Rate-monotonic scheduling

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In computer science, rate-monotonic scheduling ^[1] is an optimal, preemptive, static-priority, scheduling algorithm used in real-time operating systems. The inputs to the algorithm are processes (tasks, threads) with:

No resource sharing (processes do not share resources, e.g. a hardware resource, a queue, or a semaphore)
Deterministic deadlines exactly equal to periods
Static priorities (whenever a processor is free or a new task period begins, the task with the highest static priority is selected to preempt all other tasks)
Static priorities assigned according to the rate monotonic principle (tasks with shorter periods/deadlines are given higher priorities)

It is a statistical model. which contains a record of events. It is used in a closed environment where only a predictable number of inputs can come in. In real time systems Round robin and time sharing do not work. Whereas, Real-Monotonic Scheduling looks at the History and determines how much time a particular process generally takes , and the time of the day when these processes are expected to enter the CPU.

In 1973, Liu and Layland 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:

$U = \sum_{i=1}^{n} C_i / T_i \leq n(\sqrt[n]{2} - 1)$

Where $C i$ is the computation time, and $T i$ is the release period (with deadline one period later.) For example $U \leq 0.8284$ for $n = 2\,$ . When the number of processes tends towards infinity this expression will tend towards:

$\lim_{n \rightarrow \infty} n(\sqrt[n]{2} - 1) = \ln 2 \approx 0.693147\ldots$

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 ^[2], however this fact depends on knowing the exact task statistics (periods, deadlines) and 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 in the situation where periods and deadlines are identical, in fact in this case the algorithms are identical; in addition, deadline monotonic scheduling is also optimal when deadlines are less than periods ^[3].

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

1 Resource preemption, priority inheritance
2 Example
3 See also
4 References
5 External links

[edit] Resource preemption, priority inheritance

In many practical applications, resources are shared and the unmodified RMS will be subject to priority inversion and deadlock hazards. In practice, this is solved by introducing priority inheritance.

Examples of priority inheritance algorithms include (from simplest to most complex):

The OSIntEnter() and OSIntExit() primitives that lock CPU interrupts in a real-time kernel (uC-OS II kernel),
The splx() family of primitives which nest the locking of device interrupts (Linux kernel),
The Highest Locker protocol which requires off-line analysis of all task conflicts to find a "ceiling priority" (see below),
The Basic Priority Inheritance Protocol ^[4] which waits until a high priority task requests a lock held by a low-priority task, and then boosts the low priority task priority up to the high priority level until the lock is released, and
The Priority Ceiling Protocol ^[5] which is an enhancement of Basic Priority Inheritance which assigns a "ceiling priority" to each semaphore, which is the priority of the highest job that will ever access that semaphore. A job then cannot preempt a lower priority critical section if its priority is lower than the ceiling priority for that section.

Priority inheritance algorithms can be characterized by two parameters. First, is the inheritance lazy (only when essential) or immediate (boost priority before there is a conflict). Second is the inheritance optimistic (boost a minimum amount) or pessimistic (boost by more than the minimum amount):

	pessimistic	optimistic
immediate	`OSIntEnter/Exit()`	`splx()`, Highest Locker
lazy		Priority Ceiling Protocol, Basic Priority Inheritance Protocol

In practice there is no mathematical difference (in terms of the Liu-Layland system utilization bound) between the lazy and immediate algorithms, and the immediate algorithms are more efficient to implement, and so they are the ones used by most practical systems.

The Basic Priority Inheritance protocol can produce chained blocking, i.e. an almost arbitrarily long delay from the time a high priority task requests a critical section, until it is served. The other protocols guarantee that at most one lower priority critical section must finish before the high priority task gets its critical section.

The Priority Ceiling Protocol is available in the VxWorks real-time kernel but is seldom enabled.

An example of usage of the Basic Priority Inheritance is related to the "Mars Pathfinder Bug" which was fixed on Mars by changing the creation flags for the semaphore so as to enable the priority inheritance.

[edit] Example

Process	Execution Time	Period
P1	1	8
P2	2	5
P3	2	10

The utilization will be:

$\frac{1}{8} + \frac{2}{5} + \frac{2}{10} = 0.725$

The theoretical limit for $3\,$ processes will be:

$U = 3(2^\frac{1}{3} - 1) = 0.77976\ldots$

Since $0.725 < 0.77976\ldots$ the system is schedulable!

[edit] See also

Dynamic priority scheduling
Earliest deadline first scheduling
RTEMS, an open source real-time operating containing a working Rate Monotonic Scheduler.
Scheduling

[edit] References

↑ C. L. Liu and J. Layland. Scheduling algorithms for multiprogramming in a hard real-time environment, Journal of the ACM, 10(1), 1973.
M. Joseph and P. Pandya. Finding Response times in Real-time systems, BCS Computer Journal, 29(5), pp. 390-395, October 1986.
↑ J. Lehoczky, L. Sha and Y. Ding, The Rate monotonic scheduling algorithm: exact characterization and average case behavior, IEEE Real-Time Systems Symposium, pp. 166-171, December 1989.
↑ J. Y. Leung and J. Whitehead. On the complexity of fixed-priority scheduling of periodic, real-time tasks. Performance Evaluation, 2(4):237--250, December 1982.
↑ B.W. Lampson, and D. D. Redell. Experience with Processes and Monitors in Mesa. Communications of the ACM, Vol. 23, No. 2 (Feb 1980), pp. 105-117.
↑ L. Sha, R. Rajkumar and J. P. Lehoczky, Priority inheritance protocols: an approach to real-time synchronization, IEEE Transactions on Computers, vol. 39 no. 9, September 1990, pp. 1175-1185.