Optimal & Real Time Scheduling. using Model Checking Technology. Kim G. Larsen

Transcription

1 Optimal & Real Time Scheduling using Model Checking Technology Kim G. Larsen

2 AIPS a partial observation New community New faces New techniques

3 April 2002 June 2005 IST Academic partners: Nijmegen Aalborg Dortmund Grenoble Marseille Twente Weizmann Industrial Partners Axxom Bosch Cybernetix Terma

4 SIDMAR Overview Crane A Machine 1 Machine OBJECTIVES powerful, unifying mathematical modelling efficient computerized problem-solving tools distributed real-time systems time-dependent behaviour and dynamic resource allocation TIMED AUTOMATA INPUT sequence of steel loads ( pigs ) GOAL: Maximize utilization of of plant Machine 4 Machine Crane Lane 2 Buffer Storage Continuos Casting Machine Lane 1 OUTPUT sequence of higher quality steel LTR Project VHS (Verification of Hybrid systems) SIDMAR Modelling Crane A Machine 1 Machine 2 Machine 3 Machine 4 Machine 5 Crane B Lane 2 Buffer Storage Place Continuos Casting Machine Lane 1 A Single Load UPPAAL Model

5 Case Studies Cybernetix: Smart Card Personalization Terma: Memory Interface Bosch: Car Periphery Sensing AXXOM: Lacquer Production Benchmarks Smart Memory Card Management CPS: Informal description Personalization Radar Video Processing Subsystem Advanced CPS obtains Piles of blank Nois and cards e makes Reduction available for Techniques other systems information about environment of a car. This information may be used for: Parking assistance Pre-crash detection Blind spot supervision Lane change Test assistance Stop & go and possibly reject Etc Adder 1 S' S = A + S' - A' 16 (100 MHz) CORA B A' Adder 2 T = B + T' - B' Cybenetix, France Personalisation Input A Input B Product flow of a Product 8 (100MHz) 8 (100 MHz) Buffer (100 MHz) Output S Airport Surveillance Based on Short Range CLASSIC Radar (SRR) technology B' Output T 8 (100MHz) 16 (100 MHz) 8 (100MHz) 8 (100MHz) T' 16 (100 MHz) T 16 (100 MHz) S Cos tal Surveillance Maximize detection, throughput parking 1 Kbytes Buffer (100 MHz) 1 Kbytes Buffer bytes Buffer 4 2 Kbytes Buffer bytes Buffer bytes Buffer 7 2 Kbytes 3,2 0,5 e e this 0,4 case 1,5 study e e 1,4 e 0,3 e 1,3 e 2,5 echo Combiner e 0,2 e 2,4 e 1,2 e 2,3 (VP3) e (currently 2,2 e 3,5 2 sensors) e3,3 e e 3,4 Sweep Integration 256 (100 MHz) 256 (100 MHz) 256 (100 MHz) 256 (100 MHz) 256 (100 MHz) Buffer (100 MHz) 2 Kbytes Buffer (100 MHz) 2 Kbytes Arbiter GHz GHz Ed Brinksma Car Periphery Supervision System: Case Study 3 Überschrift Dispersion Dose Spinner Mixing Vessel Laboratory 128 (200 MHz) SDRAM Frequency Diversity The CPS considered in One sensor group only Only the front sensors combiner and corresponding controllers Application: pre-crash assistance, stop & go Storage Filling Stations UC b 2 UC b Axxom Software AG Seite: 3

6 Overview Timed Automata & Scheduling Priced Timed Automata and Optimal Scheduling AMETIST Results Beyond Static Scheduling

7 Overview Timed Automata & Scheduling Priced Timed Automata and Optimal Scheduling AMETIST Results Beyond Static Scheduling

8 Timed Automata [Alur & Dill 89] Reset Resource Invariant Synchronization Guard

9 Timed Automata [Alur & Dill 89] Resource Reset Invariant Synchronization Guard Semantics: ( Idle, x=0 ) ( Idle, x=2.5) ( InUse, x=0 ) ( InUse, x=5) ( Idle, x=5) ( Idle, x=8) ( InUse, x=0 ) d(2.5) use? d(5) done! d(3) use?

