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System-Wide Energy Minimization for Real-Time Tasks: Lower Bound and Approximation

System-Wide Energy Minimization for Real-Time Tasks: Lower Bound and Approximation. Xiliang Zhong and Cheng-Zhong Xu Dept. of Electrical & Computer Engg. Wayne State University Detroit, Michigan http://www.cic.eng.wayne.edu. Outline. Introduction Processor and system energy model

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System-Wide Energy Minimization for Real-Time Tasks: Lower Bound and Approximation

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  1. System-Wide Energy Minimization for Real-Time Tasks: Lower Bound and Approximation Xiliang Zhong and Cheng-Zhong Xu Dept. of Electrical & Computer Engg. Wayne State University Detroit, Michigan http://www.cic.eng.wayne.edu

  2. Outline • Introduction • Processor and system energy model • Related Work • System-Wide Energy Optimization for periodic tasks • The optimal algorithm • A fully polynomial time approximation scheme • Performance Evaluation • System-Wide Energy Optimization for sporadic Tasks • Solution and evaluation • Conclusions

  3. Introduction • Mobile/Embedded devices are power critical, with limited battery capacity • Software assisted power management • Dynamic power management (DPM) • Resource shutdown after a timeout • Dynamic voltage/frequency scaling (DVS) • Processing speed designed for peak performance • Slowdown the processor voltage / speed when not fully utilized

  4. Dynamic voltage scaling (DVS) • The dynamic CPU power is , P ∝ v2f • Reducing v also reduce the maximum processors frequency • Approximately, energy per cycle∝ f2 Energy per cycle of PXA processor • Processor slowdown leads to super-linear energy savings, while linear execution time increase

  5. System-Wide Energy • Processor also has leakage power • Applications may use other components such as memory and peripheral devices • Can be in active, standby, sleep, and shutdown states • System-wide energy consumed in running a task • CPU, resource standby and active energy • Lowering CPU frequency can increase overall energy expenditure due to prolonged resource standby time of other components

  6. System-Wide Energy (cont.) • critical speed,thespeed with minimum energy per cycle • Not energy efficient using lower speed Energy per cycle of PXA processor with different standby power • Execute a task at speed no lower than its critical speed, then put the devices into low power state • A combined use of slowdown and shutdown

  7. Related Work • CPU energy minimization for periodic tasks: • Heuristics [Mejia-Alvarez’04], approximations [Chen and Kuo’05] • Few studies on system-wide energy minimization • Applications w/o deadlines • Subject to a performance loss [Choi et al.’04] • Real-time periodic tasks on CPU w/ continuous speed levels • Heuristics [Zhuo and Chakrabarti’05] • Real-time periodic tasks on CPU w/ discrete speed levels • Heuristics [Jejurikar and Gupta’04] • This work • Pseudo-polynomial algorithm for optimal solutions and polynomial approximated schemes • Applicable to both offline periodic tasks and online sporadic tasks in processors with practical discrete levels

  8. System-wide energy optimization • Periodic Tasks (Offline) • : worst case execution time under max speed • : task period and deadline • : normalized speed of task • Sporadic Tasks (Online) • Task releases have irregular intervals • Online scheduling based on uncompleted tasks, no assumption about future task releases • The objective is to minimize • overall energy consumption including CPU and all other system components while meeting deadline constraints of all the tasks

  9. Energy Minimization for PeriodicTasks • Minimization of energy consumption for n periodic tasks in a hyper-period, Feasible constraint under EDF Boundary constraint • Practical processors with discrete speed levels • The minimization is an NP-hard Multiple Choice KnapSack (MCKP) problem • There exist pseudo-polynomial solutions to MCKP with integer coefficients, not applicable in this problem

  10. An Example • Basic idea: first solve subprobs with fewer #tasks • A system with an PXA processor with 5 normalized speed [0.15 0.4 0.6 0.8 1] • System with memory, flash, and WNIC • An example real-time workload w/ 4 periodic tasks

  11. f: pruned by feasibility condtion e: pruned by energy condition (utilization, energy) Solution to task 1 • Task 1, execution time 6.4; deadline 16; utilization 0.4 • Branch on four normalized speeds [0.4 0.6 0.8 1] • State pruning • Feasibility condition: • The 1st node at speed 0.4 removed with utilization already 1 • Energy condition • Task 1 at the smallest speed (2nd , 0.6); tasks 2-4 at the max. Total Energy=7.6 (upper bound) • Task 1 at 3rd or 4th speed (0.8 or 1); tasks 2-4 at the min. The required energy exceeds 7.6. The two states can be removed

