This paper presents scheduling algorithms for multiprogramming in a hard-real-time environment. It discusses two main scheduling techniques: fixed priority scheduling and dynamic priority scheduling. The fixed priority scheduler has an upper bound on processor utilization, which can be as low as 70% for large task sets. In contrast, dynamic scheduling can achieve full processor utilization by dynamically assigning priorities based on deadlines. A combination of these two techniques is also explored. The paper begins by introducing the problem of scheduling time-critical tasks in a hard-real-time environment, where tasks must be completed within strict deadlines. It then reviews existing literature on scheduling and discusses the characteristics of hard-real-time systems. The paper defines key assumptions about the environment, including periodic task requests, deadlines, and constant run-times. The paper then presents a fixed priority scheduling algorithm, which assigns priorities to tasks based on their request rates. It shows that the rate-monotonic priority assignment is optimal for fixed priority scheduling. The paper also derives an upper bound on processor utilization for fixed priority scheduling, which is approximately 70% for large task sets. The paper then presents a dynamic scheduling algorithm, which assigns priorities based on deadlines. This algorithm is shown to be globally optimal and capable of achieving full processor utilization. The paper also discusses a mixed scheduling algorithm that combines fixed priority and dynamic scheduling techniques. This algorithm is shown to be effective for many applications. The paper concludes by comparing the two scheduling techniques and discussing their implications for real-time systems. It emphasizes the importance of scheduling algorithms in ensuring the timely execution of tasks in hard-real-time environments. The paper also highlights the need for further research into scheduling algorithms for real-time systems.This paper presents scheduling algorithms for multiprogramming in a hard-real-time environment. It discusses two main scheduling techniques: fixed priority scheduling and dynamic priority scheduling. The fixed priority scheduler has an upper bound on processor utilization, which can be as low as 70% for large task sets. In contrast, dynamic scheduling can achieve full processor utilization by dynamically assigning priorities based on deadlines. A combination of these two techniques is also explored. The paper begins by introducing the problem of scheduling time-critical tasks in a hard-real-time environment, where tasks must be completed within strict deadlines. It then reviews existing literature on scheduling and discusses the characteristics of hard-real-time systems. The paper defines key assumptions about the environment, including periodic task requests, deadlines, and constant run-times. The paper then presents a fixed priority scheduling algorithm, which assigns priorities to tasks based on their request rates. It shows that the rate-monotonic priority assignment is optimal for fixed priority scheduling. The paper also derives an upper bound on processor utilization for fixed priority scheduling, which is approximately 70% for large task sets. The paper then presents a dynamic scheduling algorithm, which assigns priorities based on deadlines. This algorithm is shown to be globally optimal and capable of achieving full processor utilization. The paper also discusses a mixed scheduling algorithm that combines fixed priority and dynamic scheduling techniques. This algorithm is shown to be effective for many applications. The paper concludes by comparing the two scheduling techniques and discussing their implications for real-time systems. It emphasizes the importance of scheduling algorithms in ensuring the timely execution of tasks in hard-real-time environments. The paper also highlights the need for further research into scheduling algorithms for real-time systems.