System wide energy minimization for real time tasks lower bound and approximation
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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

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
Outline Bound and Approximation

  • 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


Introduction
Introduction Bound and Approximation

  • 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


Dynamic voltage scaling dvs
Dynamic voltage scaling (DVS) Bound and Approximation

  • 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


System wide energy
System-Wide Energy Bound and Approximation

  • 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


System wide energy cont
System-Wide Energy (cont.) Bound and Approximation

  • 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


Related work
Related Work Bound and Approximation

  • 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


  • System wide energy optimization
    System-wide energy optimization Bound and Approximation

    • 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


    Energy minimization for periodic tasks
    Energy Minimization for Bound and ApproximationPeriodicTasks

    • 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


    An example
    An Example Bound and Approximation

    • 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


    Solution to task 1

    f: pruned by feasibility condtion Bound and Approximation

    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


    Solution to the first three tasks
    Solution to the first three tasks Bound and Approximation

    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


    (utilization, energy) Bound and Approximation

    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 %


    A fully polynomial approximation scheme fptas
    A fully polynomial Bound and Approximationapproximation 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


    A fully polynomial approximation scheme cont
    A fully polynomial Bound and Approximationapproximation 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


    Performance evaluation
    Performance Evaluation Bound and Approximation

    • 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


    Performance evaluation periodic tasks
    Performance Evaluation (Periodic tasks) Bound and Approximation

    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


    Energy minimization for sporadic tasks
    Energy Minimization for Bound and ApproximationSporadicTasks

    • 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)


    Sporadic tasks cont
    Sporadic Tasks (cont.) Bound and Approximation

    • 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)


    Performance evaluation sporadic tasks
    Performance Evaluation (Sporadic tasks) Bound and Approximation

    • 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


    Energy consumption sporadic tasks
    Energy consumption (Sporadic tasks) Bound and Approximation

    56%

    23%

    • Small task number: Energy consumption up to 56% more efficient than TVDVS and DVSST

    • Large task number: 23% more efficient


    Conclusion
    Conclusion Bound and Approximation

    • 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


    Thank you
    Thank you! Bound and Approximation

    System-Wide Energy Minimization for Real-Time Tasks:

    Lower Bound and Approximation


    Algorithm running time
    Algorithm running time Bound and Approximation

    • 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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