11.2. Background¶

The rate monotonic manager provides facilities to manage the execution of periodic tasks. This manager was designed to support application designers who utilize the Rate Monotonic Scheduling Algorithm (RMS) to ensure that their periodic tasks will meet their deadlines, even under transient overload conditions. Although designed for hard real-time systems, the services provided by the rate monotonic manager may be used by any application which requires periodic tasks.

11.2.1. Rate Monotonic Manager Required Support¶

A clock tick is required to support the functionality provided by this manager.

11.2.2. Period Statistics¶

This manager maintains a set of statistics on each period object. These statistics are reset implictly at period creation time and may be reset or obtained at any time by the application. The following is a list of the information kept:

owner: is the id of the thread that owns this period.
count: is the total number of periods executed.
missed_count: is the number of periods that were missed.
min_cpu_time: is the minimum amount of CPU execution time consumed on any execution of the periodic loop.
max_cpu_time: is the maximum amount of CPU execution time consumed on any execution of the periodic loop.
total_cpu_time: is the total amount of CPU execution time consumed by executions of the periodic loop.
min_wall_time: is the minimum amount of wall time that passed on any execution of the periodic loop.
max_wall_time: is the maximum amount of wall time that passed on any execution of the periodic loop.
total_wall_time: is the total amount of wall time that passed during executions of the periodic loop.

Each period is divided into two consecutive phases. The period starts with the active phase of the task and is followed by the inactive phase of the task. In the inactive phase the task is blocked and waits for the start of the next period. The inactive phase is skipped in case of a period miss. The wall time includes the time during the active phase of the task on which the task is not executing on a processor. The task is either blocked (for example it waits for a resource) or a higher priority tasks executes, thus preventing it from executing. In case the wall time exceeds the period time, then this is a period miss. The gap between the wall time and the period time is the margin between a period miss or success.

The period statistics information is inexpensive to maintain and can provide very useful insights into the execution characteristics of a periodic task loop. But it is just information. The period statistics reported must be analyzed by the user in terms of what the applications is. For example, in an application where priorities are assigned by the Rate Monotonic Algorithm, it would be very undesirable for high priority (i.e. frequency) tasks to miss their period. Similarly, in nearly any application, if a task were supposed to execute its periodic loop every 10 milliseconds and it averaged 11 milliseconds, then application requirements are not being met.

The information reported can be used to determine the “hot spots” in the application. Given a period’s id, the user can determine the length of that period. From that information and the CPU usage, the user can calculate the percentage of CPU time consumed by that periodic task. For example, a task executing for 20 milliseconds every 200 milliseconds is consuming 10 percent of the processor’s execution time. This is usually enough to make it a good candidate for optimization.

However, execution time alone is not enough to gauge the value of optimizing a particular task. It is more important to optimize a task executing 2 millisecond every 10 milliseconds (20 percent of the CPU) than one executing 10 milliseconds every 100 (10 percent of the CPU). As a general rule of thumb, the higher frequency at which a task executes, the more important it is to optimize that task.

11.2.3. Periodicity Definitions¶

A periodic task is one which must be executed at a regular interval. The interval between successive iterations of the task is referred to as its period. Periodic tasks can be characterized by the length of their period and execution time. The period and execution time of a task can be used to determine the processor utilization for that task. Processor utilization is the percentage of processor time used and can be calculated on a per-task or system-wide basis. Typically, the task’s worst-case execution time will be less than its period. For example, a periodic task’s requirements may state that it should execute for 10 milliseconds every 100 milliseconds. Although the execution time may be the average, worst, or best case, the worst-case execution time is more appropriate for use when analyzing system behavior under transient overload conditions.

In contrast, an aperiodic task executes at irregular intervals and has only a soft deadline. In other words, the deadlines for aperiodic tasks are not rigid, but adequate response times are desirable. For example, an aperiodic task may process user input from a terminal.

Finally, a sporadic task is an aperiodic task with a hard deadline and minimum interarrival time. The minimum interarrival time is the minimum period of time which exists between successive iterations of the task. For example, a sporadic task could be used to process the pressing of a fire button on a joystick. The mechanical action of the fire button ensures a minimum time period between successive activations, but the missile must be launched by a hard deadline.

