[PDF] - Scheduling Optim al & Real Tim e using CORA CORA CORA PDF Document

SLIDE 1

Optim al & Real Tim e Scheduling

Model Checking Technology using

CORA CORA CORA

SLIDE 2

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Overview

Timed Automata & Scheduling Priced Timed Automata and Optimal

Scheduling

Optimal Infinite Scheduling Optimal Conditional Scheduling

Optimal Scheduling Using Priced Timed Automata.

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

ACM SIGMETRICS Performance Evaluation Review

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Real Tim e Model Checking

sensors actuators

a c b 1 2 4 3 a c b 1 2 4 3 1 2 4 3 1 2 4 3 a c b

UPPAAL Model

Model

f

environment (user-supplied / non-determinism) Model

f

tasks (automatic?)

Plant

Continuous

Controller Program

Discrete

SAT φ ?? SAT φ ??

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??

Real Tim e Scheduling & Control Synthesis Plant

Continuous

Controller Program

Discrete

sensors actuators

a c b 1 2 4 3 a c b 1 2 4 3 1 2 4 3 1 2 4 3 a c b

Partial UPPAAL Model

Model

f

environment (user-supplied)

Synthesis

f

tasks/scheduler (automatic)

SAT φ !! SAT φ !!

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Rush Hour

OBJECTI VE: Get your CAR out OBJECTI VE: Get your CAR out Your CAR EXI T

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Rush Hour

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Real Tim e Scheduling

5 10 20 25

UNSAFE SAFE

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 Problem CAN THEY MAKE I T TO SAFE WI THI N 70 MI NUTES ???

Crossing Times

Pass

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Real Tim e Scheduling

SAFE

5 10 20 25

UNSAFE

Solve Scheduling Problem using UPPAAL Solve Scheduling Problem using UPPAAL

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Tim ed Autom ata

Synchronization Guard Invariant Reset

[ Alur & Dill’89]

Resource Transitions: ( Idle , x= 0 ) d(2.5) ( Idle , x= 2.5) use? ( InUse , x= 0 ) d(5) ( InUse , x= 5) done! ( Idle , x= 5) d(3) ( Idle , x= 8) use? ( InUse , x= 0 ) Transitions: ( Idle , x= 0 ) d(2.5) ( Idle , x= 2.5) use? ( InUse , x= 0 ) d(5) ( InUse , x= 5) done! ( Idle , x= 5) d(3) ( Idle , x= 8) use? ( InUse , x= 0 ) States: ( location , x= v) where v∈R States: ( location , x= v) where v∈R

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Com position

Resource Task Shared variable Synchronization Transitions: ( Idle , Init , B= 0, x= 0) d(3.1415) ( Idle , Init , B= 0 , x= 3.1415 ) use ( InUse , Using , B= 6, x= 0 ) d(6) ( InUse , Using , B= 6, x= 6 done ( Idle , Done , B= 6 , x= 6 Transitions: ( Idle , Init , B= 0, x= 0) d(3.1415) ( Idle , Init , B= 0 , x= 3.1415 ) use ( InUse , Using , B= 6, x= 0 ) d(6) ( InUse , Using , B= 6, x= 6 done ( Idle , Done , B= 6 , x= 6

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Task Graph Scheduling

Optim al 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 completed

Each machine can process

at most one task at a time

Task cannot be preempted. Compute schedule with

minimum completion-time!

P2 P1 P6 P3 P4 P7 P5

1 6 ,1 0 2 ,3 2 ,3 6 ,6 1 0 ,1 6 2 ,2 8 ,2

M = { M1,M2}

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Task Graph Scheduling

Optim al Static Task Scheduling

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

P2 P1 P6 P3 P4 P7 P5

1 6 ,1 0 2 ,3 2 ,3 6 ,6 1 0 ,1 6 2 ,2 8 ,2

M = { M1,M2}

E<> (Task1.End and … and Task7.End)

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Experim ental Results

Abdeddaïm, Kerbaa, Maler

Symbolic A* Brand-&-Bound 60 sec

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Optim al Task Graph Scheduling

Pow er-Optim ality

Energy-rates:

C : M → N

Compute schedule with

minimum completion-cost!

