Optimal Scheduling. Kim G. Larsen DENMARK
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1 Priced Timed Automata Optimal Scheduling Kim G. Larsen Aalborg University it DENMARK
2 Overview Timed Automata Scheduling Priced Timed Automata Optimal Reachability Optimal Infinite Scheduling Multi Objectives Energy Automata CLASSIC CORA TIGA ECDAR TRON SMC ARTIST Design PhD School, Beijing, 2011 Kim Larsen [2]
3 Real Time Scheduling Only 1 Pass Cheat is possible (drive close to car with Pass ) UNSAFE 5 10 Pass SAFE CAN The THEY Car & MAKE Bridge IT Problem TO SAFE WITHIN 70 MINUTES??? ARTIST Design PhD School, Beijing, 2011 Kim Larsen [3]
4 Let us play! ARTIST Design PhD School, Beijing, 2011 Kim Larsen [4]
5 Real Time Scheduling Solve Scheduling Problem using UPPAAL UNSAFE 5 10 SAFE ARTIST Design PhD School, Beijing, 2011 Kim Larsen [5]
6 Resources & Tasks Resource Synchronization Task Shared variable ARTIST Design PhD School, Beijing, 2011 Kim Larsen [6]
7 Task Graph Scheduling Example A 1 B C D 2 Compute : (D ( C ( A B )) (( A B ) ( C D )) using 2 processors C 5 D P1 6 P2 1 P1 (fast) P2 (slow) 2ps 3ps 5ps 7ps ARTIST Design PhD School, Beijing, 2011 Kim Larsen [7] time
8 Task Graph Scheduling Example A 1 B C D 2 Compute : (D ( C ( A B )) (( A B ) ( C D )) using 2 processors 3 4 C 5 D P1 P2 6 P1 (fast) P2 (slow) 2ps 3ps 5ps 7ps ARTIST Design PhD School, Beijing, 2011 Kim Larsen [8] time
9 Task Graph Scheduling Example A 1 B C D 2 Compute : (D ( C ( A B )) (( A B ) ( C D )) using 2 processors 3 4 C 5 D P1 P2 6 P1 (fast) P2 (slow) 2ps 3ps 5ps 7ps ARTIST Design PhD School, Beijing, 2011 Kim Larsen [9] time
10 Task Graph Scheduling Example A 1 B C D 2 Compute : (D ( C ( A B )) (( A B ) ( C D )) using 2 processors 3 4 C 5 D P1 P2 6 P1 (fast) P2 (slow) 2ps 3ps 5ps 7ps E<> (Task1.End and and Task6.End) ARTIST Design PhD School, Beijing, 2011 Kim Larsen [10] time
11 Experimental Results Symbolic A Branch-&-Bound 60 sec Abdeddaïm, Kerbaa, Maler ARTIST Design PhD School, Beijing, 2011 Kim Larsen [11]
12 Priced Timed Automata
13 EXAMPLE: Optimal rescue plan for cars with different subscription rates for city driving! 5 Golf SAFE Citroen BMW Datsun 3 10 OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65! ARTIST Design PhD School, Beijing, 2011 Kim Larsen [13]
14 Experiments COST-rates G C B D SCHEDULE COST TIME #Expl #Pop d Min Time CG> G< BD> C< CG> CG> G< BG> G< GD> GD> G< CG> G< BG> CG> G< BD> C< CG> CD> C< CB> C< CG> BD> B< CB> C< CG> time< ARTIST Design PhD School, Beijing, 2011 Kim Larsen [14]
15 Task Graph Scheduling Revisited A 1 B C D 2 Compute : (D ( C ( A B )) (( A B ) ( C D )) using 2 processors 3 4 C 5 D P1 P2 6 P1 (fast) P2 (slow) Idle 2ps 3ps 1oW Idle 5ps 7ps 20W ENERGY: 5 10 In use 90W 30W In use ARTIST Design PhD School, Beijing, 2011 Kim Larsen [15] time
16 Task Graph Scheduling Revisited A 1 B C D 2 Compute : (D ( C ( A B )) (( A B ) ( C D )) using 2 processors 3 4 C 5 D P1 P2 6 P1 (fast) P2 (slow) Idle 2ps 3ps 10W Idle 5ps 7ps 20W ENERGY: 5 10 In use 90W 30W In use ARTIST Design PhD School, Beijing, 2011 Kim Larsen [16] time
17 Task Graph Scheduling Revisited A 1 B C D 2 Compute : (D ( C ( A B )) (( A B ) ( C D )) using 2 processors 3 4 C 5 D P1 P2 6 P1 (fast) P2 (slow) Idle 2ps 3ps 10W Idle 5ps 7ps 20W ENERGY: 5 10 In use 90W 30W In use ARTIST Design PhD School, Beijing, 2011 Kim Larsen [17] time
18 A simple example ARTIST Design PhD School, Beijing, 2011 Kim Larsen [18]
