rate monotonic analysis

Rate Monotonic Analysis

Rob Oshana

Southern Methodist University

Scheduling policies for real-time systems

Scheduling Policies in RT systems

• Two general categories– fixed or static scheduling policies– dynamic scheduling policies

• Many commercial RTOSs today support fixed priority scheduling policies

• Fixed priority scheduling algorithms do not modify a job’s priority while the task is running


– The task itself is allowed to modify its own priority for reasons

– approach requires very little support code on the scheduler to implement this functionality

– scheduler is fast and predictable with this approach

– scheduling is mostly done off-line (before the system runs)


• Requires system designer to know the task set a-priori (ahead of time)– not suitable for tasks that are created

dynamically during run time

• The priority of the task set must be determined beforehand and cannot change when the system runs unless the task itself changes its own priority


• Dynamic scheduling algorithms allow a scheduler to modify a jobs priority based on one of several scheduling algorithms or policies– more complicated approach and

requires more code in the scheduler to implement


– leads to more overhead in managing a task set

– scheduler must spend more time dynamically sorting through the system task set and prioritize tasks for execution based on the scheduling policy

– leads to non-determinism which is not favorable, especially for hard real-time systems


• Dynamic scheduling algorithms are on-line scheduling algorithms– scheduling policy is applied to the task

set during the execution of the system– active task set changes dynamically as

the system runs– priority of the tasks can also change

dynamically

Examples

• Static scheduling policies– rate monotonic scheduling– deadline monotonic scheduling

• Dynamic scheduling policies– earliest deadline first– least slack scheduling

Rate Monotonic Scheduling

• Optimal fixed priority policy– the higher the frequency (1/period) of a task,

the higher is its priority

• Approach can be implemented in any OS supporting the fixed priority preemptive scheme

• Rate monotonic scheduling assumes the deadline of a periodic task is the same as its period

Deadline Monotonic Scheduling

• Generalization of the Rate-Monotonic scheduling policy

• Deadline of a task is a fixed (relative) point in time from the beginning of the period

• The shorter this (fixed) deadline, the higher the priority

Earliest Deadline First• Dynamic priority preemptive policy• Deadline of a task instance is the absolute

point in time by which the instance must complete

• Task deadline is computed when the instance is created

• OS scheduler picks the task with the earliest deadline to run

• A task with an earlier deadline preempts a task with a later deadline

Least Slack• Dynamic priority preemptive policy• Slack of a task instance is the absolute

deadline minus the remaining execution time for the instance to complete

• OS scheduler picks the task with the shortest slack to run first

• Task with a smaller slack preempts a task with a larger slack

• This approach maximizes the minimum lateness of tasks

More on dynamic policies

• Priority of a task can change from instance to instance or within the execution of an instance

• Higher priority task preempts a lower priority task

• Very few commercial RTOS support such policies– systems that are hard to analyze for real-time

and determinism properties

Periodic tasks

• Many systems are multi-rate systems– multiple tasks in the system running at

different periodic rates

• Muli-rate systems can be managed using non-preemptive as well as preemptive scheduling techniques

• Non-preemptive techniques include using state machines as well as cyclic executives

Examples of periodic tasks

Audio sampling in hardware Audio sample processing Video capture and processing Feedback control (sensing and

processing) Navigation Temperature and speed monitoring

Scheduling periodic tasks

• Preemptive scheduling is an effective approach for scheduling real-time DSP systems– modularity simplifies the overall design

• Application can be viewed as a collection of independent tasks or jobs– complexity is reduced as the functionality

becomes encapsulated into a set of well defined tasks


• Systems designed using preemptive scheduling are also more maintainable– issue of changes to one task in the

system affecting other jobs in the system is removed

– New functionality can easily be added by adding a new task


• Preemptive scheduling approach also makes the system more efficient– preemptive scheduling is more efficient

at utilizing time slots that may not be fully utilized

• Scheduling algorithms– rate monotonic scheduling– deadline monotonic scheduling

cost of handling event C = 4

periodic arrivals. period T = 10

---- 10 ----

4 4 4

System Utilization = C/T = .40

Periodic Arrivals with Fixed Cost of Processing

System will be able to meet all deadlines. It can finish processing arrivals before the next arrival occurs.

1. periodic arrival, period T = 10 and C=4

2. periodic arrival, T=10 and C=3 ??

