This paper addresses the scheduling of multiprogramming on a single processor in a hard-real-time environment, focusing on the characteristics of program functions that require guaranteed service. The authors present two scheduling algorithms: a fixed priority scheduler and a dynamic priority scheduler based on current deadlines. The fixed priority scheduler achieves processor utilization around 70%, while the dynamic priority scheduler can achieve full utilization. The paper also discusses a combination of these two algorithms. The authors derive theorems to determine the feasibility and optimal utilization of these scheduling algorithms, showing that the least upper bound to processor utilization for the fixed priority scheduler is approximately 70%, and for the dynamic priority scheduler, it can be as high as 100%. The paper concludes by discussing the implications of the assumptions made and the potential benefits of the proposed scheduling algorithms in real-time applications.This paper addresses the scheduling of multiprogramming on a single processor in a hard-real-time environment, focusing on the characteristics of program functions that require guaranteed service. The authors present two scheduling algorithms: a fixed priority scheduler and a dynamic priority scheduler based on current deadlines. The fixed priority scheduler achieves processor utilization around 70%, while the dynamic priority scheduler can achieve full utilization. The paper also discusses a combination of these two algorithms. The authors derive theorems to determine the feasibility and optimal utilization of these scheduling algorithms, showing that the least upper bound to processor utilization for the fixed priority scheduler is approximately 70%, and for the dynamic priority scheduler, it can be as high as 100%. The paper concludes by discussing the implications of the assumptions made and the potential benefits of the proposed scheduling algorithms in real-time applications.