19 A simple example Q: What is cheapest cost for reaching? ARTIST Design PhD School, Beijing, 2011 Kim Larsen [19]
20 Corner Point Regions THM [Behrmann, Fehnker..01] [Alur,Torre,Pappas 01] Optimal reachability is decidable for PTA THM [Bouyer, Brojaue, Briuere, Raskin 07] Optimal reachability is PSPACE-complete for PTA ARTIST Design PhD School, Beijing, 2011 Kim Larsen [20]
![PTA THM [Bouyer, Brojaue, Briuere, Raskin 07] Optimal reachability](/docs-images/47/20101959/images/page_20.jpg "is PSPACE-complete for PTA 3 0 3 0 0 0 0 0 ARTIST Design PhD School,")
21 Priced Zones [CAV01] A zone Z: 1 x 2 Æ 0 y 2 Æ x - y 0 A cost function C C(x,y)= 2 x - 1 y 3 ARTIST Design PhD School, Beijing, 2011 Kim Larsen [21]
22 Priced Zones Reset [CAV01] Z[x=0]: x=0 Æ 0 y 2 A zone Z: 1 x 2 Æ 0 y 2 Æ x - y 0 C = 1 y 3 C= -1 y 5 A cost function C C(x,y) = 2 x - 1 y 3 ARTIST Design PhD School, Beijing, 2011 Kim Larsen [22]
23 Symbolic Branch & Bound Algorithm Z' Z Z is bigger & cheaper than Z is a well-quasi ordering which guarantees termination! ARTIST Design PhD School, Beijing, 2011 Kim Larsen [23]
24 Example: Aircraft Landing cost e(t-t) E dl(t-t) 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 Planes have to keep separation distance to avoid turbulences caused by preceding planes ARTIST Design PhD School, Beijing, 2011 Kim Larsen [24] Runway
25 Example: Aircraft Landing x <= 5 x >= 4 x=5 4 earliest landing time land! cost=2 5 target time x <= 5 9 latest time x <= 9 cost =3 cost =1 3 cost rate for being early 1 cost rate for being late x=5 land! 2 fixed cost for being late Planes have to keep separation distance to avoid turbulences caused by preceding planes ARTIST Design PhD School, Beijing, 2011 Kim Larsen [25] Runway
26 Aircraft Landing Source of examples: Baesley et al 2000 ARTIST Design PhD School, Beijing, 2011 Kim Larsen [26]
27 Symbolic Branch & Bound Algorithm Zone based Linear Programming Problems (dualize) Min Cost Flow ARTIST Design PhD School, Beijing, 2011 Kim Larsen [27]
28 Aircraft Landing (revisited) Using MCF/Netsimplex [TACAS04] ARTIST Design PhD School, Beijing, 2011 Kim Larsen [28]
29 Optimal Infinite Schedule ARTIST Design PhD School, Beijing, 2011 Kim Larsen [29]
30 EXAMPLE: Optimal WORK plan for cars with different subscription rates for city driving! Infor rmatio onstek knolog gi Golf Citroen 9 2 BMW Datsun 3 10 maximal 100 min. at each location UCb
31 Workplan I Infor rmatio onstek knolog gi UCb Golf Citroen U U BMW Datsun U U Golf Citroen S U BMW Datsun (25) Golf Citroen U U BMW Datsun (25) 275 S S 275 (20) Golf Citroen U U BMW Datsun (25) Golf Citroen U U BMW Datsun U U (25) 300 Golf Citroen U S BMW Datsun S U U U U S (20) 300 (25) 300 Golf Citroen U U BMW Datsun (25) Golf Citroen U S BMW Datsun U U 300 U S Value of workplan: (4 x 300) / 90 = 13.33
32 Workplan II Infor rmatio onstek knolog gi UCb Golf Citroen 25/125 Golf Citroen Golf Citroen Golf Citroen 5/25 20/180 BMW Datsun BMW Datsun BMW Datsun BMW Datsun Golf BMW Golf BMW Citroen Datsun 10/130 5/65 25/225 Citroen Datsun Value of workplan: 560 / 100 = 5.6 Golf Citroen BMW Datsun 25/125 Golf Citroen BMW Datsun 5/10 10/90 Golf Citroen BMW Datsun Golf Citroen Golf Citroen Golf Citroen 10/90 10/0 BMW Datsun BMW Datsun BMW Datsun 25/50 Golf Citroen BMW Datsun 5/10 10/0 Golf Citroen BMW Datsun