10 Timed Automata [Alur & Dill 89] Resource Reset Invariant Synchronization Guard Semantics: ( Idle, x=0 ) ( Idle, x=2.5) ( InUse, x=0 ) ( InUse, x=5) ( Idle, x=5) ( Idle, x=8) ( InUse, x=0 ) d(2.5) use? d(5) done! d(3) use?

11 Timed Automata [Alur & Dill 89] Resource Reset Invariant Synchronization Guard Semantics: ( Idle, x=0 ) ( Idle, x=2.5) ( InUse, x=0 ) ( InUse, x=5) ( Idle, x=5) ( Idle, x=8) ( InUse, x=0 ) d(2.5) use? d(5) done! d(3) use?

12 Timed Automata [Alur & Dill 89] Resource Reset Invariant Synchronization Guard Semantics: ( Idle, x=0 ) ( Idle, x=2.5) ( InUse, x=0 ) ( InUse, x=5) ( Idle, x=5) ( Idle, x=8) ( InUse, x=0 ) d(2.5) use? d(5) done! d(3) use?

13 Timed Automata [Alur & Dill 89] Resource Reset Invariant Synchronization Guard Semantics: ( Idle, x=0 ) ( Idle, x=2.5) ( InUse, x=0 ) ( InUse, x=5) ( Idle, x=5) ( Idle, x=8) ( InUse, x=0 ) d(2.5) use? d(5) done! d(3) use?

14 Composition Resource Synchronization Shared variable Semantics: ( Idle, Init, B=0, x=0) ( Idle, Init, B=0, x= ) ( InUse, Using, B=6, x=0 ) ( InUse, Using, B=6, x=6 ) ( Idle, Done, B=6, x=6 ) Task d(3) use d(6) done

15 Composition Resource Synchronization Shared variable Semantics: ( Idle, Init, B=0, x=0) ( Idle, Init, B=0, x= ) ( InUse, Using, B=6, x=0 ) ( InUse, Using, B=6, x=6 ) ( Idle, Done, B=6, x=6 ) Task d(3) use d(6) done

16 Composition Resource Synchronization Shared variable Semantics: ( Idle, Init, B=0, x=0) ( Idle, Init, B=0, x= ) ( InUse, Using, B=6, x=0 ) ( InUse, Using, B=6, x=6 ) ( Idle, Done, B=6, x=6 ) Task d(3) use d(6) done

17 Composition Resource Synchronization Shared variable Semantics: ( Idle, Init, B=0, x=0) ( Idle, Init, B=0, x= ) ( InUse, Using, B=6, x=0 ) ( InUse, Using, B=6, x=6 ) ( Idle, Done, B=6, x=6 ) Task d(3) use d(6) done

18 Jobshop Scheduling J O B s Kim Maria Nicola Sport 2. 5 min min 4. 1 min R E S O U R C E S Economy 4. 1 min min min Local News 3. 3 min 3. 1 min min Problem: compute the minimal MAKESPAN Comic Stip min 4. 1 min min

19 Jobshop Scheduling in UPPAAL Kim Maria Sport 2. 5 min min Economy 4. 1 min min Local News 3. 3 min 3. 1 min Comic Stip min 4. 1 min Nicola 4. 1 min min min min

20 Experiments [TACAS 2001] B-&-B algorithm running for 60 sec. m=5 m=10 j=10 j=15 j=20 j=10 j=15 Lawrence Lawrence Job Job Shop Shop Problems Problems

21 Symbolic Transitions using Zones x>3 a y:=0 n m y y x x 1<=x<=4 1<=y<=3 delays to conjuncts to projects to y y x x 1<=x, 1<=y -2<=x-y<=3 3<x, 1<=y -2<=x-y<=3 3<x, y=0 Thus Thus (n,1<=x<=4,1<=y<=3) =a =a => => (m,3<x, y=0) y=0)

22 Forward Rechability Init -> Final? Waiting n,z Init n,z m,u Passed Final INITIAL Passed := Ø; Waiting := {(n0,z0)} REPEAT - pick (n,z) in Waiting - if for some Z Z (n,z ) in Passed then STOP -else/explore/ add { (m,u) : (n,z) => (m,u) } to Waiting; Add (n,z) to Passed UNTIL Waiting = Ø or Final is in Waiting

