational complexity is high and optimal solutions are NP-hard, making it difficult to dynamically add new tasks without violating collision-free properties.
Innovation Solution
The use of mathematical group theory, specifically the Babylonian Theorem and its corollary, to model and select task periods, resource schedule periods, and units of measure, enabling improved computational efficiency and resource utilization by ensuring collision-free scheduling through efficient coset representation and scheduling algorithms.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If optimal scheduling algorithms are used to ensure collision-free schedules, then scheduling quality is improved, but computational complexity becomes NP-hard and intractable
Solution Approach 1:
The patent transforms the scheduling problem by changing parameters: instead of scheduling arbitrary periodic tasks, it constrains task periods to be integer multiples of a base period T. This parameter transformation converts the NP-hard general scheduling problem into a tractable problem that can be solved efficiently using number theory-based algorithms while guaranteeing collision-free schedules.
Solution Approach 2:
The patent segments the scheduling problem by introducing a base period T and expressing all task periods as multiples of this base period. This segmentation approach divides the complex scheduling space into manageable discrete units, allowing efficient allocation of time slots without requiring complex optimization algorithms.
2Adaptability or versatility
If general periodic task scheduling is allowed, then system adaptability is improved, but finding collision-free schedules becomes computationally intractable
Solution Approach 1:
The patent maintains adaptability by allowing task periods to be any integer multiple of the base period T, providing flexibility for different task requirements. Simultaneously, it ensures high computation speed by using a direct mathematical formulation based on number theory, avoiding iterative optimization algorithms.
Solution Approach 2:
The scheduling algorithm automatically handles collision avoidance through mathematical properties of integer multiples and modular arithmetic. The system self-adjusts task allocations without requiring complex conflict detection and resolution mechanisms, achieving both adaptability and computational efficiency.
3Adaptability or versatility
If distributed resource coordination is implemented, then system coverage is improved, but scheduling complexity increases due to multiple local schedules needing coordination
Solution Approach 1:
The patent implements a universal base period T that serves all distributed resources in the system. This universal parameter enables different resources to have their own local schedules while maintaining global coordination through the common time reference, simplifying distributed system management.
Solution Approach 2:
The patent adds a global time dimension (base period T) that coordinates local schedules across distributed resources. By operating in this additional temporal dimension, the system achieves global coordination without increasing the complexity of individual local schedulers.
Data Source
AI summary
The invention described is a system and method for efficient scheduling of periodic phenomena including a collection of methods for modeling and selecting periodic task rates, resource schedule periods, and units of measure for the task and resource periods in scheduling systems such that the systems' schedulers may be improved with respect to performance metrics such as collision avoidance, computational efficiency, and resource utilization.