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CprE 458/558: Real-Time Systems (G. Manimaran) 1

CprE 458/558: Real-Time Systems

Scheduling Results &

RMS and EDF Schedulers

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Understanding Fundamentals

Understanding the boundary betweenpolynomial and NP-complete problems

can provide insights into developing

useful heuristics.

Understanding the algorithms that

achieve some of the polynomial results

can again provide basis for developingheuristics.

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Understanding Fundamentals (cont.)

Understanding the fundamental

limitations of on-line algorithms will help

designers avoid scheduling anomalies

and misconceptions.

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Performance Metrics

Minimizing Schedule Length.

Minimizing Sum of Completion Times.

Maximizing Weighted Sum of Values (Useful

in RT systems).

Minimizing the Maximum Lateness (useful inRT systems).

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Uniprocessor - some results

One processor, Non-preemptive, Minimizingthe Max. Lateness (Polynomial).

One processor, Non-preemptive, release time

constraint, Minimizing the Max. Lateness (NP-

hard).

One processor, Preemptive, release timeconstraint, Minimizing the Max. Lateness

(Polynomial).

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Uniprocessor - more results

Result: When there are mutual exclusion

constraints, it is impossible to find a totally on-line optimal scheduler.

Result: The problem of deciding whether it is

possible to schedule a set of periodic tasksthat use semaphores only to enforce mutualexclusion in NP-hard.

Overload Result: There does not exist an on-

line scheduling algorithm with a competitivefactor greater than 0.25. (this is for generalcase: arbitrary number of processors).

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Multiprocessor Some Results

Result: The multiprocessor scheduling on P

processors with task preemption allowed and

with minimization of the number of late tasks

is NP-hard.

Result: For two or more processors, no

deadline scheduling algorithm can be optimal

without complete a prior knowledge of

deadlines, computation times, and task readytimes.

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Multiprocessor more results

EDF is not optimal in the multiprocessor case.

No on-line scheduling algorithm can guarantee

a cumulative value greater than one half for

the dual processor case. (A special case ofoverload result)

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Multiprocessor; Single Deadline;

Non-premptive

NP-completeness is mainly due to non-uniformtask execution time and resource constraints.

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Multiprocessing Anomalies

Assume that a set of tasks is optimallyschedulable on a multiprocessor with somepriority order, a fixed number of processors,fixed computation times, and precedenceconstraints.

Result: For the stated problem, changing thepriority list, increasing the number ofprocessors, reducing the computation times,or weakening the precedence constraints can

increase the schedule length.

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Multiprocessing Anomalies (cont.)

These anomalies may cause some of thealready guaranteed tasks to miss theirdeadlines.

It can be shown that run-time anomaliescannot occur in a multiprocessor schedulewhen the tasks are independent.

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Run-time Anomaly

Run-time anomaly may occur when the actualcomputation time of a task differs from itsworst case computation time in a non-preemptive multiprocessor schedule withresource constraints.

A processor is said to be work conserving if it isnever idle when there is a task to execute. Anywork conserving scheme may lead to run-time

anomaly.

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Run-time Anomaly Example

Example: Ti=(ai,ci,di)

T1=(0,20,22); T2=(0,12,25); T3=(10,8,26);T4=(8,10,30).

T3 and T4 have resource conflicts;

Actual computation time ofT1 is 10.

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Priority-drivenPreemptive Scheduling

Assumptions & Definitions

Tasks are periodic No aperiodic or sporadic tasks

Job (instance) deadline = end of period

No resource constraints

Tasks are preemptable

Laxity of a Task

Ti = di (t + c i)

where di: deadline;

t : current time;

ci : remaining computation time.

t

Ci

Laxity

di

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Rate Monotonic Scheduling (RMS)

Schedulability check (off-line)

- A set ofn tasks is schedulable on auniprocessor by the RMS algorithm if theprocessor utilization (utilization test):

The term n(21/n-1) approaches ln 2,(}0.69 as np g).

- This condition is sufficient, but not necessary.

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RMS (cont.)

Schedule construction (online)- Task with the smallest period is assigned the

highest priority.

- At any time, the highest priority task is

executed.

RMS is an optimal preemptive scheduling

algorithm with fixed priorities.Static/fixed priority algorithm assigns the same

priority to all the jobs (instances) in each task.

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RMS Scheduler -- Example 1

Task set: Ti = (ci, pi)

T1 = (2,4) and T2 = (1,8)

Schedulability check:

2/4 + 1/8 = 0.5 + 0.125 = 0.625 2(2 -1) = 0. 82

T11 T2

1 T12

0 2 3 4 6 8

ActiveTasks :

{T1, T2}

ActiveTasks :

{T2}

ActiveTasks :

{T1}

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RMS scheduler -- Example-2

Task set: Ti = (ci, pi)

T1 = (2,4) and T2 = (4,8)


2/4 + 4/8 = 0.5 + 0.5 = 1.0 > 2(2 -1) = 0. 82

T11 T2

1 T12

0 2 3 4 6 8

ActiveTasks :

{T1, T2}

ActiveTasks :

{T2}

ActiveTasks :

{T2, T1}

T21

ActiveTasks :

{T2}

Some task sets that FAIL the utilization-based schedulability test are alsoschedulable under RMS We need exact analysis (necessary & sufficient)

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Earliest Deadline First (EDF)

Schedulability check (off-line)

-A set ofn tasks is schedulable on a

uniprocessor by the EDF algorithm if the

processor utilization.

This condition is both necessary and sufficient.

- Least Laxity First (LLF) algorithm has the

same schedulability check.

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EDF/LLF (cont.)

Schedule construction (online)

EDF/LLF: Task with the smallestdeadline/laxity is assigned the highestpriority.

At any time, the highest priority task is

executed.

EDF/LLF is an optimal preemptive schedulingalgorithm with dynamic priorities.

Dynamic priority algorithm assigns differentpriorities to the individual jobs (instances) ineach task.

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EDF scheduler -- Example

Task set: Ti = (ci, pi, di)

T1 = (1,3,3) and T2 = (4,6,6)


1/3 + 4/6 = 0.33 + 0.67 = 1.0

T11 T2

1 T12

0 1 5 6

ActiveTasks :

{T1, T2}

ActiveTasks :

{T2}

ActiveTasks :

{T2, T1}

ActiveTasks :

{T1}

Unlike RMS, Only those task sets which pass the schedulability test areschedulable under EDF

3

T21

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RMS vs. EDF/LLF

RMS is an optimal preemptive scheduling

algorithm with fixed priorities.

EDF/LLF is an optimal preemptive schedulingalgorithm with dynamic priorities.

RMS schedulability properties can beanalyzed; rich theory exists and it is widelyused in practice.

EDF/LLF offers higher schedulability thanRMS, but it is more difficult to implement.

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RMS & EDF -- Example

0 5 10 15 20 25 30 35

0 7 14 21 28 35

T1

T2

RMS schedule

0 5 10 15 20 25 30 35

0 7 14 21 28 35

T1

T2

EDF schedule

Deadline miss

Process Period, T WCET, C

T1 5 2T2 7 4

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