23 Canonical Datastructures for Zones x1-x2<=4 x1-x2<=4 x2-x1<=10 x2-x1<=10 x3-x1<=2 x3-x1<=2 x2-x3<=2 x2-x3<=2 x0-x1<=3 x0-x1<=3 x3-x0<=5 x3-x0<=5 3 x1 x x2 x3 Shortest Path Reduction O(n^3) x1 2 Shortest Path Closure O(n^3) -4 x2 2 3 x1 DBMs Bellmann x x0 1 x3 5 2 Space worst O(n^2) practice O(n) x0 x3 Minimal Constraint Form RTSS 97

24 Datastructures for Zones DBM package -4 Minimal Constraint Form [RTSS97] Clock Difference Diagrams [CAV99] x1 x x0 1 5 x3 2

25 Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration : (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. 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

26 Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration : (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. 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

27 Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration : (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. 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

28 Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration : (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. 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

29 Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration : (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. 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

30 Task Graph Scheduling Optimal Static Task Scheduling Task P={P 1,.., P m } Machines M={M 1,..,M n } Duration : (P M) N < : p.o. on P (pred.) 16,10 2,3 P 2 P 1 2,3 6,6 10,16 P 6 P 3 P 4 P 7 P 5 2,2 8,2 M = {M 1,M 2 }

31 Experimental Results Abdeddaïm, Kerbaa, Maler

32 Optimal Task Graph Scheduling Power-Optimality Energy-rates: C : M NxN P 2 P 1 16,10 2,3 2,3 cost ==1 6,6 10,16 P 6 P 3 P 4 cost ==4 P 7 P 5 2,2 8,2 1W 4W 2W 3W

33 Overview Timed Automata & Scheduling Priced Timed Automata and Optimal Scheduling AMETIST Results Beyond Static Scheduling

34 Priced Timed Automata Timed Automata + COST variable Behrmann, Fehnker, et all (HSCC 01) Alur, Torre, Pappas (HSCC 01) cost rate l l 1 2 x 2 l 3 3 y 0 y 4 c =4 c =2 x:=0 c+=4 c+=1 x:=0 y 4 cost update

35 Priced Timed Automata Timed Automata + COST variable Behrmann, Fehnker, et all (HSCC 01) Alur, Torre, Pappas (HSCC 01) cost rate TRACES l l 1 2 x 2 l 3 3 y 0 y 4 c =4 c =2 x:=0 c+=4 c+=1 x:=0 y 4 cost update ε(3) (l 1,x=y=0) (l 1,x=y=3) (l 2,x=0,y=3) (l 3,_,_) c=17

36 Priced Timed Automata Timed Automata + COST variable Behrmann, Fehnker, et all (HSCC 01) Alur, Torre, Pappas (HSCC 01) cost rate TRACES l l 1 2 x 2 l 3 3 y 0 y 4 c =4 c =2 x:=0 c+=4 c+=1 ε(3) x:=0 y 4 (l 1,x=y=0) (l 1,x=y=3) (l 2,x=0,y=3) (l 3,_,_) ε(2.5) cost update Problem : Find the minimum cost of of reaching location l 3 (l 1,x=y=0) (l 1,x=y=2.5) (l 2,x=0,y=2.5) (l 2,x=0.5,y=3) (l 3,_,_) ε(3) (l 1,x=y=0) (l 2,x=0,y=0) (l 2,x=3,y=3) (l 2,x=0,y=3) (l 3,_,_) ε(.5) Efficient Implementation: CAV 01 and and TACAS c=17 c=16 c=11

37 Aircraft Landing Problem cost e*(t-t) E d+l*(t-t) Planes have to keep separation distance to avoid turbulences caused by preceding planes T 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 Runway

38 Modeling ALP with PTA Planes have to keep separation distance to avoid turbulences caused by preceding planes 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 Runway

39 Symbolic A*

40 Zones Operations y Z x

41 Priced Zone CAV 01 y 4 2 Cost ( x, y) -1 Z x = 2 y x + 2

42 Branch & Bound Algorithm

43 Branch & Bound Algorithm

44 Branch & Bound Algorithm

45 Branch & Bound Algorithm

46 Branch & Bound Algorithm

47 Branch & Bound Algorithm Z' Z Z Z is isbigger & cheaper cheaperthan thanzz is is a well-quasi well-quasi ordering orderingwhich guarantees guarantees termination! termination!

