Pr Priced iced Ti Time med Au Automata mata and Ti Time med - PowerPoint PPT Presentation

Pr Priced iced Ti Time med Au Automata mata and Ti Time med Ga Game mes Ki Kim m G. . La Lars rsen Aa Aalborg org Unive versity rsity, , DENMAR NMARK

Sc Sche heduling uling Pric iced Tim imed Automa mata and Sy Synt nthe hesis sis Tim imed Ga Games es Ki Kim m G. . La Lars rsen Aa Aalborg org Unive versity rsity, , DENMAR NMARK

Ov Overview view  Timed med Automata & UPPAAL  Symb mboli olic Verification & UPPAAL Engine, Options CLASSIC  Priced iced Timed Automata and Timed Game ames CORA TIGA  Stochastic chastic Timed Automata Statist tistical ical Model Checking ECDAR SMC (Lecture+Exercise) 4 TRON VTSA Summer r School, l, 2013 2013. Kim Larse sen [3]

Resourc ources s & Ta Tasks sks Resource Synchronization Task Shared variable VTSA Summer r School, l, 2013 2013. Kim Larse sen [4]

Task sk Graph aph Sched heduling uling – Example mple Compute : C D B (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) 1 2 * + A 4 using 2 processors P2 (slow) ‏ P1 (fast) ‏ 4 3 * + C + + 2ps 5ps * * 3ps 7ps 5 6 * + D 5 10 15 20 25 P1 2 3 5 6 P2 1 4 time VTSA Summer r School, l, 2013 2013. Kim Larse sen [5]

Task sk Graph aph Sched heduling uling – Example mple Compute : C D B (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) 1 2 * + A using 2 processors P2 (slow) ‏ P1 (fast) ‏ 4 3 * + C + + 2ps 5ps * * 3ps 7ps 5 6 * + D 5 10 15 20 25 P1 5 4 6 1 3 P2 2 time VTSA Summer r School, l, 2013 2013. Kim Larse sen [6]

Task sk Graph aph Sched heduling uling – Example mple Compute : C D B (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) 1 2 * + A using 2 processors P2 (slow) ‏ P1 (fast) ‏ 4 3 * + C + + 2ps 5ps * * 3ps 7ps 5 6 * + D 5 10 15 20 25 P1 5 4 6 1 3 P2 2 time VTSA Summer r School, l, 2013 2013. Kim Larse sen [7]

Task sk Graph aph Sched heduling uling – Example mple Compute : C D B (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) 1 2 * + A using 2 processors P2 (slow) ‏ P1 (fast) ‏ 4 3 * + C + + 2ps 5ps * * 3ps 7ps 5 6 * + D 5 10 15 20 25 P1 5 4 6 1 3 P2 2 E<> (Task1 .End‏and‏…‏and‏ Task6.End) time VTSA Summer r School, l, 2013 2013. Kim Larse sen [8]

Experimenta perimental l Results ults Symbolic A* Branch-&-Bound 60 sec Abdeddaïm, Kerbaa, Maler VTSA Summer r School, l, 2013 2013. Kim Larse sen [9]

Jo Jobshop hop Sched heduling uling [TACAS’ 2001] Sport Economy Local News Comic Stip Kim 2 . 5 min 4 . 1 min 3 . 3 min 1 . 10 min Jüri 1. 10 min 2 . 20 min 3 . 1 min 4 . 1 min Jan 4 . 1 min 1 . 13 min 3 . 11 min 2 . 11 min Wang 1 . 1 min 2 . 1 min 3 . 1 min 4 . 1 min NP-hard Problem: compute the minimal MAKESPAN Simulated annealing Shiffted bottleneck Branch-and-Bound VTSA Summer r School, l, 2013. Kim Larse sen [10 10] Gentic Algorithms

Jo Jobshop hop Sched heduling uling in n UPPAAL AAL VTSA Summer r School, l, 2013 2013. Kim Larse sen [11 11]

Pr Pric iced ed Tim imed ed Aut Autom omata ta

Tas ask Gra raph ph Scheduling heduling – Revis visited ited Compute : C D B (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) 1 2 * + A using 2 processors P2 (slow) P1 (fast) 4 3 * + C + + 2ps 5ps * * 3ps 7ps 5 6 * + Idle Idle 1oW 20W D ENERGY: In use In use 90W 30W 5 10 15 20 25 P1 5 4 6 1 3 P2 2 time VTSA Summer r School, l, 2013 2013. Kim Larse sen [13 13]

Tas ask Gra raph ph Scheduling heduling – Revis visited ited Compute : C D B (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) 1 2 * + A using 2 processors P2 (slow) ‏ P1 (fast) ‏ 4 3 * + C + + 2ps 5ps * * 3ps 7ps 5 6 * + Idle Idle 10W 20W D ENERGY: In use In use 90W 30W 5 10 15 20 25 P1 4 1 3 P2 2 5 6 time VTSA Summer r School, l, 2013 2013. Kim Larse sen [14 14]

Tas ask Gra raph ph Scheduling heduling – Revis visited ited Compute : C D B (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) 1 2 * A + using 2 processors P2 (slow) ‏ P1 (fast) ‏ 4 3 * + C + + 2ps 5ps * * 3ps 7ps 5 6 * + Idle Idle 10W 20W D ENERGY: In use In use 90W 30W 5 10 15 20 25 P1 4 1 3 P2 2 5 6 time VTSA Summer r School, l, 2013 2013. Kim Larse sen [15 15]

