Energy-efficient scheduling on volatile platforms Guillaume Aupy based on a work with Anne Benoit
Energy, Energy: a crucial issue Reliability, Makespan G. Aupy Introduction • Data centers Model • 330 , 000 , 000 , 000 Watts hour in 2007: more than France Reliability Makespan • 533 , 000 , 000 tons of CO 2 : in the top ten countries Energy Approximation • Exascale computers (10 18 floating operations per second) for linear chains Intractability • Need effort for feasibility FPTAS • 1% of power saved � 1 million dollar per year Approximation for independent • Lambda user tasks Inapproximability • 1 billion personal computers Approximation • 500 , 000 , 000 , 000 , 000 Watts hour per year Conclusion • � crucial for both environmental and economical reasons 1.0
Energy, Energy: a crucial issue Reliability, Makespan G. Aupy Introduction • Data centers Model • 330 , 000 , 000 , 000 Watts hour in 2007: more than France Reliability Makespan • 533 , 000 , 000 tons of CO 2 : in the top ten countries Energy Approximation • Exascale computers (10 18 floating operations per second) for linear chains Intractability • Need effort for feasibility FPTAS • 1% of power saved � 1 million dollar per year Approximation for independent • Lambda user tasks Inapproximability • 1 billion personal computers Approximation • 500 , 000 , 000 , 000 , 000 Watts hour per year Conclusion • � crucial for both environmental and economical reasons 1.0
Energy, Reliability, Makespan 1 Model G. Aupy Reliability Introduction Makespan Model Energy Reliability Makespan Energy Approximation 2 Approximation for linear chains for linear chains Intractability Intractability FPTAS FPTAS Approximation for independent tasks 3 Approximation for independent tasks Inapproximability Inapproximability Approximation Conclusion Approximation 4 Conclusion 2.0
Energy, Reliability, Makespan 1 Model G. Aupy Reliability Introduction Makespan Model Energy Reliability Makespan Energy Approximation 2 Approximation for linear chains for linear chains Intractability Intractability FPTAS FPTAS Approximation for independent tasks 3 Approximation for independent tasks Inapproximability Inapproximability Approximation Conclusion Approximation 4 Conclusion 3.0
Energy, Architecture Reliability, Makespan G. Aupy Introduction Model Reliability Makespan Energy Approximation p identical processors. for linear chains Intractability FPTAS Approximation Speed Scaling: one can modify the execution speed f of any for independent task, f ∈ [ f min , f max ]. tasks Inapproximability Approximation Conclusion 4.0
Energy, Into details Reliability, Makespan G. Aupy Introduction Model Reliability Makespan Let T i of weight w i executed on processor p j : Energy Approximation E xe ( w i , f i ) for linear chains Intractability FPTAS p j · · · f i · · · Approximation for independent tasks Inapproximability Approximation time Conclusion 5.0
Energy, Into details Reliability, Makespan G. Aupy Introduction Model Reliability Makespan Let T i of weight w i executed on processor p j : Energy Approximation for linear chains E xe ( w i , f i ) Intractability FPTAS p j · · · · · · f i Approximation for independent tasks Inapproximability Approximation time Conclusion 5.0
Energy, Hypothesis Reliability, Makespan G. Aupy Introduction Model Reliability Makespan In this model, we further suppose that there is a speed f rel , Energy such that, for any T i executed at speed f i : Approximation for linear chains Intractability FPTAS Approximation If f i < f rel then we need to execute T i a second time. for independent tasks Inapproximability Approximation Conclusion 6.0
Energy, Hypothesis Reliability, Makespan G. Aupy Introduction Model Reliability Makespan In this model, we further suppose that there is a speed f rel , Energy such that, for any T i executed at speed f i : Approximation for linear chains Intractability FPTAS Approximation If f i < f rel then we need to execute T i a second time. for independent tasks Inapproximability How come? Approximation Conclusion 6.0
Energy, Reliability Reliability, Makespan Transient failure = local failure (no impact on the system, the G. Aupy processor impacted can restart to work immediately after the Introduction failure) Model Reliability Makespan Energy The rate of transient failures follow a Poisson Law of Approximation parameter: for linear chains f max − f λ ( f ) = λ 0 e d f max − f min Intractability FPTAS Approximation for independent The reliability of T i executed at speed f i : tasks Inapproximability R i ( f i ) = e − λ ( f i ) E xe ( w i , f i ) Approximation Conclusion ≈ 1 − λ ( f i ) E xe ( w i , f i ) 7.0