---- 10 ----

4 4 4

Can a second periodic event be accommodated?



---- 10 ----

4 4 4

System Utilization C/T = .70

Can a second periodic event be accommodated?



---- 10 ----

4 4 4

How about 2nd periodic event with T=6 and C=3?



---- 10 ----

4 4 4

System Utilization C/T = .90

How about 2nd periodic event with T=6 and C=3?

---- 10 ----

4 4 4

--6--

If we process Event #1 before Event #2 then,

2nd event processing will not complete before the next comparable event occurs

Can’t Meet Deadline!

Event #1

Event #2

Task #1

Task #2

---- 10 ----

4

--6--

Event #1

Event #2

Try Event #2 before Event #1-

We still cannot complete task 1 before the next task 2 event occurs at t=6

unless...

Task #1

Task #2

---- 10 ----

4

--6--

Event #1

Event #2



unless…we Interrupt task 1

Task #1

Task #2

---- 10 ----

4

--6--

Event #1

Event #2



unless…we Interrupt task 1

Giving event #2 priority means that we can meet our deadline IF we preempt the processing of event #1 when event #2 occurs

Task #1

Task #2


• Assume a set of “n” periodic tasks– period Ti– worst case execution time Ci

• Rate-monotonic priority assignment– task with a shorter period (higher rate)

assigned a fixed higher priority


• Rate Monotonic scheduling addresses how to determine whether a group of tasks, whose individual CPU utilization is known, will meet their deadlines– assumes a priority preemption

scheduling algorithm– assumes independent tasks (no

communication or synchronization)


– restriction of no communication or synchronization may appear to be unrealistic, but there are techniques for dealing with this

– Each task is a periodic task which has a period T, which is the frequency with which it executes


An execution time C, which is the CPU time required during the period

A utilization U, which is the ratio C/T• A task is schedulable if all its deadlines

are met (i.e., the task completes its execution before its period elapses.)– A group of tasks is considered to be

schedulable if each task can meet its deadlines


• RMA is a mathematical solution to the scheduling problem for periodic tasks with known cost– assumption is that the total utilization

must always be less than or equal to 100%

• Any more and you are exceeding the capacity of the CPU

• Are you asking for more computing power than you have? IF so, forget it!


• For a set of independent periodic tasks, the rate monotonic algorithm assigns each task a fixed priority based on its period, such that the shorter the period of a task, the higher the priority

Rate Monotonic Analysis• For three tasks T1, T2, and T3 with periods

of 5, 15 and 40 msec respectively the highest priority is given to the task, T1, as it has the shortest period, the medium priority to task T2, and the lowest priority to task T3– priority assignment is independent of the

applications “priority” i.e. how important meeting this deadline is to the functioning of the system or user concerns

Rate Monotonic Analysis• A mathematical solution to the scheduling

problem for Periodic Tasks with known Cost

• Tasks will have:– Cost (Time to complete a task)– Period (Time between events)– Utilization ( Cost/Period)

• Assumption– Total Utilization must always be <= 100%

3 levels of analysis using RMA

• Utilization bound test

• Completion time test

• Response time test

Utilization bound test

• If this simple rule is followed, then all tasks are guaranteed to meet their requirements if the following holds true;

)()12(/..../ /111 nUnTCTC n

nn where and are the execution time andperiod of task , respectively.

iCiT

it


• In this rule, the bound is 1.0 for harmonic task sets

• A task set is said to be harmonic if the periods of all its tasks are either integral multiples or sub-multiples of one another– On the average, for random Cs and Ts,

this number will be about 0.88.


• Theory is a worst case approximation

• For a randomly chosen group of tasks, it has been shown that the likely upper bound is 88%– Harmonic periods can give even higher

upper bounds– The algorithm is stable in conditions

where there is a transient overload


• In this case, there is a subset of the total number of tasks, namely those with the highest priorities that will still meet their deadlines

Example of UB test

Task t1: C1=20; T1= 100; U1 = .2 Task t2: C2=30; T2= 150; U2 = .2 Task t3: C3=60; T3= 200; U3 = .3

– The total utilization for this task set is .2 + .2 + .3 = .7. Since this is less than the 0.779 utilization bound for this task set, all deadlines will be met.

ExampleCan these 4 tasks be

scheduled?

– Can the system run and meet all hard deadlines?

