Optimal and Adaptive Multiprocessor

Real-Time Scheduling:The Quasi-Partitioned Approach

Γ = { { E.Massa, G.Lima, P.Regnier }, { G.Levin, S.Brandt } }

Real-Time Systems Andrei Petrov

Padua, October 13th, 2015

Outline

+ Motivation + QPS Algorithm+ Evaluation+ Conclusion

2

Introduction

3

QPS#optimal

#partitioned proportionate fairness#preserves space & time locality of tasks

#gentle off-line bin-packing

Motivation

4

QPS#on-line adaptation

#efficient switch mechanism P-EDF <--> G-Scheduling Rules#low preemption and migration overhead

QPS Algorithm

5

Off-line phase On-line phase

❏ Major/minor execution partitions

❏ Allocates processors to execution sets

❏ Generates the schedule❏ Manages the

activation/deactivation of QPS servers

Quasi-Partition cornerstone in dealing

with sporadic tasks

QPS Algorithm /1

6

System Model❏ Γ = {

i:(R

i, D

i) } for i=1,..,n

sporadic set ❏ independent, implicit-

deadlines tasks❏ Ri≤ 1 execution rate

❏ Di infinite set of deadlines

❏ m identical processors❏ migrations and preemptions

allowed

❏ A job J:(r,c,d)∈i

❏ J.r∈{ 0 U Di }

❏ J.d = min{d∈D, d>J.r}❏ J.c = R

i *(J.d-J.r)

QPS Algorithm /2

7

Definition❏ A server S is a scheduling

abstraction ❏ acts like a proxy for a

collection of client tasks/servers

❏ schedules its clients according to the EDF policy

❏ Fixed-Rate EDF Server S:(RS,D

S)

❏ RS = Σ

∈TR( )

❏ RS ≤ 1

❏ D

S = ⋃

∈TD( )

QPS Algorithm /3Definition

❏ Quasi-partition Q(Γ,m) is a partition of Γ on m processors s.t.

1. | Q(Γ,m)|≤m ➢ restricts the cardinality of execution sets to the

number of available processors

2. ∀P∈Q(Γ,m), 0<R(P)<2➢ Each element P in Q(Γ,m) is

○ (if R(P) <=1) a minor execution set ○ (if R(P) >1) a major execution set

➢ P at maximum requires two-processors for being correctly scheduled

3. ∀P∈Q(Γ,m), ∀σ∈P, R(P)>1 => R(σ)>R(P)-1

➢ P is a major execution➢ the P’s extra ratio must be

less than the ration of any other server σ in P

➢ keystone for the QPS’s flexibility

N.B. A graphical example follows

8

QPS Algorithm /4

9

➢ are given 2 processors and a task set P

P1:(0.6)

3:(0.8)

4:(0.2)

2:(0.3)

➔ R(P)=1.9≤2 ◆ 1st cond. is OK◆ 2 execution sets

Proc1 and Proc

2◆ max(capacity( Proc

i∈{1,2} ))=1

3

0

1

2

◆� � 3

Proc1 Proc2

➔ R(P)=1.9>1◆ 2nd condition is OK◆ more than 1 processor is required

to schedule P

➔ Let P = {PA, PB} such that ◆ PA={

3,

4 }, PB={

1,

2}

◆ R(PA)=1, R(PB)=0.9● no excess ratio,fully partitioned● the slack 1-R(PB) may be

an idle task on Proc2➔ if modifying the

4’s ratio to 0.3 then the

processor system becomes 100% utilized◆ the 3rd cond. holds◆ the excess rate (red color) is less than

the rate of all tasks in Proc1 execution set● otherwise a better re-allocation

of tasks exists

4

3

1

Example

QPS Algorithm /5

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❏ QPS servers➔ P is a major execution set with rate R(P)

=1+x➔ P={PA, PB} is a bi-partition➔ σA:(R(PA)-x, PA)

σB:(R(PB)-x, PB)σM:(x, P)σS:(x, P)

Dedicated servers

Master/Slave servers

Excess ratio

➢ At any time t, all QPS servers associated with P share the same deadline D(P,t)

➢ Each QPS server σ is a Fixed-Rate EDF Server

Definitions

QPS Algorithm /6

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❏ Γ = { i:(2/3,3) } for i=1,

2,3 ❏ two-processors❏ P

1 = {

1,

2}, P

2 = {

3}

❏ R(P1)= 4/3 , R(P

2)= 2/3

❏ [2,3] time interval denotes parallel execution

Fixed-Rate Servers

Master/Slave servers

QPS Algorithm /7

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Off-line phase❏ Γ = { σ

i } for i = 1,2,...,5

a server set, three-processors

❏ R(σi ) = 0.6

❏ Proc.1 and Proc.2 dedicated processors

❏ Proc.3 shared processor

➢ σ6, σ

7 external servers reserve computer

capacity for exceeding parts

Processor Hierarchy

Lemma IV.1.Any server set Γ

0 with ceiling(R(Γ

0))≤m will be

allocated no more than m processors

QPS Algorithm /8

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On-line phase Scheduling❏ servers and tasks are scheduled according to

the following rules:❏ visit each processor in revers order

to that of its allocation ❏ select via EDF the highest priority

server/task❏ if a master server is selected,

❏ select also its slave associated server

❏ for all selected servers ❏ select all highest priority client

❏ dispatch all selected tasks to execute

QPS Algorithm /9

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On-line phase Scheduling❏ Γ ={