Energy, Reliability Reliability, Makespan G. Aupy R i ( f i ) Introduction Model Reliability Makespan Energy Approximation for linear chains Intractability FPTAS Approximation for independent tasks f i Inapproximability Approximation Conclusion 7.0
Energy, Reliability Reliability, Makespan G. Aupy R i ( f i ) Introduction R i ( f rel ) Model Reliability Makespan Energy Approximation for linear chains Intractability FPTAS Approximation for independent tasks f i Inapproximability f rel Approximation Conclusion 7.0
Energy, Reliability Reliability, Makespan G. Aupy R i ( f i ) Introduction R i ( f rel ) Model Reliability Makespan Energy Approximation for linear chains Intractability FPTAS Approximation for independent tasks f i Inapproximability f rel Approximation Conclusion 7.0
Energy, Reliability Reliability, Makespan G. Aupy Introduction The reliability of T i executed at speed f i : Model Reliability Makespan R i ( f i ) = e − λ ( f i ) E xe ( w i , f i ) Energy Approximation ≈ 1 − λ ( f i ) E xe ( w i , f i ) for linear chains Intractability FPTAS Approximation for Reliability for two executions independent tasks R i = 1 − (1 − R i ( f (1) ))(1 − R i ( f (2) )) Inapproximability i i Approximation Conclusion 7.0
Energy, Reliability Reliability, Makespan G. Aupy Introduction The reliability of T i executed at speed f i : Model Reliability Makespan R i ( f i ) = e − λ ( f i ) E xe ( w i , f i ) Energy Approximation ≈ 1 − λ ( f i ) E xe ( w i , f i ) for linear chains Intractability FPTAS Approximation for Reliability for two executions independent tasks R i = 1 − (1 − R i ( f (1) ))(1 − R i ( f (2) )) Inapproximability i i Approximation Conclusion 1 − (1 − R i ( f min )) 2 ≥ R i ( f rel ) ֒ → Let’s suppose that two execution are enough to match the reliability constraint. 7.0
Energy, Makespan Reliability, Makespan G. Aupy The execution time for T i at speed f i is: Introduction E xe ( w i , f i ) = w i Model Reliability f i Makespan Energy Approximation for linear chains Intractability If we call t i the end of the last execution of T i : FPTAS Approximation for independent f (2) tasks T (2) p 2 i i Inapproximability Approximation Conclusion f (1) p 1 T (1) i i t i time 8.0
Energy, Makespan Reliability, Makespan G. Aupy The execution time for T i at speed f i is: Introduction E xe ( w i , f i ) = w i Model Reliability f i Makespan Energy Approximation for linear chains Intractability If we call t i the end of the last execution of T i : FPTAS Approximation for independent f (2) tasks T (2) p 2 i i Inapproximability Approximation Conclusion f (1) p 1 T (1) i i t i time 8.0
Energy, Makespan Reliability, Makespan G. Aupy Introduction Model Reliability Makespan Energy Approximation for linear chains Constraint on the makespan: Intractability FPTAS We ask ∀ i , t i ≤ D (the deadline D is fixed by the user) Approximation for independent tasks Inapproximability Approximation Conclusion 8.0
Energy, Reliability, Makespan G. Aupy Introduction Model Reliability Makespan Energy Why should we decrease the speed then? Approximation • Loss in reliability for linear chains Intractability • Loss on the makespan FPTAS Approximation for independent tasks Inapproximability Approximation Conclusion 9.0
Energy, Energy Reliability, Makespan G. Aupy Introduction Model Reliability The execution of task T i at speed f i : Makespan Energy Approximation E i ( f i ) = E xe ( w i , f i ) f 3 i = w i f 2 for linear i chains Intractability FPTAS → (Dynamic part of the classical energy model) Approximation for independent tasks Energy consumption with two executions: Inapproximability � 2 � 2 Approximation � � f (1) f (2) E i = w i + w i Conclusion i i 10.0
Energy, Energy Reliability, Makespan G. Aupy Introduction Model Reliability The execution of task T i at speed f i : Makespan Energy Approximation E i ( f i ) = E xe ( w i , f i ) f 3 i = w i f 2 for linear i chains Intractability FPTAS → (Dynamic part of the classical energy model) Approximation for independent tasks Energy consumption with two executions: Inapproximability � 2 � 2 Approximation � � f (1) f (2) E i = w i + w i Conclusion i i 10.0
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