隨著Multiprocessor系統的日漸普及，如何針對在其中執行的Task做Schedule，也逐漸成為一個重要的議題；為了確保這些Task都能獲得公平的待遇，Schedule的Fairness必須被考慮。Task對執行量的需求，可以使用weight=(E/P)來表示，其中P是Task的Period長度，Ｅ則是此Period長度中Task應該要執行的時間。給定每個Task一個Minimum Weight 和一個Maximum Weight，所謂的「Min-to-Max-Weight Fairness」就是在Multiprocessor的環境下，要求每個Task的執行時間，都得被限制在由此下限和上限所導出的範圍內。
在先前的相關研究中，「Min-to-Max-Weight Fairness」從未被明確定義過，也沒有Optimal（或Near-Optimal）的Scheduling Algorithm可以滿足它。此外，先前的研究成果也無法確保每個Task都有執行到其理想上限值的可能；也就是說，每個Task的實際可能最大執行時間，不但無法保證與其理想上限值相等，其間的差距也從未被明確定義過。
此篇論文明確定義了「Min-to-Max-Weight Fairness」，並針對之提出一個Near-Optimal的Scheduling Algorithm，同時提供相關的証明來佐證。此Algorithm包含了三個步驟：Imprecise Computation分解、Network Flow Graph轉換、Schedule建構；這三步驟能夠確保每個Task的實際可能最大執行時間，都會等於其理想上限值（或最多少1）。除了這個暫時性的「少1」情況，其餘所使用的方法都能夠Optimally地減少Task的實際執行時間與其理想上限值間的差距；因此，我們提出的Scheduling Algorithm不但能永遠滿足「Min-to-Max-Weight Fairness」，其與Optimal的情況相比，最多也只會差1而已。

Relate a task to a weight = (E/P), where E is its required execution during its period length P. Min-to-Max-Weight Fairness is a fair scheduling issue on multiprocessors, defining each task’s execution as a value within a specified range (implied by its minimum weight and maximum weight). In prior works, this problem was stated as a result rather than a specific problem to solve. No optimal (or near-optimal) scheduling algorithm for Min-to-Max-Weight Fairness was proposed, in the sense of minimizing the difference between tasks’ actual execution time and their maximum bounds. Moreover, tasks were not guaranteed to have the chance to reach their own maximum bounds. In other words, the utmost values of tasks’ actual execution were neither guaranteed nor defined with formal descriptions.
In this paper, we formally define Min-to-Max-Weight Fairness, propose a near-optimal scheduling algorithm for it, and provide related proofs. Our algorithm contains three steps: imprecise computation decomposition, network flow graph formulation, and schedule construction. Comprising these three steps, our algorithm always schedule tasks to satisfy Min-to-Max-Weight Fairness, and assure each task of a possible utmost execution time equal to (or 1 unit less than) its maximum bound. Despite this 1-unit limitation, our algorithm optimally minimizes the difference between tasks’ actual execution time and their maximum bounds. In conclusion, we provide a near-optimal algorithm with 1 unit difference at most.

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