Task Ci Ti Ui

1 3 10 .302 3 12 .253 4 16 .254 7 20 .35

ExampleExampleCan these 4 tasks be Can these 4 tasks be

scheduled?scheduled?

– Can the system run and meet all hard deadlines?

– NO! The Total Utilization = 115%

Task Ci Ti Ui

1 3 10 .302 3 12 .253 4 16 .254 7 20 .35

Example

Can these tasks always meet their deadlines?Total Utilization = 80%It MAY be possible - Rate Monotonic Scheduling applies!

Task Ci Ti Ui

1 6 20 .302 4 16 .253 3 12 .25

Rate Monotonic Theorem• For PERIODIC Tasks• Most frequent task gets highest

priority• THEOREM (Simple Version)

– IF the utilization of all tasks is less than or equal to 69%, then all tasks will ALWAYS meet their deadlines

Are These Tasks Schedulable?

Task Ci Ti Ui

1 2 20 .102 4 16 .253 3 12 .254 1 20 .05

Are These Tasks Are These Tasks Schedulable?Schedulable?

Task Ci Ti Ui

1 2 20 .102 4 16 .253 3 12 .254 1 20 .05

Yes. Total CPU Utilization is 65% < 69%


Task Ci Ti Ui

1 2 20 .102 4 16 .253 3 12 .254 3 20 .15


Task Ci Ti Ui

1 2 20 .102 4 16 .253 3 12 .254 1 20 .05

Total CPU Utilization is 65%

???

Priority Inversion

Taskh

Taskmed

Tasklow

Normalexecution

Execution incritical section

Priorityinversion

Unbounded Priority InversionUnbounded Priority Inversion

Taskh

Taskmed

Tasklow

Normalexecution

Execution incritical section

Priorityinversion

Taskmed

Uncontrolled priority inversion

Motor control example

• Single DSP will be used to control a motor• DSP will also be responsible for

interfacing to an operator using a keypad, updating a simple display device, and sending data out one of the DSP ports

• The operator uses the keypad to control the system


• The motor speed must be sampled at a 1 kHz rate

• A timer will be used to interrupt processing at this rate to allow the DSP to execute some simple motor control algorithms

• At each interrupt, the DSP will read the motor speed, run some simple calculations, and adjust the speed of the motor accordingly


• Diagnostic data is transmitted out the RS232 port when selected by the operator using the keypad

Simple motor control

Motor Drive

Motor Tach

RS232 Data Out


• The first step in developing a multitasking system is to architect the application into isolated independent execution threads– tools available to help the system designer

during this phase

• This architecture definition will produce data flow diagrams, system block diagrams, or finite state machines


• Four independent threads in this design;– main motor control algorithm which is a

periodic task, running at a 1kHz rate– keypad control thread which is an

aperiodic task controlled by the operator

– display update thread which is a periodic task executing at a 2 Hz rate


– data output thread which runs as a background task and outputting data when there is no other processing required

Requirements

RequirementsControl motor speed (1 Khz sampling rate – dV/dT)Accept keyboard commands to control the motor, change the display, or send data out the

RS232 portDrive a simple display and refresh 2 times per secondSend data out the RS232 port when there is nothing else to do

Independent execution threads

ADC DACMotorDrive

Controlalgorithm

MotorTach

Motorcontrol

Keypadcontrol

Displaycontrol

Remoteoutput

SystemControl algorithm

Keypadcontrol

Display interfaceDisplayalgorithm

RS-232Remotealgorithm McBSP


• Relative priority of the threads must now be determined

• Since this motor control example is a real-time system (there are hard real-time deadlines for critical operations to complete), there must be a priority assigned to the thread execution

Motor control example• One hard real-time thread

– motor control algorithm– must execute at a 1 kHz rate

• Soft real-time tasks in the system as well– display update at a two hertz rate is a soft-real

time task (this is a soft real-time task because although the display update is a requirement, the system will not fail if the display is not updated precisely twice per second.)