1:(6,15),

2:(12,30),

3:(5,10),

4:(3.5,5)}

set of sporadic tasks❏ to be scheduled by QPS

servers on two-processors❏ all tasks release their first

job at time 0 ❏ the second job of

3

arrives at time t=16 whereas the other tasks behave periodically

❏ Q(Γ,2)={P1,P

2} with P

1={

1,

2,

3 } and P

2={

4}

❏ R(P1)=1.3>1 ⟹ σA:(0.1, {

1}), σB:(0.6, {

2,

3 }),

σM:(0.3, P1), σS:(0.3, P

1)

QPS Correctness /1

15

Assumptions Time Definitions❏ Δ = [t,t*) time interval such that❏ P is major execution set

❏ Δ is complete EDF interval if Manager activates P at time t, next activates QPS mode at t*

❏ Δ is complete QPS interval if all tasks in P are active

❏ Δ is a QPS job interval if some task in P releases a job at time t and has the t*=D(Γ,t)

QPS

The Partitoner

The Manager The Dispatcher

QPS Correctness /2

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Theorem V.1QPS produces a valid schedule for a set of implicit-deadline sporadic tasks Γ on m ≥ ceiling(R(Γ)) identical processors

Proof: Let P be a major execution set Lemma V.1.

Consider a complete QPS interval Δ. If the master server σM in charge of P is scheduled on its

shared processor so that it meets all its deadlines in Δ, then the other three QPS servers will also meet theirs on P’s dedicated processor

Lemma V.2. The individual tasks and servers in P will meet all their deadlines provided that the master server in charge of P meets its deadlines

Implementation

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QPSimplemented 1

on top of LITMUSRT

1 Compagnin D., Mezzetti E., Vardanega T., “Experimental evaluation of optimal schedulers based on partitioned proportionate fairness ” ECRTS15

#off-line decisions may influence run-time performance #RUN vs QPS comparison

#empirical evaluation

Evaluation /1

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QPS RUN1 U-EDF2

1 Regnier P., Lima G., Massa E., Levin G., Brandt S., “RUN: Optimal Multiprocessor Real-Time Scheduling via Reduction to Uniprocessor”, 2011 32nd IEEE Real-Time Systems Symposium

2 Nelissen G., Berten V., Nelis V., Goossens J., Milojevic D., “U-EDF: An Unfair but Optimal Multiprocessor Scheduling Algorithm for Sporadic Tasks”, 2012 ECRTS

#optimal scheduling algorithms

#the performance of the algorithms was assessed via simulation

#QPS’s performance is influenced by the “processor hierarchy”

tasks running on kth processor may migrate to any of m-k processors

Evaluation /2

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QPS RUN U-EDFDeveloping intuition about run-time overhead

❏ given a set of ❏ m processors ❏ m+1 periodic tasks that fully utilize the m processors

❏ the average hierarchy size is “(m-1)/2” ❏ for instance:

❏ if m=5 then 2 is the average hierarchy size❏ m+1=5+1=6 periodic tasks❏ in the worst case the hierarchy levels are as

many as the number of available m processors (i.e. 6 processor levels ) ❏ bin over-packing propagation phenomenon

Evaluation /3

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QPS RUN U-EDFContinuing the case study ❏ whenever the size of the task set grows up

❏ for example from m+1 tasks to 2m❏ the run-time overhead drops down due of

❏ more lighter tasks❏ more linear task set partitioning ❏ EDF mode which behaves like a

Partitioned EDF

Conclusion

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❏ presents itself in a fashionable way against other state-of-the-art global schedulers★ while showing a manageable run-time overhead

❏ outperforms similar schedulers★ when the processor hierarchy size is gently low★ in presence of a fully partitioned task system (i.e P-EDF bevavior) ➔ off-line phase may influence at run-time significant overhead

◆ inter-server coordination my be quite not trivial

❏ depicts a simple scheduling abstraction model based on servers★ dynamic adaptation as a function of system load variations★ parallel execution needs are assured by master/slave relation

QPS

Future Work

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❏ is related with the extension of QPS to broader problem constraints❏ arbitrarily deadlines, heterogeneous multiprocessors, etc

❏ different adaptation strategies to extend the current master/slave abstraction model to enhance less expensive inter-processor communication mechanism

❏ improve the quasi-partioning implementation ❏ extend it with different implementations ❏ evaluate the performance differences between them