CORACORA Optimal & Real Time Scheduling · 2007-09-26 · Informationsteknologi UCb Steel...

Optimal & Real TimeScheduling

Model Checking Technologyusing

CORACORACORA

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Academic partners:

− Nijmegen− Aalborg− Dortmund− Grenoble− Marseille− Twente− Weizmann

Industrial Partners

− Axxom− Bosch− Cybernetix− Terma

April 2002 – June 2005 IST-2001-35304

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SIDMAR Overview

Machine 1 Machine 2 Machine 3

Machine 4 Machine 5

Buffer

Continuos Casting Machine

Storage Place

Crane B

Crane A

@10 @20 @10

Lane 1

Lane 2

INPUTsequenceof steel loads(“pigs”)

OUTPUTsequence of higherquality steel

GOAL: Maximize utilization

of plant

GOAL: Maximize utilization

of plant

SIDMAR Modelling

Machine 4 Machine 5

Buffer

Storage Place

Crane B

Crane A

Lane 1

Lane 2

A Single Load

UPPAAL Model

OBJECTIVES

powerful, unifying mathematical modellingefficient computerized problem-solving tools

distributed real-time systemstime-dependent behaviourand dynamic resource allocation

TIMED AUTOMATA

LTR Project VHS (Verification of Hybrid systems)

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Smart Card PersonalizationCybenetix, France

Maximize throughputMaximize throughput

Piles of blank cards Personalisation

Test and possiblyreject

Adder 1

S = A + S' - A'

Adder 2

T = B + T' - B'

Buffer 1

1 Kbytes

Buffer 2

1 Kbytes

Buffer 9

2 Kbytes

Buffer 8

2 Kbytes

Buffer 7

2 Kbytes

Buffer 6

512 bytes

Buffer 4

2 Kbytes

Buffer 3

512 bytes

Input A8 (100MHz)

S' 16 (100 MHz)

Input B8 (100 MHz)

Output S Output T

256 (100 MHz)

128 (200 MHz)

Buffer 5

512 bytes

8 (100MHz)

16 (100 MHz)

256 (100 MHz)

Arbiter

Memory Management Radar Video Processing Subsystem

Advanced Noise Advanced Noise Reduction TechniquesReduction Techniques

e0,2 e2,4

e3,4e3,3

Sweep Integration

Costal Surveillance

9.170 GHz9.438 GHz

Combiner(VP3) Fr

combiner

2Ed Brinksma Car Periphery Supervision System: Case Study 3

CPS obtains and makes available for other systems information about environment of a car. This information may be used for:

Parking assistancePre-crash detectionBlind spot supervisionLane change assistanceStop & goEtc

Based on Short Range Radar (SRR) technology

The CPS considered in this case study

One sensor group only(currently 2 sensors)Only the front sensors and corresponding controllersApplication: pre-crash detection, parking assistance, stop & go

CPS: Informal description

Cybernetix:− Smart Card

Personalization Terma:

− Memory InterfaceBosch:

− Car Periphery Sensing

AXXOM:− Lacquer Production

Benchmarks

Case Studies

ÜberschriftÜberschrift

05.06.2005 Axxom Software AG Seite: 3

Product flow of a Product

Laboratory

Dispersion

Dose Spinner

Mixing Vessel Filling Stations

Storage

CORACORACORA

CLASSICCLASSICCLASSIC

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Overview

Timed Automata & Scheduling

Priced Timed Automata and Optimal Scheduling

Optimal Infinite Scheduling

Optimal Conditional Scheduling

ApplicationsOptimal Scheduling Using Priced Timed Automata.

G. Behrmann, K. G. Larsen, J. I. Rasmussen,

ACM SIGMETRICS Performance Evaluation Review

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Real Time Model Checking

sensors

actuators

UPPAAL Model

Modelofenvironment(user-supplied /non-determinism)

Model oftasks(automatic?)

PlantContinuous

Controller ProgramDiscrete

SAT φ ??SAT φ ??

