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Computing the Maximum Blocking Time for Scheduling with Deferred Preemption

Sebastian Altmeyer, Claire Burgui` ere, Reinhard Wilhelm

Saarland University

STFSSD, Tokyo 2009

Motivation

preemptive schedule good performance/bad predictability non-preemptive schedule bad performance/good predictability tradeoff: deferred preemption

deferred preemption: preemption only at predefined, selected preemption points

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Schedulability Analysis for Deferred Preemption

Schedulability needs:

• Periods
• Worst Case Execution Time
• Context Switch Costs
• Maximum Blocking Time

T1 T2

BT BT

∗ ∗ ∗ ∗ ∗

Context Switch Costs Task Activation

∗

Preemption Point

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Overview

Maximum Blocking Time (MBT) = longest non-preemptive path

1 Basics

WCET Analysis Implicit Path Enumeration Technique Cache-Related Preemption Delay

2 Computing Maximum Blocking Time 3 Conclusion

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

WCET Analysis

CFG Value Analysis Loop Analysis Low-Level Analysis Path Analysis WCET

Value Analysis determines values for registers and memory cells Loop Analysis determines loop-bounds Low-Level Analysis determines WCET for basic blocks Path Analysis determines longest path trough the program

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Implicit Path Enumeration Technique

1 2 y1 3 y2 4 y3 y4 5 y5 y6 1 = x1 x1 = y1 + y2; y1 + y4 = x2 x2 = y3 + y5; y2 = x3 x3 = y6; y3 = x4 x4 = y4; y5 + y6 = x5 x5 = 1; y3 ≤ bl · y1; max

i xici

• node counter xi, edge counter yi
• first node x1 entered once, last node x5 left once.
• nodes executed as often as entered/left
• loop l executed bl times as often as entered
• WCET given by max. sum of xi times ci ( = WCET of node i)
• set of constraints solved using ILP

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Cache-Related Preemption Delay (CRPD)

a

1

b

2

c

3

d

4

a

5

At basic block (2):

• a may be cached,

may be reused at (5)

• d may be cached,

may be reused at (4) Preemption at (2) ⇒ 2 additional misses may occur

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Useful Cache Blocks

A memory block is a useful cache block (UCB) at program point P, if it

• may be cached at P, and
• may be reused at program point P′ reached from P.

CRPD at basic block i: CRPDi ≤ costreload × |UCBi|

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Maximum Blocking Time

Maximum Blocking Time (MBT) = Longest Path from preemption point or program start to preemption point or program end longest path given by: WCET of the basic blocks + CRPD for the sake of simplicity: preemption points only at the beginning or end of a basic block

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Maximum Blocking Time

• 1. partition preemption points into two sets:

Up = {i| preemption point at beginning of Bi} Lo = {i| preemption point at end of Bi}

• 2. introduce artificial nodes:

1 2 3 4 5 1 2

3u

3 4

4l

5

Up = {3} Lo = {4}

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

1 2 y1

3u

3 y2 4 y3

4l

y4 5 y5 y6

• theoretical start node S, connected to beginnings of all paths
• theoretical end node E, connected to ends of all paths
• structural constraints as before (but with additional nodes)
• MBT given by max. xici + possible CRPD Ci

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

1 2 y1

3u

3 y2 4 y3

4l

y4 5 y5 y6 S x1 + x3 + xl

4 = 1;

• theoretical start node S, connected to beginnings of all paths
• theoretical end node E, connected to ends of all paths
• structural constraints as before (but with additional nodes)
• MBT given by max. xici + possible CRPD Ci

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

1 2 y1

3u

3 y2 4 y3

4l

y4 5 y5 y6 E x1 + x3 + xl

4 = 1;

1 = x4 + x5 + xu

3 ;

• theoretical start node S, connected to beginnings of all paths
• theoretical end node E, connected to ends of all paths
• structural constraints as before (but with additional nodes)
• MBT given by max. xici + possible CRPD Ci

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

1 2 y1

3u

3 y2 4 y3

4l

y4 5 y5 y6 x1 + x3 + xl

4 = 1;

1 = x4 + x5 + xu

3 ;

x1 = y1 + y2; y1 + y4 = x2 x2 = y3 + y5; y2 = xu

3 ;

x3 = y6; y3 = x4; xl

4 = y4;

y5 + y6 = x5; y3 ≤ by1;

• theoretical start node S, connected to beginnings of all paths
• theoretical end node E, connected to ends of all paths
• structural constraints as before (but with additional nodes)
• MBT given by max. xici + possible CRPD Ci

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

1 2 y1

3u

3 y2 4 y3

4l

y4 5 y5 y6 x1 + x3 + xl

4 = 1;

1 = x4 + x5 + xu

3 ;

x1 = y1 + y2; y1 + y4 = x2 x2 = y3 + y5; y2 = xu

3 ;

x3 = y6; y3 = x4; xl

4 = y4;

y5 + y6 = x5; y3 ≤ by1; max

i xici + xl 4C l 4 + x3C u 3

• theoretical start node S, connected to beginnings of all paths
• theoretical end node E, connected to ends of all paths
• structural constraints as before (but with additional nodes)
• MBT given by max. xici + possible CRPD Ci

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Conclusion

• Essential ingredient for the analysis of deferred preemption
• Only minor extensions to IPET:
• one new variable per preemption point
• no additional constraints
• All data available from WCET / UCB analysis

& Future Work

• implementation and evaluation
• automatic derivation of good preemption points

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Implicit Path Enumeration - Equations

max

• i

xici (1) xi = 1, where Bi = s (2) xj = 1, where Bj = e (3)

• j∈S

yj = xk, where S = {j|Ej = ( , Bk)} (4) xk =

• j∈S

yj, where S = {j|Ej = (Bk, )} (5) yl ≤ bl(

• j∈S

yj) where S = {j|Ej loop entry edge} (6)

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Maximum Blocking Time - Equations 1

max

n

• i=1

(xici) +

• Bi∈Lo

xl

i C l i +

• Bi∈Up

xiC u

i

(7) xs +

• Bi∈Lo

xl

i +

• Bi∈Up

xi = 1 (8) xe +

• Bi∈Up

xu

i +

• Bi∈Lo

xi = 1 (9)

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009

Maximum Blocking Time - Equations 2

• j∈S

yj = xu

k

if Bk ∈ Up xk

• therwise

(10) where S = {j|Ej = ( , Bk)}

• j∈S

yj = xl

k

if Bk ∈ Lo xk

• therwise

(11) where S = {j|Ej = (Bk, )} yl ≤ bl  

j∈S

yj   (12) where S = {j|Ej loop entry edge}

Computing Maximum Blocking for Deferred Preemption Altmeyer, Burgui` ere, Wilhelm STFSSD, Tokyo 2009