P2 P1 P6 P3 P4 P7 P5

1 6 ,1 0 2 ,3 2 ,3 6 ,6 1 0 ,1 6 2 ,2 8 ,2

4 W 3 W

SLIDE 15

Priced Tim ed Autom ata

Optim al Scheduling

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

SLIDE 16

Informationsteknologi

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different subscription rates for city driving !

SAFE

Golf Citroen BMW Datsun

9 2 3 10 OPTI MAL PLAN HAS ACCUMULATED COST= 195 and TOTAL TI ME= 65!

5 10 20 25

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Experim ents

447 406

85

time< 85

975

1085

BD> B< CB> C< CG>

40 30 20 1 378 252 65 55

CG> G< BG> G< GD>

1 1 1 1

# Pop’d # Expl

TI ME COST

SCHEDULE

COST-rates 408 263 65 170

CD> C< CB> C< CG>

10 3 2 1 350 232 60 140

CG> G< BD> C< CG>

4 3 2 1 233 149 65 195

GD> G< CG> G< BG>

10 3 2 9 2638 1762

1538

60

CG> G< BD> C< CG>

Min Tim e

D B C G

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Priced Tim ed Autom ata

Alur, Torre, Pappas (HSCC’01) Behrmann, Fehnker, et all (HSCC’01)

l1 l2 l3 x: = 0 c+ = 1 x · 2 3 · y c+ = 4 c’= 4 c’= 2

☺

0 · y · 4 y · 4 x: = 0

Timed Automata + COST variable

cost rate cost update

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Priced Tim ed Autom ata

Alur, Torre, Pappas (HSCC’01) Behrmann, Fehnker, et all (HSCC’01)

l1 l2 l3 x: = 0 c+ = 1 x · 2 3 · y c+ = 4 c’= 4 c’= 2

☺

0 · y · 4 y · 4 x: = 0

Timed Automata + COST variable

cost rate cost update

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

ε(3) 12 1 4

∑ c= 1 7

TRACES

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TRACES

Priced Tim ed Autom ata

Alur, Torre, Pappas (HSCC’01) Behrmann, Fehnker, et all (HSCC’01)

l1 l2 l3 x: = 0 c+ = 1 x · 2 3 · y c+ = 4 c’= 4 c’= 2

☺

0 · y · 4 y · 4 x: = 0

Timed Automata + COST variable

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(3) ε(2.5) ε(.5) ε(3) 12 1 4 10 1 1 4 1 6 4

∑ c= 1 7 ∑ c= 1 6 ∑ c= 1 1

P r

b

l e m :

F i n d t h e m i n i m u m c

s

t

f

r e a c h i n g l

c

a t i

n

l

3

P r

b

l e m :

F i n d t h e m i n i m u m c

s

t

f

r e a c h i n g l

c

a t i

n

l

3

Efficient Implementation: CAV’0 1 and TACAS’0 4 Efficient Implementation: CAV’0 1 and TACAS’0 4

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Aircraft Landing Problem

cost t E L T E earliest landing time T target time L latest time e cost rate for being early l

cost rate for being late

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

Runway 129: Earliest landing time 153: Target landing time 559: Latest landing time 10: Cost rate for early 20: Cost rate for late Runway handles 2 types of planes

Modeling ALP with PTA

SLIDE 23

Sym bolic ”A* ”

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Zones

Operations

x y Z

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Priced Zone

x y

Δ4

2

1

Z

2 2 + − = x y y x Cost ) , (

CAV’01

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Reset

x y

Δ4

2

1

Z

y:= 0 2

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Reset

x y

Δ4

2

1

Z

{ y} Z y:= 0 2

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Reset

x y

Δ4

2

1

Z

{ y} Z 6 y:= 0 2

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Reset

x y

Δ4

2

1

Z

{ y} Z 6

1

1

4 A split of { y} Z y:= 0 2

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Delay

x y

Δ4

3

1

Z

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Delay

x y

Δ4

3

1

Z

↑ Z

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Delay

x y

Δ4

3

1

Z

↑ Z

3 2

3

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Delay

x y

Δ4

3

1

Z

3 4

1

A split of

↑ Z

3

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Branch & Bound Algorithm

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Branch & Bound Algorithm

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Branch & Bound Algorithm

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Branch & Bound Algorithm

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Branch & Bound Algorithm

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Branch & Bound Algorithm

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

Z Z ≤ '

≤

· is a well-quasi

rdering which

guarantees termination! · is a well-quasi

rdering which

guarantees termination!