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Real Time Scheduling &Control Synthesis

PlantContinuous

Controller ProgramDiscrete

sensors

actuators

Partial UPPAAL Model

Modelofenvironment(user-supplied)

Synthesisoftasks/scheduler(automatic)

SAT φ !!SAT φ !!

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Rush Hour

OBJECTIVE:Get yourCAR out

Your CAR

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Rush Hour

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Rush Hour

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Real Time Scheduling

UNSAFE

• Only 1 “Pass”• Cheat is possible

(drive close to car with “Pass”)

• Only 1 “Pass”• Cheat is possible

(drive close to car with “Pass”)

The Car & Bridge ProblemCAN THEY MAKE IT TO SAFEWITHIN 70 MINUTES ???

Crossing

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Real Time Scheduling

UNSAFE

Solve Scheduling Problem

using UPPAAL

Solve Scheduling Problem

using UPPAAL

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Steel Production Plant

Machine 4 Machine 5

Buffer

Storage Place

Crane B

Crane AA. Fehnker, T. Hune, K. G. Larsen, P. PetterssonCase study of Esprit-LTRproject 26270 VHSPhysical plant of SIDMARlocated in Gent, Belgium.Part between blast furnace and hot rolling mill.

Objective: model the plant, obtain schedule and control programfor plant.

Lane 1

Lane 2

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Steel Production Plant

Machine 4 Machine 5

Buffer

Storage Place

Crane B

Crane A

Input: sequence of steel loads (“pigs”). @10 @20 @10

Load follows Recipe to obtain certain quality,

e.g:start; T1@10; T2@20; T3@10; T2@10; end within 120.

Output: sequence of higher quality steel.

Lane 1

Lane 2

∑=127

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A single load(part of)

A single load(part of) Crane BCrane B

UPPAALUPPAALUPPAAL

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Controller Synthesis for LEGO Model

LEGO RCX Mindstorms.Local controllers with control programs.IR protocol for remote invocation of programs.Central controller.

m1 m2 m3

crane a

crane b

casting

storage

buffer

centralcontroller

Synthesis1971 lines of RCX code (n=5),24860 - “ - (n=60).

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Timed Automata

Synchronization

Invariant

[Alur & Dill’89]

Resource

Semantics:( Idle , x=0 )

( Idle , x=2.5) d(2.5)( InUse , x=0 ) use?( InUse , x=5) d(5)( Idle , x=5) done!( Idle , x=8) d(3)( InUse , x=0 ) use?

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CompositionResource Task

Shared variable

Synchronization

Semantics:( Idle , Init , B=0, x=0)

( Idle , Init , B=0 , x=3.1415 ) d(3)( InUse , Using , B=6, x=0 ) use( InUse , Using , B=6, x=6 ) d(6)( Idle , Done , B=6 , x=6 ) done

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Jobshop Scheduling

4. 1 min3. 1 min2. 20 min1. 10 minMaria

2. 11 min3. 11 min1. 13 min4. 1 minNicola

1. 10 min3. 3 min4. 1 min2. 5 min Kim

Comic StipLocalNews

EconomySport

Problem: compute the minimal MAKESPAN

R E S O U R C E S

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Jobshop Scheduling in UPPAAL

4. 1 min3. 1 min2. 20 min1. 10 minMaria

2. 11 min3. 11 min1. 13 min4. 1 minNicola

1. 10 min3. 3 min4. 1 min2. 5 min Kim

Comic StipLocal NewsEconomySport

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Experiments B-&-B algorithm runningfor 60 sec.

Lawrence Job Shop ProblemsLawrence Job Shop Problems

5[TACAS’2001]

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Task Graph SchedulingOptimal Static Task Scheduling

Task P={P1,.., Pm}Machines M={M1,..,Mn}Duration Δ : (P×M) → N∞

< : p.o. on P (pred.)

A task can be executed only if all predecessors have completedEach machine can process at most one task at a timeTask cannot be preempted.