11.2.4. Rate Monotonic Scheduling Algorithm¶

The Rate Monotonic Scheduling Algorithm (RMS) is important to real-time systems designers because it allows one to sufficiently guarantee that a set of tasks is schedulable (see [LL73], [LSD89], [SG90], [Bur91]).

A set of tasks is said to be schedulable if all of the tasks can meet their deadlines. RMS provides a set of rules which can be used to perform a guaranteed schedulability analysis for a task set. This analysis determines whether a task set is schedulable under worst-case conditions and emphasizes the predictability of the system’s behavior. It has been proven that:

RMS is optimal in the sense that if a set of tasks can be scheduled by any fixed-priority algorithm, then RMS will be able to schedule that task set. RMS bases it schedulability analysis on the processor utilization level below which all deadlines can be met.

RMS calls for the static assignment of task priorities based upon their period. The shorter a task’s period, the higher its priority. For example, a task with a 1 millisecond period has higher priority than a task with a 100 millisecond period. If two tasks have the same period, then RMS does not distinguish between the tasks. However, RTEMS specifies that when given tasks of equal priority, the task which has been ready longest will execute first. RMS’s priority assignment scheme does not provide one with exact numeric values for task priorities. For example, consider the following task set and priority assignments:

Task	Period (in milliseconds)	Priority
1	100	Low
2	50	Medium
3	50	Medium
4	25	High

RMS only calls for task 1 to have the lowest priority, task 4 to have the highest priority, and tasks 2 and 3 to have an equal priority between that of tasks 1 and 4. The actual RTEMS priorities assigned to the tasks must only adhere to those guidelines.

Many applications have tasks with both hard and soft deadlines. The tasks with hard deadlines are typically referred to as the critical task set, with the soft deadline tasks being the non-critical task set. The critical task set can be scheduled using RMS, with the non-critical tasks not executing under transient overload, by simply assigning priorities such that the lowest priority critical task (i.e. longest period) has a higher priority than the highest priority non-critical task. Although RMS may be used to assign priorities to the non-critical tasks, it is not necessary. In this instance, schedulability is only guaranteed for the critical task set.

11.2.5. Schedulability Analysis¶

RMS allows application designers to ensure that tasks can meet all deadlines under fixed-priority assignment, even under transient overload, without knowing exactly when any given task will execute by applying proven schedulability analysis rules.

11.2.5.1. Assumptions¶

The schedulability analysis rules for RMS were developed based on the following assumptions:

The requests for all tasks for which hard deadlines exist are periodic, with a constant interval between requests.
Each task must complete before the next request for it occurs.
The tasks are independent in that a task does not depend on the initiation or completion of requests for other tasks.
The execution time for each task without preemption or interruption is constant and does not vary.
Any non-periodic tasks in the system are special. These tasks should not displace periodic tasks while executing and do not have hard, critical deadlines.

Once the basic schedulability analysis is understood, some of the above assumptions can be relaxed and the side-effects accounted for.

11.2.5.2. Processor Utilization Rule¶

The Processor Utilization Rule requires that processor utilization be calculated based upon the period and execution time of each task. The fraction of processor time spent executing task index is Time(i) / Period(i). The processor utilization can be calculated as follows where n is the number of tasks in the set being analyzed:

\[Utilization = \sum_{i=1}^{n} Time_i/Period_i\]

To ensure schedulability even under transient overload, the processor utilization must adhere to the following rule:

\[maximumUtilization = n * (2^{\frac{1}{n}} - 1)\]

As the number of tasks increases, the above formula approaches ln(2) for a worst-case utilization factor of approximately 0.693. Many tasks sets can be scheduled with a greater utilization factor. In fact, the average processor utilization threshold for a randomly generated task set is approximately 0.88. See more detail in [LL73].

11.2.5.3. Processor Utilization Rule Example¶

This example illustrates the application of the Processor Utilization Rule to an application with three critical periodic tasks. The following table details the RMS priority, period, execution time, and processor utilization for each task:

Task	RMS Priority	Period	Execution Time	Processor Utilization
1	High	100	15	0.15
2	Medium	200	50	0.25
3	Low	300	100	0.33

The total processor utilization for this task set is 0.73 which is below the upper bound of 3 * (2**(1/3) - 1), or 0.779, imposed by the Processor Utilization Rule. Therefore, this task set is guaranteed to be schedulable using RMS.