SLIDE 40

Experim ental Results

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Experim ents

MC Order

447 406

85

time< 85

975

1085

BD> B< CB> C< CG>

40 30 20 1 378 252 65 55

CG> G< BG> G< GD>

1 1 1 1

# Pop’d # Expl

TI ME COST

SCHEDULE

COST-rates 408 263 65 170

CD> C< CB> C< CG>

10 3 2 1 350 232 60 140

CG> G< BD> C< CG>

4 3 2 1 233 149 65 195

GD> G< CG> G< BG>

10 3 2 9 2638 1762

1538

60

CG> G< BD> C< CG>

Min Tim e

D2

5

B2 C1 G5

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Example: Aircraft Landing

cost t E L T E earliest landing time T target time L latest time e cost rate for being early l

cost rate for being late

d 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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Example: Aircraft Landing

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

land! x >= 4 x=5 x <= 5 x=5 x <= 5 land! x <= 9 cost+=2 cost’=3 cost’=1 4 earliest landing time 5 target time 9 latest time 3 cost rate for being early 1 cost rate for being late 2 fixed cost for being late

Runway

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Aircraft Landing

Source of examples: Baesley et al’2000

CAV’01

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Branch & Bound Algorithm

Zone based Linear Program m ing Problem s

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Zone LP Min Cost Flow

Exploiting duality

minimize 3 x 1-2 x 2+ 7 when x1-x2· 1 1· x2 · 3 x2≥ 1 minimize 3 y2 ,0-y0 ,2+ y1 ,2 – y0 ,1 when y2,0-y0,1-y0,2= 1 y0,2+ y1,2= 2 y0,1-y1,2= -3

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Zone LP Min Cost Flow

Exploiting duality

minimize 3 x 1-2 x 2+ 7 when x1-x2· 1 1· x2 · 3 x2≥ 1 minimize 3 y2 ,0-y0 ,2+ y1 ,2 – y0 ,1 when y2,0-y0,1-y0,2= 1 y0,2+ y1,2= 2 y0,1-y1,2= -3

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Aircraft Landing

Using MCF/ Netsim plex

Rasm ussen, Larsen, Subram ani TACAS0 4

SLIDE 49

Optim al Conditional Reachability

with Jacob I. Rasmussen

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Informationsteknologi

UC UCb EXAMPLE: Optimal rescue plan for cars with

different subscription rates for city driving !

SAFE

Golf Citroen BMW Datsun

9 2 3 10

5 10 20 25

UNSAFE

My CAR!

Minimizes Cost MYCAR subject to

Cost Citroen · 6 0 Cost BMW · 9 0 Cost Datsun · 1 0

min Cost MYCAR= 2 7 0 tim e = 70

CONDI TI ONAL

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Optim al Conditional Reachability

c’ = 1 d’ = 4 c’ = 2 d’ = 1

☺

l1 l2 l3 x · 2 y: = 0 d+ = 1 x ≥ 2 y ≥ 1 x · 3 y · 2 y: = 0

PROBLEM: Reach l3 in a way which minimizes c subject to d · 4 PROBLEM: Reach l3 in a way which minimizes c subject to d · 4

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

Dual-priced TA

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Discrete Trajectories

c’ = 1 d’ = 4 c’ = 2 d’ = 1 ☺ l1 l2 l3 x · 2 y: = 0 d+ = 1 x ≥ 2 y ≥ 1 x · 3 y · 2 y: = 0

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 1,4 2,1 2,1 2,1 2,1 2,1 0,1 0,1 0,1 0,0 0,0 0,0 0,0

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Discrete Trajectories

c’ = 1 d’ = 4 c’ = 2 d’ = 1 ☺ l1 l2 l3 x · 2 y: = 0 d+ = 1 x ≥ 2 y ≥ 1 x · 3 y · 2 y: = 0

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 1,4 2,1 2,1 2,1 2,1 2,1 0,1 0,1 0,1 0,0 0,0 0,0 0,0