P6 P3 P4

6,6 10,16

2,2 8,2

M = {M1,M2}

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P6 P3 P4

6,6 10,16

2,2 8,2

M = {M1,M2}

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P6 P3 P4

6,6 10,16

2,2 8,2

M = {M1,M2}

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P6 P3 P4

6,6 10,16

2,2 8,2

M = {M1,M2}

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P6 P3 P4

6,6 10,16

2,2 8,2

M = {M1,M2}

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P6 P3 P4

6,6 10,16

2,2 8,2

M = {M1,M2}

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Experimental Results

Abdeddaïm, Kerbaa, Maler

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Optimal Task Graph SchedulingPower-Optimality

Energy-rates: C : M → NxN P2 P1

P6 P3 P4

6,6 10,16

2,2 8,2

cost’==1

cost’==4

Priced Timed AutomataOptimal Scheduling

with Paul Pettersson, Thomas Hune, Judi Romijn, Ansgar Fehnker, Ed Brinksma, Frits Vaandrager, Patricia Bouyer, Franck Cassez, Henning Dierks

Emmanuel Fleury, Jacob Rasmussen,..

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EXAMPLE: Optimal rescue plan for cars withdifferent subscription rates for city driving !

SAFEGolf Citroen

BMW Datsun

OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65!

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Experiments

447406-0-0000

--85time<85

9751085

BD> B< CB> C< CG>4030201

3782526555CG> G< BG> G< GD>1111

#Pop’d#ExplTIMECOSTSCHEDULECOST-rates

40826365170CD> C< CB> C< CG>10321

35023260140CG> G< BD> C< CG>4321

23314965195GD> G< CG> G< BG>10329

263817621538

60CG> G< BD> C< CG>Min Time

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Priced Timed AutomataAlur, Torre, Pappas (HSCC’01)

Behrmann, Fehnker, et all (HSCC’01)

l1l2 l3

x:=0c+=1

x · 23 · y

c+=4c’=4 c’=2 ☺

0 · y · 4

y · 4x:=0

Timed Automata + COST variable

cost rate

cost update

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l1l2 l3

x:=0c+=1

x · 23 · y

c+=4c’=4 c’=2 ☺

0 · y · 4

y · 4x:=0

cost rate

cost update

(l1,x=y=0) (l1,x=y=3) (l2,x=0,y=3) (l3,_,_)ε(3)

12 1 4 ∑ c=17

TRACES

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TRACES

l1l2 l3

x:=0c+=1

x · 23 · y

c+=4c’=4 c’=2 ☺

0 · y · 4

y · 4x:=0

cost rate

cost update

(l1,x=y=0) (l1,x=y=3) (l2,x=0,y=3) (l3,_,_)

(l1,x=y=0) (l1,x=y=2.5) (l2,x=0,y=2.5) (l2,x=0.5,y=3) (l3,_,_)

(l1,x=y=0) (l2,x=0,y=0) (l2,x=3,y=3) (l2,x=0,y=3) (l3,_,_)

ε(2.5) ε(.5)

12 1 4

10 1 1 4

1 6 0 4

∑ c=17

∑ c=16

∑ c=11

Problem :

Find the minimum cost of reaching location l3Problem :

Find the minimum cost of reaching location l3

Efficient Implementation:

CAV’01 and TACAS’04Efficient Implementation:

CAV’01 and TACAS’04

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Aircraft Landing Problem

E earliest landing timeT target timeL latest timee cost rate for being earlyl cost rate for being lated fixed cost for being late

e*(T-t)

d+l*(t-T)

Planes have to keep separation distance to avoid turbulences caused by preceding planes

Runway

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Runway

129: Earliest landing time153: Target landing time559: Latest landing time10: Cost rate for early20: Cost rate for late

Runway handles 2 types of planes

Modeling ALP with PTA

Symbolic ”A*”

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Operations

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Priced Zone

22 +−= xyyxCost ),(

CAV’01

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A split of {y}Z

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A split of

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Branch & Bound Algorithm

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Z’ is bigger & cheaper than ZZ’ is bigger &

cheaper than Z

ZZ ≤'

≤· is a well-quasiordering which

guaranteestermination!