11.2.5.4. First Deadline Rule¶

If a given set of tasks do exceed the processor utilization upper limit imposed by the Processor Utilization Rule, they can still be guaranteed to meet all their deadlines by application of the First Deadline Rule. This rule can be stated as follows:

For a given set of independent periodic tasks, if each task meets its first deadline when all tasks are started at the same time, then the deadlines will always be met for any combination of start times.

A key point with this rule is that ALL periodic tasks are assumed to start at the exact same instant in time. Although this assumption may seem to be invalid, RTEMS makes it quite easy to ensure. By having a non-preemptible user initialization task, all application tasks, regardless of priority, can be created and started before the initialization deletes itself. This technique ensures that all tasks begin to compete for execution time at the same instant - when the user initialization task deletes itself. See more detail in [LSD89].

11.2.5.5. First Deadline Rule Example¶

The First Deadline Rule can ensure schedulability even when the Processor Utilization Rule fails. The example below is a modification of the Processor Utilization Rule example where task execution time has been increased from 15 to 25 units. The following table details the RMS priority, period, execution time, and processor utilization for each task:

Task	RMS Priority	Period	Execution Time	Processor Utilization
1	High	100	25	0.25
2	Medium	200	50	0.25
3	Low	300	100	0.33

The total processor utilization for the modified task set is 0.83 which is above the upper bound of 3 * (2**(1/3) - 1), or 0.779, imposed by the Processor Utilization Rule. Therefore, this task set is not guaranteed to be schedulable using RMS. However, the First Deadline Rule can guarantee the schedulability of this task set. This rule calls for one to examine each occurrence of deadline until either all tasks have met their deadline or one task failed to meet its first deadline. The following table details the time of each deadline occurrence, the maximum number of times each task may have run, the total execution time, and whether all the deadlines have been met:

Deadline Time	Task 1	Task 2	Task 3	Total Execution Time	All Deadlines Met?
100	1	1	1	25 + 50 + 100 = 175	NO
200	2	1	1	50 + 50 + 100 = 200	YES

The key to this analysis is to recognize when each task will execute. For example at time 100, task 1 must have met its first deadline, but tasks 2 and 3 may also have begun execution. In this example, at time 100 tasks 1 and 2 have completed execution and thus have met their first deadline. Tasks 1 and 2 have used (25 + 50) = 75 time units, leaving (100 - 75) = 25 time units for task 3 to begin. Because task 3 takes 100 ticks to execute, it will not have completed execution at time 100. Thus at time 100, all of the tasks except task 3 have met their first deadline.

At time 200, task 1 must have met its second deadline and task 2 its first deadline. As a result, of the first 200 time units, task 1 uses (2 * 25) = 50 and task 2 uses 50, leaving (200 - 100) time units for task 3. Task 3 requires 100 time units to execute, thus it will have completed execution at time 200. Thus, all of the tasks have met their first deadlines at time 200, and the task set is schedulable using the First Deadline Rule.

11.2.5.6. Relaxation of Assumptions¶

The assumptions used to develop the RMS schedulability rules are uncommon in most real-time systems. For example, it was assumed that tasks have constant unvarying execution time. It is possible to relax this assumption, simply by using the worst-case execution time of each task.

Another assumption is that the tasks are independent. This means that the tasks do not wait for one another or contend for resources. This assumption can be relaxed by accounting for the amount of time a task spends waiting to acquire resources. Similarly, each task’s execution time must account for any I/O performed and any RTEMS directive calls.

In addition, the assumptions did not account for the time spent executing interrupt service routines. This can be accounted for by including all the processor utilization by interrupt service routines in the utilization calculation. Similarly, one should also account for the impact of delays in accessing local memory caused by direct memory access and other processors accessing local dual-ported memory.

The assumption that nonperiodic tasks are used only for initialization or failure-recovery can be relaxed by placing all periodic tasks in the critical task set. This task set can be scheduled and analyzed using RMS. All nonperiodic tasks are placed in the non-critical task set. Although the critical task set can be guaranteed to execute even under transient overload, the non-critical task set is not guaranteed to execute.

In conclusion, the application designer must be fully cognizant of the system and its run-time behavior when performing schedulability analysis for a system using RMS. Every hardware and software factor which impacts the execution time of each task must be accounted for in the schedulability analysis.