4 2 6 10 8 6 4 2 4

d c

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54

Multiple Objective Scheduling

P2 P1 P6 P3 P4 P7 P5

1 6 ,1 0 2 ,3 2 ,3 6 ,6 1 0 ,1 6 2 ,2 8 ,2

4 W 3 W

cost1’==4 cost2’==3

3 W

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55

Multiple Objective Scheduling

P2 P1 P6 P3 P4 P7 P5

1 6 ,1 0 2 ,3 2 ,3 6 ,6 1 0 ,1 6 2 ,2 8 ,2

4 W 3 W

cost1’==4 cost2’==3 cost1 cost2

Pareto Frontier

T h e P a r e t

F

r

n

t i e r f

r

R e a c h a b i l i t y i n M u l t i P r i c e d T i m e d A u t

m

a t a i s c

m

p u t a b l e

[Illum, Larsen FoSSaCS05]

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Dual Priced Zones

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Dual Priced Zones

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Reset

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Reset

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Reset

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Reset

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Exploration

c’ = 1 d’ = 4 c’ = 2 d’ = 1 ☺ l1 l2 l3 x · 2 y: = 0 d+ = 1 x ≥ 2 y ≥ 1 x · 3 y · 2 y: = 0

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Exploration

c’ = 1 d’ = 4 c’ = 2 d’ = 1 ☺ l1 l2 l3 x · 2 y: = 0 d+ = 1 x ≥ 2 y ≥ 1 x · 3 y · 2 y: = 0

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Exploration

c’ = 1 d’ = 4 c’ = 2 d’ = 1 ☺ l1 l2 l3 x · 2 y: = 0 d+ = 1 x ≥ 2 y ≥ 1 x · 3 y · 2 y: = 0

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Term ination

T H E O R E M O p t i m a l c

n

d i t i

n

a l r e a c h a b i l i t y f

r

m u l t i

p

r i c e d T A i s c

m

p u t a b l e . T H E O R E M O p t i m a l c

n

d i t i

n

a l r e a c h a b i l i t y f

r

m u l t i

p

r i c e d T A i s c

m

p u t a b l e .

SLIDE 66

BRI CS@Aalborg FMT@Tw ente

Optimal Infinite Scheduling

w ith Ed Brinksm a Patricia Bouyer Arne Skou

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68

Optim al I nfinite Scheduling

Maximize throughput: i.e. maximize Reward / Time in the long run!

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Optim al I nfinite Scheduling

Minimize Energy Consumption: i.e. minimize Cost / Time in the long run

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70

Optim al I nfinite Scheduling

Maximize throughput: i.e. maximize Reward / Cost in the long run

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

Golf Citroen BMW Datsun

9 2 3 10

5 10 20 25

maximal 100 min. at each location

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W orkplan I

Datsun

U

BMW

U

Citroen

U

Golf

U

Datsun

S

BMW

S

Citroen

U

Golf

U

Datsun

U

BMW

U

Citroen

U

Golf

U

Datsun

U

BMW

S

Citroen

U

Golf

S

Datsun

U

BMW

U

Citroen

U

Golf

U

Datsun

S

BMW

U

Citroen

S

Golf

U

Datsun

U

BMW

U

Citroen

U

Golf

U

Datsun

S

BMW

U

Citroen

S

Golf

U ε(25) ε(25) ε(25) ε(25) ε(20) ε(20) ε(25) ε(25)

275 275 300 300 300 300 300 300 Value of workplan: (4 x 3 0 0 ) / 90 = 1 3 .3 3

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W orkplan I I

Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf Datsun BMW Citroen Golf

25/ 125 5/ 25 20/ 180 10/ 90 5/ 10 25/ 125 10/ 130 5/ 65 25/ 225 10/ 90 10/ 0 10/ 0 5/ 10 25/ 50

Value of workplan: 5 6 0 / 100 = 5 .6

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Cost Optim al Scheduling = Optim al I nfinite Path

c1 c2 c3 cn t1 t2 t3 tn

σ

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

Accumulated cost Accumulated time

¬(Car0.Err or Car1.Err or …)

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Cost Optim al Scheduling = Optim al I nfinite Path

c1 c2 c3 cn t1 t2 t3 tn

σ

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

Accumulated cost Accumulated time

¬(Car0.Err or Car1.Err or …)

THEOREM: σ* is computable THEOREM: σ* is computable

Bouyer, Brinksma, Larsen HSCC’04

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Application

Dynam ic Voltage Scaling

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Approxim ate Optim al Schedule

E[] (not (Golf.Err or Datsun.Err or BMW.Err or Citroen.Err) 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> = N T< N X

X X

Optimal infinite schedule modulo cost-horizon C= M

T< N T< N T> = N