33 Optimal Infinite Scheduling Maximize throughput: i.e. maximize Reward / Time in the long run! ARTIST Design PhD School, Beijing, 2011 Kim Larsen [33]
34 Optimal Infinite Scheduling Minimize Energy Consumption: i.e. minimize Cost / Time in the long run ARTIST Design PhD School, Beijing, 2011 Kim Larsen [34]
35 Optimal Infinite Scheduling Maximize throughput: i.e. maximize Reward / Cost in the long run ARTIST Design PhD School, Beijing, 2011 Kim Larsen [35]
36 Mean Pay-Off Optimality Bouyer, Brinksma, Larsen: HSCC04,FMSD07 Accumulated cost c 3 c n c 1 c 2 r 3 r n r 1 r 2 BAD Accumulated reward Value of path : val( ) = lim n c n /r n Optimal Schedule : val( ) = inf val( ) ARTIST Design PhD School, Beijing, 2011 Kim Larsen [36]
37 Discount Optimality 1 : discounting factor Larsen, Fahrenberg: INFINITY 08 Cost of time t n ) c(t 3) c(t n ) c(t 1 ) c(t 2) t 1 t 2 t 3 t n BAD Time of step n Value of path : val( ) = Optimal Schedule : val( ) = inf val( ) ARTIST Design PhD School, Beijing, 2011 Kim Larsen [37]
38 Soundness of Corner Point Abstraction ARTIST Design PhD School, Beijing, 2011 Kim Larsen [38]
39 Application Dynamic Voltage Scaling Informationsteknologi UCb
40 Multiple Objective Scheduling 16,10 P 2 P 1 2,3 2,3 cost 1 ==4 cost 2 ==3 1 cost 2 6,6 10,16 P 6 P 3 P 4 Pareto Frontier P 7 P 5 2,22 8,2 4W 3W cost1 1 ARTIST Design PhD School, Beijing, 2011 Kim Larsen [40]
41 Experimental Results Warehouse itunes ARTIST Design PhD School, Beijing, 2011 Kim Larsen [41]
42 Experimental Results ARTIST Design PhD School, Beijing, 2011 Kim Larsen [42]
43 Energy Automata
44 Managing Resources ARTIST Design PhD School, Beijing, 2011 Kim Larsen [44]
45 Consuming & Harvesting Energy Maximize throughput while respecting: 0 E MAX ARTIST Design PhD School, Beijing, 2011 Kim Larsen [45]
46 Energy Constrains Energy is not only consumed but may also be regained The aim is to continously satisfy some energy constriants ARTIST Design PhD School, Beijing, 2011 Kim Larsen [46]
47 Results (so far) Bouyer, Fahrenberg, Larsen, Markey, Srba: FORMATS 2008 ARTIST Design PhD School, Beijing, 2011 Kim Larsen [47]
48 Discrete Updates on Edges ARTIST Design PhD School, Beijing, 2011 Kim Larsen [48]
49 New Approach: Energy Functions Maximize energy along paths Use this information to solve general problem ARTIST Design PhD School, Beijing, 2011 Kim Larsen [49]
50 Energy Function General Strategy Spend just enough time to survive the next negative update ARTIST Design PhD School, Beijing, 2011 Kim Larsen [50]
51 Exponential PTA General Strategy Spend just enough time to survive the next negative update so that after next negative update there is a certain positive amount! Minimal Fixpoint: ARTIST Design PhD School, Beijing, 2011 Kim Larsen [51]
52 Exponential PTA Thm [HSCC10]: Lower-bound problem is decidable for linear and exponential 1-clock PTAs with negative discrete Energy updates. Function ARTIST Design PhD School, Beijing, 2011 Kim Larsen [52]
53 Conclusion Priced Timed Automata a uniform framework for modeling and solving dynamic ressource allocation problems! Not mentioned here: Model Checking Issues (ext. of CTL and LTL). Future work: Zone-based algorithm for optimal infinite i runs. Approximate solutions for priced timed games to circumvent undecidablity issues. Open problems for Energy Automata. Approximate algorithms for optimal reachability ARTIST Design PhD School, Beijing, 2011 Kim Larsen [53]
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