# Rate-monotonic Scheduling - Introduction

Introduction

A simple version of rate-monotonic analysis assumes that threads have the following properties:

• No resource sharing (processes do not share resources, e.g. a hardware resource, a queue, or any kind of semaphore blocking or non-blocking (busy-waits))
• Deterministic deadlines are exactly equal to periods
• Static priorities (the task with the highest static priority that is runnable immediately preempts all other tasks)
• Static priorities assigned according to the rate monotonic conventions (tasks with shorter periods/deadlines are given higher priorities)
• Context switch times and other thread operations are free and have no impact on the model

It is a mathematical model that contains a calculated simulation of periods in a closed system, where round-robin and time-sharing schedulers fail to meet the scheduling needs otherwise. Rate monotonic scheduling looks at a run modeling of all threads in the system and determines how much time is needed to meet the guarantees for the set of threads in question.

Liu & Layland (1973) proved that for a set of n periodic tasks with unique periods, a feasible schedule that will always meet deadlines exists if the CPU utilization is below a specific bound (depending on the number of tasks). The schedulability test for RMS is:

where Ci is the computation time, Ti is the release period (with deadline one period later), and n is the number of processes to be scheduled. For example U ≤ 0.8284 for n = 2. When the number of processes tends towards infinity this expression will tend towards:

So a rough estimate is that RMS in the general case can meet all the deadlines if CPU utilization is 69.3%. The other 30.7% of the CPU can be dedicated to lower-priority non real-time tasks. It is known that a randomly generated periodic task system will meet all deadlines when the utilization is 85% or less, however this fact depends on knowing the exact task statistics (periods, deadlines) which cannot be guaranteed for all task sets.

The rate monotonic priority assignment is optimal meaning that if any static priority scheduling algorithm can meet all the deadlines, then the rate monotonic algorithm can too. The deadline-monotonic scheduling algorithm is also optimal with equal periods and deadlines, in fact in this case the algorithms are identical; in addition, deadline monotonic scheduling is optimal when deadlines are less than periods.

An optimal static-priority scheduling algorithm when deadlines are greater than periods is an open problem.

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