A si simple mple examp mple le VTSA Summer r School, l, 2013 2013. Kim Larse sen [16 16]

A si simple mple examp mple le Q : What is cheapest cost for reaching ? VTSA Summer r School, l, 2013 2013. Kim Larse sen [17 17]

Cor orner ner Poi oint nt Regi gions ons 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 3 0 3 0 0 0 0 0 VTSA Summer r School, l, 2013 2013. Kim Larse sen [18 18]

Priced iced Zo Zone nes [CAV01 01] 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 VTSA Summer r School, l, 2013 2013. Kim Larse sen [19 19]

Priced iced Zo Zone nes – Reset [CAV01 01] A zone Z : 1 · x · 2 Æ Z [x=0] : 0 · y · 2 Æ x=0 Æ x - y ¸ 0 0 · y · 2 C = 1 ¢ y + 3 A cost function C C(x,y) = 2 ¢ x - 1 ¢ y + 3 C = -1 ¢ y + 5 VTSA Summer r School, l, 2013 2013. Kim Larse sen [20 20]

Sym ymbolic bolic Bra ranch nch & & Bound ound Al Algo gorithm rithm 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 Z  ' Z Z’ is bigger & cheaper than Z · is a well-quasi ordering which guarantees termination! VTSA Summer r School, l, 2013 2013. Kim Larse sen [21 21]

Example ample: : Aircraft craft Land nding ing cost E earliest landing time d + l *(t-T) T target time e *(T-t) L latest time e cost rate for being early l cost rate for being late d fixed cost for being late t E T L Planes have to keep separation distance to avoid turbulences caused by preceding planes VTSA Summer r School, l, 2013 2013. Kim Larse sen [22 22] Runway

Example ample: : Aircraft craft Land nding ing x <= 5 x >= 4 x=5 4 earliest landing time 5 target time land! cost+=2 9 latest time x <= 5 x <= 9 3 cost rate for being early cost’= 3 cost’= 1 1 cost rate for being late x=5 2 fixed cost for being late land! Planes have to keep separation distance to avoid turbulences caused by preceding planes VTSA Summer r School, l, 2013 2013. Kim Larse sen [23 23] Runway

Aircraft craft Land nding ing Source of examples: Baesley et al’ 2000 VTSA Summer r School, l, 2013 2013. Kim Larse sen [24 24]

Op Opti timal mal In Infi finite nite Sched hedule ule VTSA Summer r School, l, 2013 2013. Kim Larse sen [25 25]

Op Opti timal mal In Infi finite nite Sched heduling uling Maximize throughput: i.e. maximize Reward / Time in the long run! VTSA Summer r School, l, 2013 2013. Kim Larse sen [26 26]

Op Opti timal mal In Infi finite nite Sched heduling uling Minimize Energy Consumption: i.e. minimize Cost / Time in the long run VTSA Summer r School, l, 2013 2013. Kim Larse sen [27 27]

Op Opti timal mal In Infi finite nite Sched heduling uling Maximize throughput: i.e. maximize Reward / Cost in the long run VTSA Summer r School, l, 2013 2013. Kim Larse sen [28 28]

Mean an Pay ay-Off Off Op Optimality imality Bouyer, Brinksma, Larsen: HSCC04,FMSD07 Accumulated cost c 3 c n c 1 c 2 r 3 r n s r 1 r 2 Accumulated reward : BAD Value of path s : val( s ) = lim n !1 c n /r n Optimal Schedule s * : val( s * ) = inf s val( s ) VTSA Summer r School, l, 2013 2013. Kim Larse sen [29 29]

Discount count Op Optimality imality  < 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 3 t n s t 1 t 2 Time of step n : BAD Value of path s : val( s ) = Optimal Schedule s * : val( s * ) = inf s val( s ) VTSA Summer r School, l, 2013 2013. Kim Larse sen [30 30]

Soundness undness of f Corner orner Point int Ab Abstr traction action VTSA Summer r School, l, 2013 2013. Kim Larse sen [31 31]

Mu Multiple tiple Ob Obje jective ctive Sched heduling uling P 2 P 1 2,3 16,10 cost 2 6,6 10,16 P 6 P 3 P 4 2,3 Pareto Frontier cost 2 ’== 3 cost 1 ’== 4 P 7 P 5 2,2 8,2 3W 4W cost 1 VTSA Summer r School, l, 2013 2013. Kim Larse sen [32 32]

Ene Energy rgy Aut Automata omata

Ma Mana naging ging Resourc ources VTSA Summer r School, l, 2013 2013. Kim Larse sen [34 34]

Con onsuming suming & Ha Harve vesting sting Ene nergy gy Maximize throughput while respecting: 0 · E · MAX VTSA Summer r School, l, 2013 2013. Kim Larse sen [35 35]

Ene nergy gy Con onstr strains ains  Energy is not only consumed but may also be regained  The aim is to continously satisfy some energy constriants VTSA Summer r School, l, 2013 2013. Kim Larse sen [36 36]