· is a well-quasiordering which

guaranteestermination!

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Experiments MC Order

447406-0-0000

--85time<85

9751085

BD> B< CB> C< CG>4030201

3782526555CG> G< BG> G< GD>1111

#Pop’d#ExplTIMECOSTSCHEDULE

COST-rates

40826365170CD> C< CB> C< CG>10321

35023260140CG> G< BD> C< CG>4321

23314965195GD> G< CG> G< BG>10329

263817621538

60CG> G< BD> C< CG>Min Time

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Example: Aircraft Landing

E earliest landing timeT target timeL latest timee cost rate for being earlyl cost rate for being lated fixed cost for being late

e*(T-t)

d+l*(t-T)

Runway

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Example: Aircraft Landing

land!x >= 4

x <= 5x=5

x <= 5

x <= 9cost+=2

cost’=3 cost’=1

4 earliest landing time5 target time9 latest time3 cost rate for being early1 cost rate for being late2 fixed cost for being late

Runway

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Aircraft LandingSource of examples:

Baesley et al’2000

CAV’01

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Zone basedLinear ProgrammingProblems

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Zone LP Min Cost FlowExploiting duality

minimize 3x1-2x2+7when x1-x2· 1

1· x2 · 3x2≥ 1

minimize 3y2,0-y0,2+y1,2 – y0,1

when y2,0-y0,1-y0,2=1y0,2+y1,2=2y0,1-y1,2=-3

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Zone LP Min Cost FlowExploiting duality

minimize 3x1-2x2+7when x1-x2· 1

1· x2 · 3x2≥ 1

minimize 3y2,0-y0,2+y1,2 – y0,1

when y2,0-y0,1-y0,2=1y0,2+y1,2=2y0,1-y1,2=-3

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Aircraft LandingUsing MCF/Netsimplex

Rasmussen, Larsen, SubramaniTACAS04

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AXXOM Case study

3 types of recipes− for uni/metallic/bronce− use of resources,

processing times, timing

29 (73, 219) orders:− start time, due date,

recipe

extensions:− delay cost,

storage cost,setup cost

− weekend, nights

Laboratory

Dispersion

Dose Spinner

Storage

Laquer Production Scheduling

Behrmann, Brinksma, Hendriks, Mader16th IFAC World Congress

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Resources

2 mixing vessels for uni lacquers3 mixing vessels for metallic/bronce2 dose spinners1 dose spinner bronce1 disperging line1 predisperser1 bronce mixer2 filling lineslab (unlimitted)

Laboratory

Dispersion

Dose Spinner

Storage

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Recipes UPPAAL templatefor metal

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Orders

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Instantiated Model

State SpaceExplosion

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Heuristics

Nice heuristics− non-overtaking

orders of the same recipe cannot overtake each other

− non-lazinessa process that needs an available resource will not waste time if its is not claimed by others (a.k.a. active scheduling)

Cut-and-Pray heuristics− greediness

a process that needs an available resource will claim this resource immediately

− reducing active ordersthe number of concurrent orders is restricted (number of critical resources can give an indication)

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8 s4g, no219

7 s3nl, no73

--nl, no73

1 s-nl29

term.time

max.ordersheuristic#jobs

uses clock optimization &optimized successor

calculation

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Results Extended Case

avail.

workhrs

192,881,1294no29

2,263,496-es, no, g29

1,714,875-es, no, nl29

647,410-es, no, g29

530,771-es, no, nl29

min. costfound in 60 s

max.ordersheuristic#jobs

Competitivewith

Orion-piresults

Order of magnitude faster than

MILP, GAMS/CPLEX

storage, delay and setup costs, working hours

Optimal ConditionalReachability

with Jacob I. Rasmussen

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EXAMPLE: Optimal rescue plan for cars withdifferent subscription rates for city driving !