  12. Solution to the first three tasks pairs of (utilization, energy) f: pruned by feasibility condtion e: pruned by energy condition d: pruned by dominance • Dominance condition • The states (0.867, 9.107) and (0.87, 9.4) of task 3 • First one leads to smaller utilization • Any feasible schedule by the second can also be satisfied by the first • First one uses less energy; the second can be removed

  13. (utilization, energy) f: pruned by feasibility condtion e: pruned by energy condition d: pruned by dominance optimal state Maximum state number reduced to 6/4*4*3*3 = 0.4 %

  14. A fully polynomial approximation scheme (FPTAS) • State # is pseudo-polynomial in task number. • can be reduced by providing approximated solutions • Approximated with worst case perf. guarantee • An algorithm is said to be an approximation scheme if for a given in (0,1), we have • A more desirable approximation scheme (FPTAS) has a polynomial running time in both the number of tasks and the performance ratio

  15. A fully polynomial approximation scheme (cont.) • Divide the energy values into a number of groups each of size r, • Each value scaled and rounded to • Energy values in the same group are treated equally • Find the group size r, subject to a given performance bound • Energy value of each task introduces an error no larger than group size r • Accumulated errors of n tasks no larger than n*r • A lower bound of E* is when all tasks run at their critical speeds (Emin), i.e., E*≥ Emin Solving derives group size r

  16. Performance Evaluation • Simulation Settings • A system with an PXA processor • memory: standby power 0.2W, standby time 20%~60% of task execution • flash drive: 0.4W and 10%~25% • wireless interface: 1W and 5%~20% • Periodic Tasks • Randomly generated deadlines w/ utilization from 0.1~1 • Each task randomly chooses a subset of resources • Algorithms implemented • CPU-DVS, speed control for CPU energy consumption • CS-DVS, a heuristic algorithm for system-wide energy savings [Jejurikar and Gupta ISLPED2004], • OPT-P, the proposed optimal solution • Approximated scheme with perf. bounds 0.01, 0.1, 0.5

  17. Performance Evaluation (Periodic tasks) 23% 16% 8% • Proposed algorithms 23% less energy than CPU-only solutions • Energy consumption up to 16% more efficient than CS-DVS • Approximation algorithms effectively bound the performance errors

  18. Energy Minimization for SporadicTasks • Online energy minimization for all uncompleted tasks n feasible constraints under EDF boundary constraint • On a processor with discrete speed levels • Prove the problem is an instance of Multi-dimensional MCKP (NP-hard in the strong sense, any optimal solution has exponential running time)

  19. Sporadic Tasks (cont.) • Consider three tasks released at time 0 with deadlines 3, 5, 7 • Feasibility of a task (e.g. J2) is not affected by tasks finished later (tasks in a non-decreasing order of deadlines) • Satisfy one constraint (e.g. J3) at each iteration • Can be solved by a pseudo-polynomial algorithm for the optimal solution and an approximation scheme (FPTAS)

  20. Performance Evaluation (Sporadic tasks) • Experimental Settings • Varied number of tasks • Task inter-release times generated by an exponential dist. • Algorithms implemented • TV-DVS, adaptive speed scaling for CPU energy consumption on processors w/ continuous levels [Zhong and Xu RTSS2005] • DVSST, CPU energy consumption with only frequency scaling available (continuous levels) [Qadi et al. RTSS2003] • OPT-S, the proposed optimal solution • 0.1, 0.5-approximation, approximated solutions with different performance settings

  21. Energy consumption (Sporadic tasks) 56% 23% • Small task number: Energy consumption up to 56% more efficient than TVDVS and DVSST • Large task number: 23% more efficient

  22. Conclusion • System-wide energy minimization for periodictasks • pseudo-polynomial algorithm for the optimal solution • approximated solution in moderate running time with bounded performance degradation (FPTAS) • Minimization for onlinesporadic tasks • Pseudo-polynomial algorithm and an FPTAS by exploiting inherent properties of online task scheduling • On-going work • Implementation of the policies in an embedded system with PXA270 processor • Energy/Time overhead voltage and speed switches; overhead in putting a resource into low power state

  23. Thank you! System-Wide Energy Minimization for Real-Time Tasks: Lower Bound and Approximation

  24. Algorithm running time • Running time measured in a Pentium 4 machine with 2 GHz processor • OPT-P has a higher complexity than CS-DVS • Below 90 ms for systems with up to 50 tasks • All approximation algorithms require no more than 0.4 s to finish • Algorithm running time for schedules in a 10-minutes run • OPT-S has higher running time, but <1% task execution time • Comparable time for approximation algorithms with TV-DVS

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