Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Flow Shop for Dual CPUs in Dynamic Voltage Scaling Vincent Chau, Ken C.K. Fong, Minming Li and Kai Wang Department of Computer Science, City University of Hong Kong March 2016 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 1 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Outline Introduction 1 Flowshop on m machines 2 Discrete Speed (fixed order) Continuous Speed (arbitrary order) Sense-And-Aggregate Model 3 Conclusion 4 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 2 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Outline Introduction 1 Flowshop on m machines 2 Discrete Speed (fixed order) Continuous Speed (arbitrary order) 3 Sense-And-Aggregate Model Conclusion 4 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 3 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Models We are given a set of n jobs and m machines: each job j has a processing requirement p i , j on machine i Flowshop on 2 machines a job j can start on machine 2 only when it is completed on machine 1 0 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 4 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Models We are given a set of n jobs and m machines: each job j has a processing requirement p i , j on machine i Flowshop on 2 machines a job j can start on machine 2 only when it is completed on machine 1 Speed-Scaling setting � s ( t ) α dt with α > 1 Cost is s ✻ 4 p = 10 2 s 3 × 2 . 5 α = 3 s 3 × 5 ✲ time Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 4 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Models We are given a set of n jobs and m machines: each job j has a processing requirement p i , j on machine i Flowshop on 2 machines a job j can start on machine 2 only when it is completed on machine 1 Speed-Scaling setting � s ( t ) α dt with α > 1 Cost is s ✻ 4 p = 10 2 160 α = 3 40 ✲ time Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 4 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Example When order of jobs is given, there exists a O ( n 3 ) algorithm Z. Mu, M. Li, Journal of Combinatorial Optimization, 2015 √ p 1 , 4 α + p 2 , 3 α + p 2 , 4 ( p 1 , 2 + p 1 , 3 ) α + ( p 2 , 1 + p 2 , 2 ) α + � α � � p 1 , 1 + α α E = D speed 1 2 3 4 4 3 time 2 1 D 0 critical interval Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 5 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Our contributions Flowshop on m machines Fixed order, Discrete speeds, a Linear Program Formulation Arbitrary order, Continuous speeds, an approximation algorithm Sense-And-Aggregate Model Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 6 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion Outline Introduction 1 Flowshop on m machines 2 Discrete Speed (fixed order) Continuous Speed (arbitrary order) 3 Sense-And-Aggregate Model Conclusion 4 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 7 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion Continuous to Discrete Jobs order is given with processing requirement of 10 Set of speeds S = { 1 , 2 } speed 2 1 2 1 time 0 2.9289 7.071 10 Energy : 19.7994 α = 2 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 8 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion Continuous to Discrete Jobs order is given with processing requirement of 10 Set of speeds S = { 1 , 2 } speed 2 1 2 1 time 0 2.9289 7.071 10 Energy : 19.7994 Energy : 31.7157 α = 2 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 8 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion Continuous to Discrete Jobs order is given with processing requirement of 10 Set of speeds S = { 1 , 2 } speed 2 1 2 1 time 0 2.5 7.5 10 Energy : 19.7994 Energy : 31.7157 Energy : 30 α = 2 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 8 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion A Linear Program Let x i , j , v be the workload done for job j on machine i at speed v . Let s i , j (resp. c i , j ) be the starting time (resp. completion time) of job j on machine i . � � � v α − 1 x i , j , v min v ∈ S i j � s.t. x i , j , v = p i , j ∀ i , j all jobs must be scheduled v ∈ S x i , j , v � s i , j + = c i , j ∀ i , j Processing time v v ∈ S c m , n ≤ D c i , j ≤ s i , j +1 ∀ i , j Precedence const. between jobs c i , j ≤ s i +1 , j ∀ i , j between machines x i , j , v , s i , j , c i , j ≥ 0 ∀ i , j , v Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 9 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion A Linear Time Approximation Algorithm Recall that for fixed order, a polynomial time algorithm exists Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 10 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion A Linear Time Approximation Algorithm Recall that for fixed order, a polynomial time algorithm exists � i , j p i , j We schedule jobs at speed in any order on each D machine speed time 0 D Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 10 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion Approximation algorithm Theorem This algorithm is a m α − 1 -approximation Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 11 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion Approximation algorithm Theorem This algorithm is a m α − 1 -approximation Proof Let V i = � j p i , j ∀ i i V i ) α D 1 − α ( � ( � i V i ) α ALG = � α D 1 − α = � α V i V i LB �� �� m m i i m m i V i ) α ( � � α = m α − 1 = i V i ) α � 1 m ( � m Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 11 / 25
Introduction Flowshop on m machines Discrete Speed (fixed order) Sense-And-Aggregate Model Continuous Speed (arbitrary order) Conclusion Approximation algorithm Theorem This algorithm is a m α − 1 -approximation Proof Let V i = � j p i , j ∀ i i V i ) α D 1 − α ( � ( � i V i ) α ALG = � α D 1 − α = � α V i V i LB �� �� m m i i m m i V i ) α ( � � α = m α − 1 = i V i ) α � 1 m ( � m Note that if we fix an arbitrary order and compute the minimum energy consumption, the approximation cannot be larger than m α − 1 but takes O ( n 3 ) time. Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 11 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Outline Introduction 1 Flowshop on m machines 2 Discrete Speed (fixed order) Continuous Speed (arbitrary order) 3 Sense-And-Aggregate Model Conclusion 4 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 12 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion Sense-And-Aggregate Model Rules: Sensor collects one unit of data at each time Computation can decide to process now or wait for more data Outputs one unit of data for each aggregation Common deadline D Sensor Computation Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 13 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion speed sensor computation output time D 0 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 14 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion speed sensor computation output time D 0 Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 14 / 25
Introduction Flowshop on m machines Sense-And-Aggregate Model Conclusion speed sensor computation output time D 0 Observations The more we wait, the less workload there is on the computation machine Decide to compute earlier allows to speed down the processing, and potentially the energy consumption Minming Li Flow Shop for Dual CPUs in Dynamic Voltage Scaling March 2016 14 / 25
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