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S YSTEM -W IDE E NERGY M ANAGEMENT FOR R EAL -T IME T ASKS : L OWER B OUND AND A PPROXIMATION Xiliang Zhong and Cheng-Zhong Xu ICCAD 2006, ACM Trans. on Embedded Computing Systems, 2008. Presenter: Mohammad A Haque

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M OTIVATION Energy cost is becoming one of the greatest concern for Data Centers and large computer systems. The total energy and cooling cost for servers in the USA are expected to rise to $50 Billion per year by the end of this year. Generated heat is proportional to the energy consumption. Overheating affects device performance and lifetime. CS, GMU cluster HYDRA has more than 50 nodes. Only 18 nodes are kept active at the same time due to overheating problem. Introduction System Model Algorithm Simulation Results Conclusion

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B ACKGROUND System-wide energy management means the power model considers energy consumed by different system components (memory, wireless interface, …) along with CPU. In Real-Time Systems, every task has an associated deadline. Deadline could be either Hard or Soft. This paper deals with system-wide energy management of Periodic and Sporadic (Aperiodic ) tasks in hard real-time system. Introduction System Model Algorithm Simulation Results Conclusion

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P ROBLEM S ETTINGS We are given a set of periodic/sporadic tasks. The tasks need to be executed on a single processor machine with discrete speed levels. Our goal is to find speed assignments for the tasks, such that total system-level energy consumption is minimized while meeting the deadline constraints. Introduction System Model Algorithm Simulation Results Conclusion

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T ASK M ODEL A set of n tasks are scheduled using Earliest Deadline First (EDF) policy. A task is characterized by a tuple {C i, T i, D i }, where C i is the execution time under maximum speed. T i and D i are task period and deadline respectively. Its common to assume T i = D i. A task utilization is defined by the ratio C i /T i. A well known result in real-time system is, a task set can be feasibly scheduled using EDF if and only if the following condition holds. Introduction System Model Algorithm Simulation Results Conclusion

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E NERGY M ODEL CPU speed can be adjusted in discrete steps in range [ f min, f max ]. We define slowdown factor, S = f/f max. At S i, a task takes C i /S i time to complete and the new utilization is C i /(T i S i ). The system-level energy consumption for a task i at slowdown factor S i is, power consumed by j -th device in standby state. energy consumed by all the devices in active state. Introduction System Model Algorithm Simulation Results Conclusion

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P ROBLEM F ORMULATION minimize subject to Introduction System Model Algorithm Simulation Results Conclusion

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R EDUCTION TO 0-1 MCKP For each task, i and for each speed level, j we have a corresponding (utilization, energy) pair denoted by (u ij, e ij ). So the problem can be thought as choosing one pair from each set, {(u 11, e 11 ), (u 12, e 12 ), …, (u 1m, e 1m )}, {(u 21, e 21 ), (u 22, e 22 ), …, (u 2m, e 2m )}, …, {(u n1, e n1 ), (u n2, e n2 ), …, (u nm, e nm )}. to minimize the sum of e ij, while keeping the sum of u ij ’s below 1. 0-1 MCKP is NP hard. Therefore, so is our problem. Introduction System Model Algorithm Simulation Results Conclusion

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We know pseudo-polynomial time algorithm for solving 0-1 MCKP problem. But the traditional algorithms do not work here, due to a limitation of the traditional 0-1 Knapsack problem solutions. What is the biggest limitation with those solutions?? The traditional solutions for 0-1 Knapsack problem works only with integer coefficients. Introduction System Model Algorithm Simulation Results Conclusion

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O PTIMAL A LGORITHM Start with an empty list, L 0 =. For each task i and for each speed level j, perform component wise addition of the pair (u ij, e ij ) with the list L i-1. Merge the lists and keep it sorted in non-decreasing order of energy. Delete all infeasible solution from the list in each step. Prune the dominated states from the list. A state is dominated by another state, if it has a larger utilization and a larger energy consumption than the other. Introduction System Model Algorithm Simulation Results Conclusion

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E XAMPLE Consider a task set with 2 tasks and 2 speed levels. The algorithm works as follows. Initialization: L 0 = 1 st step: L 1,1 = L 1,2 = L 1 = 2 nd step: L 2,1 = L 2,2 = L 2 = Introduction System Model Algorithm Simulation Results Conclusion

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A NALYSIS The runtime of the algorithm is bounded by where U is the upper bound for the number of elements in the list and m i is the number speed levels for task i. Since the size of the list can grow very large, the runtime of the algorithm is not polynomially bounded. The space requirement for the algorithm is O(U). Introduction System Model Algorithm Simulation Results Conclusion

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A PPROXIMATION A LGORITHM (FPTAS) Choose ε and replace e ij with The maximum error we can have is nε. For a given relative error r, we choose, where, E* is the energy consumption of the optimal algorithm. The number of undominated states in the list reduces to (E max /E min - 1)(n/r). Introduction System Model Algorithm Simulation Results Conclusion

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S PORADIC T ASKS Sporadic tasks are released at random time instants and the task parameters are only known after release. Whenever a task arrives, we perform an admission control test to determine whether we can accept the task or not. The problem reduces to Multi-dimensional MCKP (MMKP), which is NP-hard in the strong sense. Similar optimal and approximation algorithm works for the sporadic task as well. A key assumption is the feasibility of a task does not depend on the task that finishes later. Introduction System Model Algorithm Simulation Results Conclusion

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S IMULATION R ESULT (P ERIODIC ) Introduction System Model Algorithm Simulation Results Conclusion

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S IMULATION R ESULT (S PORADIC ) Introduction System Model Algorithm Simulation Results Conclusion

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C ONCLUSION The paper provides pseudo-polynomial optimal algorithm and FPTAS for both periodic and sporadic tasks. The algorithm is simple yet interesting. However some more simulation results, specially the effect of number of speed levels and ratio of E max /E min will be worth experimenting. Introduction System Model Algorithm Simulation Results Conclusion

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T HANK Y OU

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