48 Experimental Results

49 Modeling APL with PDDL 3 [Dierks VVPS05]

50 Modeling APL with PDDL 3 [Dierks VVPS05]

51 Experimental Results Translating PDDL 3 to PTA [Dierks VVPS05+]

52 Experimental Results PTA versus MILP [Beasley 00]

53 Experimental Results OPTOP PDDL 3 To be provided with guidance from Maria Fox & Drew McDermott

54 Overview Timed Automata & Scheduling Priced Timed Automata and Optimal Scheduling AMETIST Result Beyond Static Scheduling

55 AXXOM Case study Laquer Production Scheduling 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 Product flow of a Product Überschrift Dispersion Dose Spinner Mixing Vessel Laboratory Storage Filling Stations Axxom Software AG Seite: 3 Behrmann, Brinksma, Hendriks, Mader 16 th IFAC World Congress

56 Resources 2 mixing vessels for uni lacquers 3 mixing vessels for metallic/bronce 2 dose spinners 1 dose spinner bronce 1 disperging line 1 predisperser 1 bronce mixer 2 filling lines lab (unlimitted) Product flow of a Product Überschrift Dispersion Storage Dose Spinner Mixing Vessel Filling Stations Laboratory Axxom Software AG Seite: 3

57 Recipes UPPAAL template for metal

58 Orders

59 Instantiated Model State Space Explosion

60 Heuristics Nice heuristics non-overtaking orders of the same recipe cannot overtake each other non-laziness a 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 orders the number of concurrent orders is restricted (number of critical resources can give an indication)

61 Experimental Results #jobs heuristic nl nl, no nl, no g, no uses clock optimization & optimized successor calculation - max. orders term. time - 1 s - 7 s 8 s

62 Results Extended Case storage, delay and setup costs, working hours #jobs work heuristic hrs - Competitive - with Orion-pi results avail. es, no, nl es, no, g es, no, nl avail. es, no, g expl. no max. orders min. cost Order of magnitude faster than MILP, - GAMS/CPLEX 530,771 found in 60 s - 647,410-1,714,875-2,263, ,881,129

63 Overview Timed Automata & Scheduling Priced Timed Automata and Optimal Scheduling AMETIST Result Beyond Static Scheduling

64 Infinite Scheduling E[] ( Kim.x 100 and Jacob.x 90 and Gerd.x 100 )

65 Infinite Optimal Scheduling E[] ( Kim.x 100 and Jacob.x 90 and Gerd.x 100 ) + most minimal limit of cost/time

66 Infinite Optimal Scheduling σ c 3 c n c 1 c 2 t 3 t n t 1 t 2 BAD Accumulated cost Accumulated time Value of path σ: val(σ) = lim n c n /t n Optimal Schedule σ * : val(σ * ) = inf σ val(σ)

67 Infinite Optimal Scheduling σ c 3 c n t 1 t 2 t 3 t n BAD Accumulated cost Accumulated time THEOREM: THEOREM: σ σ * * is is computable computable Bouyer, Brinksma, Larsen HSCC 04 Value of path σ: val(σ) = lim n c n /t n Optimal Schedule σ * : val(σ * ) = inf σ val(σ)

68 Application Dynamic Voltage Scaling

69 Dynamic Scheduling E<> ( Kim.Done and Jacob.Done and Gerd.Done ) + winning / optimal strategy Cost-Optimal Reachability Strategies for Priced Timed Game Automata [Alur et all, ICALP 04] [Bouyer, Cassez. Fleury, Larsen, FSTTCS 04] Time-Optimal Reachability Strategies for Timed Games [Cassez, David, Fleury, Larsen, Lime, CONCUR 05] Uncontrollable timing uncertainty

70 Future Work & Conclusion PDDL 3 versus PTA OPTOP versus UPPAAL Cora Planning versus Model Checking Improved heuristic search FF s relaxed planning approach pattern databases beam search externalization Please visit

71 Thank you for your attention!