– keypad control is also a soft real-time task but since it is the primary control input, it should have a higher priority than the display update

– remote output thread is a background task that will run when no other task is running


• Motor control system will be designed to use a hardware interrupt to control the motor control thread– Interrupts have fast context switching

times (faster than that of thread context switch) and can be generated from the timer on the DSP

Assignment of priorities

Task Rate Priority Periodic oraperiodic

Activationmechanism

Motor control 1 kHz 1 Periodic Hardware InterruptKeypadcontrol

5 hertz 2 Aperiodic Hardware Interrupt

Displaycontrol

2 hertz 3 Periodic Software interrupt

Remote outputl

Background 4 Aperiodic Idle loop (runscontinuously in the

background)


• Example of a rate monotonic priority assignment; – the shorter the period of the thread, the higher

the priority

• Along with the priority, the activation mechanism is described

• The highest priority motor control thread will use a hardware interrupt to trigger execution


• Hardware interrupts are the highest priority scheduling mechanism in most real time operating systems

• The keypad control function is an interface to the environment (operator console) and will use a hardware interrupt to signal the keypad control actions– This priority will be at a lower priority than the

motor control interrupt


• Display control thread will use a software interrupt to schedule a display update at a two hertz rate– Software interrupts operate similar to

hardware interrupts but at a lower priority than hardware interrupts (but at a higher priority than threads)

• Lowest priority task, the remote output task, will be executed as an idle loop


– idle loop will runs continuously in the background while there are no other higher priority threads to run

• The highest priority thread, the motor control thread, is a periodic thread– Like many DSP applications this thread

processes data periodically, in this case at a 1 kHz rate


• This motor control example is actually a multi-rate system– multiple periodic operations in the

system (motor control and display control, for example)

– threads operate at different rates– DSP RTOSs allow for multiple periodic

threads to run


• For each of these threads, the DSP system designer must determine the period the specific operation must run and the time required to complete the operation

• The DSP developer will then program the DSP timer functions in such a way to produce an interrupt or other scheduling approach to enable the thread to run at the desired rate


• Most DSP RTOSs have a standard clock manager and API function to perform this setup operation

Periodic threads

DSPtimer

UserDefinedclocking

Selectablesource

Clock source

PeriodicFunctionmanager

Motor controlDSP function

Display controlDSP function

Periodic clock Periodic functions

tick

tick

n ticks

m ticks

What Happened on Mars ?

What happened on Mars ?

• Mars pathfinder “flawless” in early days of mission– unconventional landing with airbags– deployment of Sojourner rover– gathering and transmitting data back to earth

• A few days into the mission the Pathfinder began experiencing total system resets, each including losses of data

What happened on Mars ?

• Press reported these as “software glitches”

• VxWorks RTOS provides preemptive priority scheduling of tasks– tasks executed as threads– priorities assigned reflecting relative

urgency of the tasks

What happened on Mars ?What happened on Mars ?

busmanagementtask

information bus

mutex

meteorologicaldatagatheringtask

communicationtask

high priority -frequent execution

low priority -infrequent execution

medium priority

• Combination worked fine most of the time• Possible for interrupt to occur that caused

the long running medium priority task to be scheduled during the short interval while the high priority task was blocked waiting on the semaphore that the low priority task had.


• Watchdog timer would go off, notice data bus task not in use for some time, conclude that something bad went wrong, and initiate a total system reset

• Classic case of priority inversion Classic case of priority inversion


How was this debugged ?

• VxWorks can run in trace mode, recording interesting events.

• JPL engineers spent hours in lab trying to reproduce the problem on the ground.

• When finally reproduced, the trace data indicated the priority inversion problem

How was this problem corrected?

• Mutex object accepts boolean parameter indicating whether priority inheritance should be used

• Initialized with parameter off– if on, the low-pri thread would have

inherited the pri of the high-pri thread– medium pri thread would never have

been executed

How was this problem corrected?

• VxWorks has a C language interpreter that allows C commands to be executed on the fly

• JPL engineers left this in the software• Changed global variables by uploading a

short program to the spacecraft• No more system resets occurred after re-

programming!

Analysis and Lessons

• Diagnosing this problem as a black box would have been impossible– trace data was required

• Leaving debugging facilities in the system saved the day

• Time critical situations requires additional correctness measures even at the expense of some performance

Human nature, Deadline Pressures

• One or two system resets had occurred on the ground prior to launch

• Never reproducable or explainable• “it was probably caused by a hardware

glitch”• Engineer focus caused part of the problem

– extremely focused on ensuring quality and flawless operation of landing software

– the occasional glitch was dismissed

Importance of good Theory/Algorithms

• Some of the heros were people from CMU who published a paper years ago on the priority inversion problem– “An Approach to Real-Time

Synchronization” IEEE Transaction on Computers, Vol39, pp1175-1185, September 1990

rate monotonic analysis

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