SAFEGolf Citroen

BMW Datsun

UNSAFE

My CAR!

MinimizesCostMYCAR

subject toCostCitroen · 60CostBMW · 90CostDatsun · 10

min CostMYCAR= 270time = 70

CONDITIONAL

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Optimal Conditional Reachability

c’ = 1d’ = 4

c’ = 2d’ = 1 ☺

l1l2 l3

x · 2

d+=1x ≥ 2y ≥ 1

x · 3y · 2

PROBLEM:Reach l3 in a way which

minimizes csubject to d · 4

PROBLEM:Reach l3 in a way which

minimizes csubject to d · 4

SOLUTION: c = 11/3wait 1/3 in l1; goto l2; wait 5/3 in l2; goto l3

Dual-priced TA

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Discrete Trajectories

c’ = 1d’ = 4

c’ = 2d’ = 1

l1l2 l3

x · 2

x ≥ 2y ≥ 1

x · 3y · 2

l1,0,0

l1,1,1

l1,2,2

l2,0,0 l2,1,0 l2,2,0

l2,1,1

l2,2,2

l2,2,1

l2,3,2

l2,3,1

1,4 2,1

2,1 2,1

0,00,0

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Discrete Trajectories

c’ = 1d’ = 4

c’ = 2d’ = 1

l1l2 l3

x · 2

x ≥ 2y ≥ 1

x · 3y · 2

l1,0,0

l1,1,1

l1,2,2

l2,0,0 l2,1,0 l2,2,0

l2,1,1

l2,2,2

l2,2,1

l2,3,2

l2,3,1

1,4 2,1

2,1 2,1

0,00,0

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Dual Priced Zones

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Dual Priced Zones

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Dual Priced Zones

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Exploration c’ = 1d’ = 4

c’ = 2d’ = 1

l1l2 l3

x · 2

x ≥ 2y ≥ 1

x · 3y · 2

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c’ = 2d’ = 1

l1l2 l3

x · 2

x ≥ 2y ≥ 1

x · 3y · 2

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c’ = 2d’ = 1

l1l2 l3

x · 2

x ≥ 2y ≥ 1

x · 3y · 2

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Termination

THEOREM

Optimal conditional

reachability for multi-priced TA

is computable.

THEOREM

Optimal conditional

reachability for multi-priced TA

is computable.

BRICS@AalborgFMT@Twente

Optimal Infinite Schedulingwith Ed Brinksma

Patricia BouyerArne Skou

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EXAMPLE: Optimal WORK plan for cars withdifferent subscription rates for city driving !

Golf Citroen

BMW Datsun

maximal 100 min. at each location

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Workplan I

Datsun

Citroen

UDatsun

Citroen

Datsun

Citroen

Datsun

Citroen

SDatsun

Citroen

UDatsun

Citroen

Datsun

Citroen

UDatsun

Citroen

ε(25) ε(25)

ε(25)

ε(25)ε(20)

ε(20)

ε(25)

275 275

300300

Value of workplan:

(4 x 300) / 90 = 13.33

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Workplan II

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

DatsunBMW

CitroenGolf

25/1255/25 20/180

5/1025/12510/130

25/225 10/90 10/0

5/1025/50

Value of workplan:

560 / 100 = 5.6

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Infinite Scheduling

E[] ( Kim.x · 100 and Jacob.x · 90 andGerd.x · 100 )

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Infinite Optimal Scheduling

E[] ( Kim.x · 100 and Jacob.x · 90 andGerd.x · 100 )

+ most minimal limitof cost/time

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Cost Optimal Scheduling =Optimal Infinite Path

t3 tnσ

Value of path σ: val(σ) = limn→∞ cn/tnOptimal Schedule σ*: val(σ*) = infσ val(σ)

Accumulated cost

Accumulated time¬(Car0.Err or Car1.Err or …)

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t3 tnσ

Accumulated cost

THEOREM: σ* is computable

Bouyer, Brinksma, Larsen HSCC’04

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Optimal Infinite Schedule

x · 2

x ≥ 4

x ≥ 3

x ≥ 4

x ≥ 5

y:=0y=0

c’=5 c’=1

c´=10

r’=1

r’=2

r’=5

Corner Point Automaton

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x · 2

x ≥ 4

x ≥ 3

x ≥ 4

x ≥ 5

y:=0y=0

c’=5 c’=1

c´=10

r’=1

r’=2

r’=5

Corner Point Automaton

Finite Behaviours

! Optimal behaviours are NOT prefixed closed

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x · 2

x ≥ 4

x ≥ 3

x ≥ 4

x ≥ 5

y:=0y=0

c’=5 c’=1

c´=10

r’=1

r’=2

r’=5

Dynamic Voltage Scaling

Performance vs Ressource-Efficiency ?

Consumer constantly demandbetter functionality, flexibility, availability, …

.. increase in resourcesneeded:

TimeEnergyMemoryBandwidth..

Application of CUPPAAL to modeling, analysis and synthesis of resource-efficient schedules for real-time systems.

Power Management Dynamic Voltage Scaling

Dynamically lower voltage supply (frequency)to utilize free CPU time

Task Scheduling

T2 is running{ T4 , T1 , T3 } readyordered according to somegiven priority:(e.g. Fixed Priority, Earliest Deadline,..)

SchedulerScheduler

2 14 3

readydone

stoprun

P(i), [E(i), L(i)], .. : period or earliest/latest arrival or .. for Ti

C(i): execution time for TiD(i): deadline for Ti

utilization of CPU

Modeling Task

SchedulerScheduler

2 14 3

readydone

stoprun

Modeling Scheduler

SchedulerScheduler

2 14 3

readydone

stoprun

Modeling Queue

SchedulerScheduler

2 14 3

readydone

stoprun

Modeling Queue

SchedulerScheduler

2 14 3

readydone

stoprun

In UPPAAL 4.0User Defined Function

Schedulability = Safety Property

A ¬(Task0.Error or Task1.Error or …)

¬(Task0.Error or Task1.Error or …)

May be extended with preemption

Energy Optimal Scheduling

“Choose”Scaling/Cost

(Freq/Voltage)

Using Priced Timed Automata

SchedulerScheduler

2 14 3

readydone

stoprun F:= ?? ; V:= ??

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t3 tnσ

Accumulated cost

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t3 tnσ

Accumulated cost

THEOREM: σ* is computable

Bouyer, Brinksma, Larsen HSCC’04

Approximate Optimal Schedule

E[] (not (Task0.Error or Task1.Error or Task2.Error) and(cost>=M imply time >= N))

=E[] φ(M ,N)

σ ² [] φ(M,N) imply val(σ)· M/N

C=M C=M C=M

T<NX X

Optimal infinite schedulemodulo cost-horizon

T<NT<N

Preliminary Results

EDF w preemption no DVS: avr.: 48P1=D1=32 C1=6P2=D2=48 C2=18P3=D3=64 C3=12

Preliminary Results

EDF w preemption w DVS: avr.: 43.37

Cost horizon: 2,000

P1=D1=32 C1=6P2=D2=48 C2=18P3=D3=64 C3=12

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Dynamic SchedulingE<> ( Kim.Done and

Jacob.Done andGerd.Done )

+ winning / optimalstrategy

Uncontrollabletiming uncertainty

Time-Optimal Reachability Strategies for Timed Games[Cassez, David, Fleury, Larsen, Lime, CONCUR’05]

Cost-Optimal Reachability Strategies for Priced Timed Game Automata

[Alur et all, ICALP’04][Bouyer, Cassez. Fleury, Larsen, FSTTCS’04]

CORACORA Optimal & Real Time Scheduling · 2007-09-26 · Informationsteknologi UCb Steel...

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